Suppression of post-buckling behavior using added artificial forces in optimization

Abstract

Embodiments determine an optimized design of a real-world object. A computer-based model representing a real-world object is defined, and the computer-based model is modified to include at least one labor force. The at least one labor force is defined as varying with a physics-based behavior. The real-world object is iteratively optimized relative to a load using the modified computer-based model. The result of the iterative optimization is the optimal design of the real-world object.

Description

[0001] Related applications

[0002] This application claims the benefit of U.S. Provisional Application No. 63 / 570,035, filed on March 26, 2024. The entire teachings of the above application are incorporated herein by reference. Background Art

[0003] There are a number of existing products and simulation systems on the market for designing and simulating objects (e.g., components, parts, and assemblies of components or parts, as well as other examples) in a modeling space (e.g., a three-dimensional (3D) modeling space). Such systems typically employ computer-aided design (CAD) and computer-aided engineering (CAE) programs. These systems enable users to construct, manipulate, and simulate objects or assemblies of objects, such as complex 3D models of real-world objects. CAD and CAE systems therefore provide representations of modeled objects using edges, lines, faces, polygons, or closed volumes. Lines, edges, faces, polygons, and closed volumes can be represented in various ways, such as non-uniform rational basis splines (NURBS).

[0004] CAD systems manage the parts or assemblies of modeled objects, which are primarily geometric specifications. Specifically, CAD files contain specifications from which geometric shapes are generated. Representations are generated from these geometric shapes. The specifications, geometry, and representations can be stored in a single CAD file or in multiple CAD files. CAD systems include graphical tools for representing modeled objects to designers; these tools are specialized for displaying complex objects. For example, an assembly can contain thousands of parts, i.e., components. CAD systems can be used to manage object models stored in electronic files.

[0005] CAD and CAE systems use computer-based models, such as CAD models, CAE models, and finite element models to represent objects. A computer-based model can be programmed in a way that gives the model the properties of one or more underlying real-world objects (e.g., physical, material, or other physics-based) that the model represents. When a computer-based model is programmed in this way, it can be used to perform simulations of the objects represented by the model. For example, a finite element model (FEM) can be used to represent the interior of a vehicle, the acoustic fluid around a structure, and any number of real-world objects and systems. When a given model represents an object and is programmed accordingly, it can be used to simulate the real-world object itself. For example, a FEM representing a stent can be used to simulate the use of the stent in a real-life medical setting.

[0006] Computer-based models can be used to improve the design of the object represented by the model. Design improvements can be identified by using optimization techniques that run a series of simulations to identify changes in the model design and, therefore, the underlying object represented by the model. Summary of the Invention

[0007] While optimization methods exist for designing and optimizing real-world objects, these existing methods can benefit from improvements. Embodiments provide such improvements. Embodiments relate to functionality for determining an optimal design of a real-world object by adding one or more artificial forces to a computer-based model representing the real-world object.

[0008] One such example embodiment relates to a computer-implemented method for determining an optimized design for a real-world object. The method is executed by a processor and begins by defining a computer-based model representing the real-world object in a memory of the processor. The processor modifies the computer-based model to include at least one artificial force, wherein the at least one artificial force is defined as varying with a physics-based behavior. The method continues by iteratively optimizing the real-world object relative to a load using the modified computer-based model. The result of the iterative optimization is an optimized design for the real-world object.

[0009] In an embodiment, the computer-based model can be any computer-based model known in the art. For example, according to an embodiment, the computer-based model is a finite element model, a boundary element method, a finite difference method, a finite volume method, or a discrete element method.

[0010] According to an embodiment, iteratively optimizing includes, a processor performing a simulation using the modified computer-based model, and evaluating, based on results of the simulation, a value of a property of the real-world object for consistency with design parameters. In response to the evaluation determining that the value of the property is consistent with the design parameters, the method determines that the modified computer-based model represents an optimized design of the real-world object. In response to the evaluation determining that the value of the property is inconsistent with the design parameters, the method iteratively: (i) updating the modified computer-based model, (ii) performing a simulation using the updated modified computer-based model, and (iii) evaluating, based on results of the simulation performed using the updated modified computer-based model, the value of the property of the real-world object for consistency with the design parameters until the evaluation determines that the value of the property is consistent with the design parameters.

[0011] In an embodiment, iteratively optimizing includes executing, by a processor, a simulation using the modified computer-based model. According to one such embodiment, executing the simulation includes applying a load to the modified computer-based model, determining a value of a physics-based behavior of the computer-based model resulting from the applied load, and applying at least one artificial force to the modified computer-based model based on the determined value of the physics-based behavior. In an embodiment, applying the at least one artificial force includes resisting deformation in the modified computer-based model in response to the determined value of the physics-based behavior exceeding a threshold.

[0012] In another embodiment, iteratively optimizing includes a processor solving for an equilibrium primitive of a modified computer-based model, determining a design response and at least one corresponding sensitivity for a design variable of the modified computer-based model, and optimizing the real-world object using the modified computer-based model, the determined design response, and the at least one corresponding sensitivity. In such embodiments, a result of the optimization is either a converged design representing an optimized design of the real-world object or a non-compliant solution. In response to the result of the optimization being a non-compliant solution, the embodiment modifies the at least one artificial force and iteratively solves, determines, and optimizes using the modified at least one artificial force.

[0013] In an embodiment, the at least one artificial force is configured to oppose deformation in the modified computer-based model in response to a value of the physics-based behavior of the real-world object exceeding a threshold.

[0014] In another embodiment, the physics-based behavior includes at least one of: displacement, velocity, and acceleration.

[0015] Yet another embodiment includes determining a point of inflection within a real-world object.

[0016] According to an embodiment, the at least one artificial force is configured to suppress a post-buckling response of the real-world object.

[0017] Another embodiment relates to a system for automatically determining an optimized design for a real-world object. The system includes a processor and a memory having computer code instructions stored thereon. In such embodiments, the processor and the memory having the computer code instructions are configured to cause the system to implement any embodiment or combination of embodiments described herein.

[0018] In another embodiment, a computer program product includes a non-transitory computer-readable medium having computer program instructions stored thereon. The instructions, when executed by a processor, cause the processor to implement any embodiment or combination of embodiments described herein.

[0019] It should be noted that the method, system and computer program product embodiments can be configured to implement any embodiment or combination of embodiments described herein. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] The foregoing will be apparent from the following more particular description of exemplary embodiments as illustrated in the accompanying drawings in which like reference numerals refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the embodiments.

[0021] Figure 1A and 1B Shows the overall buckling of the physical system.

[0022] Figure 1C and 1D Shown is the overall buckling of a physical system with an artificial force added, according to an embodiment.

[0023] Figure 2 is a flow chart illustrating a computer-implemented method for determining an optimized design of a real-world object by adding one or more artificial forces, according to an embodiment.

[0024] Figure 3A -C is a graph showing suppression of physical post-buckling of a physical system using added internal artificial forces, according to an embodiment.

[0025] Figure 4 A generalized continuum of physical systems and the corresponding equilibrium applied to external loads according to an embodiment is shown.

[0026] Figure 5 A generalized continuum of physical systems with added artificial forces for suppressing post-buckling response is presented according to an embodiment.

[0027] Figure 6 FIG. 1 is a flow chart illustrating an iterative design process for adding artificial forces according to an embodiment.

[0028] Figure 7 Equations for determining a solution to a physical system with added artificial forces are shown, according to an embodiment.

[0029] Figure 8A -C is a graph showing example characteristics of added artificial force in the post-buckling range according to an embodiment.

[0030] Figure 9 Equations for determining the original solution according to an embodiment are shown.

[0031] Figure 10A Equations showing an adjoint method for sensitivity of a design response according to an embodiment are presented.

[0032] Figure 10BEquations showing the adjoint method for sensitivity of design responses with respect to design variables according to an embodiment.

[0033] Figure 11 The equations of the adjoint system of Lagrange multipliers {λ} according to an embodiment are shown.

[0034] Figure 12 An equation showing a direct method for determining the sensitivity of a design response in an embodiment is shown.

[0035] Figure 13 Equations for determining direct derivatives according to an embodiment are presented.

[0036] Figure 14A and 14B Graph showing optimized settings according to an embodiment.

[0037] Figure 15A -D shows an example model for topology optimization of a physical beam system according to an embodiment.

[0038] Figure 16 Demonstrating the Figures 15A-15D Model of a beam system with artificial forces added.

[0039] Figure 17A is a graph of force versus displacement for a beam system where no artificial force is applied and where an artificial force is applied, according to an embodiment.

[0040] Figure 17B and 17C The resulting structural responses are shown for an initial beam design and an optimized beam design, according to an embodiment.

[0041] Figure 18A -C is a graph showing the optimization iteration history of an embodiment implemented with added artificial force.

[0042] Figure 19A Shows the results of the geometric linear optimization design.

[0043] Figure 19B The results of the geometric nonlinear optimization design according to the embodiment are shown.

[0044] Figure 20A is a graph of force versus displacement for a beam system where no artificial force is applied and where an artificial force is applied, according to an embodiment.

[0045] Figure 20B and 20C The resulting structural responses are shown for an initial beam design and an optimized beam design, according to an embodiment.

[0046] Figure 21A and 21BFIG. 4 is a graph showing the optimization iteration history of an embodiment.

[0047] Figure 22A Shows the results of the geometric linear optimization design.

[0048] Figure 22B The results of the geometric nonlinear optimization design according to the embodiment are shown.

[0049] Figure 23 is a simplified block diagram of a computer system in which embodiments may be implemented.

[0050] Figure 24 is a simplified block diagram of a computer network environment in which embodiments may be implemented. DETAILED DESCRIPTION

[0051] The following is a description of exemplary embodiments.

[0052] Embodiments provide functionality for determining an optimized design for a real-world object. Embodiments suppress post-buckling behavior during optimization by using artificial forces added, for example, by adding the artificial forces to a computer-based model representing the real-world object. According to embodiments, the artificial forces are configured to counteract deformations in the computer-based model representing the real-world object. In embodiments, counteracting the deformations enables the optimization procedure to converge to a solution that provides an optimized design for the real-world object.

[0053] For many optimization applications (e.g., thin-walled structures in aerospace applications), the design driver for optimization is the global buckling load carrying capacity, i.e., how much load the structure can handle before it begins to buckle under the load. This buckling load carrying capacity often depends on small imperfections in the structure (e.g., imperfections caused by manufacturing uncertainties), material nonlinearities (e.g., plasticity), and pre-buckling modes. Therefore, geometrically nonlinear modeling helps capture these effects, and determining the original solution in the post-buckling range is challenging. Therefore, embodiments provide a novel approach for sensitivity-based optimization that captures the global buckling point in the modeling for optimization, but artificially suppresses the post-buckling range for optimization by using added, artificially imposed forces.

[0054] In other examples, embodiments may benefit simulations in the aerospace and defense industries, as these industries often employ large-deformation nonlinear finite element modeling to obtain physically correct results for the pre- and maximum-buckling points. Furthermore, many models used in these applications are exposed to significant external loads. The embodiments disclosed herein enable these industrial applications to simulate external loads and further enable the determination of the pre- and maximum-buckling points. Embodiments provide an iterative optimization workflow for simulating the pre- and maximum-buckling points of a primitive solution.

[0055] An advantage of the embodiments disclosed herein is that geometrically nonlinear modeling for large displacements can simulate force-displacement curves for the pre-buckling range, global buckling point, and post-buckling range. Global buckling point calculations often include calculating various small defects, material nonlinearities, and pre-buckling modes of the structure. Typically, the industry goal is to increase the global buckling point while taking these modeling features into account. Post-buckling optimization is challenging both theoretically and numerically due to the highly nonlinear behavior and lack of uniqueness of the forces in the force-displacement curve. Furthermore, global buckling optimization is challenging due to the challenges in both modeling and solving post-buckling. Thus, the embodiments present a method of adding artificial forces to suppress responses in the post-buckling range, wherein the pre-buckling response and the global buckling response are not suppressed, so these responses can still be applied in the optimization. The embodiments can also be applied to any sensitivity-based structural optimization principles known to those skilled in the art, such as topology optimization, shape optimization, size optimization, and rib optimization, as well as other examples.

[0056] Buckling and Geometric Nonlinear Optimization Using the Arc Length Method

[0057] Existing solutions for addressing buckling during optimization use the arc-length method to solve the postbuckling response of geometrically nonlinear analysis optimizations. The arc-length method forces equilibrium to converge along arcs. This allows the original solution to be determined for the postbuckling range even when the stiffness or global operator slope is zero or negative.

[0058] When the stiffness slope is zero or negative, the arc length method will reduce the external load in the postbuckling range, as shown below Figure 1B Thus, although the embodiment suppresses post-buckling, the arc length method still cannot achieve the final artificial load in the optimization, as described below Figure 1C and Figure 1D Described in .

[0059] Buckling and Geometrically Nonlinear Optimization Using Prescribed Displacements

[0060] Another existing solution for buckling and geometric nonlinear optimization uses prescribed displacements. To implement this functionality, a post-buckling method for geometric nonlinear analysis optimization is solved using prescribed displacements instead of using external load manipulation primitives. Thus, for a given prescribed displacement, the solution containing the following parameters is uniquely determined. Figure 1B The original solution of the force in .

[0061] In contrast, embodiments enable unique determination of the loads comprising the following description for a given external load. Figure 1D The problem is that for many modeling applications, prescribed displacement modeling is not of interest because many real-world applications are subject to external loads that are actually modeled.

[0062] Buckling Optimization Using Koiter's Method

[0063] Another existing approach implements gradient postbuckling optimization by applying Koiter's asymptotic method, where the so-called "Koiter factor" is optimized. This method can determine the postbuckling of a system based on either linear or nonlinear prebuckling responses.

[0064] Koiter's method is fundamentally different from the modeling, analysis, and optimization implemented by the embodiments. In the embodiments, the equalization residual can be determined directly for geometric nonlinear modeling. Applying Koiter's method for optimization generally requires new and different implementations for calculating sensitivity.

[0065] Linear Buckling Optimization

[0066] Another existing approach applies linear buckling modeling to optimization. Linear buckling often overestimates the buckling load because it estimates local buckling. Therefore, optimization based on linear buckling may not optimize the structure's overall buckling resistance.

[0067] The embodiments use geometrically nonlinear modeling for large displacements, while linear buckling analysis uses geometrically linear modeling assuming linear displacements. Furthermore, linear buckling optimization has no imperfections for the structure, while the methods presented by the embodiments disclosed herein enable different imperfection sets to be used to analyze the structure.

[0068] In summary, existing methods for buckling and geometric nonlinear optimization using the arc-length method and the prescribed displacement method are suitable for large-displacement modeling for nonlinear finite element analysis, and can predict the pre-buckling range and buckling point. In contrast, the solution presented in the embodiments artificially suppresses post-buckling under external loads. No existing method for global buckling point optimization can address problems involving large-displacement and geometric nonlinear modeling.

[0069] Embodiments use artificial forces for large displacement modeling of nonlinear finite element analysis to suppress physical post-buckling in order to predict pre-buckling and post-buckling ranges. Embodiments provide many advantages of sensitivity-based optimization. For example, embodiments suppress post-buckling ranges in both modeling and optimization iterations. Additionally, through embodiments, structural modeling using continuously monotonically increasing loads can be directly applied to optimized modeling, which has several advantages of its own. The continuous monotonic increase in load provides user-friendly and easy modeling for industrial applications. In contrast, prescribed displacement modeling is generally not suitable for practical industrial applications because many industrial applications are subject to external force loads that are actually modeled.

[0070] Embodiments also reduce the runtime and computational cost associated with simulations when post-buckling responses are suppressed. Embodiments further provide rigorous optimization of the pre-buckling range and increase the buckling load. Existing optimization implementations and methods can be reused and applied by embodiments, as existing design responses (e.g., displacements) can be directly applied to the objective functions and / or constraints of sensitivity-based optimization. Additionally, existing numerical implementations and theories for sensitivity calculations (e.g., adjoint sensitivity to displacements) can be directly applied to optimizations in embodiments.

[0071] Overview of global buckling and stability including postbuckling

[0072] Figure 1A and 1B shows the overall buckling of the physical system, and Figure 1C and 1D The overall buckling of a physical system in which an artificial force is added according to an embodiment is shown.

[0073] Figure 1A The overall buckling of the physical system (column 101) is shown when a force 102 is applied to the column 101. The applied load 102 compresses the column 101 and causes buckling 103 of the column 101. Figure 1B The external force P111 is relative to Figure 1A Graph 110 of displacement 112 of column 101 is shown in FIG. Curve 113 in graph 110 illustrates the pre-buckling 115 and post-buckling 116 ranges, as well as the buckling point 114. It should be understood that the buckling point 114 may also be referred to as the maximum load point, bifurcation point, limit point, or load carrying capacity, among other examples.

[0074] Figure 1C The global buckling of the physical system (column 122) is shown when a force 121 is applied to column 122 to compress column 122. According to embodiments described herein and compared to column 101, column 122 has an added artificial internal force 123 that is configured to be applied after the buckling point 133 so that the physical global buckling point 133 of the physical system (i.e., beam 122) can still be consistently captured. Figure 1D is the external force 131 relative to Figure 1C Graph 130 of displacement 132 of column 122. Graph 130 shows the added artificial force ( Figure 1C 123 and Figure 1D 134) how to stabilize the displacement 132 and suppress physical post-buckling of the physical system after the buckling point 133, thereby allowing the embodiment to reach a converged solution and determine the optimal design.

[0075] This novel optimization method provided by the embodiments still captures the global buckling point in the modeling for optimization, but artificially suppresses the post-buckling range for optimization by using artificially added forces. While existing methods present sensitivity-based optimization for linear buckling, the linear buckling in these existing methods is inaccurate for global buckling and requires a fully nonlinear sensitivity-based optimization solution for global buckling. By suppressing post-buckling from geometrically nonlinear analysis, the embodiments are able to directly optimize the pre-buckling response as well as the global buckling response. By using sensitivity-based buckling optimization, the embodiments offer numerous advantages over geometrically nonlinear optimization.

[0076] Return to Figures 1A-1D , Figure 1A A typical example of a column 101 with a compressive load causing buckling 103 is shown. Large deformation modeling and geometrically nonlinear modeling are crucial for practical modeling of many applications that aim to simulate the structural response in both the pre- and post-buckling ranges as well as the global buckling point, such as 114 [1.1-1.5]. Large deformation and geometrically nonlinear modeling introduce bifurcation points in the modeled response. For example, these bifurcation points can be crucial for practical modeling of the pre- and post-buckling ranges and the global buckling point, and therefore for predicting the stability of the system. This bifurcation between the pre- and post-buckling ranges 115 and 116 is shown in Figure 1B 110 . Characteristically, these physical bifurcation points produce ill-conditioned stiffness operators at the global buckling point and in the post-buckling range. This can be numerically solved using the arc-length method and prescribed displacement loads to obtain a physical primitive solution in the post-buckling range, however, as described above, these existing methods are not adequate.

[0077] See now Figure 1C Instead of modeling the post-buckling range, such as 116, embodiments add artificial forces 123 to suppress the responses in the post-buckling range. The pre-buckling response and the overall buckling response are not suppressed, so these responses (e.g., the displacement 132 in the pre-buckling range up to and including the buckling point 133) can still be used for validation or optimization. Adding artificial forces 123 to suppress post-buckling also allows for reaching the final artificial load 134 in the analysis used for optimization, as embodiments suppress post-buckling. Thus, embodiments allow for a simpler modeling and optimization setup compared to the prior art methods described above.

[0078] Figure 22 is a flow chart illustrating another embodiment 200. The method 200 is a computer-implemented method for determining an optimal design of a real-world object by suppressing post-buckling behavior using one or more added artificial forces. The method 200 is computer-implemented such that the functions and effective operations (e.g., steps 201-203) can be automatically performed by one or more digital processors. Furthermore, the method 200 can be implemented using any computer device or combination of computing devices known in the art. In other examples, the method 200 can be implemented using the method described below with respect to Figure 23 Computer system 2320 described and described below Figure 24 The described computer network environment 2420 is implemented.

[0079] Method 200 begins at step 201 by defining a computer-based model representing a real-world object in a memory of a processor. Next, at step 202, the computer-based model (defined at step 201) is modified to include at least one artificial force. The at least one artificial force is defined to vary with physics-based behavior. Then, at step 203, the computer-based model modified to include the at least one artificial force is used to iteratively optimize the real-world object with respect to a load. The result of the iterative optimization is an optimized design of the real-world object.

[0080] According to an embodiment of method 200, a computer-based model is defined in response to user input at step 201. Furthermore, in an embodiment, the computer-based model is defined at step 201 according to principles known to those skilled in the art. For example, defining the model at step 201 may include discretization, i.e., creating a finite element model of the real-world object. Furthermore, defining the model at step 201 may include assigning appropriate properties, such as loads and boundary conditions, to the computer-based model, such as a FEM. Furthermore, at step 201, the computer-based model may be configured to include geometrically nonlinear modeling for capturing the buckling behavior of the computer-based model representing the real-world object. Furthermore, the model may be configured at step 201 to include constitutive nonlinear material modeling and contact.

[0081] In one embodiment, modifying the computer-based model at step 202 to include at least one artificial force may include adding an element to the computer-based model such that the artificial force results from movement of a load application point. Such a function (artificial force) may require a force response. In such an embodiment, modifying the computer-based model at step 202 to include at least one artificial force includes defining such a force response. According to an embodiment, the force response is modeled using highly nonlinear connector elements, where the artificial force function may be described, for example, using a nonlinear spring stiffness that varies with displacement, a damping viscosity that varies with velocity, and an inertial mechanism mass that varies with acceleration. Furthermore, in one embodiment, modifying the model at step 202 may include, for example, receiving an indication of the at least one artificial force from a user. This received indication may include at least one force property, including the force magnitude and a property of the force, such as the force magnitude as a function of physics-based behavior.

[0082] In an embodiment of the method 200, the artificial force may be defined according to any physics-based behavior known to those skilled in the art. For example, in an embodiment, the physics-based behavior is at least one of the following: displacement, velocity, and acceleration.

[0083] In an embodiment of method 200, iteratively optimizing the real-world object at step 203 may include performing simulations using the modified computer-based model and, based on the results of the simulations, evaluating the consistency of the values ​​of the properties of the real-world object with the design parameters. This evaluation may include determining whether the values ​​of the properties, such as the force before buckling, meet the requirements. Such embodiments may include, in response to evaluating that the values ​​of the properties meet the design parameters, determining that the modified computer-based model represents an optimized design of the real-world object. Furthermore, in response to evaluating that the values ​​of the properties do not meet the design parameters, such embodiments of method 200 iteratively: (i) updating the modified computer-based model, such as by changing the dimensions of the model elements; (ii) performing simulations using the updated modified computer-based model; and (iii) evaluating the values ​​of the properties of the real-world object for consistency with the design parameters based on the results of the simulations performed using the updated modified computer-based model, until the values ​​of the properties are evaluated to meet the design parameters.

[0084] In another embodiment of method 200, iteratively optimizing the real-world object at step 203 may include, by the processor, performing a simulation using the modified computer-based model. According to an example embodiment, performing the simulation may include (i) applying a load to the modified computer-based model, (ii) determining a value of a physics-based behavior of the computer-based model resulting from the applied load, and (iii) applying at least one artificial force to the modified computer-based model based on the determined value of the physics-based behavior. In one such example embodiment, applying the at least one artificial force includes resisting deformation in the modified computer-based model in response to the determined value of the physics-based behavior exceeding a threshold.

[0085] In yet another embodiment of method 200, iteratively optimizing the real-world object at step 203 may include the processor solving for an equilibrium primitive solution of the modified computer-based model and determining a design response and at least one corresponding sensitivity with respect to a design variable of the modified computer-based model. Such embodiments then optimize the real-world object using the modified computer-based model, the determined design response, and the at least one corresponding sensitivity. In this embodiment, the result of the optimization may be a converged design representing an optimized design of the real-world object or a non-compliant solution (e.g., a solution in which the values ​​of properties do not conform to requirements). In response to the result of the optimization being a non-compliant solution, such embodiments modify at least one artificial force and iteratively solve, determine, and optimize using the modified at least one artificial force. In this manner, such embodiments allow for modification of the artificial force during optimization iterations.

[0086] According to an embodiment of method 200, at least one artificial force may be configured to oppose deformation in the modified computer-based model in response to a value of the physics-based behavior of the real-world object exceeding a threshold. In another embodiment, at least one artificial force is configured to suppress a post-buckling response of the real-world object.

[0087] In yet another embodiment, the method 200 may determine a buckling point within a real-world object.

[0088] Embodiments of method 200 can be used to analyze objects existing in the real world, such as a real-world object. In such example embodiments, measurements are taken from the real-world object, for example using one or more sensors or other known measurement devices, and these measurements are used to define a computer-based model in step 201. Thus, the model defined in step 201 represents the object existing in the real world. Therefore, such embodiments can determine an optimized design for the real-world object in step 203 and, based on the results of the embodiment, can improve the object. For example, a failure point in an object, such as a bridge, can be identified, and additional supports can be added to the bridge to prevent the failure. Similarly, embodiments can be used to determine an optimized design for a real-world object that has not yet been constructed. In such embodiments, a computer-based model is defined in step 201 and represents an initial object design. This model is then used in steps 202 and 203, and after the optimized design is determined in step 203, the real-world object can be constructed / manufactured according to the determined optimized design.

[0089] Figure 3A -C shows how an embodiment adds artificial forces to suppress the response in the post-buckling range. Figure 3A As shown in Figure 2-C, the pre-buckling response and the global buckling response are not suppressed, and therefore, these responses can still be used in the optimization.

[0090] Specifically, Figure 3A -C are graphs 310, 320 and 330, respectively, depicting the suppression of physical post-buckling of the physical system 311 by adding internal artificial forces. Figure 3A The graph 310 depicts an external force P 312 applied to a physical system 311 relative to a displacement u 313 in the physical system 311. In the graph 310, line 315 indicates the displacement pattern in the physical system 311 in response to the force 312. Figure 3A As can be seen in graph 310 , as the external force P 312 increases, the displacement u 313 of the physical system 311 (e.g., a beam) also increases until a bifurcation point 314. After the bifurcation point 314, the physical system enters a phase known as the "post-buckling response." In graph 310 , there are no added artificial forces, so a converged design cannot be achieved after the buckling point (bifurcation point) 314 of the physical system 311, i.e., within the post-buckling response / range.

[0091] Figure 3B The graph 320 in FIG. 1 shows the artificial force I according to an embodiment. art,stab 321 relative to the displacement u 313. The graph 320 also shows the displacement value U of the physical system 311 art,add 322, of which artificial force I art,stab 321 is added to the physical system 311. According to an embodiment, this displacement value Uart,add 322 is after the flexion point 314 . Figure 3B As shown, the manual force 321 is zero until the predetermined displacement u313 is reached, that is, the displacement value U art,add 322.

[0092] Figure 3C is a graph 330 showing the external force P resulting from adding the artificial force 321 to the physical system 311. stab 332 and the relationship between the displacement u 313. After the buckling point 314 with a given displacement value U art,add 322 Adding Artificial Internal Force I art,stab 321, thus still consistently captures the physical overall buckling point 314 (bifurcation point) of the physical system 311. At a given displacement value U for the original solution art,add 322, added artificial force I art,stab 321 increases significantly, thus suppressing the response of the physical system 311 during the post-buckling response. This can be seen by point 334 of the graph 330. At a given displacement value U art,add Before 322, there was no artificial force added. art,stab 321, so the response of the original physical system 311 is not modified for the pre-buckling range and the global buckling point 314. In the post-buckling range, i.e., after point 334, the deformation of the physical system 311 is suppressed, thereby preventing the physical system from further deformation. This can be observed by line 331, which indicates that the additional external force 332 no longer causes the displacement 313 to increase.

[0093] Structural Analysis: Adding Artificial Forces to Suppress Postbuckling

[0094] Figure 4 A generalized continuum 401 of a physical system 411 and a corresponding equilibrium 402 applied to an external load {P} 403 is shown, with the original solution {u} 404 and the corresponding reaction response {P} determined by the residual of the equilibrium 402. p 405 is zero {R}={0}. Figure 4 The internal force {I}406 varies with the design variable {φ}407, the original solution {u}404 and the external boundary condition {u} p 408. For example, the physical system 411 may represent the above Figure 1A -D described column 101.

[0095] In the generalized continuum 401 of the physical system 411, the original solution {u} 404 may contain translations, rotations, and other degrees of freedom (DOF) types. In addition, the reaction response {P} p405 may contain both reaction forces and reaction moments. In addition, the original solution {u} 404 may represent displacement, velocity, and acceleration for static, quasi-static, and transient modeling, respectively.

[0096] Figure 5 Display and Figure 4 The physical system 411 is similar to the generalized continuum 501 of the physical system 511, but with the addition of an artificial force {I} to suppress the post-buckling response art,stab 521, and the corresponding residuals of equilibrium 508. The internal force {I} 515 varies with the design variable {φ} 510, the original solution {u} 516, and the external boundary condition {u} p 512. For the artificial force {I} added to suppress the post-buckling response art,stab 521 physical system 511, added artificial force {I} art,stab 521 Follow the original solution {u} f,stab External load {P} ext,stab 502 and original solution {u} f,stab 503 is the added artificial force {I} art,stab 521. Therefore, the DOF of the external load {P} 513 and the original solution {u} 516 are classified as {P} with the added artificial force ext,stab 502 and {u} f,stab 503 and {P} without added artificial force ext 506 and {u} f 507. Usually, the artificial force {I} art,stab 521 is added to the position or DOF where there is also an external load {P} 513. In an embodiment, the artificial internal force {I} art,stab 521 is added after the buckling point, so the physical overall buckling point of the physical system 511 is still consistently captured.

[0097] In the generalized continuum 501 of the physical system 511, the original solution {u} 516 may contain translation, rotation and DOF types. In addition, the reaction response {P} p 514 may contain both reaction forces and reaction moments. In addition, the original solution {u} 516 may represent displacement, velocity, and acceleration for static, quasi-static, and transient modeling, respectively.

[0098] Structural Optimization Workflow: Adding Artificial Forces to Suppress Postbuckling

[0099] Figure 6A flow chart illustrating a computer-implemented method 600 of an iterative design process in which artificial forces are added to a computer-based model representing a real-world object. In method 600, the artificial forces are configured to suppress a post-buckling response of the object while solving an equilibrium primitive for the model representing the object.

[0100] Method 600 is a sensitivity-based iterative design process. Process 600 illustrates the design response (DRESP) and corresponding sensitivity with respect to a design variable {φ}, such as 407, where equalization, such as 508, has been added to suppress post-buckling modeling using artificial forces.

[0101] The iterative design process method 600 can be implemented, for example, within a predefined workflow of an existing CAE or CAD system / platform, where the existing workflow is modified to include functionality of the method 600, such as step 603. The method 600 begins at step 601 by creating an initial model for optimization, for example, representing a physical system. This initial model may include various loads and boundary conditions. At step 603, the model created at step 601 is modified to include internal artificial forces for suppressing physical postbuckling of the physical system. Following step 603, the modified model with the added artificial forces is employed in the iterative design process (steps 605-617).

[0102] Each design iteration begins at step 605, where method 600 solves for an equilibrium primitive solution of the model with artificial forces for suppressing post-buckling. Next, at step 607, method 600 uses the equilibrium of the model determined at step 605 to determine the design response (DRESP) with respect to the design variables {φ} and its sensitivity. According to an embodiment, DRESP defines the response of the current analysis of the model for a given optimization iteration. Thus, DRESP extracts a scalar value from the model, which can be a direct measure (e.g., mass, center of gravity, etc.) or determined from the results of the equilibrium primitive solution of the model (e.g., stress, displacement, reaction force, etc.).

[0103] Continuing at step 609, DRESP is applied to define an optimization problem to be solved by mathematical programming. The optimization problem consists of constraints that must be satisfied and an optimization objective function. The mathematical programming is based on, for example, a user-defined design goal, DRESP, and the values ​​of DRESP's sensitivities used to update the design variables. Therefore, if DRESP and the corresponding sensitivities cannot be determined for the postbuckling range (numerical instability), the mathematical programming cannot update the design variables, and the optimization terminates prematurely. Therefore, modeling the artificial forces used to suppress postbuckling is added as a factor in the numerical stability of the optimization workflow of method 600. The results of solving the mathematical programming at step 609 are the values ​​of the design variables.

[0104] At step 611 of method 600, a new model is generated for the next optimization iteration based on the design variables determined in step 609. Sometimes, the design variables and model variables may be the same, such as the thickness design variables used for size optimization; but they may also be different, such as for density topology optimization [6.1], where the design variables are relative density mapped to physical density. Optionally, at step 613, method 600 may also modify the modeling of the artificial forces used to suppress postbuckling. If, for example, the modified variables from step 611 fundamentally change the behavior of the postbuckling response, the artificial forces may be modified at step 613.

[0105] At step 615, method 600 determines whether the optimization has converged. If the optimization does not produce a converged design, i.e., "No" at step 615, method 600 returns to step 605 and begins a new optimization iteration. If the optimization produces a converged design, i.e., "Yes" at step 615, method 600 outputs the final design at step 617. For a converged design, the objective function can be optimized, and modifications in the design should converge.

[0106] The method 600 has been implemented and tested [6.2] for structural modeling based on the modeling method [6.3] and the workflow has been optimized using existing methods [6.4] known in the art. Numerical results and numerical experiments of this embodiment are described in detail below.

[0107] Determine the solution of a system that suppresses the postbuckling response using added artificial forces

[0108] Figure 7 and 8A -8C shows how the embodiment can be used by artificial force I art,stab The residual and corresponding operator matrix term K added to the partial derivatives for the artificial force with respect to the original solution art 701 to artificially suppress post-buckling in the solution.

[0109] Figure 7 Equation 700 is shown for determining a solution for a physical system with an added artificial force to suppress post-buckling, according to an embodiment. Equation 700 includes the artificial force I for the original solution term art,stab Operator K of partial derivatives of 721 art 701. Equation 700 shows that embodiments support all types of physical modeling including large deformations as static, quasi-static, and transient modeling. Additionally, equation 700 shows that embodiments support all types and combinations of numerical solver methods applied to determine the original finite element solution using partial derivatives of the residual 702 with respect to the original solution (also called a global operator).

[0110] Figures 8A-8C To show the added artificial force I art,stabThree graphs 810, 820, and 830 (a non-continuous linear graph 810, a bi-linear graph 820, and a smooth non-linear graph 830) of examples of the characteristics of 821a-c in the post-buckling range. The exemplary graphs 810, 820, and 830 show an example artificial force I art,stab 821a-c are relative to displacements 804a-c, where the displacement u is defined art,add 822a-c Implementing Manual Force I art,stab 821a-c. Operator K art 801a-c are the derivatives of the artificial force curves with respect to the original solution of DOF with added artificial forces, e.g., for K art 801a-c generate displacements of the stiffness operator type.

[0111] exist Figures 8A-8C In the original solution, given value u art,add 822a-c, the added manual force 821a-c is significantly increased (I art,stab >>0,K art >>0), so the physical response to the postbuckling response is suppressed. art,add Before 822a-c is zero or close to zero (depending on the characteristics of the artificial force), additional artificial forces 821a-c (I art,stab =0 or 1 art,stab ≈0)(I art,stab <<0,K art <<0), so that the response of the original physical system is not modified for the pre-buckling range and global buckling point.

[0112] Since the embodiments apply almost no artificial forces before buckling, the numerical solvers supported for applying global operators are implicit, explicit, and hybrid methods. For example, the following are non-limiting examples of supported solver methods: factorization of operators using direct solvers or iterative solvers using preconditioners, Newton-Raphson procedures with incremental iteration techniques, incremental loads, and time integration using first-order (e.g., backward Euler), second-order methods (e.g., Newmark-β), and higher-order methods (e.g., Runge-Kutta).

[0113] Figure 9 Equation 900 is shown for determining the original solution, where the original solution u f,stab 903 is higher than the displacement value u art,add (u f,stab >u art,add ), so the added artificial force I art,stab Post-buckling of the physical system is suppressed, and therefore, operator 901 also contains derivatives of the supplementary forces that suppress post-processing.

[0114] Solver solution conclusion

[0115] If the added artificial force I art,stab and the artificial operator matrix K art , for example, if 701 is set too low, then physical post-buckling is not suppressed. Therefore, according to an embodiment, the artificial force I art,stab and the artificial operator matrix K art Greater than some given minimum value: I art,stab >I art,min And K art,stab >K art,min .

[0116] If the artificial operator matrix K art If the post-buckling range is too high, the global operator will be ill-conditioned and cause numerical instability or zero solution when solving the artificial solution. art,stab is chosen so that the artificial operator matrix K art Less than a given maximum value: K art,stab <K art,min .

[0117] In an embodiment, a method is to apply the operator K art , the operator produces an operator that is similar to or within the span of the pre-buckling range, depending on the optimization application.

[0118] Determine the sensitivity of a system using added artificial forces to suppress postbuckling responses

[0119] If the artificial operator matrix K art Too low, and the physical postbuckling sensitivity will not be suppressed (see Figures 10 and 11 discussed below). Figure 12 ). Therefore, in an embodiment, the artificial operator matrix K art Greater than some given minimum value: K art,stab >K art,min If the artificial operator matrix K art If the post-buckling range is too high, the sensitivity of the global operator in the post-buckling range will be zero. art,stab , so that the artificial operator matrix K art Less than a given maximum value: K art,stab <K art,min .

[0120] The operator K is used to select the sensitivity in the postbuckling range. art The potential methods for (see Figures 10-13) can be divided into three groups. For the first group, if the artificial postbuckling sensitivity should be numerically eliminated in the optimization, then the artificial operator K artis set to be much higher than the physical operator used for the pre-buckling range. For the second group, if the artificial post-buckling sensitivity should be partially considered in the optimization range, then the artificial operator K art For the third group, if the artificial post-buckling sensitivity should be fully considered in the optimization range, then the artificial operator K art Set below the physical operator used for the forward buckling range.

[0121] Accompanying sensitivity

[0122] The calculation of adjoint sensitivities for objective terms and / or constraints is typically part of a sensitivity-based optimization problem with many design variables. According to an embodiment, adjoint sensitivities for a system that suppresses post-buckling response using added artificial forces are derived, as shown in FIG. 10 discussed below. Figure 11 The effect of the Lagrange multiplier on the concomitant sensitivity of the postbuckling range is shown in FIG.

[0123] Figure 10A Display for Design Response (DRESP) 1002 and Lagrange multiplier {λ} T 1003 (where {λ} T By {λ} f,stab (i.e., the Lagrange multiplier of the DOF in the model with added artificial forces), {λ} f (i.e., the Lagrange multiplier of the DOF without artificial forces) and {λ} p Equation 1001 of the adjoint method for the sensitivity of the DOF (i.e., composed of Lagrange multipliers of DOF with external boundary conditions similar to 408 and 512), where the derivative 1004 contains a supplementary term that suppresses post-buckling of the physical system using added artificial forces.

[0124] Figure 10B Equation 1011 of the adjoint method showing the sensitivity of the design response 1012 with respect to the design variable φ 1013. The operator matrix term 1014 ({u} f,stab The residual error {R} of the DOF f,stab Relative to {u} f,stab The derivatives of the original solution of ) contain the derivatives of the artificial forces added to the physical system to suppress post-buckling of the physical system.

[0125] Figure 11 Show that for the Lagrange multiplier {λ} T Equation 1100, {λ} T is the transpose of {λ}1103 for the adjoint system from FIG10 , where the original solution u f,stab 1123 is higher than uart,add 1122(u f,stab >u art,add ), thus using an added artificial force to suppress post-buckling of the physical system, and thus, operator 1101 also contains the derivative of the supplementary force that suppresses post-buckling. {λ} T By {λ} f,stab (i.e., the Lagrange multiplier of the DOF in the model with added artificial forces), {λ} f (i.e., the Lagrange multiplier of the DOF without artificial forces) and {λ} p (ie, Lagrange multipliers of DOF with external boundary conditions similar to 408 and 512).

[0126] Direct sensitivity

[0127] The computation of direct sensitivities to design variables is often part of a sensitivity-based optimization problem with many objective terms and / or constraints.

[0128] In one embodiment, as discussed below Figure 12 As shown, direct sensitivity is derived for a system that suppresses the post-buckling response using an added artificial force. Figure 13 Figure 2 shows the effect of the direct derivative of the direct system for the postbuckling range, i.e., the sensitivity, when high artificial forces are used to suppress postbuckling.

[0129] Figure 12 Equation 1200, showing a direct method for calculating the sensitivity of DRESP according to an embodiment, includes derivatives of complementary terms that suppress post-buckling of the physical system using added artificial forces. The operator matrix terms (i.e., {u} f,stab The residual error {R} of the DOF f,stab Relative to {u} f,stab The derivatives of the original solution of ) contain the derivatives of the artificial forces added to the physical system to suppress post-buckling of the physical system.

[0130] Figure 13 Show from Figure 12 The direct derivative of the system of equations 1300, where the original solution u f,stab 1323 is higher than u art,add 1322(u f,stab >u art,add ), thus an added artificial force is used to suppress post-buckling of the physical system, and therefore, operator 1301 also contains the derivative of the supplementary force that suppresses post-processing.

[0131] Optimization application for systems that suppress post-buckling response using added artificial forces

[0132] The embodiments may be implemented using existing geometrically nonlinear topology optimization methods [2.1-2.3], [2.7], [3.1], [3.2], [6.1], [6.4], where the existing methods are modified so that the post-buckling response is suppressed using added artificial forces.

[0133] Figure 14A and 14B Graphs 1401 and 1402 are shown, respectively, showing the relationship between the external force P stab Graph 1401 shows an optimization for minimizing total displacements, and graph 1402 shows an optimization for constrained displacements. Both graphs 1401 and 1402 illustrate a buckling optimization in which internal artificial forces are added to suppress physical post-buckling of the physical system. Graph 1401 shows displacements 1412a-g of the initial design 1410 and displacements 1413a-g of the optimized design 1411, which have been minimized as applied to the objective function.

[0134] Graph 1402 shows displacement 1434 b of geometrically nonlinear optimized design 1405 subject to displacement 1404 constraints compared to geometrically nonlinear initial design 1409 .

[0135] refer to Figure 14A Graph 1401 compares the total displacements of the initial design 1410 and the optimized design 1411. For the first topology optimization shown in graph 1401, the displacements 1413a-g of the optimized design 1411, where the post-buckling response is suppressed using added artificial forces, are used in the objective function. Therefore, when the objective function is subject to mass constraints, the sum of the displacements is minimized. The refined optimization formula of 1401 implicitly maximizes the overall buckling load carrying capacity for a given external load 1432a.

[0136] refer to Figure 14B Graph 1402 compares the total displacement of the initial design 1409 with the total displacement of the optimized design 1405. For the second topology optimization shown in graph 1402, the displacement 1434b of the system using the post-buckling response suppressed by the added artificial force is constrained to be below a given value. Therefore, mass is minimized because the objective function is constrained by the displacement. The refined optimization formulation of graph 1402 implicitly constrains the overall buckling load-carrying capacity for a given external load. Note that both the objective function optimization formulation of graph 1401 and the constrained optimization formulation of graph 1402 have high external loads in the post-buckling range. Due to the added artificial force, the last load increment for this high external load only has a numerical solution and corresponding sensitivity for the optimization setup.

[0137] Model setup for suppressing postbuckling response using added artificial forces

[0138] Figures 15A-15D Example models 1500a-d are shown for topology optimization of a physical beam system having a length dimension of 700 millimeters (mm), a height dimension of 170 mm, a width dimension of 5 mm, a T (flange) dimension of 5 mm, and a T (web) dimension of 3 mm.

[0139] Figure 15A Loads 1511 and 1512 and boundary conditions 1510 applied to model 1500a are shown. Boundary condition 1510 requires that the right side of beam 1510a be fully clamped and that beam 1510a be subjected to both bending load 1511 and compressive load 1512 on the left side to buckle. In this model 1500a, dynamic couplings are used to connect the load points to the end faces.

[0140] Figure 15B Shown based on Figure 15A The finite element model 1500b is generated from the model 1500a. The finite element model 1500a contains 296384 C3D10 ten-node tetrahedral elements [6.2]. In addition, both elastic and elastoplastic material properties are defined for the geometrically nonlinear model 1500b.

[0141] Figure 15C Model 1500c is shown in which a geometric defect 1513 is introduced into Figure 15A The initial model 1500a and Figure 15B Finite element model 1500b is shown in FIG. Model 1500c introduces geometric imperfections 1513 into finite element model 1502 for buckling analysis. The introduction of imperfections 1500c enables embodiments to analyze actual real-world beams, as imperfections are commonly found in real-world objects to create uncertainty and imperfections.

[0142] The physical bifurcation point used for geometrically nonlinear modeling produces an ill-conditioned stiffness operator before the postbuckling range, but this can be numerically stabilized using quasi-static modeling. [6.2, 6.3] is used to obtain a physically primitive solution in the prebuckling range and around the buckling point. Therefore, quasi-static modeling is applied to the model in 1500a-d.

[0143] For example, embodiments of the optimization workflow 600 can be applied to density-based topology optimization, using the so-called solid isotropic material penalty (SIMP) method for material interpolation, and using design variable filtering to regularize the density-based topology optimization problem in order to ensure mesh independence and prevent checkerboard effects [6.1, 6.4]. Figure 15D In the model 1500d, the design variables are shown defined as four design regions 1514, where each finite element is a design variable in the topology optimization used to determine a new optimal concept material layout for the beam model 1500d.

[0144] Figure 16 Display based on Figures 15A-15D 16. The model 1600 of the original physical beam system is shown, but with artificial forces 1601a-d added to suppress the post-buckling response. The added artificial forces 1601a-d vary strictly with the original solution response. These added artificial forces include: artificial elastic movement 1601a to suppress rotation, artificial axial elastic force 1601b to suppress axial displacement, artificial adhesive force 1601c to suppress velocity, and artificial bending elastic force 1601d to suppress bending displacement. Forces 1601a-d are defined according to graphs 1602a-d, respectively. Artificial elastic forces 1601b (axial) and 1601d (bending), elastic moment 1601a, and adhesive force 1601c are added at the location of external load 1602. These artificial forces 1601a-d vary with rotation, axial displacement, velocity, and bending displacement, respectively, as shown in graphs 1602a-d. It should be noted that before and at the buckling point, the velocity is numerically zero. Therefore, a linear function is sufficient for the viscous force to vary with velocity.

[0145] Model 1600 is used to obtain the following Figure 17A -C, 18A-C, 19A-B, 20A-C, 21A-B, and 22A-B.

[0146] Targeted optimization of the response using added artificial forces to suppress the post-buckling response

[0147] Figure 17A A graph 1710 is shown showing the initial design 1714 (see Figure 17B )(50% material) and topology optimization design 1713 (see Figure 17C ) The structural response of both forces 1711 and displacements 1712 is calculated, where the sum of the displacements is the objective function, and the objective function is minimized for 20 external load increments using artificial forces added to suppress the post-buckling response. The relative mass is constrained to be less than or equal to 50%. The initial design 1714 buckles at approximately 78% of the external load, but the displacement 1712 solution is artificially stabilized in the post-buckling range using artificial forces added to suppress the post-buckling response. Thus, a displacement 1712 solution exists for optimizing all load 1711 increments independently of the load 1711 level and the displacement 1712 response within the post-buckling range. The optimized design 1713 does not buckle globally under full load.

[0148] As pointed out above, Figure 17BAn initial design 1714 is shown, where beam 1715a is a model of the initial design in which no load was applied, and beam 1715b is a model of the design that buckled under load at point 1702a during optimization. The relative density of the design variables in each element is shown in beam 1715a, which is colored according to key 1720, and the deformation displacement is shown in beam 1715b, which is colored according to key 1721. This relative density may be scaled up or down between approximately 0 and 1 during optimization. In an embodiment, the optimization process uses a SIMP penalty to obtain a 0-1 density distribution, so the final optimized design consists primarily of void (i.e., close to 0) or rigid (i.e., 1) material, however, there may still be very few intermediate density elements in the final optimized design. Figure 17A In FIG. 1 , line 1714 shows the displacement from beam 1715a, resulting in beam 1715b.

[0149] Figure 17C Optimized design 1713 is shown, where beam 1716a is a beam model with no load applied, and beam 1716b is a model with load applied at point 1702b. The relative density of the design variables in each element is shown in beam 1716a, which is colored according to legend 1720, and the deformation displacement is shown in beam 1716b, which is colored according to legend 1721. Figure 17C Shows the optimized beam geometry resulting from geometrically nonlinear topology optimization under a relative mass constraint of 50% of the design domain, which is used to Figure 14A The objective function defined in minimizes the sum of the displacement magnitudes at the load point 1702b.

[0150] Figure 18A Graph 1810 is shown, which shows Figure 14A The total displacement is minimized by Graph 1800 shows an optimization iteration history 1811a for an optimization of displacements 1812a relative to an objective function for minimizing the sum of displacements, where the optimization is performed by suppressing post-buckling responses using added artificial forces. According to an embodiment, the objective function value is a scalar sum of a finite number of design responses representing the load point deflections at different points in time during the nonlinear load sequence. As can be observed from graph 1810, the switch from the initial buckled configuration to the non-buckled configuration is indicated by a rather sharp drop 1815 in the objective function value.

[0151] Figure 18BGraph 1820 is shown, illustrating the results of an optimization performed according to a defined volume constraint (e.g., the optimized design should have a relative material volume or relative mass of less than or equal to 50%). Graph 1820 shows the optimization iteration history 1811b for relative mass 1812b relative to an optimization performed subject to a 50% relative mass constraint, where the optimization utilizes artificial forces as described herein. In graph 1820, series 1813 shows the displacement versus optimization history when a 50% mass constraint is employed, and series 1814 shows the displacement versus optimization history for relative mass.

[0152] Figure 18C Graph 1830 shows the design response for displacement 1812c versus the optimization history for optimizations performed at 100% load 1834, 95% load 1833, 90% load 1832, and 85% load 1831. Graph 1830 shows the switch from the initial design buckling to the non-buckling design configuration at full load 1834, as can be seen by the rapid decrease in displacement 1812c between optimization iterations 1811c 10 and 20. Graph 1840 shows the load point displacement 1841 response in the initial design 1842 and the optimized 1843 designs. Graph 1840 highlights that the initial design 1842 would buckle, and that artificial forces would support the loading point, as indicated by the sharp inclination 1844. However, it can also be observed that in this example, the optimized design 1843 is sufficiently stiff to carry the applied load without buckling, and therefore no artificial forces are required.

[0153] Graph 1810( Figure 18A ) and graph 1830( Figure 18C ) shows that the global buckling of the initial design switches to an optimized design that does not buckle for the full load. In addition, conventional optimization convergence is shown in 18A-18C, indicating that the existing optimization implementation (objective function) and method can be reused and applied to optimize the response in the pre-buckling range and used to indirectly optimize the global buckling load carrying capacity. This conventional optimization convergence is shown by the consistent and stable optimization iteration history convergence of graphs 1810, 1820 and 1830, approximately past iteration 20, where the objective function of the sum of displacements is minimized and all optimization iterations meet the relative quality constraint. In addition, it can be observed in graph 1830 that the displacement term ( Figure 18C , the maximum displacement 1812c) is initially in the post-buckling range and transitions to the pre-buckling range during the optimization iterations.

[0154] As referenced below Figure 19A and 19B As discussed, verification is used to confirm that an optimized design will function as expected. Figure 19A and 19B, validation is performed on both a geometrically linear optimized design 1901a and a geometrically nonlinear optimized design 1902a. Validation of designs 1901a and 1902a is performed using a full set of nonlinear effects (e.g., large deformation modeling enabling both buckling and elastic-plastic material constitutive modeling). Figure 19A and 19B The validation model does not contain any stability in order to find the true load carrying capacity and also determine the post-buckling range. Figure 19A It is observed that the geometrically linear optimized design 1901a will buckle at approximately 90% of the design load and thus can be considered a design failure. However, it can be observed that the geometrically nonlinear optimized design 1902a is verified to successfully carry the entire design load without structural instability or failure until the full design load is achieved, thereby indicating a well-verified optimized design.

[0155] Figure 19A and 19B The geometric nonlinear verification for suppressing the post-buckling response of both the geometrically linear optimized design 1901a and the geometrically nonlinear optimized design 1902a is shown in the unstable case without adding artificial forces. Figure 19A In , beam 1920a is a model of a beam in which no load is applied, and beam 1920b is a model in which a load is applied in which the beam has buckled. Figure 19B In Figure 1, beam 1940a is a model of a beam with no applied load, and beam 1940b is a model of a beam with load applied but not yet buckling. The relative density of design variables in each element is shown in beams 1920a and 1940a, colored according to legend 1930, and the deformation displacement is shown in beams 1920b and 1940b, colored according to legend 1931. Both designs 1901a and 1902a were implemented using displacement minimization as the objective function under a relative mass constraint of 50%. Both designs 1901a and 1902a were verified using geometrically nonlinear modeling in an unstable state without adding artificial forces to suppress post-buckling response. Graph 1901b compares external force 1903a with displacement 1904a, showing that geometrically linear optimized design 1901a buckles at approximately 90% of the full load. Graph 1902b compares external force 1903b to displacement 1904b, showing that the geometrically nonlinear optimized design 1902a does not buckle under full load.

[0156] Constrained optimization of responses for suppressing post-buckling responses using added artificial forces

[0157] Figure 20A2000 shows the structural response using curves 2003 and 2004 of force 2001 versus displacement 2002. Curve 2003 shows the response for the initial beam design (50% material) 2003. Curve 2004 shows the response for the topology optimized design 2004, where the objective function is to minimize mass, and the displacement 2002 is constrained 2005 to be less than or equal to 14 mm for 20 load increments using added artificial forces to suppress the post-buckling response. Figure 20A In , beam 2020a is a model of a beam in which no load is applied, and beam 2020b is a model in which a load is applied in which the beam has buckled. Figure 20B , beam 2040a is a model of a beam without applied load, and beam 2040b is a model of a beam with applied load but not yet buckled. The relative density of the design variables corresponding to each element is shown in beams 2020a and 2040a, which are colored according to legend 2030, and the deformation displacement is shown in beams 2020b and 2040b, which are colored according to legend 2031.

[0158] The initial design 2003 buckles at approximately 78% of the external load, but the displacement solution is artificially stabilized in the post-buckling range using an added artificial force to suppress the post-buckling response. Thus, there is a displacement solution that optimizes all load increments independently of the load level and displacement response in the post-buckling range. The optimized design 2004 does not have global buckling at full load. Figure 20C The geometric nonlinear topology optimization results 2004 are shown in FIG, where the objective function is to minimize the mass of an object subject to 20 displacement constraints, where each constraint is defined at a load increment of 5% relative to the full load of 100%. This is achieved by using an added artificial force to suppress the post-buckling response, forcing the displacement 2002 at the load point 2022 to be less than or equal to 14 mm to achieve the 20 load increments, as shown in FIG. Figure 14B As defined in .

[0159] Figure 21A and 21B Graphs 2100 and 2110 are shown, respectively, each illustrating an optimization iteration to achieve a design constraint. In graph 2100, the objective function is to minimize relative mass. Graph 2110 shows 20 individual displacement constraints, where the objective function is to minimize displacement. Figure 21A A graph 2100 is shown illustrating relative quality 2102 versus optimization iterations 2101 for an objective function configured to minimize relative quality. Figure 21BA graph 2110 is shown comparing displacement 2104 versus optimization iteration 2103. Graph 2110 illustrates displacement constraints 2105 for load increments of the post-buckling response with added artificial force suppression. The optimization iteration history 2103 for the displacement constraints 2105 shows that the full load of the initial buckled design switches to a non-buckling design configuration under full load in less than 10 optimization iterations, as can be seen from the displacement 2104 remaining at or below the displacement constraint 2015 after 10 iterations. Figure 21B The displacement response in 2104 shows that the initial design switched to the optimized design showed no buckling under full load. Figure 21A and 21B Conventional optimization convergence is demonstrated in , indicating that existing optimization implementations (constraints) and methods can be reused and applied for optimizing the response in the pre-buckling range and for indirectly optimizing the global buckling load carrying capacity.

[0160] Figure 22A and 22B The geometric nonlinearity verification for suppressing the post-buckling response of both the geometrically linear optimized design 2201a and the geometrically nonlinear optimized design 2202a is shown in the unstable case without adding artificial forces. Figure 22A In , beam 2220a is a model of a beam in which no load is applied, and beam 2220b is a model in which a load is applied in which the beam has buckled. Figure 22B In Figure 2, beam 2240a is a model of a beam with no applied load, and beam 2240b is a model of a beam with load applied but not yet buckled. The relative density of design variables corresponding to each element is shown in beams 2220a and 2240a, colored according to legend 2230, and the deformation displacement is shown in beams 2220b and 2240b, colored according to legend 2231. Designs 2201a and 2201b were implemented using mass minimization as the objective function, subject to a 20 displacement constraint of 14 mm. Designs 2201a and 2202a were verified using geometrically nonlinear modeling in an unstable state without adding artificial forces to suppress the post-buckling response. Graph 2201b compares external force 2203a with displacement 2204a, showing that geometrically linearly optimized design 2201a buckles at approximately 60% of the full load. Graph 2202b compares external force 2203b to displacement 2204b, showing that the geometrically nonlinear optimized design 2202a does not buckle under full load.

[0161] Technical Observation

[0162] The embodiments propose a novel sensitivity-based optimization method that optimizes global buckling for external loads. This ensures stable convergence and accurate solution of the original solution in the pre-buckling range and for global buckling, while artificially suppressing post-buckling. The embodiments also allow for the reuse of existing software and solutions for modeling and optimization. For example, the embodiments achieve convergence in the solution of all optimization iterations and promote rapid numerical convergence in the post-buckling range, thereby reducing computational costs.

[0163] Embodiments support post-buckling suppression through the use of added artificial forces. Embodiments can be implemented for any type of constitutive model and any type of shape function used in large-deformation finite element modeling. Thus, embodiments can be used for all finite element types; all physical modeling, including large deformations such as static, quasi-static, and transient modeling; and all types and combinations of numerical solver methods used to determine the original finite element solution. Embodiments can address any sensitivity-based structural optimization principle, such as topology optimization, shape optimization, size optimization, and rib optimization, among other examples.

[0164] Example Benefits

[0165] Embodiments suppress the physical postbuckling response of computer-based models representing real-world objects by using artificial forces added for large-displacement modeling using nonlinear finite element analysis to predict the pre-buckling and post-buckling ranges. Embodiments have several important applications, among other examples. For example, embodiments can facilitate rigorous numerical computation of numerical solutions to geometrically nonlinear finite element models. This facilitates obtaining physically accurate solutions for the pre-buckling range and global buckling point, as well as artificial solutions for the post-buckling range, requiring fewer solver iterations for solutions in the post-buckling range and, therefore, requiring less computational cost.

[0166] As another example, embodiments benefit geometrically nonlinear optimization. Specifically, embodiments suppress the postbuckling range in optimization-targeted modeling. Furthermore, structural modeling using continuous and monotonically increasing loads can be directly applied in optimization-targeted modeling, offering several advantages. For example, embodiments provide user-friendly, simple modeling that can determine a solution (and thus DRESP and corresponding sensitivity) for a given external load, and achieve a design because the solution does not fail when solving for a numerical solution in the postbuckling range (and thus achieving convergence in the solution for all optimization iterations).

[0167] Furthermore, for practical industrial applications, prescribed displacement modeling is often not of interest because many industrial applications are subject to external force loads for practical modeling. Prescribed displacement driven modeling generally does not capture the correct physical modeling of the system. Many industrial applications may choose to utilize force driven modeling methods for optimization settings, which may not be easily solved when the tangential stiffness approaches 0, thereby indicating stability limits. Force driven modeling generally cannot be converted into equivalent prescribed displacement driven modeling. Therefore, force driven modeling must be applied to obtain practical modeling and therefore practical results. Thus, previous work has provided workarounds to this problem.

[0168] However, embodiments make it possible to pass the instability limit and then "capture" the load point in the post-buckling range through artificial forces. From an optimization iteration perspective, when the design fails, the solution has been "lost," and in embodiments, this allows the current iteration to be completed in a controlled manner. Ultimately, the optimization will reach a material distribution with sufficient stiffness to carry the design load without the need for artificial forces for stabilization. Furthermore, the stabilization method provided by the embodiments disclosed herein is "non-invasive," meaning that while the structure itself has sufficient load-bearing capacity, the artificial stabilization forces are negligible and will only reach significant levels in the post-buckling range.

[0169] Additionally, in embodiments, design responses can be applied in the objective function and / or constraints, and corresponding sensitivities can be determined at the same load point. Embodiments also reduce runtime and computational cost because detailed analysis of the post-buckling range is not included in the modeling, yet the method still enables the design response in pre-buckling to be applied in optimization for increasing buckling loads. Existing optimization implementations and methods can be reused and applied by embodiments because existing design responses (e.g., displacements) can be directly applied in both the objective function and / or constraints for sensitivity-based optimization. Furthermore, existing numerical implementations and theories for sensitivity calculations (e.g., concomitant sensitivity to displacements) can be directly applied in optimizations of embodiments.

[0170] Computer Support

[0171] Figure 23This is a simplified block diagram of a computer-based system 2320 that can be used to determine the optimal design of a real-world object according to any of the various embodiments described herein. System 2320 includes a bus 2323. Bus 2323 serves as an interconnect between the various components of system 2320. Connected to bus 2323 is an input / output device interface 2326 for connecting various input and output devices, such as a keyboard, mouse, display, and speakers, to system 2320. A central processing unit (CPU) 2322 is connected to bus 2323 and provides execution of computer instructions that implement the embodiments. Memory 2325 provides volatile storage of data for executing computer instructions that implement the embodiments described herein. Storage 2324 provides non-volatile storage for software instructions, such as an operating system (not shown) and embodiment configurations. System 2320 also includes a network interface 2321 for connecting to any of a variety of networks known in the art, including wide area networks (WANs) and local area networks (LANs).

[0172] It should be understood that the exemplary embodiments described herein can be implemented in many different ways. In some cases, the various methods and machines described herein can each be implemented by a physical, virtual, or hybrid general-purpose computer, such as the computer system 2320 or a computer network environment, such as described below with respect to Figure 24 The computer environment 2420 described herein. The computer system 2320 can be transformed into a machine that performs the methods described herein, for example, by loading software instructions into the memory 2325 or non-volatile memory 2324 for execution by the CPU 2322. It will be further understood by those skilled in the art that the system 2320 and its various components can be configured to perform any embodiment or combination of embodiments described herein. In addition, the system 2320 can implement the various embodiments described herein using any combination of hardware, software, and firmware modules operatively coupled internally or externally to the system 2320. In addition, the system 2320 can be communicatively coupled to a manufacturing device or embedded within the manufacturing device to control the device to create a physical object using an optimized design as described herein.

[0173] Figure 24 A computer network environment 2420 is shown in which embodiments of the present invention may be implemented. In the computer network environment 2420, a server 2421 is linked to clients 2423a-n via a communication network 2422. The environment 2420 can be used to enable the clients 2423a-n, alone or in combination with the server 2421, to perform any of the methods described herein. For non-limiting examples, the computer network environment 2420 provides a cloud computing embodiment, a software as a service (SAAS) embodiment, and the like.

[0174] The embodiments or aspects thereof may be implemented in the form of hardware, firmware, or software. If implemented in software, the software may be stored on any non-transitory computer-readable medium that is configured to enable a processor to load the software or a subset of its instructions. The processor then executes the instructions and is configured to operate or cause the device to operate in the manner described herein.

[0175] Further, firmware, software, routines, or instructions may be described herein as performing certain actions and / or functions of a data processor. However, it should be understood that such descriptions are included herein only for convenience and that such actions are actually caused by a computing device, processor, controller, or other device executing the firmware, software, routines, or instructions.

[0176] It should be understood that the flow charts, block diagrams, and network diagrams may include more or fewer elements, be arranged in different ways, or be presented in different ways. However, it should be further understood that certain embodiments may indicate the number of block diagrams and network diagrams and block diagrams and network diagrams that illustrate the execution of an embodiment implemented in a particular manner.

[0177] Thus, additional embodiments may also be implemented in various computer architectures, physical, virtual, cloud computers, and / or some combination thereof, and therefore, the data processors described herein are intended to be illustrative only and not limiting of the embodiments.

[0178] While example embodiments have been particularly shown and described, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the embodiments as encompassed by the appended claims.

[0179] The teachings of all patents, published applications, and references cited herein are incorporated by reference in their entirety.

[0180] References:

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Claims

1. A computer-implemented method for determining an optimal design of a real-world object, the method comprising: defining a computer-based model representing a real-world object in a memory of said processor; modifying the computer-based model to include at least one artificial force, wherein the at least one artificial force is defined as a function of physics-based behavior; and The real-world object is iteratively optimized with respect to loads using the modified computer-based model, wherein a result of the iterative optimization is an optimized design of the real-world object.

2. The method of claim 1 , wherein the iteratively optimizing comprises: performing a simulation using the modified computer-based model; evaluating consistency of a value of a property of the real-world object with a design parameter based on a result of the simulation; responsive to said evaluating determining that said value of said property complies with said design parameters, determining that said modified computer-based model represents said optimized design of said real-world object; as well as In response to the evaluating determining that the value of the property does not conform to the design parameters, iteratively performing the following operations: (i) updating the modified computer-based model, (ii) performing a simulation using the updated modified computer-based model, and (iii) evaluating the value of the property of the real-world object for consistency with the design parameters based on results of the simulation performed using the updated modified computer-based model until the evaluating determines that the value of the property conforms to the design parameters.

3. The method of claim 1 , wherein the iteratively optimizing comprises, by the processor: Simulations were performed using the modified computer-based model.

4. The method of claim 3, wherein performing the simulation comprises: applying the load to the modified computer-based model; determining a value of the physics-based behavior of the computer-based model resulting from the applied load; as well as The at least one artificial force is applied to the modified computer-based model based on the determined value of the physics-based behavior.

5. The method of claim 4, wherein applying the at least one artificial force comprises: In response to the determined value of the physics-based behavior exceeding a threshold, deformation in the modified computer-based model is countered.

6. The method of claim 1 , wherein the iteratively optimizing comprises, by the processor: solving the modified computer-based model for an equilibrium primitive solution; determining a design response and at least one corresponding sensitivity for a design variable of the modified computer-based model; optimizing the real-world object using the modified computer-based model, the determined design response, and the at least one corresponding sensitivity, wherein a result of the optimization is a converged design or a non-compliant solution representing the optimized design of the real-world object; as well as In response to the result of the optimization being a non-compliant solution, the at least one artificial force is modified and the solving, determining, and optimizing are iterated using the modified at least one artificial force.

7. The method of claim 1, wherein the at least one artificial force is configured to oppose deformation in the modified computer-based model in response to a value of the physics-based behavior of the real-world object exceeding a threshold. The method of claim 1 , wherein the physics-based behavior comprises at least one of: displacement, velocity, and acceleration.

9. The method of claim 1, further comprising determining a point of inflection within the real-world object.

10. The method of claim 1, wherein the at least one artificial force is configured to suppress a post-buckling response of the real-world object.

11. A computer-implemented system for determining an optimal design of a real-world object, the system comprising: processor; as well as a memory having computer code instructions stored thereon, the processor and the memory having the computer code instructions stored thereon being configured to cause the system to: defining in said memory a computer-based model representing a real-world object; modifying the computer-based model to include at least one artificial force, wherein the at least one artificial force is defined as a function of physics-based behavior; as well as The real-world object is iteratively optimized with respect to loads using the modified computer-based model, wherein a result of the iterative optimization is an optimized design of the real-world object.

12. The system of claim 11, wherein, in iteratively optimizing the real-world object, the processor and the memory having the computer code instructions are configured to cause the system to: performing a simulation using the modified computer-based model; evaluating consistency of a value of a property of the real-world object with a design parameter based on a result of the simulation; responsive to said evaluating determining that said value of said property complies with said design parameters, determining that said modified computer-based model represents said optimized design of said real-world object; as well as In response to the evaluating determining that the value of the property does not conform to the design parameters, iteratively performing the following operations: (i) updating the modified computer-based model, (ii) performing a simulation using the updated modified computer-based model, and (iii) evaluating the value of the property of the real-world object for consistency with the design parameters based on results of the simulation performed using the updated modified computer-based model until the evaluating determines that the value of the property conforms to the design parameters.

13. The system of claim 11, wherein the processor and the memory having the computer code instructions are configured to cause the system to perform a simulation using the modified computer-based model while iteratively optimizing the real-world object.

14. The system of claim 13, wherein when performing the simulation, the processor and the memory having the computer code instructions are configured to cause the system to: applying the load to the modified computer-based model; determining a value of the physics-based behavior of the computer-based model resulting from the applied load; as well as The at least one artificial force is applied to the modified computer-based model based on the determined value of the physics-based behavior.

15. The system of claim 14, wherein upon applying the at least one artificial force, the processor and the memory having the computer code instructions are configured to cause the system to: In response to the determined value of the physics-based behavior exceeding a threshold, deformation in the modified computer-based model is countered.

16. The system of claim 11, wherein, in iteratively optimizing the real-world object, the processor and the memory having the computer code instructions are configured to cause the system to: solving the modified computer-based model for an equilibrium primitive solution; determining a design response and at least one corresponding sensitivity for a design variable of the modified computer-based model; optimizing the real-world object using the modified computer-based model, the determined design response, and the at least one corresponding sensitivity, wherein a result of the optimization is a converged design or a non-compliant solution representing the optimized design of the real-world object; as well as In response to the result of the optimization being a non-compliant solution, the at least one artificial force is modified and the solving, determining, and optimizing are iterated using the modified at least one artificial force.

17. The system of claim 11, wherein the at least one artificial force is configured to oppose deformation in the modified computer-based model in response to a value of the physics-based behavior of the real-world object exceeding a threshold.

18. The system of claim 11, wherein the processor and the memory having the computer code instructions stored thereon are further configured to cause the system to determine a buckling point within the real-world object.

19. The system of claim 11, wherein the at least one artificial force is configured to suppress a post-buckling response of the real-world object.

20. A computer program product for determining an optimized design of a real-world object, the computer program product being executed by a server in communication with one or more clients across a network and comprising: A non-transitory computer-readable medium comprising program instructions that, when executed by a processor, cause the processor to: defining a computer-based model in memory representing a real-world object; modifying the computer-based model to include at least one artificial force, wherein the at least one artificial force is defined as a function of physics-based behavior; as well as The real-world object is iteratively optimized with respect to loads using the modified computer-based model, wherein a result of the iterative optimization is an optimized design of the real-world object.

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