Geometric Dimension Control in Optimization

Abstract

To automatically determine an optimized design for manufacturing real-world objects.SOLUTION: A computer implementation method is to define a finite element model representing real-world objects within a memory for a computer-based processor, and comprises the steps of: defining that finite elements include a plurality of elements; utilizing singular value decomposition (SVD) at the computer-based processor for evaluating distribution of design variables across a whole neighborhood of the finite element model, and generating singular values for the design variables in each element present in the neighborhood of the finite element model; defining optimal constraint against the neighborhood of the finite element model based on the singular values generated from the SVD; and optimizing the finite element model pertaining to the design variables by locally forcing geometry for the real-world objects present in the neighborhood, based on the defined optimized constraint.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS This application claims the benefit of priority to U.S. Provisional Application No. 62 / 945,961, entitled Geometrical Dimensionality Control in Optimization, filed December 10, 2019. The disclosure of the prior application is incorporated herein by reference in its entirety.

[0002] The present disclosure relates to the field of geometric dimension control, and more particularly, to systems and methods for geometric dimension control in optimization. [Background technology]

[0003] Several computer systems and programs are available that provide virtual environments for the design of real-world parts or assemblies of parts (e.g., objects). These systems, some of which are called computer-aided design (CAD) systems, allow users to build, view, and manipulate complex three-dimensional virtual models of the physical, real-world object(s) being designed. CAD systems are typically configured to provide a visual representation of such modeled objects on a computer-based visual display device using, for example, edges, lines, faces, etc.

[0004] Modeled objects are typically represented in CAD systems in CAD files that contain computer-readable specifications of physical, real-world geometry associated with one or more underlying objects. A particular object may be represented by computer-readable code stored in either a single CAD file or multiple CAD files. In one example, a CAD file may contain computer-readable code that represents specifications (e.g., characteristics) associated with a real-world version of a real-world physical object, from which associated geometries of the real-world physical object may be identified (e.g., by the CAD system). From these geometries, the CAD system may generate one or more visual representations of the associated objects and display the objects on a computer display device. CAD systems include graphic tools for representing modeled objects to designers (e.g., on a computer display device), and these tools may be configured to display objects that may be highly complex.

[0005] Design optimization is an engineering design technique that uses a mathematical formulation of the design problem to support the selection of the optimal design among many alternatives. For example, topology optimization (TO) refers to the technical field associated with optimizing the material layout within a given design space for a given set of loads, boundary conditions, and constraints with the goal of maximizing the performance of the system.

[0006] Improvements are needed in the field of design optimization in general, and topology optimization in particular. Summary of the Invention

[0007] In one aspect, a computer-implemented method for automatically determining an optimized design for manufacturing a real-world object includes defining, in a memory of a computer-based processor, a finite element model representing the real-world object, the finite elements including a plurality of elements; evaluating, in the computer-based processor, a distribution of design variables throughout a neighborhood of the finite element model using singular value decomposition (SVD) to generate singular values ​​of the design variables at each respective element in 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 finite element model with respect to the design variables by locally enforcing a geometry of the real-world object in the neighborhood based on the defined optimization constraints.

[0008] In another aspect, a computer-based system for automatically determining an optimized design for manufacturing a real-world object is disclosed. The computer-based system includes a computer-based processor and a computer-based memory coupled to the computer-based processor. The computer-based memory stores data defining a finite element model representing the real-world object, the finite element model having a plurality of elements. The computer-based memory also stores computer-readable instructions that, when executed by the computer-based processor, cause the computer-based processor to: evaluate a distribution of design variables throughout a neighborhood of the finite element model using singular value decomposition (SVD) to generate singular values ​​of the design variables at each respective element in 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 finite element model with respect to the design variables by locally enforcing the geometry of the real-world object in the neighborhood based on the defined optimization constraints.

[0009] In yet another aspect, a non-transitory computer readable medium having computer readable instructions stored thereon that, when executed by a computer-based processor, cause the computer-based processor to: evaluate distributions of design variables over a neighborhood of a finite element model using singular value decomposition (SVD) to generate singular values ​​of the design variables at each respective element in 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 finite element model with respect to the design variables by locally enforcing the geometry of real-world objects in the neighborhood based on the defined optimization constraints.

[0010] Yet another aspect is directed to a computer-based process including identifying non-parametric design variables in a finite element model, evaluating the distribution of design variables around a given finite element design variable using SVD, obtaining singular values ​​for each design variable, calculating optimized design response(s) from the singular values ​​for each design variable element, aggregating each type of optimized design response for the design variable elements into a single constraint design response value for the entire group of design variables, and implementing the aggregated design response values ​​as constraints in a structural optimization solution.

[0011] In some implementations, one or more of the following advantages are present.

[0012] For example, various implementations provide highly effective and optimized designs in a highly efficient manner. Also, this approach can be implemented in addition to imposing other design requirements, e.g. stiffness and strength. In addition to this, this approach potentially provides one implementation for multiple tasks (2D constraints, 1D constraints, local material amount limitations, curvature control, thin / thick control, ratio control, orientation control). This approach is generally applicable to any finite element mesh (structured / unstructured, any type of element, 1D-3D, etc.). This approach enjoys an intuitive setup and the interpretation and relationships of all applied scales are relatively straightforward.

[0013] Also, many manufacturing processes for 3D printing (also known as additive manufacturing) do not allow for powder capture during the printing process. Therefore, hollow enclosed structural parts are not allowed as feasible manufacturable design solution parts. In some implementations, hollow enclosed structural parts are not allowed as feasible manufacturable design solution parts due to certain singular values ​​(e.g., s min and s mid ) to be very small (e.g., close to zero), thereby forcing the structural layout of the parts to be a lattice.

[0014] In addition, many metal parts for automotive structures are sheet-form structures, because the manufacturing process of sheet-form structures is very cost-competitive. However, designs optimized using TO often have a large volume of material layout that is more suitable for casting. Therefore, the optimized sheet-form designs are often optimized for certain singular values ​​(e.g., s max ands mid This can be achieved by forcing the layout of parts to have a membrane layout locally via geometric dimensional control that constrains {overscore (r)}(the relationship between

[0015] Other features and advantages will become apparent from the description and drawings, and from the claims. [Brief description of the drawings]

[0016] [Figure 1] FIG. 1 is a simplified block diagram of a computer system for implementing an embodiment of the design process disclosed herein. [Diagram 2] FIG. 1 is a simplified block diagram of a computer network environment in which an embodiment of the design process disclosed herein may be implemented. [Figure 3A-3B] 1 depicts a flowchart illustrating an example implementation of a design optimization process that involves constraining design values ​​based on a singular value decomposition (SVD) of the design variables. [Figure 4A] 1 shows a structuring element in the form of a sphere with a given radius. [Figure 4B] 4B shows the relative density field within the structural element of FIG. 4A. [Figure 4C] 4C shows the element centroids of the relative density field of FIG. 4B. [Figure 4D] 4C scaled by relative density. [Figure 4E] 4D shows the singular value decomposition (SVD) values ​​(smax, smid, smin) associated with the three dimensions of the object and the corresponding directions / vectors of the point cloud. [Diagram 5] For example, we show a contrast between solids (3D object representation), voids (0D object representation), and intermediate bodies (between 3D object representation and 0D object representation) for the normalized singular values ​​of the relative density field as a design variable for topology optimization. [Figure 6] We show the conversion from solids (3D object representation) to shell, plate or membrane layouts (2D object representation) for example for normalized singular values ​​of relative density fields as design variables for topology optimization. [Figure 7]We show the transformation from a shell, plate or membrane layout (2D object representation) to a beam, bar or lattice layout (1D object representation), e.g., for the normalized singular values ​​of the relative density field as design variables for topology optimization. [Figure 8] We show the transformation from a beam, bar or lattice layout (1D object representation) to a near-void layout (0D object representation) for normalized singular values ​​of the relative density field as design variables for e.g. topology optimization. [Figure 9] We show the transformation from solids (3D object representation) to void layouts (0D object representation) for normalized singular values ​​of the relative density field as design variables for example for topology optimization. [Figure 10] We show the all-ones normalized singular values ​​for a relative density field as a design variable for topology optimization representing, for example, an assembly of 1D or 0D parts in 3D space. [Figure 11] We show that for a relative density field as a design variable for topology optimization representing an assembly of 1D parts or 0D parts in 2D space, it has two normalized singular values ​​that are unity and one that is close to zero. [Figure 12A] Figure 1 shows the design space for topology optimization subdivided into finite elements where the relative density of each element is the design variable. The beam is fully clamped on the left side and on the right bottom end under loading. [Figure 12B] FIG. 12B shows topology optimization results based on the design space of FIG. 12A without geometric dimension control. [Figure 12C] FIG. 12C shows the smoothed topology optimization result based on the design of FIG. 12B. [Figure 12D] The topology optimization results are shown where the dimensional control forcing the 2D subcomponent is a structural plate by constraining the lowest singular value σ3 to be smaller than 0.15 of the design variable (DV) decomposition (SVD) at a diameter of 10.0. [Figure 12E]FIG. 12C shows the results of a smoothed topology optimization based on the design of FIG. 12D. [Figure 13A-13B] The optimization iteration history for the objective function compliance, which minimizes compliance subject to a relative mass constraint of 15%, is shown. [Figures 14A-14C] We present the optimization iteration convergence history for minimizing compliance under a 15% relative mass constraint and an approach that constrains the minimum singular value σ3 to be less than 0.15 of the singular value decomposition (SVD) of the design variables (DV) at a diameter of 10.0. [Figure 15] 1 is a flow chart involving the use of the design response (DRESP) and the sensitivity of the computed singular values ​​(SVD) for geometric dimension control with respect to the design variables. [Figure 16A] FIG. 1 shows the design space for topology optimization partitioned into finite elements, where the relative density of each element is the design variable with each lower corner fully clamped and a load applied to the central top point. [Figure 16B] Topology optimization without geometric dimension control is presented. [Figure 16C] We demonstrate dimensionality control in topology optimization constraining 1D structural subcomponents. [Figure 16D] We demonstrate dimensional control in topology optimization constraining 2D structural subcomponents. [Figure 17A] We show dimensional control in topology optimization that enforces 1D structural subcomponents representing fibers that do not cross each other, and the topology optimization results are represented by finite elements with relative densities of solids and intermediates. [Figure 17B] Dimensional control in topology optimization is shown to enforce 1D structural subcomponents representing fibers that do not cross each other, and the results of the topology optimization are represented by an isosurface description of the relative density field. [Figure 18] A sphere of radius r is shown in a finite element model of a real-world physical object being designed, with the center of gravity identified for each respective finite element in the model. [Figure 19A]A rectangular 2D design domain is shown discretized by a 400x200 uniform grid. [Figure 19B] A classical stiffness topology optimization is shown using material volume fraction f=60%. [Figure 19C] We present the classical topology optimization of 19B, but with an additional constraint for the proposed approach defined in equation (17) that constrains the lowest singular value to be less than or equal to 0.6 using a diameter D = 12, corresponding to 12 finite elements. [Figure 20A] 4 shows the optimization iteration convergence history for minimizing compliance. [Figure 20B] We show that compliance is minimized under a 15% relative mass constraint. [Figure 20C] We show that compliance is minimized under the conditions of an implementation of the techniques presented herein with constraining the lowest singular value to be smaller than 0.60 of the design variable decomposition (SVD) at diameter D=12. [Figure 21] 1 shows an exemplary optimization result. [Fig. 22A-22D] 4 illustrates another exemplary optimization result. [Fig. 23A-23D] Further exemplary optimization results are shown. [Figure 24] 1 illustrates an exemplary femur finite element model. [Fig. 25A-25E] 1 illustrates an exemplary topology optimized filling design for a femur. Like reference characters refer to like elements. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0017] Design optimization generally refers to an engineering design methodology that involves the mathematical formulation of a design problem to support the selection of an optimal design from among many alternatives.

[0018] Optimization methodologies generally involve the selection of one or more elements that are best (with respect to some criterion) from a set of available alternatives. Optimization problems arise in a variety of fields, even beyond product design optimization. In simple cases, optimization problems involve maximizing or minimizing a real function by systematically choosing input values ​​of the real function from within a permitted set and calculating the value of the function.

[0019] Design optimization generally involves a set of design variables (which can be used to describe various design alternatives), design objectives (combinations of individual design variable values ​​to be maximized or minimized), constraints (conditions that must be satisfied for a particular design alternative to be acceptable; e.g., variables or combinations of variables represented by equalities, inequalities, etc.), and feasibility (represented by the set of values ​​of the design variables that satisfy all the constraints and / or design objectives).

[0020]

number

[0021] One particular type of optimization is topology optimization, which generally involves mathematical methods of optimizing material layouts for a given set of loads, boundary conditions, and / or constraints with the goal of maximizing the performance of a system, for example, within a given design space. Topology optimization differs from shape optimization and sizing optimization in that in topology optimization, the design can obtain any shape within the design space instead of working with predefined configurations.

[0022] Topology optimization formulations typically use the finite element method to evaluate design performance. Designs can be optimized using either gradient-based mathematical programming methods, such as optimality-criteria algorithms and the method of moving asymptotes, or non-gradient-based algorithms, such as generic algorithms.

[0023] Topology optimization has a wide range of 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 free-form nature that naturally occurs in topology optimization, the results can be difficult to manufacture. In certain implementations, the techniques disclosed herein can help overcome limitations of the prior art.

[0024] Leading Optimization Methods Some prior optimization techniques, which are described in turn below, include filter techniques (sometimes called regularization techniques) and projection techniques of design variables, direct parameterization of design variables, penalty or projection functions on design variables, local volume constraints, and heuristic techniques. The techniques disclosed in this application differ from those prior techniques. For example, the techniques disclosed in this application involve utilizing singular value decomposition (SVD) of design variables to optimize certain aspects of the design (e.g., topology). In some implementations, the resulting singular values ​​may be applied to an objective function for the optimization process as terms to be optimized. Alternatively, in some implementations, the resulting singular values ​​may be utilized as optimization constraints to locally constrain the geometry of the optimized structure. None of the prior optimization techniques described below utilize singular value decomposition (SVD) of design variables in a similar manner.

[0025] 1. Filtering techniques (sometimes called regularization techniques) and design variable projection techniques. A common approach is density variable filtering for topology optimization, where the relative density is a design variable and is filtered. (For example, MP Bendsoe 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) 68-75, BSLazarov, F. Wang and O.Sigmund,Length scale and manufacturability in density-based topology optimization,Archive of Applied Mechanics 86(1-2)(2016)189-218, M.Zhou,BSLazarov,F.Wang and O.Sigmund,Minimum length scale in topology optimization by geometric constraints,Computer Methods in Applied Mechanics and Engineering 293(2015), 266-282, BSLazarov and (F. Wang, Maximum length scale in density based topology optimization, Journal Computer Methods in Applied Mechanics and Engineering, 318 (2017) 826-844, and other references mentioned herein.) Combining these filter techniques and projection methods to enforce length scales can help ensure manufacturable structures relative to part size requirements, for example, for cast manufactured structures, 3D printed structures, milled structures, etc.

[0026] The techniques disclosed herein can enforce a geometric length scale through the diameter. However, the techniques disclosed herein are fundamentally different from design variable filter techniques and projection techniques. For example, as noted above, the techniques disclosed herein involve utilizing singular value decomposition (SVD) of the design variables in a particular way to optimize a particular aspect of the design (e.g., topology), whereas the design variable filter techniques and projection techniques do not utilize singular value decomposition (SVD) of the design variables.

[0027] 2. Direct parameterization of design variables Direct parameterization of design variables can be used to obtain feasible designs for casting and plate manufacturing structures, for example (see, for example, S. Zhang, J. Orato, A. L. Gain, et al., A geometry projection method for the topology optimization of plate structures, Structural and Multidisciplinary Optimization 54(5)(2016)1173-1190; JP Leiva, B. C. Watson† and I. Kosaka, An Analytical Bi-Directional Growth Parameterization to Obtain Optimal Castable Topology Designs, 10th AIAA / ISSMO Multidisciplinary Analysis and Optimization Conference, 30 August-1 September 2004, Albany, New York; and A. Roulund-Gersborg and C. Andreasen, An explicit parameterization for casting constraints in gradient driven topology optimization, Structural and Multidisciplinary Optimization 44(6)(2010)875-881).

[0028] The techniques disclosed herein can also be applied to obtain feasible designs for plate manufacturing structures. However, the techniques disclosed herein are different from direct parameterization of design variables. For example, the techniques disclosed herein do not involve applying a new parameterization or mapping of design variables. Also, as noted above, the techniques disclosed herein involve utilizing singular value decomposition (SVD) of design variables in a particular way to optimize certain aspects of the design (e.g., topology), whereas direct parameterization of design variables does not utilize singular value decomposition (SVD) of design variables.

[0029] 3. Penalty or projection functions for design variables A penalty function can be applied to ensure geometrically feasible designs for additive manufacturing (3D printing) and multi-axis manufacturing. (For example, M. Langelaar, An additive manufacturing filter for topology optimization of print-ready designs, Structural and Multidisciplinary Optimization 55(3) (2017) 871-883, M. Langelaar, Topology optimization for multi-axis machining, Computer Methods in Applied Mechanics and Engineering, Structural and Multidisciplinary Optimization 351(2019)226-252, M. Hoffarth, N. Gerzen, CBWPedersen, ALM Overhang Constraint in Topology Optimization for Industrial Applications, 12th World Congress on Structural and Multidisciplinary Optimization 05th-09th, June 2017, Braunschweig, Germany, and “Penalty Function On Design Variables For Designing Variables For Designing Cost Beneficially Additive Manufacturable See U.S. Patent Application Publication No. 2017 / 0176975 A1, entitled "Metal-Based Tube Structures."

[0030] Projection schemes have the ability to apply geometric control for optimization so that designs are geometrically feasible for manufacturing. (See, for example, SLVatanabe, TNLippi, CRLima, GHPaulino and ECNSilva, Topology optimization with manufacturing constraints, A unified projection-based approach, Advances in Engineering Software 100 (2016) 97-112, and JVCarstensen and JKGuest, Projection-based two-phase minimum and maximum length scale control in topology optimization, Structural and Multidisciplinary Optimization, 58(5) (2018) 1845-1860.) Certain works use geometry projection methods to project analytical descriptions of sets of geometric primitives and fixed thickness plates, respectively. These can constrain structural members to super shapes or plates for manufacturability. (See, for example, J. Orato, Topology optimization with supershapes, Structural and Multidisciplinary Optimization 58(3)(2018)1-20, and S. Zhang, J. Orato, A. L. Gain and N. Lyu, A geometry projection method for the topology optimization of plate structures, Structural and Multidisciplinary Optimization 54(5)(2016),1173-1190.)

[0031] The techniques disclosed herein can enforce plate-like structures, but are fundamentally different from penalty or projection functions. For example, as noted above, the techniques disclosed herein involve utilizing singular value decomposition (SVD) of design variables in a particular way to optimize certain aspects of the design (e.g., topology), whereas penalty or projection functions do not utilize singular value decomposition (SVD) of design variables.

[0032] 4. Local volume constraints Some studies have applied local volume constraint approaches to geometrically enforce porous structures, and local volume constraints can also be applied to mimic lattice structures (see, for example, Wu, N. Aage, R. Westermann, O. Sigmund, Infill optimization for additive manufacturing Approaching bone-like porous structures, IEEE Transactions on Visualization and Computer Graphics 24(2)(2018)1127-1140; Wu, A. Clausen, O. Sigmund, Minimum compliance topology optimization of shell-infill composites for additive manufacturing, Computer Methods in Applied Mechanics and Engineering 326(2017)358-375; and M. Schmidt, C.B.W. Pedersen, C.Gout, On structural topology optimization using graded porosity control, Structural and Multidisciplinary Optimization 60(4)(2019)1437-1453).

[0033] The techniques disclosed herein can enforce a lattice-like structure, but are fundamentally different from local volume constraint approaches: for example, as noted above, the techniques disclosed herein involve utilizing singular value decomposition (SVD) of the design variables in a particular way to optimize certain aspects of the design (e.g., topology), whereas local volume constraint approaches do not utilize singular value decomposition (SVD) of the design variables.

[0034] 5. Heuristics Some work involves heuristic approaches. For example, the work of N. Stromberg, Topology optimization of structures with manufacturing and unilateral contact constraints by minimizing an adjustable compliance-volume product, Structural and Multidisciplinary Optimization 42(3)(2010)341-350, applies a heuristic method that updates the movement constraints at each optimization iteration defined such that the drawing constraints are satisfied. Thereby, no explicit constraints appear in the nested formulation, except for lower and upper density bounds, but the updates of these movement constraints are entirely heuristic.

[0035] The work of R. Dienemann, A. Schumacher and S. Fiebig, Topology optimization for finding shell structures manufactured by deep drawing, Structural and Multidisciplinary Optimization 56(2)(2017)473-485 applies a mid-surface approach by calculating the average of element coordinates in the punch direction to achieve deep-drawn manufactured structures. The implementation of this approach is more heuristic, since the mid-surface movement at constant wall thickness is not consistently included in the mathematical sensitivity calculations. This is seen by an increase in the number of optimization iterations.

[0036] These heuristic methods are partially mathematically inconsistent because they are heuristic, whereas the techniques disclosed herein are mathematically consistent and can also enforce structures suitable for deep drawing. However, the techniques disclosed herein are fundamentally different from the heuristic methods. For example, as noted above, the techniques disclosed herein involve utilizing singular value decomposition (SVD) of design variables in a particular way to optimize certain aspects of the design (e.g., topology), whereas heuristic methods do not utilize singular value decomposition (SVD) of the design variables.

[0037] The new techniques disclosed herein differ from the prior approaches disclosed above, generally involving utilizing singular value decomposition (SVD) of design variables in a particular way to optimize certain aspects of the design (e.g., topology). These differences are not trivial. For example, in some implementations, the new techniques disclosed herein result in optimized designs that are less bulky and more feasible to manufacture than prior approaches (e.g., using a structure of a single stamped metal sheet, or several stamped metal sheets welded together).

[0038] Case in point: Automotive companies (e.g., General Motors®, Volkswagen®, AUDI®, BMW®, Ford®, Jaguar Land Rover™) and aerospace companies (e.g., Airbus®, Boeing®) use optimization processes intensively, especially for structural parts. These optimization processes often result in structures with sub-components whose designs are bulky. Often, the practical and industrial design requirements from a manufacturing process standpoint are that the structure is manufactured using a single stamped metal sheet, or a structure in which several stamped metal sheets are welded together.

[0039] In some implementations, the techniques disclosed in this application can generate and advantageously constrain designs to be viable for manufacturing, for example, for pressed plate structures. (See, e.g., FIG. 12). The topology optimization model of FIG. 12 is a fully clamped beam on the left side and a lower end of the right side under load. The model shown in FIG. 12A can be composed of 375,000 linear 8-node continuum finite elements, each with a size of 1×1×1, with a total beam size of 50×50×150. Each finite element in the illustrated example is a design variable that can be defined using the SIMP approach (Solid Isotropic Material with Penalties), and topology optimization can be performed using topology sensitivity and mathematical programming (see, e.g., M.P. Bendsoe and O. Sigmund, Topology Optimization-Theory, Methods, and Applications, 2004).

[0040] The finite element model is linear with linear constituent materials with Poisson's ratio of 0.30 and a Young's modulus of 210,000 is applied. The optimization objective in this example is to maximize stiffness by minimizing compliance for a given fixed relative mass of 15% of the design space of FIG. 12A. FIG. 12B shows an example topology optimization result without geometric dimensionality control. FIG. 12C shows an example of a resulting real-world object based on the design of FIG. 12B. FIG. 12D shows the lowest singular value s min We present an example topology optimization result with dimensional control forcing a 2D subcomponent that is a tectonic plate by constraining s to be less than 0.15. This improves the manufacturability of the optimized design, especially with tectonic plates. In this example, the value s min is calculated by performing a design variable (DV) decomposition (SVD) on 10 diameters (see FIG. 12D) in the neighborhood of each finite element. FIG. 12E shows an example of the resulting real-world object based on the design of FIG. 12D.

[0041] As shown, the traditional topology optimization result without geometric dimension control consists of subcomponents that are strictly bars and beams, whereas the present topology optimization result with dimensional control enforcement 2D consists of subcomponents that are mainly structural plates. As a result, two different designs, usually with very different processes, are applied for manufacturing. Notably, both designs in FIG. 12B and FIG. 12D have a relative mass of 15% of the design domain in FIG. 12A. However, the design in FIG. 12B without geometric dimension control has a lower compliance of 0.00750, whereas the design in FIG. 12D with geometric dimension control has a higher compliance of 0.00896. As a result, by enforcing geometric dimension control as shown here for 2D subcomponents, the manufacturing feasibility for pressed plate structures affects the objective function value, which in this example results in a 16% reduction in stiffness for the same mass.

[0042] 13A and 13B show the optimization iteration history of the design of FIG. 12B obtained using a traditional topology optimization workflow without geometric dimension control. More specifically, FIG. 13A shows how the objective function compliance varies with respect to the number of optimization iterations, and FIG. 13B shows how the 15% relative mass constraint varies with respect to the optimization iterations.

[0043] 14A-14C show the optimization iteration history of the design of FIG. 12D obtained using the topology optimization workflow with geometric dimension control. More specifically, FIG. 14A shows how the objective function compliance varies with the number of optimization iterations, FIG. 14B shows how the 15% relative mass constraint varies with the optimization iterations, and FIG. 14C shows how the lowest singular value (below 0.15) varies with the optimization iterations.

[0044] Comparing the optimization histories in Fig. 13A / B and Fig. 14A / B / C, the lowest singular value s min It can be observed that additionally constraining to be smaller than 0.15 does not significantly change the optimization history. Second, the approach of geometric dimension control in optimization that constrains the singular values ​​of the design variable decomposition (SVD) can be used in the optimization workflow based on sensitivity and mathematical programming and does not result in optimization convergence problems.

[0045] FIG. 1 is a schematic representation of an example computer-based system 100 that may be used to generate optimized designs for physical, real-world objects in accordance with one or more techniques disclosed herein, including, for example, techniques that utilize singular value decomposition (SVD) of design variables to optimize particular aspects of the design, such as topology.

[0046] The computer-based system 100 has a central processing unit 102, a computer-based memory 104, a computer-based storage 106, a network interface 108, an input / output device interface 110, and a bus that serves as an interconnect between 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 one another.

[0047] Central processing unit (CPU) 102 is configured to perform various computer-based functionalities, including those disclosed herein. Typically, the CPU performs these functions by executing computer-readable instructions from non-transitory computer-readable media, such as computer-based memory 104 and / or computer-based storage 106, and / or from computer-readable data received from external sources, for example, via I / O device interface 110 and / or network interface 108.

[0048] Computer-based memory 104 provides a form of volatile storage for computer-readable instructions that, when executed by CPU 102, cause the CPU to perform various computer-based functionalities, including, for example, those disclosed herein.

[0049] Computer-based storage 106 provides a form of non-volatile storage for software instructions, such as an operating system (not shown) and configuration for an embodiment.

[0050] The network interface 108 is for connecting to any of a variety of computer-based or communications networks, including, for example, a local area network (LAN) and / or a wide area network (WAN) such as the Internet.

[0051] The 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, a mouse, a display, speakers, and the like. In a typical embodiment, the computer system 100 may be configured to display (e.g., on a display device connected to the I / O device interface 110) a visual representation of an optimized design optimized according to the techniques disclosed herein. In some implementations, the design may appear, for example, as shown in FIG. 12D or 12E, FIG. 16C or 16D, or FIG. 17A or 17B. The display in such implementations may also allow a user to view other information (e.g., performance characteristics, manufacturing information, etc.) related to the displayed optimized design.

[0052] In various implementations, computer system 100 may have additional elements, such as controllers, buffers (caches), drivers, repeaters, and receivers, to facilitate communication and other computer-based functionality. Additionally, interfaces may include address, control, and / or data connections to facilitate communication between the illustrated components.

[0053] FIG. 2 is a schematic representation of an exemplary computer network environment 200 that may be used to determine optimized designs for physical, real-world objects in accordance with one or more techniques disclosed herein, including, for example, those that utilize singular value decomposition (SVD) of design variables to optimize certain aspects of the design (such as topology).

[0054] The computer network environment 200 includes a server 202 and clients 204a, 204b, ... 204n, which are coupled to each other via a communication network 206 that allows the server 202 and the clients 204a to communicate and interact with each other. In a typical implementation, all the clients 204a, 204b, ... 204n and the server may include the same types of components as in the computer system 100 of FIG. 1. In some implementations, each client 204a, 204b, ... 204n may be configured to perform the functionality disclosed herein to optimize the design without requiring the involvement of the server 202. In some implementations, the functionality disclosed herein to optimize the design may be distributed between the server 202 and the clients (e.g., 204b), with the server 202 providing cloud computing functionality, for example, in a Software as a Service (SAAS) context.

[0055] 3A and 3B , taken together, show a flowchart illustrating an example implementation of a method 300 for automatically determining an optimized design for a real-world object, the method including constraining values ​​from a singular value decomposition (SVD) of design variables.

[0056] The illustrated method 300 begins with defining 302, in the memory of a computer-based processor, a finite element model that is based on a CAD model of a real-world physical object whose design is to be optimized.

[0057] For example, CAD and computer-aided engineering (CAE) systems offer a wide range of possibilities for how to represent real-world physical objects in virtual, computer-based environments. Representing real-world physical objects as finite element models is one such possibility. The phrase finite element model (FEM) refers to a type of model associated with and utilized in and related to finite element analysis.

[0058] Finite element analysis refers to the practical application of the finite element method, which is a method of solving engineering or design problems that involves subdividing a system (or an object or a CAD model of an object) into smaller and simpler parts called finite elements. This subdivision may be achieved by a specific spatial discretization in the spatial dimensions, which is implemented by constructing a mesh (continuous domain) of the object in question into a set of distinct subdomains, usually called elements. In some implementations, the finite element method may involve the formulation of a boundary value problem that results in a system of algebraic equations. The finite element method approximates an unknown function over the domain. The simple equations that model these finite elements are then folded into a larger system of equations that models the entire problem. The finite element method may then utilize a calculus of variations from the calculus of variations to approximate the solution (e.g., by minimizing an associated error function, etc.).

[0059] There are a variety of software applications for performing or facilitating finite element analysis and for generating finite element models. One such software application is Abaqus FEA™, a software suite available from Dassault Systemes Simulia Corp., which includes computer-readable instructions that, when executed by a processor in a computer system such as that depicted in FIG. 1, perform computer-based finite element analysis functionality. In a typical implementation, the Abaqus FEA™ software may be incorporated into or provided as part of a version of SIMULIA® Tosca Structure, also available from Dassault Systemes Simulia Corp., that is adapted to support various optimization-related functionalities disclosed herein. In a typical implementation, such computer-readable instructions (e.g., the Abaqus FEA™ software incorporated into or provided as part of an adapted version of SIMULIA® Tosca Structure) may be stored on a non-transitory computer-readable medium (e.g., in computer-based memory 104 or computer-based storage 106 of FIG. 1 ) or may be executable by a computer-based processor (e.g., central processing unit 102 of FIG. 1 ) coupled to the non-transitory computer-readable medium.

[0060] In a typical implementation, the finite element model includes a system of points, called nodes, interconnected to create a grid, called a mesh. In a typical implementation, the finite element model is represented by a collection of data defining the finite element model stored in a computer-readable format in a computer-based memory device (e.g., 104 or 106 in FIG. 1). The stored data associated with the finite element model may reflect one or more properties of the underlying object or objects. When the finite element model is so configured, it may be used to perform a simulation of the object it represents. In this regard, the finite element model may represent any one of a variety of different objects or combinations of objects, including, for example, a vehicle, a type of physical hinge, a structural support (e.g., a beam, etc.), or any number of other real-world physical objects or combinations of objects. When the finite element model represents an object, and the underlying data associated with the finite elements so indicates, the finite element model may 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 may be used to simulate the use of the stent in an actual medical environment.

[0061] In a typical implementation, the finite element model includes a plurality of elements. According to an embodiment, the elements may be tessellated elements of the finite element model. However, the embodiment is not limited and may be used in connection with any element, such as a triangular, quadrilateral membrane or shell. According to an embodiment of the method 300, the finite element model may be defined in step 302 according to any method known in the art. Furthermore, in an embodiment, the finite element model defined in step 302 may be any such type of finite element as known in the art. For example, in an exemplary embodiment, the finite element model defined in step 302 may be a sheet model of a surface of a real-world object. According to an embodiment, defining the model in step 302 may include defining a finite element model in a computer-based memory that reflects all (or some subset) of known characteristics, such as dimensions, materials, etc., of the object that the finite element model represents, the model including some design variables (e.g., plate thickness). In some implementations, the design variables may represent object characteristics for which values ​​(e.g., inches or centimeters) are determined (or optimized). Furthermore, the behavior of the finite element model may be governed by equations, including respective sensitivity equations for the design variables, e.g., plate thickness. In such embodiments, defining the finite element model may include defining equations that govern the behavior of the model, i.e., the object that the model represents.

[0062] The method 300 continues at 304 with identifying a neighborhood or region (e.g., a sphere having a given radius, see, e.g., FIG. 18 ) within the real-world model and / or the corresponding finite element model to be considered in connection with the associated optimization process.

[0063] The dimension of an object calculated using singular values ​​generally describes how much space the object requires within the considered neighborhood (e.g., sphere). Therefore, the size of the neighborhood (e.g., sphere) may be appropriately selected depending on the individual application. In some implementations, a fairly small sphere (radius = 2-5 x finite element size) may be applied to locally control the dimension of the structure. In some implementations, using a larger sphere for practical applications may result in a misinterpretation of the structure dimension for optimization. For example, as shown in Figures 10 and 11, an assembly of 1D parts or 0D parts can be interpreted as a 3D object or as a 2D object. Whether such a geometric interpretation is intended or not generally depends on the individual design application.

[0064] 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 the nodes and the corresponding coordinates of each node. These matrices can be stored, for example, in the computer-based memory 104.

[0065] The illustrated method 300 then includes evaluating 306 the distribution of the design variables across a neighborhood of the real-world object based on each finite element in the neighborhood of the object / finite element model using singular value decomposition (SVD). In some implementations, singular value decomposition (SVD) may refer to a factorization of a real or complex matrix that generalizes, for example, the eigenvalue decomposition of a square normal matrix to any m×n matrix via an extension of polar decomposition. Specifically, the singular value decomposition of an m×n real or complex matrix M is a factorization of the form UΣV*, where U is an m×m real or complex unitary matrix, Σ is an m×n rectangular diagonal matrix with nonnegative real numbers on the diagonal, and V is an n×n real or complex unitary matrix.

[0066] Next, the illustrated method 300 finds singular values ​​(e.g., s, alternatively referred to as s1, s2, and s3, respectively) for the design variables within a neighborhood of the real-world object based on the SVD evaluation. max , s mid , and s min ) and corresponding orientations at 308. In the topology optimization aspect, for example, the distribution of design variables (e.g., relative density) around each finite element (sphere with a given radius, see FIG. 4A or FIG. 18) can be evaluated using SVD. The obtained singular values ​​(s max , s mid , s min , see Figure 4E) are related to the three geometric dimensions of the object, described by the density field in the considered neighborhood (sphere). These measured values ​​can be used, for example, to construct a design response for structural optimization.

[0067] Referring again to FIG. 3A, the method 300 includes defining 310 optimization constraints for the neighborhood based on the singular values.

[0068] FIG. 3B illustrates an exemplary method 300a for defining 310 optimization constraints for neighborhoods based on singular values.

[0069] In a typical implementation, the measures s1, s2, and s3 are discontinuous between optimization iterations and cannot be used directly in a gradient-based optimization algorithm. This may be resolved in some examples using smooth approximations. In this regard, the method 300a includes applying 310a a smooth approximation to each of the singular values ​​associated with the design variables within a neighborhood of the real-world object based on the SVD evaluation.

[0070] 3B method 300a then includes identifying an additional value at 310b by performing an SVD of the matrix with all element design variable values ​​set to 1.0. This value generally represents the maximum possible dimensionality of the object (or a neighborhood of the object) being considered and can be used to calculate relative singular values.

[0071] The method 300b then includes defining, in a computer-based memory, optimization constraints for each neighborhood at 310c utilizing the smoothing approximation and additional values.

[0072] Referring again to FIG. 3A, the method 300 includes iteratively optimizing (at 312) the finite element model. In a typical implementation, this includes limiting the dimensionality of the finite element model based on defined optimization constraints (defined based on singular values, SVD). The iterative optimization at step 312 results in an optimized topology associated with a given element of the finite element model. According to one embodiment, the iterative optimization of step 312, i.e., the iterative design process, is a process of selecting a new improved solution (with respect to the objectives and constraints defined through the design response) for each optimization iteration.

[0073] After the iterative optimization in step 312, the properties of the given element of the finite element model are automatically updated to reflect the optimized values ​​of the sizing design variables in step 314. In this manner, the method 665 creates an optimized model of the 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 then be converted into a format and subjected to a real manufacturing process, where local control of the design pattern of the surface is important to satisfy manufacturing constraints, and the obtained design is directly manufactured. In an embodiment, by controlling the local volumetric constraints, design constraints can be set that comply with the real-world limitations of the manufacturing process, and thus the embodiment determines an optimized design for manufacturing the real-world object. An embodiment of the method 300, for example, determines an optimized topology for multiple elements of the finite element model. In this manner, such an embodiment can determine an optimized design for manufacturing the entire real-world object.

[0074] Next, in the illustrated embodiment, the method 300 includes causing (at 316) the real-world object to be manufactured according to the optimized model. In one such embodiment, after the model is updated in step 314, the updated model is digitally communicated to a manufacturing machine (e.g., an additive manufacturing machine such as a 3D printer) that can generate the real-world object according to the optimized model. In one embodiment, the digital model is communicated using any communication method known in the art and may be communicated to any manufacturing machine known in the art, such as an additive manufacturing machine, a stamping machine, or a computer numerically controlled (CNC) machine.

[0075] The following is a description of the techniques disclosed herein applied in the context of topology optimization, where the distribution of design variables (relative density) around each finite element (e.g., a sphere with a given radius, see FIG. 4a or FIG. 18) is evaluated using singular value decomposition (SVD) and the obtained singular values ​​(e.g., s max , s mid、 s min , see Fig. 4e) are related to the three geometric dimensions of the object described by the density field of the considered sphere. These measured values ​​are used in a typical implementation to construct a design response for structural optimization.

[0076] For a finite element model described in 3D space, the finite element E of the structure i is the node N ij and each node N ij The corresponding coordinates (x l , y l , z l ) and the matrix E(N ij ) and N(x l , y l , z l ) can be used to express

number

[0077] Each element E i For evaluation in a given neighborhood of radius R (see Fig. 4a or Fig. 18) around the neighboring elements and the centroids of the neighboring elements (see Fig. 4c),

number

number

[0078] In addition, for each neighborhood, the relative element density

number

[0079] In a typical implementation, to perform SVD, the centroids need to be centered around the mean of the centroids.

number

[0080] Then, the singular values ​​(s1, s2, s3) of each neighborhood are stored in the matrix X j It can be obtained by SVD of

number

[0081] A graphical representation of this matrix is ​​shown in Fig. 4d, where all centroids with low relative density have been moved to the center of the sphere. As a by-product, singular vectors can also be calculated (shown as vectors in Fig. 4e), which represent the directions of the main dimensions of the object described by the considered design variable density field. The scales s1, s2, s3 are also max , s mid , s min They can be referred to by their size (s max≧s mid ≧s min ) because they are ordered by

[0082] In Figures 5 to 11, the normalized values ​​s max , s mid , s min Correspondingly, all normalized singular values ​​are equal to 1.0 for solid spheres and 0.0 for void spheres, as shown in Figure 5. For all other configurations, the singular values ​​are between 0.0 and 1.0.

[0083] Figure 6 shows the transformation from a solid (3D object representation) to a shell, plate or membrane layout (2D object representation) for the normalized singular values ​​of the relative density field as a design variable for example for topology optimization. As shown in Figure 6, when the 3D object is transformed to a 2D object, one of the singular values ​​approaches zero.

[0084] Figure 7 shows the transformation from a shell, plate or membrane layout (2D object representation) to a beam, bar or lattice layout (1D object representation) for the normalized singular values ​​of the relative density field as a design variable for example for topology optimization. As shown in Figure 7, when the 2D object is being transformed to a 1D object, two singular values ​​approach zero.

[0085] Figure 8 shows the transformation from a beam, bar, or lattice layout (1D object representation) to a near-void layout (0D object representation) for the normalized singular values ​​of the relative density field, e.g., as a design variable for topology optimization. As shown in Figure 8, when the 1D object is transformed to an empty sphere (void), all the singular values ​​approach zero.

[0086] Figure 9 shows the transformation from a solid (3D object representation) to a void layout (0D object representation) for the normalized singular values ​​of a relative density field as a design variable for example for topology optimization. As shown in Figure 9, when the 3D object is transformed to an empty sphere (void), all the singular values ​​approach zero.

[0087] FIG. 10 shows the all-ones normalized singular values ​​for a relative density field as a design variable for a topology optimization representing, for example, an assembly of 1D or 0D parts in 3D space.

[0088] FIG. 11 shows that for a relative density field as a design variable for a topology optimization representing an assembly of 1D parts or 0D parts in 2D space, we have two normalized singular values ​​that are unity and one that is close to zero.

[0089] As mentioned above, the dimension of an object calculated using singular values ​​describes how much space the object requires within the considered neighborhood (sphere). The size of the sphere must therefore be chosen appropriately depending on the individual application. In many cases, rather small spheres (radius = 2-5 x finite element size) are applied to locally control the structural dimension. In many cases, real applications using larger spheres may result in a misinterpretation of the structural dimension for optimization. For example, as shown in Figures 10 and 11, an assembly of 1D or 0D parts can be interpreted as a 3D or 2D object. Whether such a geometric interpretation is intended or not may depend, for example, on the individual design application.

[0090] 5-11 thus illustrate that singular values ​​can be applied to control and enforce a given type of form and shape of a design variable layout. In an exemplary implementation, system 100 is configured to apply singular values ​​in exactly this manner, for example, to control and enforce a variety of different types of form and shape in topology optimization. This has never been applied to control the form, shape, and layout of design variables for non-parametric optimization, and is a unique aspect of the present approach.

[0091] The measures s1, s2, s3 are discontinuous between optimization iterations and cannot be used directly in gradient-based optimization algorithms. In a typical implementation, this is solved using a smooth approximation as follows:

number

[0092] Also, in a typical implementation, an additional value,

number

[0093] The optimization constraints for each neighborhood (the sphere around each finite element is equivalent to the design variable field) can then be defined as follows:

number

[0094] quantity q A limits the minimum local dimension of the structure, while q B limits the second largest value of q C sets the relationship between the largest and second largest dimensions. In order to reduce the number of constraints for the optimization (to avoid having constraints for each finite element equal to the design variables), the left-hand sides of the above constraints are aggregated using a smooth max / min approximated aggregation function (to obtain a single constraint of type A, B, and C that covers all or a subset of the finite elements and therefore all or a subset of the design variables). The right-hand sides of the constraints are the constrained values. Thus, the following constraints can be used locally to enforce that the optimized design has a certain dimension: 1) 1D or 2D:q A Constraint of type A with << 1.0 (e.g. 0.1) 2) 1D:q A <<1.0 and q B << Constraints of types A and B in 1.0 3) 2D:q A <<1.0 (e.g., 0.1) and q C Constraints of types A and C at >>0.0 (e.g., 0.6)

[0095] In some implementations, the left hand side of the above constraint can be implemented as a design response in an existing structural optimization solution to define the constraint, or as an objective term. Typically, the derivative of the smoothed constraint with respect to the design variable (relative density) may be required during the optimization process and can be obtained by differentiating the above equation.

[0096] In general, the techniques disclosed herein use sensitivity optimization for geometric dimension control of design variables. This can be done by looping through all design variables (DVs), identifying other adjacent design variables for which geometric dimension control is enforced for a given design variable (DV), calculating (405) singular values ​​(SVDs) and sensitivities (also called derivatives) for each given design variable (DV), and calculating (407) design responses (DRESPs) and sensitivities using one or more aggregation functions for the design variables (DVs). An aggregation function is a function that groups multiple different values ​​together to form, for example, one summary value.

[0097] FIG. 15 illustrates an iterative design process 1550 that involves constraining design values ​​based on a singular value decomposition (SVD) of the design variables, according to one embodiment.

[0098] According to the illustrated embodiment, the process 1550 meets the design response (DRESP) and sensitivity of the calculated singular values ​​(SVD) for geometric dimension control with respect to the design variables. The iterative design process method 1550 can be implemented, for example, in a predefined workflow of a computer-aided engineering (CAE) system. The method 1550 starts at step 1501, creating an initial model including various applied loads and boundary conditions for balancing for optimization. In one embodiment, the model is defined by a designer (e.g., the designer inputs data into a computer, such as the computer 100 depicted in FIG. 1, via one or more I / O devices connected to the I / O device interface 110). The model is then subjected to the iterative design process via steps 1503-1513.

[0099] According to an illustrated embodiment, the design iteration cycle may include a processor (e.g., CPU 102 of FIG. 1 ) reading (at 1503) data representing a geometric description of the model from a memory (e.g., 104 from FIG. 1 ) and solving equilibrium of the model. Then, for example, the processor may model (at 1505) the design responses and the equilibrium design responses, and the sensitivities of the design responses and the equilibrium design responses with respect to the design variables.

[0100] According to an illustrated embodiment, each design iteration cycle may include (at 1507) the processor reading a geometric description of the model (e.g., a finite element model). The processor may then determine (at 1509), for example, design responses and sensitivities of calculated singular values ​​(SVDs) for geometric dimension control with respect to the design variables.

[0101] In a typical implementation, the design response defines the response to the current analytical model for a given optimization iteration, whereby in a typical implementation, the design response may be extracted as a single scalar value (e.g., mass, center of mass, etc.) that can be measured directly from the model, or may be determined by the outcome of a primary solution of the model's equilibrium (e.g., stresses, displacements, reaction forces, etc.).

[0102] The design responses are then applied to define an optimization problem consisting of constraints that need to be satisfied and an optimized objective function. The optimization problem is solved using mathematical programming in step 1511. In a typical implementation, the mathematical programming, i.e., optimization calculation, is strictly based on the values ​​of user-defined design objectives, the design responses, and the sensitivities of the design responses. Thus, if the calculated singular value (SVD) design responses and sensitivities for geometric dimension control with respect to the design variables are included in the iterative design process, in some implementations, the design responses and sensitivities can help enforce local control of the design characteristics while also enhancing the physical characteristics for other design responses applied in the optimization setup.

[0103] In mathematics, computer science, and operations research, mathematical programming, alternatively referred to as mathematical optimization or simply optimization, is the process of selecting the best solution (with respect to some criteria) from a set of available alternatives. Embodiments of method 1550 may use any such mathematical programming method as known in the art. In a typical implementation, a processor may implement functionality associated with mathematical programming by executing computer-readable mathematical programming instructions stored in memory.

[0104] After the mathematical programming in step 1511, a new physical model for the next optimization iteration is generated in step 1513 based on the design variables determined in step 1511. The iterative design process is a process of selecting a new improved solution (with respect to some objectives and constraints) for each optimization iteration. The design variables determined in step 1511 and the physical model variables updated in step 1513 may be the same. If the physical model variables and the design domain variables are the same, the physical model in step 1513 is simply obtained as the output of the mathematical programming. Otherwise, an additional step is required to interpret the design variables as physical model variables using filters, as known in the art.

[0105] To proceed, it is determined whether the optimization has converged. Convergence is determined numerically when the change in the objective function from one optimization iteration to the next is less than an abrupt threshold, and the change in the relative sum of the design variables from one optimization iteration to the next is less than an abrupt threshold. If the optimization has not converged, a new optimization cycle is started and the method 1550 returns to steps 1503 and 1507. If the optimization has converged, a final design is created in step 1515. For a converged design, the constraints on the design responses should be satisfied and the objective function should be optimized.

[0106] The method 1550 and the output final design 1515 of the method 1550 can be used on a variety of real-world objects to determine optimized designs for manufacturing.

[0107] A practical application of these techniques can be seen in the context of introducing local constraints for gradient-based topology optimization to locally enforce predefined dimensionality of structural components. The constraints in this example are based on principal values ​​(singular values) from the singular value decomposition (SVD) of the point cloud represented by the element centroids and corresponding non-parametric design variables (e.g., relative density) to enforce dimensional control in design optimization of real-world objects.

[0108] As an example, FIG. 16A is a schematic diagram showing a very coarse finite element (FE) model of a simple cube clamped at its bottom corners and with a downward force (applied load) applied to the center of the top face of the cube. A topology optimization to minimize compliance given a volume constraint may result in the design of FIG. 16B. In that design, no additional constraints on the geometric dimensions are applied. The structural components shown in that design are rather bulky and thick within a given radius of the search neighborhood (see, for example, the sphere outlined for the object design shown in FIG. 16B). In some implementations, the geometric properties of the optimized structure can be locally explored considering a neighborhood with some predefined radius (see, for example, the sphere shown in FIG. 16B-D) at a specific location depending on the design requirements. Compared to this radius, the structural components considered are rather bulky and thick in 3D space. Adding an additional constraint based on the SVD value to the optimization problem allows locally enforcing 1D and 2D structures on the optimized result, as shown in Figures 16C and 16D, respectively. Again, the dimensionality evaluation in typical implementations is relative to a predefined radius of the considered neighborhood. In this regard, Figure 16C shows a locally enforced 1D lattice structure, and Figure 16D shows a locally enforced 2D membrane structure.

[0109] The computer-based methods disclosed herein are applicable to a variety of real-world design applications.

[0110] 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) thin-walled structures for deep drawing using topology optimization with enforced 2D SVD constraints (see, e.g., FIG. 16D); 2) thin-walled welded structures with enforced 2D SVD constraints; 3) avoiding bulky areas for cooling control of metal porous cast components with enforced 1D and 2D SVD constraints; 4) ensuring structural component orientation, e.g., rib and lattice design, enforced orientation, or molded forms, with enforced 1D (see, e.g., FIG. 16C) and 2D SVD constraints (see, e.g., FIG. 16D), respectively; 5) ensuring removability of powder for 3D printing with enforced 1D SVD constraints (see, e.g., FIG. 16C); and 6) 3D continuous fiber printing (see, e.g., FIG. 17A and FIG. 17B), where subcomponents are attachments or splints at branches with no crossing fibers, with 1D SVD constraints.

[0111] The exemplary schematic representations of Figures 17A and 17B represent dimensional control in topology that enforces 1D structural subcomponents representing fibers that do not cross each other. The topology optimized result is represented in Figure 17A by finite elements with solid and intermediate relative densities. The topology optimized result is represented in Figure 17B by an isosurface description of the relative density field.

[0112] In some implementations, these methods can help ensure easy removability of powders in objects produced using additive manufacturing techniques such as 3D printing.

[0113] Geometric Dimension Control in Topology Optimization 1.General This section describes a specific implementation that involves introducing additional constraints to the gradient-based topology optimization to locally enforce predefined dimensions of the structural components. These constraints are based on the principal values ​​(singular values) from the singular value decomposition (SVD) of the point cloud represented by the element centroids and the corresponding relative density design variables.

[0114] 2. Topology Optimization Implementation and Workflow In an exemplary implementation, the geometric control techniques disclosed herein (e.g., imposed by constraining singular values ​​of the decomposition (SVD) for the design variables) are implemented in SIMULIA Tosca Structure, an optimization software using mathematical programming to update the design variables and adjoint sensitivities implemented in SIMULIA Abaqus for structural finite element modeling. The direct solver of SIMULIA Abaqus is applied to solve the equilibrium R=0 between the finite element model and the adjoint solution, except for the numerical results shown in Sections 5.2 and 5.3, where the iterative algebraic multigrid solver by SIMULIA Abaqus is applied. It should be noted that both the direct finite element solver and the iterative algebraic multigrid solver can solve nonlinear structural modeling such as contact, large deformation, and constitutive nonlinear material models. However, linear applications are shown here.

[0115] Furthermore, all the present numerical topology optimization results use the so-called SIMP model (Solid Isotropic Material with Penalties) to model the construction material model as proportional to a power law of the relative element density ρ, which is a vector that encapsulates the design variables of all structural design elements in the design domain.

[0116] In addition, first, the present numerical optimization results (Sections 5.1, 5.2, and 5.4) result in the following optimization framework that minimizes compliance C (maximizes stiffness) with respect to an external applied load P and the resulting displacement U:

number

[0117] The total mass of the design domain, m(ρ), is the product of the relative material fraction, f, and the mass of the design domain with full material, m full The optimization is constrained to meet a specific weight target defined by and. A sensitivity filter is applied for regularization to introduce length scales and checkerboard suppression. The radius of the sensitivity filter for all present optimization results is 1:3 compared to the average element size for all elements specified in the design domain.

[0118] Second, mass is the design response that is minimized as the objective function for the strength optimization applied in Section 5.3. Furthermore, the constrained design response is subject to a single aggregate constraint A(s v The von Mises stress s at the element integration points is applied as (ρ,U(ρ)) v (ρ,U(ρ)),

number

[0119] 3. Calculating Geometric Properties of Local Structural Components Using SVD The present approach applies the definition of new measures to be used as design responses for topology optimization. These measures can be applied either within the objective function or as optimization constraints to locally enforce a preferred geometric layout of the optimized structure. Mathematically, the measures are based on the singular values ​​(or corresponding vectors) of a matrix describing the design variable distribution within a given neighborhood (e.g., a sphere with a given radius). In the topology optimization aspect, the distribution of the design variables (relative density) around each finite element (sphere with a given radius, see Figure 4A) is evaluated using singular value decomposition (SVD). The obtained singular values ​​(s max , s mid , and s min , see Figure 4E) are related to the three geometric dimensions of the object described by the density field of the considered sphere. These measures can be used to construct design responses for structural optimization that enforce local geometric control.

[0120] 3.1 Singular values ​​of the relative density field In the following, then, for a given finite element mesh with n finite elements,

number

number

number

number

number

[0121] The quantities Ce and Cj correspond to the e-th and j-th lines of C. The corresponding relative densities are

number

[0122]

number

number

[0123] This corresponds to a shift of the considered point of a given cloud to the center of mass of the global coordinate system, where we use the relative density values ​​to

number

number

[0124] The centroid coordinates of the elements representing solid materials are not changed by this operation, except for the elements representing intermediate materials, whose corresponding centroids are moved towards the center of mass of the considered point in the group (Figure 4D). min ≦s mid ≦s max The singular value s defined as min and s mid and s max is obtained by SVD of the matrix as

number

number

number

[0125] The quantities in the formula represent the main local dimensions of the object described by the considered relative density field, see FIG. 4E. The corresponding singular vectors v k The directions of the leading dimensions are mutually orthogonal. The matrix W contains singular vectors that are not considered within this contribution. In the following sections, the singular values ​​s k Let us use the matrix

number

number

number

[0126] This makes the singular values ​​invariant with respect to the absolute size of the neighborhood considered.

number

number

number

[0127] 3.2 Relationship between singular values ​​and eigenvalues If numerous algorithms for eigenvalue decomposition are already available in the finite element software environment, these algorithms can be reused to perform the SVD computation of a matrix.

number

number

[0128] Next, decomposing the matrix in Eq. (9)

number

number

[0129] 3.3 Geometric interpretation of singular values 5 to 11 show normalized values ​​for different example structural layouts of the design variables (relative density) in a finite element setup for topology optimization.

number

[0130] When a fully solid 3D object is converted to a 2D object, as shown in Figure 6, one of the singular values ​​approaches zero. In this case, 2D means a shell, plate or membrane-like material layout. When a 2D object is converted to a 1D object, as shown in Figure 7, two singular values ​​approach zero. A beam, bar or lattice-like material layout corresponds to a 1D object.

[0131] As shown in FIG. 8, when a 1D object is transformed into an empty sphere (void), all singular values ​​approach zero.

[0132] As a result, singular values ​​such as those illustrated in Figures 5-8 can be used to formulate new types of constraints to control and enforce the geometric features of the optimized structure. Such constraints are described in the following sections.

[0133] 4 Local constraints on the dimensions of structural components The singular values ​​derived in the previous section are now used to construct new types of design responses for topology optimization. Constraints for locally controlling the dimensionality of the structure are formulated in terms of these responses.

[0134] 4.1 Eliminating Singular Value Discontinuities The singular values ​​s of the neighborhood under consideration k are discontinuous because the order of their singular values ​​may switch between optimization iterations. Therefore, they cannot be used directly in gradient-based optimization algorithms. This is solved using smooth approximations of the maximum and minimum singular values.

number

[0135] The quantity r is used here to overcome numerical problems for high exponent values, and is usually taken to be r=s, which approximates the maximum or minimum value, respectively. max or r=s min is set to.

[0136] 4.2 Singular Value-Based Design Responses For a given neighborhood of a finite element i ∈ {1,...,n}, we consider three measures:

number

[0137] The quantities on the left hand side of the first two equations in (13) locally describe the smallest and second smallest dimensions of the structuring element, respectively. These quantities are normalized by the largest possible dimension of the considered neighborhood, and are therefore invariant with respect to absolute neighborhood size. The third quantity on the left hand side of the bottom equation represents the relationship between the second largest and largest dimensions. Note that this third quantity is a relative quantity per definition and does not directly constrain the denominators in the first two equations in (13). Assuming that the design domain consists of all finite elements, the quantities on the left hand side of the first two equations in (13) are aggregated over all finite elements using a smooth maximum approximation when using these quantities to formulate unequal constraints. The quantity represented by the left hand side of the third equation in (13) is aggregated using a smooth minimum approximation when using these quantities to formulate more equal constraints. With q∈{min,mid,mid2max}, three design responses are introduced:

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[0138] Note that here we use smooth maximum and minimum approximations to obtain only three scalar design responses for the entire design domain, which are applied in the following sections to formulate constraints for the optimization.

[0139] 4.3 Constraints on Gradient-Based Optimization To locally control the dimensions of the structuring elements, the following constraints can be formulated:

[0140] By constraining the smallest dimension, 1D or 2D structural elements are locally forced. As a result, 3D fully solid material objects are excluded for a given neighborhood. This is achieved by applying the following constraints,

Number

Number

[0141] By constraining the smallest dimension and the second smallest dimension, 1D structural elements are locally forced. As a result, 2D and 3D objects are excluded for a given neighborhood. This is achieved by applying the following constraints.

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[0142] These constraints cause the design to become a lattice structure, avoiding bulky material concentrations and membrane-like structures. The value

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[0143] We locally enforce the 2D structuring element by constraining its smallest dimension and the relationship between its second largest and largest dimension. This excludes 1D and 3D objects. This is obtained by applying the following constraints:

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[0144] These constraints lead to a membrane-like design, avoiding bulky material concentrations and lattice-like structures. Again, the values ​​are:

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[0145] 4.4 Relationship between relative material fraction and geometric control 4 to 11 show the relative material fraction f of a sphere with radius R. s is partly related to the geometric control imposed by constraining the singular values ​​of the singular values ​​of the decomposition (SVD) for the design variables inside the sphere. The following guidelines for lattice and membrane structures apply in the present numerical application in 3D.

[0146] Lattice - Figures 7 and 8 show that the singular values ​​s min and s mid , which shows a particular exemplary lattice structure obtained by constraining , to be smaller than a selected value.

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[0147] Membrane - Figures 6 and 7 show the ratio of singular values

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[0148] A practical membrane structure would yield a low volume fraction for the design domain, e.g., R = 15t, s = 5%.

[0149] As a result, if the finite element model for the optimization specifies two elements across the thickness, the radius contains 30 elements making the radius quite large compared to the element size for the geometric controls imposed on the membrane layout. Note that, as described in Section 4.3, the thickness t can be forced by constraining:

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[0150] 4.5 Sensitivity analysis of SVD design response

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[0151] The derivative of the aggregated design response with respect to the dimensional measures defined in equations (13) and (14) yields:

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[0152] The smoothed singular value s defined in (12) smin , s smid , and s smax The derivative of the dimensional measure with respect to yields:

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[0153] The singular value s defined in (11) min , smid , and s max The derivative of the smoothed singular values ​​with respect to yields:

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[0154] Derivatives of singular values ​​with respect to relative density

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[0155] amount,

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[0156] 5. Working Example 5.1 2D cantilever beam To illustrate the properties of the proposed approach in comparison with classical topology optimization, a simple 2D example is applied below. Figure 19A shows a rectangular 2D design domain discretized by a 400x200 uniform grid using fully integrated 4-node plane stress elements in SIMULIA Abaqus (CP4S) of dimension 1x1. 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 shows the results of classical topology optimization using a material volume fraction of f=60%. Figure 19C shows the results of the lowest singular value using a diameter D=12 equivalent to 12 finite elements.

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[0157] FIG. 21 shows the optimization results when constraining the lowest singular value to 0.60, 0.50, and 0.40 when the diameter of the included design variables for decomposition (SVD) is D=12 and D=24, respectively. In FIG. 21, the geometric dimensions (a-c) are constrained by reducing the lowest singular value from 0.6 to 0.4 using a diameter of D=12 equivalent to 12 finite elements. This reduces the size of the members resulting in an increase in compliance C as well as a decrease in the relative material volume f. Structures (d-f) are optimized using the same lowest singular value constraint as for structures (a-c), but the optimization results are generated using larger diameters when calculating the lowest singular value of the decomposition (SVD) for the design variables. Comparing structures (a-c) with structures (d-e), it can be seen that the increase in diameter for calculating the lowest singular value causes larger member sizes with an expanded distance between members, which not only results in higher stiffness but also in lower mass. The model mesh, applied loads, and boundary conditions applied are defined as shown in FIG. 19A. Note that although symmetry is not enforced, the solution is still remarkably symmetric. Also note that no volume constraints are applied to the optimization in FIG. 21. The maximum member size obtained is given by multiplying the "diameter of the included design variables for the decomposition" by the "constraint value for the lowest singular value". However, small members appear and the length of the sensitivity filter gives the minimum member size.

[0158] The present approach appears to have an advantage over local volume constraint approaches: lattice members obtained using an approach based solely on local volume constraints typically suffer from size reduction when approaching intersections and junctions that make up the lattice, but the results of the present approach do not appear to suffer from this drawback.

[0159] In FIG. 22, all designs are optimized using a diameter D=12 and constraining the minimum singular value to be less than or equal to 0.6. The model mesh, applied loads, and boundary conditions applied are shown in FIG. 19A. In FIG. 22A, the designs have no volume constraint applied, whereas in FIG. 22B-D, the designs have a volume constraint that decreases from 60% to 40%. In FIG. 23, all designs are optimized using a diameter D=24 and constraining the minimum singular value to be less than or equal to 0.6. The model mesh, applied loads, and boundary conditions applied are shown in FIG. 19A. In FIG. 23A, the designs have no volume constraint applied, whereas in FIG. 23B-D, the designs have a volume constraint that decreases from 60% to 40%.

[0160] Figures 22 and 23 show the effect of adding a global volume constraint to the optimization formulation compared to the designs in Figures 22(a) and 23(a), respectively. In general, it can be seen that classical volume constraints for topology optimization work well even when constraining the lowest singular value. The material volumes of the designs shown in Figures 22(a) and 23(a), optimized without material volume constraints, are only slightly higher than the lowest singular value constraint. However, these designs have some intermediate densities relative to the low strain energy density region of the design domain. Figures 22(b) and 23(b) show that these intermediate densities are eliminated by adding a volume constraint similar to the lowest singular value constraint.

[0161] Figures 22 and 23 show the results of the proposed approach defined in equation (17) constraining the lowest singular value, which in this case results in a design with a satisfactory solid\void representation in a significantly fewer optimization iterations and a most important objective function value of compliance that is almost similar to the target value without the volume constraint. Thus, the present results in Figures 22 and 23, as well as additional numerical experiments, show that to obtain a solid\void 2D optimization solution, it is beneficial to apply a volume constraint that is the same as or lower than the constraint on the lowest singular value.

[0162] 5.2 Bone filling design This section considers the possibility of generating lattice-filled and membrane-filled structures for the femur, respectively. The femur finite element model depicted in FIG. 24 consists of 1,090,793 hexahedral elements (C3D8) each with a size of 1×1×1, resulting in 3,420,603 DOFs that are solved using an iterative algebraic multigrid solver. An elastic material with a Poisson's ratio of 0.3 has been applied. The 3D femur model is fully clamped at the bottom and two-point forces applied at the top are shown in the initial state. In this example, the elements on the outer surface are not part of the design domain.

[0163] 25A-E show topology optimized infills for the femur using a relative material mass fraction of 0.50. The outer surface is not part of the design domain. FIG. 25A represents a classical stiffness optimization under the condition of a mass constraint. FIG. 25B and FIG. 25C show the results for radii of 2.5 (in FIG. 25B) and 4.0 (in FIG. 25C), respectively.

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[0164] 25D and 25E show the results for radii of 2.5 (in FIG. 25D) and 4.0 (in FIG. 25E), respectively.

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[0165] These constraint singular values ​​are estimated to be around 0.58 using the lattice approximation with f=0.50 as outlined in Section 4.4.

[0166] For the optimization results shown for R=2.5 in FIG. 25B and R=4.0 in FIG. 25C, respectively, only the first singular value is constrained to be 0.50. The results shown in FIG. 25B and FIG. 25C are mainly due to the fact that the membrane component is expected to be stiffer than the lattice structure as a filler in the femur, and the thickness

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[0167] Subsequently, the first singular value is constrained to be 0.50 and the second singular value is constrained to be 0.60 to force a design with a lattice-filled component, as shown in FIG. 15(d) and FIG. 15(e), respectively. The optimized results in FIG. 15D and FIG. 15E have a separate lattice component as a filler, compared to the membrane-filled results in FIG. 15B and FIG. 15C. However, the membrane-filled structure has about 20% higher stiffness than the lattice-filled structure for both radii R=2.5 and R=4.0, respectively. As a result, the present geometric dimension control allows obtaining both membrane designs similar to the solutions obtained using local volume constraints, but also, importantly, a design that is a true lattice design with an open cell structure. The present approach for geometric dimension control thereby achieves an open lattice cell structure that is viable for additive manufacturing (AM) in terms of removing powder after fabrication as well as allowing another medium to flow through the optimized structure.

[0168] Although several embodiments of the present invention have been described, it will be understood that various modifications may be made without departing from the spirit and scope of the invention.

[0169] For example, some of the implementations disclosed herein may be directed to considering sensitivity-based solutions for structural optimization disciplines such as topology optimization, shape optimization, sizing optimization, and bead optimization. However, the technique is not theoretically limited to the field of structural optimization, but is also valid for multiphysics optimization, e.g., computational fluid dynamics (CFD), thermomechanical, electromechanical, fluid-structural, etc., where values ​​from a decomposition (SVD) of design variables (DVs) can be determined and used to define additional constraints to enforce geometric features.

[0170] It should be understood that the exemplary embodiments described herein may be implemented in many different ways. In some cases, the various methods and machines described herein may each be implemented by a physical, virtual, or hybrid general-purpose computer, such as the computer system or computer network environment described herein. The computer system may be converted into a machine that executes the methods described herein, for example, by loading software instructions into either memory or non-volatile storage for execution by the CPU. Those skilled in the art should further appreciate that the system and various components of the system may be configured to execute any embodiment or combination of embodiments of the invention described herein. Furthermore, the system may implement the various embodiments described herein utilizing any combination of hardware, software, and firmware modules internally or externally operably coupled to or embedded in the system. Furthermore, the system may be communicatively coupled to or embedded within a manufacturing device and configured to control the device to create physical objects as described herein.

[0171] Various aspects of the subject matter disclosed herein can be implemented in digital electronic circuitry, 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 subject matter disclosed herein can be implemented in one or more computer programs, i.e., one or more modules of computer program instructions encoded on a computer storage medium for execution by or for control of the operation of one or more data processing devices (e.g., processors). Alternatively, or in addition, the program instructions can be encoded on an artificially generated propagated signal, e.g., a machine-generated electrical, optical, or electromagnetic signal generated to encode information transmitted to a receiving device suitable for execution by a data processing device. The computer storage medium can be or can be included 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 is not considered to be merely a propagated signal, the computer storage medium may be a source or destination of computer program instructions encoded within an artificially generated propagated signal. The computer storage medium can also be, or be included in, one or more separate physical components or media, for example, multiple CDs, computer disks, and / or other storage devices.

[0172] Certain operations described herein (e.g., those depicted in Figures 3A, 3B, 15, and otherwise disclosed herein) can be implemented as operations performed by a data processing apparatus (e.g., a processor / specially programmed processor) on data stored on one or more computer-readable storage devices or received from other sources, such as the computer systems and / or network environments of Figures 1 and 2. The term "processor" (or the like) encompasses any kind of apparatus, device, and machine for processing data, including, as examples, programmable processors, computers, systems-on-chips, or a combination or combinations of the above. An apparatus can also include special purpose logic circuitry, such as a field programmable gate array (FPGA) or an application specific integrated circuit (ASIC). An apparatus can also include, in addition to hardware, code that creates an execution environment for the computer program, such as code that configures a processor firmware, a protocol stack, a database management system, an operating system, a cross-platform runtime environment, a virtual machine, or one or more combinations thereof. The apparatus and execution environment can implement a variety of different computing model infrastructures, such as web services, distributed computing, and grid computing infrastructures.

[0173] Although the present specification contains details of many individual implementations, these are not to be construed as limiting the scope of the invention or the scope of what may be claimed, but as descriptions of features specific to particular embodiments of a particular invention. Certain features described herein in the context of separate embodiments may also be implemented in combination in a single embodiment. Conversely, various features described herein in the context of a single embodiment may also be implemented in multiple embodiments separately or in any suitable subcombination. Also, although features may be described above as acting in a particular combination and may initially be claimed as such, in some cases one or more features from a claimed combination may be deleted from the combination, and the claimed combination may be directed to a subcombination or a variation of the subcombination.

[0174] Similarly, although operations may be described herein as occurring in a particular order or manner, this should not be understood as requiring that such operations be performed in the particular order shown, or in sequence, or that all illustrated operations be performed, to achieve a desired result. In certain situations, multitasking and parallel processing may be advantageous. Additionally, the separation of various system components in the embodiments described above should not be understood as requiring such separation in all embodiments, and the program components and systems described may generally be integrated together in a single software product or packaged into multiple software products.

[0175] Other embodiments are within the scope of the following claims.

Claims

1. 1. A computer-implemented method for automatically determining an optimized design for manufacturing a real-world object, comprising: defining, within a memory of a computer-based processor, a finite element model representing a real world object, the finite element model including a plurality of elements; evaluating, in a computer-based processor, a distribution of design variables across a neighborhood of the finite element model using singular value decomposition (SVD) to generate singular values ​​of the design variables at each respective element in 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 to generate defined optimization constraints; optimizing the finite element model with respect to the design variables by locally enforcing geometry of the real-world objects in the neighborhood based on the defined optimization constraints.

2. 2. The computer-implemented method of claim 1, further comprising: updating a property of a given one of the elements in the finite element model to reflect the optimized design variables to create an optimized model of the real-world object for manufacturing.

3. determining, at the computer-based processor, whether the optimization has converged; In response to the result of the determination of the computer-based processor, Start a new optimization cycle, or generating a final optimized design; 3. The computer-implemented method of claim 2, wherein the computer-based processor determines that the optimization has converged when constraints on associated design responses are satisfied and when associated objective functions are optimized.

4. The computer-implemented method of claim 3 , further comprising causing the real-world object to be manufactured according to the final optimized design.

5. The computer-implemented method of claim 1 , wherein the design variables represent relative density and the design responses represent compliance, stiffness, stress, strain, force, modal natural frequencies, and / or other typical structural design responses.

6. Defining the optimization constraints applying, in the computer-based processor, a smooth approximation function to each of the singular values ​​generated from the SVD to generate a smoothed approximation; identifying additional values ​​by running an SVD with all component design variable values ​​set to predetermined values ​​in the computer-based processor; and calculating the optimization constraints utilizing the smoothed approximation and the additional value.

7. evaluating the distribution of the design variables across the neighborhood of the finite element model using singular value decomposition (SVD), using the computer-based processor; reading, from the memory of the computer-based processor, data representing a geometric description of the finite element model from the memory; solving the equilibrium of the finite element model based on the data; modeling a first design response and an equilibrium design response and associated sensitivities with respect to the design variables based on the data; The computer-implemented method of claim 1 , further comprising determining a second design response and sensitivity of the calculated singular values ​​for geometric dimension control with respect to the design variables.

8. Defining the optimization constraints 8. The computer-implemented method of claim 7, further comprising applying, at the computer-based processor, the first and second design responses to define an optimization problem comprising constraints to be satisfied and an objective function to be optimized.

9. Defining the optimization constraints 10. The computer-implemented method of claim 8, further comprising solving the optimization problem with the computer-based processor using an optimization calculation based on user-defined values ​​of design objectives, the first and second design responses, and sensitivities of the first and second design responses.

10. 1. A computer-based system for automatically determining an optimized design for manufacturing a real-world object, comprising: a computer-based processor; a computer-based memory coupled to the computer-based processor; a computer-based memory storing data defining a finite element model representing a real world object, the finite element model including a plurality of elements; The computer-based memory stores computer-readable instructions that, when executed by the computer-based processor, cause the computer-based processor to: evaluating a distribution of design variables across a neighborhood of the finite element model using singular value decomposition (SVD) to generate singular values ​​of the design variables at each respective element in 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 to generate defined optimization constraints; optimizing the finite element model with respect to the design variables by locally enforcing the geometry of the real-world objects in the vicinity based on the defined optimization constraints.

11. The computer-based memory stores further computer-readable instructions that, when executed by the computer-based processor, cause the computer-based processor to:

11. The computer-based system of claim 10, further comprising: updating a property of a given one of the elements in the finite element model to reflect the optimized design variables to create an optimized model of the real-world object for manufacturing.

12. The computer-based memory stores further computer-readable instructions that, when executed by the computer-based processor, cause the computer-based processor to: determining whether the optimization has converged; Depending on the result of the determination, Start a new optimization cycle, or generating a final optimized design; 12. The computer-based system of claim 11, wherein the computer-based processor determines that the optimization has converged when constraints on associated design responses are satisfied and when associated objective functions are optimized.

13. The computer-based memory stores further computer-readable instructions that, when executed by the computer-based processor, cause the computer-based processor to: The computer-based system of claim 12 , further comprising: causing the real-world object to be manufactured according to the final optimized design.

14. The computer-based system of claim 10 , wherein the design variables represent relative density and the design responses represent compliance, stiffness, stress, strain, force, modal natural frequencies, and / or other typical structural design responses.

15. The computer-readable instructions for causing the computer-based processor to define the optimization constraints include: applying a smooth approximation function to each of the singular values ​​generated from the SVD to generate a smoothed approximation; identifying additional values ​​by running an SVD with all component design variable values ​​set to predetermined values ​​in the computer-based processor; 11. The computer-based system of claim 10, further comprising instructions to: calculate the optimization constraints utilizing the smoothed approximation and the additional value.

16. The computer-readable instructions for causing the computer-based processor to evaluate the distribution of the design variables across the neighborhoods of the finite element model using singular value decomposition (SVD), comprising: reading, from the memory of the computer-based processor, data representing a geometric description of the finite element model from the memory; solving an equilibrium of the finite element model based on the data; and modeling first and equilibrium design responses and associated sensitivities with respect to the design variables based on the data; and determining a second design response and a sensitivity of the calculated singular values ​​for geometric dimension control with respect to the design variables.

17. The instructions for causing the computer-based processor to define the optimization constraints include instructions for causing the computer-based processor to:

17. The computer-based system of claim 16, further comprising instructions for applying the first and second design responses to define an optimization problem comprising constraints to be satisfied and an objective function to be optimized.

18. The instructions for causing the computer-based processor to define the optimization constraints include instructions for causing the computer-based processor to:

20. The computer-based system of claim 17, further comprising instructions for causing the computer-based processor to solve the optimization problem using an optimization calculation based on user-defined values ​​of design objectives, the first and second design responses, and sensitivities of the first and second design responses.

19. A non-transitory computer-readable medium having computer-readable instructions stored thereon, the computer-readable instructions, when executed by a computer-based processor, causing the computer-based processor to: evaluating a distribution of a design variable across a neighborhood of a finite element model including a plurality of elements using singular value decomposition (SVD) to generate singular values ​​of the design variable at each respective element in 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; optimizing the finite element model with respect to the design variables by locally enforcing geometry of real-world objects in the vicinity based on the defined optimization constraints.

20. The non-transitory computer-readable medium further includes computer-readable instructions that, when executed by the computer-based processor, cause the processor to:

20. The non-transitory computer readable medium of claim 19, further comprising updating a property of a given one of the elements in the finite element model to reflect the optimized design variables, thereby creating an optimized model of the real-world object for manufacturing.

21. The computer-implemented method of claim 1, wherein the finite element model having an optimized topology is capable of digitally communicating with a real-world machine capable of manufacturing the real-world object based on the finite element model having the optimized topology.

22. The computer-based system of claim 10, wherein the finite element model having an optimized topology is capable of digitally communicating with a real-world machine capable of manufacturing the real-world object based on the finite element model having the optimized topology.

23. The non-transitory computer-readable medium of claim 19, wherein the finite element model having an optimized topology is capable of digitally communicating with a real-world machine capable of manufacturing the real-world object based on the finite element model having the optimized topology.

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