Method for accelerating numerical calibration of computationally expensive constitutive response parameters using surrogate models

Abstract

To provide a method for accelerating numerical calibration of computationally expensive constitutive response parameters using surrogate models.SOLUTION: A method for constitutive response model calibration employs a neural network-based surrogate model in place of output from a corresponding finite element simulation. An application calibrates the constitutive response model upon receipt of user selections including test data output from an experiment, a finite element model simulating the experiment, a constitutive response model selection, a history output quantity, a numerical minimization algorithm, an error measure, constitutive response model parameters, an error measure for test data vs finite element simulation output, constitutive response model parameters, a parameter space sampling technique, architecture settings of a surrogate model, and a training algorithm for the surrogate model that trains an algorithm.SELECTED DRAWING: Figure 4

Description

Detailed Description of the Invention

[0001] [Field of the Invention] The present invention relates to parameterization of objects, and more particularly to improving the efficiency of calibration of constituent response parameters. [Background of the invention] A Constitutive Response Model (CRM) is a mathematical model used to describe the behavior of a real object, structure, or interaction between structures. CRMs are a fundamental part of finite element (FE) simulation procedures. CRMs depend on certain parameters being accurately specified. As a simple example, when simulating a steel structure in the elastic regime, the user may select isotropic elasticity as the constitutive response model and specify accurate values ​​for the elastic moduli and Poisson's ratio.

[0002] While constitutive models frequently refer to object behavior, the concept of constitutive response may extend beyond object behavior. For example, mechanical parts may be approximated during finite element simulation using simplified mathematical models by using specially formulated finite elements, such as spring elements, link elements, etc. Such special elements use special constitutive response models designed to reproduce structural behavior. Another type of CRM may be specialized in modeling the interaction between contacting surfaces during finite element simulation, such as friction and wear models and cohesive / adhesive contact models (used to simulate bonded or cohesive surfaces). Some constitutive response models can be complex and span multiple physical domains, such as coupled electro-thermal displacement response.

[0003] In many situations, real-world objects and structures need to be experimentally tested in the laboratory to obtain parameters for the constitutive response models used in finite element simulations. Typically, multiple experimental setups can be used to obtain sufficient experimental data. Each experimental setup attempts to elicit the behavior of the object or structure under a specific deformation mode or a specific set of loading conditions. During each experiment, the observed behavior of the object or structure is recorded as a function of time, load, or deformation. Such recordings are referred to as a test dataset. For example, a typical test dataset may include experimentally measured stress as a function of time and / or applied strain, or may include experimentally measured force as a function of time and / or applied displacement, etc., as shown by FIGS. 1A and 1B.

[0004] As shown by Figure 1A, experiments can be used to measure and record strain-stress test data sets for a particular deformation mode, such as uniaxial, biaxial, planar (pure shear), bulk, and simple shear. Figure 2 depicts an example of an experimentally recorded strain-stress test data set.

[0005] Such experimentally recorded test data sets cannot be used directly in finite element simulation models. Instead, the typical approach is to select an appropriate constitutive response model and calibrate its parameters so that certain simulation output quantities closely match the experimentally recorded test data.

[0006] If an experimental test data set is available, the user typically follows the following main steps to obtain a calibrated constituent response: 1. Generate a finite element model for each experimental device. For example, the user may generate separate models for uniaxial deformation, biaxial deformation, etc. Such finite element models may be simple (containing a single element) or complex, with many interacting mesh sections, each containing many elements. 2. Select an initial guess for the parameters of the constituent response. 3. Run the finite element simulation and collect the resulting output in a simulation output database. In some cases, the time required to run the simulation can be very long (especially for complex models), e.g., tens of hours. 4. Extracting result data from the simulation output database and comparing the simulation output with the corresponding experimentally recorded test data. 5. If the simulation results and experimental results match well (i.e., the curves corresponding to the simulation output and the curves representing the experimental test data overlap or are sufficiently close to each other), the calibration process can be stopped. The match between the experimental test data and the simulation output is typically quantified mathematically using an error norm function, such as the mean square error (MSE) function. These error norms reach a certain minimum value when the simulation output matches the experimental test data. In particular, the MSE value will be very close to zero when the simulation output is very close to the experimental test data. If the simulation results do not match the experimental results, the parameters of the constituent response must be modified, and the simulation is re-run (with the modified parameters) until a sufficiently good match is achieved (e.g., the MSE approaches zero).

[0007] Initially, this approach was typically implemented manually, but more recently, numerical minimization algorithms have been employed, using one of several well-known minimization algorithms. For numerical minimization processes, the minimization algorithm automatically tries new parameter values ​​for the constitutive response model such that the error between the simulation results and the corresponding experimental data is iteratively reduced until the simulation results agree satisfactorily with the experimental data.

[0008] Figure 3 shows an example of a calibrated strain-stress response compared to input test data for uniaxial, biaxial, and planar deformation modes, where each deformation mode corresponds to a different set of loading conditions (i.e., different experimental setups as shown schematically in Figure 1A). Here, the process of determining parameter values ​​of a constitutive response by means of a manual or automated iterative approach using a numerical minimization algorithm is referred to as calibrating the parameters of that constitutive response. Calibration workflows employing numerical minimization algorithms can be found in existing software, including the 3DX Material Calibration App.

[0009] The time required for an automated calibration workflow depends strongly on the complexity and duration required to run the finite element simulation. Computation time can range from a few seconds to several hours or even days, and can require the use of significant computational resources. A user may calibrate multiple constituent-response models until a suitable one is found, in which case the entire calibration process must be repeated. Therefore, there is a need in the industry to reduce the time required to perform constituent-response calibrations. [Summary of the Invention]

[0003] Embodiments of the present invention provide a method for accelerating the computationally expensive numerical calibration of constituent response parameters using surrogate models. Briefly, the present invention is directed to a method for constitutive response model calibration that employs a neural network-based surrogate model in place of the output from a corresponding finite element simulation. An application calibrates a constitutive response model upon receiving user selections including test data output from an experiment, a finite element model simulating the experiment, a constitutive response model selection, historical output quantities, a numerical minimization algorithm, an error metric, constitutive response model parameters, an error metric for the test data versus the finite element simulation output, the constitutive response model parameters, a parameter space sampling technique, an architecture for the surrogate model architecture configuration, and a training algorithm for the surrogate model training algorithm.

[0010] Other systems, methods, and features of the invention will be or become apparent to one with skill in the art upon examination of the following figures and detailed description. All such additional systems, methods, and features are intended to be included within this specification, be within the scope of the invention, and be protected by the accompanying claims. [Brief explanation of the drawings]

[0011] The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.

[0012] The accompanying drawings are included to provide a further understanding of the present invention, and are incorporated in and constitute a part of this specification. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the present invention. The drawings illustrate embodiments of the present invention and, together with the description, serve to explain the principles of the present invention. [Figure 1A] 1 is a schematic diagram of a typical experiment for providing experimentally measured stress as a function of time and / or applied strain, or experimentally measured force as a function of time and / or applied displacement. [Figure 1B] 1 is a schematic diagram of an experiment for providing experimentally measured shear stresses. [Figure 2] 1 is a plot of stress versus strain overlaid with uniaxial, biaxial, and planar test data. [Figure 3] Separate plots of experimental test data and simulated responses corresponding to three deformation modes: uniaxial, biaxial, and planar are shown. [Figure 4] 1 is a flowchart of an exemplary first method embodiment. [Figure 5] 5 is a detailed flowchart of the first exemplary method embodiment of FIG. 4. [Figure 6A]10 is a screenshot showing an example of experimental test data to be imported under the first embodiment. [Figure 6B] 1 is a screenshot of an example experimental test dataset imported by a user into the 3DX object calibration application. [Figure 6C] 10 is a plot showing test data selected during the import process, plotting experimental nominal stress as a function of time. [Figure 6D] 1 shows the imported experimental test data for editing and viewing after importing into the application. [Figure 6E] Shows a number of experimental test datasets rendered within the application. After import, the user can select which test datasets will be used for calibration. [Figure 7A] An example of a set of three finite element models used together during the same plasticity calibration of a damage constitutive response model is shown. [Figure 7B] An example of an Abaqus finite element model used to simulate a double cantilever beam experiment is shown. [Figure 7C] An example of a set of three finite element models used together during the same calibration of anisotropic hyperelastic constitutive response is shown. [Figure 7D] 1 shows an exemplary set of two finite element models used together to calibrate the coupling behavior. [Figure 8] 10 is a screenshot showing an exemplary list of configuration response models for user selection. [Figure 9] 1 is an annotated screenshot showing an example calibration setup using three finite element models. [Figure 10] 1 is a screenshot showing user selection of one of several available numerical minimization algorithms to be used for calibration. [Figure 11] 10 is a screenshot showing how a user may select one of the available error metrics. [Figure 12]1 is a screenshot showing an example scenario of a user selected construction object model. [Figure 13] 10 is a screenshot illustrating an exemplary scenario in which a user is presented with options to modify default settings regarding the initial sampling of parameter space. [Figure 14]

[0047] Figure 14 highlights a portion of the screenshot in Figure 13 showing user selection of settings for the architecture of the surrogate model that is trained to reproduce the finite element output. [Figure 15] 10 is a screenshot and plot comparing the FE response using calibrated parameters with experimental test data. [Figure 16A] 10 is a plot from a first experiment showing results based on the previous calibration approach. [Figure 16B] 1 is a plot from a first experiment showing results according to the present embodiment. [Figure 17] 10 is a graph showing three plots from a second experiment showing results according to the present embodiment. [Figure 18] 10 is a plot from a third experiment showing results according to the present embodiment. [Figure 19] 10 is a plot from a fourth experiment showing results according to the present embodiments. [Figure 20] 10 is a plot from a fifth experiment showing results according to the present embodiments. [Figure 21] 1 is a schematic diagram illustrating an example of a system for performing the functions of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0013] [Detailed explanation] The following definitions are useful for interpreting terms applied to features of the embodiments disclosed herein and are intended solely to define elements within the present disclosure.

[0014] In this disclosure, "artificial neural network" (ANN) refers to a mathematical model commonly used to solve (i) classification problems and (ii) regression problems. This document addresses using neural networks to solve regression problems. When used to approximate or reproduce an existing response or set of data, an ANN may be referred to as a surrogate model. A surrogate model is a mathematical model that can be generated in various ways and does not necessarily have to be a neural network. A surrogate model may be thought of as a map between a set of inputs and a set of outputs. There are many types of ANN architectures, such as feedforward neural networks (FFNNs), recurrent neural networks (RNNs), and transformers. ANNs contain many internal parameters, including "weights" and "biases." The number of weights and biases depends on the size of the ANN, which can range from tens to trillions of parameters.

[0015] In this disclosure, "ANN training" refers to the process of calibrating an ANN. Regardless of the ANN architecture, the internal ANN parameters (i.e., weights and biases) need to be calibrated before the ANN can be used in a practically meaningful way. While this calibration process is referred to as "training" in the ANN literature, this ANN training process is conceptually similar to constitutive response model calibration. ANN training also uses numerical minimization algorithms that modify the internal ANN parameters (i.e., weights and biases) until the ANN predictions match to a certain level of accuracy for the specific data it needs to reproduce.

[0016] In this disclosure, "Bayesian minimization" refers to the refinement of design parameters and optimization frameworks targeted at costly "black-box" objective functions (e.g., time-consuming simulations). The primary goal of Bayesian minimization is to obtain the best set of design parameters in a relatively small number of objective function evaluations (e.g., fewer than 100 objective function evaluations). The method relies on (i) generating and using a response surface to fit the objective function and (ii) a gain function to predict the next sampling point to be attempted with the costly objective function. The method includes evaluating a measure of "uncertainty" when using the response surface to fit the costly objective function.

[0017] In this disclosure, "Long Short-Term Memory Neural Network (LSTM-NN)" refers to a special category of recurrent neural networks. Various variants of LSTM neural networks exist in the literature. An LSTM neural network can be designed as a stack of two or more LSTM cells.

[0018] As used herein, "historical output" refers to a quantity provided by a simulation (based on user request) as a function of time, such as stress as a function of time at a point in a finite element model, or reaction forces recorded as a function of time during a simulation.

[0019] Embodiments herein describe a method for using a long-short-term memory (LSTM) based NN architecture to generate surrogate models for reproducing historical outputs extracted from finite element simulations used during constitutive response calibration.

[0020] DETAILED DESCRIPTION OF THE INVENTION Embodiments of the present invention will now be described in detail, examples of which are illustrated in the accompanying drawings. Wherever possible, the same reference numbers are used in the drawings and the description to refer to the same or like parts.

[0021] Exemplary embodiments of the present invention are used to speed up the calibration of constitutive models employed during finite element simulations. This embodiment significantly reduces the time required to determine parameters for constitutive response models, especially when time-consuming finite element simulations are used to replicate experimental tests. Thus, this embodiment provides a practical approach for scenarios where even a single simulation can take several hours or more to complete.

[0022] The calibration workflow embodiments described herein provide an iterative process using a numerical minimization algorithm. A user supplies an experimental test data set along with a finite element model as part of the input to the calibration workflow. The finite element model is used to simulate a test experiment. During the iterative calibration process, historical outputs produced by the finite element simulation are repeatedly compared with corresponding experimental test data until they match. This process may involve running the finite element simulation many times (e.g., hundreds to tens of thousands or more) and can take a long time.

[0023] The present embodiment provides an approach that can significantly reduce the time required to perform constituent-response model calibration by automatically employing a neural network-based surrogate model that is used to replace otherwise required outputs from the corresponding finite element simulation. As a non-limiting example, the present embodiment can generate a surrogate model using a neural network, particularly one based on long-short-term memory. Depending on the specific constituent-response model, the neural network selected can be any neural network architecture or network of mathematical functions that can be concatenated and / or pruned to process sequence data, including recurrent neural networks / recurrent nets, feedforward neural networks, and transformers with various customized / combined architectures.

[0024] 4 is a flowchart of an exemplary object calibration workflow method. Any process description or block in the flowchart should be understood as representing a module, segment, portion of code, or step that includes one or more instructions for implementing a particular logical function during processing, and it should be noted that alternative implementations, as understood by those skilled in the art of the present invention, in which functions may be performed out of order from that shown or discussed, including substantially concurrently or in reverse order, depending on the functionality involved, are within the scope of the present invention.

[0025] Embodiments of the method described herein extend and enhance object calibration workflows using finite element models, such as those implemented in 3DX object calibration applications. While this embodiment is directed to calibrating object behavior, alternative embodiments may also be used to calibrate the constitutive response of special elements (e.g., cohesive elements, link elements, spring elements, gasket elements, etc.), constitutive interface response of contact surfaces (e.g., cohesive contact, wear, friction, etc.), and even failure models.

[0026] In the following description, terms such as "optimization" and "minimization" may be used interchangeably. While mathematically, optimization can refer to either "minimization" or "maximization," it is trivial to convert between minimization and maximization by multiplying the objective function being minimized / maximized by (minus) -1. Furthermore, terms such as "finite element (FE) response" or "FE simulation response" refer to output quantities, which are part of the results produced by a simulation.

[0027] According to an exemplary calibration method embodiment, test data for an object is received, as indicated by block 410. The test data may be, for example, output from an experiment. A finite element model simulating the experiment is received, as indicated by block 420. A user selection of a constitutive response model is received, as indicated by block 430. A user selection of historical output quantities to collect during the simulation is received, as indicated by block 440. A user selection of a numerical minimization algorithm for calibrating the constitutive response model is received, as indicated by block 450. A user selection of an error metric for quantifying differences between the test data and the finite element simulation is received, as indicated by block 460. A user selection of parameters for the constitutive response model is received, as indicated by block 470. A first technique for parameter space sampling during an initial calibration stage and a second technique for collecting finite element output for calculating an objective function are received, as indicated by block 480. A user selection of settings for the architecture of a surrogate model to be trained to reproduce the finite element output, and a user selection of a training algorithm for the surrogate model, are received, as indicated by block 490. Automated constitutive response model calibration is performed, as indicated by block 500. Each of blocks 410-500 is described in further detail below.

[0028] As indicated by block 410, test data for the object is received. A user of the simulation application provides a relevant test dataset previously obtained, for example, through experimentation. It should be noted that the experimental work required to obtain this test dataset is beyond the scope of this embodiment. FIG. 6A is a screenshot illustrating example experimental test data to be imported into the 3DX object calibration application. The test dataset may include, for example, stress as a function of time and deformation, force as a function of time and deformation, and / or any relevant quantity of interest recorded as a function of time, where the quantities described below are obtainable from the simulation, either as direct output or based on post-processed simulation output. The test dataset may be obtained in a laboratory using object samples or may be measured using sensors placed throughout parts and structures. Existing experimental techniques for obtaining test datasets as input to the methods described herein include, for example, universal testing machines, environmental control chambers, and digital image correlation. FIG. 6B is a screenshot of an example experimental test dataset imported by a user into the 3DX object calibration application. Here, the user has selected the test data of interest to be imported (from an Excel file). In this particular example, the selected experimental test data set includes time, nominal stress, and nominal strain. A user may import multiple test data sets from the same file or from multiple files.

[0029] The test data may be represented in a tabular format. On a computer, the test dataset may be stored, for example, in a spreadsheet file, a CSV (comma separated values) list, or a text file. For example, the 3DX object calibration application allows a user to import test datasets from spreadsheet files, CSV files, and text files. Multiple test datasets may be stored in a single file or in separate files. While the present embodiment describes and illustrates a 3DX object calibration application, alternative embodiments may involve other platforms, such as (but not limited to) Abaqus / CAE, or launching via a command line prompt.

[0030] Once the test data set is imported into the 3DX object calibration application, the user may optionally edit the contents of the test data set as needed to suit calibration purposes. These editing operations may include removing duplicate data points, reducing the number of test data points by decimation or deletion, smoothing the test data set to remove noise in the test data set, and regularizing the test data set (reproducing the data at equal intervals).

[0031] FIG. 6C shows a plot illustrating test data selected during the import process, plotting experimental nominal stress (recorded during a prior experiment) as a function of time. FIG. 6D shows the imported experimental test data for editing and display after import into the application. Here, a user may manually delete points or use one of the available operations (e.g., decimation, smoothing, regularization) to improve the quality of the test data set before it is used for calibration. FIG. 6E shows multiple experimental test data sets rendered within the application. After import, a user may select which test data sets to use for calibration.

[0032] As indicated by block 420 (FIG. 4), a finite element model simulating an experiment is received. A user generates and provides a finite element model as input to simulate a laboratory experiment. Calibration can use one or more finite element models simultaneously. The finite element model generation process is not part of the present invention and must be performed in advance. Existing pre-processing software allows a user to generate the finite element model. For example, the finite element model may be generated in a 3DX Mechanical Scenario App or in Abaqus / CAE.

[0033] In the context of this embodiment, exemplary formats used to describe finite element models include the Abaqus input file format (.inp) and the (internal) 3DX Simulation Object Format. The Abaqus input file format (.inp) is a textual representation of a finite element model for the Abaqus software, and is publicly documented. Existing translators between different finite element model formats may be used, for example, to convert a finite element model generated in a format specific to a particular commercial software to the Abaqus input file format (.inp). For example, Abaqus input files can be converted from various commercial software (other than Simulia) as shown in Table 1.

[0034] [Table 1] Abaqus input files (.inp) may be converted to and from other formats (other than DS / Simulia).

[0035] FIG. 7A shows an example of a set of three finite element models used together during the same plasticity calibration of a damage constitutive response model (finite element meshes are not shown in this figure).

[0036] Figure 7B shows an example of an Abaqus finite element model used to simulate a double cantilever beam experiment. This type of FE model can be used to calibrate the adhesive interface constitutive response.

[0037] Figure 7C shows an example set of three finite element models used together during the same calibration of anisotropic hyperelastic constitutive response, which can be used (among other applications) to model the response of human tissue (e.g., cardiac tissue) in FE simulations.

[0038] 7D shows an exemplary set of two finite element models used together to calibrate a constitutive response model specific to link behavior, e.g., a link element refers to an element used to model a connection between parts, e.g., a clinch joint, a self-piercing rivet, a spot weld, etc.

[0039] As indicated by block 430 (FIG. 4), a user selection of a constitutive response model is received. The user selects a constitutive response model of interest (e.g., available in Abaqus or on the 3DX platform). In the context of the present embodiment, constitutive response model may refer to object behavior (e.g., elastic, hyperelastic, viscoelastic, or plastic object behavior). However, constitutive response model here may also refer to a mathematical model used to describe interactions between contacting surfaces, such as adhesive or cohesive interaction models, friction between surfaces, wear, etc. Constitutive response model may also refer to a specific mathematical model used to determine the behavior of special types of finite elements, such as link elements, spring elements, and cohesive elements. For example, cohesive elements may be used to simulate fracture and delamination. The constitutive response for a cohesive element may be represented by a traction-separation law. FIG. 8 shows an exemplary list of constitutive response models for user selection. Each constitutive response model has a specific set of parameters that may be calibrated using an associated experimental test dataset. A mathematical description of each of these constitutive models is provided in the software documentation.

[0040] As indicated by block 440 (FIG. 4), a user selection of historical output quantities to collect during the simulation is received. For each finite element model used during calibration, the user may specify one or more historical output quantities to collect during the simulation. Historical output refers to the output stored as a function of time at a specific model location or for the entire model. Each experimental test data quantity may be used, along with a corresponding quantity obtained from one or more collected FE simulation historical outputs, to determine the objective function.

[0041] Figure 9 is an annotated screenshot showing an example calibration setup using three FE models. For each FE model, the user can select one or more FE outputs to match during calibration with corresponding experimental quantities contained in one of the imported test data sets. Figure 9 shows that for one of the FE models named "adventitia_axial.inp," the user wants to extract an FE history output representing displacement U1 at the node location defined by the node set named "nLoad" during FE simulation. During calibration, this FE output (U1) will match the experimental displacement value extracted from the imported test data set named "axial_small." In Figure 9, the user indicated that the FE outputs should first be scaled by a factor of two before matching. The error metrics between the scaled FE outputs and the experimental quantities can be weighted by a user-specified weight value.

[0042] As indicated by block 450, a user selection of a numerical minimization algorithm for calibration of the constituent response model is received. The user-selected numerical minimization algorithm is used, along with any associated settings, during calibration of the constituent response model. FIG. 10 is a screenshot showing a user selection of one of several available numerical minimization algorithms to be used for calibration. The user may modify the default settings as desired. Some settings are common to all algorithms, however, some settings are specific to individual algorithms. The terms "minimization" and "optimization" are used interchangeably in the dialog window.

[0043] A user selection of an error metric for quantifying the difference between the test data and the finite element simulation is received, as indicated by block 460. The user selects the error metric (or "error norm") used to quantify the difference between the test data set and the corresponding finite element simulation ("mean squared error"). The error metric is a function used to form the objective function.

[0044] Equation 1 shows how the objective function F(X) can be derived for a typical calibration case employing multiple finite element (FE) models. For each FE model, any number of historical outputs can be collected from the simulation. The collected outputs can be combined using a post-processing function. The values ​​produced by the post-processing function are compared to the experimentally recorded values ​​of the corresponding test data volume using a user-specified error metric. The error metric is a function that quantifies the difference between two sets of values—in this case, experimental values ​​versus simulated values. Examples of error metrics are mean squared error, mean absolute error, coefficient of determination, etc. The values ​​returned by the error metric can optionally be scaled by user-specified weights. Penalty values ​​may be added to the objective function to account for failed finite element simulations (e.g., due to convergence issues), various constraint violations, etc. The time t in Equation 1 can be an actual time or an artificial time used to index the applied load increment.

[0045]

number

[0046]

number

[0047] In the simplest case, there is a single output quantity (m=1) and no output processing, i.e.,

[0048]

number

[0049] Figure 11 is a screenshot showing the mechanism for the user to select from the available error metrics, which are functions used to quantify the difference between the experimental test data and the finite element response.

[0050] As indicated by block 470, a user selection of a parameter set for the constituent response model is received. The user selects the parameters of the constituent response model to be calibrated. Additionally, the user may optionally indicate minimum and maximum bounds for each parameter. The user may specify initial values ​​for each parameter. The user may optionally perform an evaluation based on a single FE model using the initial values ​​of the constituent response parameters.

[0051] FIG. 12 shows an example scenario of a user-selected anisotropic Holzapfel-Gasser-Ogden hyperelastic constitutive object model used to simulate cardiac tissue. This constitutive response has five parameters that can be calibrated: "C10," "D," "K1," "K2," and "fiber_dispersion_parameter." In this setup, parameter D is set to a fixed, known value (0 1 / Pa) and is not calibrated. The remaining parameters are calibrated. For each parameter to be calibrated, the user provides an initial guess and range (defined by minimum and maximum bounds). The plot (bottom) shows a comparison of the experimental test data versus the FE output (i.e., the FE response) when using the initial guess for the calibrated parameter. In this case, the initial values ​​are far from the "real" values, and the FE and experimental curves do not match well.

[0052] As indicated by block 480 (FIG. 4), the user selects a technique to be used in the initial stages of calibration to sample the parameter space and to collect the finite element outputs needed to calculate the objective function. The selected technique can be either a sampling technique (design of experiments technique) or an existing minimization algorithm. Examples of sampling techniques include random sampling and Latin hypercube sampling. A suitable minimization algorithm that can be used in this step is Bayesian minimization. Bayesian minimization is a numerical algorithm for objective function evaluation that is expensive when attempting to obtain an acceptable solution with a small number of function evaluations. Bayesian minimization can improve sampling using the point where the greatest improvement is expected. Optionally, the user can reuse saved FE responses evaluated during a previous calibration.

[0053] 13 illustrates a scenario in which a user is presented with options to modify the default settings for the initial sampling of parameter space. As shown, the UI presents the case of a favored approach using Latin Hypercube followed by Bayesian iterations. In this state, the user may specify the number of sampling points and the number of Bayesian iterations. However, other sampling approaches (not shown) may easily be adopted in alternative embodiments. The user may optionally reuse previous sampling data (collected during a previous calibration).

[0054] As indicated by block 490 (FIG. 4), a user selection of settings related to the architecture of a surrogate model to be trained to reproduce the finite element output and a user selection of a training algorithm for the surrogate model are received. While not limited to the approach described herein, an architecture of interest is based on a stack of long-short-term memory (LSTM) neural networks (NNs) concatenated with an output feedforward network. Non-default settings may include the number of LSTM cells used in the NN stack, the number of units per cell, and the activation function for the output feedforward NN. Settings specific to NN training may include settings related to the minimization algorithm used to train the NN.

[0055] Figure 14 shows an example of how a user can modify the default settings that control the NN architecture. For usability purposes, Figure 14 shows that only a few settings are visible to the user in the UI. In alternative embodiments, other features, such as settings specific to NN training, may be displayed.

[0056] 5 is a flowchart 500 that further expands upon block 500 of FIG. 4, which describes actions of an embodiment for configuration response calibration. Block 400 represents receiving user input, as per blocks 410-490 of FIG. 4.

[0057] The parameter space is sampled by evaluating the objective function based on the FE model (see the description of block 480 above), and the required FE responses are collected and stored internally. Additionally, as indicated by block 510, previous FE responses (collected during a previous calibration) can optionally be loaded from an external file. A small number (typically 20-40, generally fewer than 100) of finite element simulations need to be performed at the sampling locations using the finite element model provided by the user. The sampling locations can be obtained using established sampling techniques, such as random sampling or Latin hypercube sampling. A particularly advantageous minimization algorithm that can be used in this step (to obtain the sampling locations) is Bayesian minimization. The FE history output quantities of interest (i.e., used to evaluate the objective function) resulting from the simulation are collected and stored internally. The collected FE history output quantities can be saved, for example, to disk and reused for subsequent surrogate-based calibrations (based on user selection).

[0058] As indicated by block 520, one or more neural networks are trained (e.g., using typical NN training approaches that rely on numerical minimization algorithms such as stochastic gradient descent, Adam, or BFGS) to accurately reproduce the FE output of interest without having to perform a finite element simulation. This step is important because, depending on the actual neural network architecture, the surrogate model may or may not be able to accurately reproduce the output from the finite element simulation. In particular, a stack of LSTM cells followed by a feedforward NN is used to reproduce the individual historical outputs. However, it is possible that other surrogate types may also produce accurate results. Once trained, the trained neural network is referred to as a surrogate model. The user can switch between using a surrogate model (which evaluates very quickly) and a finite element model (which evaluates slowly but guarantees accuracy).

[0059] As indicated by block 530, the surrogate model replaces more computationally expensive finite element simulations in the iterative minimization process to provide historical output quantities that are used to calculate the objective function (during minimization). The result is a set of calibrated parameters for the selected constitutive response.

[0060] As indicated by block 540, the calibrated constituent response parameters are evaluated using the evaluation FE model (instead of the surrogate) to evaluate the accuracy of the surrogate model and to provide the user with FE history outputs corresponding to the calibrated constituent response parameters.

[0061] Figure 15 is a screenshot and plot comparing the FE response using calibrated parameters with experimental test data, where the calibrated parameters of the constituent response are reported to the user along with the FE response. In the example shown by Figure 15, the calibrated parameters for the FE response closely match the experimental test data.

[0062] As indicated by block 550, the surrogate predicted response is compared to the FE response, and the user is provided with a diagnosis of the FE response and the agreement between the FE response and the surrogate prediction of the calibrated parameter set. If the FE response and the surrogate prediction agree, the user may approve the calibrated parameters (block 570). Otherwise, the user may make one or more modifications (block 560) before re-executing blocks 510-540. For example, the user may modify the surrogate architecture (e.g., change the number of LSTM cells, modify the number of connections between cells), change the surrogate training settings and training algorithm, modify the sampled FE positions by adding more locations sampled with the FE model and / or removing previously sampled FE positions that are far from the calibrated parameter set, collect FE responses for the newly sampled locations, and / or change the minimization settings used during calibration.

[0063] Example 1: A specific finite element model is used to calibrate the hyperelastic YEOH model available in Abaqus. In this example, the recorded force = function (time) matches between the test data and the simulation. The Nelder-Mead minimization algorithm is used. For the purposes of this example, the number of simulations is limited to 30; that is, the minimization is stopped after only 30 objective function evaluations, where each objective evaluation involves a previously run finite element model. As shown by Figure 16A, using existing calibration approaches, the minimization algorithm is unable to find a set of object parameters that can result in good agreement between the simulation results and the test data set. Instead, the data collected during the 30 finite element simulations is used to train a neural network-based surrogate model that predicts the force response as a function of time and object parameter values. Once trained, the surrogate model is automatically used in place of the finite element model during the minimization process. The evaluation time of the surrogate model is negligible (on the order of milliseconds) compared to the time previously required to run the finite element model. Therefore, when using a surrogate model, minimization can use multiple objective evaluations to find a set of parameter values ​​that result in a good match with the experimental data (Figure 16B). The coefficient of determination, R2, is used to quantify the goodness of fit. The coefficient of determination has a maximum value of R2 = 1 when the compared curves are in perfect agreement. (The smaller R2 is below 1, the worse the fit.) As shown in Figure 16B, 30 objective evaluations are performed using the collected simulation results to train an LSTM-NN surrogate to predict force = function (time, object parameters). The trained surrogate model is then automatically used in place of the finite element model during calibration. At the end of calibration, we run a final simulation using the final object properties. The simulation results (dashed curve) are in excellent agreement with the test data (dots). The goodness of fit is R2 = 0.99 using a coefficient of determination.

[0064] For the second example, three separate finite element models are used to calibrate (in Abaqus) the Holzapfel-Gasser-Ogden anisotropic hyperelastic object model, which is sometimes used to simulate the behavior of cardiac tissue. Here, separate neural network surrogate models are trained to reproduce the displacement = function (time, object properties) for each of the three finite element models.

[0065] As shown in Figure 17, 30 objective evaluations were performed, and the collected simulation results were used to train an LSTM-NN surrogate to predict displacement = function (time, object parameters). The trained surrogate model was then automatically used in place of the finite element model during calibration. At the end of calibration, we performed a final simulation using the finite element model with the final object properties. The simulation results (dashed lines) are in good agreement with the test data (dots). For all three datasets, the goodness of fit is R2 > 0.99 using the coefficient of determination.

[0066] In the third example, a single finite element model is used to calibrate the friction coefficient required by the Coulomb friction model used to describe the interaction between two contacting parts. As shown in Figure 18, 30 objective evaluations were performed, and the collected simulation results were used to train an LSTM-NN surrogate to predict the force = function (time, friction coefficient). The trained surrogate model was used in place of the finite element model during calibration. At the end of calibration, a final simulation was performed using the final value of the friction coefficient. The simulation results (dashed line) match the test data (dots). The goodness of fit is R2 > 0.999 using the coefficient of determination.

[0067] In the fourth example, a single finite element model was used to calibrate the parameters used by the failure criterion used to initiate and propagate a crack at the interface between two bonded parts. In this example, 30 objective evaluations were performed, and the collected simulation results were used to train an LSTM-NN surrogate to predict the force = function (time, failure criterion parameters). The trained surrogate model was then automatically substituted for the finite element model during failure model calibration. At the end of the calibration, we performed a final simulation using the final values ​​of the failure criterion parameters. The simulation results (blue curve) closely matched the test data (red dots). As shown in Figure 19, the goodness of fit was R2 > 0.87 using the coefficient of determination.

[0068] In the fifth example, a single finite element model was used to calibrate a Johnson-Cook plasticity model in Abaqus / Standard. Here, 30 objective evaluations were performed, and the collected simulation results were used to train an LSTM-NN surrogate to predict the nominal stress as a function of (time, object model parameters). The trained surrogate model was then automatically substituted for the finite element model during object behavior model calibration. At the end of the calibration, we performed a final simulation using the final values ​​of the object model parameters. The simulation results (blue curve) matched the test data (red dots). As shown in Figure 20, the goodness of fit was R2 = 0.999 using the coefficient of determination.

[0069] Existing literature provides examples of neural networks being used as new structural response models, with ANN models replacing existing traditional object models. For example, neural network models can be used to model hyperplastic objects. This type of application for neural networks has been developed in recent years. However, this is not how neural networks are used in the context of the above-described embodiment.

[0070] The system for performing the functions detailed above may be a computer, an example of which is shown in the schematic diagram of FIG. 21. The system 2100 includes a processor 2102, a storage device 2104, a memory 2106 with software 2108 stored therein defining the functions described above, input and output (I / O) devices (or peripherals), and a local bus or interface 2112 for enabling communication within the system 2100. The local interface 2112 may be, for example, but not limited to, one or more buses or other wired or wireless connections known in the art. The local interface 2112 may include additional elements for enabling communication, such as controllers, buffers (caches), drivers, repeaters, and receivers, which are omitted for simplicity. Furthermore, the local interface 2112 may include addresses, control means, and / or data connections for enabling appropriate communication between the above-mentioned components.

[0071] The processor 2102 is a hardware device for executing software, particularly software stored in the memory 2106. The processor 2102 may be any custom or commercially available single-core or multi-core processor, a central processing unit (CPU), a coprocessor among several processors associated with the system 2100, a semiconductor-based microprocessor (in the form of a microchip or chipset), a microprocessor, or generally any device that executes software instructions.

[0072] The memory 2106 may include any one or combination of volatile memory elements (e.g., random access memory (RAM, e.g., DRAM, SRAM, SDRAM, etc.)) and non-volatile memory elements (e.g., ROM, hard drive, tape, CD-ROM, etc.). Furthermore, the memory 2106 may include electronic, magnetic, optical, and / or other types of storage media. It should be noted that the memory 2106 may have a distributed architecture, where various components are located remotely from each other yet can be accessed by the processor 2102.

[0073] Software 2108 defines the functions performed by system 2100 in accordance with the present invention. Software 2108 in memory 2106 may include one or more individual programs, each of which includes an ordered list of executable instructions for implementing the logical functions of system 2100, as described below. Memory 2106 may also include operating system (O / S) 2120. An operating system essentially controls the execution of programs in system 2100 and provides scheduling, input / output control, file and data management, memory management, and communication control and related services.

[0074] The I / O devices 2110 may include input devices such as, but not limited to, a keyboard, a mouse, a scanner, a microphone, etc. Additionally, the I / O devices 2110 may also include output devices such as, but not limited to, a printer, a display, etc. Finally, the I / O devices 2110 may further include devices that communicate via both input and output, such as, but not limited to, a modulator / demodulator (a modem for accessing another device, system, or network), a radio frequency (RF) or other transceiver, a telephone interface, a bridge, a router, or other device.

[0075] When system 2100 is operating, processor 2102 is configured to execute software 2108 stored in memory 2106, to communicate data to and from memory 2106, and to generally control the operation of system 2100 in accordance with software 2108, as described above.

[0076] During operation of the functions of the system 2100, the processor 2102 is configured to execute software 2108 stored in the memory 2106, to communicate data to and from the memory 2106, and to generally control the operation of the system 2100 in accordance with the software 2108. The operating system 2120 is loaded by the processor 2102, possibly stored in a buffer within the processor 2102, and then executed.

[0077] It should be noted that when system 2100 is implemented in software 2108, instructions for implementing system 2100 may be stored on any computer-readable medium for use by or in connection with any computer-related device, system, or method. Such computer-readable medium may, in some embodiments, correspond to either memory 2106 or storage device 2104, or both. In the context of this specification, a computer-readable medium is an electronic, magnetic, optical, or other physical device or means that may contain or store a computer program for use by or in connection with a computer-related device, system, or method. Instructions for implementing the system may be embodied on any computer-readable medium for use by or in connection with a processor or other such instruction execution system, apparatus, or device. While processor 2102 has been described as an example, such an instruction execution system, apparatus, or device may, in some embodiments, be any computer-based system, processor-including system, or other system that may fetch instructions from and execute the instructions from the instruction execution system, apparatus, or device. In the context of this specification, a "computer-readable medium" may be any means that can store, communicate, propagate, or transport a program for use by or in connection with a processor or other such instruction execution system, apparatus, or device.

[0078] Such a computer-readable medium may be, for example, but not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, device, or propagation medium. More specific examples (a non-exhaustive list) of computer-readable media would include the following: an electrical connection having one or more wires (electronic), a portable computer diskette (magnetic), a random access memory (RAM) (electronic), a read-only memory (ROM) (electronic), an erasable programmable read-only memory (EPROM, EEPROM, or flash memory) (electronic), an optical fiber (optical), and a portable compact disc read-only memory (CDROM) (optical). It should be noted that the computer-readable medium may even be paper or another suitable medium on which the program is printed, since the program is captured electronically, for example, via optical scanning of paper or other medium, and then compiled, interpreted, or otherwise processed in an appropriate manner as needed, and then stored in computer memory.

[0079] In alternative embodiments, system 2100 is implemented in hardware and may be implemented using any one or combination of the following technologies, each of which is well known in the art: discrete logic circuits having logic gates that perform logical functions in response to data signals, application specific integrated circuits (ASICs) having appropriate combinatorial logic gates, programmable gate arrays (PGAs), field programmable gate arrays (FPGAs), etc.

[0080] The present embodiment relates to a substantial performance enhancement of existing finite element-based calibration workflows, where surrogates can be automatically generated and trained during a calibration run to reproduce specific Abaqus history outputs indicated by a user to be matched, for example, for a test data volume. For example, a particular type of tested surrogate model was based on using a neural network formed from a stack of one, two, or more LSTM cells followed by a feedforward neural network. Test data time and sampled parameter values ​​of a constitutive model can be provided as inputs to the surrogate model, the output of which was a prediction of the Abaqus history output as a function of time.

[0081] In this embodiment, multiple NNs (one for each test data set) can be trained independently. If any one of the NNs is not accurate, it can be removed from the calibration. The objective function used by the minimization algorithm can be switched on the fly between using finite element simulations and using surrogate models during evaluation of the objective function. In this embodiment, the parameters of the constituent response can be sampled using a variety of approaches. A particularly advantageous approach is to use iterative Bayesian optimization.

[0082] The illustrative embodiments may significantly reduce the time required to determine the parameters of a constitutive response model when computationally expensive finite element models are used during the calibration process.

[0083] As a simple example, if a finite element simulation used during the calibration process requires 15 minutes to run once, then the estimated total time for calibration, which requires running the same simulation 200 times, would be approximately 3000 minutes. However, using this embodiment, the same simulation only needs to be run a significantly fewer number of times to obtain acceptable results, e.g., 20-50 times instead of 200 times.

[0084] In some cases, finite element models can take hours or more to run once, and calibration may require running the same simulation thousands of times. In such cases, the present embodiment may provide the only practical way to calibrate the constitutive response model.

[0085] An alternative embodiment may employ a Transformer neural network. The Transformer neural network architecture avoids repetition and instead relies solely on attention mechanisms, which allows the network to capture dependencies across variable-length sequences, regardless of their distance within the input or output sequences. Transformers may be used in a manner similar to LSTMs and LSTM-NNs to generate surrogate models to reproduce historical outputs extracted from finite element simulations used during the calibration of constitutive responses with history-dependent object responses.

[0086] It will be apparent to those skilled in the art that various modifications and variations can be made to the structure of the present invention without departing from the scope or spirit of the invention. In view of the foregoing, it is intended that the present invention cover modifications and variations of this invention provided they come within the scope of the following claims and their equivalents.

Claims

1. 1. A method for reducing the time to perform constitutive response model calibration by automatically employing a neural network based surrogate model in place of an output from a corresponding finite element simulation, comprising: receiving test data about the object, the test data including output from an experiment; receiving a finite element model that simulates the experiment; receiving a user selection of the configured response model; receiving a user selection of historical output quantities for collection during the simulation; receiving a user selection of a numerical minimization algorithm for calibration of the constitutive response model; receiving a user selection of an error metric for quantifying the difference between the test data and the output of the finite element simulation; receiving a user selection of parameters of the constituent response model; receiving a user selection of a parameter space sampling technique during an initial calibration phase when finite element outputs for calculating an objective function have been collected; receiving a user selection of settings for the architecture of the surrogate model to be trained to reproduce the collected finite element outputs; receiving a user selection of a training algorithm for the surrogate model; performing an automated calibration of the constituent response model; A method comprising:

2. The method of claim 1 , wherein the surrogate model is generated with a long short-term memory neural network.

3. The method of claim 1 , wherein the received test data includes at least one of the group of: stress data as a function of time and / or deformation; and force as a function of time and / or deformation.

4. performing a plurality of said finite element simulations using said finite element model at locations determined by said selected sampling or said minimization algorithm; collecting finite element history outputs from the plurality of simulations; training a neural network to approximate the finite element output in place of the finite element simulation; The method of claim 1 further comprising:

5. replacing the finite element simulation with a surrogate model that includes the trained neural network; and generating a surrogate history output based on the surrogate model to produce a set of calibrated parameters for a selected constituent response; The method of claim 4 further comprising:

6. Returning to using the finite element model; evaluating the results of the finite element model against the surrogate model; The method of claim 5 further comprising:

7. The method of claim 4 further comprising storing the finite element history.

8. The method of claim 4 , wherein the neural network comprises a stack of LSTM cells followed by a feedforward neural network.

9. The method of claim 1 , wherein the user selection of a parameter of the constitutive response model indicates minimum and / or maximum bounds for the parameter.

10. The method of claim 1 , wherein the parameter space sampling technique comprises Bayesian minimization.

11. The method of claim 10 , further comprising receiving a user-specified number of Bayesian iterations.

12. The method of claim 1 , further comprising receiving a user-specified number of sampling points.

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