Method and system for simulating objects and measuring the uncertainty of simulation results

Abstract

This invention provides a method and system for simulating objects and measuring the uncertainty of the simulation results. [Solution] The embodiment receives a mesh-based model containing nodes representing real-world objects, performs a simulation of real-world objects using the mesh-based model, and as a result obtains (i) the behavior of real-world objects and (ii) the sensitivity of the values ​​of one or more properties at each node to the geometric coordinates of the nodes. For each node, geometrically uncorrelated uncertainty is determined based on the sensitivity of the values ​​of one or more properties at the node and the deviation of the geometric coordinates of the node. Using the summation function and uncertainty characterization parameters of the geometrically uncorrelated uncertainty determined for each node, uncertainty in the behavior of real-world objects is determined and outputs (i) a representation of the behavior of real-world objects and (ii) a representation of the determined uncertainty in one or more behaviors of real-world objects.

Description

[Technical Field]

[0001] The present invention relates to a method and system for simulating an object and measuring the uncertainty of the simulation results. [Background technology]

[0002] Many existing products and simulation systems are available on the market for the design and simulation of objects, such as vehicles. These systems typically employ computer-aided design (CAD) programs and computer-aided engineering (CAE) programs. These systems allow users to construct, manipulate, and simulate complex three-dimensional models of objects or assemblies of objects. These CAD and CAE systems represent objects, such as real-world objects, using edges, lines, faces, polygons, or closed volumes. Lines, edges, faces, polygons, and closed volumes can be represented in various ways, for example, as non-uniform rational B-splines (NURBS).

[0003] These systems primarily manage the specifications of the shape of modeled objects, such as parts or assemblies of parts. In particular, CAD files contain the specifications from which the shape is generated. From this shape, a representation, such as a three-dimensional (3D) CAD model, is generated.

[0004] The advent of CAD and CAE systems enables a wide range of representation possibilities for objects. Exemplary computer-based models used by CAD and CAE systems include CAD models and finite element (FE) models (i.e., meshes). Computer-based models may be programmed so that the model possesses properties (e.g., physical, material, or other physics-based) of the underlying real-world object or the object represented by the model. Exemplary properties, among other embodiments, include stiffness (the ratio of force to displacement), plasticity (irreversible strain), and viscosity (resistance to flow between adjacent layers). When a CAD model or other such computer-based model known in the art is programmed in this manner, it can be used to perform simulations of the object represented by the model. For example, a mesh-based model can be used to represent the internal cavities of a vehicle, the acoustic fluid surrounding a structure, or any number of real-world objects. Furthermore, CAD and CAE systems, along with computer-based models, can be used, among other embodiments, to simulate real-world physical systems, such as engineering systems like automobiles, airplanes, buildings, and bridges. Furthermore, CAE systems can be used to simulate any variety and combination of the behavior of these physics-based systems, such as noise and vibration. [Overview of the project]

[0005] Computer-based simulations can be performed using existing CAD and CAE methods. However, existing methods have numerical problems, such as those present in CAE, and meshing on a software platform presents epistemological uncertainty. This embodiment solves this problem.

[0006] The mesh uncertainty problem resolved by the embodiments disclosed herein may exist in itself when the same CAD model is meshed several times using the same preprocessing mesh settings for numerical modeling. Many of these meshing implementations often produce results where the number of nodes in the mesh and / or the location of the meshed nodes may differ slightly due to numerical uncertainty in the meshing algorithm or due to different software compilers. The subsequent problem arises when these slightly different meshed models are explicitly applied in a deterministic optimization setting. Specifically, differences in the modeling of the principal solution (e.g., due to different meshes) accumulate numerically over optimization iterations, thereby resulting in significantly different optimized designs. This can be problematic for users because they expect the same optimized design for each optimization run, assuming that slightly different meshed models should solve the same system. Thus, users may observe significantly different optimized designs in numerical modeling for essentially the same optimization, due to small geometric differences in nodes (i.e., geometric uncertainty). The embodiment solves this problem by providing a novel approach to uncertainty that can be applied as a design response to sensitivity-based optimization, using an uncertainty measure for the nodal distribution of uncorrelated geometric shapes.

[0007] One such embodiment relates to a computer implementation method. In this embodiment, the processor implements the method by receiving a mesh-based model representing a real-world object into the processor's memory, the mesh-based model comprising a plurality of nodes. The method continues by performing a simulation of the real-world object using the received mesh-based model, the results of which include (i) the behavior of one or more real-world objects, and (ii) the sensitivity of the values ​​of one or more properties at each node of a subset of the plurality of nodes to the geometric coordinates of the nodes. Furthermore, for each node in the subset of the plurality of nodes, the geometrically uncorrelated uncertainty of the node is determined based on the sensitivity of the values ​​of one or more properties at the node and the deviation of the geometric coordinates of the node. To continue, the embodiment determines the uncertainty in one or more behaviors of the real-world objects, which is determined using a summation function of the geometrically uncorrelated uncertainty and uncertainty characterization parameters determined for each node in the subset. The embodiment then outputs (i) a representation of one or more behaviors of the real-world objects, and (ii) a representation of the determined uncertainty in one or more behaviors of the real-world objects.

[0008] One embodiment further includes receiving a representation of the value of an uncertainty characterization parameter.

[0009] In one embodiment, in response to the uncertainty characterization parameter exceeding a threshold, the representation of the determined uncertainty includes a representation of the contribution to the determined uncertainty by one or more nodes from a subset of nodes. In such embodiments, the representation of the contribution includes a representation of at least one of magnitude and position.

[0010] According to one embodiment, in response to the uncertainty characterization parameter falling below a threshold, the representation of the determined uncertainty does not include a representation of the contribution to the determined uncertainty.

[0011] In another embodiment, determining uncertainty in the behavior of one or more real-world objects includes: determining regularized geometrically uncorrelated uncertainty for two or more nodes of a subset of nodes as a function of a weighted average of geometrically uncorrelated uncertainties determined for each of two or more nodes; and determining uncertainty in the behavior of one or more real-world objects using (i) an aggregate function of the geometrically uncorrelated uncertainties determined for each node of the subset, (ii) an uncertainty characterization parameter, and (iii) the determined, regularized geometrically uncorrelated uncertainties.

[0012] According to one embodiment, the behavior of one or more objects in the real world includes at least one of stress, strain, force, displacement, velocity, acceleration, natural frequency, temperature, and magnetic flux.

[0013] One embodiment further includes defining the design response of a real-world object as the objective function. In such embodiments, the design response is based on one or more behaviors of the real-world object and determined uncertainties in one or more behaviors of the real-world object. Such embodiments may further include defining the optimization problem as a function of the defined design response and at least one constraint. Furthermore, such embodiments may include iteratively modifying at least one design variable in a mesh-based model representing the real-world object, and using the mesh-based model with the modified at least one design variable to determine the simulation, geometrically uncorrelated uncertainties, and uncertainties in one or more behaviors of the real-world object until the optimization problem is satisfied.

[0014] In one embodiment, for a given node in a subset of nodes, the sensitivity of the value of one or more properties at a given node to the geometric coordinates at that node is the magnitude of the vector.

[0015] According to one embodiment, the determined uncertainty is at least one of accidental uncertainty and epistemological uncertainty.

[0016] Another exemplary embodiment relates to a computer-based system. According to one embodiment, the system includes a processor and memory having computer code instructions stored thereon. In such an embodiment, the processor and memory are configured to use the computer code instructions to cause the system to implement any embodiment or combination of embodiments described herein.

[0017] Another embodiment relates to a computer program product. The computer program product comprises a non-temporary computer-readable medium containing program instructions that, when executed by a processor, cause the processor to implement any embodiment or combination of embodiments described herein.

[0018] It should be noted that embodiments of methods, systems, and computer program products may be configured to implement any embodiment or combination of embodiments described herein.

[0019] The foregoing will become clear from the following more specific description of the exemplary embodiments, as similar reference letters throughout the different figures are illustrated in the accompanying drawings to refer to the same parts. The drawings are not necessarily to exact scale and are intended to emphasize that they illustrate embodiments. [Brief explanation of the drawing]

[0020] [Figure 1A] Figure 1A shows an exemplary computer-based model of an L-shaped bracket. [Figure 1B] Figure 1B shows the optimized design for compliance topology optimization of the L-shaped model in Figure 1A, using two meshes. [Figure 2] Figure 2 shows a workflow diagram of a computer implementation method according to one embodiment. [Figure 3] Figure 3 shows the local, uncorrelated geometric distribution at the node level in discrete coordinates for the mesh nodes. [Figure 4A] Figure 4A shows the deformation of the magnitude of the displacement as a response to geometric perturbations in the plane in the x and y directions of the nodal points. [Figure 4B] Figure 4B shows the deformation of a four-node plane stress element due to geometric perturbation. [Figure 5] Figure 5 shows the standard deviations of a coarse mesh and a fine mesh, as well as the standard deviations of the coarse mesh and the fine mesh for different values ​​of the uncertainty characterization parameter p, according to one embodiment. [Figure 6A] Figure 6A shows the design space and material layout of a topology-optimized model of an inverter mechanism that may illustrate an implementation of an embodiment disclosed herein. [Figure 6B] Figure 6B shows the geometric perturbation field and corresponding sensitivity for various geometric uncertainty measures for the element topology model in Figure 6A. Figure 6C shows the geometric perturbation field and corresponding sensitivity for various geometric uncertainty measures for the element topology model in Figure 6A. Figure 6D shows the geometric perturbation field and corresponding sensitivity for various geometric uncertainty measures for the element topology model in Figure 6A. [Figure 7] Figure 7 is a flowchart of a method for deterministic optimization, extended for robustness against nodal geometric uncertainty, according to one embodiment. [Figure 8] Figure 8A shows the design space and non-design space of an exemplary L-shaped bracket in computer-based mode, which may illustrate embodiments disclosed herein. Figure 8B shows the pressure-loaded region and clamped boundary conditions of the L-shaped bracket model of Figure 8A. Figure 8C shows two different mesh models of the L-shaped bracket model of Figure 8A. Figure 8D shows an overlay of the two mesh models of Figure 8C, highlighting the differences between the two mesh models. [Figure 9]Figure 9A shows the topology optimization results using sensitivity filters for the two mesh models defined in Figures 8A-8C. Figure 9B shows the topology optimization results using sensitivity filters for the two mesh models defined in Figures 8A-8C. Figure 9C shows the topology optimization results using sensitivity filters for the two mesh models defined in Figures 8A-8C. Figure 9D shows the topology optimization results using sensitivity filters for the two mesh models defined in Figures 8A-8C. Figure 9E shows the topology optimization results using sensitivity filters for the two mesh models defined in Figures 8A-8C. Figure 9F shows the topology optimization results using sensitivity filters for the two mesh models defined in Figures 8A-8C. [Figure 10] Figure 10A shows the topology optimization results using design variable filters for the two mesh models defined in Figures 8A-8C. Figure 10B shows the topology optimization results using design variable filters for the two mesh models defined in Figures 8A-8C. Figure 10C shows the topology optimization results using design variable filters for the two mesh models defined in Figures 8A-8C. Figure 10D shows the topology optimization results using design variable filters for the two mesh models defined in Figures 8A-8C. [Figure 11] Figure 11A shows the topology optimization results using design variable filters for the two mesh models defined in Figures 8A-8C. Figure 11B shows the topology optimization results using design variable filters for the two mesh models defined in Figures 8A-8C. Figure 11C shows the topology optimization results using design variable filters for the two mesh models defined in Figures 8A-8C. Figure 11D shows the topology optimization results using design variable filters for the two mesh models defined in Figures 8A-8C. [Figure 12]Figure 12A shows the thickness size optimization results using sensitivity filters for the two mesh models defined in Figures 8A-8C. Figure 12B shows the thickness size optimization results using sensitivity filters for the two mesh models defined in Figures 8A-8C. Figure 12C shows the thickness size optimization results using sensitivity filters for the two mesh models defined in Figures 8A-8C. Figure 12D shows the thickness size optimization results using sensitivity filters for the two mesh models defined in Figures 8A-8C. [Figure 13A] Figure 13A shows the design space and non-design space for an exemplary computer-based model of a crane hook that can be analyzed using the embodiment. [Figure 13B] Figure 13B shows the clamped boundary conditions and distributed force region of the crane hook model in Figure 13A. [Figure 13C] Figure 13C shows two different mesh models of the crane hook model shown in Figure 13A. [Figure 13D] Figure 13D shows an overlay of the two mesh models in Figure 13C, highlighting the differences between the two mesh models. [Figure 14] Figure 14A shows the topology optimization results for two mesh models defined in relation to Figures 13A-13D. Figure 14B shows the topology optimization results for two mesh models defined in relation to Figures 13A-13D. Figure 14C shows the topology optimization results for two mesh models defined in relation to Figures 13A-13D. Figure 14D shows the topology optimization results for two mesh models defined in relation to Figures 13A-13D. [Figure 15A] Figure 15A shows the modeling and design space of a computer-based model of the inverter mechanism, and also includes additional design variables used to demonstrate the implementation of the embodiments disclosed herein. [Figure 15B] Figure 15B shows the deterministically optimized design of the inverter mechanism shown in Figure 15A. [Figure 16]Figure 16A shows the topology optimization results generated using a structured coarse mesh. Figure 16B shows the topology optimization results generated using an unstructured coarse mesh. Figure 16C shows the topology optimization results generated using a structured fine mesh. Figure 16D shows the topology optimization results generated using an unstructured fine mesh. [Figure 17A] Figure 17A shows the design space and mesh for a computer-based model of an aircraft bracket to demonstrate an embodiment. [Figure 17B] Figure 17B shows the deterministically optimized design of the aircraft bracket model in Figure 17A. [Figure 18] Figure 18 shows the optimized design of the aircraft bracket defined in relation to Figures 17A and 17B, generated using different optimization conditions. [Figure 19A] Figure 19A shows the optimization iteration history for the relative density of design variables, using a novel uncertainty characterization parameter for output displacement as the objective function to be minimized, according to one embodiment. [Figure 19B] Figure 19B shows the vector of nodal standard deviations for each optimization iteration in Figure 19A. [Figure 20] Figure 20A shows the topology optimization results of the model defined in relation to Figures 15A and 15B, generated using different optimization conditions. Figure 20B shows the topology optimization results of the model defined in relation to Figures 15A and 15B, generated using different optimization conditions. Figure 20C shows the topology optimization results of the model defined in relation to Figures 15A and 15B, generated using different optimization conditions. [Figure 21A] Figure 21A illustrates a deterministically optimized adaptive mechanism design with an adaptive hinge. [Figure 21B] Figure 21B shows an embodiment in which uncertainty is determined using a geometrically uncorrelated nodal distribution. [Figure 22]Figure 22A shows the design space, mesh, and optimized design for each L-shaped beam, determined using the respective embodiments. Figure 22B shows the design space, mesh, and optimized design for each L-shaped beam, determined using the respective embodiments. Figure 22C shows the design space, mesh, and optimized design for each L-shaped beam, determined using the respective embodiments. [Figure 23] Figure 23A shows the topology optimization results for the L-shaped beam defined in relation to Figures 22A-C. Figure 23B shows the topology optimization results for the L-shaped beam defined in relation to Figures 22A-C. Figure 23C shows the topology optimization results for the L-shaped beam defined in relation to Figures 22A-C. Figure 23D shows the topology optimization results for the L-shaped beam defined in relation to Figures 22A-C. [Figure 24A] Figure 24A shows the central node representing the geometric perturbation field, as well as the variation in sensitivity. [Figure 24B] Figure 24B shows a four-node plane stress element for the design variable element in Figure 24A. [Figure 24C] Figure 24C shows a four-node shell element for the design variable element in Figure 24A. [Figure 25] Figure 25 is a schematic diagram of a computer network in which the embodiment may be implemented. [Figure 26] Figure 25 is a block diagram illustrating an exemplary embodiment of a computer node in a computer network. [Modes for carrying out the invention]

[0021] A description of an exemplary embodiment is provided below.

[0022] Deterministic optimization (i.e., optimization of computer-based models without considering geometric uncertainty), when applied to non-parameter optimizations such as topology, sizing, or shape optimization, can result in mesh-dependent or local minima-dependent optimized designs. Traditional robust design optimization methods that consider global geometric perturbations demonstrate the ability to suppress many local minima in the response function, thereby resulting in improved optimized designs. However, different mesh discretizations can sometimes result in fundamentally different optimized designs that cannot be suppressed using the global uncertainty formula. This is problematic because users expect consistent optimization results when evaluating the same object. Furthermore, users cannot be confident in which design is appropriate for real-world use and application from these fundamentally different optimized designs. Embodiments address this problem.

[0023] One embodiment implements a novel uncertainty measure based on a nodal-level uncorrelated local geometric random distribution. An exemplary embodiment discloses a computer-implemented, computationally efficient, and generalized first-order method that ensures improved numerical mesh independence for optimized designs. The embodiment allows for semi-intrusive implementation, regardless of the number and type of design variables. In the discussion herein, embodiments implementing local uncertainty measures (e.g., uncertainty characterization parameter "p") are detailed in various numerical examples addressing both numerical and physical geometric robustness, including the design of a conforming hinge mechanism, as well as formulas for stiffness and stress-based optimization.

[0024] Deterministic optimization can result in structurally optimized designs that are highly sensitive even to small geometric defects in specific regions of the design. Therefore, minor modifications to the model caused by, for example, manufacturing uncertainties, material uncertainties, or uncertainties to numerical discretization (e.g., meshing of CAD models) can lead to a significant decrease in the structural performance of real-world objects. Furthermore, gradient-based optimization algorithms typically yield solutions corresponding to local minima. Obtaining local minima in topology optimization is a generally well-known phenomenon [1]. Accordingly, embodiments disclosed herein also suppress some local minima by applying conventional robustness methods customized to model uncertainties associated with different numerical discretizations. Thus, local minima can be classified as either physical or numerical (mesh-induced). Papadopoulos et al. [2] show that even simple optimization problems can yield highly non-convex response functions with local minima that are independent of the physical mesh.

[0025] In contrast, numerical local minima can also be caused by slight geometric differences in numerical discretization (mesh). Talischi et al. [3] and [4] show that structured meshes can influence the optimized design by introducing a preferred orientation to the mesh that is consistent with the design features of the optimal solution. Therefore, Talischi et al. [3] and [4] apply an unstructured polygonal mesh to prevent the mesh from influencing the optimized design. Furthermore, polygonal meshes do not exhibit checkerboarding and cannot form single-node connections. However, polygonal meshes are not commonly used and are not widely available in finite element implementations. Generally, optimization requires regularization to obtain both mesh-independent designs and ensure a well-placed optimization formula [5]. The most common approach for regularization is to use filtering [1], thereby forcing a length scale. Therefore, using a finer mesh for the same regularization does not provide additional design degrees of freedom and does not allow for smaller design features in the optimal solution.

[0026] Although unstructured meshes and filtering for regularization are applied, the numerical experiments disclosed herein show that for nearly identical meshes, even for filter radii much larger than the element size, minimums of numerical mesh dependence are frequently obtained.

[0027] Thus, a function is needed to provide users with the ability to measure the "uncertainty" of simulation results generated using a mesh model. The embodiment provides such a solution.

[0028] Figures 1A and 1B show the optimization results of the problem from existing methods. Figure 1A shows the design space 130 and a model of an exemplary L-shaped bracket 100. Figure 1A shows the components of the L-shaped bracket 100 specified in the design space 130. Specifically, Figure 1A shows the pressure-loaded area 101 and the clamped boundary conditions 102, both of which are defined as the non-design space 103. The design space 130 also includes a scale 104 indicating the relative density of the bracket 100.

[0029] Figure 1B shows two optimized designs 110 (Figures 111a-b) and two optimized designs 120 (Figures 121a-b) of bracket 100 resulting from compliance topology optimization implemented using two similar but different meshes. Although all parameters of the optimization formula used to determine designs 110 and 120, such as filter size, were the same, significantly different optimized designs 110 and 120 were obtained. Furthermore, optimized design 120 was shown to be a mesh-induced local minimum, which can be seen in the deterministic optimization iteration history plot 1021 in Figure 10B, which will be discussed later in this specification.

[0030] By forcing a minimum length scale through filtering, the manufacturability of the optimized design is ensured [6]. However, simple filtering methods do not necessarily force a length scale, such as the optimization of the conforming mechanism [7]. As a result of these filtering methods, non-physical hinges occur, or clumpy regions with intermediate density elements occur at single nodes in the mesh, mimicking conforming hinges [8]. Thus, these hinge regions are highly mesh-dependent. A comparison of different filtering techniques is provided in [7]. More advanced approaches, such as considerations of projection filters, are provided in [6], including robust optimization techniques for material position with respect to under-etching and over-etching. Similar approaches are also applied to implement a minimum length scale. Thus, this disclosure also investigates conforming force inverters, which are introduced in [8] and proposed as benchmark examples of the embodiments disclosed herein (see Figures 6A-D, 15A-B, 16A-D, 19A-B, 20A-C, and 21A-B) in reference benchmarks of [7]. A conforming force converter is a conforming mechanism device that not only transmits force by reversing the input force (bringing the output force in the opposite direction), but also uses an elastic deformation mechanism to reverse the output force from compression into tension. A conforming force inverter used as a benchmark for conforming force inverters typically has a non-physical hinge for deterministic optimization. In the benchmark embodiment of the conforming force inverter, the "hinge" is an elastic conforming object subjected to large deformation, while the rest of the mechanism (other than the hinge) has practical stiffness and practical inelasticity compared to the overall deformation during activation of the mechanism.

[0031] Numerical mesh dependency can be viewed as epistemological uncertainty, as it is possible to reduce uncertainty by using improved modeling [9], i.e., the application of finer meshes. Typically, the mesh size (fineness) required to achieve mesh independence is unknown. Generally, robust design optimization approaches take into account chance uncertainty, but can also address uncertainty arising from numerical mesh discretization, which is therefore briefly discussed herein.

[0032] Robust design optimization may take into account not only deterministic parameters but also several indeterminate parameters and some probabilistic measures of the response. Thus, the response to the optimization may yield a smoother surface, thereby reducing the likelihood of ending at a local minimum

[10] . To quantify the probabilistic measures of the response (e.g., mean and variance), an approximation method of uncertainty propagation is needed. The simplest method is the Monte Carlo method

[11] . Although the Monte Carlo method is generally applicable to non-parametric optimization problems such as topology optimization

[12] , it is computationally very expensive. Improving computational efficiency using alternative models may lead to dimensional issues

[13] . Therefore, methods based on local Taylor expansions are often applied to quantifying uncertainty in the context of robust design optimization [10, 14, 15, 16, 17, 18, 19, 20, 21, 22]. A drawback of methods based on local Taylor expansions is that it is not easy to evaluate the gradient of the probabilistic measures with respect to the design variables. Doltsinis and Kang [14, 15] apply a discretized static equilibrium and adjacent method Taylor expansion to determine the mean and variance of the response and the corresponding gradient. This method has been applied to topology optimization by Lazarov et al.

[16] and shows advantages over sampling-based methods. Instead of perturbing the static equilibrium, Lazarov et al.

[16] locally approximates the response function using a Taylor expansion, thereby significantly reducing the computational cost. Using a first-order Taylor expansion requires up to three additional solutions of the adjacent equation system

[17] . For the second-order method, the computational cost scales linearly with the number of random parameters [18, 19]. While the aforementioned methods (

[16] ,

[17] ,

[18] ) require highly intrusive numerical implementations, Steltner et al. [10, 20] and Kruger et al. [19, 21] propose a semi-intrusive method for robust optimization based on Taylor expansions.

[0033] Existing methods for robust optimization account for random uncertainties in applied loads, material properties, or geometry. Robust optimization methods have not been specifically used to address epistemological uncertainty. However, approaches that consider geometric deviations may also be suitable for capturing mesh discretization uncertainty. Different parameterization approaches for geometric uncertainty are presented in the references. Jansen et al.

[23] implement geometric uncertainty using randomly perturbed density filters, where the mesh is unaffected by uncertainty. On the other hand,

[18] implements geometric uncertainty using direct geometric perturbations of the nodes of a finite element mesh. Another method describes geometric uncertainty assuming a random distribution for projection parameters of a heavy side projection

[12] . Uncorrelated node geometric uncertainty is applied in

[24] . However, uncorrelated uncertainty in

[24] is used to model manufacturing uncertainty for individual truss structures. Uncertainty formulas for continuous models generally assume spatial correlations associated with a specific correlation length, forcing the uncertainty formulas to be independent of the nodes in the finite element model.

[0034] While considering mesh perturbations in the same way as geometric uncertainty is explored in the aforementioned references, these references do not solve the problem of obtaining optimized results for mesh dependence. Specifically, existing methods do not solve the problem of obtaining optimized results for mesh dependence because the applied geometric uncertainty is modeled using a stochastic field where spatial correlations mask local effects. Therefore, the embodiment perturbs the nodal coordinates geometrically independently, i.e., without considering spatial correlations. Such a method presents two challenges.

[0035] Firstly, without spatial correlation for geometric uncertainty, it is not possible to reduce the number of random parameters by using techniques such as the Carunen-Loebe expansion or the Expanded Optimal Linear Estimator (EOLE) method

[25] . This first problem is avoided by using an approach in which the computational cost is independent of the number of random parameters. Secondly, without spatial correlation for uncertainty, the standard deviation of the response is mesh-dependent. To solve an embodiment of this second problem, instead of using the standard deviation, L p A norm-based uncertainty measure (i.e., an uncertainty characterization parameter "p") is disclosed. As described herein, embodiments show that for p=1, the uncertainty measure is independent of mesh discretization and, when embedded in a formula for robust design optimization, allows the uncertainty measure to determine a mesh-independent minimum. Furthermore, embodiments show that for higher values ​​of p, the uncertainty measure allows for the suppression of highly localized phenomena (such as fitted hinges or stress singularities).

[0036] Figure 2 is a flowchart of a computer implementation method 200 for simulating an object and determining the uncertainty of the simulation results, according to one embodiment. The method begins in step 201 by receiving a mesh-based model representing a real-world object in the processor's memory, the mesh-based model containing multiple nodes. In step 202, a simulation of the real-world object is performed using the received mesh-based model. The results of performing the simulation may include (i) the behavior of one or more real-world objects, and (ii) the sensitivity of one or more characteristic values ​​at each node of a subset of multiple nodes to the geometric coordinates of the nodes. According to one embodiment, the “sensitivity” of a value referring to a node characteristic may reflect a change in the value based on a variation in the geometric coordinates of the node. Then, in step 203, for each node of the subset of multiple nodes, the method 200 determines the geometrically uncorrelated uncertainty of the node based on the sensitivity of one or more characteristic values ​​at the node and the deviation of the geometric coordinates of the node. As described herein, the term “uncorrelated” means that the uncertainty of each node is independent of the uncertainty of each other. Next, in step 204, uncertainties (e.g., random uncertainty and / or epistemological uncertainty) in the behavior of one or more real-world objects are determined. In method 200, uncertainties in the behavior of one or more real-world objects are determined using a summation function of geometrically uncorrelated uncertainties determined for each node of the subset and an uncertainty characterization parameter (p). To continue, in step 205, method 200 outputs (i) a representation of the behavior of one or more real-world objects and (ii) a representation of the determined uncertainties in the behavior of one or more real-world objects.

[0037] As previously stated, Method 200 is computer-implemented, and therefore its functions and effective operation, such as receiving 201, performing 202, determining 203, determining 204, and outputting 205, can be performed automatically by one or more digital processors. Method 200 can be implemented using any computer device or combination of computing devices known in the art. In particular among other embodiments, Method 200 can be implemented using the computer / device 50 and / or 60 described below herein in relation to Figures 25 and 26.

[0038] In one embodiment of Method 200, the mesh-based model representing the real-world object received in step 201 may be any mesh-based model known in the Art. For example, according to one embodiment, the mesh-based model may be a finite element model, a boundary element model, a finite difference model, or a finite volume model, among many others. In the embodiment, the real-world object represented by the mesh-based model may be any object. For example, in one embodiment, the real-world object may be a bracket, a compliance force inverter (see Figures 6A-D, 15A-B, 16A-D, 19A-B, 20A-C, 21A-B), a crane hook (see Figures 13A-D and 14A-D), an aircraft bracket (see Figures 17A-B and 18), or a Nordic sock (see Figure 22A-C).

[0039] Furthermore, in one embodiment of Method 200, the simulation of a real-world object may be performed in step 202, or may be implemented in another way with any suitable software solution having the ability to include one or more behaviors of a real-world object and the sensitivity of one or more properties at each node to the geometric coordinates of the nodes. Such a simulation can, for example, specify any load scenarios and boundary conditions related to the model and simulate a real-world object subjected to the load scenarios and boundary conditions. Furthermore, the simulation can be performed in step 202 in accordance with principles well known in the art. For example, in one embodiment, the optimization framework and mathematical programming can be performed using the optimization tool Tosca® structure

[33] , while the finite element solver for the principal solution and neighbor sensitivity can be obtained using both the structural solver Abaqus®

[31] and the Tosca® structure

[33] .

[0040] In step 203 of Method 200, determining the geometrically uncorrelated uncertainty of a node based on the sensitivity of one or more properties at the node and the deviation of the node's geometric coordinates can be done by using a 2-norm function of the node's sensitivity to produce a first-order approximation of the standard deviation of the geometrically uncorrelated uncertainty of the node (e.g., the FOSM method for uncertainty propagation).

[0041] In step 204 of Method 200, determining the uncertainty in the behavior of one or more real-world objects using a summation function of geometrically uncorrelated uncertainties and uncertainty characterization parameters determined for each node of the subset may be done according to equations (11) and (12) described below herein.

[0042] In step 205 of Method 200, outputting a representation of the behavior of one or more real-world objects may include providing the user with representations of stress, strain, force, displacement, velocity, acceleration, natural frequency, temperature, and magnetic flux associated with the real-world objects via a graphical user interface (GUI) coupled to the processor. Furthermore, in step 205, outputting a representation of determined uncertainty may provide the user with a representation of areas of a mesh-based model that have been determined to have a high level of uncertainty. For example, one embodiment may identify areas of a mesh that have been determined to have high uncertainty based on nodal uncertainty, i.e., the standard deviation with respect to uncorrelated geometric uncertainty. The embodiment may identify such areas in the output visualization by highlighting the representation of the object or by notifying the user via the GUI that a particular area has been identified as having high uncertainty. Furthermore, one embodiment may provide the user with a summation uncertainty measure using aggregate functions, as shown in the following equations (11) and (12) herein.

[0043] Furthermore, the embodiments provide real-world improvements in the design and modeling of real-world objects, as well as advantages in the manufacturing of real-world objects. By determining and indicating areas of an object in a mesh-based model that are identified as having a high degree of uncertainty, the embodiments show designers or engineers areas of the model that are most susceptible to, for example, tolerance surface concerns in the manufacturing process. By presenting designers and engineers with a display of one or more behaviors of real-world objects, and the determined uncertainties in one or more behaviors of real-world objects, designers and engineers can implement any necessary redesigns to mitigate areas of the model with high uncertainty. The redesigned object may then be manufactured.

[0044] In further embodiments, the mesh-based model obtained in step 201 can represent candidate designs for real-world objects, such as planes, cars, buildings, and bridges. In such embodiments, the determined uncertainties can be used not only to influence manufacturing tolerances but also to influence design safety standards. Furthermore, in such exemplary implementations, method 200 can be used to evaluate candidate designs and determine which designs meet the requirements. A real-world object that meets the design requirements, as determined through the use of method 200, can then be manufactured.

[0045] Furthermore, embodiments of Method 200 may be used to improve and redesign real-world objects. In one such exemplary embodiment, measurements of an existing real-world object may be taken and input into a computer processing environment to generate a mesh-based model representing the real-world object. This mesh-based model may then be received in step 201, and Method 200 may be implemented to determine the behavior and behavioral uncertainties of the real-world object. The determined behavior may be used, for example, by iterating Method 200 to evaluate modifications to the design of the real-world object and determine an optimized design for the real-world object. The existing real-world object may then be modified to correspond to the determined optimized design, or a real-world object having the optimized design may be manufactured.

[0046] One embodiment may include, for example, receiving a display of the value of an uncertainty characterization parameter (p) via user input.

[0047] In one embodiment, in response to the uncertainty characterization parameter exceeding a threshold (for example, the uncertainty characterization parameter is a number (p) such that p>1), the representation of the determined uncertainty may include a representation of the contribution to the determined uncertainty by one or more nodes from a subset of multiple nodes. In such embodiments, the representation of the contribution may include a representation of at least one of the following: magnitude (e.g., magnitude of a vector) and position (e.g., geometric position of a node).

[0048] In another embodiment, in response to the uncertainty characterization parameter being below a threshold (e.g., the uncertainty characterization parameter is p<2, e.g., p=1), the representation of the determined uncertainty does not include a representation of the contribution to the determined uncertainty.

[0049] According to embodiments of Method 200, determining uncertainty in the behavior of one or more real-world objects may involve determining regularized geometrically uncorrelated uncertainty for two or more nodes of a subset of nodes as a function of a weighted average of geometrically uncorrelated uncertainties determined for each of two or more nodes. To continue, such embodiments determine uncertainty in the behavior of one or more real-world objects using (i) an aggregate function of the sum of geometrically uncorrelated uncertainties determined for each node of the subset, (ii) an uncertainty characterization parameter, and (iii) a determined regularized geometrically uncorrelated uncertainty.

[0050] In one embodiment of Method 200, one or more behaviors of a real-world object (e.g., resulting from the simulation performed in step 202) may include at least one of stress, strain, force, displacement, velocity, acceleration, natural frequency, temperature, and magnetic flux.

[0051] Embodiments of Method 200 further include defining a design response of a real-world object as an objective function, the design response based on one or more behaviors of the real-world object and determined uncertainties in one or more behaviors of the real-world object. Such embodiments may continue by defining the optimization problem as a function of the defined design response and at least one constraint. Furthermore, such embodiments may iteratively modify at least one design variable in a mesh-based model representing a real-world object and use the mesh-based model with the modified at least one design variable to perform simulations (step 202), determine geometrically uncorrelated uncertainties (step 203), and determine uncertainties in one or more behaviors of the real-world object (step 204) until the optimization problem is satisfied.

[0052] In another embodiment of Method 200, for a given node in a subset of nodes, the sensitivity of the value of one or more properties at a given node to the geometric coordinates of the given node is the magnitude of the vector.

[0053] In this specification, Method 200 described above is an exemplary method for simulating an object and determining the uncertainty in the results. Embodiments, for example, examples of implementing and applying Method 200 to an exemplary real-world object, and the results of such embodiments are discussed below.

[0054] Formulas for optimization problems The optimization methods described below (e.g., objectives and constraints) introduce a Taylor problem of design, but one embodiment of the uncertainty method is applied to obtain an optimized design that is robust with respect to geometric uncertainty. Non-parameter structural optimization is an iterative process of determining the optimized structural properties of an object represented by a model, for example, a model representing a mechanical object such as a car. A general optimization formula is given by the following equation (1).

[0055]

number

[0056] In the formula, f0 is the system f i The objective function of the finite element residual R(φ, u(φ)) that implicitly defines the response is f i is a general set of inequality constraints, and Φ is the design vector. The design vector φ is the lower boundary φ, respectively. L and upper boundary φ U It is constrained by . For topology optimization, according to one embodiment, the design variable φ defines the relative density of each design element and applies solid isotropic material interpolation by a penaltying (SIMP) approach

[26] . Thus, φ is bounded between 0 and 1. Compliance is often selected as the objective function f0 to be minimized, which is given a relative volume or mass constraint fi. In thickness size optimization, for example, the design variable is the thickness of each shell element. It should be noted that the numerical examples described herein assume linear finite element modeling. However, this is not a requirement of the embodiments, and instead, the embodiments can be directly applied to nonlinear finite element modeling, among many other examples.

[0057] For the topology optimization problem of equation (1) to be properly arranged, regularization of the design variables is necessary. Thus, the filtering introduced in [1] is applied in one embodiment to achieve mesh independence, introduce length scaling, and prevent checkerboarding. An overview of different filter types and their characteristics can be found in [7]. In one embodiment, a conical sensitivity filter

[27] and a design variable filter

[28] are applied based on continuous meandering and expanding motion. Subsequently, the optimization problem is solved iteratively using a moving asymptotic method based on mathematical programming proposed in

[29] .

[0058] For robust design optimization, it is assumed that some of the input parameters are randomly distributed. Therefore, any response is a random variable given by an unknown probability distribution. Thus, the deterministic response function is given by the mean μ, as shown by equation (2) below.f and the standard deviation σ f is replaced by the weight function of f .

[0059]

Equation

[0060] where Φ is a vector containing design variables, α is a vector containing random variables, and κ is a weight coefficient defining the influence of the standard deviation. The robust response function f r can be applied as either an objective function or a constraint.

[0061] Robustness approach based on Taylor expansion The robustness method based on Taylor expansion described below provides a method for determining uncertainty propagation through implicit functions, such as finite element formulas. The mean μ α,i and the standard deviation σ α,i for an uncorrelated random distribution characterized by μ f ​​​​​​​​​​​​​​​​​​​​​​​​​

number

[0066]

number

[0067] For linear approximations, the mean value is equal to the deterministic response function. Next, the gradient of the standard deviation with respect to the design variables is calculated using the chain rule, as shown in equation (7) below.

[0068]

number

[0069] The calculation of the mixed second derivatives in equation (6) may be required in terms of computation time and memory. The neighboring method eliminates the calculation of the second derivative, but the neighboring system is different for all combinations of the response function and random variables.

[0070] The first-order approximate second-moment method of principal sensitivity (ps-FOSM, see

[21] for detailed derivation) re-determines the gradient of the variance such that only one directional derivative is required in addition to the deterministic gradient df / dφ and gradient for a random vector df / dα. Thus, if the gradient for a random vector df / dα is available in the finite element implementation, robust sensitivity can be computed using only one single non-intrusive finite difference evaluation. For example, when geometric uncertainty is considered, the derivative with respect to nodal positions can be mapped to arbitrary uncertainty parameterization. This gradient df / dα is available in the finite element solver Simulia® Abaqus®

[31] , where nodal gradients are conventionally applied for gradient-based shape optimization.

[0071] Assuming equal and uncorrelated random distributions of random variables, the principal sensitivity direction is given by the following equation (8).

[0072]

number

[0073] By inserting finite difference steps in direction (8) and rearranging, we obtain the gradient of the variance as shown in equation (9) below.

[0074]

number

[0075] Note that ε is a scaling factor that needs to be selected appropriately. See

[21] .

[0076] Proposed numerical robustness method The proposed numerical robustness methods described below outline exemplary methods for defining geometric uncertainty in combination with novel uncertainty measures that can be implemented by embodiments. Embodiments achieve robust optimal solutions with respect to the geometric nodal coordinates of a finite element mesh. A potential approach to achieving robust optimal solutions might be to consider the mesh as a discretized random field, assume a correlation function, and then use equation (2) to solve the robust design optimization problem. Nevertheless, numerical experiments show that such approaches do not consistently solve the optimization problem, as using different meshes rather leads to different optimized designs. Furthermore, in the absence of physical uncertainty to be modeled, the selection of a correlation function cannot be determined or justified; that is, a “correct” correlation function cannot be observed in measurement. This observation motivates the approaches presented by the embodiments disclosed herein. Firstly, embodiments may define parameterization of uncertainty in which geometric nodal positions are modeled without assumptions regarding correlations between nodal uncertainties. Secondly, embodiments may introduce a novel uncertainty measure based on uncorrelated nodal uncertainties. Finally, gradients of geometric uncertainty measures are derived, and several aspects of numerical implementations are disclosed.

[0077] Geometrically uncorrelated local distributions at the node level To enforce as few assumptions as possible, one embodiment treats each coordinate of each node as an independent random variable α. In such embodiments, the mean of each variable is equal to the nominal value, and the standard deviation is σ α This is shown by σ. As will be shown later, α The selection is omitted. No specific distribution type is assumed.

[0078] Figure 3 shows an exemplary mesh 300 having a local uncorrelated geometric distribution α at the node level in discrete coordinates for nodes i(n)301 and i+1(n+1)320 (i.e., the uncertainty formula from step 203 of Method 200) according to one embodiment. Each coordinate direction (exemplified by the coordinate system compass 310 (x311, y312, z313)) has the same probability distribution and standard deviation σ α It is considered an independent random variable characterized by the following. For example, as shown in mesh 300, the discrete nodal coordinates 301 are the magnitude 302α of the associated vector representing the x-axis deviation. x,n (σ α ), the magnitude of the related vector representing the y-axis deviation is 303α y,n (σ α ), and the magnitude of the related vector representing the z-axis deviation is 304α z,n (σ α ) has the same property. Similarly, discrete nodal coordinates 320 have the magnitude 322α of the associated vector representing the x-axis deviation. x,n+1 (σ α ), the magnitude of the related vector representing the y-axis deviation is 323α y,n+1 (σ α ), and the magnitude of the related vector representing the z-axis deviation is 324α z,n+1 (σ α ) has.

[0079] The choice to have no correlation between geometric nodal locations may seem counterintuitive. If geometric variation is modeled using scattering of nodal locations as in

[18] , the mesh is a discretization of a stochastic field and would account for geometric uncertainty. Thus, some spatial correlations are expected. However, embodiments model epistemological uncertainty, and arbitrary spatial correlations may fill in local effects, thereby failing to achieve the desired numerical robustness for the geometric mesh distribution.

[0080] There are two drawbacks to not choosing spatial correlation. First, by using correlation, the number of random variables can be reduced, for example, by using the Carunen-Loebe expansion or the Expanded Optimal Linear Estimator (EOLE) method

[25] . Typically, the computational cost of the most robust optimization approach increases or decreases with the number of random variables, resulting in computational costs that are practically unfeasible for most approaches. However, one embodiment avoids this increase or decrease in computational cost with the number of random variables by using a first-order approximation such as the ps-FOSM method, which imposes a computational cost independent of the number of random variables. The effectiveness of using a linear (e.g., first-order) approximation for the proposed geometric uncertainty is shown in Figures 4A and 4B. Figures 4A and 4B show the magnitude of the displacement of a single mesh element 400 as a function of geometric change of the x and y coordinates of loaded nodes.

[0081] Figure 4A shows the x-direction α of the upper left node 402a. x and y-direction α y The magnitude of the displacement f=u as the response to a geometric perturbation in the plane mag Figure 4B shows the representation of mesh element 400 showing variation 401. In mesh element 400, the nodal load P403 causes displacement of the magnitude of the node. L404 represents the element length, and boundary conditions 405a~b are fully clamped. Figure 4B shows the geometric perturbation α of a four-node (402a~d) plane stress element (CPS4

[31] ). x 411a~b and α y f=u for 412a~b magThe expression 410 shows the variation of 401, where α = [-0.25, 0.25]·L413. Figure 4B shows the linear dependence of the magnitude of displacement 401 on the geometric perturbations 411a~b and 412a~b of the small nodes.

[0082] A second disadvantage of not utilizing spatial correlation is that when the mesh is modified or refined, the shape of the local geometric perturbations changes and is regularized when a spatial correlation function is used. Although this result (mesh modification or refinement that changes the shape of local geometric perturbations) is intended by the embodiments, the mesh modeling or refinement by one embodiment causes a change in the standard deviation of the response function as the number and location of the mesh nodes change. Therefore, a new uncertainty measure different from the standard deviation is disclosed herein.

[0083] Novel uncertainty measure for geometrically uncorrelated distributions A novel generalized first-order uncertainty measure using the proposed uncertainty parameterization introduced above in relation to Figures 3 and 4 is disclosed below in this specification. The novel generalized first-order uncertainty measure described herein may be implemented in embodiments, for example, in step 204 of Method 200.

[0084] The standard deviation is commonly chosen as a measure of uncertainty for geometric scattering [10, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24] due to its mathematical properties. Geometric scattering can be understood as the geometrically uncertain location of the mesh nodes, i.e., the uncorrelated geometric uncertainty of the nodes. However, when using uncorrelated nodal uncertainty as described herein, the standard deviation of the response function depends on discretization, i.e., mesh refinement. Therefore, embodiments disclose alternative measures of geometric scattering.

[0085] Kanno

[32] explored different concepts for robust design optimization and defined different robustness measures as L, as shown by the following equation (10). p- We will show that this can be viewed as a variation of norm-based uncertainty.

[0086]

number

[0087] In the formula, the scattering amount α is a scalar δ. α This is shown by f is all α i It is a linear function of width

[0088]

number

[0089] The coefficient α is bounded at intervals of i Assuming this, L 1 -The norm is α i Provides the worst-case combination of all α. i semi-axis length

[0090]

number

[0091] Assuming it is bounded by a hyperellipsoid having L, 2 -The norm is α i Provides the worst-case combination.

[0092]

number

[0093] is α i This is the standard deviation of all α i If we assume that they are uncorrelated, then L 2 -The norm is equal to the standard deviation of the response. Kanno

[32] also approximates it using a high exponent p in equation (10), L ∞ - This refers to work in which the norm was used as an end in itself.

[0094] For robustness optimization with respect to discretization uncertainty, the embodiment uses three spatial L values ​​per node. 2 - Norm, then L p - We propose using the norm to combine the contributions of all nodes. L 2 - It may be necessary to apply the norm at the nodal level; otherwise, the isotropy of the proposed scale (i.e., the scale being independent of the selected coordinate system) may not be satisfied.

[0095] In particular, one embodiment uses the following equation (11) to express the L of the single-node uncertainty. 2 Calculate the norm.

[0096]

number

[0097] This measure of single-node uncertainty given by equation 11 corresponds to the standard deviation of the response f induced by the single-node uncertainty n and approximated by the FOSM method, where σ α σ is the standard deviation of the nodal positions, and this is the same for all nodes and in all directions, i.e., σ α,i =σ α It is assumed that... Subsequently, one embodiment is defined as L as the following equation (12) p We use the norm to aggregate the scale of equation (11) over all geometric discrete points.

[0098]

number

[0099] When p=1, and assuming that f is linearly approximated, the uncertainty scale

[0100]

number

[0101] This can be seen as a measure of the worst-case combination of single-node perturbations affecting response f. When p=2,

[0102]

number

[0103] This is equal to the standard deviation, and if p is a larger value,

[0104]

number

[0105] It approaches the maximum norm and describes only the largest nodal standard deviations that exist within the mesh, which are equivalent to the most important locations within the mesh with respect to geometric variation. Further in this specification, the generalized uncertainty measure proposed in equation (12) is used.

[0106]

number

[0107] We will further investigate this and prove its superiority by comparing it with the standard deviation for the proposed parameterization of geometric uncertainty.

[0108] For larger values ​​of p, a local uncertainty measure of mesh dependence that approximates the largest nodal standard deviation is obtained, but mesh independence is restored for the uncertainty measure proposed using p=1. The response applied to the topology optimization formula is generally mesh-independent and unaffected by the rigid body motion of the domain, thus for the domain

[0109]

number

[0110] The integral of is equal to zero. The integral across the domain is equivalent to the sum of the nodal values. As a result,

[0111]

number

[0112] The sum of the yield magnitudes yields a mesh-independent constant.

[0113] Figure 5 shows the two different meshes, as discussed below in this specification.

[0114]

number

[0115] The proposed uncertainty scale is shown for compliance response values ​​and standard deviations at p=1 and p=2. In Figure 5, the mesh independence of the proposed uncertainty scale is shown for p=1 using the deterministic response function μ f However, this shows a 4.1% change between the two different meshes, and a 3.6% change in the uncertainty scale, which can be considered numerically the same.

[0116] Figure 5 shows sensitivity studies using coarse mesh 500 and fine mesh 510. Coarse mesh 500 includes void regions 501a-b and solid region 502, and similarly, fine mesh 510 includes void regions 511a-b and void region 512. Figure 5 also shows the standard deviation results 520a for coarse mesh 500 (determined using equation 521) and 520b for fine mesh 510 when p=2. Furthermore, Figure 5 shows the aggregate uncertainty results 530a for coarse mesh 500 (determined using equation 530) and 530b for fine mesh 510 when p=1. For sensitivity studies using coarse mesh 500 and fine mesh 510, both models are equally attached to the same boundary conditions and subjected to equivalent loadings P503 and 513, respectively. The solid regions 502 and 512 and the void regions 501a-b and 511a-b of models 500 and 510 are modeled using a linear elastic isotropic material with a Poisson's ratio of 0.30. The solid regions 502 and 512 have a Young's modulus of 1.0, and the void regions 501a-b and 511a-b have a Young's modulus of 1.0 × 10⁻¹⁰. -9 The coarse model 500 is meshed with 80 four-node plane stress elements (CPS4

[31] ) using 99 nodes, and the fine model 510 is meshed with 320 four-node plane stress elements (CPS4

[31] ) using 357 nodes. Geometric standard deviations 520a~b are calculated node by node using the derivative of the response function 540 to the nodal finite element coordinates using FOSM. In addition, compliance responses 541 and uncertainty measures of equation (12) for standard deviations of p=1 and p=2 are calculated, respectively.

[0117] According to one embodiment, the fixed standard deviation σ α This must be selected for the uncertainty scale in equation (12) when assuming uncorrelated parameters. For the formula for robust design optimization, a new uncertainty scale in equation (12)

[0118]

number

[0119] However, this replaces the standard deviation of the objective function given by equation (2). Furthermore, the weight coefficient κ is selected. As shown in equation (12), σ α teeth,

[0120]

number

[0121] It is a linear coefficient of and then multiplied by κ. Therefore, σ α And selecting κ is equivalent. Therefore, one embodiment can be expressed without loss of generality σ α Set =1, leaving only κ as the only parameter to be selected. Numerical studies show that the recommended choice of κ for p=1 is the initial optimization iteration.

[0122]

number

[0123] This indicates that... Regarding other values ​​of p, no general recommendations are provided because the values ​​of p are often model-dependent.

[0124] Furthermore, as can be seen in the vectors of plots 522 and 526 in Figure 5, the nodal gradients generally coincide with the direction of the density gradients at all nodes not associated with boundary conditions or loads. This is expected, as the location of internal nodes should not significantly affect the structural response of a fine mesh. However, adding or removing material on the surface does affect the value of the structural response. Combined with the mesh independence of the uncertainty measure proposed for p=1, we can conclude that the proposed uncertainty measure is a measure of how sensitive the response function is to adding or removing material on the surface. Moreover, recognizing that the uncertainty is generally tied to volume changes and that perturbations at a single node inside the checkerboard pattern are volume-conserving, we can then conclude that the proposed method cannot impose a minimum length scale. Thus, the proposed method (i.e., applying the uncertainty measure using a weighted version of the objective function in structural optimization) is applied in combination with regularization to the objective function and / or constraints, as discussed herein.

[0125] Semi-intrusive gradients related to design variables Uncertainty scale of equation (12) for design variables

[0126]

number

[0127] The gradient is derived as shown in equation (13) below.

[0128]

number

[0129] σ α Note that is set to 1. According to one implementation, the principal sensitivity calculation is similar to the derivation in

[21] based on finite difference approximation and is generalized using a first-order Taylor expansion, as shown by equation (14) below.

[0130]

number

[0131] Rearranging equation (14), which is solved using finite differences, we obtain the following equation (15).

[0132]

number

[0133] In the equation, ε is the scaling factor. Rewriting equation (13), which is calculated using a typical principal sensitivity workflow, as in

[21] , yields the following equation (16).

[0134]

number

[0135] The scaling parameter ε is selected using the size of the generalized finite difference step, as shown by equation (17) below.

[0136]

number

[0137] This applies to p of various choices. According to one embodiment, the size of the finite difference step

[0138]

number

[0139] Generally, it is selected to be two orders of magnitude smaller than the characteristic element length. Generally, according to one embodiment,

[0140]

number

[0141] However, it is selected such that it is significantly smaller than the shortest element end.

[0142] Figures 6A-D show the geometric perturbation field Δα and sensitivity given by the equation (16) of the geometric uncertainty scale with respect to the element topology design variable field, according to one embodiment.

[0143]

number

[0144] However, it demonstrates functionality, which is visualized in relation to the optimized structure.

[0145] Figure 6A shows the design space 601 of the topology optimization model 602 of the inverter mechanism (i.e., the space that optimizes the objective, which is the design domain where the optimization algorithm places materials and determines the layout of the optimized materials). Figure 6B shows the benchmark model 602 of the compliance force inverter and the geometric perturbation field Δα 610 (and the related mathematical description of the perturbation field 610 given by equation 611) and uncertainty sensitivity 620 determined using equation 16 with p=1.

[0146]

number

[0147] This demonstrates both. The compliance force inverter benchmark example reverses the input force, thereby generating an output force in the opposite direction. Furthermore, the compliance force inverter uses elastic deformation not only to transmit force but also to reverse the output force from compression into tension. The compliance force inverter is used as a benchmark example to illustrate the embodiment because the compliance force inverter typically has a non-physical hinge for deterministic optimization methods. Figure 6C shows the geometric perturbation field Δα630 (and the related mathematical description of the perturbation field 630 given by equation 631) and uncertainty sensitivity 640 determined using Model 602 and equation 16 with p=2.

[0148]

number

[0149] Figure 6D shows the geometric perturbation field 650Δα (and the related mathematical description of the perturbation field 650 given by equation 651) and uncertainty sensitivity 660, determined using Model 602 and equation 16 with p=6.

[0150]

number

[0151] This indicates.

[0152] Generally speaking, the proposed L p Norm-based uncertainty measures may have numerical challenges for p=1 and p>>1. Therefore, L p Regarding the implementation details of the norm-based uncertainty scale, additional implementation details are provided below. Specifically, for p=1, the denominator of the principal sensitivity perturbation is approximately zero, resulting in a numerical overflow as visualized in the denominator 612 of equation 611 in Figure 6B. Therefore, regarding the minimum threshold for the geometric uncertainty scale calculation and the corresponding sensitivity, a threshold scheme is proposed below herein. For large values ​​of p, the numerical overflow is Lp -Regarding the regularization of the set of norms, prevention is achieved using the regularization derived below herein. In addition, numerical regularization using nodal gradient filtering is proposed herein in relation to nodal gradient filtering and improves the convergence of local uncertainty measures with large values ​​of p, as demonstrated below.

[0153] To effectively model the uncertainty of the geometric mesh for the shell element model, the parameterization of uncertainty should be strictly in-plane, as discussed herein in relation to in-plane projection. In addition, the response of the shell element to the parameterization of uncertainty is not perfectly linear. Therefore, a central finite difference can be applied instead of a forward finite difference in sensitivity calculations, as discussed below with respect to the implementation of a finite difference approach.

[0154] Numerical implementation example Figure 7 shows a deterministic optimization workflow 700 that includes geometric robustness (with respect to nodal geometric uncertainty) in the optimization formula using a semi-intrusive implementation.

[0155] Method 700 begins with an initial design 711 (for example, obtaining a finite element model representing the real-world object to be optimized). From there, a first finite element analysis is performed in deterministic design step 701, i.e., performed on the mesh without considering sensitivity. From the finite element analysis in step 701, Method 700 moves to step 710, performing an adjacent sensitivity analysis of the deterministic design to obtain the deterministic design used in step 707, as discussed below.

[0156]

number

[0157] Determine the sensitivity.

[0158] Also, from step 701, method 700 may move to step 702 and perform a calculation of geometric uncertainty (steps 702-707). At step 702, an adjacent node sensitivity analysis of the deterministic design

[0159]

Number

[0160] is performed. Following step 702, at step 703, method 700 continues by using equation (12) to determine a new geometric first-order approximation uncertainty measure

[0161]

Number

[0162] . Thereafter, at step 704, a geometric perturbation Δα of the finite element mesh with respect to uncertainty sensitivity is performed using equation (16). After step 704, at step 705, a finite element analysis of the perturbed design is performed by the first-order finite difference method. At step 706, method 700 performs an adjacent sensitivity analysis of the perturbed design

[0163]

Number

[0164] . To continue, at step 707, a robust response function f r is executed using equation (2), and the corresponding sensitivity of the response function is obtained from equation (16)

[0165]

Number

[0166] Determined using finite differences from , the sensitivity of the deterministic design is determined at step 710. Thereafter, at step 708, method 700 updates the design variables using not only the leakage-free response function and the corresponding sensitivity, but also the constraint functions and the corresponding sensitivity. At step 709, the design is evaluated to assess whether a converged design has been determined. If the optimization converges, i.e., the answer is "yes" at step 709, the method determines that the design has been optimized and ends at step 712. Alternatively, if at step 709 the design has not converged, i.e., the answer is "no" at step 709, method 700 returns to step 701 and performs another finite element analysis of the deterministic design using the design updated at step 708.

[0167] In workflow 700, an additional adjacency system is solved for the nodal gradients of the deterministic design (step 710), and thereafter an additional finite element analysis is performed on the geometric perturbation mesh (step 705) to determine the robust response and the corresponding gradients for the design variables.

[0168] According to one embodiment, the proposed numerical robustness method 700 can be implemented using, for example, a software solution from Dassault Systemes Americas Corp. For example, in one embodiment, the optimization tool Tosca® Structure

[33] can be utilized to implement the optimization framework and the mathematical programming, while the finite element solver for the primary solution and the adjoint sensitivity can be obtained using the structural solver Abaqus®

[31] , and workflow 700 can be implemented using both Tosca® Structure

[33] and Abaqus®

[31] . The additional calculations and interface modifications can be processed using a JSON script.

[0169] In one embodiment, Abaqus provides nodal derivatives of the response with respect to the coordinate direction, which are, by definition, derivatives with respect to geometric random variables. Finite difference evaluation, e.g., step 705, is performed using additional Abaqus finite element analysis on a perturbed mesh. Robust response function f with respect to design variables r And the corresponding sensitivity is analyzed into a Tosca structure, for example, in step 707, where the moving asymptotic method

[29] modifies the design variables in each optimization iteration.

[0170] Numerical results of mesh robustness The following sections of this specification describe exemplary implementations of the embodiments and illustrate the advantages of the embodiments.

[0171] Setting p=2 is equivalent to calculating the standard deviation using the original FOSM method, as described above herein. Therefore, the embodiments discussed below are applied first to the L-shaped embodiment, using p=2 for completeness (Figures 1 and 8A-D, 9A-F, 10A-D, 11A-D, and 12A-D). Conversely, selecting the uncertainty characterization parameter p=1, where p>2, indicates the full potential and uncertainty parameterization of the embodiment. Setting p=1 significantly improves the numerical robustness with respect to numerical mesh uncertainty. On the other hand, as discussed below herein, larger values ​​of p increase the manufacturability of the conforming hinge and indicate that the compliance-optimized design is equivalent to the stress-constrained-optimized design.

[0172] First, two examples demonstrating numerical mesh dependency with two very similar meshes are investigated. Specifically, L-shaped brackets (see Figures 8A-D, 9A-F, 10A-D, 11A-D, and 12A-D) and crane hooks (Figures 13A-D, 14A-D), modeled using shell elements, are considered as 3D continuous models. Second, examples of fitted force inverters (Figures 15A-B and 16A-D) are analyzed to demonstrate mesh sensitivity across different mesh types and element sizes. Finally, an aircraft bracket is investigated (Figures 17A-B and 18A-D). This aircraft bracket case study provides insights into distinguishing between numerical and physical local minima.

[0173] Figure 8A shows an exemplary L-shaped bracket model 800 having a design space 801 defined for topology and size optimization, as well as a non-design space 802. The material of the bracket 800 is modeled as a linear elastic isotropic material with a Poisson's ratio of 0.3 and a Young's modulus of 147 GPa, and the filter radius is set to 3.88 mm. The L-shaped bracket 800 has a height 803 of 0.2 m, a length 804 of 0.1 m, and a width 805 of 0.1 m.

[0174] Referring to Figure 8B for an optimization example, the bracket 800 is clamped through two holes 821a and 821b and mounted on top through the pressure-loaded region 810. Furthermore, in the optimization example, the fit of a given pressure load of 100 Pa (within the pressure-loaded region 810) and the clamped boundary conditions 820 (within holes 821a and 821b) is minimized as an objective function subject to a 50% relative mass constraint. Referring to Figure 8C, the bracket 800 (i.e., the CAD model of the bracket) is meshed twice to obtain two different meshes 830 and 840. Mesh model 830 is meshed with 13 S3 (i.e., three-node triangular shell elements) three-node elements and 3275 S4 (i.e., four-node visual system shell elements) four-node 3D shell elements, each having 3426 nodes, and mesh model 840 is meshed with 19 S3 three-node elements and 3265 S4 four-node 3D shell elements, each having 3418 nodes. Figure 8D illustrates the overlay 850 of the two meshes 830 and 840 on the bottom portion of the L-shaped bracket 800, showing the geometric differences in nodes between mesh 830 and mesh 840.

[0175] Figures 9A–F, 10A–D, 11A–D, and 12A–D, discussed below in this specification, show the results of optimization performed using meshes 830 and 840.

[0176] Figures 9A-9F show the topology optimization results determined using sensitivity filters for the two mesh models 830 and 840 shown in Figure 8C. Figures 9A and 9B show the deterministic optimization results 910 and 920 determined using meshes 830 and 840, respectively, with density given by scale 909. Figures 9C-F show the robust optimization results 930, 940, 950, and 960, respectively, with density given by scale 909. Results 930 and 950 were determined using mesh 830, and results 950 and 960 were determined using mesh 840. Deterministic designs 910 (Figure 9A) and 920 (Figure 9B) were determined using f0901 as the objective function to be minimized. Robust designs 930 (Figure 9C), 940 (Figure 9D), 950 (Figure 9E), and 960 (Figure 9F) were determined using f02 as the objective function to be minimized. r It has, and p903 has a value of 2, and σ α 904 used a value of 1. The objective function to be minimized, 902, uses κ=0.01 to obtain results 930 and 940, and κ=0.001 to obtain results 950 and 960. For the results in Figures 9A-F, the sensitivity filter radius is the same for the optimized designs 910 (Figure 9A), 920 (Figure 9B), 930 (Figure 9C), 940 (Figure 9D), 950 (Figure 9E), and 960 (Figure 9F). The deterministic optimization results 910 (Figure 9A) and 920 (Figure 9B), and the robust optimization results 930 (Figure 9C), 940 (Figure 9D), 950 (Figure 9E), and 960 (Figure 9F) are σ α Apply the standard deviation (p=2) of 903 to =1 904. Furthermore, Figures 9A to 9F show the optimization iteration history of the objective function in plots 911, 921, 931, 941, 951, and 961, respectively, for designs 910, 920, 930, 940, 950, and 960.

[0177] The robust optimized designs 930 (Figure 9C) and 940 (Figure 9D) with κ=0.01 have numerically identical optimized response functions, while optimized designs 930 and 940 are mirrored around the centerline. The robust optimized designs 930 (Figure 9C), 940 (Figure 9D), 950 (Figure 9E), and 960 (Figure 9F) have robust compliance f r Despite being optimized for 902, design 910 has a lower deterministic compliance f0 than the deterministically optimized design 920 (Figure 9B). However, design 910 (Figure 9A) remains the design with the lowest deterministic compliance f0.

[0178] The robust optimized designs 950 (Figure 9E) and 960 (Figure 9F) have numerically the same deterministic response function as the deterministically optimized design in 920 (Figure 9B). However, the optimized density distributions 950 and 960 are more similar to the deterministically optimized design 910.

[0179] In conclusion, the proposed node-level uncorrelated uncertainty parameterization using the FOSM method (implemented in embodiments and shown by results 930, 940, 950, and 960) suppresses the possibility of ending in a mesh-induced minimum (shown by result 920). However, the method also, according to embodiments, using larger values ​​for the aggregate parameter p leads to the side effects investigated and characterized below herein. Thus, a robustly optimized design does not have an optimal deterministic response function value. The desired mesh independence, isolated from the described side effects, is obtained using p=1 and is therefore applicable to the following study.

[0180] The proposed uncertainty measure using p=1 is the sum of the standard deviations of the nodes, where each standard deviation has the same weight and satisfies mesh independence as introduced above.

[0181] Figures 10A-D and 11A-D show topology-optimized designs, and Figures 12A-D show thickness-size optimized designs where the thickness of each shell element is a design variable. Sensitivity filters are applied for topology optimization shown in Figures 10A-10D and for thickness-size optimization shown in Figures 12A-12D, where design variable filters are applied for topology optimization in Figures 11A-11D. Note that the filter radii of the design variable filters and sensitivity filters are the same for all 12 optimized designs shown in Figures 10A-D, 11A-D, and 12A-D.

[0182] Figures 10A–10D show the topology optimization results determined using sensitivity filters for the two mesh models 830 and 840 shown in Figure 8C. Figures 10A and 10B show the deterministic optimization results 1010 and 1020 determined using meshes 830 and 840, respectively, with relative densities given by scale 1009. Each of Figures 10A–D includes two representations / visualizations for each optimization result 1010, 1020, 1030, and 1040. Each representation is a blueprint from a specific viewpoint. Figures 10C and 10D show the robust optimization results 1030 and 1040, respectively, with relative densities given by scale 1009. Results 1010 and 1030 were determined using mesh 830, and results 1020 and 1040 were determined using mesh 840. Deterministic designs 1010 (Figure 10A) and 1020 (Figure 10B) were determined using f01001 as the objective function to be minimized. Robust designs 1030 (Figure 10C) and 1040 (Figure 10D) were determined using f as the objective function to be minimized. r It has 1002, with a value of 1 for p1003, a value of 0.001 for κ1004, and σ αUsing the value of 1005 as 1, results 1030 and 1040 are obtained. For the results in Figures 10A to D, the sensitivity filter radius is the same for the optimized designs 1010 (Figure 10A), 1020 (Figure 10B), 1030 (Figure 10C), and 1040 (Figure 10D). The deterministic optimization results 1010 (Figure 10A) and 1020 (Figure 10B), and the robust optimization results 1030 (Figure 10C) and 1040 (Figure 10D) are σ α An uncertainty scale (p=1) 1003 is applied to =1 1005. Furthermore, Figures 10A-D show the optimization iteration history of the objective function used to obtain the results 1010, 1020, 1030, and 1040, respectively, for plots 1011, 1021, 1031, and 1041.

[0183] The topology optimization results 1010 (Figure 10A), 1020 (Figure 10B), 1030 (Figure 10C), and 1040 (Figure 10D) in 10A-D using sensitivity filters demonstrate that the two robust optimized designs 1030 (Figure 10C) and 1040 (Figure 10D) are numerically identical optimized designs. Therefore, the robust topology optimization results 1030 (Figure 10C) and 1040 (Figure 10D) are numerically mesh-independent when the uncertainty of the new sum for p=1 (1003) is included as numerical robustness in the optimization formula. Furthermore, comparing plots 1031 and 1041 with plots 1011 and 1021 shows that robust optimizations 1030 (Figure 10C) and 1040 (Figure 10D) using sensitivity filters require fewer optimization iterations than deterministic optimizations 1010 (Figure 10A) and 1020 (Figure 10B).

[0184] Figures 11A–11D show the topology optimization results determined using design variable filters for two mesh models 830 and 840 shown in Figure 8C. Figures 11A and 11B show the deterministic optimization results 1110 and 1120 determined using meshes 830 and 840, respectively, with relative densities given by scale 1109. Each of Figures 11A–D includes two representations / visualizations for each optimization result 1110, 1120, 1130, and 1140. Each representation is a blueprint from a specific viewpoint. Figures 11C and 11D show robust optimization results 1130 and 1140, respectively, with relative densities given by scale 1109. Results 1110 and 1130 were determined using mesh 830, and results 1120 and 1140 were determined using mesh 840. Deterministic designs 1110 (Figure 11A) and 1120 (Figure 11B) were determined using f01101 as the objective function to be minimized. Robust designs 1130 (Figure 11C) and 1140 (Figure 11D) were determined using f r The objective function to be minimized is 1102, with the value of p1103 set to 1, the value of κ1104 set to 0.001, and σ α Using the value of 1105 as 1, results 1130 and 1140 are obtained. For the results in Figures 11A to D, the design variable filter radius is the same for the optimized designs 1110 (Figure 11A), 1120 (Figure 11B), 1130 (Figure 11C), and 1140 (Figure 11D). The deterministic optimization results 1110 (Figure 11A) and 1120 (Figure 11B), and the robust optimization results 1130 (Figure 11C) and 1040 (Figure 10D) are σ α The uncertainty method (p=1) 1103 is applied to =1 1105. Furthermore, Figures 11A-D show the optimization iteration history of the objective function used to obtain the results 1110, 1120, 1130, and 1140, respectively, for plots 1111, 1121, 1131, and 1141.

[0185] The topology optimizations shown in Figures 11A-D using design variable filters demonstrate that the two deterministically optimized designs 1110 and 1120, as well as the two robustly optimized designs 1130 and 1140, are similar, but the robustly optimized designs 1130 and 1140 have better structural performance.

[0186] Figures 12A–D show the thickness size optimization results determined using sensitivity filters for two mesh models 830 and 840 shown in Figure 8C. Figures 12A and 12B show the deterministic optimization results 1210 and 1220 determined using meshes 830 and 840, respectively, with relative densities given by scale 1209. Each of Figures 12A–D includes two representations / visualizations for each optimization result 1210, 1220, 1230, and 1240. Each representation is a blueprint from a specific viewpoint. Figures 12C and 12D show robust optimization results 1230 and 1240, respectively, with relative densities given by scale 1209. Results 1210 and 1230 were determined using mesh 830, and results 1220 and 1240 were determined using mesh 840. Deterministic designs 1210 (Figure 12A) and 1220 (Figure 12B) were determined using f01201 as the objective function to be minimized. Robust designs 1230 (Figure 12C) and 1240 (Figure 12D) were determined using f r The objective function to be minimized is 1202, with the value of p1203 set to 1, the value of κ1204 set to 0.001, and σ α Using the value of 1205 as 1, results 1230 and 1240 are obtained. For the results in Figures 12A to 12D, the sensitivity filter radius is the same for the optimized designs 1210 (Figure 12A), 1220 (Figure 12B), 1230 (Figure 12C), and 1240 (Figure 12D). The deterministic optimization results 1210 (Figure 12A) and 1220 (Figure 12B), and the robust optimization results 1230 (Figure 12C) and 1240 (Figure 12D) are σ αApply the uncertainty measure (p = 1) 1203 to 11205. Further, FIGS. 12A - D show the optimization iteration history of the objective functions used to obtain the results 1210, 1220, 1230, and 1240 of the plots 1211, 1221, 1231, and 1241, respectively.

[0187] The thickness size optimization results shown in FIGS. 12A - D lead to the same conclusion as the topology optimization using the design variable filters of FIGS. 11A - D. Therefore, the method of the embodiment is not only effective for topology optimization, but also shows that it can generalize the total layer thickness, total layer angle, beads, shape, etc.

[0188] FIGS. 9A - F, FIGS. 10A - D, and FIGS. 11A - D show the deterministic topology - optimized designs using two meshes 830 and 840, and apply the sensitivity filter and design variables for regularization to the optimization formula. The deterministically optimized designs 910 and 920 of FIGS. 9A and 9B, and 1010 and 1020 of FIGS. 10A - B for the sensitivity filter are mesh - dependent, and at least the optimal solution of mesh 2 840 is a minimum value because the compliance is significantly larger than that of the optimized design of mesh 1 830. The deterministically optimized designs 1110 and 1120 of FIGS. 11A and 11B are mesh - independent. However, in contrast to the optimal solution using mesh 1 830 in FIG. 9A, the appearance of the two arms and the optimized compliance values suggest that the optimized designs achieve the same minimum value. FIGS. 12A and 12B show the deterministic thickness - size - optimized designs using two different meshes. The optimized designs seem to be mesh - independent. However, the optimized designs consist of two arms similar to the minimum value obtained using topology optimization.

[0189] The robust, optimized designs 1030 (Figure 10C), 1040 (Figure 10D), 1130 (Figure 11C), 1140 (Figure 11D), 1230 (Figure 12C), and 1240 (Figure 12D) also ensure robust compliance. r Despite being optimized for (1002, 1102, and 1202, respectively), they have lower deterministic compliance f0 (1001, 1101, and 1201, respectively) than the deterministically optimized designs 1010 (Figure 10A), 1020 (Figure 10B), 1110 (Figure 11A), and 1120 (Figure 11B). Therefore, the novel aggregate uncertainty using p=1 in the robust optimization formula of this application suppresses the possibility of ending at a local minimum.

[0190] The deterministic and robust optimization results show that the same conclusions were reached for topology optimization and thickness size optimization, and that the robust optimization formula using novel total uncertainty for p=1 is independent of the design variable type for non-parameter optimization (e.g., other design variable types may be nodal shape design variables, bead design variables, composite layer thickness, composite layer angle, etc.). Secondly, it can be observed that robust optimization requires a similar number of optimization iterations as deterministic optimization.

[0191] Crane hook Figures 13A–D show an embodiment of applying the crane hook optimization, in which the crane hook is clamped to an upper hole and loaded through a distributed force using two load cases. In this example, compliance is minimized to be subject to a 50% volume constraint.

[0192] Figure 13A shows the design space of the crane hook model 1300 in an example of topology optimization. The crane hook model 1300 has a design space 1301 and non-design spaces 1302a-b. The crane hook model 1300 has a height 1303 of 0.21 m, a length 1304 of 0.13 m, and a width 1305 of 0.04 m. The material of the crane hook 1300 is modeled as a linear elastic isotropic material with a Poisson's ratio of 0.3, a Young's modulus of 210 GPa, and a filter radius of 9.73 mm.

[0193] Figure 13B shows two meshes 1310 and 1320 of the crane hook model 1300 subjected to two different load cases (e.g., distributed force 1306a for mesh 1310 and distributed force 1306b for mesh 1320). In the load case of mesh 1310, the crane hook 1300 is subjected to a distributed force 1306a (15.43 N / mm²). 2 In the load case of mesh 1320, the crane hook 1300 receives a distributed force 1306b (11.23 N / mm²). 2 ) is subjected to. As shown in Figures 13A-D, the total compliance is minimized as an objective function subject to a 50% relative mass constraint. In meshes 1310 and 1320 of Figure 13B, boundary condition 1307 is located on the inner portion of the crane hook hole and is clamped.

[0194] Figure 13C shows mesh models 1330 and 1340 generated based on model 1300. Mesh model 1330 (mesh 1) is meshed with 22,335 four-node tetrahedron 3D continuum elements (C3D4

[31] ) having 6,843 nodes, and mesh model 1340 (mesh 2) is meshed with 22,212 four-node tetrahedron 3D continuum elements (C3D4

[31] ) having 6,810 nodes.

[0195] Figure 13D shows a mesh comparison 1350 illustrating two overlapping meshes 1330 and 1340. This comparison 1350 shows that the two meshes 1330 and 1340 were meshed for the same model 1300 and are quite similar, but there are significant small differences between the two meshes.

[0196] Figures 14A–D show the topology optimization results determined using the two mesh models 1330 and 1340 shown in Figure 13C. Figures 14A and 14B show the deterministic optimization results 1410 and 1420, respectively. Figures 14C and 14D show the robust optimization results 1430 and 1440, respectively. Each of Figures 14A–D includes two representations / visualizations for each optimization result 1410, 1420, 1430, and 1440. Each representation is a blueprint from a specific viewpoint. Results 1410 and 1430 were determined using mesh 1330, and results 1420 and 1440 were determined using mesh 1340. The deterministic designs 1410 (Figure 14A) and 1420 (Figure 14B) were determined using f01401 as the objective function to be minimized. The two deterministic optimized designs 1410 and 1420 have fundamentally different optimized material layouts in their designs, despite the very similar differences between the two finite element meshes 1330 and 1340, as shown in the comparison of 1350 in Figure 13D. The robust designs 1430 (Figure 14C) and 1440 (Figure 14D) have f as the objective function to be minimized. r It has 1402, with a value of 1 for p1403, a value of 0.001 for κ1404, and σ α Using the value of 1405 as 1, results 1430 and 1440 are obtained. For the results in Figures 14A to D, the sensitivity filter radius is the same for the optimized designs 1410 (Figure 14A), 1420 (Figure 14B), 1430 (Figure 14C), and 1440 (Figure 14D). The deterministic optimization results 1410 (Figure 14A) and 1420 (Figure 14B), and the robust optimization results 1430 (Figure 14C) and 1440 (Figure 14D) are σ αAn uncertainty scale (p=1) 1403 is applied to =1 1405. Furthermore, Figures 14A-D show the optimization iteration history of the objective function used to obtain the results 1410, 1420, 1430, and 1440, respectively, for plots 1411, 1421, 1431, and 1441.

[0197] Robust designs 1430 (Figure 14C) and 1440 (Figure 14D) have numerically identical optimized material layouts 1430 and 1440, and robust optimized designs 1430 and 1440 have robust compliance f r Despite being optimized for 1402, it also exhibits significantly lower deterministic compliance f0 (3.98 Nm for 1430 and 3.96 Nm for 1440) than two deterministically optimized designs 1410 and 1420 (4.28 Nm for 1410 and 4.38 Nm for 1420). Thus, suppression of mesh-induced minimums and mesh independence for robust optimized designs is also demonstrated for a given 3D continuum embodiment.

[0198] The adaptive force inverter, introduced in [8] and Figures 6A-D and used as a benchmark example in [7], is defined in Figures 15A-B, discussed below, and represents a deterministically optimized design featuring an adaptive hinge with a concentrated mass region of intermediate density elements that are not feasible to manufacture.

[0199] Figures 15A-B show an example of a benchmark inverter mechanism for adapted topology optimization as defined in [8] and Figures 6A-D. The design variable filter is applied to all mechanism examples, and the design variable filter radius 1508 is the same for all optimization formulas. Figure 15A shows the modeling and design space 1500 (having a width of 120 mm 1505 and a height of 40 mm 1506), as well as the actuator spring 1501, roller boundary conditions 1502, clamped boundary conditions 1503, and workpiece 1504. In Figure 15A, the finite element model for mechanism design is a linear spring k in = 1N / mm and point load 1507P inIt has an input actuator 1501 defined using 1 N. The objective function is the linear stiffness k out = 0.001 N / mm and the output displacement u at the attachment point to the workpiece 1504 out 1508 to be minimized. By minimizing the output displacement 1508 as the objective function, an optimized inverter mechanism behavior is imposed. The design is subject to a 20% relative mass constraint. The model of the design space 1500 consists of 7200 four-node continuum plane stress elements (CPS4

[31] ) with 7381 nodes. The material is modeled as a linear elastic isotropic material with a Poisson's ratio of 0.3 and a Young's modulus of 1 MPa, and the filter radius is set to 2.5 mm. FIG. 15B shows the finalized optimized design 1510, forming a hinge consisting of intermediate density elements much smaller than the defined filter radius 1508, while the relative density is indicated by the scale 1509.

[0200] In FIG. 15A, the contribution of the uncertainty of the nodes at the edge of the initial design space 1500 is not included in the calculation of the geometric uncertainty in order to avoid the contribution of the singularity uncertainty from the node boundary conditions and the load. Note that the design variable filter is applied and the design variable filter radius is the same for all optimized mechanism designs.

[0201] Figures 16A-D show topology optimization mechanisms (Figures 16A-D, respectively) using four different meshes for the model defined in Figures 15A-B. Figure 16A shows a structured coarse mesh 1610 containing 7200 elements and 7381 nodes, along with the deterministic optimization result 1611 and the robust optimization result 1612. Figure 16B shows an unstructured coarse mesh 1620 containing 8872 elements and 9054 nodes, along with the deterministic optimization result 1621 and the robust optimization result 1622. Figure 16C shows a structured fine mesh 1630 containing 28800 elements and 29161 nodes, along with the deterministic optimization result 1631 and the robust optimization result 1632. Figure 16D shows an unstructured fine mesh 1640 containing 36,143 elements and 36,505 nodes, as well as the deterministic optimization result 1641 and the robustness optimization result 1642. The relative density in Figures 16A-D is indicated by the scale 1609. For the robustness optimization in Figures 16A-D, the value of p1601 is 1, the value of κ1602 is 0.2, and σ αUsing the value 1603 as 1, results 1612, 1622, 1632, and 1642 were obtained. Two designs were determined for each mesh 1610, 1620, 1630, and 1640, specifically, deterministically optimized mechanism designs 1611 (Figure 16A), 1621 (Figure 16B), 1631 (Figure 16C), and 1641 (Figure 16D), and robust optimized mechanism designs 1612 (Figure 16A), 1622 (Figure 16B), 1632 (Figure 16C), and 1642 (Figure 16D). A design variable filter was applied, and the filter radius (1508, Figure 15B) was the same for all eight optimized designs (1611, 1612, 1621, 1622, 1631, 1632, 1641, and 1642). Numerical results show that the four robust optimized mechanisms 1612 (Figure 16A), 1622 (Figure 16B), 1632 (Figure 16C), and 1642 (Figure 16D) are similar in material layout and independent of their respective meshes (1610, 1620, 1630, and 1640). The deterministically optimized mechanisms 1611 (Figure 16A) and 1621 (Figure 16B) for the two coarse meshes 1610 (Figure 16A) and 1620 (Figure 16B) are significantly different. The robust optimized designs 1612 and 1622 (Figure 16A) and 1620 (Figure 16B) for the two coarse meshes 1610 are slightly different. The two deterministically optimized designs 1611 and 1621 for two coarse meshes 1610 (Figure 16A) and 1620 (Figure 16B) are fundamentally different from the two deterministically optimized designs 1631 and 1641 for two fine meshes 1630 (Figure 16C) and 1640 (Figure 16D). Furthermore, the deterministically optimized designs 1631 and 1641 for fine meshes 1630 (Figure 16C) and 1640 (Figure 16D) are quite similar to the four designs 1612 (Figure 16A), 1622 (Figure 16B), 1632 (Figure 16C), and 1642 (Figure 16D), and are robustly optimized. These results demonstrate that robust optimization, including the uncertainty of geometrically uncorrelated nodal distributions in the optimization formula, ensures a robust mechanism independent of acceptable variations in finite element meshes, thereby demonstrating mesh independence.

[0202] From the deterministic response values, it cannot be concluded that the obtained mesh-independent designs are generally optimal or optimal for a particular mesh. This is due to the length scaling effect in the conforming hinge. Using p=1, no significant visual difference in hinge design is observed between the deterministically optimized mechanisms (1611, 1621, 1631, 1641) and the robustly optimized mechanisms (1612, 1622, 1632, 1642). However, the compliance response is highly sensitive even to small differences in the geometric layout of the conforming hinge. Therefore, the influence of the method on the conforming hinge is further discussed below in relation to embodiments of compliance force inverters in Figures 19A-B, 20A-C, and 21A-B, using larger values ​​for the collective parameter p.

[0203] Aircraft bracket Figure 17A shows an embodiment of the aircraft bracket using aircraft bracket model 1700, which is loaded into the upper hole using four load cases and clamped to the bottom through eight bolt connections as shown in Figure 17A. In this embodiment, compliance is subject to a 12.5% ​​volume constraint, including undesigned areas around the upper and bottom holes, and is minimized.

[0204] Figures 17A and 17B show the design space and mesh of the aircraft bracket used in topology optimization, as well as the deterministically optimized design, respectively. Specifically, Figure 17A shows the design space 1701, load conditions 1702a to d, boundary conditions 1703a to d, and non-design space 1704a to e for Model 1700. Boundary conditions 1703a to d are clamped with connecting elements representing the bolt stiffness in all eight holes of the aircraft bracket (four of which are shown in Figure 17A). The total compliance of the four load cases 1702a to d (LC1(1702a)=11.27kN, LC2(1702b)=11.27kN, LC3(1702c)=6.44kN, and LC4(1702d)=6.44kN) is minimized as an objective function subject to a 12.5% ​​relative mass constraint. Model 1700 is meshed using 448,897 tetrahedral 3D continuum elements (C3D4

[31] ) with 83,481 nodes. The material is modeled as a linear elastic isotropic material with a Poisson's ratio of 0.33 and a Young's modulus of 68 GPa, and the filter radius is set to 1.3 times the characteristic element length. Figure 17B shows the deterministically optimized designs 1710 and 1720 using shift limits of 0.10 (1710) and 0.25 (1720) for the density design variable per optimization iteration.

[0205] Deterministically optimized designs depend heavily on specific parameters of the optimization algorithm. When limiting the maximum allowable change in design variables per optimization iteration, only the movement limits of the so-called movement asymptotic method

[29] change. Figure 17B shows deterministically optimized designs 1710 and 1720 using movement limits of 0.10 (1710) and 0.25 (1720), respectively. The dimensions of the optimized aircraft bracket are visualized using an isocat value of 0.3. The two deterministic designs 1710 and 1720 are fundamentally different but have the same total compliance value. Note that the compliance for different load cases is fundamentally different in both deterministically optimized designs.

[0206] Generally, to achieve numerical mesh independence for the embodiments described above, κ is selected such that the weighted uncertainty measure is much smaller than the deterministic response value. However, if κ is selected such that the weighted uncertainty measure has a value similar to the deterministic compliance value of the optimized design, no suppression of local minima is observed. Therefore, numerical studies are conducted and discussed in relation to Figure 18 using larger values ​​for the weighting parameter κ.

[0207] Figure 18 shows the deterministic optimization results 1710 and 1720 from Figure 17B, along with the robust topology optimization results 1801-1808, for the deterministic topology optimization shown in Figure 17B, determined using different values ​​for the weight coefficient κ. The deterministic designs 1710 and 1720 were determined using f01730 as the compliance value. From the deterministic optimization results 1710 and 1720, it is observed that κ=0 results in a deterministic optimized design, and different movement limits of 0.10 (1710) and 0.25 (1720) are applied, respectively. Furthermore, additional second deterministic f01813a-b are optimized compliances to the deterministic optimization, using their respective robust optimized designs as initial density distributions to prove the existence of local minima. Compared to the deterministic f01810a-b, the compliance of the second deterministic f01813a-b is improved, but this is expected due to the relaxation of the objective function (κ=0), and there is almost no visual difference observed between the design of the deterministic response f01810a-b and the second deterministic f01813a-b. Therefore, a deterministically optimized design for the second deterministic f01813a-b has not been shown.

[0208] Figure 18 also shows eight robust optimized designs 1801-1808 for two sets of movement limit values ​​(i.e., a movement limit of 0.10 for 1801-1804 and a movement limit of 0.25 for 1805-1808), and different κ values ​​(κ=0.4 for 1801 and 1805, κ=1 for 1802 and 1806, κ=2 for 1803 and 1807, and κ=3 for 1804 and 1808). Figure 18 also shows deterministic compliance f01810a-b, uncertainty, using the optimized designs as initial Φ1813a-b.

[0209]

number

[0210] 1811a~b, robust compliance f r (κ)1812a~b and deterministic compliance f0 are shown.

[0211] All optimized designs 1710, 1720, and 1801-1808 in Figure 18 differ significantly from their counterparts (i.e., designs having either the same movement limit or the same κ value, e.g., design 1801 is the counterpart to 1802, 1803, 1804, and 1805). All robust optimized designs (1801, 1802, 1803, and 1804) in Figure 18 for an applied movement limit of 0.10 have a higher deterministic compliance value 1801a compared to the deterministically optimized designs 1710 and 1720.

[0212] When the robust optimized designs 1801–1808 are applied as initial density distributions for deterministic optimization, optimized designs very similar to each applied initial density distribution are obtained. Furthermore, the corresponding deterministic compliance values ​​f01810a–b in Figure 18 are very similar for different density distributions. Therefore, it can be concluded that all obtained designs 1710, 1720, and 1801–1808 are local minima in a very flat region of deterministic compliance.

[0213] Secondly, it can be concluded that the proposed method generally fails to suppress the local minimums of this application. The differences are most likely caused by physical local minimums, which cannot be addressed using this robust approach. However, the deterministically optimized designs in Figures 17A-B and 18, obtained using a movement limit of 0.25 (i.e., 1720), are mesh-induced local minimums. The best-optimized design using the proposed method occurs at κ=0.1 (not shown), which differs significantly from the deterministic design. Furthermore, increasing κ, as shown in Figure 18, leads to the suppression of small design features, resulting in a slightly but more symmetrical optimized design for only a small increase in compliance value.

[0214] Numerical results of physical robustness The effects of using a larger value p for the aggregate parameter (uncertainty characterization parameter) are discussed below in this specification. The results discussed below in this specification provide an understanding of the intended numerical mesh independence for p=1 (i.e., uncertainty characterization parameter is below the threshold) and the standard deviation for p=2 (i.e., uncertainty characterization parameter is above the threshold). Two additional studies below are the enhanced hinge design for the fitted mechanism shown herein, which results in a design similar to the stress-constrained design, and the compliance optimization below in this specification. p The norm is L for large values ​​of p. ∞ It approximates the norm. Therefore, only the node with the largest geometric standard deviation affects the optimized design.

[0215] Single-node hinges and stress singularities can be seen as shortcomings of numerical modeling. However, optimized designs are classified as having physical robustness because they are highly sensitive to the physical realization and manufacturing of the design in unexpected places.

[0216] Compliance Power Inverter The above considerations conclude that the proposed method affects hinge design due to the standard deviation of large nodes in so-called compliant hinges. Therefore, the same example is investigated using a larger value p for the set parameter, thereby introducing a very local uncertainty measure in the compliant hinge.

[0217] Figure 19A shows visualizations 1920 - 1925 (corresponding to iteration 0, 5, 10, 20, 40, and 72 respectively) of the design variable (relative density) using the new uncertainty for σ out of α = 1, the output displacement u as the objective function to minimize the objective function, and the objective function f r is minimized. The relative density of visualizations 1920 - 1925 is indicated by scale 1910. The visualizations 1926 - 1931 in Figure 19B each show the vector of the standard deviation of the nodes, corresponding to optimization iterations 0 (1901), 5 (1902), 10 (1903), 20 (1904), 40 (1905), and 72 (1906)

[0218]

Number

[0219] in magnitude. Figure 19B shows that the uncertainty measure

[0220]

Number

[0221] significantly changes between optimization iterations 0 (1901), 5 (1902), 10 (1903), 20 (1904), 40 (1905), and 72 (1906). The overall

[0222]

Number

[0223] The location of the standard deviation of the node that contributes most to the process changes fundamentally between optimization iterations 0 (1901), 5 (1902), 10 (1903), 20 (1904), 40 (1905), and 72 (1906). Figure 19B also shows that the standard deviation of the largest node for the last optimization iterations 20 (1904), 40 (1905), and 72 (1906) is located at the so-called fitted hinge. Therefore, the proposed method, which considers an uncorrelated nodal random distribution, is an alternative approach to designing fitted hinges in topology optimization for fitted mechanisms.

[0224] Figures 20A-C show the topology optimization results of Model 1510 defined above, in relation to Figures 15A-15C. Specifically, Figure 20A shows the result 2010 of deterministic optimization. Figure 20B shows the results 2020a-c for robust optimization without nodal gradient filtering, and Figure 20C shows the results 2030a-c for robust optimization with nodal gradient filtering using radius R. The relative densities of results 2010, 2020a-c, and 2030a-c are indicated by the scale 2009. The robust optimized design is (i) p=1, κ=0.2, and σ α =1 (2020a and 2030a), (ii) p=6, κ=10, and σ α =1 (2020b and 2030b), and (iii) p=10, κ=10, and σ α =1 (for 2020c and 2030c). Applying the design variable filter, the corresponding design variable filter radius is the same for the optimized designs 2010, 2020a-c, and 2030a-c.

[0225] The optimization results in Figures 20A-C and 21A-B illustrate the effect of the parameters of the geometric uncertainty approach on so-called hinges in compliant mechanism designs. These hinges make the inverter mechanism a "compliant" design because the hinges have highly localized bends. However, some of these compliant mechanism designs have a better design than other hinges in compliant mechanisms because the deformation is not as highly localized as in deterministic designs (e.g., when robustness is involved). For example, as shown in Figure 21A, compliant mechanism design 2101 with a concentrated block region 2102 is a "poor" design. As shown in Figure 21B, compliant mechanism design 2111 with a concentrated block region 2114 is an "improved" design. Compatibility mechanism design 2112 without a concentrated block region in hinge 2115 is a "good" design. Compatibility mechanism design 2113 without a concentrated block region in hinge 2116 is a "good" design. A “good” hinge represents a hinge that is manufacturable and can be physically formed correctly, while a “bad” hinge may have intermediate density or single nodes that represent poor modeling and is not manufacturable. Both “good” and “bad” hinges may have local deformations, but in a “good” hinge, these deformations are spread to more elements than in a “bad” hinge.

[0226] Figure 21A shows a deterministically optimized fitted hinge mechanism design 2101 consisting of a concentrated clump region 2102 with intermediate density elements. Several procedures (e.g., [7] and [6]) indirectly suppress the concentrated clump region with intermediate density elements to mimic the fitted hinge 2102 (which is an extension of the visualization of the hinge mechanism design 2101). Figure 21B shows (p=1, κ=0.2, σ α =1[2111)), (p=6, κ=10, σ α =1(2112)) and (p=10, κ=10, σ αThe embodiment demonstrates an approach to uncertainty using geometrically uncorrelated nodal distributions by introducing a method of pushing the material layout of the optimized mechanism design toward distributed fitted hinges 2114, 2115, and 2116, which are extended to visualizations of designs 2111, 2112, and 2113, respectively. This approach to uncertainty using an uncorrelated distribution of geometric nodes introduces a method of pushing the optimized mechanism design toward distributed fitted hinges, improving the feasibility of manufacturing.

[0227] Furthermore, the robust optimized designs 2111–2113 in Figure 21B also show that applying a higher exponent, e.g., p=6 (designs 2112 and 2213), or including nodal gradient filtering (design 2113), forces a larger distribution of fitted hinges compared to applying a lower exponent, e.g., p=1 and no nodal gradient filtering (design 2111).

[0228] As a result, the formula for robust optimization using an uncertainty measure of geometrically uncorrelated nodal distributions may also be applied to constructively influence the physical design, which in turn is less sensitive to very local geometric distributions, for example, due to manufacturing considerations. Hence, a geometrically robust design is obtained with respect to the most sensitive locations for manufacturing.

[0229] Nordic socks Figures 22A-C show an example of an L-shaped beam, also known as a Nordic sock. Figure 22A shows the design space 2200 of the L-shaped beam 2210 with a load 2201 of f = 1.5 kN and boundary conditions 2202 for the L-shaped beam 2210. The L-shaped beam 2210 has a left-side length L2203 (L is 0.2 m) and the relative width of the upper portion 2204 is

[0230]

number

[0231] The relative length of the bottom region 2205 is 5L, and the relative length of the right side 2206 is

[0232]

number

[0233] The material has a Poisson's ratio of 0.33, a Young's modulus of 71 MPa, and a density of 1 kg / m³. 3 It is modeled as a linear elastic isotropic material. The depth of the L-shaped beam 2210 is 1 mm, and the filter radius is set to 2.6 mm. Figure 22B shows the mesh model 2220 of the design space 2220, consisting of 6400 four-node continuum plane stress elements (CPS4

[31] ) and 6601 nodes. Due to stress singularities at the interior angle, the L-shaped beam 2210 is frequently studied in references on topology optimization. Figure 22C shows the stress constraint f at 600 MPa. 1-stress Mass minimization f used in 2232 0-mass Mass constraint f having the same mass as 2231 and stress-constrained optimized design 2230 1-mass Compliance minimization f provided to 2242 0-compliance The definitively optimized designs 2230 and 2240 for 2241 are shown. In Figure 22C, the relative density is indicated by the scale 2209. The two optimized designs 2230 and 2240 differ significantly because design 2240, which minimizes compliance, has a stress singularity 2243 at the interior angle.

[0234] Figures 23A to 23D show the topology optimization results of the L-shaped beam model 2210 defined above, in relation to Figures 22A to 22C. Figure 23A shows the stress constraint f 1-stress Mass minimization f used in 2232 0-mass Figure 22C shows the deterministically optimized design 2230 for 2231. Figure 23B shows the mass constraint f 1-mass Compliance minimization f provided to 2242 0-compliance Figure 23C shows the deterministically optimized design 2240 for 2241, derived from Figure 22C. Figure 23C shows the mass constraint f 1-massCompliance minimization f provided to 2332 r-compliance A robust and optimized design 2330 for 2331 is shown. Figure 23D shows the mass constraint f 1-mass Filtered compliance minimization f provided to 2342 r-filtered-compliance A robust and optimized design 2340 is shown for 2341. Both designs 23D and 2340 are achieved by filtering out nodal gradients (i.e., nodal uncertainty) with a filter radius R2309, and with p=5, κ=1, σ α Determined using a standard deviation of =1. Optimized designs 2230, 2240, 2330, and 2340 have a movement limit of 0.1 per optimization iteration imposed on the regularized density variable. As shown in Figures 23A-23D, the relative density is indicated by the scale 2309. Furthermore, Figures 23A-23D show the f for designs 2230, 2240, 2330, and 2340. misses stress 2310a, f compliance 2310b, f r-compliance 2310c, f r-filtered-compliance 2310d, and f mass Figure 2310e is shown. Furthermore, Figures 23A to 23D show the optimization iteration history of the objective function used to obtain the results 2230, 2240, 2330, and 2340 in plots 2301, 2302, 2304, and 2305, respectively.

[0235] As expected, the compliance-optimized design 2240 in Figure 23B has twice the stress value f misses stress At the expense of having 2310a, the optimized design 2240 has a lower stress value f misses stress It has 2310b. In contrast, the robust compliance-optimized designs 2330 and 2340 in Figures 23C-D have p=5, κ=1, σ αUsing =1, with and without filtering of nodal gradients, the stress-constrained optimized design 2230 is more similar to the deterministic compliance optimized design 2240 than to the deterministic compliance optimized design 2240, and does not include the inner acute angle 2243 which causes high stress singularity for the two robust compliance optimized designs 2330 and 2340. Furthermore, the stress response value 2310a is significantly reduced for the two robust compliance optimized designs 2330 and 2340. The robust optimized design 2340 shows that when the nodal gradient is filtered, as characterized by the filter radius R2390, the stress response 2310a and mass 2310e have the same values ​​as the stress-constrained design 2230 in Figure 23A.

[0236] Embodiments using large values ​​for the uncertainty characterization parameter p can be concluded to implicitly minimize stress peaks at the expense of slightly greater deterministic compliance. However, selecting the weighting coefficient κ is not straightforward when a specific threshold for stress must be met. Furthermore, filtering nodal gradients improves convergence compared to unfiltered robust optimization. However, a robust stiffness-optimized design requires approximately twice as many optimization iterations to reach the deterministically optimal solution compared to both stiffness and strength optimizations.

[0237] Parameter values ​​of the geometric uncertainty scheme The embodiment is the parameter κ of equation (2) and the parameter of equation (17)

[0238]

number

[0239] You may use the following values ​​for this.

[0240]

number

[0241]

number

[0242] L p - Implementation details of norm-based geometric uncertainty measures This specification provides additional implementation details for exemplary robustness methods.

[0243] Calculation of geometric uncertainty scale and minimum threshold for corresponding sensitivity The geometric perturbation field Δ in equation (16) when p=1 approaches zero in the denominator (shown in Figure 6B) (division by numerical zero).

[0244]

number

[0245] When it approximates zero, there are numerical problems, and the geometric perturbation field vector Δα then has a large value. Therefore, in one embodiment, uncertainty scale

[0246]

number

[0247] And to determine the corresponding sensitivity, a threshold scheme is applied to suppress division by zero. First, the threshold scheme determines the maximum nodal standard deviation for all nodal contributions. Then, the uncertainty scale is given by equation (18) below.

[0248]

number

[0249] Furthermore, when determining the geometric perturbation field Δα, only nodal standard deviations that have a value higher than a specific proportion β of the largest nodal standard deviation are included.

[0250]

number

[0251] The threshold parameter β is set to 0.01 in all current studies and applies only to the Shell model when optimized using the aggregate parameter p=1. Numerical studies use a geometric uncertainty scale.

[0252]

number

[0253] This indicates that the value of hardly changes compared to not applying the threshold scheme.

[0254] L p - Regularization of normed sets Regularization of the standard deviation of the aggregated nodes is performed in one embodiment to prevent numerical overflow in numerical implementations that utilize agglomerative functions, for example, values ​​greater than 5 for p. In such embodiments, the maximum value of the nodal standard deviation is σ. max Using this, we regularize the term that is punished by the force p, which gives the following equation (19).

[0255]

number

[0256] This then provides the corresponding geometric perturbation for gradient calculation.

[0257]

number

[0258] This is given by the following equation (20).

[0259]

number

[0260] Filtering of nodal gradients A linear filter can be applied to the nodal gradient df / dα for additional regularization. Classical filtering allows for cancellation, which causes additional local minima, because the input to the nodal gradient is not strictly positive. Thus, according to one embodiment, the magnitude of the nodal gradient df / d can be maintained while preserving the direction of the gradient. n As a scaling method, nodal filtering can be applied. This nodal filtering can be given by the following equation (21).

[0261]

number

[0262] In the formula, the weight function H i It is given by a linearly damped conical function and is defined by the following equation (22).

[0263]

number

[0264] In the formula, R is the filter radius,

[0265]

number

[0266] is the distance between any pair of nodes.

[0267] If p=1,

[0268]

number

[0269] Since the direction of the gradient is unaffected because it is applied only as a perturbation for gradient calculation, filtering of nodal gradients has no practical effect other than linear scaling of the uncertainty measure. Therefore, according to one embodiment, nodal filtering of the gradient is applied for p>1, resulting in improved convergence toward solid void design.

[0270] Implementation details of the shell model: This specification describes the application of exemplary embodiments to shell models.

[0271] In-plane projection When applying the geometric uncertainty scale to the shell elements, the nodal gradient is calculated using the nodal normal direction n, as shown in equation (23) below.

[0272]

number

[0273] This is projected onto the plane of the shell element.

[0274]

number

[0275] The projection has the same effect as directly defining random variables using a coordinate system that is strictly defined on the plane of the shell elements. Therefore, the uncertainty of the underlying mesh is modeled more accurately with respect to the typical procedure applied to the generation of the shell mesh.

[0276] Implementation Examples of Finite Difference Methods The sensitivity in equation (16) can be viewed as a forward (or backward) finite difference sensitivity method. See

[34] . Thus, the equation for numerical sensitivity can also be modified to a central finite difference method

[34] , as shown by equation (24) below.

[0277]

number

[0278] Generally, the central finite difference method is numerically more accurate than the forward (or backward) finite difference method in sensitivity calculations, but the central finite difference method has twice the computational cost of the forward (or backward) finite difference method. Therefore, the finite difference method for sensitivity is specifically considered in Figures 24A-C below, using four 2D plane stress continuum elements (CPS4

[31] ) and four 3D shell elements (S4

[31] ).

[0279] Figure 24A shows the geometric perturbation field Δ α The geometric perturbations α in the x and y directions, respectively, of the central node 2401, which represents the central node. x and α y Sensitivity to

[0280]

number

[0281] The expression 2400 represents the variation. In Figure 24A, L2402 represents the element length, P2403 represents the load, and φ2404 represents the design variable element. Boundary conditions 2405a to c are fully clamped.

[0282] Figure 24B shows deformations 2410 and 2420, respectively. For deformation 2410 in Figure 24B, sensitivity was measured using a four-node plane stress element (CPS4

[31] ).

[0283]

number

[0284] α of 2411 ~ [-0.01, 0.01]·L, geometric perturbation α x 2414 and 2415, and α y Models 2412 and 2413 are shown. For deformation 2420, a four-node plane stress element (CPS4

[31] ) was used, and the sensitivity

[0285]

number

[0286] α ~ [-0.1, 0.1]·L of 2421, geometric perturbation α x 2424 and 2425, and α y Figures 2422 and 2423 are shown. Figure 24C shows variations 2430 and 2440, respectively. For variation 2430 in Figure 24C, sensitivity is shown using a four-node shell element (CPS4

[31] ).

[0287]

number

[0288] α of 2431 ~ [-0.01, 0.01]·L, geometric perturbation α x 2434 and 2435, and α y Models 2432 and 2433 are shown. For deformation 2440, a four-node plane stress element (S4

[31] ) is used, and sensitivity

[0289]

number

[0290] α ~ [-0.1, 0.1]·L of 2441, geometric perturbation α x 2444 and 2445, α y Numbers 2442 and 2443 are shown.

[0291] Results 2410 and 2420 in Figure 24B show that the forward (or backward) finite difference method is sufficient for 2D continuum element models because the results are linear at 0 in all directions. However, results 2430 and 2440 in Figure 24C show that the central finite difference method, which is computationally more expensive and has a twist at 0 (the gradient changes at 0, but remains linear beyond 0 in all directions), is the best for 3D shell element models.

[0292] Several numerical experiments have verified that the forward finite difference method is sufficient for element types with only translational degrees of freedom, such as all element types (e.g., 2D or 3D continuum elements), while the central finite difference method should be applied to element types with both translational and rotational degrees of freedom (e.g., shell elements and beam elements) (e.g., all element types).

[0293] Exemplary advantages of the embodiment In this embodiment, the L of the uncorrelated standard deviation of the nodes p - Based on a set of norms, we present a novel isotropic geometric uncertainty measure. When p=1, the uncertainty measure is mesh-independent and effectively suppresses mesh-induced local minima for optimized designs. For higher values ​​of the uncertainty characterization parameter p, the method reduces the effects of local geometric deformations, analogous to the properties of stress-constrained optimized designs, and constructively contributes to forcing desired length scales for hinge designs in fitted mechanisms. We implement a semi-intrusive and computationally efficient first-order principal sensitivity approach to compute the required sensitivity of the geometric uncertainty measure, utilizing only one additional finite element analysis per optimization iteration for continuum models and two additional finite element analyses per optimization iteration for shell models.

[0294] It should be noted that the selected numerical examples, characterized by symmetric design domains, simple load configurations, and linear finite element models, were deliberately chosen due to their demonstrated sensitivity to mesh-induced local minima. While the proposed method proves effective in these scenarios, embodiments may be used in relation to any real-world object / design.

[0295] Computer support Figure 25 is a schematic diagram of a computer network in which an embodiment may be implemented. The client computer / device 50 and server computer 60 provide processing, storage, and input / output (I / O) devices for running application programs and the like. The client computer / device 50 can also link to other computing devices, including other client devices / processors 50 and server computers 60, via a communication network 70. The communication network 70 can be part of a remote access network, a global network (e.g., the Internet), a collection of computers worldwide, a local area or wide area network, and a gateway that communicates with each other using its respective protocol (e.g., TCP / IP, Bluetooth®, etc.). Other electronic device / computer network architectures are also suitable.

[0296] Figure 26 is a block diagram illustrating an exemplary embodiment of computer nodes (e.g., client processor / device 50 or server computer 60) within the computer network 70 of Figure 25. Each computer node 50, 60 encompasses a system bus 79, which is a set of hardware lines used for data transfer between components of a computer or processing system. The system bus 79 is essentially a shared conduit that connects various different elements of a computer system (e.g., processor, disk storage, memory, I / O ports, network ports, etc.) and enables information transfer between these elements. An I / O device interface 82 is connected to the system bus 79 for connecting various input / output devices (e.g., keyboard, mouse, display, printer, speaker, etc.) to the computer nodes 50, 60. A network interface 86 allows the computer nodes to connect to various other devices connected to the network (e.g., network 70 in Figure 25). Memory 90 provides volatile storage for computer software instructions 92a and data 94a used to implement embodiments of the present invention (e.g., method 200 in Figure 2). The disk storage 95 provides non-volatile storage for computer software instructions 92b and data 94b used to implement embodiments of the present disclosure. A central processing unit 84 is also connected to the system bus 79 and provided for executing computer instructions.

[0297] In one embodiment, the processor routines 92a-92b and data 94a-94b are a computer program product (generally referred to as 92) comprising a computer-readable medium (e.g., a removable storage medium such as a DVD-ROM, CD-ROM, diskette, or tape) that provides at least a portion of the software instructions for the disclosed system. The computer program product 92 can be installed by any preferred software installation procedure, as is well known in the art. In another embodiment, at least a portion of the software instructions may also be downloaded via cable, communications, and / or wireless connections. In yet another embodiment, the program of the Disclosure is a computer program propagated signal product embodied in a propagated signal on a propagated medium (e.g., radio waves, infrared waves, laser waves, sound waves, or electrical waves propagated over a global network such as the Internet or other networks). Such carrier medium or signal provides at least a portion of the software instructions for the routines / programs 92 of the Disclosure.

[0298] In alternative embodiments, the propagated signal is an analog carrier wave or digital signal carried on a propagation medium. For example, the propagated signal may be a digital signal propagated over a global network (e.g., the Internet), a communication network, or another network (such as network 70 in Figure 25). In one embodiment, the propagated signal is a signal transmitted over a period of time via the propagation medium, such as instructions for a software application transmitted in packets over a network over a period of milliseconds, seconds, minutes, or more. In another embodiment, the computer-readable medium of the computer program product 92 is a propagation medium that the computer system 50 can receive and read, for example, by receiving the propagation medium and identifying the propagated signal embodied within the propagation medium, as described above for the computer program propagated signal product.

[0299] Generally, the terms "carrier medium" or "transient carrier" encompass the aforementioned transient signals, propagated signals, propagation mediums, storage mediums, and so on.

[0300] In other embodiments, the program product 92 may be implemented as so-called Software as a Service (SaaS), or as other installations or communications supporting the end user.

[0301] Embodiments or aspects thereof may be implemented in the form of hardware, including but not limited to hardware circuits, firmware, or software. When implemented in software, the software may be stored on any non-temporary computer-readable medium configured to allow a processor to read the software or a subset of its instructions. The processor is then configured to execute instructions and operate a device or to cause a device to operate in the manner described herein.

[0302] Furthermore, hardware, firmware, software, routines, or instructions may be described herein as performing specific operations and / or functions of a data processor. However, naturally, such descriptions included herein are merely for convenience, and such operations are actually the responsibility of the computing device, processor, controller, or other device that performs the firmware, software, routines, instructions, etc.

[0303] Naturally, flowcharts, block diagrams, and network diagrams may contain more or fewer elements, be arranged differently, or be represented differently. However, even more naturally, a particular implementation may carry out in a particular way the number of block diagrams and network diagrams, as well as the number of block diagrams and network diagrams illustrating the execution of the embodiment, are determined.

[0304] Therefore, other embodiments may also be implemented in various computer architectures, physical computers, virtual computers, cloud computers, and / or some combination thereof, and thus the data processors described herein are for illustrative purposes only and not to limit the embodiments.

[0305] All patents, published applications, and references cited herein are incorporated in their entirety by reference.

[0306] While exemplary embodiments have been specifically shown and described, those skilled in the art will understand that various modifications of form and detail can be made therein without departing from the scope of embodiments included in the appended claims.

[0307] For example, the foregoing description and details of the embodiments shown in the figures refer to, but are not limited to, the applicant-assignee (Dassault Systemes Americas Corp) and Dassault Systemes, tools and platforms for illustrative purposes. Other similar tools and platforms may be appropriate.

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Claims

1. A computer implementation method, which is determined by the processor, The process involves receiving a mesh-based model representing a real-world object into the memory of the processor, wherein the mesh-based model includes a plurality of nodes. Performing a simulation of the real-world object using the received mesh-based model, wherein the results of the simulation include (i) one or more behaviors of the real-world object, and (ii) the sensitivity of the values ​​of one or more properties at each node of a subset of the nodes to the geometric coordinates of the nodes, For each node in the subset of the plurality of nodes, the geometrically uncorrelated uncertainty for the node is determined based on the sensitivity of the value of one or more characteristics at the node and the deviation of the geometric coordinates of the node. Determining the uncertainty in one or more behaviors of the real-world object, wherein the uncertainty in one or more behaviors of the real-world object is determined using the summation function of geometrically uncorrelated uncertainties and uncertainty characterization parameters determined for each node of the subset, (i) outputting a display of one or more behaviors of the real-world object, and (ii) outputting a display of the determined uncertainty in the one or more behaviors of the real-world object, Computer implementation methods, including those mentioned above.

2. The computer implementation method according to claim 1, further comprising receiving a display of the value of the uncertainty characterization parameter.

3. The computer implementation method according to claim 1, wherein, in response to the uncertainty characterization parameter exceeding a threshold, the representation of the determined uncertainty includes a representation of the contribution to the determined uncertainty by one or more nodes from the subset of the plurality of nodes.

4. The computer implementation method according to claim 3, wherein the indication of the contribution includes at least one indication of size and position.

5. The computer implementation method according to claim 1, wherein, in response to the uncertainty characterization parameter falling below a threshold, the representation of the determined uncertainty does not include a representation of the contribution to the determined uncertainty.

6. Determining the uncertainty in one or more of the behaviors of the aforementioned real-world objects, The regularized geometrically uncorrelated uncertainty for two or more nodes of the subset of the plurality of nodes is determined as a function of the weighted average of the geometrically uncorrelated uncertainties determined for each of the two or more nodes, A computer implementation method according to claim 1, comprising: (i) the summation function of geometrically uncorrelated uncertainties determined for each node of the subset; (ii) the uncertainty characterization parameter; and (iii) determining the uncertainty in one or more behaviors of the real-world object using the determined regularized geometrically uncorrelated uncertainties.

7. The computer implementation method according to claim 1, wherein one or more of the behaviors of the real-world object include at least one of stress, strain, force, displacement, velocity, acceleration, natural frequency, temperature, and magnetic flux.

8. The computer implementation method according to claim 1, further comprising defining the design response of the real-world object as an objective function, wherein the design response is based on one or more behaviors of the real-world object and the determined uncertainties in the one or more behaviors of the real-world object.

9. The computer implementation method according to claim 8, further comprising defining the optimization problem as a function of the defined design response and at least one constraint.

10. The computer implementation method according to claim 9, further comprising: iteratively modifying at least one design variable in the mesh-based model representing the real-world object; and performing the simulation, determining the geometrically uncorrelated uncertainties, and determining the uncertainties in one or more behaviors of the real-world object using the mesh-based model having the modified at least one design variable until the optimization problem is satisfied.

11. The computer implementation method according to claim 1, wherein, for a given node among the subset of the plurality of nodes, the sensitivity of the value of one or more characteristics at the given node to the geometric coordinates of the given node is the magnitude of a vector.

12. The computer implementation method according to claim 1, wherein the determined uncertainty is at least one of accidental uncertainty and epistemological uncertainty.

13. A computer-based system, Processor and A processor and a memory storing computer code instructions are provided, and the processor and the memory use the computer code instructions to operate the computer-based system. The process involves receiving a mesh-based model representing a real-world object into the memory of the processor, wherein the mesh-based model includes a plurality of nodes. Performing a simulation of the real-world object using the received mesh-based model, wherein the results of the simulation include (i) one or more behaviors of the real-world object, and (ii) the sensitivity of the values ​​of one or more properties at each node of a subset of the nodes to the geometric coordinates of the nodes, For each node in the subset of the plurality of nodes, the geometrically uncorrelated uncertainty for the node is determined based on the sensitivity of the value of one or more characteristics at the node and the deviation of the geometric coordinates of the node. Determining the uncertainty in one or more behaviors of the real-world object, wherein the uncertainty in one or more behaviors of the real-world object is determined using the summation function of geometrically uncorrelated uncertainties and uncertainty characterization parameters determined for each node of the subset, (i) outputting a display of one or more behaviors of the real-world object, and (ii) outputting a display of the determined uncertainty in the one or more behaviors of the real-world object, A computer-based system configured to perform a specific action.

14. The computer-based system according to claim 13, wherein, in response to the uncertainty characterization parameter exceeding a threshold, the representation of the determined uncertainty includes a representation of the contribution to the determined uncertainty by one or more nodes from the subset of the plurality of nodes.

15. The computer-based system according to claim 13, wherein, in response to the uncertainty characterization parameter falling below a threshold, the representation of the determined uncertainty does not include a representation of the contribution to the determined uncertainty.

16. In determining the uncertainty in one or more of the behaviors of the real-world objects, the processor and the memory use computer code instructions to the computer-based system The regularized geometrically uncorrelated uncertainty for two or more nodes of the subset of the plurality of nodes is determined as a function of the weighted average of the geometrically uncorrelated uncertainties determined for each of the two or more nodes, (i) the summation function of the geometrically uncorrelated uncertainties determined for each node of the subset, (ii) the uncertainty characterization parameter, and (iii) the determined regularized geometrically uncorrelated uncertainties to determine the uncertainty in one or more behaviors of the real-world objects, A computer-based system according to claim 13, configured to perform the following:

17. The processor and the memory use the computer code instructions to configure the computer-based system. The computer-based system according to claim 13, further configured to define the design response of the real-world object as an objective function, wherein the design response is based on the one or more behaviors of the real-world object and the determined uncertainties in the one or more behaviors of the real-world object.

18. The processor and the memory use the computer code instructions to configure the computer-based system. The computer-based system according to claim 17, further configured to define the optimization problem as a function of the defined design response and at least one constraint.

19. The processor and the memory use the computer code instructions to configure the computer-based system. Iteratively modifying at least one design variable in the mesh-based model representing the real-world object, and performing the simulation, determining the geometrically uncorrelated uncertainty, and determining the uncertainty in one or more behaviors of the real-world object using the mesh-based model having the modified at least one design variable until the optimization problem is satisfied. A computer-based system according to claim 18, further configured to perform the following:

20. A computer program product, A non-temporary computer-readable medium is provided, the computer-readable medium contains program instructions, and when the program instructions are executed by the processor, the processor... The process involves receiving a mesh-based model representing a real-world object into the memory of the processor, wherein the mesh-based model includes a plurality of nodes. Performing a simulation of the real-world object using the received mesh-based model, wherein the results of the simulation include (i) one or more behaviors of the real-world object, and (ii) the sensitivity of the values ​​of one or more properties at each node of a subset of the nodes to the geometric coordinates of the nodes, For each node in the subset of the plurality of nodes, the geometrically uncorrelated uncertainty for the node is determined based on the sensitivity of the value of one or more characteristics at the node and the deviation of the geometric coordinates of the node. Determining the uncertainty in one or more behaviors of the real-world object, wherein the uncertainty in one or more behaviors of the real-world object is determined using the summation function of geometrically uncorrelated uncertainties and uncertainty characterization parameters determined for each node of the subset, (i) outputting a display of one or more behaviors of the real-world object, and (ii) outputting a display of the determined uncertainty in the one or more behaviors of the real-world object, A computer-based system configured to perform a specific action.

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