Optimization of geometric dimension control

Abstract

A computer-implemented method automatically determines an optimized design for manufacturing a real-world object, the method comprising: defining, in a memory of a computer-based processor, a finite element model representing the real-world object, the finite element comprising a plurality of elements; evaluating, by the computer-based processor, a distribution of design variables within a neighborhood of the finite element model using singular value decomposition (SVD) to generate singular values of the design variables for each respective element within the neighborhood of the finite element model; defining, based on the singular values generated from the SVD, an optimization constraint condition for the neighborhood of the finite element model; and optimizing, based on the defined optimization constraint condition, the design variables of the finite element model by locally enforcing a geometry of the real-world object within the neighborhood.

Description

[0001] Cross-reference to related applications

[0002] This application claims priority to U.S. Provisional Patent Application No. 62 / 945,961, filed December 10, 2019, entitled "Geometrical Dimensionality Control in Optimization," the disclosure of which is incorporated herein by reference in its entirety. Technical Field

[0003] This disclosure relates to the field of geometric dimension control, and more specifically, to systems and methods for optimizing geometric dimension control. Background Technology

[0004] Many computer systems and programs are available that provide a virtual environment for the design of real-world components or assemblies of components (e.g., objects). These systems (some of which are called computer-aided design (CAD) systems) enable users to build, view, and manipulate three-dimensional virtual models of real-world physical objects being designed, which are sometimes complex. CAD systems are typically configured to provide a visual representation of any such modeled object on a computer-based vision display device using, for example, edges, lines, and surfaces.

[0005] Modeled objects are typically represented in a CAD system as CAD files, which contain computer-readable specifications of the real-world physical geometry associated with the underlying object. A particular object may be represented by computer-readable code stored in a single CAD file or multiple CAD files. In one example, a CAD file may contain computer-readable code representing specifications (e.g., properties) associated with a real-world version of a real-world physical object, from which the geometry associated with the real-world physical object can be identified (e.g., by the CAD system). Based on these geometries, the CAD system can generate one or more visual representations of the associated object and display the object on a computer display device. CAD systems include (e.g., on computer display devices) graphical tools for representing modeled objects to designers; these tools can be configured to sometimes display highly complex objects.

[0006] Design optimization is an engineering design methodology that uses mathematical formulas for design problems to support the selection of the optimal design from among many alternatives. For example, topology optimization (TO) involves the technical field of optimizing the material layout within a given design space for a given set of loads, boundary conditions, and constraints to maximize the performance of the system.

[0007] Typically, there are areas for improvement in the field of design optimization, especially in the field of topology optimization. Summary of the Invention

[0008] In one aspect, a computer-implemented method for automatically determining an optimal design for manufacturing a real-world object includes: defining a finite element model representing the real-world object in the memory of a computer-based processor, the finite element comprising a plurality of elements; evaluating the distribution of design variables within a neighborhood of the finite element model using singular value decomposition (SVD) via the computer-based processor to generate singular values ​​of the design variables for each corresponding element within the neighborhood of the finite element model; defining optimization constraints for the neighborhood of the finite element model based on the singular values ​​generated from the SVD; and optimizing the design variables of the finite element model by locally enforcing the geometry of the real-world object within the neighborhood based on the defined optimization constraints.

[0009] On the other hand, a computer-based system is disclosed for automatically determining an optimal design for manufacturing real-world objects. The computer-based system includes a computer-based processor and computer-based memory coupled to the processor. The computer-based memory stores data defining a finite element model representing a real-world object; the finite element has multiple elements. The computer-based memory also stores computer-readable instructions that, when executed by the computer-based processor, cause the processor to: evaluate the distribution of design variables within a neighborhood of the finite element model using singular value decomposition (SVD) to generate singular values ​​of the design variables for each corresponding element within the neighborhood of the finite element model; define optimization constraints for the neighborhood of the finite element model based on the singular values ​​generated from the SVD; and optimize the design variables of the finite element model by locally enforcing the geometry of the real-world object within the neighborhood based on the defined optimization constraints.

[0010] In another aspect, a non-transitory computer-readable medium stores computer-readable instructions that, when executed by a computer-based processor, cause the computer-based processor to: evaluate the distribution of design variables within a neighborhood of a finite element model using singular value decomposition (SVD) to generate singular values ​​of the design variables for each corresponding element within the neighborhood of the finite element model; define optimization constraints for the neighborhood of the finite element model based on the singular values ​​generated from the SVD; and optimize the design variables of the finite element model by locally enforcing the geometry of real-world objects within the neighborhood based on the defined optimization constraints.

[0011] Another aspect relates to a computer-based approach that includes: identifying nonparametric design variables in a finite element model; using SVD to evaluate the distribution of design variables around a given finite element design variable; obtaining singular values ​​for each design variable; calculating the optimal design response from the singular values ​​for each design variable element; aggregating each type of optimal design response for each design variable element into a single constraint design response value for the entire set of design variables; and implementing the aggregated design response value as a constraint in the structural optimization solution.

[0012] In some implementations, there are one or more of the following advantages.

[0013] For example, various implementations provide efficient and optimized designs in a highly efficient manner. Furthermore, the method can be implemented in addition to adding other design requirements such as stiffness and strength. Moreover, the method provides an implementation for multiple potential tasks (enforcing 2D, enforcing 1D, limiting local material quantities, curvature control, thin / thickness control, controlling scale, controlling orientation). The method is generally applicable to any finite element mesh (structured / unstructured, any type of element, 1D to 3D, etc.). The method has an intuitive setup, and the interpretation and relationships of all applied measurements are relatively simple and clear.

[0014] Furthermore, many manufacturing processes in 3D printing (also known as additive manufacturing) do not allow powder to become trapped during the printing process. Therefore, hollow and closed structural components are not permitted as part of a viable manufacturable design. In some implementations, hollow and closed structural components can be removed from the optimized design by controlling certain singular values ​​(e.g., s) through geometric dimension control. mid and s min The constraint is set to a very small value (e.g., close to zero) to force the structural layout of the components to form a lattice.

[0015] Furthermore, many metal components for automotive structures are sheet metal fabricated because these structures have highly cost-competitive manufacturing processes. However, designs often optimized using TO (Top-Order) have larger material layouts and are more suitable for casting. Therefore, optimized designs for sheet metal fabrication can be achieved by controlling certain singular values ​​(e.g., s) through geometric dimension control. max and s mid The relationship between the components is constrained to a higher value (e.g., approximately 1) to force the layout of the components to locally have a membrane layout.

[0016] Other features and advantages will become apparent from the description and drawings, as well as from the claims. Attached Figure Description

[0017] Figure 1This is a simplified block diagram of a computer system used to perform an embodiment of the design process disclosed herein.

[0018] Figure 2 This is a simplified block diagram of a computer network environment that is an embodiment of the design process disclosed herein.

[0019] Figure 3A and Figure 3B The flowchart illustrates an exemplary implementation of a design optimization process that includes constraining design values ​​based on singular value decomposition (SVD) of design variables.

[0020] Figure 4A The structural element is shown as a sphere with a given radius.

[0021] Figure 4B It shows Figure 4A The relative density field in the structural elements.

[0022] Figure 4C It shows Figure 4B The center of mass of the element in the relative density field.

[0023] Figure 4D It shows the relative density as a measure. Figure 4C The coordinates of the centroid of the element in the array.

[0024] Figure 4E It shows the relationship with Figure 4D The singular value decomposition (SVD) values ​​(s) related to the three dimensions of the object and the corresponding orientation / vector of the point cloud. max ,s mid ,s min ).

[0025] Figure 5 The relationships between the normalized singular values ​​of the relative density field, used as design variables for topology optimization, are shown. These relationships include the entity (representation of 3D objects), the imaginary part (representation of 0D objects), and the intermediate (between 3D and 0D object representations).

[0026] Figure 6 The diagram shows how, for the normalized singular values ​​of the relative density field, which is used as a design variable, for example, for topology optimization, an entity (represented as a 3D object) is transformed into a shell, plate, or membrane layout (represented as a 2D object).

[0027] Figure 7 The diagram shows how a shell, plate, or membrane layout (represented as a 2D object) is transformed into a beam, strip, or lattice layout (represented as a 1D object) for the normalized singular values ​​of the relative density field, which is used as a design variable, for example, for topology optimization.

[0028] Figure 8The diagram shows how beam, strip, or lattice layouts (1D object representations) are transformed into nearly virtual layouts (0D object representations) for normalized singular values ​​of the relative density field, which is used as a design variable, for example, for topology optimization.

[0029] Figure 9 The diagram shows how, for the normalized singular values ​​of the relative density field, which is used as a design variable, for example, for topology optimization, entities (represented as 3D objects) are transformed into virtual layouts (represented as 0D objects).

[0030] Figure 10 The diagram shows that for a relative density field, which is used as a design variable, for example for topology optimization, all normalized singular values ​​are 1, representing an assembly of 1D or 0D components in 3D space.

[0031] Figure 11 The diagram shows an assembly of 1D or 0D components in 2D space, where the relative density field, used as a design variable for example for topology optimization, has two normalized singular values ​​of 1 and one normalized singular value approximately zero.

[0032] Figure 12A The design space, subdivided into finite element methods, is shown for topology optimization, where the relative density of each element is the design variable; the beam is completely clamped on the left side and the bottom edge on the right side bears the load.

[0033] Figure 12B It shows the basis Figure 12A The topology optimization results are obtained in the design space without geometric dimension control.

[0034] Figure 12C It shows the basis Figure 12B The design yields smooth topology optimization results.

[0035] Figure 12D The results of topology optimization with dimension control are shown, which force the 2D sub-components to become structural plates by constraining the minimum singular value σ3 to be less than 0.15 of the decomposition (SVD) of the design variable (DV) with a diameter of 10.0.

[0036] Figure 12E The results of smooth topology optimization based on the 12D design are shown.

[0037] Figure 13A and Figure 13B The optimization iterative history is shown as the objective function compliance minimizes the compliance to a relative quality constraint of 15%.

[0038] Figures 14A to 14CThe optimization iterative convergence history is shown for methods that minimize the compliance to a relative mass constraint of 15% and constrain the minimum singular value σ3 to 0.15 of the singular value decomposition (SVD) of the design variable (DV) with a diameter of 10.0.

[0039] Figure 15 It is a flowchart that includes the sensitivity of the design response (DRESP) for the design variables and the calculated singular values ​​(SVD) used for geometric dimension control.

[0040] Figure 16A The design space for topology optimization, subdivided into finite element methods, is shown, where the relative density of each element is the design variable, each lower corner is fully clamped, and the upper-middle point bears the load.

[0041] Figure 16B Topology optimization without geometric dimension control is shown.

[0042] Figure 16C This illustrates dimensionality control in topology optimization of 1D structural sub-components.

[0043] Figure 16D This illustrates dimensionality control in enforcing topology optimization for 2D structural sub-components.

[0044] Figure 17A Dimension control is shown in the forced topology optimization of 1D structural sub-components, which represent fibers that do not intersect each other; the topology optimization results are represented by finite element methods with relative densities of solids and intermediates.

[0045] Figure 17B Dimension control is shown in the forced topology optimization of 1D structural sub-components, which represent fibers that do not intersect each other; the topology optimization results are represented by isosurfaces describing the relative density field.

[0046] Figure 18 A sphere of radius r is shown in a finite element model of a real-world physical object, where the centroid is identified for each corresponding finite element in the model.

[0047] Figure 19A A rectangular 2D design domain discretized by a 400×200 uniform grid is shown.

[0048] Figure 19B The classical stiffness topology optimization using a material volume fraction f = 60% is shown.

[0049] Figure 19C It shows Figure 19BThe classic topology optimization in the model includes additional constraints on the proposed method defined in equation (17), which uses a diameter D = 12 equivalent to 12 finite elements to constrain the minimum singular value to less than or equal to 0.6.

[0050] Figure 20A The convergence history of the optimization iterations for minimizing the compliance is shown.

[0051] Figure 20B The relative quality constraint that minimizes the compliance to 15% is shown.

[0052] Figure 20C An implementation of the technique proposed herein, which minimizes the compliance, is shown, involving constraining the minimum singular value to less than 0.60 of the decomposition (SVD) of the design variables with a diameter D = 12.

[0053] Figures 21A to 21F Exemplary optimization results are shown.

[0054] Figures 22A to 22D Other exemplary optimization results are shown.

[0055] Figures 23A to 23D Several other exemplary optimization results are shown.

[0056] Figure 24 This represents an exemplary finite element model of the femur.

[0057] Figures 25A to 25E An exemplary topology-optimized filler design for the femur is shown.

[0058] Similar reference characters refer to similar components. Specific Implementation

[0059] Design optimization typically refers to engineering design methods that involve mathematical formulas for design problems to support the selection of the best design from among many alternatives.

[0060] Optimization methods typically involve selecting one or more optimal elements (for some criteria) from a number of available alternatives. Beyond product design optimization, optimization problems even appear in a variety of disciplines. In simpler cases, optimization problems involve maximizing or minimizing a real function by systematically selecting input values ​​from a set of allowed values ​​and computing the function's value.

[0061] Design optimization typically involves design variables (which can be used to describe various design alternatives), design objectives (a combination of specific design variable values ​​to be maximized or minimized), constraints (e.g., variables or combinations of variables expressed as equations, inequalities, etc., which must be satisfied for a particular design alternative to be acceptable), and feasibility (represented by a set of values ​​of design variables that satisfy all constraints and / or design objectives).

[0062] An example of a design optimization problem expressed in mathematical form is:

[0063] Minimize f(x)

[0064] Subject to h i (x) = 0, i = 1, ..., m1

[0065] g j (x)≤0, j=1,...,m2

[0066] and

[0067] Where x is a vector of n real-valued design variables, f(x) is called the objective function, and h i (x) and m1 are equality constraints, g i (x) and m2 are inequality constraints, and X is a set of constraints that include additional restrictions on x that go beyond the restrictions implied by the equality and inequality constraints.

[0068] One specific type of optimization is topology optimization, which typically involves mathematical methods that, for example, given a set of loads, boundary conditions, and / or constraints, optimize the material layout within a given design space to maximize system performance. Topology optimization differs from shape and size optimization in that, in topology optimization, the design can take any shape within the design space, rather than dealing with predefined configurations.

[0069] Topology optimization formulas typically use the finite element method to evaluate design performance. Designs can be optimized using gradient-based mathematical programming techniques (such as optimality criterion algorithms and moving asymptote methods) or non-gradient-based algorithms (such as general algorithms).

[0070] Topology optimization has wide applications in various engineering and design disciplines, such as aerospace, mechanical, biochemical, and civil engineering. Topology optimization can be used at the conceptual level of the design process. Due to the inherent free forms in topology optimization, it can sometimes be difficult to produce results. In some implementations, the techniques disclosed herein can help overcome the limitations of the prior art.

[0071] Existing optimization techniques

[0072] Some existing optimization techniques discussed below include filtering techniques (sometimes also called regularization techniques) and projection methods of design variables, direct parameterization of design variables, penalty functions or projection functions of design variables, local volume constraints, and heuristic methods. The techniques disclosed in this application differ from those prior art. For example, the techniques disclosed in this application relate to using singular value decomposition (SVD) of design variables to optimize certain aspects of a design (e.g., topology). In some embodiments, the obtained singular values ​​can be applied to the objective function as terms to be optimized in the optimization process. Alternatively, in some embodiments, the obtained singular values ​​can be used as optimization constraints to locally constrain the geometry of the optimized structure. None of the existing optimization techniques discussed below utilize singular value decomposition (SVD) of design variables in a similar manner.

[0073] 1. Filtering techniques (sometimes also called regularization techniques) and projection methods for design variables.

[0074] A common approach is density variable filtering for topology optimization, where relative density is a design variable and is filtered. (See MP for example) O. Sigmund and O. Sigmund, Topology Optimization—Theory, Methods, and Applications, 2004; O. Sigmund and J. Petersson, Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies, and local minima, Structural Optimization 16 (1998), pp. 68–75; BSLazarov, F. Wang, and O. Sigmund, Length scale and manufacturability in density-based topology optimization, Archives of Applied Mechanics 86 (1–2) (2016), pp. 189–218; M. Zhou, BSLazarov, F. Wang, and O. Sigmund, Minimum length scale in topology optimization by Geometric constraints, Minimum length scaling in topology optimization implemented through geometric constraints, *Computer Methods in Applied Mechanics and Engineering* 293 (2015), pp. 266–282; BSLazarov and F. Wang, Maximum length scale indensity based topology optimization, *Computer Methods in Applied Mechanics and Engineering*, 318 (2017), pp. 826–844, and other references mentioned herein. These filtering techniques and projection methods can be combined to enforce length scaling, thereby helping to ensure manufacturable structures that meet component dimensional requirements, such as fabricated structures for casting, 3D printed structures, milled structures, etc.

[0075] The techniques disclosed in this application can enforce geometric length scaling via diameter. However, the techniques disclosed in this application are fundamentally different from filtering techniques and projection methods of design variables. For example, as mentioned above, the techniques disclosed in this application involve using singular value decomposition (SVD) of design variables in a specific manner to optimize certain aspects of the design (e.g., topology), while filtering techniques and projection methods of design variables do not use singular value decomposition (SVD) of design variables.

[0076] 2. Direct parameterization of design variables

[0077] Direct parameterization of design variables can be used, for example, to obtain feasible designs for structures used in casting and plate fabrication. (See, for example, S. Zhang, J.A. Orato and A.L. Gain et al., A geometry projection method for the topology optimization of plate structures, Structures and Multidisciplinary Optimization 54(5)(2016), pp. 1173-1190; J.P. Leiva, BC.) And I. Kosaka, An Analytical Bi-Directional Growth Parameterization to Obtain Optimal Castable Topology Designs, 10th AIAA / ISSMO Multidisciplinary Analysis and Optimization Conference, August 30–September 1, 2004, Albany, New York; and A. Roulund-Gersborg and C. Andreasen, An explicit parameterization for casting constraints in gradient driven topology optimization, Structures and Multidisciplinary Optimization 44(6)(2010), pp. 875–881.

[0078] The techniques disclosed in this application can also be applied to obtain feasible designs for structures used in board manufacturing. However, the techniques disclosed in this application differ from the direct parameterization of design variables. For example, the techniques disclosed herein do not include the application of new parameterizations or mappings of design variables. Furthermore, as stated above, the techniques disclosed in this application relate to optimizing certain aspects of a design (e.g., topology) using singular value decomposition (SVD) of design variables in a specific manner; direct parameterization of design variables does not use singular value decomposition (SVD) of design variables.

[0079] 3. Design the penalty function or projection function of the variables.

[0080] Penalty functions can be applied to ensure geometrically feasible designs for additive manufacturing (3D printing) and multi-axis machining. (See, for example, M. Langeraar, An additive manufacturing filter for topology optimization of print-ready designs, Structural and Multidisciplinary Optimization 55(3)(2017), pp. 871-883; M. Langeraar, Topology optimization for multi-axis machining, Computer Approaches to Applied Mechanics and Engineering, Structural and Multidisciplinary Optimization 351(2019), pp. 226-252; M. Hoffarth, N. Gerzen and CBWPedersen, ALM Overhang Constraint in Topology Optimization for Industrial Applications, 12th World Congress on Structural and Multidisciplinary Optimization, 5-9 June 2017, Braunschweig, Germany; and Publication No. 2017 / 0176975A1, entitled “Penalty Function On Design Variables For Designing Variables For Designing Cost”) (A U.S. patent application for "Beneficially Additive Manufacturable Structures" which is a penalty function for design variables used to design additively manufactureable structures that save design costs.)

[0081] Projection schemes possess, to some extent, the ability to apply geometric control for optimization, thus ensuring that the designed geometry is manufacturable. (See, for example, SL Atanabe, TN Lippi, CR Lima, GH Paulino, and ECN Silva, Topology optimization with manufacturing constraints: A unified projection-based approach, Advances in Engineering Software 100 (2016), pp. 97–112; and JV Carstensen and JK Guest, Projection-based two-phase minimum and maximum length scale control in topology optimization, Structures and Multidisciplinary Optimization 58(5) (2018), pp. 1845–1860.) Some works use geometric projection methods to project a plate of fixed thickness and an analytical description of a set of geometric primitives separately. These may force structural members to be manufacturable, well-designed shapes or plates. (See, for example, JANorato, Topology optimization with supershapes, Structures and Multidisciplinary Optimization 58(3)(2018), pp. 1-20; and S. Zhang, JANorato, ALGain and N. Lyu, Geometry projection method for the topology optimization of plate structures, Structures and Multidisciplinary Optimization 54(5)(2016), pp. 1173-1190.)

[0082] The techniques disclosed in this application can enforce plate-like structures, but are fundamentally different from penalty functions or projection functions. For example, as mentioned above, the techniques disclosed in this application involve using singular value decomposition (SVD) of design variables in a specific manner to optimize certain aspects of the design (e.g., topology); penalty functions or projection functions do not use singular value decomposition (SVD) of design variables.

[0083] 4. Local volume constraints

[0084] Some works use local volume constraints to enforce geometrically porous structures. Local volume constraints can also be applied to structures that mimic lattice shapes. (See, for example, Wu, N. Aage, R. Westermann and O. Sigmund, Infill optimization for additive manufacturing Approaching bone-like porous structures, IEEE Transactions on Visualization and Computer Graphics, 24(2)(2018), pp. 1127-1140; Wu, A. Clausen and O. Sigmund, Minimum compliance topology optimization of shell-infill composites for additive manufacturing, Computer Approaches to Applied Mechanics and Engineering, 326(2017), pp. 358-375; and M. Schmidt, C.B. Wedersen and C. Gout, On structural topology optimization using graded porosity control, Structures and Multidisciplinary Optimization, 60(4)(2019), pp. 1437-1453.)

[0085] The techniques disclosed in this application can enforce lattice-shaped structures, but are fundamentally different from local volume constraint methods. For example, as mentioned above, the techniques disclosed in this application involve using singular value decomposition (SVD) of design variables in a specific manner to optimize certain aspects of the design (e.g., topology); local volume constraint methods do not utilize singular value decomposition (SVD) of design variables.

[0086] 5. Heuristic methods

[0087] Some works involve heuristic methods.

[0088] For example, see N. The work, "Topology optimization of structures with manufacturing and unilateral contact constraints by minimizing an adjustable compliance-volume product," in Structural and Multidisciplinary Optimization 42(3)(2010), pp. 341-350, applies a heuristic approach to update the movement constraints in each defined optimization iteration to satisfy drag constraints. Therefore, apart from upper and lower limits for density, there are no explicit constraints in the nested formulas, but the updating of these movement constraints is entirely heuristic.

[0089] R. Dienemann, A. Schumacher, and S. Fiebig's work, *Topology optimization for finding shell structures manufactured by deep drawing*, Structural and Multidisciplinary Optimization, 56(2)(2017), pp. 473-485, applies the mid-surface method, which calculates the average of the element coordinates in the stamping direction to realize deep-drawn structures. This method is more illuminating because, with a constant wall thickness, the movement of the mid-surface is not always included in the mathematical sensitivity calculations. This is evident from the increasing number of optimization iterations.

[0090] These heuristics are mathematically inconsistent, while the techniques disclosed in this application are mathematically consistent and can enforce structures suitable for deep drawing. However, the techniques disclosed in this application are fundamentally different from heuristics. For example, as mentioned above, the techniques disclosed in this application involve using singular value decomposition (SVD) of design variables in a specific manner to optimize certain aspects of the design (e.g., topology); heuristics do not use singular value decomposition (SVD) of design variables.

[0091] The novel techniques disclosed herein typically involve optimizing certain aspects of a design (e.g., topology) using singular value decomposition (SVD) of design variables in a specific manner, unlike the existing methods disclosed above. These differences are not trivial. For example, in some embodiments, the novel techniques disclosed herein produce optimized designs that are smaller in volume than those that could be produced by existing methods and are more feasible to manufacture (e.g., structures using a single stamped sheet or multiple stamped sheets welded together).

[0092] For example: automobile companies (e.g., General Motors) ),public Audi BMW Ford Jaguar Land Rover TM and aerospace companies (such as Airbus) Boeing In particular, optimization processes are extensively used for structural components. These optimization processes often result in structures with sub-components, thus leading to large-scale designs. Typically, from a manufacturing process perspective, practical and industrial design requirements dictate that structures should be manufactured using single stamped metal sheets or structures made by welding multiple stamped metal sheets together.

[0093] In some embodiments, the techniques disclosed in this application can produce and advantageously enforce designs that are manufacturable, for example, for stamped sheet structures. (See, for example) Figures 12A to 12E ). Figures 12A to 12E The model used for topology optimization is a beam completely clamped on the left side, with the bottom edge on the right side bearing the load. Figure 12A The model shown can include 375,000 linear eight-node continuous finite element methods, each with a size of 1×1×1, and the total beam size is 50×50×150. In the example shown, each finite element is a design variable that can be defined using the SIMP (Solid Isotropic Material Penalty) method, and topology optimization can be performed using topology sensitivity and mathematical programming (see, for example, MP). (With O. Sigmund, Topology Optimization: Theory, Methods, and Applications, 2004.)

[0094] The finite element model is linear, and the applied linear constituent material has a Poisson's ratio of 0.30 and a Young's modulus of 210,000. In this example, the optimization objective is to minimize the coefficients of the linear constituent material. Figure 12AThe stiffness is maximized by using a fixed relative mass of 15% of the given design domain. Figure 12B An exemplary topology optimization result without geometric dimension control is shown. Figure 12C It shows the basis Figure 12B Examples of real-world objects obtained from the design. Figure 12D An exemplary topology optimization result with dimension control is shown, wherein the minimum singular value s is obtained by... min The constraint is less than 0.15, forcing the 2D sub-component to be a structural plate. This enhances the manufacturability of the optimized design, particularly the manufacturability using structural plates. In this example, the value s min The calculation is performed by decomposing (SVD) the design variables (DV) with a diameter of 10 within the neighborhood of each corresponding finite element (see Figure 12D). Figure 12E It shows the basis Figure 12D Examples of real-world objects obtained from the design.

[0095] As noted, traditional topology optimization without geometric dimension control results in sub-components strictly as strips and beams, while the topology optimization with dimension control in this paper forces 2D sub-components to be included as structural plates. Therefore, two different designs with typically drastically different processes are applied to manufacturing. It should be noted that... Figure 12B and Figure 12D The two designs have Figure 12A The relative quality of the design domain is 15%. However, there is no geometric dimension control. Figure 12B The design has a low compliance of 0.00750, while featuring geometric dimension control. Figure 12D The design exhibits a high flexibility of 0.00896. Therefore, for this example, enforcing geometric dimension control, as shown here, for 2D sub-components to ensure the feasibility of manufacturing the stamped plate structure will affect the objective function value; for the same mass, the stiffness is reduced by 16%.

[0096] Figure 13A and 13B It shows Figure 12B The optimization iteration history of the design is obtained using a traditional topology optimization workflow without geometric dimension control. More specifically, Figure 13A This shows how the compliance of the objective function changes with the number of optimization iterations, and Figure 13B This shows how the 15% relative quality constraint changes with optimization iterations.

[0097] Figures 14A to 14C It shows Figure 12D The optimization iteration history of the design is obtained through a topology optimization workflow with geometric dimension control. More specifically, Figure 14AThis shows how the compliance of the objective function changes with the number of optimization iterations. Figure 14B This shows how the 15% relative quality constraint changes with each optimization iteration, and Figure 14C This shows how the minimum singular value (less than 0.15) changes with each optimization iteration.

[0098] Compare Figure 13A , Figure 13B and Figure 14A , Figure 14B and Figure 14C From the optimization history, it can be observed that the minimum singular value s is additionally included. mid Constraints less than 0.15 do not significantly alter the optimization history. Furthermore, the singular value decomposition (SVD) method, which uses geometric dimension control constraints on design variables during optimization, can be used in sensitivity-based and mathematically programmed optimization workflows without leading to optimization convergence issues.

[0099] Figure 1 This is a schematic diagram of an exemplary computer-based system 100, which can be used to generate optimized designs for real-world physical objects according to one or more techniques disclosed herein, such techniques including, for example, techniques that utilize singular value decomposition (SVD) of design variables to optimize certain aspects of the design (e.g., topology).

[0100] The computer-based system 100 includes a central processing unit 102, computer-based memory 104, computer-based storage 106, a network interface 108, an input / output device interface 110, and a bus for interconnecting these components. In a typical implementation, the bus provides a communication medium through which the various components of the computer-based system 100 can communicate and interact with each other.

[0101] The central processing unit (CPU) 102 is configured to perform various computer-based functions, including those disclosed herein. Typically, the CPU performs these functions by executing computer-readable instructions from a non-transitory computer-readable medium (e.g., computer-based memory 104 and / or computer-based storage 106), and / or by executing computer-readable instructions from computer-readable data received from external sources, such as via I / O device interface 110 and / or network interface 108.

[0102] Computer-based memory 104 provides a form of volatile memory for computer-readable instructions that, when executed by CPU 102, enable the CPU to perform various computer-based functions, including, for example, those disclosed herein.

[0103] Computer-based memory 106 provides a form of non-volatile memory for software instructions (such as operating systems (not shown) and implementation configurations).

[0104] Network interface 108 is used to connect to any kind of computer-based network or communication network, including, for example, a local area network (LAN) and / or a wide area network (WAN) such as the Internet.

[0105] Input / output device interface 110 is configured to act as a connection interface for one or more input and / or output devices (such as a keyboard, mouse, monitor, speakers, etc.). In a typical implementation, computer system 100 may be configured (e.g., on a display device connected to I / O device interface 110) to display a visual representation of an optimized design based on the techniques disclosed herein. In some implementations, the design may appear, for example, as shown below. Figure 12D or Figure 12E , Figure 16C or Figure 16D or Figure 17A or Figure 17B As shown. Furthermore, in such an implementation, the display allows the user to view additional information about the displayed, optimized design (e.g., performance characteristics, manufacturing information, etc.).

[0106] In various implementations, the computer system 100 may have additional elements, such as controllers, buffers (caches), drivers, repeaters, and receivers, to facilitate communication and other computer-based functions. Furthermore, interfaces may include address, control, and / or data connections to facilitate communication between the illustrated components.

[0107] Figure 2 This is a schematic diagram of an exemplary computer network environment 200 based on one or more techniques disclosed herein, which can be used to determine an optimal design for a real-world physical object. Such techniques include, for example, techniques that utilize singular value decomposition (SVD) of design variables to optimize certain aspects of the design (e.g., topology).

[0108] Computer network environment 200 has a server 202 and clients 204a, 204b...204n coupled to each other via a communication network 206, which enables the server 202 and clients 204a to communicate and interact with each other. In a typical implementation, each client 204a, 204b...204n and the server may include... Figure 1The components in the computer system 100 are of the same type. In some embodiments, each client 204a, 204b...204n can be configured to perform the functions disclosed herein to optimize the design without the involvement of server 202. In some embodiments, the functions disclosed herein for optimizing the design can be distributed between server 202 and clients (e.g., 204b), with server 202 providing, for example, cloud computing functions in the case of Software as a Service (SaaS).

[0109] Figure 3A and Figure 3B A flowchart is shown together, illustrating an exemplary implementation of a method 300 for automatically determining an optimized design for real-world objects; the method includes constraining the values ​​of singular value decomposition (SVD) from design variables.

[0110] The method 300 shown begins at 302: defining a finite element model in the memory of a computer-based processor, which is based on a CAD model of the real-world physical object to be designed and optimized.

[0111] For example, CAD systems and Computer-Aided Engineering (CAE) systems offer a wide range of possibilities for representing real-world physical objects in virtual, computer-based environments. Representing real-world physical objects as finite element models is one such possibility. The term finite element model (FEM) refers to a type of model associated with, used in, and related to the finite element method.

[0112] Finite element analysis refers to the practical application of the finite element method, a method for solving engineering or design problems that involves subdividing a system (or object, or a CAD model of the object) into smaller, simpler parts called finite elements. This subdivision can be achieved through specific spatial discretization in a spatial dimension, which is accomplished by constructing the mesh (continuous domain) of the object under discussion into a set of discrete subdomains (often called elements). In some implementations, the finite element method may involve forming boundary value problems that result in a system of algebraic equations. The finite element method approximates unknown functions over the domain. The simple equations modeling these finite elements are then assembled into a larger system of equations modeling the entire problem. The finite element method can then approximate the solution using variational methods derived from variational methods (e.g., by minimizing associated error functions, etc.).

[0113] There are various software applications available for performing or facilitating finite element analysis and generating finite element models. One such application is Abaqus FEA. TMThis is a software suite available from Dassault Systèmes Simulia Corp., which includes computer-readable instructions that are executed by a processor in a computer system (e.g., ...). Figure 1 The processor shown performs computer-based finite element analysis functions during execution. In a typical implementation, Abaqus FEA... TM The software can be merged into Tosca Structure version or as A portion of the Tosca architecture version is provided, and is also available from Dassault Systèmes, suitable for supporting the various optimization-related functions disclosed herein. In a typical implementation, any such computer-readable instructions (e.g., incorporated into...) A suitable version of the Tosca structure or as A suitable version of the Tosca structure is provided by Abaqus FEA TM Software can be stored in a non-transitory computer-readable medium (e.g., stored on...). Figure 1 (in computer-based memory 104 or computer-based storage 106), and may be coupled to a computer-based processor (e.g., ...) on a non-transitory computer-readable medium. Figure 1 The central processing unit 102 in the middle executes the commands.

[0114] In a typical implementation, the finite element model comprises a set of points called nodes, which are interconnected to form a lattice, called a mesh. In a typical implementation, the finite element model is represented by a set of data that defines a computer-readable format stored on a computer-based storage device (e.g., ...). Figure 1 A finite element model (FEM) is defined as either 104 or 106 in the model. The stored data associated with the finite element model can reflect one or more characteristics of one or more underlying objects represented by the finite element model. When the finite element model is configured in this way, it can be used to perform simulations of the objects represented by the finite element model. In this respect, a finite element model can represent any of a variety of different objects or combinations of objects, including, for example, vehicles, some type of physical hinge, structural supports (e.g., beams), or any number of other real-world physical objects or combinations of objects. If the finite element model represents an object, and the underlying data associated with the finite element indicates this, then the finite element model can be used to simulate one or more aspects or characteristics of the real-world physical object itself. For example, a finite element model representing a stent can be used to simulate the use of stents in a real-world medical environment.

[0115] In a typical implementation, the finite element model comprises multiple elements. According to an embodiment, the elements may be checkerboard-shaped elements of the finite element model. However, the embodiments are not limited and can be used in relation to any element such as a triangle, quadrilateral membrane, or shell. According to an embodiment of method 300, the finite element model can be defined at step 302 according to any method known in the art. Furthermore, in embodiments, the finite element model defined at step 302 can be any finite element of this type known in the art. For example, in an exemplary embodiment, the finite element model defined at step 302 can be a sheet-like model of the surface of a real-world object. According to an embodiment, defining the model at step 302 may include defining in computer-based memory all (or a subset) of known properties (e.g., dimensions, materials, etc.) of the object represented by the finite element model, and the model includes multiple design variables (e.g., plate thickness). In some implementations, the design variables may represent object characteristics whose values ​​(e.g., inches or centimeters) will be determined (or optimized). Furthermore, the behavior of the finite element model may be controlled by an equation that includes a corresponding sensitivity equation for the design variables (e.g., plate thickness). In such an embodiment, defining a finite element model may include defining equations that control the behavior of the model (i.e., the object represented by the model).

[0116] Method 300 continues at 304, incorporating the associated optimization process to identify the neighborhood or region of the real-world model and / or corresponding finite element model to be considered (e.g., a sphere with a given radius, see example...). Figure 18 ).

[0117] The dimension of an object computed using singular values ​​typically describes how much space the object requires within a considered neighborhood (e.g., a sphere). Therefore, the size of the neighborhood (e.g., the sphere) should be appropriately chosen based on the specific application. In some implementations, a smaller sphere (radius = 2–5 × finite element size) should be used to locally control the structural dimension. In some implementations, using a larger sphere for practical applications may lead to a misunderstanding of the structural dimension used for optimization. For example, as... Figure 10 and Figure 11 As shown, a combination of 1D or 0D parts can be interpreted as either a 3D object or a 2D object. Whether such a geometric interpretation is needed typically depends on the specific design application.

[0118] For a finite element model described in 3D space, the finite elements of the structure (or a selected neighborhood of the structure) can be represented using matrices containing nodes and the corresponding coordinates of each node. These matrices can be stored, for example, in computer-based memory 104.

[0119] Next, the method 300 shown includes 306: evaluating the distribution of design variables within the neighborhood of the real-world object using singular value decomposition (SVD) for each finite element in the neighborhood of the object / finite element model. In some implementations, singular value decomposition (SVD) may refer to the factorization of a real or complex matrix, which, for example, generalizes the eigenvalues ​​of a squared normal matrix to any m×n matrix via an extension of extreme decomposition. Specifically, the singular value decomposition of an m×n real or complex matrix M is a factorization of the form U∑V*, where U is an m×m real unitary or complex unitary matrix, ∑ is an m×n rectangular diagonal matrix with non-negative real numbers on the diagonal, and V is an n×n real unitary or complex unitary matrix.

[0120] Next, the method 300 shown includes 308: based on SVD evaluation, determining singular values ​​(e.g., s) of design variables within the neighborhood of a real-world object. max ,s mid and s min , or s1, s2, and s3 respectively) and their corresponding orientations. In the context of topology optimization, for example, SVD can be used to evaluate each finite element (a sphere with a given radius, see [reference]). Figure 4A or Figure 18 The distribution of design variables (e.g., relative density) around the target. The obtained singular values ​​(s) max ,s mid and s min See Figure 4E This relates to the three geometric dimensions of the object, described by the density field of the considered neighborhood (the sphere). These measurements can be used, for example, to construct a design response for structural optimization.

[0121] Refer again Figure 3A Method 300 includes 310: defining optimization constraints for the neighborhood based on singular values.

[0122] Figure 3B Representation 310: An exemplary method 300a for defining optimization constraints for a neighborhood based on singular values.

[0123] In a typical implementation, the measurements s1, s2, and s3 are discontinuous between optimization iterations and cannot be directly used in gradient-based optimization algorithms. In some cases, a smoothing approximation can be used to address this issue. In this regard, method 300a includes 310a: applying a smoothing approximation to each singular value among the singular values ​​associated with the design variables within the neighborhood of the real-world object, based on SVD evaluation.

[0124] Next, Figure 3BMethod 300a includes 310b: identifying additional values ​​by performing SVD on the matrix, where the values ​​of all design variables are set to 1.0. This value typically represents the maximum possible dimension of the object under consideration (or its neighborhood) and can be used to calculate relative singular values.

[0125] Then, method 300b includes 310c: defining optimization constraints for each neighborhood using smooth approximations and additional values ​​in computer-based memory.

[0126] Refer again Figure 3A Method 300 includes iteratively optimizing a finite element model (at 312). In a typical implementation, this includes limiting the dimensions of the finite element model based on defined optimization constraints (defined based on singular value SVD). The iterative optimization at step 312 results in an optimized topology associated with a given element of the finite element model. According to an embodiment, the iterative optimization at step 312 (i.e., the iterative design process) is a process of selecting a new improved solution (with respect to the constraints and objectives defined by the design response) in each optimization iteration.

[0127] After iterative optimization in step 312, the properties of the given elements of the finite element model are automatically updated to reflect the optimized values ​​of the dimensional design variables (at step 314). In this way, method 665 creates an optimized model of a real-world object for manufacturing. Once the optimized design variables (i.e., the optimized values ​​of the design variables) are determined, the optimized finite element model can be converted into a format and submitted to the actual manufacturing process. In the actual manufacturing process, local control of the surface design pattern is crucial for fulfilling manufacturing constraints, and the obtained design is directly manufactured. In an embodiment, by controlling local volume constraints, design-specific constraints that conform to the real-world limitations of the manufacturing process can be set; therefore, the embodiment determines an optimized design for manufacturing a real-world object. In an embodiment of method 300, an optimized topology is determined for, for example, multiple elements of the finite element model. In this way, such an embodiment can determine an optimized design for manufacturing the entire real-world object.

[0128] Next, in the illustrated embodiment, method 300 includes: manufacturing a real-world object according to an optimized model (at 316). In one such embodiment, after updating the model at step 314, the updated model is digitally transferred to a manufacturing machine (e.g., an additive manufacturing machine, such as a 3D printer) capable of producing real-world objects according to the optimized model. In the embodiment, any communication method known in the art is used to transfer the digital model, and the digital model can be transferred to any manufacturing machine known in the art, such as an additive manufacturing machine, a stamping machine, or a computer numerical control (CNC) machine.

[0129] The following is a discussion of applying the techniques disclosed herein in the context of topology optimization, where singular value decomposition (SVD) can be used to evaluate each finite element surrounding (e.g., a sphere with a given radius, see [link to relevant documentation]). Figure 4A or Figure 18 The distribution of the design variables (relative density), and among which, the obtained singular values ​​(e.g., s) max s mid s min See Figure 4E These measurements relate to the three geometric dimensions of the object described by the density field of the sphere under consideration. In a typical implementation, these measurements can be used to construct a design response for structural optimization.

[0130] For a finite element model described in 3D space, a model containing nodes N can be used. ij and each node N l The corresponding coordinates (x) l y l , z l The matrix E(N) ij ) and N(x l y l , z l The finite element E of the structure is used to represent the structure. i .

[0131]

[0132] Where m is the number of elements, n is the number of nodes per element, o is the total number of nodes, and p is the number of dimensions.

[0133] In order to each element E i A given neighborhood of radius R (see surrounding area) Figure 4A or Figure 18 The evaluation is performed in ) using the following equation to determine neighboring elements and their centroids (see Figure 4C ):

[0134]

[0135] And the center of mass is:

[0136]

[0137] Additionally, for each neighbor, the relative element density is stored. (see Figure 4B ).

[0138] In a typical implementation, the centroid needs to be its average value. Perform SVD with the center as the reference point. This typically corresponds to an offset of the global coordinate system towards the center of the sphere.

[0139] Then it can be done through matrix X j The SVD method obtains the singular values ​​(s1, s2, s3) of each neighborhood.

[0140] in,

[0141] The graphical representation of this matrix is ​​in Figure 4D As shown in the diagram. Here, all the centroids with relatively low density are moved to the center of the sphere. As a byproduct, singular vectors can also be calculated (in...). Figure 4E (Displayed as a vector in the diagram), the singular vector represents the orientation of the principal dimensions of the object described by the density field of the design variable under consideration. The measured values ​​s1, s2, and s3 can also be derived from s... max s mid and s min This is because they are usually sorted by their size (s) max ≥s mid ≥s min ).

[0142] exist Figures 5 to 11 The diagram shows normalized values ​​s for different exemplary structural layouts for design variables in a finite element setting used for topology optimization. max s mid and s min .like Figure 5 As shown, for a real sphere, all normalized singular values ​​are equal to 1.0, while for a virtual sphere, all normalized singular values ​​are equal to 0.0. For all other configurations, the singular values ​​are between 0.0 and 1.0.

[0143] Figure 6 The diagram illustrates the transformation of an entity (represented as a 3D object) into a shell, plate, or membrane layout (represented as a 2D object) for the normalized singular values ​​of the relative density field, which serves as a design variable, for example, for topology optimization. As shown in Figure 6, one of the singular values ​​approximates zero when a 3D object is transformed into a 2D object.

[0144] Figure 7 The diagram illustrates how, for the normalized singular values ​​of the relative density field—used as a design variable, for example, topology optimization—shell, plate, or membrane layouts (represented as 2D objects) are transformed into beam, strip, or lattice layouts (represented as 1D objects). Figure 7 As shown, when a 2D object is converted to a 1D object, the two singular values ​​are approximately zero.

[0145] Figure 8 This illustrates how beam, strip, or lattice layouts (represented as 1D objects) are transformed into virtual layouts (represented as 0D objects) for normalized singular values ​​of the relative density field, which are used as design variables, for example, for topology optimization. Figure 8As shown, when a 1D object is converted into a virtual sphere (virtual part), all singular values ​​are approximately zero.

[0146] Figure 9 This illustrates how, for the normalized singular values ​​of the relative density field—used as a design variable, for example, for topology optimization—the entity (representation of a 3D object) is transformed into a virtual layout (representation of a 0D object). For example... Figure 9 As shown, when a 3D object is converted into a virtual sphere (virtual part), all singular values ​​are approximately zero.

[0147] Figure 10 The diagram shows that for a relative density field, which is used as a design variable, for example for topology optimization, all normalized singular values ​​are 1, representing an assembly of 1D or 0D components in 3D space.

[0148] Figure 11 The diagram shows an assembly of 1D or 0D components in 2D space, where the relative density field, used as a design variable for example for topology optimization, has two normalized singular values ​​of 1 and one normalized singular value approximately zero.

[0149] As mentioned above, the dimension of an object computed using singular values ​​describes how much space the object requires within the considered neighborhood (sphere). Therefore, the size of the sphere must be appropriately chosen based on the specific application. Generally, a smaller sphere (radius = 2-5 × finite element size) should be used to locally control the structural dimension. Typically, using a larger sphere for practical applications can lead to a misunderstanding of the structural dimension used for optimization. For example, as... Figure 10 and Figure 11 As shown, an assembly of 1D or 0D parts can be interpreted as a 3D object or a 2D object. Whether such a geometric interpretation is needed may depend on, for example, the specific design application.

[0150] therefore, Figures 5 to 11 This illustrates how singular values ​​can be applied to control and enforce the form and shape of a given type of design variable layout. In a typical implementation, system 100 is configured to apply singular values ​​in this way to control and enforce various different types of forms and shapes, for example, in topology optimization. This has never been applied to control the form and shape of design variables and layouts used for nonparametric optimization, and this is a unique aspect of this method.

[0151] The measured values ​​s1, s2, and s3 are discontinuous between optimization iterations and cannot be directly used in gradient-based optimization algorithms. In a typical implementation, this problem can be solved using smooth approximations, as follows:

[0152] s max ≈smooth_max(s j ), s min≈smooth_min(s j )

[0153]

[0154] Furthermore, in a typical implementation, X is performed by setting the design variable density of all elements to 1.0. j SVD, to obtain additional value This value represents the maximum possible dimension of the object under consideration and can be used to calculate the relative singular value.

[0155] Then the optimization constraints for each neighborhood (the spheres surrounding each finite element are equivalent to the design variable field) can be defined as:

[0156] Constraints: A, B, and C

[0157]

[0158] Quantity q A The minimum local dimension of the structure is restricted, while the quantity q B The second maximal local dimension is constrained. The quantity q C Establish the relationship between the maximum and second maximum dimensions. To reduce the number of constraints for optimization (to avoid constraints for each finite element being equivalent to design variables), a smooth maximum / minimum approximation aggregation function is used to aggregate the left-hand side of the above constraints (to obtain single constraints of types A, B, and C that cover all finite elements or subsets of finite elements and therefore all design variables or subsets of design variables). The right-hand side of the constraints is the constraint value. Therefore, the following constraints can be used locally to force the optimized design to have a certain dimension:

[0159] 1) 1D or 2D: The constraint condition of type A is q A << 1.0 (e.g., 0.1)

[0160] 2) 1D: The constraint conditions for types A and B are q A << 1.0 and q B <<1.0

[0161] 3) 2D: The constraint conditions for types A and C are q A << 1.0 (e.g., 0.1) and q C >>0.0 (e.g., 0.6)

[0162] In some implementations, the left-hand side of the above constraints can be implemented as a design response in an existing structural optimization solution, used to define the constraints or as an objective term. Typically, a smoothed derivative of the constraints with respect to the design variables (relative density) may be required during optimization, and this derivative can be obtained by differentiating the above equation.

[0163] Typically, the techniques disclosed herein employ sensitivity optimization using geometric dimension control for design variables. This can be accomplished as follows: looping through all design variables (DV), identifying other adjacent design variables for which geometric dimension control should be enforced for a given design variable (DV), calculating the singular value (SVD) and sensitivity (also known as derivative) for each given design variable (DV) (405), and calculating the design response (DRESP) and sensitivity (407) using one or more aggregation functions for the design variables (DV). An aggregation function is a function that combines multiple distinct values ​​together to form, for example, a single summary value.

[0164] Figure 15 An iterative design process 1550 according to an embodiment is described, which involves constraining design values ​​based on singular value decomposition (SVD) of design variables.

[0165] According to the illustrated embodiment, process 1550 considers the sensitivity of the design response (DRESP) for geometric dimension control and the calculated singular values ​​(SVD) for the design variables. The iterative design process scheme 1550 can be implemented, for example, within a predetermined workflow of a computer-aided engineering (CAE) system. Method 1550 begins with step 1501: creating an initial model that includes, for example, various loads and boundary conditions for optimization equilibrium. In this embodiment, the model is defined by the designer (e.g., the designer inputs data into a computer via one or more I / O devices connected to I / O device interface 110, for example...). Figure 1 The computer 100 is depicted. Then, the model is iteratively designed through steps 1503 to 1513.

[0166] According to the illustrated embodiment, the design iteration cycle may include a processor (e.g., Figure 1 CPU 102 in the memory (e.g., Figure 1 (104) Reads the data representing the geometric description of the model and solves for the equilibrium of the model (at 1503). The processor can then, for example, model the design response and equilibrium design response for the design variables and their sensitivities (at 1505).

[0167] According to the illustrated embodiment, each design iteration cycle may include the processor reading the geometric description of the model (e.g., a finite element model) (at 1507). The processor may then, for example, determine the sensitivity of the design response to the design variables and the calculated singular values ​​(SVD) for geometric dimension control (at 1509).

[0168] In a typical implementation, the design response defines the response of the current analysis model for a given optimization iteration. Therefore, in a typical implementation, the design response can extract a scalar value, which can be a direct measurement of the model (e.g., mass, center of gravity, etc.), or the scalar value can be determined from the results of the original solution for the model's equilibrium (e.g., stress, displacement, reaction forces, etc.).

[0169] The design response is then applied to define an optimization problem, which includes constraints that must be satisfied and an objective function to be optimized. At step 1511, mathematical programming is used to solve the optimization problem. In a typical implementation, the mathematical programming (i.e., the optimization computation) is strictly based on the user-defined design objective, the design response, and the values ​​of the sensitivity of the design response. Therefore, if the iterative design process includes design responses for geometric dimension control against design variables and calculated singular value (SVD) sensitivity, in some implementations, these can help enforce local control of design features. This also enhances the physical properties of other design responses applied in the optimization setup.

[0170] In mathematics, computer science, and operations research, mathematical programming, also known as mathematical optimization or simply optimization, is the process of selecting the best solution from a set of available alternatives (for some criteria). Embodiments of method 1550 can use any such mathematical programming known in the art. In a typical implementation, a processor can implement the functionality associated with mathematical programming by executing computer-readable mathematical programming instructions stored in memory.

[0171] Following the mathematical programming at step 1511, a new physical model for the next optimization iteration is generated at step 1513 based on the design variables determined at step 1511. The iterative design process is the process of selecting new, improved solutions (with respect to certain objectives and constraints). Sometimes, the design variables determined at step 1511 are the same as the physical model variables updated at step 1513. If the physical model variables and design domain variables are the same, the physical model is simply obtained at step 1513 as the output of the mathematical programming. Otherwise, as is known in the art, additional steps are required to interpret the design variables as physical model variables using filters.

[0172] To proceed, it is determined whether the optimization has converged. Convergence is numerically determined when the change in the objective function from one optimization iteration to a subsequent iteration is less than a sudden threshold, and the relative summation change of the design variables from one optimization iteration to a subsequent iteration is less than a sudden threshold. If the optimization has not converged, a new optimization cycle begins, and method 1550 returns to steps 1503 and 1507. If the optimization has converged, the final design is created at step 1515. For a converged design, the constraints for the design response should be satisfied, and the objective function should be optimized.

[0173] Method 1550 and its output final design 1515 can be used for a variety of real-world objects to determine an optimized design for manufacturing.

[0174] The practical application of these techniques can be understood in the context of introducing local constraints for gradient-based topology optimization to locally enforce a predetermined dimension of structural components. In this example, the constraints are based on principal values ​​(singular values), which are derived from the singular value decomposition (SVD) of a point cloud represented by the centroids of the elements that enforce dimension control in the design optimization of real-world objects and the corresponding non-parametric design variables (e.g., relative density).

[0175] For example, Figure 16A This is a schematic diagram showing a very coarse finite element (FE) model of a simple cube, clamped at the base corners of the cube, with a downward force (load) applied towards the center of the cube's top surface. Topology optimization that minimizes the compliance under volume constraints can produce... Figure 16B The design does not apply any additional constraints on the geometric dimension. Within a given radius of the search neighborhood (e.g., see...),... Figure 16B The object shown is a sphere (depicted in the design), where the structural components are quite bulky and thick. In some embodiments, a neighborhood with certain predetermined radii at certain locations can be considered, depending on design requirements (e.g., see...). Figures 16B to 16D Using a sphere as shown, we can locally study the geometric properties of the optimized structure. Thus, relative to the current radius, the considered structural component is quite bulky and thick in 3D space. By adding additional constraints based on SVD values ​​to the optimization problem, we can then locally enforce the 1D and 2D structures to obtain results as shown below. Figure 16C and Figure 16D The optimization results are shown. Similarly, in a typical implementation, the dimensionality is evaluated relative to a predetermined radius of the considered neighborhood. In this respect, Figure 16C This illustrates a 1D lattice structure with localized enforcement. Figure 16D A 2D membrane structure with localized forced execution is shown.

[0176] The computer-based methods disclosed in this paper can be applied to a variety of real-world design applications.

[0177] For example, in various implementations, computer-based methods can be used to help meet design requirements for various manufacturing processes, such as those involving: 1) enforcing 2D SVD constraints (e.g., see...). Figure 16D 1) Topology optimization for deep-drawn thin-walled structures; 2) Obtaining thin-walled welded structures by forcibly applying 2D SVD constraints; 3) Avoiding cooling control of large areas of metal porosity in cast components by forcibly applying 1D SVD and 2D SVD constraints; 4) Forcibly applying 1D SVD constraints (e.g., see...) Figure 16C ) and 2D SVD constraints (for example, see Figure 16D ), to obtain (e.g., for rib and lattice designs) the orientation of structural components, enforce orientation, or ensure molded form; 5) by enforcing 1D SVD constraints (e.g., see Figure 16C 6) Ensure the removability of powder used for 3D printing; 7) Ensure continuous 3D fiber printing by enforcing 1D SVD constraints (e.g., see...). Figure 17A and Figure 17B ), in which sub-components are attached to or sandwiched in branches without cross fibers.

[0178] Figure 17A and Figure 17B The exemplary diagram illustrates the enforcement of dimensionality control in the topology of a 1D structural sub-component, which represents fibers that do not intersect each other. Figure 17A In this study, the topology optimization results are represented using finite element methods with relative densities of solids and intermediates. Figure 17B In this paper, the topology optimization results are represented by isosurfaces describing the relative density field.

[0179] In some implementations, this method can help ensure that powder in objects to be manufactured using additive manufacturing techniques (such as 3D printing) can be easily removed.

[0180] Geometric Dimension Control in Topology Optimization

[0181] 1. Overview

[0182] This section discusses specific implementations that involve introducing additional constraints for gradient-based topology optimization to locally enforce a predetermined dimension for structural components. These constraints are based on principal values ​​(singular values) derived from the singular value decomposition (SVD) of the point cloud represented by the centroids of the elements, and the corresponding relative density design variables.

[0183] 2. Implementation methods and workflow of topology optimization

[0184] In an exemplary implementation, the geometric control techniques disclosed herein are implemented using mathematical programming in the optimization software SIMULIA Tosca Structures (e.g., by applying singular value decomposition (SVD) constraints to the design variables). These geometric control techniques are used to update the design variables and associated sensitivities implemented in SIMULIA Abaqus for structural finite element modeling. The direct solver in SIMULIA Abaqus is used to solve the equilibrium R=0 and associated solutions of the finite element model, except for the numerical results shown in Sections 5.2 and 5.3, where the iterative algebraic multigrid solver of SIMULIA Abaqus is applied in Sections 5.2 and 5.3. Note that both the direct finite element solver and the iterative algebraic multigrid solver can solve nonlinear structural models, such as contact, large deformation, and constitutive nonlinear material models. However, linear applications are shown here.

[0185] Furthermore, all current numerical topology optimization results model the constitutive material model using the so-called SIMP (Solid Isotropic Material Penalty) model, which is proportional to the power law of the relative element density ρ and to the vector of design variables that contain all structural design elements in the design domain.

[0186] Furthermore, the current numerical optimization results (Sections 5.1, 5.2, and 5.4) will have the following optimization framework, which minimizes the compliance C of the external load P and the resulting displacement U (maximizing stiffness):

[0187]

[0188] The total mass m(ρ) of the design domain is constrained to satisfy a certain weight objective, which is determined by the relative material fraction f and m. full Definition, m full This represents the quality of the design domain with all materials. A sensitivity filter is used for regularization, introducing length scaling and suppressing the chessboard. The radius of the sensitivity filter for all current optimization results is scaled 1:3 compared to the average element size of all elements specified in the design domain.

[0189] Secondly, the mass is the design response that is minimized as the objective function of strength optimization applied in Section 5.3. Furthermore, the constrained design response is the von Mitsch stress S at the element integration points. v (ρ, U(ρ)) is used as a single aggregation constraint A(S) v (ρ, U(ρ))), as shown below:

[0190]

[0191] Among them, s c This is the stress constraint value. The stress design response sv(ρ, U(ρ)) in Equation 2 is applied using relaxation. Furthermore, the aggregation function A(s) in the form of the p-norm method... v (ρ, U(ρ))) is applied to the stress points of the element to avoid stress integration points of each element having stress design responses.

[0192] 3. Use SVD to calculate the geometric properties of local structural components.

[0193] This method employs a new definition of measurements that serve as the design response for topology optimization. These measurements can be applied to the objective function or used as optimization constraints to locally enforce a preferred geometric layout of the optimized structure. Mathematically, these measurements are based on the singular values ​​(or corresponding vectors) of a matrix describing the distribution of design variables within a given neighborhood (e.g., a sphere with a given radius). In the context of topology optimization, singular value decomposition (SVD) is then used to evaluate each finite element (for a sphere with a given radius, see [link to finite element analysis]). Figure 4A The distribution of design variables (relative density) around the target. The obtained singular values ​​(s) max s mid and s min See Figure 4E This relates to the three geometric dimensions of the object, described by the density field of the sphere under consideration. These measurements can be used to construct a design response for structural optimization that enforces local geometric control.

[0194] 3.1 Singular values ​​of the relative density field

[0195] Next, assume that for a given finite element mesh with n finite elements, the corresponding centroid coordinates are:

[0196]

[0197] It is defined as

[0198]

[0199] And the relative density is:

[0200] Wherein, 0.0 < ρ is provided for each optimization iteration. i ≤1.0. To evaluate the geometric properties of structural components ( Figure 4A ), which is the relative density field in a given neighborhood with radius R around the finite element e∈{1,...,n} ( Figure 4B Let ) represent the m adjacent elements j∈{1,...,m} and their centroids. (Figure 4C) is determined by the following conditions:

[0201] ||C e -C j ||≤R. (4)

[0202] Quantities Ce and Cj correspond to the e-th and j-th rows of C. The corresponding relative densities are determined by... express.

[0203] Before applying SVD, stored The data in the table must be centered on its average value, and expressed as follows:

[0204]

[0205] This corresponds to shifting the global coordinate system to the centroid of the considered point for a given cloud. Now, the stored values ​​are scaled using relative density values. The centroid coordinates of the offset in the image are as follows:

[0206]

[0207] The centroid coordinates of elements representing solid materials will not change due to this operation. However, the corresponding centroids of elements representing intermediate materials will change along the cloud ( Figure 4D The centroid of the points considered in the equation shifts in the direction of movement. For each neighborhood singular value s... min ,s mid and s max , defined as s min ≤s mid ≤s max It is obtained through the singular value decomposition of the following matrix:

[0208]

[0209] Solid materials can be defined as:

[0210] Intermediate materials can be defined as:

[0211] The quantities in the equation represent the principal local dimensions of the object described by the relative density field under consideration; see [link to relevant documentation]. Figure 4E The corresponding singular vector v k The directions of the principal dimensions are represented and are orthogonal to each other. Matrix W contains singular vectors that have not been considered in the contributions of this paper. In the following subsections, it is necessary to use the matrix... Maximum Singularity singular value s k Normalization, as follows:

[0212]

[0213] Therefore, singular values ​​are invariant with respect to the absolute size of the neighborhood under consideration. It should be noted that the matrix... Corresponding matrix In the matrix All relative densities in the range are set as follows: This represents the largest possible instance of a local structural component for a design.

[0214] 3.2 Relationship between singular values ​​and eigenvalues

[0215] If a large number of algorithms for eigenvalue decomposition are available in a finite element software environment, then these algorithms can be used again to perform matrix operations. The SVD calculation is performed. Therefore, the following matrix is ​​introduced: and Equation (9).

[0216] The matrix decomposition in equation (9) is as follows:

[0217]

[0218] This decomposition directly provides the singular vector v obtained from the eigenvalues ​​of matrix Q. k And the corresponding singular values, as follows:

[0219]

[0220] 3.3 Geometric Interpretation of Singular Values

[0221] Figures 5 to 11 Normalized values ​​are shown Different exemplary structural layouts for use as design variables (relative density) for topology optimization in finite element settings. For example... Figure 5 As shown, for a completely real sphere, all normalized singular values ​​are equal to 1.0, while for a virtual sphere, all normalized singular values ​​are equal to 0.0. For all other configurations, the singular values ​​are between 0.0 and 1.0.

[0222] like Figure 6 As shown, when a fully solid 3D object is converted to a 2D object, one of the singular values ​​approaches zero. In this case, the 2D represents a shell, plate, or membrane-like material layout. Figure 7 As shown, when a 2D object is converted to a 1D object, the two singular values ​​are approximately zero. Beam, strip, or grid-like material layouts correspond to 1D objects.

[0223] like Figure 8 As shown, when a 1D object is converted into an empty sphere (imaginary part), all singular values ​​are approximately zero.

[0224] Therefore, as Figures 5 to 8The singular values ​​shown can be used to formulate new types of constraints to control and enforce the geometric characteristics of the optimized structure. These constraints will be discussed in the following subsections.

[0225] 4. Dimensions of locally constrained structural components

[0226] The singular values ​​derived in the previous sections are now used to construct novel design responses for topology optimization. These responses are then used to formulate constraints for local control of the structural dimension.

[0227] 4.1 Eliminating the discontinuity of singular values

[0228] For the singular values ​​s of the considered neighborhood k The singular values ​​are discontinuous because their order can switch between optimization iterations. Therefore, these singular values ​​cannot be directly used in gradient-based optimization algorithms. This problem is addressed by using smooth approximations of the maximum and minimum singular values ​​as follows:

[0229]

[0230] The quantity r is used here to overcome numerical problems with high exponent values, and is usually set as r = s. max Or r = s min Each of these values ​​approximates either the maximum or minimum value.

[0231] 4.2 Design Response Based on Singular Values

[0232] For a given neighborhood of a finite element i∈{1,...,n}, consider the following three measurements:

[0233]

[0234] The quantities on the left-hand side of the first two equations in (13) locally describe the minimum and second minimum dimensions of the structuring element, respectively. These quantities are normalized by the maximum possible dimension of the considered neighborhood, thus making them invariant relative to the absolute neighborhood size. The third quantity on the left-hand side of the bottom equation represents the relationship between the second maximum dimension and the maximum dimension. It should be noted that this third quantity is the definition of each relative quantity and does not directly constrain the denominators in the first two equations in (13). Assuming the design domain includes all finite elements, the quantities on the left-hand side of the first two equations in (13) are aggregated over all finite elements using the smoothed maximum approximation, since these quantities will be used to formulate the less-than-equal-to-the-other constraint. The quantities represented on the left-hand side of the third equation in (13) are aggregated using the smoothed minimum approximation, since these quantities will be used to formulate the greater-than-equal-to-the-other constraint. Making q∈{min,mid,mid2max}, the following three design responses are then introduced:

[0235]

[0236] wherein, for the following equations: n = the total number of elements; p = 6 for q = min or q = mid; and p = -6 for q = min2max.

[0237] It should be noted that smooth maximum approximation and smooth minimum approximation are used herein to obtain three scalar design responses for the entire design domain. These design responses are used in the following subsections to formulate constraints for optimization.

[0238] 4.3 Constraints for gradient-based optimization

[0239] In order to locally control the dimensions of structural elements, the following constraints may be formulated.

[0240] 1D or 2D structural elements are locally enforced by constraining the minimum dimension. Therefore, completely solid 3D material objects are eliminated for a given neighborhood. This is obtained by applying the following constraint:

[0241]

[0242] wherein 0.0 < sˉ*min < 1.0. This constraint causes the design to be membrane-like and lattice-like structures and avoids bulky material concentration. The value implicitly specifies the maximum thickness of the lattice and membrane members relative to the diameter of the considered neighborhood.

[0243] 1D structural elements are locally enforced by constraining the minimum dimension and the second minimum dimension. Therefore, 2D and 3D objects are eliminated for a given neighborhood. This is obtained by applying the following constraints: and is obtained, wherein and

[0244] These constraints result in the design becoming a lattice structure and avoid bulky material concentration and membrane structures. The value implicitly specifies the maximum thickness of lattice members relative to the diameter of the considered neighborhood and controls the curvature. If the selected value is too small, the optimizer will not be able to generate joints between 1D components. Sometimes such behavior is intended to avoid fiber intersection in composite optimization, for example. An increase in the value allows an increase in the curvature of structural components.

[0245] 2D structural elements are locally enforced by constraining the minimum dimension and the relationship between the second maximum dimension and the maximum dimension. Accordingly, 1D and 3D objects are eliminated. This is obtained by applying the following constraints: and is obtained, wherein and

[0246] These constraints led to a membrane structure design, avoiding bulky material concentrations and lattice structures. Again, the value... The maximum thickness of the membrane module is implicitly specified, and the value is... Controls the dimensional proportions perpendicular to the thickness direction. Value It also implicitly controls the curvature of the membrane module. Smaller values ​​result in more planar structures, while larger values ​​result in curved structures.

[0247] 4.4 Relationship between relative material fraction and geometric control

[0248] Figures 4A to 11 The relative material fraction f of a sphere with radius R is shown. s This is partly related to the geometric control imposed by constraining the design variables within a sphere using singular value decomposition (SVD). The following criteria for lattice and membrane structures are applied to current numerical applications in 3D:

[0249] grid- Figure 7 and Figure 8 It shows how singular values ​​s mid and s min Certain typical lattice structures are obtained by constraining values ​​less than the chosen values. Assume... And the volume of the grid component (V) lattice The volume V of a sphere sphere =4 / 3πR 3 If it is much smaller, then the volume of the grid is:

[0250] Therefore, the relative material fraction in the sphere is:

[0251]

[0252] membrane- Figure 6 and Figure 7 It shows the ratio of singular values. Some typical membrane structures are obtained with constraints higher than a sudden value (e.g., 0.75). Assume the membrane volume V... membrane Much smaller than the volume V of a sphere sphere Then the volume of the membrane is determined by V. membrane =πR 2 t is given, where t is the expected thickness of the membrane structure. Therefore, the relative material fraction in the sphere is:

[0253] For the design domain, deep drawing with a low volume fraction is typically used to fabricate practical membrane structures, such as f s =5%, producing R=15t.

[0254] Therefore, if two elements are specified across the entire thickness for the optimized finite element model, the radius comprises 30 elements, resulting in a considerably large radius compared to the element dimensions for applying geometric control over the membrane layout. It should be noted that, as described in Section 4.3, the thickness t can be enforced by the following constraint: produce And therefore

[0255] 4.5 Sensitivity Analysis of SVD Design Response

[0256] Determine the design response introduced in equation (14) Relative to relative density ρ e The derivative is used for gradient-based optimization. The derivation of the corresponding sensitivity relationship is based on multiple applications of the chain rule to calculate the derivative. Therefore, purely geometric quantities are considered, and quantities for finite element analysis are not involved. In this section, the required partial derivatives are derived.

[0257] The derivative of the aggregate design response with respect to the dimension measure defined in equations (13) and (14) is derived as follows:

[0258]

[0259] The dimension measure relative to the smooth singular value s defined in equation (12) min ,s mid and s max The derivative is obtained as follows:

[0260]

[0261] Smoothed singular values ​​relative to the singular values ​​s defined in equation (11) min ,s mid and s max The derivative is obtained as follows:

[0262]

[0263] Singular values ​​relative to relative density The derivative is obtained as follows:

[0264]

[0265] The quantity ρ can be described using Boolean operators. e and The relationship between them is straightforward, and will not be elaborated upon here.

[0266] 5. Example

[0267] 5.1 2D Cantilever Beam

[0268] The following simple 2D example illustrates the features of the proposed method compared to classical topology optimization. Figure 19A A rectangular 2D design domain is shown, discretized using fully integrated four-node plane stress elements of dimension 1 × 1 in SIMULIA Abaqus (CP4S) with a uniform 400 × 200 grid. An elastic material with a Poisson's ratio of 0.3 is applied. The left edge of the design domain is fully clamped, and an external force is applied at the midpoint of the right edge. Figure 19B The results of classical topology optimization using a material volume fraction of f = 60% are shown. Figure 19C The proposed method for constraining minimum singular values, as defined in equation (17), is shown using a diameter D = 12, equivalent to 12 finite elements. The result. This additional constraint is added... Figure 19B In classic topology optimization applications, for... Figure 19C The optimization results shown, the optimization iteration history are in Figures 20A to 20C As shown in the figure, it can be observed that the number of optimization iterations is 73 when additional constraints are applied to the minimum singular value. Figure 19B The optimization iterations for the design in the example are 55. Therefore, for the minimum singular value... Additional constraints have almost no impact on the optimization convergence iteration, which can also be observed from other numerical experiments.

[0269] Figures 21A to 21F The optimization results are shown when the diameters of the design variables included in the SVD decomposition are D=12 and D=24, respectively, and the minimum singular value constraints are 0.60, 0.50, and 0.40. Figures 21A to 21F In this study, a diameter D = 12, equivalent to 12 finite element methods, was used to reduce the minimum singular value from 0.6 to 0.4 to constrain the geometric dimension (ac). This reduced the component size, thereby increasing the compliance C and decreasing the relative material volume f. The structure (df) was optimized using the same minimum singular value constraints as the (ac) structure, but a larger diameter was used to generate the optimization results when calculating the minimum singular value decomposition (SVD) of the design variables. Comparing the (ac) and (de) structures illustrates that increasing the diameter to calculate the minimum singular value results in larger component sizes and greater distances between them, leading to higher stiffness but lower mass. Figure 19A As shown, the applied model mesh, loads, and boundary conditions are defined. It should be noted that symmetry is not enforced, but the solution is still significantly symmetric. Also note that... Figures 21A to 21FThe optimization in this study did not apply volume constraints. It was observed that the maximum component size obtained was given by multiplying the diameter of the included design variables for the decomposition by the constraint value for the minimum singularity. However, small components appeared, and the length of the sensitivity filter gave the minimum component size.

[0270] This method appears to have advantages over local volume constraint methods. Lattice members obtained using methods based solely on local volume constraints typically decrease in size as they approach the intersections and knots that make up the lattice. However, the results of the method presented in this paper do not appear to have this drawback.

[0271] like Figures 22A to 22D As shown, by using a diameter D = 12 to constrain the minimum singular value to less than or equal to 0.6, all designs are optimized. Figure 19A The applied model mesh, loads, and boundary conditions are shown. Figure 22A In the design, no volume constraints were applied, but... Figures 22B to 22D In this design, the volume constraint was reduced from 60% to 40%. Figures 23A to 23D In this model, by using a diameter D = 24 to constrain the minimum singular value to less than or equal to 0.6, all designs were optimized. Figure 19A shows the applied model mesh, loads, and boundary conditions. Figure 23A In the design, no volume constraints were applied, but... Figures 23B to 23D In this design, the volume constraint was reduced from 60% to 40%.

[0272] Figures 22B to 22D and Figures 23B to 23D They respectively showed the same as Figure 22A and Figure 23A Compared to the design in the previous example, adding global volume constraints to the optimization formula has a significant effect. Generally, it is observed that classical volume constraints for topology optimization work well when simultaneously constraining minimum singular values. Figure 22A and Figure 23A The material volume of the optimized design shown, without material volume constraints, is just slightly above the constraint value for the minimum singularity. However, these designs exhibit some intermediate densities for the low strain energy density region of the design domain. Figure 22B and Figure 23B The diagram shows how these intermediate densities are removed by adding volume constraints similar to those for the minimum singularity.

[0273] Figures 22A to 22D and Figures 23A to 23DThe results of the proposed method defined in Equation (17) are shown, which constrains the minimum singular value. The resulting design has a satisfactory real / imaginary representation with few optimization iterations, and the objective function value of the most important compliance is almost identical to the objective value under no volume constraint. Therefore, Figures 22A to 22D and Figures 23A to 23D The results and additional numerical experiments show that, in order to obtain real / virtual 2D optimization solutions, volume constraints that are the same as or lower than the constraints of the minimum singular value should be applied.

[0274] 5.2 Bone Filler Design

[0275] This section discusses the possibility of generating lattice-filled and membrane-filled structures for the femur, respectively. For example... Figure 24 As shown, the femur finite element model comprises 1,090,793 hexahedral elements (C3D8), each with a size of 1×1×1, and is solved using an iterative algebraic multigrid solver, producing 3,420,603 DOFs. An elastic material with a Poisson's ratio of 0.3 is applied. The 3D femur model is initially shown as being completely clamped at the bottom, with two forces applied at the top. In this example, the elements on the outer surface are not part of the design domain.

[0276] Figures 25A to 25E The diagram shows a topology-optimized infill using femur material with a relative material mass fraction of 0.50. The outer surface is not part of the design domain. Figure 25A This represents classical stiffness optimization under mass constraints. Figure 25B and Figure 25C This indicates that the values ​​are relative to a radius of 2.5 (in...) Figure 25B (in the middle) and radius 4.0 (in Figure 25C (in Chinese), through geometric constraints Topology optimization of membrane filler obtained with a value less than 0.5.

[0277] Figure 25D and Figure 25E This indicates that the values ​​are relative to a radius of 2.5 (in...) Figure 25D (in the middle) and radius 4.0 (in Figure 25E (in Chinese) through geometric constraints Less than 0.5 and geometric constraints Topology optimization to obtain lattice-filled structures with a value less than 0.6.

[0278] Figure 25A The standard topology optimization results are shown, which maximize the stiffness under a relative volume constraint of 0.50, but without any additional constraints. The same relative volume constraint value was applied to the following numerical experiments, but respectively... and Add additional constraints.

[0279] As described in Section 4.4, using the lattice approximation with f = 0.50, these constrained singular values ​​are estimated to be approximately 0.58.

[0280] for Figure 25B In the case of R=2.5 and Figure 25C The optimization results shown in R=4.0 show that only the first singular value is constrained to 0.50. Figure 25B and Figure 25C The results shown mainly include those with thickness The membrane assembly is used as the desired filler for the femur because it is stiffer than the lattice structure. Additionally, in Figure 25B and Figure 25C The membrane assembly obtained is quite similar to some previous results presented using local volume constraints, and can also produce a membrane-like structure for filling the femur. Therefore, the current membrane design obtained using a single constraint for the first singularity, as well as the previously disclosed designs using local volume constraints, cannot achieve the diffusion of blood through the nutrient transport structure, and cannot be used in many powder-based additive manufacturing processes because the powder would be trapped inside the design after printing.

[0281] Subsequently, the first singular value was constrained to 0.50 and the second singular value was constrained to 0.60, respectively, to enforce the following: Figure 25D and Figure 25E The design shown features a grid-filled component. (Compared to...) Figure 25B and Figure 25C Compared to the membrane filling results in the middle, Figure 25D and Figure 25E The optimized results in this study feature unique lattice components as fillers. However, at radii of R = 2.5 and R = 4.0, the stiffness of the membrane-filled structure is approximately 20% higher than that of the lattice-filled structure. Therefore, the geometric dimension control presented in this paper allows for membrane designs similar to those obtained using local volume constraints, and importantly, also allows for designs with true lattice structures featuring open-cell structures. Consequently, the geometric dimension control method presented in this paper realizes open-cell lattice structures that can be used in additive manufacturing (AM), where powder is removed after manufacturing, allowing another medium to flow through the optimized structure.

[0282] Several embodiments of the present invention have been described. However, it should be understood that various modifications can be made without departing from the spirit and scope of the invention.

[0283] For example, some of the implementations disclosed herein can be designed to solve sensitivity-based solutions for structural optimization disciplines such as topology optimization, shape optimization, size optimization, and bead optimization. However, these techniques are not theoretically limited to structural optimization disciplines, but are also effectively used for, for example, multi-physical optimization, such as computational fluid dynamics (CFD), thermomechanical, electromechanical, and fluid structures, where the values ​​of singular value decompositions (SVD) from design variables (DV) can be determined and used to define additional constraints to enforce geometric features.

[0284] 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 be implemented by physical, virtual, or hybrid general-purpose computers (e.g., computer systems or the computer network environment described herein). For example, a computer system can be converted into a machine that performs the methods described herein by loading software instructions into memory or non-volatile memory for execution by a CPU. Those skilled in the art should further understand that the system and its various components can be configured to perform any embodiment or combination of embodiments of the invention described herein. Furthermore, the system can utilize any combination of hardware, software, and firmware modules operatively coupled to or incorporated into the system, whether internally or externally. Additionally, the system can be communicatively coupled to or embedded in a manufacturing apparatus and can be configured to control the apparatus to create physical objects as described herein.

[0285] Various aspects of the subject matter disclosed herein can be implemented in digital electronic circuits or computer-based software, firmware, or hardware, including the structures disclosed herein and / or their structural equivalents and / or combinations thereof. In some embodiments, the technical solutions disclosed herein can be implemented as one or more computer programs, i.e., one or more computer program instruction modules encoded on a computer storage medium for execution by or control of one or more data processing devices (e.g., processors). Optionally or additionally, program instructions can be encoded on artificially generated propagation signals, such as machine-generated electrical, optical, or electromagnetic signals, which are generated to encode information and transmit it to a suitable receiver device for execution by the data processing device. The computer storage medium can be a computer-readable storage device, a computer-readable storage substrate, a random or serial access memory array or device, or a combination thereof, or can be contained in a computer-readable storage device, a computer-readable storage substrate, a random or serial access memory array or device, or a combination thereof. Although the computer storage medium should not be considered merely as a propagation signal, it can be a source or destination of computer program instructions encoded as artificially generated propagation signals. Computer storage media can also be one or more separate physical components or media, or can be contained in one or more separate physical components or media such as multiple CDs, computer disks and / or other storage devices.

[0286] Some operations described in this specification (e.g., Figure 3A , Figure 3B , Figure 15 The operations described herein, and others disclosed herein, can be implemented by a data processing apparatus (e.g., a processor / specially programmed processor) on data stored in one or more computer-readable storage devices or from other sources (e.g., Figure 1 and Figure 2 The data received by the computer system and / or network environment (in the context of the computer system and / or network environment) is the operation performed. The received information. The term "processor" (or a similar term) encompasses all kinds of devices, apparatuses, and machines used to process data, including programmable processors, computers, one or more systems-on-a-chip, or combinations thereof. Devices may include special-purpose logic circuitry, such as FPGAs (Field-Programmable Gate Arrays) or ASICs (Application-Specific Integrated Circuits). In addition to hardware, devices may also include code that creates an execution environment for the computer program in question, such as code constituting processor firmware, protocol stacks, database management systems, operating systems, cross-platform runtime environments, virtual machines, or combinations thereof. Devices and execution environments can implement a variety of different computing model infrastructures, such as network services, distributed computing, and grid computing infrastructures.

[0287] Although this specification contains numerous specific implementation details, these details should not be construed as limiting the scope of any invention or claimable matter, but rather as descriptions of features characteristic of particular embodiments of a particular invention. In the context of a single embodiment, certain features described herein may also be implemented in combination in a single embodiment. Conversely, various features described in the context of a single embodiment may also be implemented separately in multiple embodiments or in any suitable sub-combination. Moreover, although the features described above may function in certain combinations and even as originally stated, in some cases one or more features of a claimed combination may be removed from the combination, and the claimed combination may be for sub-combinations or variations thereof.

[0288] Similarly, although operations may be described herein as occurring in a specific order or manner, this should not be construed as requiring such operations to be performed in the specific order shown or in a sequential order, or as requiring all shown operations to achieve the desired result. In some cases, multitasking and parallel processing may be advantageous. Furthermore, the separation of the various system components in the above embodiments should not be construed as requiring such separation in all embodiments, and it should be understood that the described program components and systems can generally be integrated into a single software product or packaged into multiple software products.

[0289] Other embodiments are within the scope of the claims.

Claims

1. A computer-implemented method, the computer-implemented method automatically determining an optimal design for manufacturing real-world objects, the method comprising: Define a finite element model representing a real-world object in the memory of a computer's processor; the finite element model includes multiple elements. Using the computer-based processor, the distribution of design variables in the neighborhood of the finite element model is evaluated using singular value decomposition (SVD) to generate singular values ​​of the design variables for each corresponding element in the neighborhood of the finite element model, wherein the design variables represent relative densities. Based on the singular values ​​generated from the SVD, optimization constraints are defined for the neighborhood of the finite element model; and Based on the defined optimization constraints, the design variables of the finite element model are optimized by locally enforcing the geometry of the real-world objects within the neighborhood.

2. The computer-implemented method according to claim 1, the method further comprising: The properties of a given element in the finite element model are updated to reflect the optimized design variables, thereby creating an optimized model for manufacturing the real-world object.

3. The computer-implemented method according to claim 2, the method further comprising: The computer-based processor determines whether the optimization has converged. as well as Based on the results determined by the computer-based processor: Begin a new optimization cycle; or Create the final optimized design. The computer-based processor determines that the optimization has converged if the constraints for the associated design response have been met and if the associated objective function has been optimized.

4. The computer-implemented method according to claim 3, the method further comprising: This enables the manufacture of the real-world object based on the final optimized design.

5. The computer-implemented method according to claim 1, wherein, Design response represents flexibility, stiffness, stress, strain, modal eigenfrequency and / or other typical structural design responses.

6. The computer-implemented method according to claim 1, wherein, The optimization constraints are defined as follows: The computer-based processor applies a smooth approximation function to each singular value among the singular values ​​generated from the SVD to generate a smooth approximation. Using the computer-based processor, additional values ​​are identified by performing SVD, wherein the values ​​of the design variables for all elements are set to predetermined values, the additional values ​​representing the maximum possible dimensions of the real-world object; and The optimization constraints are calculated using the smoothed approximation and the additional value.

7. The computer-implemented method according to claim 1, wherein, Evaluating the distribution of design variables within the neighborhood of the finite element model using Singular Value Decomposition (SVD) includes: using the computer-based processor to: Data is read from the memory of the computer-based processor, the data representing a geometric description of the finite element model from memory; The equilibrium equations of the finite element model are solved based on the data. Based on the data, the design response and equilibrium design response for the design variables, as well as the associated sensitivity, are modeled; and Determine the design response for the design variables and the sensitivity of the calculated singular values ​​for geometric dimension control.

8. The computer-implemented method according to claim 7, wherein, The optimization constraints are defined as follows: The optimization problem is defined by applying the design response to the computer-based processor, and the optimization problem includes constraints to be satisfied and an objective function to be optimized.

9. The computer-implemented method according to claim 8, wherein, The definition of optimization constraints further includes: The optimization problem is solved using the computer-based processor and user-defined values ​​of the design objective, design response, and sensitivity of the design response.

10. A computer-based system for automatically determining an optimal design for manufacturing real-world objects, the computer-based system comprising: Computer-based processors; as well as Computer-based memory, which is coupled to the computer-based processor. The computer-based memory storage contains data defining finite element models representing real-world objects. These finite element models include multiple elements, and The computer-based memory stores computer-readable instructions, which, when executed by the computer-based processor, cause the computer-based processor to: Singular value decomposition (SVD) is used to evaluate the distribution of design variables in the neighborhood of the finite element model to generate singular values ​​of the design variables for each corresponding element in the neighborhood of the finite element model, where the design variables represent relative densities. Based on the singular values ​​generated from the SVD, optimization constraints are defined for the neighborhood of the finite element model; and Based on the defined optimization constraints, the design variables of the finite element model are optimized by locally enforcing the geometry of the real-world objects within the neighborhood.

11. The computer-based system according to claim 10, wherein, The computer-based memory further stores computer-readable instructions that, when executed by the computer-based processor, cause the computer-based processor to: The properties of a given element in the finite element model are updated to reflect the optimized design variables, thereby creating an optimized model for manufacturing the real-world object.

12. The computer-based system according to claim 11, wherein, The computer-based memory further stores computer-readable instructions that, when executed by the computer-based processor, cause the computer-based processor to: Determine whether the optimization has converged; and Based on the determined results: Start a new optimization cycle; or Create the final optimized design. The computer-based processor determines that the optimization has converged if the constraints for the associated design response have been met and if the associated objective function has been optimized.

13. The computer-based system according to claim 12, wherein, The computer-based memory further stores computer-readable instructions that, when executed by the computer-based processor, cause the computer-based processor to: This enables the manufacture of the real-world object based on the final optimized design.

14. The computer-based system according to claim 10, wherein, Design response represents flexibility, stiffness, stress, strain, modal eigenfrequency and / or other typical structural design responses.

15. The computer-based system according to claim 10, wherein, The computer-readable instructions that cause the computer-based processor to define optimization constraints further include instructions that cause the computer-based processor to: The smoothing approximation function is applied to each of the singular values ​​generated from the SVD to generate a smooth approximation. The computer-based processor identifies additional values ​​by performing SVD, where the values ​​of the design variables for all elements are set to predetermined values, and the additional values ​​represent the maximum possible dimensions of the real-world object. as well as The optimization constraints are calculated using the smoothed approximation and the additional value.

16. The computer-based system according to claim 10, wherein, The computer-readable instructions that enable the computer-based processor to evaluate the distribution of design variables in the neighborhood of the finite element model using singular value decomposition (SVD) further include instructions that cause the computer-based processor to: Data is read from the memory of the computer-based processor, the data representing a geometric description of the finite element model from memory; The equilibrium equations of the finite element model are solved based on the data. Based on the data, the design response and equilibrium design response for the design variables, as well as the associated sensitivity, are modeled. as well as Determine the design response for the design variables and the sensitivity of the calculated singular values ​​for geometric dimension control.

17. The computer-based system according to claim 16, wherein, The computer-readable instructions that cause the computer-based processor to define optimization constraints further include instructions that cause the computer-based processor to: The design response is used to define an optimization problem, which includes constraints to be satisfied and an objective function to be optimized.

18. The computer-based system according to claim 17, wherein, The computer-readable instructions that cause the computer-based processor to define optimization constraints further include instructions that cause the computer-based processor to: The optimization problem is solved using optimization calculations based on user-defined values ​​of the design objective, the design response, and the sensitivity of the design response.

19. A non-transitory computer-readable medium storing computer-readable instructions that, when executed by a computer-based processor, cause the computer-based processor to: Define a finite element model representing a real-world object in the memory of a computer's processor; the finite element model includes multiple elements. Singular value decomposition (SVD) is used to evaluate the distribution of design variables within the neighborhood of the finite element model to generate singular values ​​of the design variables for each corresponding element within the neighborhood of the finite element model, wherein, The design variables represent relative density; Based on the singular values ​​generated from the SVD, optimization constraints are defined for the neighborhood of the finite element model; as well as Based on the defined optimization constraints, the design variables of the finite element model are optimized by locally enforcing the geometry of the real-world objects within the neighborhood.

20. The non-transitory computer-readable medium according to claim 19, wherein, The computer-readable medium further includes computer-readable instructions that, when executed by the computer-based processor, cause the computer-based processor to: The properties of a given element in the finite element model are updated to reflect the optimized design variables, thereby creating an optimized model for manufacturing the real-world object.

Images