Average Acceleration for Sensitivity-Based Optimization

Abstract

To provide a computer-implemented method and a system for automatically determining optimized designs of real-world objects.SOLUTION: A method includes using a computer-based model representing a real-world object, to determine equilibriums of the real-world object across a plurality of time steps. Determining the equilibriums is determining velocities. Average acceleration of the real-world object is determined for each of a plurality of time windows, using the determined velocities. Sensitivity of each determined average acceleration is calculated. The determined average accelerations are used to define a constraint and an objective function. The computer-based model representing the real-world object is iteratively optimized, using the calculated sensitivity of each determined acceleration, with respect to the constraint and the objective function. The iterative optimization results in an updated computer-based model, representing the optimized design of the real-world object.SELECTED DRAWING: Figure 4

Description

[Technical field]

[0001] The present invention relates to the design of objects that are optimized through the use of average acceleration. [Background technology]

[0002] Many systems and programs are offered on the market for the design of parts using computer-aided design (CAD) or computer-aided engineering (CAE). These so-called CAD systems allow a user to build and manipulate complex three-dimensional models of objects, or assemblies of objects. Hence, CAD systems provide a representation of the modeled object using edges, lines, or in certain cases with faces or polygons. The lines, edges, faces or polygons can be represented in various ways, for example with non-uniform rational basis splines (NURBS).

[0003] These CAD systems manage parts or assemblies of parts modeled by objects, which are primarily specifications of geometric shapes. In particular, a CAD file contains the specifications from which geometric shapes are generated. From the geometric shapes, a representation is generated. The specifications, geometric shapes, and representations may be stored in a single CAD file or in multiple CAD files. CAD systems include graphic tools to show the modeled object to the designer, and these tools are specialized in displaying complex objects. For example, an assembly may contain thousands of parts. CAD systems may be used to manage models of objects, which are stored in electronic files.

[0004] The advent of CAD and CAE systems allows for a wide range of possible representations for objects. One such representation is the finite element model (FEM). An FEM, or other such CAD, CAE, or computer-based model, can be programmed in such a way that the model has the properties of the underlying object or objects that it represents. When programmed in such a way, an FEM, or other such computer-based model, can be used to simulate the object that it represents. For example, an FEM can be used to represent the interior cavity of a vehicle, the acoustic fluid surrounding a structure, and any real-world object and system. A given model represents an object, and when programmed accordingly, can be used to simulate the real-world object itself. For example, an FEM representing a stent can be used to simulate the use of the stent in an actual medical setting.

[0005] The computer-based model may be used to improve the design of the object that the model represents. Design improvements may be identified by using computer-based optimization techniques that run a series of simulations to identify changes to the design of the model, and therefore the underlying real-world object that the model represents. [Prior art documents] [Non-patent literature]

[0006] [Non-Patent Document 1] Pedersen, CBW (2004). Crashworthiness Design of Transient Frame Structures using Topology Optimization, Computer Methods in Applied Mechanics and Engineering, 193(6-8): 653-678. [Non-Patent Document 2] Pedersen, CBW (2002). On topology design of frame structures for crashworthiness. PhD thesis, Publisher Danish Center for Applied Mathematics and Mechanics. Technical University of Denmark. [Non-Patent Document 3] Pedersen, CBW (2002). Topology Optimization of Energy Absorbing Frames. WCCM V - Fifth World Congress on Computational Mechanics, Vienna, Austria. [Non-Patent Document 4] Kim, Y., Park, G. (2010). Nonlinear dynamic response structural optimization using equivalent static loads. Computer Methods in Applied Mechanics and Engineering. 199(9-12): 660-676. [Non-Patent Document 5] Jeong S., Yoon S., Xu S., Park G. (2010). Non-linear dynamic response structural optimization of an automobile frontal structure using equivalent static loads. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering. 224(4):489-501. [Non-Patent Document 6] Yoon J., Lee Y., Park S., Han Y., Park G. (2019). Crash optimization considering the head injury criterion. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering. 233(11):2879-2890. [Non-Patent Document 7] Stolpe, M., Verbart, A., Labanda, SR (2018). The equivalent static loads method for structural optimization does not in general generate optimal designs. Structural and Multidisciplinary Optimization, 58(1): 139-154. [Non-Patent Document 8] Roux, W., Yi, G., Gandikota, I. (2020). A spatial kernel approach for topology optimization. Computer Methods in Applied Mechanics and Engineering. 361(1):112794. [Non-Patent Document 9] Gandikota, I., Roux, W., Yi, G. (2021). Topology optimization of an automotive hood for multiple load cases and disciplines. Conference: 13th European LS-DYNA Conference 2021, Ulm, Germany. [Non-Patent Document 10] Gandikota, I., Yi, G., Roux, W. (2019). Crashworthiness and lightweight optimization of an automotive crash box using LS-TaSC. FEA Information Engineering Solutions. October 2019. [Non-Patent Document 11] Bujny, M., Aulig, N., Olhofer, M., Duddeck, F. (2018). Identification of optimal topologies for crashworthiness with the evolutionary level set method. International Journal of Crashworthiness 23(4): 395-416. [Non-Patent Document 12] Pedersen, NL (1998). Shape optimizing of a sleeping policeman. Trafik dage pa Aalborg Universitet, Aalborg, Denmark. [Non-Patent Document 13] Pedersen, NL (1998). Shape Optimization of a Vehicle Speed ​​Control Bump, Journal of Structural Mechanics, 26(3): 319-342. [Non-Patent Document 14] Diehl, T., Carroll, D., Nagaraj, B. (2000). Applications of DSP to explicit dynamic FEA simulations of elastically-dominated impact problems, Shock and Vibration, 7(3):167-177. [Non-Patent Document 15] Kleiber, M., Antunez, H., Hien, TD and Kowalczyk, P. (1997). Parameter Sensitivity in Nonlinear Mechanics, Theory and Finite Element Computations, John Wiley and Sons. [Non-Patent Document 16] Choi, K. K. and Kim, N. H. (2005). Structural Sensitivity Analysis and Optimization 2: Nonlinear Systems and Applications, Springer-Verlag New York Inc. [Non-Patent Document 17] Michaleris, P., Tortorelli, DA and Vidal, CA (1994). Tangent Operators and Design Sensitivity Formulations for Transient Non-linear Coupled Problems with Applications to Elastoplasticity, International Journal for Numerical Methods in Engineering 37: 2471-2499. [Non-Patent Document 18] Taylor, JE and Bendsoe, MP (1984). An interpretation of min-max structural design problems including a method for relaxing constraints. International Journal of Solids and Structures. 20: 301-314. [Non-Patent Document 19] Olhoff. N. (1989). Multicriterion structural optimization via bound formulation and mathematical programming. Structural Optimization. 1:11-17. [Non-Patent Document 20] Abaqus. (2021). SIMULIA User Assistance. Dassault Systemes. [Non-Patent Document 21] Tosca. (2021). SIMULIA User Assistance. Dassault Systemes. Summary of the Invention

[0007] While computer-based optimization methods exist for designing real-world objects, these existing methods are computationally expensive and could benefit from improvements. Embodiments provide such improvements. Embodiments are directed to automated real-world object design, optimization, and simulation that provides a computer-based design facility that determines optimized object designs through the use of average accelerations.

[0008] One such exemplary embodiment provides a computer-implemented method for automatically determining an optimized design of a real-world object. The method begins by defining, in a memory of a processor, a computer-based model representing the real-world object. The computer-based model is then used to determine an equilibrium (associated with Newton's equilibrium) of the real-world object for each of a number of time steps (or time periods). By determining the equilibrium, a velocity (along each axis or along a common axis) of the real-world object across the number of time steps is determined. One such exemplary embodiment continues by using the determined velocities to determine an average acceleration for each of a number of time windows defined across the number of time steps. Further, sensitivities of each determined average acceleration are calculated using the sensitivities of the determined velocities. The determined average accelerations are then used to define at least one of a constraint and an objective function. Finally, the calculated sensitivities of each determined average acceleration are used to iteratively optimize the computer-based model representing the real-world object with respect to the defined at least one of the constraint and the objective function. This iterative optimization updates a computer-based model that represents the optimized design of the real-world object.

[0009] In embodiments, the computer-based model may be any computer-based model known in the art, for example, in one embodiment, the computer-based model is a finite element model, a boundary element method, a finite difference method, a finite volume method, or a discrete element method.

[0010] According to one embodiment, using the determined velocities to determine the average acceleration for a given one of the time windows includes identifying (1) a velocity at a start point of the given window, and (2) a velocity at an end point of the given time window. Further, such an embodiment calculates the average acceleration for the given time window using (1) the identified velocity at the start point, (2) the identified velocity at the end point, and (3) the length of time for the given time window. Yet another embodiment calculates the sensitivity of the average acceleration for the given time window using the identified velocity at the start point and the identified velocity at the end point with respect to a given design variable.

[0011] In a further embodiment, the calculated sensitivities of each determined average acceleration and at least one of the discrete accelerations, the structural measure sensitivities, and the multi-physics measure sensitivities are used to iteratively optimize the computer-based model. In yet another embodiment, user-defined requirements are received, and the user-defined requirements and each determined average acceleration are used to define at least one of a constraint and an objective function.

[0012] In embodiments, the real-world object represented by the computer-based model can be any object, for example, in one embodiment, the real-world object is an automobile, a plane, an electronic device, a medical instrument, or armor, or other similar protective designs / objects.

[0013] A further embodiment is directed to a system for automatically determining an optimized design of a real-world object, the system including a processor and a memory having computer code instructions stored therein, in one such embodiment, the processor and memory are configured to cause the computer code instructions to cause the system to implement any embodiment, or combination of embodiments, described herein.

[0014] Another embodiment is directed to a non-transitory computer program product for automatically determining an optimized design of a real-world object. One such embodiment is directed to a computer program product executed by a server in communication with one or more clients over a network, the computer program product including program instructions that, when executed by a processor, cause the processor to implement any embodiment or combination of embodiments described herein. [Brief description of the drawings]

[0015] The foregoing will become apparent from the following more particular description of example embodiments, as illustrated in the accompanying drawings, in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the embodiments.

[0016] [Figure 1] FIG. 1 illustrates an example system capable of calculating vertical acceleration response and sensitivity, according to one embodiment. [Figure 2A] 2A-2C are plots illustrating vertical acceleration response and sensitivity, respectively, for the system shown in FIG. [Figure 2B] 2A-2C are plots illustrating vertical acceleration response and sensitivity, respectively, for the system shown in FIG. [Diagram 3] 13 is a plot showing the response of a structure optimized using 20 design points. [Figure 4]1 is a flowchart of a method for automatically determining an optimized design of a real-world object, according to one embodiment. [Diagram 5] 1 is a plot illustrating a method of calculating continuous average accelerations a-, according to one embodiment. [Figure 6A] 1 is a plot illustrating calculation of rolling average acceleration according to one embodiment. [Figure 6B] 6B is a plot illustrating the velocities used in calculating the rolling average acceleration of FIG. 6A. [Figure 7A] 1 is a plot illustrating a method of determining average continuous acceleration using two velocities over a time window that may be used in an embodiment. [Figure 7B] 11 is a plot illustrating a method of determining average continuous acceleration using two velocities over a time window that may be used in an embodiment. [Figure 7C] 11 is a plot illustrating a method of determining average continuous acceleration using two velocities over a time window that may be used in an embodiment. [Figure 7D] 11 is a plot illustrating a method of determining average continuous acceleration using two velocities over a time window that may be used in an embodiment. [Figure 8A] 1 is a plot illustrating a comparison between optimizing a continuous average acceleration calculated using velocities at discrete times where direct accelerations are evaluated for optimization, and rigorously optimizing acceleration using additional discrete direct acceleration points for optimization. [Figure 8B] 11 is a plot illustrating a comparison between optimizing a continuous average acceleration calculated using velocities at discrete times where direct acceleration is evaluated for optimization, and rigorously optimizing acceleration using additional discrete direct acceleration points for optimization. [Figure 9]FIG. 1 is a flow diagram illustrating an iterative design process for calculating and utilizing rolling average acceleration, according to one embodiment. [Figure 10] 1 is a simplified diagram illustrating an environment including real-world objects that may be optimized using embodiments. [Figure 11] FIG. 11 is a plot showing a comparison of acceleration curves and displacement intrusion for the environment of FIG. 10. [Figure 12] FIG. 11 is a plot showing discrete displacements, discrete accelerations, and continuous average accelerations in time for optimization of the environment of FIG. 10, where the maximum of the discrete accelerations a(ti) and continuous average accelerations a−(ti) are minimized and displacement intrusion is constrained. [Figure 13A] FIG. 1 illustrates a simplified pedestrian and vehicle impact model applied for non-parametric shape optimization using an embodiment. [Figure 13B] FIG. 1 illustrates a simplified pedestrian and vehicle impact model applied for non-parametric shape optimization using an embodiment. [Figure 13C] FIG. 1 illustrates a simplified pedestrian and vehicle impact model applied for non-parametric shape optimization using an embodiment. [Figure 13D] FIG. 1 illustrates a simplified pedestrian and vehicle impact model applied for non-parametric shape optimization using an embodiment. [Figure 13E] FIG. 1 illustrates a simplified pedestrian and vehicle impact model applied for non-parametric shape optimization using an embodiment. [Figure 14A] 1 is a plot showing optimization iteration history for one embodiment that minimizes maximum acceleration. [Figure 14B]1 is a plot showing an optimization iteration history for one embodiment that minimizes maximum acceleration. [Figure 14C] 1 is a plot showing an optimization iteration history for one embodiment that minimizes maximum acceleration. [Figure 15A] 11 is a plot showing optimized acceleration determined using an embodiment as a function of both time and displacement. [Figure 15B] 11 is a plot showing optimized acceleration determined using an embodiment as a function of both time and displacement. [Figure 15C] 11 is a plot showing optimized acceleration determined using an embodiment as a function of both time and displacement. [Figure 16A] 11 is a plot illustrating an optimization design of a nodal non-parametric shape determined using an embodiment. [Figure 16B] 11 is a plot illustrating a nodal non-parametric shape optimization design determined using an embodiment. [Figure 16C] 11 is a plot illustrating a nodal non-parametric shape optimization design determined using an embodiment. [Figure 17A] FIG. 1 illustrates steps of bonnet shell model optimization implemented using an embodiment. [Figure 17B] FIG. 1 illustrates steps of bonnet shell model optimization implemented using an embodiment. [Figure 17C] FIG. 1 illustrates steps of bonnet shell model optimization implemented using an embodiment. [Figure 17D] FIG. 1 illustrates steps of bonnet shell model optimization implemented using an embodiment. [Figure 18]1 is a plot depicting optimization iteration history and accelerations for a sizing thickness optimized design and a bead nodal optimized design determined using an embodiment. [Figure 19A] FIG. 13 illustrates an optimized design with sizing element thickness and bead node determined using an embodiment. [Figure 19B] FIG. 13 illustrates an optimized design with sizing element thickness and bead node determined using an embodiment. [Figure 20] FIG. 1 is a simplified block diagram illustrating a computer system for automatically determining an optimized design of a real-world object, according to one embodiment. [Figure 21] FIG. 1 is a simplified block diagram illustrating a computer network environment in which an embodiment of the present invention may be implemented. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0017] Exemplary embodiments are described below.

[0018] Although the term "sensitivities" is used herein, it should be noted that sensitivities are mathematically equivalent to derivatives, and the term "sensitivities" is commonly used in multidisciplinary optimization.

[0019] The embodiments provide an improvement to automated product design based on optimization and simulation. Generally, sensitivity-based optimization of acceleration over time using sensitivities is difficult because acceleration is displacement differentiated twice with respect to time, and acceleration sensitivities are further differentiated once with respect to design variables. As a result, both acceleration and corresponding acceleration sensitivities are noisy, both physically and numerically. See non-patent documents 1, 2, and 3 (parenthetical numbers in this document refer to the enumerated list of non-patent documents below). The aforementioned noise is commonly observed due to higher order, highly-localized, oscillations in acceleration and its sensitivity. FIG. 1, as well as FIGS. 2A and 2B, illustrate this problematic noise. Specifically, FIG. 1 illustrates a simple fixed beam system 100, in which a beam having a cross section 104 is composed of elements 101-102 and a mass 103, the mass 103 moving at a speed of 20 m / s. 2A and 2B are plots 220 and 221 showing a comparison of acceleration sensitivity versus time and acceleration versus time, respectively, as determined in a sensitivity-based optimization of system 100. Enlarged views 222 and 223 show large and highly localized oscillations in acceleration (FIG. 222) and corresponding sensitivity (FIG. 223) in the context of optimization of beam system 100.

[0020] Therefore, directly applying these accelerations and sensitivities in a sensitivity-based optimization has many drawbacks, such as requiring additional optimization iterations and / or resulting in local suboptimal minimal optimized designs.

[0021] Among other applications, many designers are requesting current acceleration optimization and corresponding non-parametric optimization capabilities for crashworthiness designing. Competitive and academic publications have proposed various approaches in handling acceleration using sensitivity to obtain crashworthiness optimized designs, but none have the current generality and potential industrial impact for acceleration design.

[0022] Non-Patent Documents 1, 2, and 3 propose two approaches to reduce the effects of vibration and noise in acceleration and sensitivity for sensitivity-based optimization. For example, smooth acceleration and acceleration sensitivity for optimization have been achieved by applying numerical damping, which adds high damping to the main dynamic solution, thereby suppressing transient higher order, highly-localized oscillations. This has the disadvantage that the physical accuracy of the main dynamic solution may be compromised, which may cause failures in the optimization design. Furthermore, it may be difficult to select a feasible numerical damping for a particular implementation, and many trial-and-error studies need to be carried out to determine a feasible damping.

[0023] Additionally, indirect slope control has been applied to indirectly control the slope of the acceleration curve. This is done by adding acceleration design points close to existing design points relative to the design points where acceleration is measured for use in the optimization formulation. However, this formulation of acceleration in optimization does not represent or capture the acceleration characteristics between acceleration design points. Plot 330 in FIG. 3 illustrates this problem. Plot 330 illustrates a comparison of acceleration 331, i.e., deceleration and displacement 332. A close-up 333 of plot 330 shows additional design points 334a-334b and existing design points 335a-335b. In close-up 333, it can be seen that design points 334a-334b and 335a-335b do not capture the acceleration given by curve 336, for example, which does not have damping in the numerical modeling. Furthermore, curve 337 has an infeasible maximum numerical damping, yet does not capture the local accelerations between design points 334a and 335a.

[0024] Non-Patent Documents 4, 5, and 6 propose a fundamentally different approach using the so-called equivalent static loads (ESL), which maps the dynamic response of transient modeling to an approximated linear static model where sensitivities are calculated. Non-Patent Documents 4 and 5 also apply accelerations in sensitivity-based optimization, but the accelerations and their sensitivities are defined using displacements from the approximated linear static modeling. The inventive approach described here directly applies the transient responses and their sensitivities of the dynamic analysis for optimization, and no linear static approximation is applied. Non-Patent Document 7 includes concerns about whether the ESL optimization approach works in general for sensitivity-based transient optimization.

[0025] [8] proposes a fundamentally different approach compared to [9] and

[10] for acceleration optimization. The approach in [8] reduces the design modeling of the design space (reducing the number of design variables), which is defined as "having only few spatial variables". Having only a few spatial variables allows [8] to calculate the acceleration numerically using finite differences. Furthermore, the introduction of spatial variables significantly reduces the noise by averaging a very large number of design variables. However, this approach does not guarantee a reduction in the noise level in time and in terms of acceleration and its sensitivity to highly localized vibrations. Moreover, this approach is not valid for the innovative method described herein, since a large number of design variables are calculated analytically or semi-analytically, thereby producing a better performance of the optimized system with a high level of design freedom.

[0026] There are non-parametric optimization methods that are not based on acceleration sensitivity but still include acceleration in the optimization formulation, see Non-Patent Document 11. However, these methods are not related to the family of acceleration optimization approaches based on acceleration sensitivity as described herein.

[0027] The aforementioned optimization approaches have previously applied transient continuum finite element modeling for transient simulation. However, multibody simulation (MBS) is a method of numerical simulation in which a multibody system is composed of various rigid or flexible bodies. Non-Patent Documents 12 and 13 describe acceleration optimization for MBS systems using acceleration sensitivity. In Non-Patent Documents 12 and 13, acceleration is directly applied in the optimization without any additional consideration because the acceleration of the MBS systems described in Non-Patent Documents 12 and 13 appears rather smooth and has low levels of noise and highly localized vibration. Therefore, the acceleration characteristics of these MBS are different from the acceleration responses considered in the inventive method described herein, which have high levels of noise and vibration. However, if the MBS system needs to have high levels of noise and vibration in acceleration (e.g., due to contacts and bearings), the embodiments described herein can also be applied to the acceleration optimization of the MBS system.

[0028] Typically, many situations related to crashworthiness events, impact events, drop events, etc. from the transportation and mobility sector (e.g., automobiles, etc.), aeronautics and defense sector (e.g., aircraft, etc.), life science sector (e.g., medical devices, etc.), and high-tech sector (e.g., mobile phones and laptops, etc.) are applied by engineers to address the following design criteria in time.

[0029] Engineers apply conditions such as impact resistance, shock events, drop events, etc. to minimize maximum intrusions (maximum elastic or plastic displacements penetrating into a structure and thereby impacting critical components or humans). Engineers apply conditions such as impact resistance, shock events, drop events, etc. to minimize material failure at local material point levels. For example, local material tears, substances leaking from tanks, or insulating materials are torn or penetrated or shorted. Engineers apply conditions such as impact resistance, shock events, drop events, etc. to minimize maximum acceleration. Maximum acceleration is often a performance measure that deals with the overall biomechanical effects on the human body. In addition, other criteria determined by acceleration are applied to measure biomechanical effects, for example, head injury criterion (often called HIC). The use of these biomechanical criteria sometimes requires the use of detailed and realistic human body models and the knowledge of the specific masses and stiffnesses of the body parts and joints. Therefore, in the early design stages, maximum acceleration is often considered as a design measure. Typically, if the maximum acceleration in time is globally minimized, then many of these criteria are also indirectly minimized.

[0030] Among other applications, the inventive approach described herein addresses minimizing the maximum acceleration in time. The embodiments can be applied in conjunction with the design criteria of minimizing the maximum penetration into the structure, meaning that both criteria are considered in the optimization. Furthermore, the embodiments can be applied indirectly to address design criteria requirements for minimizing material failure at the local material point level. Furthermore, the maximum acceleration is easier for engineers to calculate compared to calculating and defining material failure at the local material point level.

[0031] 4 is a flow chart illustrating a method 440 for automatically determining an optimized design of a real-world object. The method 440 is computer-implemented and, as such, executed by one or more processors.

[0032] Method 440 begins by step 441 of defining, in a processor memory, a computer-based model representing a real-world object. Then, in step 442, method 440 uses the defined computer-based model to determine the equilibrium (associated with Newton's theory of equilibrium) of the real-world object for each of a number of time steps (or periods). This equilibrium determination step 442 determines the velocity (along each axis or along a common axis) of the real-world object over the number of time steps. Then, in step 443, an average acceleration for each of a number of time windows is determined. In one such embodiment of method 440, the time windows are defined over a number of time steps. To illustrate, consider an embodiment in which, for an optimization of a 30 second period, step 442 determines the equilibrium at 1 second time increments. In one such embodiment, six 5 second time windows may be defined, and in step 443, an average acceleration is determined for each of the 5 second time windows using the start and end velocities of each 5 second time window. For example, an average acceleration for a time window from 25 seconds to 30 seconds is calculated in step 443 using the 25 second velocity and the 30 second velocity. Subsequently, in step 444, a sensitivity of each determined average acceleration (determined in step 443) is calculated using the sensitivity of the determined velocity (determined in step 442). Then, in step 445, the determined average acceleration from step 443 is used to define at least one of a constraint and an objective function. In step 446, the calculated sensitivity of each determined average acceleration from step 444 is used to iteratively optimize a computer-based model representing the real-world object with respect to the defined at least one of the constraint and the objective function (from step 445). The iterative optimization step 446 updates the computer-based model representing the optimized design of the real-world object.

[0033] In an embodiment of method 440, the computer-based model defined in step 441 can be any computer-based model known in the art. For example, in one embodiment, the computer-based model is a finite element model, a boundary element method, a finite difference method, a finite volume method, or a discrete element method. In an embodiment, the real-world object represented by the computer-based model can be any object. For example, in one embodiment, the real-world object is an automobile, an airplane, an electronic device, a medical device, or an armored vehicle, or other similar protection design / object.

[0034] According to one embodiment of the method 440, using the determined velocities to determine 443 an average acceleration for a given one of the time windows includes identifying (1) a velocity at a start point of the given window, and (2) a velocity at an end point of the given time window. See steps 602 and 603 in FIG. 6B. Further, such an embodiment uses (1) the identified velocity at the start point, (2) the identified velocity at the end point, and (3) the length of time in the given time window to calculate (step 443) an average acceleration for the given time window. See FIG. 5. Yet another embodiment of the method 440 uses the identified velocity at the start point and the identified velocity at the end point to calculate the sensitivity of the average acceleration for the given time window with respect to a given design variable. See FIGS. 11 and 12. One embodiment of the method 440 determines the average acceleration in step 443 using the functionality described later in this specification in connection with FIG. 5.

[0035] In another embodiment, in step 445, user-defined requirements are received and the user-defined requirements and the determined average accelerations are used to define at least one of a constraint and an objective function. According to yet another embodiment of the method 440, the calculated sensitivities of the determined average accelerations and at least one of the discrete accelerations, structural measure sensitivities, and multi-physics measure sensitivities are used in step 446 to iteratively optimize the computer-based model.

[0036] The embodiments disclosed herein introduce an additional average acceleration for sensitivity-based optimization, i.e., “continuous average acceleration”, where accelerations at discrete time points are also applied to the design optimization in the objective function and / or constraints.

[0037] FIG. 5 shows the continuous average acceleration over a time window using the velocity at discrete times.

[0038]

number

[0039] 5 is a plot 550 showing, in one exemplary embodiment, how the average acceleration 505a-505e is calculated, where acceleration is also evaluated for optimization. Plot 550 shows time windows 504a-504e, over which the average acceleration 505a-505e is given by equation 506. To illustrate one embodiment, consider time window 505c. The length of time window 505c is (t i+1 -t i ), and the velocity 501 at the start of the time window 505c is v(t i ), and the velocity 502 at the end of the time window 505c is v(t i+1 For time window 505c, the average acceleration is given by:

[0040]

number

[0041] The method illustrated in plot 550 may be implemented in step 443 of method 440 described later in this specification. Plot 550 shows that the time discretization on acceleration can be coarse and still capture the overall acceleration characteristics.

[0042] Continuous average acceleration

[0043]

number

[0044] The measure of, and possibly other measures, may be, in an embodiment (e.g., in step 445 of method 440 described herein above in connection with FIG. 4), * can be applied to define an optimization problem consisting of constraints c that must be satisfied as: and an objective function f to be optimized (minimized or maximized). An example of an objective function and constraints is given by:

[0045]

number

[0046] Additionally, optimization of the continuous average acceleration can be combined (e.g., in step 446 of method 440) with other measures to which sensitivity exists. For example, such other measures may include direct measures of the real-world object represented by the computer-based model (e.g., mass, center of gravity, etc.) or measures determined by the results of a structural and / or multiphysics solution for equilibrium of the model (e.g., discrete accelerations, stresses, displacements, forces, modal natural frequencies, temperatures, etc.).

[0047] 6A and 6B show continuous average acceleration for optimization according to one embodiment.

[0048]

number

[0049] 6A and 6B are plots 660 and 661, respectively, illustrating the advantages of 601. Specifically, plots 660 and 661 illustrate the running average acceleration 601 calculated strictly using the velocity at a start point 602 and the velocity at an end point 603 with respect to a time window 604.

[0050] 7A-7D are plots 770a-770d illustrating average continuous accelerations 700a-700d using two velocities 701a-701d and 702a-702d over time windows 703a-703d. Plots 770a-770d show that the continuous average accelerations 700a-700d calculated using velocities 701a-701d and 702a-702d are independent of additional discrete acceleration time points, e.g., 704a-704n. As a result, the average continuous accelerations 700a-d over time windows 703a-703d are consistent. 7B-7D further show that selecting additional acceleration points 704a-704n in the time windows 703b-703d does not provide better accuracy without ensuring that the acceleration peaks, e.g., 705a-705f, in the time windows 703b-703d are captured in a consistent manner. Thereby, averaging continuous accelerations 700a-700d for a given time window 703a-703d typically reduces the effects of highly localized vibrations, noise in acceleration, and their sensitivity to the optimization. As such, better convergence for the optimization iteration history is frequently observed, and the resulting designs have improved structural properties with respect to acceleration.

[0051] More specifically, the average continuous acceleration 700a in Fig. 7A does not depend on the values ​​within the time window 703a, e.g., 707a-707b. Figs. 7B-7D show that if additional discrete acceleration points, e.g., 704a-704n, within the time windows 703b-703d are considered, these additional points 704a-704n will not capture the high or low peak accelerations, e.g., 705a-705f. Thereby, using the start point velocities 701a-701d and the end point velocities 702a-702d to determine the average continuous acceleration 700a-700d has better accuracy and optimization robustness compared to using the discrete acceleration points 704a-704n within the time windows 703b-703d.

[0052] FIG. 8A is a plot 880a showing acceleration 881a compared to time 882a, with discrete acceleration points indicated by dots 883a, and continuous average acceleration indicated by bars 884. FIG. 8B is a plot 880b showing acceleration 881b compared to time 882b, with discrete acceleration points indicated by dots 883b, and additional acceleration points indicated by open circles 885. Plot 880b illustrates a conventional approach to capture acceleration 881b over time 882b by applying an additional discrete acceleration 885. In plot 880b, the additional acceleration 885 can be used in a conventional sensitivity-based optimization in an attempt to capture all characteristics of acceleration 881b as a function of time 882b for design optimization. In comparison, plot 880a in FIG. 8A shows an enhanced optimization formulation that includes a continuous average acceleration 884 to account for all characteristics of acceleration 881a at time 882a for design optimization. By using the average acceleration 884 in the optimization formulation, such an embodiment is less dependent on time discretization since the number of time points and the location of these time points within the time window have less impact on the optimization iteration history and the final optimized design.

[0053] Another typical approach is to apply additional discrete acceleration, e.g., 885, numerical averaging or filtering (also called digital signal processing, see Non-Patent Document 14) over time to remove high frequency components, often considered as "noise". However, numerical averaging or filtering of discrete acceleration is suboptimal compared to the embodiments described herein. First, numerical acceleration averaging or filtering of discrete acceleration can be highly dependent on the time discretization within the time window, e.g., the number of time points within the time window and the location of these time points. These values ​​then affect the optimization iteration history and the final optimized design. This is especially true when the acceleration has highly localized vibrations and noise. If the acceleration has highly localized vibrations and noise, numerical acceleration averaging or filtering of discrete acceleration for a given time window can easily contain local misleading acceleration values. Second, the numerical performance of the embodiments described herein can be significantly improved because the embodiments use velocity values ​​at the start and end of each time window 703. In comparison, numerical averaging or filtering of discrete accelerations may require many discrete points in time within a time window to capture the acceleration characteristics.

[0054] Instead of using many discrete acceleration points, or instead of numerically and / or approximately computing the acceleration integral of the average acceleration over a discrete time window, embodiments may use a time window, e.g., the start t i , e.g., 50, and end t i +1, for example, continuous average acceleration using a speed of 502

[0055]

number

[0056] Define the design response of, for example, 505c. i From i+1The analytical integral of acceleration for a time interval up to t i and t i This is feasible because for discrete speeds at +1 we have the following relationship:

[0057]

number

[0058] Then, the continuous average acceleration as the design response

[0059]

number

[0060] is calculated as follows:

[0061]

number

[0062] This results in a continuous average acceleration

[0063]

number

[0064] The design response of t i From i+1 It includes the average acceleration value of the underlying acceleration curve over a given time interval up to and including , thus capturing the acceleration peak behavior.

[0065] In one embodiment, the average acceleration is the average value of the underlying acceleration curve between two consecutive time points. This average acceleration is calculated from the starting point t for a given time window. i and end point t i+1 and the time span for the time window, Δt=t i+1 -t iAs a result, such an embodiment does not require values ​​within a time window to calculate the running average acceleration.

[0066] In one embodiment, the sensitivity of the average acceleration to a given design variable x j According to one embodiment, the sensitivity of the average acceleration is applied in a mathematical iterative design optimization according to principles known to those skilled in the art.

[0067] As described herein, the continuous average acceleration is calculated from a starting point t i and end point t i+1 t is a function of the velocity at the starting point t . Thus, according to one embodiment, the sensitivity of the running average acceleration is i and end point t i+1 is a function of the sensitivity to velocity at

[0068]

number

[0069] Thus, the sensitivities are mathematically equivalent to total derivatives. Non-Patent Documents 1 to 21 are often associated with structural mechanics applications, and similarly, the embodiments can be used in structural mechanics applications. However, it should be noted that the embodiments are not limited to structural mechanics, but rather are of full generality and can be applied to any application involving acceleration. Both acceleration and velocity sensitivities can be calculated using finite difference methods, semi-analytical, and analytical approaches. Non-Patent Documents 1, 2, 3, 15, 16, and 17 address sensitivities for transient and dynamic modeling.

[0070] 9 illustrates a sensitivity-based iterative design process 900 according to one embodiment. The optimization process 900 includes running average acceleration using velocity from a transient structural analysis. In addition, velocity sensitivity is calculated, whereby running average acceleration sensitivity is also included in the process 900.

[0071] The process 900 begins with defining or otherwise obtaining a model representing an initial design, i.e., a non-optimized design, of the object, in step 901. The equilibrium of the model is solved in step 902. Subsequently, in step 903, the acceleration at a given discrete time instant, and the running average acceleration, are calculated using the equilibrium 902 velocity for a given time window.

[0072] Further, in step 90, velocity sensitivities are calculated for the design variables, and these velocity sensitivities are used to calculate the sensitivity of continuous average acceleration to the design variables. Non-Patent Documents 1, 2, 3, 15, 16, and 17 address sensitivities for transient and dynamic modeling and describe sensitivity calculation methodologies that may be used for step 904.

[0073] Subsequently, in step 905, the acceleration at given discrete times and the running average acceleration are applied to define, in whole or in part, an optimization set of objective functions and / or constraints.

[0074] Subsequently, the discrete acceleration and continuous average acceleration measures, as well as other possible measures, are applied to define an optimization problem consisting of constraints that must be satisfied and an objective function to be optimized (minimized or maximized). The optimization problem is solved using mathematical programming 907, i.e., optimization calculations. According to one embodiment, the mathematical programming is strictly based on user-defined design objectives, measures, and their sensitivity values. Furthermore, in one embodiment, the optimization of the discrete acceleration and continuous average acceleration can be combined with other measures 906, for which there are sensitivities in step 907. Examples of measures that may be utilized are direct measures of the model (e.g., mass, center of gravity, etc.) or measures determined by the results of a structural solution and / or a multiphysics solution to the equilibrium of the model (e.g., stress, displacement, force, modal natural frequencies, temperature, etc.).

[0075] In mathematics, computer science, and operations research, mathematical programming, alternatively called mathematical optimization or simply optimization, is the process of selecting the best solution (with respect to some criteria) from some set of available alternatives. Embodiments of method 900 may use any such mathematical programming method as known in the art.

[0076] After the mathematical programming in step 907, in step 908, a new physical model is generated for the next optimization iteration based on the design variables found in step 907. The iterative design process is a process of selecting a new improved solution (with respect to some objectives and constraints) for each optimization iteration. In some cases, the design variables determined in step 907 and the physical model variables updated in step 908 may be the same as, for example, thickness design variables for sizing optimization, but may be different, for example, density topology optimization, where the design variables are relative densities mapped to physical densities. In an exemplary embodiment where the physical model variables and the design domain variables are the same, the physical model in step 908 is simply obtained as the output of the mathematical programming 907. Otherwise, in step 908, the design variables are interpreted as physical model variables through the use of filters as known in the art.

[0077] If the model determined in step 908 has converged to some criteria, e.g., user-defined criteria, the process 900 proceeds to step 909 to achieve a final optimized design of the real-world object. If the optimization has not converged, a new optimization iteration is initiated and the method 900 returns to step 902 and repeats steps 902-908 with the updated model (from step 908). The optimization workflow 900 outputs a final design in step 909 when the optimization has converged. In one embodiment, the optimization workflow 900 is converged when the convergence criteria for the change in the objectives is less than, e.g., 0.1% and the change in the design variables is less than, e.g., 0.5%. If the design responses of the continuous average accelerations determined in step 905 and their sensitivities are not applicable, additional optimization iterations are required and / or local suboptimal minimum designs are frequently determined.

[0078] The continuous average acceleration and methods described herein, e.g., 440 and 900, can be applied to many different optimization scenarios. FIG. 10 is a representative example demonstrating the continuous average acceleration applied for crashworthiness optimization of a bonnet 1000 against a pedestrian head 1001 impact. The overall goal is to minimize the maximum acceleration in time for the head 1001 by optimizing the structural layout of the bonnet 1000 and at the same time suppressing the intrusion of the head 1001 into the structure (bonnet 1000) as measured by displacement in time. The following description of FIGS. 11-19 outlines an exemplary optimization of the environment in FIG. 10 implemented using the embodiments described herein, e.g., methods 440 or 900.

[0079] Figure 11 is a plot 1100 showing acceleration curves 1101a-1101c for a given design compared to the displacement intrusion 1102 of the head 1001 for three different structural designs for the bonnet 1000 shown in Figure 10. If the structural bonnet 1000 design is too stiff, the acceleration is too high for the head 1001. If the structural bonnet design 1000 is too soft and therefore highly flexible, the intrusion displacement is too high for the head. Therefore, optimization should result in a viable design where the intrusion displacement 1102 is constrained to the maximal allowable intrusion displacement and the maximum acceleration is minimized.

[0080] In the experimental optimization setup of the environment in FIG. 10, the discrete acceleration in time and the continuous average acceleration in time are formulated using the so-called MIN-MAX formulation (see Non-Patent Documents 18 and 19), and the objective function is i ) and I-1 consecutive average accelerations

[0081]

number

[0082] Using this formulation, the user can move in a discrete acceleration a(t i ) and continuous average acceleration

[0083]

number

[0084] 12 illustrates the inclusion of additional discrete design response points 1222 such that the design iterations may determine an optimized design where the targeted maximum acceleration is minimized and the targeted maximum allowable penetration 1221 is not violated. i ) ensures that the maximum permissible intrusion 1221 of the head 1001 into the structure is not exceeded. i * The design variable x k ∈{x} is the upper bound x such that the vector {x} contains all the design variables. k,max and lower bound x k,min where K is the number of design variables. Since structural non-linear equilibrium (see Non-Patent Document 20) is satisfied, the residuals {R} are i Therefore, the optimization formulation is written as follows:

[0085]

number

[0086] The above MIN-MAX formulation is solved as a standard optimization problem where the objective function is remodeled into a set of constraints, with bounds formulated as in

[18] and

[19] . The non-parametric design variables {x} are updated using an iterative optimization design workflow, e.g., 900, implemented using

[20] and

[21] , and a sensitivity-based optimization method is applied using mathematical programming. The sensitivities are calculated using the adjoint sensitivity method (see

[15] ,

[16] , and

[17] ), implemented in the kernel of

[20] .

[0087] For the finite element model of structural equilibrium {R}, nonlinearities such as large deformations, nonlinear constituent materials, and contact can be included in the finite element model and integrated into the finite element solution of

[20] .

[0088] 13A-E further illustrate the optimization of the head 1001 and bonnet 1000 environment in FIG. 10 implemented using an embodiment. FIG. 13A depicts a simplified flat plate 1330 and rigid spherical mass 1331 model, which is a representative simplified example for optimizing the design of the bonnet 1000 (represented by plate 1330) for the impact of a pedestrian's head 1001 (represented by rigid spherical mass 1331) for a given initial velocity of 10 m / s. The continuous finite model 1332 (element type C3D8 in Non-Patent Document 20) is shown in FIG. 13B. The continuous finite model 1332 includes boundary conditions and nodes, and the location of the nodes on the lower surface (opposite the contact surface for the head impact) is a nodal design variable, e.g., 1333 shown in FIG. 13C. The material modeling is elastic-plastic and is given by the plot 1334 shown in FIG. 13D for the plate 1330, while the head 1331 is modeled using rigid elements. The acceleration at the center of the head 1331 is applied for the non-parametric optimization. An introduction to standard non-parametric shape optimization is given in Non-Patent Document 21. FIG. 13E shows the representations 1335a-1335e showing the impact deformation of the head 1331 of the initial plate 1330 (i.e. before the iterative optimization process) at 0.0 seconds (1335a), 0.006 seconds (1335b), 0.008 seconds (1335c), 0.011 seconds (1335d) and 0.025 seconds (1335e).

[0089] The analytically "ideal" theoretical constant acceleration is shown in Figures 14A-14C and 15A-15C. The analytically "ideal" theoretical constant acceleration is calculated for a given displacement penetration constraint (see, for example, Non-Patent Document 2) assuming that the acceleration is constant in time since kinetic energy is absorbed by mechanical work. This analytically "ideal" theoretical constant acceleration is used herein as a reference to illustrate the performance of optimization formulations and corresponding optimized designs determined using the embodiments.

[0090] Figures 14A-14C illustrate three optimization setups for optimizing the bonnet 1000 and head 1001 impact design to minimize the maximum acceleration tested to illustrate the benefits of the embodiments. The three optimization setups were defined in the following plots: 1440: acceleration point, I=25; 1442: additional acceleration point, I=100; and 1443, including average acceleration and acceleration point (according to embodiments described herein), I=25. The results for these three different optimization setups are shown in Figures 14A-14C, 15A-15C, and 16A-16C.

[0091] Specifically, plots 1440, 1442, and 1443 illustrate the process of minimizing the maximum acceleration by the iterative optimization method described herein. Plots 1440, 1442, and 1443 depict the maximum acceleration 1444a-1444c per each design iteration 1445a-1445c, illustrating the effect of simply adding additional acceleration points (plot 1442 in FIG. 14B) versus using the average acceleration in the optimization equation (plot 1443 in FIG. 14C) on the speed of design convergence based on the number of optimization iterations. FIG. 14A depicts the maximum acceleration 1446a of a model 1440 using an acceleration point of I=25, which exhibits an iteration history 1448a that includes some high oscillations, and requires 60 design iterations 1449a to converge the maximum acceleration 1446a to the ideal acceleration 1447a for the model. FIG. 14B illustrates that adding additional acceleration points to plot 1442 reduces the oscillations in the optimization iteration history 1448b, but requires a similar number of design iterations 1449b as plot 1443 for the maximum acceleration 1446b to converge to the ideal acceleration 1447b.

[0092] 14C illustrates, however, in plot 1443, that including the average acceleration in the optimization equation significantly reduces the oscillations in the iteration history 1448c, while requiring a much fewer number of design iterations 1449c. Moreover, this leads to a much faster design convergence, with the maximum acceleration 1446c converging to the ideal acceleration 1447c within fewer design iterations 1449c.

[0093] In summary, Figures 14A-14C illustrate that an improved optimization iteration history leads to an improved simulation. Comparing the optimization iteration history of Figures 14A-14B with the optimization iteration history of Figure 14C, it is clear that the approach including continuous average acceleration results in an optimization iteration history with less and / or smaller oscillations in acceleration during the optimization iterations, resulting in fewer and / or the same number of optimization iterations and thereby faster convergence to achieve an optimized design.

[0094] 15A-15C further illustrate the benefit of including average acceleration in the design optimization by plotting the acceleration of the model against time in plots 1550a-1550c and against displacement in plots 1551a-1551c. FIGs. 15A-15C illustrate the effect between increasing the number of acceleration points and using average acceleration instead of simply adding additional acceleration points on the design convergence rate in seconds as well as displacement. The three optimization settings utilized to achieve the results depicted in FIGs. 15A-15C were defined in the following plots: 1550a and 1551a: acceleration points, I=25; 1550b and 1551b: additional acceleration points, I=100; and 1550c and 1551c, I=25, including average acceleration and acceleration points (according to embodiments described herein).

[0095] Specifically, plots 1550a and 1551a depict maximum acceleration in time and displacement, respectively, with acceleration point I=25 represented by discrete time instants 1555a and 1559a, respectively. Plots 1550a and 1551a show initial accelerations 1552a and 1556a, and optimized accelerations 1553a and 1557a, compared to ideal designs 1554a and 1558a, where the optimized accelerations 1553a and 1557a are stuck in highly localized regions of optimal minima 1561a-1561b and never reach the ideal designs 1554a and 1558a. Additionally, optimized designs 1557a never reach the maximum allowable intrusion 1560a.

[0096] The addition of further acceleration points to I=100 1555b and 1559b is illustrated in plots 1550b and 1551b. Plots 1550b and 1551b show the initial accelerations 1552b and 1556b, and the optimized accelerations 1553b and 1557b, compared to the ideal designs 1554b and 1558b. The addition of additional acceleration points 1555b and 1559b helps the model reach the ideal designs 1554b and 1558b, but the optimized designs 1553b and 1557b are still stuck in the highly localized regions of optimal minima 1561c-1561d. Furthermore, the optimized design 1557b never reaches the maximum allowable penetration 1560b.

[0097] However, plots 1550c and 1551c in FIG. 15C illustrate that including the average acceleration in the optimization equation results in optimized designs 1553c and 1557c that are ideal in terms of acceleration. Plots 1550c and 1551c show initial accelerations 1552c and 1556c, and optimized accelerations 1553c and 1557c, compared to ideal designs 1554c and 1558c. Furthermore, using the same number of acceleration points 1555c and 1559c as plots 1550a and 1551a, but without the average acceleration, the optimized designs get stuck in the highly localized minima 1561a-1561b regions, whereas optimized designs 1553c and 1557c converge quickly with ideal designs 1554c and 1558c and do not get stuck in highly localized minima such as 1561a-1561d.

[0098] In summary, comparing the accelerations of the optimized designs in Figures 15A and 15B with those of the optimized designs in Figure 15C, it can be seen that the approach including continuous average acceleration results in superior optimized designs 1553c and 1557c in terms of acceleration, and the optimized designs 1553c and 1557c do not get stuck in highly localized minima 1561a-d. The acceleration results in Figure 15C are thereby much closer to the analytically "ideal" constant accelerations 1554a-1554c. Furthermore, plot 1551c shows an intrusion displacement constraint 1560c, while plots 1551a-b do not have an intrusion displacement constraint (the designs do not reach constraints 1560a-1560b). The designs 1553c and 1557c for the optimized designs using continuous average acceleration thereby have better energy absorption characteristics for the head impact event of Figure 10.

[0099] 16A-16C further illustrate the advantages of applying the embodiments, i.e., using velocity to calculate the average acceleration and including the average acceleration in the optimization equation. Plots 1660a-1660c illustrate nodal non-parametric shapes for optimized designs determined using respective optimization settings defined as follows: 1660a: acceleration point, I=25; 1660b: additional acceleration point, I=100; and 1660c: 1660c including average acceleration and acceleration point (according to embodiments described herein), I=25.

[0100] Improvements in the material layout of the optimized design lead to improvements in the optimized design. Based on the acceleration and displacement design responses from Figures 15A and 15B, Figures 16A and 16B illustrate the nodal non-parametric shapes of the optimized designs using acceleration points I = 25 and I = 100, respectively. As shown in Figures 16A and 16B, these designs have highly localized lumped material design allocations 1661a-1661b just below where the head is impacting the plate, which is a highly localized minimum. However, Figure 16C illustrates the nodal non-parametric shape 1660c of the optimized design including the average acceleration. Plot 1660c does not have lumped material design allocated to location 1661c where the head is impacting the plate, and is therefore an overall superior design. As a result, Figures 14C, 15C, and 16C show that the approach of including continuous average acceleration in the formulation of the transient optimization for head impact avoids many local minima during the optimization history and leads to superior designs in terms of impact resistance properties such as acceleration and intrusion, thereby allowing the inventive approach described herein to reduce the impact on the optimization of physical and numerical noise (higher order and highly localized vibrations) in acceleration and corresponding acceleration sensitivity.

[0101] FIG. 17A shows an industrial bonnet 1770 for head impact optimization, which consists of shell elements (element types S4 and S3R in non-patent document 20). The material modeling is elastic-plastic and is given by the plot 1771 shown in FIG. 17B for a plate bonnet 1770. Two non-parametric optimization types are applied as shown in FIG. 17C. The first non-parametric optimization type uses a sizing shell thickness optimization 1772a (e.g. manufactured using a tailored blank), where the thickness of each finite element (t1 and t1) is the design variable. The second non-parametric optimization type uses a bead shell nodal optimization 1772b (e.g. manufactured using stamping), where the location of the nodes in the finite element model is the design variable. Non-patent document 21 provides an introduction to standard non-parametric sizing and bead optimization.

[0102] FIG. 17D is depictions 1772a-1772g showing the impact deformation of head 1773 of initial plate 1770, with the acceleration at the center of the head (I=80) and running average acceleration applied to both thickness sizing 1772a and nodal bead optimization 1772b at times 0.0 seconds (1772a), 0.001 seconds (1772b), 0.003 seconds (1772c), 0.006 seconds (1772d), 0.008 seconds (1772e), 0.011 seconds (1772f), and 0.020 seconds (1772g).

[0103] FIG. 18 shows the optimization iteration history when minimizing the maximum acceleration for each of sizing 1880 and bead optimization 1881. A stable optimization iteration history is shown for both optimization types 1880 and 1881. Plots 1880a (shell sizing thickness optimization) and 1881a (bead shell nodal optimization) show the maximum acceleration 1882 at each optimization design iteration 1883. In addition, FIG. 18 shows the optimized acceleration 1884 as a function of both time (plots 1880b and 1881b) and displacement (plots 1880c and 1881c). The optimized acceleration 1884 for both optimization types 1880b-1880c and 1881b-1881c is very close to the "ideal" theoretical constant acceleration 1885 calculated using the approach in Non-Patent Document 2. Additionally, while the intrusion displacement constraint is violated for the initial design 1886, the constraint is later satisfied during the optimization iterations and is valid for both final optimized designs 1887. The approach including continuous average acceleration for both designs 1880 and 1881 thereby has a valid intrusion displacement constraint 1888 and therefore has beneficial energy absorption properties for a head impact event.

[0104] The optimized design 1990 for the sizing element thickness design variable is shown in Figure 19A and the optimized design 1991 for the bead node design variable is shown in Figure 19B. Hence the approach for continuous average acceleration in sensitivity optimization works well for different design variable types. Plates 1990 and 1991 do not have a chunk of material assigned to location 1992 where the head is impacting the plate, thereby resulting in an overall superior design compared to designs that can be achieved using existing methods.

[0105] Embodiments provide important advantages in optimization formulations through the calculation and inclusion of continuous average acceleration using velocity in the optimization problems, and the solution of these problems using an iterative design process based on sensitivity and mathematical programming.

[0106] For example, one advantage is that it facilitates pre-processing and post-processing in the user interface. The design response to the average acceleration is defined by the user only at the start and end time points for the time window for which the average acceleration is calculated. This means that there are fewer pre-processing considerations by the user, i.e., fewer considerations by the user regarding the number of acceleration points for the simulation to be modeled in a given time window to determine the average acceleration, and easier post-processing of the optimized results.

[0107] Another example is improved simulation, which is indicated by a better or similar optimization iteration history, i.e., less oscillations for the optimization iterations. Furthermore, fewer or the same number of optimization iterations are required to achieve a converged design, meaning faster convergence and less computational resource usage. Furthermore, the velocity and its sensitivity need only be calculated at the beginning and end time points for the time window in which the average acceleration is calculated. This means that less computational storage and execution time is used to calculate the design response values ​​and sensitivities, and the optimization is less affected by the number of discrete time points of the simulation where the acceleration is calculated within the acceleration average and their location in time. Therefore, the time discretization as the number of time points in the time window and the location of these time points has less impact on the optimization iteration history and the final optimized design.

[0108] Another example is an improved optimized design: avoiding local minima during the optimization history means that the objective function value for the mean acceleration is improved and the mean acceleration constraint is easier to enforce. Furthermore, the improved optimized design leads to a better or similar design in terms of crashworthiness properties, which means that the overall acceleration of the final optimized design is lower and the penetration requirements of the final optimized design are better met.

[0109] FIG. 20 is a simplified block diagram illustrating a computer-based system 2020 that may be used to determine a running average acceleration of a computer-based model using velocities at the beginning and end of a given time window according to any of the various embodiments described herein. The system 2020 comprises a bus 2023. The bus 2023 serves as an interconnect between various components of the system 2020. Connected to the bus 2023 is an input / output device interface 2026 for connecting various input / output devices, such as a keyboard, mouse, display, speakers, etc., to the system 2020. A central processing unit (CPU) 2022 is connected to the bus 2023 and provides for the execution of computer instructions implementing the embodiments. The memory 2025 provides volatile storage for data used to execute computer instructions implementing the embodiments described herein, such as the methods 440 and 900 previously described with respect to FIGS. 4 and 9, respectively. The storage 2024 provides non-volatile storage for software instructions, such as an operating system (not shown) and configuration of the embodiments. System 2020 also includes a network interface 2021 for connecting to any of a variety of networks known in the art, including wide area networks (WANs) and local area networks (LANs).

[0110] It should be understood that the exemplary embodiments described herein may be implemented in many different ways. In some implementations, the various methods and machines described herein may each be implemented by a physical, virtual, or hybrid general-purpose computer, such as computer system 2020, or a computer network environment, such as computer environment 2120, described herein below in connection with FIG. 21. Computer system 2020 may be converted into a machine that performs the methods described herein, for example, by loading software instructions into either memory 2025 or non-volatile storage 2024 for execution by CPU 2022. Those skilled in the art should further appreciate that system 2020 and its various components may be configured to perform any embodiment or combination of embodiments described herein. Furthermore, system 2020 may utilize any combination of hardware, software, and firmware modules operably coupled, internally or externally to system 2020, to perform the various embodiments described herein. Additionally, system 2020 may be communicatively coupled to or embedded within a manufacturing device to control the device to create physical objects, as described herein.

[0111] 21 illustrates a computer network environment 2120 in which an embodiment of the present invention may be implemented. In the computer network environment 2120, a server 2121 is linked to clients 2123a-2123n via a communication network 2122. The environment 2120 may be used to enable the clients 2123a-2123n, alone or in combination with the server 2121, to perform any of the methods described herein. As non-limiting examples, the computer network environment 2120 may provide for cloud computing embodiments, Software as a Service (SAAS) embodiments, and the like.

[0112] The embodiments or aspects thereof may be implemented in the form of hardware, firmware, or software. If implemented in software, the software may be stored on any non-transitory computer-readable medium configured to enable a processor to load the software, or a subset of its instructions. The processor is then configured to execute the instructions to operate or cause an apparatus to operate in the manner as described herein.

[0113] Furthermore, firmware, software, routines, or instructions may be described herein as performing certain operations and / or functions of a data processor, however, it should be understood that such descriptions contained herein are merely for convenience and that such operations actually result from a computing device, processor, controller, or other device executing the firmware, software, routines, instructions, etc.

[0114] It should be understood that the flow diagrams, block diagrams, and network diagrams may include more or fewer elements, or be arranged differently, or represented differently, however, it should be further understood that a particular implementation may dictate that the block diagrams and network diagrams illustrating the execution of an embodiment, and the number of block diagrams and network diagrams, be implemented in a particular manner.

[0115] Accordingly, further embodiments may also be implemented in various computer architectures, physical computers, virtual computers, cloud computers, and / or some combination thereof, and therefore the data processors described herein are intended for illustrative purposes only and not as limitations of the embodiments.

[0116] Although exemplary embodiments have been particularly shown and described, it will be understood by those skilled in the art that various changes in form and detail may be made therein without departing from the scope of the embodiments as encompassed by the appended claims.

[0117] The teachings of all patents, published applications, and non-patent literature cited herein are hereby incorporated by reference in their entirety.

Claims

1. 1. A computer-implemented method for automatically determining an optimized design for a real-world object, comprising: defining, in a memory of the processor, a computer-based model representing the real-world object; determining an equilibrium of the real-world object for each of a plurality of time steps using the defined computer-based model, the determining of the equilibrium determining a velocity of the real-world object over the plurality of time steps; determining an average acceleration using the determined velocity for each of a plurality of time windows defined over the plurality of time steps; calculating a sensitivity of each determined average acceleration using the determined velocity sensitivities; using each determined average acceleration to define at least one of a constraint and an objective function; using the calculated sensitivities for each determined average acceleration, iteratively optimizing the computer-based model representing the real-world object with respect to the defined at least one of the constraints and the objective function, wherein the iterative optimization updates the computer-based model representing the optimized design of the real-world object; A method comprising:

2. The method of claim 1 , wherein the computer-based model is a finite element model, a boundary element method, a finite difference method, a finite volume method, or a discrete element method.

3. For a given one of the plurality of time windows, determining the average acceleration using the determined velocity comprises: identifying from among the determined velocities: (1) a velocity at a start of the given time window; and (2) a velocity at an end of the given time window; calculating the average acceleration for the given time window using (1) the identified velocity at the start point, (2) the identified velocity at the end point, and (3) the length of time in the given time window; The method of claim 1 , comprising:

4. Calculating the sensitivity of the average acceleration for the given time window comprises: calculating a sensitivity of the average acceleration for the given time window using the identified speeds at the start point and the identified speeds at the end point for a given design variable. The method of claim 3 , comprising:

5. using the calculated sensitivities for each determined average acceleration and at least one of discrete accelerations, structural scale sensitivities, and multi-physics scale sensitivities in the step of iteratively optimizing the computer-based model. The method of claim 1 further comprising:

6. receiving user defined requirements; using both the user-defined requirements and the determined average accelerations in the step of defining at least one of the constraints and the objective function; The method of claim 1 further comprising:

7. The method of claim 1 , wherein the real-world object represented by the computer-based model is an automobile, an airplane, an electronic device, a medical device, or an armored vehicle.

8. 1. A system for automatically determining an optimized design for a real-world object, comprising: A processor; a memory having computer code instructions stored therein, said processor and said memory using said computer code instructions to cause said system to: defining in said memory a computer-based model representing said real-world object; determining an equilibrium of the real-world object for each of a plurality of time steps using the defined computer-based model, the determining of the equilibrium determining a velocity of the real-world object over the plurality of time steps; determining an average acceleration using the determined velocity for each of a plurality of time windows defined over the plurality of time steps; calculating the sensitivity of each determined average acceleration using the determined velocity sensitivity; defining at least one of a constraint and an objective function using each determined average acceleration; iteratively optimizing the computer-based model representing the real-world object with respect to the defined at least one of the constraints and the objective function using the calculated sensitivity of each determined average acceleration, the iterative optimization updating the computer-based model representing the optimized design of the real-world object; a memory configured to cause A system comprising:

9. The system of claim 8 , wherein the computer-based model is a finite element model, a boundary element method, a finite difference method, a finite volume method, or a discrete element method.

10. In determining the average acceleration using the determined velocity for a given one of the plurality of time windows, the processor and the memory use the computer code instructions to cause the system to: identifying from the determined velocities: (1) a velocity at a start of the given time window; and (2) a velocity at an end of the given time window; (1) calculating the average acceleration for the given time window using the identified velocity at the start point, (2) the identified velocity at the end point, and (3) the length of time in the given time window; The system of claim 8 , further configured to:

11. To calculate the average acceleration sensitivity for the given time window, the processor and the memory use the computer code instructions to provide the system with: calculating the sensitivity of the average acceleration for the given time window using the identified speed at the start point and the identified speed at the end point for a given design variable; The system of claim 10 , further configured to:

12. The processor and the memory use the computer code instructions to cause the system to: using the calculated sensitivities for each determined average acceleration and at least one of discrete accelerations, structural scale sensitivities, and multi-physics scale sensitivities in the iterative optimization of the computer-based model. The system of claim 8 , further configured to:

13. The processor and the memory use the computer code instructions to cause the system to: Receiving user-defined requirements; using both the user-defined requirements and the determined average accelerations in the definition of at least one of the constraints and the objective function; The system of claim 8 , further configured to:

14. The system of claim 8 , wherein the real-world object represented by the computer-based model is an automobile, an airplane, an electronic device, a medical device, or an armored vehicle.

15. A non-transitory computer program product for automatically determining an optimized design of a real-world object, the non-transitory computer program product being executed by a server in communication with one or more clients over a network, the non-transitory computer program product comprising: A computer-readable medium that, when executed by a processor, causes the processor to: defining, in a memory of the processor, a computer-based model representing the real-world object; determining an equilibrium of the real-world object for each of a plurality of time steps using the defined computer-based model, the determining of the equilibrium determining a velocity of the real-world object over the plurality of time steps; determining an average acceleration using the determined velocity for each of a plurality of time windows defined over the plurality of time steps; calculating the sensitivity of each determined average acceleration using the determined velocity sensitivity; defining at least one of a constraint and an objective function using each determined average acceleration; iteratively optimizing the computer-based model representing the real-world object with respect to the defined at least one of the constraints and the objective function using the calculated sensitivity of each determined average acceleration, the iterative optimization causing the computer-based model representing the optimized design of the real-world object to be updated; 23. A computer readable medium further comprising program instructions for causing 1. A non-transitory computer program product comprising:

16. 16. The non-transitory computer program product of claim 15, wherein the computer-based model is a finite element model, a boundary element method, a finite difference method, a finite volume method, or a discrete element method.

17. When executed by the processor, in determining the average acceleration using the determined velocity for a given one of the plurality of time windows, the program instructions cause the processor to: identifying from the determined velocities: (1) a velocity at a start point of the given time window; and (2) a velocity at an end point of the given time window; (1) calculating the average acceleration for the given time window using the identified velocity at the start point, (2) the identified velocity at the end point, and (3) the length of time in the given time window; 20. The non-transitory computer program product of claim 15,

18. The program instructions, when executed by the processor, to calculate the average acceleration sensitivity for the given time window, cause the processor to: calculating the sensitivity of the average acceleration for the given time window using the identified speed at the start point and the identified speed at the end point for a given design variable; 20. The non-transitory computer program product of claim 17,

19. The program instructions, when executed by the processor, cause the processor to: further using the calculated sensitivities for each determined average acceleration and at least one of discrete accelerations, structural scale sensitivities, and multi-physics scale sensitivities in the iterative optimization of the computer-based model.

20. The non-transitory computer program product of claim 15, further comprising:

20. The program instructions, when executed by the processor, cause the processor to: Receiving user-defined requirements; using both the user-defined requirements and each determined average acceleration in the definition of at least one of the constraints and the objective function; 20. The non-transitory computer program product of claim 15, further comprising:

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