Method for accelerating numerical calibration of constitutive response parameters using proxy model

Abstract

A method for constitutive response model calibration employs a neural network-based proxy model in place of the output from a corresponding finite element simulation. When the application program receives user selection, the constitutive response model is calibrated, user selection comprises test data output from an experiment, a finite element model for simulating the experiment, constitutive response model selection, historical output quantity, a numerical value minimization algorithm, error measurement, constitutive response model parameters, error measurement of test data relative to finite element simulation output, constitutive response model parameters and a parameter space sampling technology. Architecture setting of an architecture of the proxy model, and a training algorithm for a proxy model training algorithm.

Description

Technical Field

[0001] The present invention relates to parameterization of materials, and more particularly to improving the efficiency for calibrating constitutive response parameters. Background Art

[0002] A constitutive response model (CRM) is a mathematical model used to describe the behavior of real materials, structures, or interactions between structures. CRMs represent an essential part of the finite element (FE) simulation process. CRMs rely on certain parameters being accurately specified. As a simple example, when simulating a steel structure in an elastic state, the user might select isotropic elasticity as the constitutive response model and specify accurate values ​​for the elastic modulus and Poisson's ratio.

[0003] Although constitutive models often refer to material behavior, the concept of constitutive response can be extended beyond material behavior. For example, a simplified mathematical model can be used to approximate a mechanical component during a finite element simulation, for example, by using specially formulated finite elements (e.g., spring elements, connector elements, etc.). Such specialized elements use special constitutive response models designed to replicate the behavior of the structure. Another type of CRM can be specifically used to model the interaction between contacting surfaces during a finite element simulation, such as friction and wear models, and adhesive / adhesive contact models (for simulating glued or sticky surfaces). Some constitutive response models may be more complex, spanning multiple physical domains, such as coupled electrical-thermal-displacement responses.

[0004] In many cases, real materials and structures need to be tested experimentally in the laboratory to obtain the parameters of the constitutive response model that will be used for finite element simulation. Typically, multiple experimental setups are used to obtain sufficient experimental data. Each experimental setup attempts to isolate the behavior of a material or structure under a specific deformation mode or a specific set of loading conditions. During each experiment, the observed behavior of the material or structure is recorded as a function of time, load or deformation. Such a record is called a test data set. For example, a typical test data set may include experimentally measured stresses as a function of time and / or applied strain, or may include experimentally measured forces as a function of time and / or applied displacement, etc. Figure 1A and Figure 1B shown.

[0005] like Figure 1A As shown, the experiment can be used to measure and record strain-stress test data sets for certain deformation modes (e.g., uniaxial, biaxial, planar (pure shear), volumetric, and simple shear). Figure 2 Depicted is an example of an experimentally recorded strain-stress test data set.

[0006] Such experimentally recorded test data sets cannot be directly used in finite element simulation models. Instead, the typical approach is to select an appropriate constitutive response model and calibrate the parameters of the constitutive response model so that certain simulation output quantities are closely consistent with the experimentally recorded test data.

[0007] In cases where an experimental test dataset is available, typically the user will follow these broad steps to obtain a calibrated constitutive response:

[0008] 1. Create a finite element model for each experimental setup. For example, you can create separate models for uniaxial deformation, biaxial deformation, and so on. Such finite element models can be simple (containing a single element) or complex (having multiple interacting mesh sections, each containing a large number of elements).

[0009] 2. Select initial guesses for the constitutive response parameters.

[0010] 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 (especially for complex models) may be very long, such as on the order of tens of hours.

[0011] 4. Extract the obtained data from the simulation output database and compare the simulation output with the corresponding test data recorded by the experiment.

[0012] 5. If the simulation results and experimental results have good agreement (i.e., the curve corresponding to the simulation output and the curve representing the experimental test data overlap or are sufficiently close to each other), the calibration process can be stopped. The agreement between the experimental test data and the simulation output is usually quantified mathematically using an error norm function (e.g., a mean squared error (MSE) function). If the simulation output matches the experimental test data, these error norms take a certain minimum value. In particular, if the simulation output is very close to the experimental test data, the MSE value will be very close to zero. If the simulation results do not match the experimental results, the constitutive response parameters must be modified and the simulation rerun (using the modified parameters) until sufficiently good agreement is obtained (e.g., the MSE is close to zero).

[0013] Initially, this approach was often performed manually, but more recently, numerical minimization algorithms have been employed, using one of several known minimization algorithms. For the numerical minimization process, the minimization algorithm automatically tries new parameter values ​​for the constitutive response model so that the error between the simulation results and the corresponding experimental data is iteratively reduced until the simulation results are sufficiently consistent with the experimental data.

[0014] Figure 3An example of comparing the calibrated strain-stress response with input test data for uniaxial, biaxial, and planar deformation modes is shown, where each deformation mode corresponds to a different set of loading conditions (i.e., Figure 1A ( Schematic representation of different experimental setups in ). In this paper, the process of identifying the parameter values ​​of a constitutive response using a numerical minimization algorithm, either manually or through an automated iterative approach, is referred to as parameter calibration of that constitutive response. Calibration workflows using numerical minimization algorithms can be found in existing software, including the 3DX Material Calibration application.

[0015] The time required for an automated calibration workflow depends largely on the complexity and duration of the finite element simulations required. Computation times can range from a few seconds to many hours or even days, potentially requiring the use of significant computing resources. Users may also calibrate multiple constitutive response models until a suitable one is identified, in which case the entire calibration process must be repeated. Therefore, there is a need in the industry to reduce the time required to run constitutive response calibrations. Summary of the Invention

[0016] Embodiments of the present invention provide a method for accelerating the numerical calibration of computationally expensive constitutive response parameters using a proxy model. Briefly, the present invention relates to a method for calibrating a constitutive response model that employs a neural network-based proxy model to replace outputs from corresponding finite element simulations. An application calibrates the constitutive response model upon receiving user selections, wherein the user selections include test data output from an experiment, a finite element model that simulates the experiment, a constitutive response model selection, historical output quantities, a numerical minimization algorithm, an error metric, constitutive response model parameters, an error metric of the test data relative to the finite element simulation output, the constitutive response model parameters, a parameter space sampling technique, an architectural setup for the proxy model architecture, and a training algorithm for the proxy model training algorithm.

[0017] By examining the following drawings and detailed description, other systems, methods and features of the present invention will be or become apparent to one of ordinary skill in the art. All such additional systems, methods and features are intended to be included within this description, fall within the scope of the present invention and be protected by the following claims. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] This 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.

[0019] The accompanying drawings are included to provide a further understanding of the present invention and are incorporated into and constitute a part of this specification. The components in the drawings are not necessarily drawn to scale, emphasis instead being placed upon clearly illustrating the principles of the present invention. The accompanying drawings illustrate embodiments of the present invention and, together with the description, serve to explain the principles of the present invention.

[0020] Figure 1A is a graphical representation of a typical experiment 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.

[0021] Figure 1B is a diagrammatic representation of an experiment used to provide experimentally measured shear stress.

[0022] Figure 2 is a graph covering uniaxial, biaxial, and planar test data showing stress versus strain.

[0023] Figure 3 Graphs of experimental test data and simulation responses corresponding to three deformation modes: uniaxial, biaxial, and planar are shown.

[0024] Figure 4 is a flow chart of an exemplary first method embodiment.

[0025] Figure 5 yes Figure 4 Flowchart of details of an exemplary first method embodiment.

[0026] Figure 6A are screenshots illustrating an example of importing experimental test data according to the first embodiment.

[0027] Figure 6B is a screenshot of an example of a user importing an experimental test dataset into the 3DX Material Calibration application.

[0028] Figure 6C is a graph showing selected test data during the import process, where the experimental nominal stress is plotted as a function of time.

[0029] Figure 6D Imported experimental test data is shown after importing into the application for editing and display.

[0030] Figure 6E Shown are multiple experimental test data sets plotted inside the application. After importing, the user can select which test data sets to use for calibration.

[0031] Figure 7AAn example of a set of three finite element models used together during the same calibration of a plastic part using a damage constitutive response model is shown.

[0032] Figure 7B An example of an Abaqus finite element model used to simulate a double cantilever beam experiment is shown.

[0033] 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.

[0034] Figure 7D An exemplary set of two finite element models used together to calibrate connector behavior is shown.

[0035] Figure 8 is a screenshot showing an exemplary list of constitutive response models for user selection.

[0036] Figure 9 is an annotated screenshot showing an exemplary calibration setup using three finite element models.

[0037] Figure 10 is a screenshot showing a user selecting one of several available numerical minimization algorithms to be used for calibration.

[0038] Figure 11 is a screenshot showing how a user may select one of the available error metrics.

[0039] Figure 12 is a screenshot of an exemplary scene showing a user-selected constitutive material model.

[0040] Figure 13 is a screenshot illustrating an exemplary scenario in which a user is presented with options for modifying default settings related to initial sampling of parameter space.

[0041] Figure 14 Highlight Figure 13 Portion of a screenshot indicating user selection of settings related to the architecture of the surrogate model to be trained to replicate finite element output.

[0042] Figure 15 are screenshots and graphs comparing the finite element response using the calibration parameters with the experimental test data.

[0043] Figure 16A is a graph from a first experiment showing the results based on the previous calibration technique.

[0044] Figure 16Bis a graph from the first experiment showing the results based on this example.

[0045] Figure 17 are three graphs from a second experiment showing the results based on this example.

[0046] Figure 18 is a graph from the third experiment, which shows the results based on this example.

[0047] Figure 19 is a graph from the fourth experiment, which shows the results based on this example.

[0048] Figure 20 is a graph from the fifth experiment, which shows the results based on this example.

[0049] Figure 21 is a schematic diagram illustrating an example of a system for performing the functions of the present invention. DETAILED DESCRIPTION

[0050] The following definitions may be used to explain terms applied to features of the embodiments disclosed herein and are meant only to define elements within the present disclosure.

[0051] As used in this disclosure, "artificial neural network" (ANN) refers to a mathematical model that is typically used to solve (i) classification problems and (ii) regression problems. This document describes the use of neural networks to solve regression problems. When an ANN is used to approximate or replicate an existing response or data set, the ANN may be called a surrogate model. A surrogate model is a mathematical model that can be created in different ways and may not necessarily be a neural network. A surrogate model can be viewed as a mapping between a set of inputs and a set of outputs. There are many types of ANN architectures, for example, feedforward neural networks (FFNNs), recurrent neural networks (RNNs), and transformers, among others. An ANN contains multiple internal parameters, including "weights" and "biases." The number of weights and biases depends on the size of the ANN and can range from tens of parameters to trillions of parameters.

[0052] As used in this disclosure, "ANN training" refers to the process of calibrating an ANN. Regardless of the ANN architecture, before the ANN can actually be used in a meaningful way, the internal ANN parameters (i.e., weights and biases) need to be calibrated. Although this calibration process is referred to as "training" in the ANN literature, the ANN training process is conceptually similar to constitutive response model calibration. ANN training also uses numerical minimization algorithms that modify internal ANN parameters (i.e., weights and biases) until the ANN predictions match a certain level of accuracy of some data it needs to replicate.

[0053] As used in this disclosure, "Bayesian minimization" refers to a design parameter improvement and optimization framework intended for costly "black box" type objective functions (e.g., time-consuming simulations). The primary goal of a Bayesian minimizer is to obtain the best possible set of design parameters using a relatively small number of objective function evaluations (e.g., less than 100 objective function evaluations). The method relies on (i) creating and using a response surface to approximate the objective function, and (ii) an acquisition function for predicting the next sampling point to be attempted using the costly objective function. The method involves evaluating a measure of "uncertainty" when using a response surface to approximate a costly objective function.

[0054] As used in this disclosure, "Long Short-Term Memory Neural Network (LSTM-NN)" refers to a specific class of recurrent neural networks. There are different variations of LSTM neural networks in the literature. An LSTM neural network can be architecturally a stack of two or more LSTM units.

[0055] As used herein, "historical output" is a quantity provided by a simulation (based on user request) as a function of time, for example, stress as a function of time at a point in a finite element model, reaction forces recorded as a function of time during a simulation, etc.

[0056] Embodiments herein describe methods for creating a surrogate model using a long short-term memory (LSTM)-based NN architecture to replicate historical outputs extracted from finite element simulations used during constitutive response calibration.

[0057] Reference will now be made in detail to the embodiments of the present invention, examples of which are illustrated in the accompanying drawings. Wherever possible, the same reference numerals are used in the drawings and the description to refer to the same or like parts.

[0058] Exemplary embodiments of the present invention are used to accelerate the calibration of constitutive models employed during finite element simulations. Embodiments significantly reduce the time required to identify parameters for constitutive response models, particularly for scenarios where time-consuming finite element simulations are used to replicate experimental tests. Thus, embodiments provide a practical approach for scenarios where even a single simulation can take hours or longer to complete.

[0059] The calibration workflow embodiment described herein provides an iterative process using a numerical minimization algorithm. The user provides an experimental test data set and a finite element model as part of the input to the calibration workflow. The finite element model is used to simulate the test experiment. During the iterative calibration process, the historical output provided by the finite element simulation is compared with the corresponding experimental test data in an iterative manner until the two are consistent. This process may involve running the finite element simulation multiple times (e.g., from hundreds to tens of thousands of times or more) and may take a long time.

[0060] Embodiments provide a method in which the time required to perform constitutive response model calibration can be significantly reduced by automatically employing a neural network-based proxy model that replaces the otherwise required output from a corresponding finite element simulation. As a non-limiting example, embodiments may use a specific long short-term memory (LSTM)-based neural network to create the proxy model. Depending on the specific constitutive response model, the choice of neural network can be any neural network architecture, or a network of mathematical functions connected and / or tailored to process sequence data, including recurrent neural networks / recurrent networks, feedforward neural networks, and customized / combined transformers of different architectures.

[0061] Figure 4 is a flow chart of an exemplary material calibration workflow method. It should be noted that any process description or block in the flow chart should be understood to represent a module, segment, code portion or step that includes one or more instructions to implement the specific logical functions in the process, and alternative implementations are included within the scope of the present invention, wherein, depending on the functions involved, the functions may not be performed in the order shown or discussed, including being performed substantially simultaneously or in reverse order, as will be understood by those with reasonable skill in the art of the present invention.

[0062] The method embodiments described herein extend and enhance material calibration workflows using finite element models, such as implemented in the 3DX material calibration application. While the embodiments relate to calibration of material behavior, alternative embodiments can also be used to calibrate the constitutive response of special elements (e.g., adhesive elements, connector elements, spring elements, gasket elements, etc.) and the constitutive interface response of contact surfaces (e.g., adhesive contact, wear, friction, etc.), and even calibrate fracture models.

[0063] In the following description, the terms "optimization" and "minimization" are used interchangeably. Although optimization can refer to "minimization" or "maximization" mathematically, it is trivial to switch between minimization and maximization by multiplying the objective function to be minimized / maximized by (negative) -1. In addition, the term "finite element (FE) response" or "FE simulation response" refers to an output quantity that is part of the results provided by a simulation.

[0064] In an exemplary calibration method embodiment, as shown in block 410, test data related to a material is received. The test data can be, for example, output from an experiment. As shown in block 420, a finite element model for simulating the experiment is received. As shown in block 430, a user selection of a constitutive response model is received. As shown in block 440, a user selection of historical output quantities collected during the simulation is received. As shown in block 450, a user selection of a numerical minimization algorithm for calibrating the constitutive response model is received. As shown in block 460, a user selection of an error metric for quantifying the difference between the test data and the finite element simulation is received. As shown in block 470, a user selection of parameters for the constitutive response model is received. As shown in block 480, a user selection of a first technique for sampling the parameter space during an initial calibration phase and a second technique for collecting finite element outputs for computing the objective function are received. As shown in block 490, a user selection of settings related to the architecture of a surrogate model to be trained to replicate the finite element outputs is received, as well as a user selection of a training algorithm for the surrogate model. As shown in block 500, automatic constitutive response model calibration is performed. Each of blocks 410 through 500 is described in further detail below.

[0065] As shown in block 410, test data related to the material is received. A user of the simulation application provides a relevant test data set previously obtained, for example, through experimentation. It should be noted that the experimental operations required to obtain the test data set are outside the scope of this embodiment. Figure 6A is a screenshot showing an example of importing experimental test data into the 3DX Material Calibration application. The test data set 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, provided that such quantities can be obtained from the simulation as direct output or based on post-processing of the simulation output. The test data set can be obtained in the laboratory using material samples or measured using sensors placed on intact components and structures. Existing experimental techniques for obtaining test data sets as input to the methods described herein include, for example, universal testing machines, environmentally controlled chambers, digital image correlation, and the like. Figure 6B This screenshot shows an example of a user importing an experimental test dataset into the 3DX Material Calibration application. Here, the user has selected the test data of interest (from an Excel file) to import. In this particular example, the selected experimental test dataset includes time, nominal stress, and nominal strain. The user can import multiple test datasets from the same file or from multiple files.

[0066] The test data can be presented in a tabular format. On a computer, the test data set can be stored, for example, in a spreadsheet file, a CSV (comma-separated values) list, or even a text file. For example, the 3DX Material Calibration application enables users to import test data sets from spreadsheet files, CSV files, and TXT files. Multiple test data sets can be stored in a single file or in separate files. While this embodiment describes and depicts the 3DX Material Calibration application, alternative embodiments may involve other platforms, such as (but not limited to) Abaqus / CAE, or be invoked through a command line prompt.

[0067] Once the test dataset has been imported into the 3DX Material Calibration Application, the user can optionally edit the contents of the test dataset to suit the calibration purposes. These edits may include removing duplicate data points, reducing the number of test data points by decimation or deletion, smoothing the test dataset to remove noise, and regularizing the test dataset (regenerating the data at equally spaced intervals).

[0068] Figure 6C A graph is shown which illustrates selected test data during the lead-in process, where the experimental nominal stress is plotted as a function of time (recorded during a previous experiment). Figure 6D The imported experimental test data is shown for editing and display after being imported into the application. Here, the user can manually delete points, or can use one of the available operations (e.g., decimation, smoothing, regularization) to improve the quality of the test dataset before it is used for calibration. Figure 6E Shown are multiple experimental test data sets plotted inside the application. After importing, the user can select which test data sets to use for calibration.

[0069] As shown in block 420( Figure 4 ), a finite element model for simulating an experiment is received. The user creates and provides a finite element model as input for simulating the laboratory experiment. Calibration can use one or more finite element models simultaneously. The process of creating the finite element model is not part of the present invention and needs to be performed in advance. Existing pre-processing software allows users to create finite element models. For example, the finite element model can be created in the 3DX Mechanical Scene application or Abaqus / CAE.

[0070] Exemplary formats for describing finite element models in the context of the embodiments 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, for public record. Existing translators between different finite element model formats can be used, for example, to translate a finite element model generated in a format specific to a commercial software into the Abaqus input file format (.inp). For example, Abaqus input files can be translated from various commercial (non-Simulia) software, as shown in Table 1:

[0071] Table 1

[0072]

[0073] Conversely, Abaqus input files (.inp) can be converted to other (non-DS / Simulia) formats.

[0074] Figure 7A Shown is an example of a set of three finite element models used together during the same calibration of a plastic part using a damage constitutive response model (the finite element mesh is not shown in this figure).

[0075] Figure 7B An example of an Abaqus finite element model used to simulate a double cantilever beam experiment is shown. This type of FE model can be used to calibrate the constitutive response of the bonded interface.

[0076] 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. Anisotropic hyperelastic response can be used (among other applications) to model the response of human tissue (e.g., cardiac tissue) in FE simulations.

[0077] Figure 7D An exemplary set of two finite element models used together to calibrate connector behavior (e.g., a constitutive response model specific to connector elements) is shown. Connector elements herein refer to elements used to model connections between components (e.g., clinching joints, self-piercing rivets, spot welds, etc.).

[0078] As shown in block 430 ( Figure 4), a user selection of a constitutive response model is received. The user selects the constitutive response model of interest (e.g., available on the Abaqus or 3DX platform). In the context of this embodiment, the constitutive response model can be used in the sense of material behavior (e.g., elastic, hyperelastic, viscoelastic, plastic material behavior). However, here, the constitutive response model can also refer to a mathematical model used to describe the interaction between contact surfaces, such as an adhesive or adhesion interaction model, friction between surfaces, wear, etc. The constitutive response model can also mean a special mathematical model used to define the behavior of a special type of finite element (e.g., connector element, spring element, adhesive element, etc.). For example, an adhesive element can be used to simulate fracture and delamination. The constitutive response of an adhesive element can be represented by a traction-separation law. Figure 8 An exemplary list of constitutive response models for the user to select is shown. Each constitutive response model has a specific set of parameters that can be calibrated using an associated experimental test data set. A mathematical description of each of these constitutive models is provided in the software documentation.

[0079] As shown in block 440( Figure 4 ), receives a user selection of historical output quantities collected during simulation. For each finite element model to be used during calibration, the user can specify one or more historical output quantities to be collected during simulation. Historical output refers to output stored as a function of time at a specific model location or for the entire model. Each experimental test data quantity can be used with a corresponding quantity obtained from one or more collected FE simulation historical outputs to define an objective function.

[0080] Figure 9 is an annotated screenshot showing an exemplary calibration setup using three FE models. For each FE model, the user can select one or more FE outputs that will be matched during calibration to corresponding experimental quantities contained in one of the imported test datasets. Figure 9 Indicates that for the FE model named "adventitia_axial.inp", the user wishes to extract the FE history output representing the displacements U1 at the nodal locations defined by the node set named "nLoad" during FE simulation. During calibration, this FE output (U1) will be matched to the experimental displacement values ​​extracted from the imported test dataset named "axial_small". Figure 9 In [ 15 ], the user indicates that the FE output first needs to be scaled by a factor of 2 before matching. The error metric between the scaled FE output and the experimental quantity can be weighted by a user-specified weight value.

[0081] A user selection of a numerical minimization algorithm for calibrating the constitutive response model is received, as shown in block 450. The user-selected numerical minimization algorithm is used during calibration of the constitutive response model, along with any associated settings. Figure 10 is a screenshot showing the user selecting one of several available numerical minimization algorithms to be used for calibration. The user can modify the default settings if desired. Some settings are common to all algorithms, but some settings may be specific to individual algorithms. The terms "minimize" and "optimize" are used interchangeably in the dialog window.

[0082] As shown in block 460, a user selection of an error metric for quantifying the difference between the test data and the finite element simulation is received. The user selects an error metric (or "error norm") to be used to quantify the difference ("mean squared error") between the test data set and the corresponding finite element simulation. The error metric is a function used to form the objective function.

[0083] Equation 1 illustrates how the objective function F(X) can be obtained for a general calibration case employing multiple finite element (FE) models. For each FE model, any number of historical outputs can be collected from the simulation. A post-processing function can be used to combine the collected outputs. The value provided by the post-processing function is compared with the experimentally recorded values ​​for the corresponding amount of test data using a user-indicated error metric. The error metric is a function that quantifies the difference between two sets of values ​​(in this example, the difference between the experimental values ​​and the simulated values). Examples of error metrics are mean squared error, mean absolute error, coefficient of determination, etc. The value returned by the error metric can optionally be scaled by a user-assigned weight. Penalty values ​​can be added to the objective function to account for failed finite element simulations (e.g., due to convergence issues), violations of various constraints, etc. The time t in Equation 1 can be real time or artificial time used to index the applied load increments.

[0084]

[0085] in:

[0086] t = time, X = vector containing the values ​​of the constitutive parameters,

[0087]

[0088] E = error metric function (e.g., mean square error, etc.),

[0089]

[0090] w j；i = weight value for the ith comparison and the ith FE model.

[0091] For the simplest case, there is a single output quantity (m=1) and no output processing is performed, that is,

[0092] Figure 11 is a screenshot showing the mechanism for the user to select the available error metrics. The error metric is a function used to quantify the difference between the experimental test data and the finite element response.

[0093] As shown in block 470, a user selection of a parameter set for the constitutive response model is received. The user selects the parameters of the constitutive response model to be calibrated. In addition, the user may optionally indicate minimum and maximum limits for each parameter. The user may specify initial values ​​for each parameter. The user may optionally use the initial values ​​of the constitutive response parameters to perform an evaluation based on a single FE model.

[0094] Figure 12 An example scenario is shown for a user-selected anisotropic Holzapfel-Gasser-Ogden hyperelastic constitutive material model for simulating cardiac tissue. The 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 (01 / Pa) and is not calibrated. The remaining parameters will be calibrated. For each parameter to be calibrated, the user provides an initial guess and a range (defined by minimum and maximum limits). The graph (bottom) shows a comparison between the experimental test data and the FE output (i.e., the FE response) when using the initial guess for the parameter to be calibrated. In this case, the initial values ​​are far from the "true" values, and the FE curve and the experimental curve do not match well.

[0095] As shown in box 480 ( Figure 4 ), the user selects the technique to be used in the initial phase of calibration to sample the parameter space and collect the finite element outputs required to calculate the objective function. The selected technique can be a sampling technique (design of experiment technique) or an existing minimization algorithm. Examples of sampling techniques include random sampling and Latin hypercube sampling. The preferred minimization algorithm that can be used in this step is Bayesian minimization. Bayesian minimization is a numerical algorithm that is intended to be used for expensive objective function evaluations when trying to obtain an acceptable solution using a small number of function evaluations. Bayesian minimization can improve the sampling using points where the greatest improvement is expected to be achieved. Optionally, the user can reuse a saved FE response evaluated during a previous calibration.

[0096] Figure 13A scenario is shown in which the user is presented with the option to modify default settings related to the initial sampling of parameter space. As shown, the UI presents a scenario where a Latin hypercube method followed by Bayesian iterations is advantageous. In this case, the user can specify the number of sampling points and the number of Bayesian iterations. However, in alternative embodiments, other sampling methods (not shown) can be readily employed. The user can optionally reuse previously sampled data (data collected during a previous calibration).

[0097] As shown in box 490( Figure 4 ), receives a user selection of settings related to the architecture of a proxy model to be trained to replicate the finite element output, and receives a user selection of a training algorithm for the proxy model. The architecture of interest (but not limited to the methods described herein) is based on a stack of long short-term memory (LSTM) neural networks (NNs) connected to an output feedforward network. Non-default settings may include the number of LSTM units used in the NN stack, the number of units per unit, the activation function used for the output feedforward NN, etc. Settings specific to NN training may include settings related to the minimization algorithm to be used to train the NN.

[0098] Figure 14 An example of how a user can modify the default settings of the control NN architecture is shown. For usability purposes, Figure 14 It is shown that only a small number of settings are exposed to the user in the UI. Alternative embodiments may display other features, for example, settings specific to NN training.

[0099] Figure 5 is Figure 4 500, which shows the actions of an embodiment for constitutive response calibration. Figure 4 Blocks 410 through 490 receive user input.

[0100] By sampling the parameter space based on the evaluation of the objective function using the FE model (see the description of block 480 above), the required FE responses are collected and stored internally. In addition, as shown in block 510, previous FE responses (collected during a previous calibration) can optionally be loaded from an external file. A small number (typically 20 to 40, typically less than 100) of finite element simulations need to be performed using the finite element model provided by the user at the sampling locations. 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 historical output quantities of interest provided by the simulation (i.e., for evaluating the objective function) are collected and stored internally in stages. The collected FE historical output quantities can be saved to disk, for example, and (based on user selection) can be reused for subsequent proxy-based calibrations.

[0101] As shown in block 520, one or more neural networks are trained (using typical NN training methods that rely on numerical minimization algorithms (e.g., stochastic gradient descent, Adam, BFGS, etc.) to accurately replicate the FE outputs of interest without the need to run a finite element simulation. This stage is important because, depending on the actual neural network architecture, the proxy model may or may not accurately replicate the outputs from the finite element simulation. In particular, a stack of LSTM units is followed by a feed-forward NN to replicate a single historical output. However, it is possible that other proxy types may also provide accurate results. Once trained, the trained neural network is called a proxy model. The user can switch between using a proxy model (very fast evaluation) and using a finite element model (slower evaluation, but guaranteed to be accurate).

[0102] As shown in block 530, during the iterative minimization process, the surrogate model replaces the more computationally expensive finite element simulation to provide historical output quantities (during the minimization) used to calculate the objective function. The result is a calibrated parameter set for the selected constitutive response.

[0103] As shown in block 540 , the calibrated constitutive response parameters are evaluated using the FE model for evaluation (instead of the proxy) to assess the accuracy of the proxy model and provide the user with FE history output corresponding to the calibrated constitutive response parameters.

[0104] Figure 15 Here are screenshots and graphs comparing the FE response using the calibration parameters with the experimental test data. Here, the calibration parameters for the constitutive response are reported to the user along with the FE response. Figure 15 In the example shown, the calibration parameters of the FE response are very close to the experimental test data.

[0105] As shown in block 550, the response predicted by the agent is compared to the FE response, and the user is provided with the FE response and a diagnostic related to the consistency between the FE response and the agent prediction for the calibration parameter set. If the FE response and the agent prediction are consistent, the user can accept the calibration parameters (block 570). Otherwise, the user can make one or more modifications (block 560) before rerunning blocks 510 to 540. For example, the user can modify the agent architecture (e.g., change the number of LSTM units, modify the number of connections between units), change the agent training settings and training algorithm, modify the sampled FE positions by adding more positions to be sampled using the FE model and / or removing previously sampled positions that are far from the calibration parameter set, collect FE responses for the newly sampled positions, and change the minimization settings used during calibration.

[0106] 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) is matched between the test data and the simulation. The Nelder-Mead minimization algorithm is used. For the purpose of this example, the number of simulations is limited to 30, i.e., the minimization is stopped after only 30 objective function evaluations, where each objective evaluation consists of one run of the finite element model. Figure 16A As shown, using the pre-existing calibration method, the minimization algorithm was unable to find a set of material parameters that could provide a good match between the simulation results and the test data set. Instead, the data collected during the 30 finite element simulations were used to train a NN-based surrogate model that predicts the force response as a function of time and material parameter values. Once trained, the surrogate model is automatically used during the minimization process instead of the finite element model. The evaluation time of the surrogate model is negligible (on the order of a few milliseconds) compared to the time required to run the finite element model once. Therefore, when using the surrogate model, the minimization can use a large number of objective evaluations, and the minimization can find a set of parameter values ​​that provides a good match with the experimental data ( Figure 16B The coefficient of determination, R2, is used to quantify the fit. If the compared curves match perfectly, the coefficient of determination has a maximum value of R2 = 1. (The smaller the R2 < 1, the worse the fit.)

[0107] like Figure 16B As shown, 30 objective evaluations were run using the collected simulation results to train an LSTM-NN agent to predict Force = Function (time, material parameters). The trained agent model was then automatically used during calibration instead of the finite element model. At the end of calibration, a final simulation was performed using the final material properties. The simulation results (dashed curve) matched the test data (points). With a coefficient of determination of R² = 0.99, the fit was good.

[0108] For the second example, three separate finite element models were used to calibrate the HOLZAPFEL-GASSER-OGDEN anisotropic hyperelastic material model (in Abaqus). This material model is sometimes used to simulate the behavior of cardiac tissue. Here, a separate NN agent model was trained to replicate the displacement = function (time, material properties) for each of the three finite element models.

[0109] like Figure 17 As shown, 30 objective evaluations were run, using the collected simulation results to train an LSTM-NN agent to predict displacement = Function (time, material parameters). The trained agent model was then automatically used during calibration instead of the finite element model. At the end of calibration, the final simulation was performed using the finite element model using the final material properties. The simulation results (dashed line) matched the test data (points) well. A coefficient of determination R2 > 0.99 was used for all three data sets, indicating a good fit.

[0110] In a third example, a single finite element model is used to calibrate the friction coefficient required for the Coulomb friction model used to describe the interaction between two contacting parts. Figure 18 As shown, 30 objective evaluation runs were performed, using the collected simulation results to train an LSTM-NN agent to predict force = function (time, friction coefficient). The trained agent model was used during calibration instead of the finite element model. 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). With a coefficient of determination (R²) > 0.999, the fit is good.

[0111] In the fourth example, a single finite element model is used to calibrate the parameters used by the fracture criterion for the initiation and propagation of cracks at the interface between two bonded parts. In this example, 30 objective evaluations are run and the collected simulation results are used to train an LSTM-NN agent to predict force = Function (time, fracture criterion parameters). The trained agent model is then automatically used during the fracture model calibration instead of the finite element model. At the end of the calibration, the final simulation is performed using the final values ​​of the fracture criterion parameters. The simulation results (blue curve) match the test data (red dots). Using the coefficient of determination R2>0.87, the fit is good, as shown in Figure 2. Figure 19 shown.

[0112] In the fifth example, a single finite element model is used to calibrate the Johnson-Cook plasticity model in Abaqus / Standard. Here, 30 objective evaluations are run and the collected simulation results are used to train an LSTM-NN agent to predict nominal stress = Function (time, material model parameters). The trained agent model is then automatically used during the calibration of the material behavior model instead of the finite element model. At the end of the calibration, the final simulation is performed using the final values ​​of the material model parameters. The simulation results (blue curve) are consistent with the test data (red dots). Using the coefficient of determination R2 = 0.999, the fit is good, as shown in Figure 2. Figure 20 shown.

[0113] Existing literature provides examples of neural networks being used as new constitutive response models, where ANN models replace existing classical material models. For example, neural network models can be used to model superplastic materials. This type of application of neural networks has been under development in recent years. However, this is not how neural networks are used in the context of the aforementioned examples.

[0114] The present system for performing the functions described in detail above may be a computer, an example of which is shown in FIG. Figure 21 Schematic diagram of system 2100 is shown. System 2100 comprises processor 2102, storage device 2104, memory 2106, input and output (I / O) device (or peripheral device), local bus or local interface 2112, and software 2108 of the function mentioned above is stored in memory 2106, and local bus or local interface 2112 make it possible to communicate within system 2100. Local interface 2112 can be, for example but not limited to, one or more buses or other wired or wireless connections, as known in the art. Local interface 2112 can have additional elements (for example, controller, buffer (cache), driver, repeater and receiver) to be able to communicate, and these elements are omitted for simplicity. In addition, local interface 2112 may include address, control and / or data connection, so that appropriate communication can be carried out between the components mentioned above.

[0115] The processor 2102 is a hardware device for executing software, particularly software stored in the memory 2106. The processor 2102 can be any custom or commercially available single-core or multi-core processor, a central processing unit (CPU), a secondary processor among several processors associated with the present system 2100, a semiconductor-based microprocessor (in the form of a microchip or chipset), a microprocessor, or generally any device for executing software instructions.

[0116] The memory 2106 may include any one or a combination of volatile memory elements (e.g., random access memory (RAM, such as DRAM, SRAM, SDRAM, etc.) and non-volatile memory elements (e.g., ROM, hard drive, magnetic tape, CDROM, etc.). In addition, 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 in which components are located remotely from each other but can be accessed by the processor 2102.

[0117] According to the present invention, the software 2108 defines the functions performed by the system 2100. The software 2108 in the memory 2106 may include one or more separate programs, each of which contains an ordered listing of executable instructions for implementing the logical functions of the system 2100, as described below. The memory 2106 may contain an operating system (O / S) 2120. The operating system generally controls the execution of programs within the system 2100 and provides scheduling, input and output control, file and data management, memory management, communication control, and related services.

[0118] I / O devices 2110 may include input devices such as, but not limited to, a keyboard, a mouse, a scanner, a microphone, etc. Furthermore, I / O devices 2110 may also include output devices such as, but not limited to, a printer, a display, etc. Finally, I / O devices 2110 may also include devices that communicate via input and output, such as, but not limited to, a modulator / demodulator (modem; used to access another device, system, or network), a radio frequency (RF) or other transceiver, a telephone interface, a bridge, a router, or other devices.

[0119] When the system 2100 is in operation, the processor 2102 is configured to execute software 2108 stored within the memory 2106 , transfer data to and from the memory 2106 , and generally control the operation of the system 2100 according to the software 2108 , as explained above.

[0120] When the functionality of the system 2100 is in operation, the processor 2102 is configured to execute software 2108 stored in the memory 2106, transfer data to and from the memory 2106, and generally control the operation of the system 2100 according to the software 2108. The operating system 2120 is read by the processor 2102, possibly buffered within the processor 2102, and then executed.

[0121] When system 2100 is implemented in software 2108, it should be noted that the instructions for implementing system 2100 may be stored on any computer-readable medium for use by, or in conjunction with, any computer-related device, system, or method. In some embodiments, such computer-readable medium may correspond to either or both of memory 2106 and storage device 2104. In the context of this document, a computer-readable medium is an electronic, magnetic, optical, or other physical device or apparatus that can contain or store a computer program for use by, or in conjunction with, a computer-related device, system, or method. The instructions for implementing the system may be embodied in any computer-readable medium for use by, or in conjunction with, a processor or other such instruction execution system, device, or apparatus. While processor 2102 is mentioned by way of example, in some embodiments, such an instruction execution system, device, or apparatus may be any computer-based system, system containing a processor, or other system that can retrieve and execute instructions from an instruction execution system, device, or apparatus. In the context of this document, a "computer-readable medium" can be any means that can store, communicate, propagate, or transport the program for use by or in connection with a processor or other such instruction execution system, apparatus, or device.

[0122] Such a computer readable medium can 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 may include the following: an electrical connection having one or more conductors (electronic), a portable computer floppy disk (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 can even be the paper or another suitable medium on which the program is printed, as the program can be electronically captured, for example, by optically scanning the paper or other medium, and then compiled, interpreted, or otherwise processed in a suitable manner if desired, and then stored in a computer memory.

[0123] In an alternative embodiment where the system 2100 is implemented in hardware, the system 2100 may be implemented using any one or a combination of the following technologies, all of which are well known in the art: discrete logic circuits having logic gates for implementing logic functions based on data signals, application specific integrated circuits (ASICs) having appropriate combinations of logic gates, programmable gate arrays (PGAs), field programmable gate arrays (FPGAs), etc.

[0124] This embodiment involves a practical performance enhancement of existing finite element-based calibration workflows, wherein a proxy can be automatically created and trained during calibration execution to replicate a specific Abaqus historical output, e.g., as specified by the user, to be matched to the volume of test data. For example, a particular type of test proxy model is based on the use of a neural network formed by stacking one, two, or more LSTM cells followed by a feedforward neural network. Sampled parameter values ​​of the constitutive model and the test data time are provided as input to the proxy model, and the output is a prediction of the Abaqus historical output as a function of time.

[0125] In this embodiment, multiple NNs can be trained independently (one for each test data set). If any one NN is inaccurate, that NN can be removed from the calibration. The objective function used by the minimization algorithm can be switched on the fly between using finite element simulation and using a surrogate model during the evaluation of the objective function. In embodiments, different methods can be used to sample the parameters of the constitutive response. One particularly useful method is to use Bayesian optimization iterations.

[0126] When computationally expensive finite element models are used during the calibration process, exemplary embodiments may significantly reduce the time required to identify parameters of the constitutive response model.

[0127] As a simple example, assuming that the finite element simulation used during the calibration process takes 15 minutes to run once, the estimated total time for calibration requiring running the same simulation 200 times can be estimated to be approximately 3000 minutes. However, using the present embodiment, the same simulation can be run significantly fewer times to obtain acceptable results, for example, only 20 to 50 times instead of 200 times.

[0128] In some cases, a finite element model may take hours or longer to run a single time, and calibration may require running the same simulation thousands of times. In such cases, embodiments may provide the only practical way to calibrate a constitutive response model.

[0129] An alternative embodiment may employ a Transformer neural network. The Transformer neural network architecture avoids recurrence and instead relies solely on an attention mechanism, which enables the network to capture dependencies between variable-length sequences without having to consider their distance in the input or output sequences. Transformers can be used in a manner similar to LSTMs and LSTM-NNs to create proxy models that replicate historical outputs extracted from finite element simulations used during constitutive response calibration using historically correlated material responses.

[0130] It will be apparent to those skilled in the art that various modifications and variations may be made to the structure of the present invention without departing from the scope or spirit of the present invention. In view of the foregoing, it is intended that the present invention encompass modifications and variations of the present invention as long as these modifications and variations fall within the scope of the appended claims and their equivalents.

Claims

1. A method for reducing the time it takes to run a constitutive response model calibration, the method automatically employing a neural network-based proxy model to replace output from a corresponding finite element simulation, thereby reducing the time it takes to run a constitutive response model calibration, the method comprising the following steps: receiving test data associated with the material, wherein the test data comprises output from an experiment; receiving a finite element model for simulating the experiment; receiving a user selection of a constitutive response model; receiving a user selection of historical output quantities to collect during the simulation; receiving a user selection of a numerical minimization algorithm for calibrating the constitutive response model; receiving a user selection of an error metric for quantifying a difference between the test data and an output of the finite element simulation; receiving a user selection of parameters of the constitutive response model; receiving user selections of techniques for sampling parameter space during an initial calibration phase and for collecting finite element outputs for computing an objective function; receiving a user selection of settings related to an architecture of a surrogate model to be trained to replicate the collected finite element outputs; receiving a user selection of a training algorithm for the proxy model; and Perform automatic constitutive response model calibration.

2. The method according to claim 1, wherein The proxy model is created using a long short-term memory neural network.

3. The method according to claim 1, wherein The received test data includes at least one of the following: a set of stress data as a function of time and / or deformation; and A set of forces as a function of time and / or deformation.

4. The method according to claim 1, further comprising: performing a plurality of finite element simulations using the finite element model at locations determined by the selected sampling or minimization algorithm; collecting finite element history output from the plurality of simulations; as well as A neural network is trained to approximate the finite element output, instead of the finite element simulation.

5. The method according to claim 4, further comprising: replacing the finite element simulation with a surrogate model comprising a trained neural network; as well as A proxy history output is generated based on the proxy model to produce a set of calibration parameters for a selected constitutive response.

6. The method according to claim 5, further comprising: resuming use of the finite element model; as well as Results of the finite element model are evaluated relative to the surrogate model.

7. The method according to claim 4, further comprising: The finite element history is stored.

8. The method according to claim 4, wherein The neural network comprises a stack of long short-term memory (LSTM) units followed by a feed-forward neural network.

9. The method according to claim 1, wherein: User selections of parameters of the constitutive response model dictate minimum and / or maximum bounds on the parameters.

10. The method according to claim 1, wherein The techniques for sampling parameter space include Bayesian minimization.

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

12. The method according to claim 1, further comprising the step of receiving a number of sampling points specified by a user.

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