Predicting macrophysical properties of multi-scale materials

Abstract

The present disclosure relates to a method for training a deep material network (Deep Material Network)-based neural network, the neural network being configured to predict macro physical properties of a multi-scale material. The multi-scale material includes one or more components. The method includes obtaining a data set, each entry of the data set corresponding to a respective multi-scale material object. The entries include tensors that describe physical properties of the object at a macroscopic level, one or more tensors that each describe physical properties of components of the object at a microscopic level, and one or more morphological parameters that each describe a morphology of the object. The method further includes training a neural network based on the data set to predict a tensor describing a physical property of the multi-scale material object at a macroscopic level based on the one or more tensors of the object and the one or more morphological parameters of the object.

Description

Technical Field

[0001] The present disclosure relates to the field of computer programs and systems, and more particularly to methods, systems and programs for predicting macroscopic physical properties of multi-scale materials. Background Art

[0002] Several systems and programs for design, engineering and manufacturing of objects are provided on the market. CAD (Computer-Aided Design) is an acronym for Computer-Aided Design, for example, it relates to software solutions for designing objects. CAE (Computer-Aided Engineering) is an acronym for Computer-Aided Engineering, for example, it relates to software solutions for simulating the physical behavior of future products. CAM (Computer-Aided Manufacturing) is an acronym for Computer-Aided Manufacturing, for example, it relates to software solutions for defining manufacturing processes and operations. In such computer-aided design systems, graphical user interfaces play an important role in technical efficiency. These technologies can be embedded in Product Lifecycle Management (PLM) systems. PLM refers to a business strategy under the concept of cross-extended enterprise, which helps companies share product data, apply common processes, and adopt company knowledge to develop the product from product concept to the end of the product life cycle. The PLM solution provided by Dassault Systèmes (using trademarks such as CATIA, ENOVIA and DELMIA) provides an engineering center (Engineering Hub) that integrates product engineering knowledge, a manufacturing center (Manufacturing Hub) that manages manufacturing engineering knowledge, and an enterprise center (Enterprise Hub) that integrates and connects enterprises to both engineering and manufacturing centers. These systems jointly deliver an open object model that links products, processes, and resources to achieve dynamic, knowledge-based product creation and decision support, which promotes the optimization of product definition, manufacturing preparation, production, and services.

[0003] Several solutions and products have been proposed to determine the mechanical properties of materials used in different products and processes. These solutions and products include material constitutive models and experimental methods. Due to the increasing demand for multi-scale materials in various industries, several specific solutions and products have been proposed to predict the mechanical properties of such materials.

[0004] The paper by Liu, Wu, and Koishi, "A Deep Material Network for Multiscale Topological Learning and Accelerated Nonlinear Modeling of Heterogeneous Materials." Computer Methods in Applied Mechanics and Engineering, 345, 1138-1168, 2019, teaches a data-driven multiscale material modeling approach based on the mechanical homogenization theory of representative volume elements (RVE) and advanced machine learning techniques. The approach uses a collection of connected mechanical building blocks with an analytical homogenization scheme to describe complex overall material responses.

[0005] Based on this, there is still a need to improve the prediction of the macroscopic physical properties of multi-scale materials. Summary of the invention

[0006] Based on this, a computer-implemented method for training a neural network based on a deep material network (DMN) is provided herein, wherein the neural network is configured to predict the macroscopic physical properties of a multiscale material. The multiscale material includes one or more components. The method includes: acquiring a data set, each entry of the data set corresponding to a corresponding multiscale material object. The entry includes a tensor, and the tensor describes the physical properties of the multiscale material object at a macroscopic level. The entry includes one or more tensors, each tensor describing the physical properties of a component of the multiscale material object at a microscopic level. The entry further includes one or more morphological parameters, each morphological parameter describing the morphology of the multiscale material object. The method also includes training a neural network based on the acquired data set to predict a tensor describing the physical properties of the multiscale material object at a macroscopic level based on one or more tensors of the object and one or more morphological parameters of the object.

[0007] The method may include one or more of the following:

[0008] The neural network based on the deep material network DMN includes a first block and a second block; wherein the first block is configured to receive one or more morphological parameters of the object as input and output values ​​of multiple network parameters; the second block is configured to receive the values ​​of multiple network parameters and one or more tensors of the object as input, and output a prediction of the physical properties of the multi-scale material object at a macroscopic level;

[0009] The first block includes a feedforward neural network; and / or, the second block includes a DMN architecture having a plurality of network parameters;

[0010] -The first block is a fully connected neural network;

[0011] The first block includes a first sub-block and a second sub-block; wherein: the first sub-block is configured to receive values ​​of a first subset of the one or more morphological parameters as input, and output values ​​of a corresponding subset of the plurality of network parameters; the second sub-block is configured to receive values ​​of a second subset of the one or more morphological parameters as input, and output values ​​of a corresponding subset of the plurality of network parameters;

[0012] The first subset of the one or more morphological parameters comprises a set of volume fractions of each component; and / or the second subset of the one or more morphological parameters comprises one or more orientation parameters.

[0013] The training comprises: minimizing a loss function that penalizes, for each entry, the difference between: an acquired tensor that describes the physical properties of the multi-scale material object at a macroscopic level; and a predicted tensor that describes the physical properties of the multi-scale material object at a macroscopic level; and / or

[0014] The loss function also penalizes: failure to satisfy a volume fraction constraint of the multi-scale material object, and / or failure to satisfy a material orientation constraint of the multi-scale material object.

[0015] Also provided is a neural network that can be trained (i.e., learnable) according to the training method, i.e., a computer-implemented neural network data structure having weights of a neural network trained by the method. The provided neural network can, for example, be directly learnable by the method, and its weights have been fixed by the training step of the method.

[0016] A computer-implemented method for using a neural network trainable according to a training method is also provided. The method of using includes obtaining one or more tensors and one or more morphological parameters, each tensor describing a physical property of a component of a multi-scale material object at a microscopic level, and each morphological parameter describing a morphology of the multi-scale material object. The method of using also includes predicting a macroscopic physical property of the multi-scale material object by applying the neural network to the one or more tensors and the one or more parameters.

[0017] A computer-implemented method for using a neural network trainable according to a training method is also provided. The method of using includes obtaining a tensor, the tensor describing a physical property of the multi-scale material object at a macroscopic level. The method further includes determining one or more tensors and one or more morphological parameters, each tensor describing a physical property of a component of the multi-scale material object at a microscopic level, and each morphological parameter describing the morphology of the multi-scale material object. The determining includes minimizing a difference between the obtained tensor and a candidate prediction value of the obtained tensor, the candidate prediction value determined by applying the neural network to a candidate one or more tensors and a candidate one or more morphological parameters.

[0018] In the method of use, minimizing the difference may include calculating the gradient of the candidate prediction value relative to each corresponding candidate tensor in the one or more tensors, and the gradient of the candidate prediction value relative to each corresponding candidate morphological parameter in the one or more morphological parameters.

[0019] A database of multiscale materials is also provided. Each entry of the data set corresponds to a corresponding multiscale material. The entry includes one or more morphological parameters, each morphological parameter describing the morphology of the multiscale material object; and a neural network trained according to the training method for predicting the macroscopic physical properties of the multiscale material.

[0020] Also provided is a computer program comprising instructions for performing the method and / or using the method.

[0021] A computer-readable storage medium is also provided, on which a computer program and / or a neural network and / or a database are recorded.

[0022] A system is also provided, the system comprising a processor coupled to a memory having recorded thereon a computer program and / or a neural network and / or a database, and a graphical user interface.

[0023] Also provided is a device including a computer-readable storage medium having a computer program recorded thereon.

[0024] The device may form or be used as a non-transitory computer-readable medium, such as on a SaaS (Software as a service) or other server or cloud-based platform, etc. The device may alternatively include a processor coupled to a data storage medium. Thus, the device may form a computer system in whole or in part (e.g., the device is a subsystem of the entire system). The system may also include a graphical user interface coupled to the processor. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Non-limiting examples will now be described with reference to the accompanying drawings, in which:

[0026] Figure 1 An example of a system is shown.

[0027] Figure 2-32 An example of this method is shown. DETAILED DESCRIPTION

[0028] A computer-implemented method for training (i.e., learning) a neural network based on a deep material network (DMN) is proposed. The DMN-based neural network is configured to (i.e., trained to) predict macroscopic physical properties of a multiscale material. The multiscale material includes one or more components. The method includes obtaining a data set. Each entry of the data set corresponds to a corresponding multiscale material object. The entry includes a tensor (i.e., a first tensor), which describes the physical properties of the multiscale material object at a macroscopic level. The entry also includes one or more tensors (i.e., a second tensor). Each entry includes one or more tensors, each of which describes the physical properties of a component of the multiscale material object at a microscopic level. The entry also includes one or more morphological parameters. Each of the one or more morphological parameters describes the morphology of the multiscale material object. The method also includes training a neural network based on the obtained data set to predict a tensor (i.e., a third tensor). The (predicted) tensor describes the macroscopic physical properties of the multiscale material object at a macroscopic level based on the one or more tensors of the object and the one or more morphological parameters of the object.

[0029] The method may be referred to as a "training method" or a "learning method". The training method trains the neural network to output a third tensor as a prediction (e.g., an approximation) of a physical property of a multi-scale material object at a macroscopic level. The training method performs training by providing a set of first tensors. The first tensor is the ground truth of the physical property of the multi-scale material object at a macroscopic level. The training method further performs training by providing a set of second tensors. The second tensor is the ground truth of the physical property of each component of the multi-scale material object at a microscopic level.

[0030] The training method improves the prediction of the macroscopic physical properties of multiscale materials given the physical properties of the components of the multiscale materials. The method obtains the improvement by training a neural network architecture based on a deep material network (DMN). This architecture follows the mechanism of multiscale materials and can provide more accurate prediction results. In addition, the neural network is also configured to output predictions based on one or more morphological parameters of the multiscale material. The morphology of the multiscale material plays an important role in its macroscopic physical properties. Therefore, the method is able to train the neural network for a wide range of materials defined by the one or more morphological parameters.

[0031] Compared to the cited prior art, the method constitutes an improved solution for training DMN-type neural networks. For example, the prior art discloses training for a given (i.e., fixed) morphology (which is defined by one or more morphological parameters). Since the training parameters of the neural network depend on the given morphology, it is necessary to retrain the neural network for each new morphology, which makes the computational cost high, or to interpolate / extrapolate the output of the trained network for the new morphology, which reduces the accuracy of the prediction. However, the method disclosed in the present invention integrates the morphological parameters into the training phase (offline), thereby significantly reducing the computational cost of each new morphology at the deployment / inference phase (online). The method provides these advantages without reducing the accuracy of the prediction.

[0032] Therefore, the method is used to train a neural network that can predict the macroscopic physical properties of multi-scale materials composed of various components and in different forms. This prediction is an advantageous solution compared to the traditional mechanical testing that experimentally determines the mechanical properties. This trained neural network provides designers with the ability to predict mechanical properties in a non-destructive manner with high accuracy and significantly shorter time consumption compared to the mechanical testing. This enables designers to evaluate a wider range of materials for specific use cases, thereby improving the final design. It can be applied to mechanical objects that are at least partially formed by multi-scale materials. It can also be applied to various load conditions applied to the mechanical object. The various load conditions can represent the forces, torques and / or stresses to which the mechanical object is subjected during operation.

[0033] The method and / or the trained neural network can be used to design a multi-scale material object or product and / or, after design, to physically manufacture the object or product. The object can be an additively manufacturable part, i.e., a part manufactured by additive manufacturing (i.e., 3D printing). Additive manufacturing methods are particularly suitable for manufacturing multi-scale material objects, since such methods can accurately manufacture the internal microstructure of the object.

[0034] For example, according to the first aspect of the method of use discussed below, a trained neural network can be used to predict macroscopic physical properties (e.g., mechanical properties of the object at the macroscopic level, such as compliance or stiffness) of a given multiscale material object based on one or more tensors and one or more morphological parameters (e.g., volume fraction, fiber misalignment, and / or fiber aspect ratio), each tensor describing a physical property of a component of the multiscale material object at the microscopic level (e.g., mechanical properties of each component at the microscopic level, such as compliance or stiffness), and each morphological parameter describing the morphology of the multiscale material object. This can be done during the design of the object (e.g., repeatedly for various inputs, and / or, repeatedly to predict different properties), for example, testing various inputs to converge to the desired microscopic physical properties of the object.

[0035] Alternatively, for example according to the second aspect of the method of use discussed below, a trained neural network can be used to optimize inputs (i.e., one or more tensors and one or more morphological parameters (e.g., volume fraction, fiber misalignment and / or aspect ratio of fibers), each tensor describing a physical property of a component of a multiscale material object at a microscopic level (such as a mechanical property of each component at a macroscopic level, e.g., compliance or stiffness), each morphological parameter describing the morphology of the multiscale material object) to arrive at a desired (or provided) tensor describing a physical property of the multiscale material object at a macroscopic level (such as a mechanical property of the object at a macroscopic level, e.g., compliance or stiffness). This allows finding the optimal input to the network to arrive at the provided optimal tensor describing the physical properties of the multiscale material object at a macroscopic level.

[0036] The method and / or use of a neural network may be included in a design and manufacturing process. Such a process may include:

[0037] Optionally, executing the training method and / or providing a neural network obtained according to the training method;

[0038] Designing a multiscale material object, for example, according to a trained neural network using the first aspect or the second aspect of the method (e.g., repeatedly or iteratively) to test various inputs to the network (i.e., tensors describing macroscopic physical properties, and / or, morphological parameters), and / or determining the best input to obtain a desired output (i.e., optimal parameters describing physical properties of components at the microscopic level and / or optimal tensors for obtaining a desired tensor describing physical properties at the microscopic level); and

[0039] Use the results of the design to manufacture physical objects.

[0040] Depending on the type of system used for design, the result of the design can be defined by different kinds of data. The system can actually be any combination of a CAD system, a CAE system, a CAM system, a PDM (Product Data Management) system and / or a PLM (Product Lifecycle Management) system, as known in the art. Designing a CAD model is the first step towards computer-aided manufacturing. In fact, CAD solutions provide key functions such as feature-based modeling and boundary representation (Boundary representation, B-Rep) to reduce the risk of errors and loss of precision during the manufacturing process handled with CAM solutions. In fact, CAD models are intended to be manufactured. Therefore, a CAD model is a virtual twin of the object to be manufactured, also called a digital twin, with two goals:

[0041] Checking the correct properties of an object to be manufactured in a particular environment (e.g. using the first or second aspects of the method in combination to ensure the physical properties desired by the design); and

[0042] Ensure the manufacturability of the object to be manufactured.

[0043] The design results can be applied in the production process, which may include: after obtaining the design results, producing the corresponding physical products for the design results input into the production process. The production process may include the following steps:

[0044] - (e.g. automatically) obtaining a CAD model or CAE model of the design (e.g. after one or more transformations / conversions);

[0045] - optionally, (e.g. automatically) converting the obtained CAE model into a CAD model using a CAE to CAD conversion process (e.g. an automatic process);

[0046] -Use the obtained CAD model to manufacture the design result.

[0047] Manufacturing using the CAD model may, for example, include the following steps:

[0048] Edit the acquired CAD model;

[0049] performing simulations based on the CAD model or a corresponding CAE model (e.g., the CAE model from which the CAD model is branched after a CAE to CAD conversion process), such as simulations for verifying mechanical, usage and / or manufacturing properties and / or constraints (e.g., structural simulations, thermodynamic simulations, aerodynamic simulations);

[0050] Edit CAD models based on simulation results;

[0051] Optionally (i.e., depending on the manufacturing process used, the production of the mechanical product may or may not include this step), determining (e.g., automatically) a manufacturing file / CAM file based on the (e.g., edited) CAD model for producing / manufacturing the manufactured product;

[0052] Send CAD files and / or manufacturing files / CAM files to the factory; and / or

[0053] (e.g., automatically) produce / manufacture a mechanical product represented by the model output according to the design method based on the determined manufacturing file / CAM file or CAD model. This may include feeding (e.g., automatically) the manufacturing file / CAM file and / or CAD file to a machine that performs the manufacturing process.

[0054] The last step of the production / manufacturing can be referred to as a manufacturing step or a production step. This step produces / manufactures the part / product based on the CAD model and / or CAM file, for example, when the CAD model and / or CAD file is fed to one or more production machines or a computer system that controls the production machine. The manufacturing step may include: performing any known manufacturing process or a series of manufacturing processes, such as one or more additive manufacturing steps, one or more cutting steps (e.g., laser cutting or plasma cutting steps), one or more stamping steps, one or more forging steps, one or more bending steps, one or more deep drawing steps, one or more molding steps, one or more machining steps (e.g., milling steps) and / or one or more perforating steps. Since the design method improves the design of the model (CAE or CAD) presenting the part / product, manufacturing and its productivity are also improved.

[0055] Now let's discuss the training methods.

[0056] As is known in the field of machine learning, a neural network is a function that includes several operations based on an architecture, each operation being defined by data including parameters (also referred to as weight values). The architecture of a neural network defines the operands of each operation and the relationship between the parameters. Therefore, learning of a neural network includes determining the value of the weight based on a data set configured for such learning and corresponding to the neural network architecture. That is, the neural network is trained to obtain a deterministic function that associates the input of the network with the corresponding output. This training is performed using a training data set that includes a series of inputs and corresponding real-valued outputs. The training data set includes several data segments, each of which forms a respective training sample. The training samples present a diversity of various situations in which the neural network will be applied after learning is completed. Any data set mentioned herein may include more than 1,000, 10,000, 100,000, or 1,000,000 training samples.

[0057] A "deep material network (DMN)-based neural network" means a neural network whose architecture includes a DMN-based sub-architecture (i.e., a portion of the architecture). Such a DMN sub-architecture can be any DMN architecture known in the art, for example, the DMN sub-architecture can be the deep material network architecture in the document "A Deep Material Network for Multi-Scale Topological Learning and Accelerated Nonlinear Modeling of Heterogeneous Materials" by Liu et al. (2019), Computer Methods in Applied Mechanics and Engineering, 345, 1138-1168, which is incorporated herein by reference.

[0058] As known in the field of mechanics and materials science, "multiscale materials" (also called composite materials, constructed materials or metamaterials) mean heterogeneous materials containing microstructures. Heterogeneous materials include multiple materials that are spatially distributed in the form of an internal structure (i.e., microstructure). Multiscale materials include one or more components. That is, the microstructure of the multiscale material object defines the topology and geometry of the corresponding parts of the object formed in each component. Each of the one or more components can be a heterogeneous or homogeneous material. Multiscale materials can be parameterized multiscale materials, that is, their morphology / microstructure is defined by (multiple sets of) one or more (morphological) parameters. Each set of values ​​of one or more parameters defines a multiscale material (realization). That is, each multiscale material object is a realization of a parameterized multiscale material of multiple sets of parameter values. "Multiscale material object" means a tangible object formed by multiscale materials. Such a multiscale material object can have a standard geometry and size, for example based on a test standard.

[0059] The method includes obtaining a data set. Each entry of the data set corresponds to a corresponding multi-scale material object. Obtaining the data set means providing the data set. The method may obtain the data set by accessing a local storage device storing the data set (e.g., stored in a database template), or by remotely accessing a data storage service (e.g., via a cloud storage device). The method may include: before the method step of providing the data set, first creating the data set (e.g., in a data set forming step).

[0060] Each entry includes a tensor that describes the physical properties of the multiscale material object at the macroscopic level. Each entry further includes one or more tensors, each of which describes the physical properties of the components of the multiscale material object at the microscopic level. "Physical properties of a multiscale material object" means any component property that defines the physical properties of the object formed by the multiscale material. Physical properties can be mechanical properties, thermal properties, magnetic properties, or electrical properties. Mechanical properties can be flexibility or stiffness. Thermal / electrical properties can be thermal conductivity / electrical conductivity, and magnetic properties can be magnetic permeability. "Physical properties of the components of a multiscale material object at the microscopic level" means each physical property that can describe the physical properties of the component at the microscopic level. "Physical properties of a multiscale material object at the macroscopic level" means each equivalent or effective physical property that can describe the physical properties of the object (i.e., the entire object) at the macroscopic level. As known in mathematics (see https: / en.wikipedia.org / wiki / Tensor), a tensor is an algebraic object that describes a multilinear relationship between a set of algebraic objects related to a vector space. Tensors can map between different objects (such as vectors, scalars, or other tensors). A tensor can be represented as an array (e.g., a multidimensional array) consisting of one or more components. Tensors can be of various types. For example, a scalar or a vector is an example of a tensor. Therefore, a tensor that describes a physical property (at a macroscopic or microscopic level) means that the components of the tensor represent the physical property. For example, when the physical property is flexibility or stiffness, each component of the tensor describing the flexibility or stiffness may include one or more linear elastic properties. Linear plasticity can be Young's modulus (which describes the stiffness required per unit extension in a certain direction), shear modulus (which describes the shear stiffness required per unit shear deformation) and / or Poisson's ratio (which describes the relative deformation between two orthogonal directions).

[0061] Each of the macroscopic tensor and one or more microscopic tensors can be obtained according to experiments (e.g., under corresponding standard conditions) or high-fidelity numerical methods (e.g., finite element methods). The tensor can be calculated before executing the method and stored in a database to be provided to the method.

[0062] The entry may also include one or more morphological parameters. Each of the one or more morphological parameters describes the morphology of the multi-scale material object. The morphology of the multi-scale material object refers to a qualitative description of the microstructure of the multi-scale material.

[0063] Then, the method trains a neural network based on the obtained data set. "Based on the obtained data set" means that the data set is a training data set for the neural network. The trained neural network is configured to predict a tensor describing the physical properties of a multi-scale material object at a macroscopic level based on one or more tensors describing the physical properties at a microscopic level and based on one or more morphological parameters of the object. That is, the (trained) neural network receives as input one or more tensors describing the physical properties at a microscopic level, and receives as input one or more morphological parameters of the object, and predicts (i.e., outputs) the tensor. "Configured to predict the tensor" means that the neural network outputs a tensor that approximates the (true value) physical properties of a multi-scale material object at a macroscopic level. The method can train the neural network according to any known method in the field of machine learning and artificial intelligence.

[0064] Now let's discuss examples of training methods.

[0065] In an example, a neural network based on a deep material network (DMN) may consist of a first block and a second block. That is, the first block and the second block do not overlap and form a block based on a DMN neural network. The first block is configured to receive one or more morphological parameters of the object as input and output values ​​of multiple network parameters (also referred to as DMN fitting parameters). The second block is configured to receive the values ​​of multiple network parameters (i.e., the output of the first block) and one or more tensors of the object as input. The second block is also configured to output a prediction of the physical properties of a multi-scale material object at a macro level. "Multiple network parameters" refers to multiple parameters that define the relationship between the output of the second block and one or more tensors input in the second block. Therefore, the architecture of the network provides for the dependence of the parameters of the second block on the output of the first block, which itself depends on one or more morphological parameters. In a standard DMN, the network parameters are trained by given (fixed) morphological parameters and are therefore not suitable for different morphologies of multi-scale material objects, which constitutes an improved training compared to the standard DMN. In addition, these examples provide a separation between the first block and the second block. That is, in these examples, the first block is configured to output network parameters without inputting one or more tensors of the object and using only morphological parameters. On the other hand, the second block is configured to output physical properties at a macro level without inputting one or more morphological parameters. Such separation enables the neural network to use a more efficient architecture for the unknowns in each block (i.e., network parameters in the first block and predicted tensors of physical properties in the second block). For example, the second block can include a DMN architecture that has higher accuracy for such physical property predictions.

[0066] In the example, the first block can be a feedforward neural network. As the literal meaning of "feedforward neural network", it means a neural network in which the flow direction flows from the input node through the hidden node (if any) to the output node in one direction without any cycles or loops (see https: / / en.wikipedia.org / wiki / Feedforward_neural_network). Optionally or additionally, the second block may include a DMN architecture with multiple network parameters (i.e., composed of the architecture). In this case, the multiple network parameters may include multiple weights and multiple rotations of the DMN architecture. Each rotation of the neural network may include a rotation tensor (e.g., a 3D rotation tensor). Such rotations and weights can be, for example, based on the literature cited by Liu et al., "A Deep Material Network for Multi-Scale Topological Learning and Accelerated Nonlinear Modeling of Heterogeneous Materials" (2019), Computer Methods in Applied Mechanics and Engineering, 345, 1138-1168.

[0067] In an example, the first block may be a fully connected neural network. That is, in such an example, each of the plurality of network parameters is defined on an aggregate of one or more morphological parameters.

[0068] Optionally, and in other examples, the first block may consist of a first sub-block and a second sub-block. The first sub-block is configured to receive as input the value of a first subset of the one or more morphological parameters and output the value of a corresponding subset of the plurality of network parameters. The second sub-block is configured to receive as input the value of a second subset of the one or more morphological parameters and output the value of a corresponding subset of the plurality of network parameters. That is, in such examples, each subset of the plurality of network parameters is defined on a corresponding subset of the one or more morphological parameters. The first subset of the one or more morphological parameters and the second subset of the one or more morphological parameters form a decomposition or partition of (a set of) one or more morphological parameters. Similarly, a subset of the plurality of network parameters corresponding to the first subset and a subset of the plurality of network parameters corresponding to the second subset form a decomposition or partition of the plurality of network parameters. That is, for each morphological parameter or network parameter, the union of the first subset and the second subset is equal to the entire parameter set, while the intersection of the first subset and the second subset is empty.

[0069] In an example, the first subset of the one or more morphological parameters includes a set of volume fractions for each component. Optionally or additionally, the second subset of the one or more morphological parameters includes one or more orientation parameters. An orientation parameter refers to a parameter that defines one or more orientations of a microstructure of a multiscale material, such as an orientation tensor, such as a relative orientation between components. In an example where the second block has a DMN architecture with multiple network parameters, a subset of the multiple network parameters corresponding to the first subset includes weights of the DMN. Further optionally or additionally, a subset of the multiple network parameters corresponding to the second subset includes a rotation of the DMN.

[0070] In an example, training includes minimizing a loss function. The loss function penalizes, for each entry, the difference between: an acquired tensor that describes the physical properties of the multi-scale material object at a macroscopic level; and a predicted tensor that describes the physical properties of the multi-scale material object at a macroscopic level. The difference can be measured by a mathematical norm. The loss function can be of the following type:

[0071]

[0072] Loss i Tensor-based predictions The real value tensor generated (obtained) by FE-RVE (Finite Element-Representative Volume Element) simulation Mean Squared Error between . Symbol represents the number of samples in parameter space, i.e., one or more morphological parameters. represents the number of samples in the material space, i.e. one or more obtained tensors. Then The errors (i.e., differences) between the two are averaged to calculate the loss for a fixed microstructural parameter value. Finally, we average the samples in parameter space to produce the total loss function Any sample in material space and any sample in parameter space can be obtained according to any sampling method, such as the Latin hypercube method. A mathematical formulation of this type of identity means that the identity is given exactly by the formula, or is given by scaling and / or regularizing the formula.

[0073] The loss function also penalizes: not satisfying the volume fraction constraints of the multiscale material object. This penalty means that the neural network is trained to match the volume fractions in the DMN framework with the true (i.e., real-valued) volume fraction values ​​of the microstructure for arbitrary volume fraction values. Optionally or additionally, the loss function can also penalize: not satisfying the material orientation constraints of the multiscale material object. This penalty means that the neural network is trained to match the average material frame orientation predicted by the DMN with the real microstructure.

[0074] In an example where the loss function penalizes both failure to satisfy the volume fraction constraint of the multi-scale material object and failure to satisfy the material orientation constraint of the multi-scale material object, the loss function may be of the following type:

[0075]

[0076] represents the loss function defined above, represents the corresponding term of the loss function, which penalizes the volume fraction that does not meet the requirement (denoted as vf), represents the corresponding term in the loss function, which penalizes unsatisfactory material orientations.

[0077] The present application also provides a neural network that can be trained according to the training method, that is, a neural network having weight values ​​that are the same as the weight values ​​obtained by training according to the training method. Such a neural network can be any DMN-based neural network that is configured to predict tensors describing physical properties of a multiscale material object at a macroscopic level based on one or more tensors of physical properties of components of the multiscale material object at a microscopic level and based on one or more morphological parameters of the object. Such a neural network may have been trained according to an example of any training method, that is, a neural network having weight values ​​directly generated by training according to the training method.

[0078] The present application also provides a computer-implemented method using a neural network. The method includes obtaining one or more tensors and one or more morphological parameters, each tensor describing the physical properties of a component of a multiscale material object at a microscopic level, and each morphological parameter describing the morphology of the multiscale material object. The method also includes: predicting the macroscopic physical properties of the multiscale material object by applying a neural network (e.g., a neural network trained according to a training method) to the one or more tensors and the one or more parameters. In this first aspect, the method forms a method for directly predicting the physical properties of a multiscale material at a macroscopic level. Optionally or additionally, the method includes: obtaining a tensor describing the physical properties of a multiscale material object at a macroscopic level, and determining one or more tensors, each tensor describing the physical properties of a component of a multiscale material object at a microscopic level; and one or more morphological parameters, each morphological parameter describing the morphology of a multiscale material object. In this second aspect, the method forms an inverse identification method. In such an inverse identification method, the method determines the physical properties and one or more morphological parameters of one or more components of a multiscale material based on a given physical property of a multiscale material object at a macroscopic level (via the obtained tensor). The determination includes minimizing the difference between the obtained tensor and the candidate prediction value of the obtained tensor. The candidate prediction value is determined by applying a neural network (e.g., a neural network trained according to a training method) to the candidate one or more tensors and the candidate one or more morphological parameters. That is, the method performs minimization by (e.g., iteratively) searching for (a set of) candidates in one or more tensors and (a set of) candidates in one or more morphological parameters, applying the trained neural network to each set of candidates in the one or more tensors and the one or more morphological parameters to obtain the candidate prediction value of the obtained tensor, thereby minimizing the difference. That is, the trained neural network is part of the objective function (i.e., the difference) of such minimization, and the one or more tensors and the one or more morphological parameters are free variables of such minimization. The difference can equivalently be referred to as a loss function. In practice, this aspect can form a recognition process or be part of a recognition process where the macroscopic physical properties of a multi-scale material object are known (e.g., via high-fidelity numerical simulations, such as finite element methods or via experiments).

[0079] The method of use may include all or any of the above aspects. That is, when the neural network is trained according to the training method, it can be used to directly predict the macroscopic physical properties of multi-scale material objects, and / or for reverse identification of the physical properties of the components.

[0080] In either of the two aspects of the method of use discussed above, the inputs and outputs of the network during the online use phase (i.e., inference or deployment) are of the same type as the corresponding parts of the entries in the obtained dataset used in the offline training phase according to the training method. That is, the obtained tensor at the macro level, the one or more tensors at the micro level, and the one or more morphological parameters obtained are of the same type (i.e., physically and / or in units) as the tensor describing the physical properties of the multi-scale material object at the macro level, the one or more tensors at the micro level, and the one or more morphological parameters for each entry of the dataset provided in the training method, respectively.

[0081] The difference can be the norm of the difference between the obtained tensor and its predicted value. In the example, the difference can be of the following type:

[0082]

[0083] Indicates the difference, represents the obtained tensor, Here, the norm ‖.‖ represents the standard Frobenius norm.

[0084] Minimization of the difference may include calculating the gradient of the candidate prediction value relative to each corresponding candidate tensor in the one or more tensors, and the gradient of the candidate prediction value relative to each corresponding candidate morphological parameter in the one or more morphological parameters. That is, the minimization is a gradient-based minimization. The method may use automatic differentiation according to the architecture of the DMN-based framework to calculate the gradient of the candidate prediction value relative to each candidate tensor in the one or more tensors and the gradient of the candidate prediction value relative to each candidate morphological parameter in the one or more morphological parameters.

[0085] The present application also provides a database of multiscale materials. Each entry in such a database corresponds to a corresponding multiscale material. Each entry includes one or more morphological parameters, each morphological parameter describing the morphology of a multiscale material object. Each entry also includes a neural network trained according to the above-mentioned training method, and the neural network is used to predict the macroscopic physical properties of the multiscale material. Each entry can be particularly related to a multiscale material object formed in the corresponding multiscale material. Such a multiscale material object can have a standard geometry and size, for example based on a test standard.

[0086] This constitutes an improved solution to store a database of multiscale materials. As mentioned above, multiscale materials have microstructures that need to be represented in memory. Known solutions in the art use unstructured grids that spatially discretize the microstructures to store multiscale materials. The actual storage range for the microscopic properties of the constituent parts and for the microstructures of different morphologies is extremely memory intensive. Therefore, the provided database is an improved solution because it only stores the trained network for each multiscale material.

[0087] "Database" refers to any collection of data (information) organized for search and retrieval (e.g., a relational database, such as based on a predetermined structured language, such as SQL (Structured Query Language)). When stored on a memory, a database enables a computer to quickly search and retrieve. Databases are constructed to facilitate the storage, retrieval, modification, and deletion of data, as well as various data processing operations. A database may include a file or set of files, which may be broken down into a number of records, each record consisting of one or more fields. A field is the basic unit of data storage. Users can retrieve data primarily through queries. Using keywords and sorting commands, users can quickly search, rearrange, group, and select fields from many records to retrieve or create reports on specific data sets according to the rules of the database management system used.

[0088] As described above, in an example where the DMN-based neural network consists of a first block and a second block, the first block is configured to output network parameters without inputting one or more tensors of an object and using only morphological parameters. That is, the parameters of the trained second block (i.e., network parameters) can be obtained from a plurality of morphological parameters and the first block. In such an example, the database of multi-scale materials can store only the first block, rather than storing the entire trained neural network (i.e., the first block and the second block). This significantly improves the required storage space.

[0089] The method is computer-implemented. This means that the steps (or substantially all steps) of the method are performed by at least one computer or any system. Therefore, the steps of the method are performed by a computer and can be performed fully automatically or semi-automatically. In an example, at least some of the steps of the method can be triggered by interaction between a user and a computer. The level of user-computer interaction required can depend on the level of automation foreseen and weighed against achieving user needs. In an example, the interaction level can be user-defined and / or predefined.

[0090] A typical example of a computer-implemented method is to perform the method using a system suitable for performing the method. The system may include a processor coupled to a memory and a graphical user interface (GUI), the memory having recorded thereon a computer program including instructions for performing the method. The memory may also store a database. The memory is any hardware suitable for such storage and may include several physically distinct parts (e.g., one for the program and one for the database).

[0091] Figure 1 An example of a system is shown, where the system is a client computer system, such as a user's workstation.

[0092] The client computer of this example includes a central processing unit (CPU) 1010 connected to an internal communication bus, to which a random-access memory (RAM) 1070 is also connected. The client computer is also provided with a graphical processing unit (GPU) 1110, which is associated with a video random-access memory 1100 connected to the bus. The video RAM 1100 is also referred to in the art as a frame buffer. A mass storage device controller 1020 manages access to a mass storage device such as a hard disk drive 1030. Mass storage devices suitable for tangibly embodying computer program instructions and data include all forms of non-volatile memory, including, for example, semiconductor memory devices, such as Erasable Programmable Read Only Memory (EPROM), Electrically Erasable Programmable Read Only Memory (EEPROM), and flash memory devices; disks, such as internal hard disks and removable disks; magneto-optical disks. Any of the foregoing may be supplemented or incorporated by a specially designed ASIC (application-specific integrated circuits). The network adapter 1050 manages access to the network 1060. The client computer may also include a tactile device 1090, such as a cursor control device, a keyboard, and the like. The cursor control device is used in the client computer to allow the user to selectively position the cursor at any desired location on the display 1080. In addition, the cursor control device allows the user to select various commands and input control signals. The cursor control device includes a plurality of signal generating devices for inputting control signals to the system. Typically, the cursor control device may be a mouse, the buttons of which are used to generate signals. Alternatively or additionally, the client computer system may include a sensitive board and / or a sensitive screen.

[0093] The computer program may include instructions executable by a computer, the instructions including means for causing the above-mentioned system to perform the method. The program may be recorded on any data storage medium, including the memory of the system. The program may be implemented, for example, in a digital electronic circuit, or in computer hardware, firmware, software, or a combination thereof. The program may be implemented as a device, such as a product tangibly embodied in a machine-readable storage device for execution by a programmable processor. The method steps may be performed by a programmable processor executing an instruction program to perform the functions of the method by operating on input data and generating output. Therefore, the processor may be programmable and coupled to receive data and instructions from a data storage system, at least one input device, and at least one output device, and to send data and instructions thereto. The application may be implemented in a high-level procedural or object-oriented programming language, or in an assembly or machine language if necessary. In any case, the language may be a compiled or interpreted language. The program may be a fully installed program or an update program. The application of the program on the system generates any situation in which instructions for executing the method are generated. The computer program may alternatively be stored and executed on a server in a cloud computing environment, which communicates with one or more clients via a network. In this case, the processing unit executes instructions included by the program, thereby causing the method to be performed on a cloud computing environment.

[0094] The implementation of the above method is now discussed.

[0095] This embodiment involves an AI agent with embedded physics knowledge that can predict the effective properties of materials with a specific fixed microstructure. More specifically, this embodiment can design predictions of the effective physical properties of these materials when the microstructure itself can vary.

[0096] This embodiment relates to numerical simulation of effective macroscopic physical properties of heterogeneous materials containing microstructures. Heterogeneity means that multiple materials are spatially arranged into an internal structure, i.e., a microstructure. Such materials are generally referred to as composite materials, constructed materials, metamaterials, or multiscale materials. Such constructed materials or metamaterials also include grid-like or spin-like structures, in which only one material forms a specific structure. The other material is a void phase. These multiscale materials are often used in various industrial applications, such as automotive, aerospace, medical, civil engineering, etc.

[0097] Figure 2 Examples of constructed materials are shown. Example 210 is a unidirectional fiber composite, Example 220 is a woven composite, Example 230 is a lattice structure, and Example 40 is a spinodal structure.

[0098] The multi-scale nature of these materials arises from the fact that the macroscopic properties are determined by both the compositional properties of the microstructure (mechanical, thermal, or other physical properties) and the morphology of the microstructure (geometry, topology, etc.). For example, mechanical properties are often described by stress-strain curves, which measure the stress required to deform a material by a certain amount.

[0099] Figure 3 An example of a strain-stress curve is shown. The initial slope is called Young's modulus, which characterizes the elastic stiffness. In the case of multiscale materials, the stress-strain curve depends on the material properties of its components as well as the microstructural morphology.

[0100] One of such heterogeneous materials is fiber-reinforced composites, in which glass or carbon fibers are used as reinforcements in a polymer matrix. These reinforcements can guarantee high mechanical resistance specifications while achieving overall part weight reduction. Figure 4 Examples of such materials are shown.

[0101] The microstructure of such multiscale materials can be described by several microstructural parameters that characterize their morphology.

[0102] For unidirectional fiber composites, Figure 5 represents the volume fraction vf (of fibers) varying from 0.2 to 0.8.

[0103] Another morphological parameter of such unidirectional fiber composites is the fiber misalignment. Due to the manufacturing process, the fibers are not perfectly aligned but there is some misalignment. This misalignment can be described by the initial misalignment angle.

[0104] Figure 6 Examples of such misalignments between fibers are presented. Figure 6 Three fiber composites with different misalignment angles are presented: (upper) 0°, (middle) 1°, and (lower) 2°. In embodiments, the morphology parameter may be considered if its influence on the effective physical properties is to be studied.

[0105] Short fiber reinforced plastic composites can be described by the volume fraction of the fibers, the aspect ratio of the fibers, the fiber orientation, etc. The aspect ratio (ar) is a scalar defined as the ratio between the length and the diameter. This parameter characterizes the elongation of the fiber and usually varies from 1 (nearly spherical particles) to more than 100 (very elongated fibers). Figure 7 and 8 Two examples of aspect ratio values ​​are shown. Figure 7 In , ar = 1 and the fiber is spherical, while in Figure 8 In the example, ar=10 and the fibers are thinner and longer.

[0106] The fiber orientation can be expressed by the vector (a x ,a y ,a z ) indicates that, and satisfies a x +a y +a z = 1. Each of these three components a i describes the statistical probability of finding a fiber in direction i. Fig. 9 and 10 An example is shown. Fig. 9 In , all fibers are oriented in the direction X, and the fiber orientation is described by the vector (1,0,0). Fig.10 In the 3D space, the fibers are randomly oriented and the fiber orientation is determined by the vector describe.

[0107] Such multiscale materials with correlated microstructural parameters are called parametric multiscale materials. This embodiment involves predicting the effective mechanical properties of such parametric multiscale materials. To design optimal metamaterials for industrial applications, accurate and fast predictions of effective physical properties are needed to quantify the uncertainty in material properties and inversely identify the properties of its constituent parts. Efficient scale-bridging techniques between macro- and micro-level simulations are also needed to design and verify industrial components made of multiscale materials.

[0108] This embodiment proposes a new neural network architecture for general parametric microstructures with varying volume fractions and possibly other geometric parameters. In this embodiment, the DMN formulation is used for a fixed microstructure, while being able to accurately and efficiently predict effective linear and nonlinear properties at any given microstructure parameters.

[0109] In this embodiment, a deep material network (DMN) formulation is used. This formulation can be viewed as a neural network-like architecture that, given a component elastic tensor, In the case of a non-homogeneous material, the effective elastic tensor The fitting parameters of the DMN are the weights and rotations of the internal neurons, denoted as (w, θ). During the offline training process, these parameters are adjusted to make the DMN's predictions It matches the values ​​calculated by computational homogenization techniques such as the finite element method applied to representative volume elements of the microstructure (FE-RVE). Fig.11 A schematic comparison between the DMN representation and the finite element solution is shown.

[0110] The original DMN formulation is limited to a specific fixed microstructure. The trained fitting parameters are learned as a compact representation of a specific microstructure. If the morphology of the microstructure varies with some microstructure parameter p, then it can be expected that (w,θ) should also vary with p in some way. However, with the original DMN formulation, the fitting parameters (w,θ) are fixed.

[0111] To solve this problem, this embodiment proposes to use a standard feedforward neural network to consider the dependence of the DMN fitting parameters (w, θ) on the microstructure parameters p. The neural network includes an input layer and an output layer, wherein the input layer uses the microstructure parameters p and the output layer calculates the DMN fitting parameters (w, θ). The hidden layer includes an affine function and a nonlinear activation function. Fig.12 A simple example of such a feed-forward neural network is shown.

[0112] Therefore, the neural network architecture of the embodiment includes two parts. An example of such a two-part neural network is Fig.13 In this example and implementation, a parameter vector p is provided to the first neural network, which represents the microstructural features of the multi-scale material (volume fraction, fiber length, fiber diameter, fiber orientation, etc.). The first neural network is configured to predict a compact representation (w,θ) of a specific microstructure. In this implementation, the effective elastic tensor of the microstructure is calculated using the original formula of the DMN using the calculated compact representation.

[0113] Predicting effective properties The feedforward function of can be described by the following process:

[0114] - Providing a parameter vector p that characterizes the microstructure of the multiscale material (e.g. volume fraction or fiber orientation).

[0115] - Using this parameter vector p as input, the first neural network predicts a compact representation (w,θ) of this specific microstructure with the given parameters.

[0116] - Provided by the elastic tensor The linear elastic properties (Young's modulus, shear modulus and Poisson's ratio) of each component are represented.

[0117] -The original DMN model takes (w,θ) as input to predict the macroscopic properties of multi-scale materials

[0118] This embodiment uses a general feedforward neural network to consider the dependence of DMN parameters (i.e., a compact representation of microstructure) on microstructure. Therefore, it can be generalized to arbitrary parametric microstructures with multiple parameters of different properties. Specifically, this embodiment uses a Physics-Informed Neural Network (PINN). In this network, compared with a standard feedforward neural network, the architecture itself is designed so that it satisfies certain properties proven by physics. In addition, micromechanics-based constraints can be incorporated to improve the generalization ability of the neural network architecture. Therefore, the obtained neural network is able to predict effective characteristics with high accuracy and efficiency in parameter space.

[0119] In this embodiment, the offline training process can be completed simultaneously by using data sets generated from multiple instances of parameter microstructures (different parameter values). This ensures that the proposed neural network can be efficiently trained using all currently available information. The offline training process can be described as follows:

[0120] - Provide an experimental design (sampling points) in a parameter space that describes different microstructural parameters of the microstructure. In the example of a microstructure containing only volume fraction parameters, the parameter space is a one-dimensional interval [vf min ,vf max ].

[0121] - Provides experimental designs (sampling points) in a material space that describes the properties of different input materials

[0122] -Perform finite element simulation for each combined sampling point in parameter space and material space.

[0123] - Defines the loss function that compares the predictions of the parametric DMN model with those of the previous dataset.

[0124] -Adjust the fitting parameters of the parametric DMN model so that the DMN can approximate the previous data set well. The above process is done in an iterative manner. For each cycle (epoch) (i.e., iteration), the fitting parameters are incrementally adjusted according to the gradient of the loss function relative to each fitting parameter.

[0125] The architecture according to an embodiment also provides a new compact representation of parametric microstructures with an arbitrary number of parameters. The reduction in storage requirements is significant, since a finite number of parameters can now represent an unlimited number of complex microstructured three-dimensional meshes.

[0126] This embodiment can also be used for the inverse identification problem set. For the effective properties of the heterogeneous material that can be measured experimentally, the properties of the components of its input can now be effectively calibrated or identified using this embodiment. The process of this inverse identification problem can be described as follows:

[0127] - Train a parametric DMN that represents a given parametric microstructure.

[0128] - Provide effective properties for composite materials.

[0129] - Inverse identification of input components and microstructural parameters.

[0130] Fig.14 An example flow chart of such inverse identification according to this embodiment is shown.

[0131] DMN Architecture

[0132] We now discuss the basic structure of the DMN formula and its fitting parameters (w,θ).

[0133] DMNs are multi-level laminated microstructures consisting of hierarchical nested laminates of different length scales or levels of lamination.

[0134] Fig.15 An example of a DMN with three levels (i.e., level 0, level 1, and level 2) is shown. The number of levels n of this multi-level stack (also called the number of DMN layers) characterizes the number of nested layers. Its architecture corresponds topologically to a perfect binary tree. Each "node" is a level 1 stacked microstructure that serves as a "building block" or "neuron" in the neural network-like architecture. Fig.16 An example of one neuron of the DMN, discussed in detail later, is shown.

[0135] The DMN is based on the stiffness tensor of its two components To calculate the effective stiffness tensor of the metamaterial At the lowest level, the input is fed into the DMN architecture. At the highest level (i.e., level 0), the output is calculated

[0136] Stiffness Tensor are given in their respective material frames. The stiffness tensor It can be represented by several material parameters that describe its elastic properties: Young's modulus (which describes the stiffness required per unit extension in a specific direction), Poisson's ratio (which describes the relative deformation between two orthogonal directions), and shear modulus (which describes the shear stiffness required per unit shear deformation).

[0137] For DMN offline training, the implementation method will The material is set to be an orthotropic material as known in the art, for example according to the already cited literature by Liu et al., A Deep Material Network for Multiscale Topology Learning and Accelerated Nonlinear Modeling of Heterogeneous Materials (2019), Computer Methods in Applied Mechanics and Engineering, 345, 1138-1168. In this case, the stiffness tensor is represented by 9 material parameters: (E1, E2, E3, v 12 ,v 13 ,v 23 ,μ 12 ,μ 13 ,μ 23 ). The flexibility matrix, which is the inverse of the stiffness matrix, is given by the following symmetric matrix:

[0138]

[0139] The empty element represents 0.

[0140] Neuron definition

[0141] In this implementation, by analyzing the function To define each neuron, the analysis function calculates the effective homogeneous stiffness tensor based on the following elements

[0142] - Stiffness tensors of the two components that make up the multilayer microstructure

[0143] - the volume fraction f′ of one component in the multilayer microstructure, and

[0144] - The rotation θ′ of the current neuron relative to other neurons in the upper layer.

[0145] Without loss of generality, this embodiment uses the volume fraction of the first component.

[0146] The exact formula of the function Lam depends on the spatial dimensionality and the physics considered: mechanical, thermal, etc. Examples of formulas for mechanical properties can be found in the already cited reference Liu et al. A deep material network for multiscale topological learning and accelerated nonlinear modeling of heterogeneous materials (2019), Computer Methods in Applied Mechanics and Engineering, 345, 1138-1168. Fig.16 Examples are given for 2D mechanical properties (ie, plane strain or plane stress conditions).

[0147] In this example, it is assumed that the interface between the two components is defined by a plane with a normal vector aligned with direction 2. Due to mechanical static equilibrium, the stress component σ 22 and σ 12 Along the normal direction and is continuous across the interface. The strain component ε parallel to the interface11 These interface conditions can be expressed by the following equations, including the strain tensor ε = (ε 11 ,ε 22 ,ε 12 ), and the modified stiffness tensor, given by and express:

[0148]

[0149] Using the definition of the effective strain tensor Where f is the volume fraction of component 1, the following formula can be obtained to calculate the local strain tensor of component 1 from the effective stress tensor:

[0150]

[0151] Finally, the definition of the effective stress tensor is adopted The final formula contains the effective stiffness tensor to be calculated:

[0152]

[0153] Finally, the rotation is applied using the following rotation matrix:

[0154]

[0155] get Where R T is the transpose of the rotation matrix.

[0156] In other examples, the embodiment may define a neuron to predict the effective thermal conductivity of a metamaterial. In this case, represents the thermal conductivity tensor of the two components, which is a 2×2 symmetric tensor in the 2D case. According to the heat balance equation, the interface equation becomes:

[0157]

[0158] In the equation, the strain tensor is now replaced by the temperature gradient vector. The effective thermal conductivity tensor is The final formula for is of the same type as above:

[0159]

[0160] The rotation matrix becomes:

[0161]

[0162] DMN fitting parameters

[0163] The fitting parameters of DMN are weights w and rotations θ. Weights are defined on leaf nodes and characterize the corresponding volume fraction of each material node. Weights need to be non-negative. Rotations are defined for each node. Rotations can be described by quaternions respectively.

[0164] In the original DMN formulation, it is assumed that these fitting parameters (w,θ) are fixed. Therefore, this is not sufficient for parametric microstructures p, for which the fitting parameters should vary with the microstructure parameters. Therefore, this embodiment considers the dependence of (w,θ) on p.

[0165] Neural Network with Embedded Physical Knowledge for DMN Fitting Parameters

[0166] In this embodiment, the functional dependence of DMN parameters (w, θ) on microstructure parameters p is directly considered through a feedforward neural network composed of multiple layers of affine transformations and activation functions. Due to the DMN's architecture of embedded physics knowledge, the architecture of the hidden layer does not significantly change the performance.

[0167] exist Fig.17 In the option represented in , this embodiment uses an embedded physics framework where the DMN weight vector w depends only on the volume fraction parameter, while the DMN rotation vector θ varies with all other parameters that do not change the volume fraction. Therefore, the microstructure parameters are divided into:

[0168]

[0169] where vf is the volume fraction of the second stage, and q represents all other independent parameters q that are orthogonal to vf. Based on this partitioning, the Physics-Informed Neural Network (PINN) for DMN parameters is now given by:

[0170] w(p)=w(vf)=ρ(vf·w1+w0),

[0171] θ(p)=θ(q)=θ1q+θ0,

[0172] Where w0 and w1 represent vectors of length n, θ0 represents (2 n -1)×4 matrix, and Θ1 represents (2 n -1)×4×q tensors. These tensors (w0,w1,θ0,θ1) are the fitting parameters of the proposed architecture. The architecture used in this option is referred to as PINN-DMN in the following. In the following, vf (scalar) can be considered as a function of the DMN weight w (represented as a vector), denoted by vf(w).

[0173] exist Fig.18In another option represented by , embodiments employ a fully connected architecture where the DMN fitting parameters w and θ depend on the total parameter vector p.

[0174] Compared to fully connected, the PINN architecture requires fewer fitting parameters due to partitioning. This improves computational efficiency. In addition, numerical simulations show that the PINN architecture provides comparable accuracy compared to fully connected architectures and can also enhance generalization capabilities (interpolation and extrapolation) in parameter space. This means that the proposed PINN method maintains high accuracy in predicting effective properties while changing microstructural parameters.

[0175] Fig.19 Show Fig.17 An example of the application of the PINN method in , where the method is used to perform predictions for different values ​​of the parameters p0, p1, p2.

[0176] Compared with the transfer learning method, only offline training is now required to optimize the fitting parameters of PINN-DMN. Now, for each p in the parameter space, i The neural network can be jointly evaluated with linear elastic property data. Furthermore, the neural network functional dependencies naturally define interpolation and extrapolation inside or outside the training domain and are easily extended to higher parameter dimensions.

[0177] In addition to functional dependencies In addition to the embedded physics knowledge architecture, some physical constraints can be specified to further improve its generalization ability, similar to the embedded physics knowledge machine learning method for nonlinear partial differential equations. This so-called embedded physics knowledge method consists of a standard feedforward neural network and a loss function that considers the physical equations or constraints defined by partial differential equations (PDEs). An example of such an embedded physics knowledge network is Fig. 20 , where the PDE equation is specified as an additional term in the loss function.

[0178] In this embodiment, when calculating Two additional physical constraints are imposed on the neural network. The first constraint is the volume fraction constraint That is, for any volume fraction value, the DMN volume fraction should match the true volume fraction value of the microstructure. The second constraint is the orientation constraint This requires that the average material framework orientation predicted by DMN should also match the orientation of the real microstructure.

[0179] We now discuss these two constraints.

[0180] Volume fraction constraint

[0181] The DMN volume fraction can be calculated from the DMN weight vector using the following formula:

[0182]

[0183] In this formula, w i Represents the weight defined on each leaf node (material node) of the DMN architecture. The weight is divided into two parts: The weights defined for component 1 and for the index set The union of these two index sets is The above formula calculates the volume fraction of phase 2 in the entire microstructure. Due to the embedded physics-informed design of DMN, the actual volume fraction vf of the actual metamaterial is Ω can be well approximated to the DMN volume fraction.

[0184] Using the PINN-DMN framework, currently these DMN weights w i Calculated by a neural network. Therefore, this embodiment can specify the following volume fraction constraints for all volume fraction values:

[0185]

[0186] This formula means that the DMN volume fraction always matches the actual volume fraction of the microstructure. To enforce this constraint during offline training, a loss function based on this volume fraction constraint is defined as follows:

[0187]

[0188] where vf i denotes collocation points that weakly prescribe the volume fraction constraint, and n denotes the number of such collocation points. Since the volume fraction is a scalar between 0 and 1, uniform sampling is used.

[0189] Fig.21 and 22 An example of a volume fraction constraint prescribed on a PINN is shown. Fig.21 shows the minimization of the volume fraction constrained loss function, while Fig. 22 Predicted DMN volume fractions at an early stage (ie, epoch=10) and at the end of training (ie, convergence) are shown.

[0190] As the loss function decreases during the optimization iterations, the DMN volume fraction better matches the actual volume fraction of the microstructure.

[0191] Orientation Constraints

[0192] The DMN not only captures the microstructural morphology. The DMN also learns the material orientation distribution function in the microstructure. Using the orientation tensor, this implementation proposes orientation constraints imposed on the DMN weights w and rotations θ so that the parametric DMN generalizes this material orientation knowledge throughout the parameter space.

[0193] The orientation tensor briefly describes the statistical information of the orientation distribution of a unit vector. Given an orientation distribution function f: in Representing the two-dimensional surface of a unit sphere, the 3×3 second-order orientation tensor is defined by:

[0194]

[0195] Due to the normalization constraint of e and the probability density f, it can be easily shown that a is symmetric. Although higher-order orientation tensors exist, the second-order orientation tensor is the most commonly used to characterize the local fiber orientation due to the manufacturing process and its impact on material properties.

[0196] In this implementation, this generalization of the (second-order) orientation tensor is used for the orientation distribution of rotations. In contrast to transversely isotropic fibers, where a single unit vector is sufficient to characterize their material frame, general anisotropic materials (such as orthotropic materials) require a rotation matrix R to describe the transformation from their material frame (e1, e2, e3) to the global frame. In this case, the orientation distribution function f is now defined for the rotation matrix: The domain of this function SO(3) is the 3D rotation group, which contains all rotation matrices in 3D space that are orthogonal to the unit determinant. Assume that each column of R with 1≤i≤3 is essentially represented in the global frame Then it defines the orientation distribution of the material framework.

[0197] Figure 23 to Figure 25 Examples of such orientation distributions are shown for each of the three material axes provided for the tows in the braided microstructure. Fig.23 shows the direction e1, Fig.24 shows the direction e2, Fig.25 Direction e3 is shown.

[0198] According to this interpretation, three (second-order) orientation tensors can be defined for each axis of the material frame:

[0199]

[0200] where does not represent an Einstein summation. The tensor a can be interpreted as a third-order material framework orientation tensor. For discrete probability functions defined on a grid, the integral can be interpreted as a weighted average using the element volumes as weights. Note that in the case of a two-phase microstructure, this can be calculated for each phase.

[0201] In the example above, the material framework orientation tensor of the tow is given by:

[0202]

[0203] Therefore, the tows are isotropically oriented to e1 and e2 in the XY plane and unidirectionally oriented to e3 in the Z axis. The material framework of the matrix phase is consistent with the global framework. In this case, a is similar to the unidirectional orientation tensor and satisfies:

[0204]

[0205] Such orientation tensors can also be defined for DMN. A DMN rotation θ defines a rotation matrix between the local frame of the current stack and the local frame of the next nested layer. Therefore, a DMN rotation can be constructed to obtain an effective rotation matrix from the material frame (e1, e2, e3) (leaf stack) to the global frame Ω of the microstructure (root stack). p(i) represents the parent node of stack i in the DMN binary tree architecture. For example, in Fig.15 In the 3-layer DMN example shown in , for leaf layer 4, p(4) = 2, and for the root layer, p 2 (4) = p(p(4)) = p(2) = 1. For each leaf stack i that hosts a DMN material node, the effective rotation matrix from the material frame to the global frame is given by:

[0206]

[0207] Note that for an n-layer DMN, for any leaf stack i, p n-1 (i) = 1. Since there is no Input rotation matrix, material nodes sharing the same leaf stack also get the same effective rotation matrix. For example, for material nodes 3 and 4 contained in the leaf stack, their effective rotation is:

[0208]

[0209] Using these effective rotations on the leaf stack, the DMN material framework orientation tensor for phase 1 can be calculated as follows:

[0210]

[0211] A similar formula can be defined for phase 2. Compared to the DMN volume fraction, the calculation of the DMN orientation tensor requires both the DMN weights w and the rotation θ. Using an embedded physics-informed neural network for the DMN parameters, this embodiment exploits the following orientation constraint by defining a loss function:

[0212]

[0213] A similar loss function is defined for phase 2 and then summed together. In this equation, p i is a configuration point in the microstructure parameter space, and a(p i ) is the actual orientation tensor of the parametric microstructure. Similar to the volume fraction constraint, the number n represents the number of these configuration points. These configuration points can be sampled using the Latin hypercube sampling method.

[0214] Material sampling

[0215] This implementation follows the original material sampling method proposed by the already cited Liu et al., "A Deep Material Network for Multiscale Topological Learning and Accelerated Nonlinear Modeling of Heterogeneous Materials", Computer Methods in Applied Mechanics and Engineering, 345, 1138-116. In this implementation, it is assumed that for their respective material frameworks, All are orthotropic. A total of 9+9=18 material parameters are required to characterize its orthotropic elastic behavior, and 1 additional scaling parameter is used to introduce a contrast in the elastic moduli between the two phases. In this work, Latin hypercube sampling is used to sample this 19-dimensional space.

[0216] Total loss function

[0217] These two constraints are added to the overall loss function based on the mean squared error between the homogenized elastic tensor predicted by the DMN and the elastic tensor generated by the FE-RVE simulation. These errors are averaged to calculate the loss for fixed microstructural parameter values Finally, the samples in parameter space , which are averaged into the total loss function.

[0218]

[0219] The total loss function is minimized using a gradient-based approach. In this embodiment, the PINN-DMN architecture is implemented using PyTorch. Therefore, the derivative of the loss function with respect to the fitting parameters can be easily calculated using automatic differentiation.

[0220] Examples of Specific Parameter Microstructures

[0221] The implementation was tested on a specific parametric microstructure: a 2×2 twill woven composite with different tow volume fractions. The goal was to evaluate the PINN-DMN for such microstructures in terms of elastic modulus predictions as a function of volume fraction. The inverse identification of material and microstructural parameters using the PINN-DMN was also considered as an accurate and efficient surrogate for parametric microstructures.

[0222] The 3D finite element model was constructed using TexGen with 4 body integral values ​​of the tow. Three of them (vf=0.459, vf=0.608 and vf=0.729) were used to generate the training data set, while vf=0.537 was used to test the interpolation accuracy. FE-RVE analysis was performed under Abaqus to obtain 6×6 linear elastic stiffness tensors. Each FE-RVE model contains 75,000 voxel elements and 241,899 degrees of freedom. The simulation was performed using 24 threads of an Intel(R) Xeon(R) Gold5220R CPU@2.20GHz, which took about 11 seconds.

[0223] Fig.26 (a)-(d) present the FE-RVE models of woven composites with different fiber volume fractions. Fig.26 (a) vf = 0.459, Fig.26 (b) vf = 0.608, Fig.26 (c) vf = 0.729, Fig.26 In (d), vf = 0.537. The first three models are used to generate training datasets, while the last model is used to test interpolation accuracy.

[0224] The true linear elastic properties of the two phases can be obtained as follows. The matrix is ​​isotropic, while the carbon fiber tows are assumed to be transversely isotropic in the local material framework. For the training microstructures, vf = 0.459, vf = 0.608 and vf = 0.729, 400 of the 500 material samples were used as the training dataset, while the other 100 were retained for validation. For vf = 0.537, all 500 samples were used to test the interpolation accuracy.

[0225] Table 1. Real linear elastic properties of the two phases of the woven composites.

[0226] E(MPa) v Matrix 3800 0.387

[0227] Table 2 Real linear elastic properties of the two phases of the woven composites

[0228] <![CDATA[E1(GPa)]]> <![CDATA[E2(GPa)]]> v ν <![CDATA[μ 12 (GPa)]]> Tow 78.8 6.24 0.35 0.6 2.39

[0229] The PINN-DMN is then trained according to the two physical constraints. Compare the loss histories for the 7-layer DMN and the 9-layer DMN. As the number of layers increases, the neural network becomes more accurate, resulting in smaller (converged) loss values. The total loss can also be divided into the FE-RVE data part and the physical constraint part. These two parts are monotonically decreasing. The physical constraints are well satisfied, as the corresponding 9-layer loss value is close to 10 -5 Subsequently, results using 7 layers are reported as it provides satisfactory accuracy. Fig. 27 PINN-DMN training of a braided composite is shown. Fig. 27 (a) shows the loss history of 7-layer DMN and 9-layer DMN, while Fig. 27 (b) shows the partitioning of the total loss into the FE-RVE data part and the physical constraint part.

[0230] The volume fraction of the tow and the material framework orientation tensor are compared to their prescribed values. The DMN volume fraction matches the FE-RVE data points and agrees well with the theoretical straight line even when evaluated outside the training region. The components of the DMN orientation tensor As shown below. Excellent consistency was achieved.

[0231] Fig.28 Demonstrated verification of the physical constraints of a braided composite material. Fig.28 (a) shows the volume fraction of the tow, Fig.28 (b) shows the material framework orientation tensor of the tow.

[0232] Training, validation, and test errors were calculated at different volume fractions. The median error for all four microstructures was about 2%. The test error was only slightly larger than the training error.

[0233] Fig.29 Shown are the training, validation, and test errors of PINN-DMN at different volume fractions represented by their respective 0.1, 0.5 (median), and 0.9-quantities.

[0234] Homogenized elastic moduli are calculated with varying volume fractions using true linear elastic properties. For comparison, a fully connected architecture is also tested with physical constraints. As mentioned above, a fully connected architecture means that the DMN rotation also becomes a function of volume fraction. Both models capture the nonlinear vf dependence of these elastic moduli well. The fully connected architecture not only increases the number of fitting parameters (addition of Θ1), but also does not improve the prediction accuracy compared to PINN.

[0235] Fig.30 The parametric DMN model is shown: Fig.30 (a) is the Young's modulus in the plane Fig.30 (b) is the transverse Poisson's ratio and Fig.30 (c) is the in-plane shear modulus FE-RVE training and testing data are indicated.

[0236] DMN calculates the given homogenized elastic tensor and The computational efficiency of DMN in terms of uniform elastic tensor prediction is compared with FE-RVE simulations. Due to this significant speedup, DMN can be further used for parameter analysis, uncertainty quantification and material property calibration. The following table shows the computational speedup of DMN compared to FE-RVE in terms of uniform elastic tensor prediction. For both cases, the experiments have been performed on 24 cores of an Intel(R) Xeon(R) Gold5220R CPU @ 2.20GHz.

[0237] Table 3 Computational speedup of DMN compared with FE-RVE

[0238] FE-RVE 7-layer DMN Time consumption (speed up) 11s 6.62ms(1662)

[0239] Inverse identification of input material and microstructural parameters

[0240] The trained PINN-DMN can be used as an accurate and computationally efficient surrogate for parametric microstructures. It can be used not only for the forward prediction of effective properties under different microstructure parameters, but also for identifying material and microstructure properties in the inverse identification problem.

[0241] In this application, a homogenized elastic tensor is provided and the goal is to identify the elastic properties of the matrix Elastic properties of tow and the volume fraction vf of the tow. In practice, it can be measured experimentally Here, this embodiment uses the FE-RVE simulation results on the test microstructure vf=0.537 The results are obtained using the required real characteristics.

[0242] Due to the PINN-DMN architecture, automatic differentiation can be used to easily compute Driven by the fact that the gradient of , this embodiment adopts a gradient-based optimization method based on the loss function, which combines the DMN prediction and the input homogenized data Compare with the standard Frobenius norm.

[0243]

[0244] Optimization iterations require unknowns Therefore, the true properties and the true volume fraction vf = 0.537 are randomly perturbed using a normal distribution with a coefficient of variation of 20%. The initial values ​​generated by the two random realizations are indicated in the two tables below.

[0245] Table 4. Initial linear elastic properties of the two phases of the inverse identification problem.

[0246] Matrix E(MPa) v Implementation 1 4491 0.400 Implementation 2 3908 0.437

[0247] Table 5. Initial linear elastic properties of the two phases of the inverse identification problem.

[0248]

[0249] The loss function converges quickly and can reach 10 within 1000 optimization iterations. -5 The input homogenized elastic tensor is For both sets of initial values, the converged homogenized elastic tensor is similar to the input data The relative errors between them are 0.42% and 0.26%, respectively.

[0250] Fig.31 The calibration loss history using two initial values ​​is shown: (a) Implementation 1; (b) Implementation 2.

[0251] Table 6 Comparison of converged homogenized elastic tensor based on initial values ​​and true values ​​according to two different implementations

[0252]

[0253] Input parameters for reverse identification Report. The material and microstructural parameters found by implementation 1 are similar to the actual properties. However, with implementation 2, the identified volume fraction of 0.667 is higher than the data of 0.537. At the same time, the longitudinal Young's modulus of the tow (64.6 GPa) is also smaller than the data (78.8 GPa). This illustrates the non-uniqueness of the inverse identification problem. The user can provide additional conditions to further constrain the inverse problem. For example, some input properties (such as the properties of the matrix) can be assumed to be fixed. However, this inverse identification problem can now be efficiently solved using PINN-DMN in seconds.

[0254] Table 7 Linear elastic properties and volume fractions of the two phases identified in the inverse identification problem

[0255] Matrix E(MPa) ν Implementation 1 4309 0.385 Implementation 2 4624 0.375

[0256] Table 8 Linear elastic properties and volume fractions of the two phases identified in the inverse identification problem

[0257] Tow vf <![CDATA[E1(GPa)]]> <![CDATA[E2(GPa)]]> <![CDATA[v 12 ]]> <![CDATA[ν 23 ]]> <![CDATA[μ 12 (GPa)]]> Implementation 1 0.533 80.7 5.53 0.340 0.614 2.09 Implementation 2 0.667 64.6 5.27 0.364 0.617 1.90

[0258] Compressed representation of microstructures with arbitrary parameters

[0259] Traditionally, parametric microstructures can be computationally described using an unstructured grid that discretizes the microstructure in space. For example, for a woven microstructure, Fig.32A typical grid can be set up with a fixed volume fraction value. For different parameter values, another grid is provided.

[0260] For each mesh, the node coordinates and the cell connectivity matrix need to be stored. In this case, about 4MB (Mega Bytes) are required. It should be noted that this only involves one specific volume fraction value. If five volume fractions are required, five microstructure meshes need to be stored. When many parameter values ​​are required, the grid-based representation may require a lot of resources.

[0261] Table 9

[0262] Grid-based representation node unit Total Bytes Data Structure 80000×3 floating point numbers 75000×8 integer <![CDATA[4.32×10 6 ]]>

[0263] The parameter DMN architecture of this embodiment is also used as a new compressed representation of the parameter microstructure. In contrast to the grid-based representation, the parameter DMN approach describes the entire parameter microstructure. For a 7-layer parameter DMN architecture, only 6KB (Kilo Bytes) is required to represent the parameter microstructure with arbitrary parameter value changes.

[0264] Table 10

[0265] Implementation Weight Rotation Total Bytes Data Structure 28 = 256 floating point numbers 4×27-1=508 floating point numbers 6.112×103

[0266] This compressed representation can be used to construct a database of parameter multi-scale materials. Each entry corresponds to a specific parameter multi-scale material with corresponding morphological parameters that are allowed to vary. Only the optimized neural network parameters need to be stored. The following table presents a tabular example of such an entry.

[0267] Table 11

[0268]

Claims

1. A computer-implemented method for training a neural network based on a deep material network (DMN), the neural network being configured to predict macroscopic physical properties of a multiscale material, the multiscale material comprising one or more components, the method comprising: - Obtain a data set, each entry of the data set corresponds to a multi-scale material object, and the entry includes: ○Tensor The tensor describes the physical properties of the multi-scale material object at a macroscopic level; ○One or more tensors Each tensor describes a physical property of a component of the multiscale material object at a microscopic level; and o one or more morphological parameters (p), each morphological parameter describing the morphology of the multi-scale material object; and - training the neural network based on the acquired data set to generate one or more tensors of the object and one or more morphological parameters of the object to predict a tensor that describes the physical properties of the multi-scale material object at a macroscopic level.

2. The method according to claim 1, wherein: The neural network based on the deep material network DMN includes a first block and a second block; wherein The first block is configured to receive as input one or more morphological parameters of the object and output values ​​of a plurality of network parameters ((w,θ)), and The second block is configured to receive the values ​​of the plurality of network parameters ((w,θ)) and one or more tensors of the object As input, it outputs a prediction of the physical properties of the multi-scale material object at a macroscopic level.

3. The method according to claim 2, wherein: The first block comprises a feed-forward neural network; and / or The second block includes a DMN architecture having the plurality of network parameters ((w,θ)).

4. The method according to claim 2 or 3, wherein: The first block includes a fully connected neural network.

5. The method according to claim 2 or 3, wherein: The first block includes a first sub-block and a second sub-block; wherein: The first sub-block is configured to receive as input values ​​of a first subset (vf) of the one or more morphological parameters (p) and to output values ​​of a corresponding subset (w) of the plurality of network parameters ((w,θ)), and The second sub-block is configured to receive as input values ​​of a second subset (q) of the one or more morphological parameters (p) and to output values ​​of a corresponding subset (θ) of the plurality of network parameters ((w,θ)).

6. The method according to claim 5, wherein: said first subset of said one or more morphological parameters comprises a set of volume fractions for each component; and / or The second subset of the one or more morphological parameters comprises one or more orientation parameters.

7. The method according to any one of claims 1 to 6, wherein: The training involves minimizing a loss function that penalizes, for each entry, the difference between: Obtaining tensors describing physical properties of the multiscale material object at a macroscopic level; and, The predicted tensors describe the physical properties of the multiscale material object at the macroscopic level.

8. The method according to claim 7, wherein: The loss function also penalizes: failure to satisfy a volume fraction constraint of the multi-scale material object, and / or failure to satisfy a material orientation constraint of the multi-scale material object.

9. A trainable neural network according to any one of the preceding claims.

10. A computer-implemented method using the neural network of claim 9, the method comprising: - Get: o one or more tensors, each tensor describing a physical property of a component of the multiscale material object at a microscopic level; as well as o one or more morphological parameters, each morphological parameter describing the morphology of the multi-scale material object; and - predicting macroscopic physical properties of the multiscale material object by applying a neural network to the one or more tensors and the one or more parameters, and / or, - Get: ○Tensor The tensor describes the physical properties of the multi-scale material object at a macroscopic level; -Sure: o one or more tensors, each tensor describing a physical property of a component of the multiscale material object at a microscopic level; as well as o one or more morphological parameters, each morphological parameter describing the morphology of the multi-scale material object; as well as The determining comprises minimizing a difference between the acquired tensor and a candidate prediction value of the acquired tensor, the candidate prediction value being determined by applying the neural network to candidate one or more tensors and candidate one or more morphological parameters.

11. The method according to claim 10, wherein: The minimization of the difference involves computing: the gradient of the candidate prediction value with respect to each corresponding candidate tensor of the one or more tensors, and The gradient of the candidate prediction value relative to each corresponding candidate morphological parameter of the one or more morphological parameters.

12. A database of multi-scale materials, wherein each entry of the data set corresponds to a corresponding multi-scale material, and the entry comprises: one or more morphological parameters, each morphological parameter describing the morphology of the multi-scale material object; as well as A neural network trained according to any one of claims 1 to 8, used to predict the macroscopic physical properties of multi-scale materials.

13. A computer program comprising instructions for executing the method according to any one of claims 1-8 and / or 10-11.

14. A computer-readable storage medium having recorded thereon the computer program according to claim 13, and / or the neural network according to claim 9, and / or the database according to claim 12.

15. A system comprising a processor coupled to a memory, the memory having recorded thereon the computer program according to claim 13, and / or the neural network according to claim 9, and / or the database according to claim 12.

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