Predicting macroscopical physical properties of multi-scale material

Abstract

To provide: a method, data base, program, storage medium and system for training a Deep Material Network-based neural network configured to predict a macroscopical physical property of a multi-scale material.SOLUTION: The method according to the invention comprises: training, based on a dataset, a Deep Material Network-based neural network, each entry of the dataset including a tensor C describing a physical property of an object at a macroscopical level, one or more tensors C1 and the like each describing a physical property of a component of the object at the microscopical level, and one or more morphological parameters P each describing a morphology of the object; and predicting a tensor C describing the physical property of the object at a macroscopical level based on the one or more tensors for the multi-scale material object and based on the one or more morphological parameters for the object.SELECTED DRAWING: Figure 13

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 prediction of macroscopic physical properties of multi-scale materials. [Background technology]

[0002] For the design, engineering and manufacturing of objects, many systems and programs are offered on the market. CAD is an abbreviation for Computer-Aided Design and relates, for example, to software solutions for designing objects. CAE is an abbreviation for Computer-Aided Engineering and relates, for example, to software solutions for simulating the physical behavior of future products. CAM is an abbreviation for Computer-Aided Manufacturing and relates, for example, to software solutions for defining manufacturing processes and operations. In such computer-aided design systems, the graphical user interface plays a key role with regard to the efficiency of the techniques. These techniques can be incorporated into Product Lifecycle Management (PLM) systems. PLM refers to a business strategy that helps companies share product data, apply common processes and leverage enterprise knowledge for the development of products across the concept of an extended enterprise, from conception to the end of its life. The PLM solutions offered by Dassault Systèmes (under the trademarks CATIA, ENOVIA and DELMIA) provide an engineering hub that organizes product engineering knowledge, a manufacturing hub that manages manufacturing engineering knowledge, and an enterprise hub that allows enterprise integration and connection to both the engineering and manufacturing hubs. Overall, the system provides an open object model that links products, processes, and resources to enable dynamic, knowledge-based product creation and decision support that drives optimized product definition, manufacturing preparation, production, and service.

[0003] Solutions and products have been proposed for determining the mechanical behavior of materials used in various products and processes. Such solutions and products include material constitutive models and experimental methods. Due to the increasing demand for multi-scale materials in various industries, certain solutions and products have been proposed for predicting the mechanical behavior of such materials.

[0004] A data-driven multiscale material modeling method based on representative volume element (RVE) mechanistic homogenization theory and advanced machine learning techniques is taught (see, for example, non-patent literature 1). The method uses a collection of connected mechanistic building blocks with analytical homogenization solutions to describe the entire complex material response.

[0005] In this context, there remains a need for improved prediction of the macroscopic physical properties of multiscale materials. [Prior art documents] [Non-patent literature]

[0006] [Non-Patent Document 1] Liu,Wu&Koishi“A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials.”Computer Methods in Applied Mechanics and Engineering,345,1138-1168,2019 Summary of the Invention

[0007] Accordingly, a computer-implemented method is provided for training a deep material network (DMN)-based neural network configured to predict macroscopic physical properties of a multiscale material. The multiscale material includes one or more components. The method includes acquiring a dataset, each entry of the dataset corresponding to a respective multiscale material object. The entries include tensors describing physical properties of the multiscale material object at a macroscopic level. The entries include one or more tensors each describing physical properties of components of the multiscale material object at a microscopic level. The entries further include one or more morphological parameters each describing a morphology of the multiscale material object. The method further includes training a neural network based on the acquired dataset to predict tensors describing physical properties of the object at a macroscopic level based on the one or more tensors for the multiscale material object and based on the one or more morphological parameters for the object.

[0008] The method may include one or more of the following. - a neural network based on a deep material network (DMN) comprising a first block and a second block, the first block configured to receive as input one or more morphological parameters of an object and to output values ​​of a plurality of network parameters, and the second block configured to receive as input values ​​of the plurality of network parameters and one or more tensors of the object and to output a prediction of a physical property of a multi-scale material object at a macroscopic level; - the first block is a feedforward neural network and / or the second block has a DMN architecture with a plurality of network parameters; - The first block is a fully connected neural network, - the first block comprises a first sub-block and a second sub-block, the first sub-block configured to receive as input values ​​of a first subset of one or more morphological parameters and to output values ​​of a respective subset of the plurality of network parameters, and the second sub-block configured to receive as input values ​​of a second subset of one or more morphological parameters and to output values ​​of a respective subset of the plurality of network parameters; - the first subset of one or more morphological parameters comprises a set of volume fractions of each of the components and / or the second subset of one or more morphological parameters comprises one or more orientation parameters; - the training includes minimizing a loss function, which penalizes, for each entry, the difference between a obtained tensor describing the physical properties of the multi-scale material object at a macroscopic level and a predicted tensor describing the physical properties of the multi-scale material object at a macroscopic level; and / or The loss function further penalizes non-compliance with volume fraction constraints of multi-scale material objects and / or non-compliance with material orientation constraints of multi-scale material objects.

[0009] There is further provided a neural network trainable (i.e. learnable) according to the above training method, i.e. a computer-implemented neural network data structure having neural network weights trained by the above method. The provided neural network may, for example, be directly trained by the above method, with its weights being fixed by the training step of the above method.

[0010] There is further provided a computer-implemented method of using a trainable neural network according to the training method. The method of use includes obtaining one or more tensors each describing physical properties of components of the multi-scale material object at a microscopic level and one or more morphological parameters each describing the morphology of the multi-scale material object. The method of use further 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.

[0011] A computer-implemented method of using a trainable neural network according to the method of training is further provided. The method of use includes obtaining tensors describing physical properties of a multi-scale material object at a macroscopic level. The method further includes determining one or more tensors each describing physical properties of components of the multi-scale material object at a microscopic level and one or more morphological parameters each describing a morphology of the multi-scale material object. The determining includes minimizing a difference between the obtained tensors and candidate predicted values ​​of the obtained tensors, the candidate predicted values ​​being determined by applying the neural network to the candidate one or more tensors and the candidate one or more morphological parameters.

[0012] In the above method of use, minimizing the difference may include computing gradients of the candidate predicted values ​​with respect to one or more tensors of each respective candidate, and computing gradients of the candidate predicted values ​​with respect to one or more morphological parameters of each respective candidate.

[0013] A database of multi-scale materials is further provided, with each entry of the dataset corresponding to a respective multi-scale material, the entries including one or more morphological parameters each describing the morphology of the multi-scale material object, and a neural network trained to predict macroscopic physical properties of the multi-scale material according to a training method.

[0014] Further provided is a computer program comprising instructions for carrying out the above methods and / or methods of use.

[0015] Further provided is a computer readable storage medium having recorded thereon the computer program and / or the neural network and / or the database.

[0016] There is further provided a system comprising a processor coupled to a memory and a graphical user interface, the memory having stored therein a computer program and / or a neural network and / or a database.

[0017] There is further provided a device comprising a computer readable storage medium having a computer program recorded thereon.

[0018] The device may form or function as a non-transitory computer-readable medium, such as in a Software as a service (SaaS) or other server or cloud-based platform. The device may alternatively comprise a processor coupled to a data storage medium. Thus, the device may form in whole or in part a computer system (e.g., the device is a subsystem of an overall system). The system may further comprise a graphical user interface coupled to the processor. [Brief description of the drawings]

[0019] Non-limiting examples will now be described with reference to the accompanying drawings.

[0020] [Figure 1] FIG. 1 illustrates an example of the present system.

[0021] [Diagram 2] FIG. 1 illustrates an example of the method. [Diagram 3] FIG. 1 illustrates an example of the method. [Figure 4] FIG. 1 illustrates an example of the method. [Diagram 5] FIG. 1 illustrates an example of the method. [Figure 6] FIG. 1 illustrates an example of the method. [Figure 7] FIG. 1 illustrates an example of the method. [Figure 8] FIG. 1 illustrates an example of the method. [Figure 9] FIG. 1 illustrates an example of the method. [Figure 10] FIG. 1 illustrates an example of the method. [Figure 11] FIG. 1 illustrates an example of the method. [Figure 12] FIG. 1 illustrates an example of the method. [Figure 13] FIG. 1 illustrates an example of the method. [Figure 14] FIG. 1 illustrates an example of the method. [Figure 15] FIG. 1 illustrates an example of the method. [Figure 16] FIG. 1 illustrates an example of the method. [Figure 17] FIG. 1 illustrates an example of the method. [Figure 18] FIG. 1 illustrates an example of the method. [Figure 19] FIG. 1 illustrates an example of the method. [Figure 20] FIG. 1 illustrates an example of the method. [Figure 21] FIG. 1 illustrates an example of the method. [Figure 22] FIG. 1 illustrates an example of the method. [Diagram 23] FIG. 1 illustrates an example of the method. [Figure 24] FIG. 1 illustrates an example of the method. [Diagram 25] FIG. 1 illustrates an example of the method. [Figure 26] FIG. 1 illustrates an example of the method. [Figure 27] FIG. 1 illustrates an example of the method. [Figure 28] FIG. 1 illustrates an example of the method. [Figure 29] FIG. 1 illustrates an example of the method. [Diagram 30] FIG. 1 illustrates an example of the method. [Diagram 31] FIG. 1 illustrates an example of the method. [Diagram 32] FIG. 1 illustrates an example of the method. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0022] 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 (i.e., trained) to predict macroscopic physical properties of a multi-scale material. The multi-scale material includes one or more components. The method includes acquiring a dataset. Each entry of the dataset corresponds to a respective multi-scale material object. The entry includes a tensor, i.e., a first tensor that describes a physical property of the multi-scale material object at a macroscopic level. The entry further includes one or more tensors, i.e., a second tensor. Each of the one or more tensors describes a physical property of a component of the multi-scale material object at a microscopic level. The entry further includes one or more morphological parameters. Each of the one or more morphological parameters describes a morphology of the multi-scale material object. The method further includes training the neural network to predict a tensor, i.e., a third tensor, based on the acquired dataset. The (predicted) tensors describe the macroscopic physical properties of the object at a macroscopic level based on one or more tensors for the multi-scale material object and based on one or more geometric parameters for the object.

[0023] 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., approximation) of a physical property of the multi-scale material object at a macroscopic level. The training method trains by being provided with a set of first tensors, which are ground truths of the physical property of the multi-scale material object at the macroscopic level. The training method trains by being further provided with a set of second tensors, which are ground truths of the physical property of each component of the multi-scale material object at the microscopic level.

[0024] The training method improves prediction of macroscopic physical properties of a multiscale material given the physical properties of its constituents. The method obtains this improvement by training a DMN-based neural network architecture. Such an architecture can follow the mechanics of the multiscale material and provide more accurate prediction results. Furthermore, the neural network is further 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. The method can thereby train the neural network for a wide range of materials that can be defined by the one or more morphological parameters.

[0025] The present method constitutes an improved solution for training neural networks of the DMN type compared to the cited prior art. The prior art discloses the training of a given (i.e. fixed) morphology (which is defined by one or more morphological parameters). Since the trained parameters of the neural network depend on this given morphology, it is necessary to retrain the neural network for each new morphology, which is computationally expensive, or to interpolate / extrapolate the output of the trained network for the new morphology, which reduces the accuracy of the prediction. However, the method of the present disclosure integrates the morphological parameters in the training phase (which is offline), thereby significantly reducing the computational cost for each new morphology in the evolution / inference phase (which is online). The present method provides these benefits without sacrificing the accuracy of the prediction.

[0026] The method serves to train a neural network capable of predicting macroscopic physical properties of a multi-scale material, consisting of various components and of different morphologies. Such prediction is an advantageous solution compared to conventional mechanical testing that experimentally determines said mechanical properties. Such trained neural networks provide designers with the ability to obtain predictions of mechanical properties in a non-destructive manner, with high accuracy and in a significantly shorter time compared to said mechanical testing. This allows designers to evaluate a broader class of materials for a particular use case, thereby improving the final design. A use case may be defined for a mechanical object formed at least in part of a multi-scale material. The use case may further be defined by a set of load cases that are assumed to be applied to said mechanical object. The set of load cases may be representations of forces, torques, and / or stresses that are exerted on the mechanical object during its functioning.

[0027] The method and / or trained neural network can be used to design a multi-scale material object or product and / or to manufacture the object or product in the real world according to the design. 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 objects in multi-scale materials because the internal microstructure of the object can be precisely machined.

[0028] The trained neural network may be used, for example, according to a first aspect of a method of use described below, to predict a macroscopic physical property of a given multi-scale material object (a mechanical property of the object at a macroscopic level, such as compliance or stiffness) based on one or more tensors each describing a physical property of the components of the multi-scale material object at a microscopic level (a mechanical property of each component at a microscopic level, such as compliance or stiffness) and one or more morphological parameters each describing the morphology of the multi-scale material object (e.g., volume fraction, fiber misalignment, and / or fiber aspect ratio). This may be done during design of the object (e.g., repeated for different inputs and / or to predict different properties), for example, to test various inputs to converge towards desired microphysical properties for the object.

[0029] Alternatively, the trained neural network can be used to optimize the inputs (i.e., one or more tensors each describing a physical property of a component of the multiscale material object at a microscopic level (e.g., a mechanical property of each component at a macroscopic level, such as compliance or stiffness, and one or more morphological parameters each describing the morphology of the multiscale material object (e.g., volume fraction, fiber misalignment, and / or fiber aspect ratio)) to arrive at a desired (given) tensor describing the physical property of the multiscale material object at a macroscopic level (e.g., a mechanical property of the object at a macroscopic level, such as compliance or stiffness, etc.). This allows for finding optimal inputs for the network to arrive at a given optimal tensor describing the physical property of the multiscale material object at a macroscopic level.

[0030] The methods and / or uses of neural networks may be included in design and manufacturing processes. Such processes include: - optionally carrying out a training method and / or providing a neural network obtained according to the training method; - designing a multi-scale material object, for example using a neural network trained according to the first or second aspect of the method of use (e.g. repeatedly or iteratively) by testing various inputs of the network (i.e. tensors describing physical properties and / or morphological parameters at a macroscopic level) and / or determining optimal inputs for a desired output (i.e. optimal parameters and / or optimal tensors describing physical properties of components at a microscopic level for a desired tensor describing physical properties at a microscopic level); - using the results of the design to manufacture objects in the real world.

[0031] Depending on the type of system used for the design, the outcome of the design may be defined by different kinds of data. The system may in fact be any combination of CAD, CAE, CAM, PDM and / or PLM systems known in the art. Designing a CAD model is the first step towards computer-aided manufacturing. In fact, CAD solutions offer important functions such as feature-based modeling and boundary representation (B-Rep) to reduce the risk of errors and loss of precision during the manufacturing process handled in CAM solutions. In fact, the CAD model is intended to be manufactured. It is therefore a virtual twin, also called digital twin, of the object to be manufactured with two objectives: - Ensuring the correct behaviour of the manufactured object in a particular environment (e.g. in combination with the first or second aspect of the method of use to ensure desired physical properties in the design). - Ensuring the manufacturability of the objects produced.

[0032] The use of the results of the design may be included in a production process, which may include producing a physical product corresponding to the results of the design input into the production process after obtaining the design. The production process may include the following steps: - (e.g. automatically) obtaining a CAD or CAE model of the design (e.g. upon one or more conversions / transformations). - Optionally, (e.g., automatically) converting the CAE model into a CAD model using a (e.g., automatic) CAE to CAD conversion process. - Using the resulting CAD model to manufacture the design.

[0033] Using a CAD model to manufacture can include, for example, the following processes. - editing the resulting CAD model, - performing simulations based on the CAD model or the corresponding CAE model (e.g. the CAE model on which the CAD model is based after a CAE to CAD conversion process), such as simulations for validation of mechanical, use and / or manufacturing properties and / or constraints (e.g. structural simulations, thermodynamic simulations, aerodynamic simulations); - Editing the CAD model based on the results of the simulation; - optionally (i.e. depending on the manufacturing process used, the manufacturing of the mechanical product may or may not include this step) determining (e.g. automatically) manufacturing / CAM files based on the (e.g. edited) CAD model in order to produce / manufacture the manufactured product; - Sending CAD and / or manufacturing / CAM files to the factory, and / or - Producing / manufacturing (e.g. automatically) the machine product originally represented by the model output by the design method based on the determined manufacturing / CAM files or on the CAD model. This may involve (e.g. automatically) feeding the manufacturing / CAM files and / or the CAD files to a machine that performs the manufacturing process.

[0034] This final step of production / manufacturing may be called the manufacturing or production step. This step produces / machines the part / product based on the CAD model and / or CAM file, for example when the CAD model and / or CAD file are fed to one or more manufacturing machines or computer systems controlling the machines. The manufacturing step may include performing any known manufacturing process or sequence of manufacturing processes, such as, for example, one or more additive manufacturing steps, one or more cutting steps (e.g., laser cutting or plasma cutting steps), one or more blanking steps, one or more forging steps, one or more bending steps, one or more deep drawing steps, one or more forming steps, one or more machining steps (e.g., milling steps) and / or one or more punching steps. Since the design method improves the design of the model (CAE or CAD) representing the part / product, the manufacturing and its productivity are also improved.

[0035] Next, the training method is described.

[0036] As known in the field of machine learning, a neural network is a function that includes operations that follow an architecture, where each operation is defined by data that includes parameters (also known as weight values). The architecture of a neural network defines the operands of each operator and the relationships between the parameters. Thus, learning a neural network includes determining the values ​​of weights based on a data set that is configured for such learning and that corresponds to its architecture. In other words, training a neural network obtains a decision function that associates the inputs of the network with corresponding outputs. Such training is performed using a training data set that is composed of a set of inputs with corresponding ground truth outputs. The training data set includes data pieces that each form a respective training sample. The training samples represent the diversity of situations in which the neural network will be used after it has been learned. Any data set referred to herein may include more than 1000, 10000, 100000, or 1000000 training samples.

[0037] "Deep Material Network (DMN) based neural network" means a neural network whose architecture includes a sub-architecture (i.e., a part of this architecture) that follows DMN. Such a DMN sub-architecture may be any DMN architecture known in the art, for example the Deep Material Network architecture of "Deep Material Network (DMN) based neural network" in ...

[0038] As known in the fields of mechanics and materials science, a "multiscale material" (sometimes referred to synonymously as a composite material, a structured material, or a metamaterial) refers to a heterogeneous material that includes a microstructure. A heterogeneous material is composed of multiple materials that are spatially arranged in an internal structure, i.e., a microstructure. A multiscale material includes one or more components. In other words, the microstructure of the multiscale material object defines the topology and geometry of each part of the object formed in each component. Each of the one or more components can be heterogeneous or homogeneous materials. A multiscale material can be a parametric multiscale material, i.e., a material with a morphology / microstructure defined by (a set of) one or more (morphological) parameters. Each set of values ​​of the one or more parameters defines (a realization of) a multiscale material. In other words, each multiscale material object is a realization of a parametric multiscale material having values ​​of a set of parameters. A "multiscale material object" refers to a physical object that is formed in a multiscale material. Such a multiscale material object can have a standard geometry and dimensions, for example, according to a test standard.

[0039] The method acquires a dataset, where each entry of the dataset corresponds to a respective multi-scale material object. Acquiring a dataset means providing a dataset. The method may acquire the dataset by accessing a local storage where the dataset is stored (e.g., in a database format) or by remote access to a data storage service, e.g., on cloud storage. The method may create the dataset (e.g., in a dataset formation step) before providing the dataset to the method.

[0040] Each entry includes a tensor that describes a physical property of the multiscale material object at the macroscopic level. Each entry further includes one or more tensors that each describe a physical property of a component of the multiscale material object at the microscopic level. By "physical property of the multiscale material object" we mean any constituent property that defines the physical behavior of an object formed in this multiscale material. The physical property can be a mechanical, thermal, magnetic, or electrical property. The mechanical property can be compliance or stiffness. The thermal / electrical property can be thermal / electrical conductivity, and the magnetic property can be permeability. By physical property of a component of the multiscale material object at the microscopic level we mean the respective physical properties that can describe the physical behavior of the component at the microscopic level. By physical property of the multiscale material object at the macroscopic level we mean the respective equivalent or effective physical properties that can describe the physical behavior of the object at the macroscopic level, i.e., the physical behavior of the object as a whole. 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. Tensors can be represented as arrays (e.g., multidimensional arrays) that consist of one or more components. Tensors can be of different types. For example, scalars or vectors are examples of tensors. Thus, a tensor that describes a physical property (macroscopic or microscopic level) means a tensor whose components represent this physical property. For example, when the physical property is compliance or stiffness, each component of a tensor that describes the compliance or stiffness can include one or more linear elastic properties. The linear plastic properties can be Young's modulus (which describes the stiffness required per unit extension in a 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).

[0041] Each of the macroscopic level tensors and the one or more microscopic level tensors may be obtained according to an experiment (e.g., at respective standard conditions) or a high-fidelity numerical method, such as, for example, a finite element method. Such tensors may be calculated prior to carrying out the method and stored in a database provided to the method.

[0042] An entry may further include one or more morphological parameters, each describing a morphology of the multi-scale material object, where morphology of the multi-scale material object refers to a qualitative description of the microstructure of the multi-scale material.

[0043] The method then trains a neural network based on the acquired dataset. "Based on the acquired dataset" means that the dataset is a training dataset for the neural network. The trained neural network is configured to predict a tensor describing a physical property of the multi-scale material object at a macroscopic level based on one or more tensors describing the physical property at a microscopic level and based on one or more morphological parameters for the object. In other words, the (trained) neural network receives as input one or more tensors describing the physical property at a microscopic level and one or more morphological parameters for the object, and predicts (i.e., outputs) the tensor. By configured to predict the tensor, it is meant that the neural network outputs a tensor that approximates the (ground truth) physical property of the multi-scale material object at a macroscopic level. The method may train the neural network according to any known method in the field of machine learning and artificial intelligence.

[0044] Next, an example of a training method is described.

[0045] In an example, a neural network based on a deep material network (DMN) may consist of a first block and a second block. In other words, the first block and the second block do not overlap and form a division of the neural network based on the DMN. The first block is configured to receive as input one or more morphological parameters for the object and to output values ​​of a plurality of network parameters (also called DMN fitting parameters). The second block is configured to receive as input values ​​of the plurality of network parameters (i.e., the output of the first block) and one or more tensors for the object. The second block is further configured to output a prediction of a physical property of the multi-scale material object at a macroscopic level. By "plurality of network parameters" we mean a plurality of parameters that define a relationship between the output of the second block and one or more tensors input to the second block. Thus, the architecture of the network provides a dependency of the parameters of the second block on the output of the first block, which itself depends on one or more morphological parameters. This constitutes an improved training compared to standard DMNs, where the network parameters are trained for predetermined (i.e., fixed) morphological parameters and therefore cannot be applied to different morphologies of multi-scale material objects. Moreover, these examples provide a separation between the first block and the second block. In other words, in these examples, the first block is configured to output the network parameters using only the morphological parameters, without inputting one or more tensors for the object. On the other hand, the second block is configured to output the physical properties at the macroscopic level, without inputting one or more morphological parameters. Such a separation allows the neural network to use a more efficient architecture for the unknowns in each block (i.e., the network parameters in the first block and the predicted tensors for the physical properties in the second block).For example, the second block may include a DMN architecture that has higher accuracy for such physical property prediction.

[0046] In an example, the first block may be a feedforward neural network. By "feedforward neural network", as known per se, is meant a neural network whose flow is unidirectional, from input nodes through hidden nodes (if they exist) to output nodes, and does not contain cycles or loops (see https: / / en.wikipedia.org / wiki / Feedforward_neural_network). Alternatively or additionally, the second block may be (i.e., consists of) a DMN architecture with multiple network parameters. In such a case, the multiple network parameters may include multiple weights and multiple rotations for the DMN architecture. Each rotation of the neural network may be a rotation tensor (e.g., a 3D rotation tensor). Such rotations and weights may, for example, follow the already cited literature (Non-Patent Document 1).

[0047] In an example, the first block may be a fully connected neural network, in other words, in such an example, each of the plurality of network parameters is defined over the entirety of one or more morphological parameters.

[0048] Alternatively, 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 values ​​of a first subset of one or more morphological parameters as input and to output values ​​of the respective subset of the plurality of network parameters. The second sub-block is configured to receive values ​​of a second subset of one or more morphological parameters as input and to output values ​​of the respective subset of the plurality of network parameters. In other words, in such examples, each subset of the plurality of network parameters is defined for a respective subset of the one or more morphological parameters. The first subset of one or more morphological parameters and the second subset of one or more morphological parameters form a decomposition or partition of the (set of) one or more morphological parameters. Similarly, the subset of the plurality of network parameters corresponding to the first subset and the subset of the plurality of network parameters corresponding to the second subset form a decomposition or partition of the plurality of network parameters. In other words, for each morphological or network parameter, the union of the first subset and the second subset is equal to the entire set of parameters, but the intersection of the first subset and the second subset is the empty set.

[0049] In an example, the first subset of the one or more morphological parameters includes a set of volume fractions for each of the components. Alternatively or additionally, the second subset of the one or more morphological parameters includes one or more orientation parameters. By orientation parameter is meant a parameter that defines one or more orientations of the microstructure of the multi-scale material, e.g., an orientation tensor, e.g., a relative orientation between the components. In an example where the second block has a DMN architecture with multiple network parameters, the subset of the multiple network parameters corresponding to the first subset includes DMN weights. Further, alternatively or additionally, the subset of the multiple network parameters corresponding to the second subset includes DMN rotations.

[0050] In an example, the training includes minimizing a loss function that penalizes, for each entry, the difference between a retrieved tensor describing the physical properties of the multi-scale material object at a macroscopic level and a predicted tensor describing the physical properties of the multi-scale material object at a macroscopic level. This difference may be measured in a mathematical norm. The loss function may be of the following type:

[0051]

number

[0052] Here, the loss e i is the predicted tensor generated (i.e. obtained) by the FE-RVE simulation

[0053]

number

[0054]

number

[0055] It is based on the mean squared error between

[0056]

number

[0057] denotes the number of samples in the parameter space, i.e., one or more morphological parameters.

[0058]

number

[0059] denotes the number of samples in the material space, i.e., the acquired tensor or tensors.

[0060]

number

[0061] The error (i.e., the difference) between the loss at a fixed microstructure parameter value L p Finally, they are averaged to compute the sample

[0062]

number

[0063] are averaged between,to obtain the total loss function,L,. Both the samples in the material space and the samples in the parametric space,can be taken according to any sampling method, e.g., the Latin Hypercube method.,By being a "type" of mathematical formula for identity, we mean that the identity is,given exactly by that formula, or up to the scaling and / or regularization terms,by that formula.

[0064] The loss function may further penalize non-adherence to volume fraction constraints for the multi-scale material object. This penalty means that training of the neural network will tend to match the volume fractions in the DMN architecture to the actual (i.e., ground truth) volume fraction values ​​of the microstructure for any volume fraction value. Alternatively or additionally, the loss function may further penalize non-adherence to material orientation constraints for the multi-scale material object. This penalty means that training of the neural network will tend to match the average material frame orientation predicted by the DMN to the actual microstructure.

[0065] In an example where the penalty function penalizes both non-compliance with volume fractions and non-compliance with material orientation constraints, the penalty function may be of the following type:

[0066]

number

[0067] Here, L p is the loss function defined above, and L νf are the respective terms of the loss function that penalizes non-compliance with the volume fraction (denoted as νf), and L a are the respective terms of a loss function that penalizes non-compliance with material orientation.

[0068] Further provided is a neural network trainable according to the training method, i.e., a neural network having weight values ​​identical to those obtained from training according to the training method. Such a neural network may be any DMN-based neural network configured to predict tensors describing physical properties of a multiscale material object at a macroscopic level based on one or more tensors for physical properties of components of the multiscale material object at a microscopic level and based on one or more morphological parameters for the object. Such a neural network may have been trained according to any example of the training method, i.e., may be a neural network having weight values ​​obtained directly from training according to the training method.

[0069] A computer-implemented method of using a neural network is further provided. Such a method of use includes obtaining one or more tensors each describing a physical property of a component of the multiscale material object at a microscopic level and one or more morphological parameters each describing a morphology of the multiscale material object. The method further includes predicting a macroscopic physical property 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 of use forms a direct prediction method for a physical property of the multiscale material at a macroscopic level. Alternatively or additionally, the method of use includes obtaining a tensor describing a physical property of the multiscale material object at a macroscopic level and determining one or more tensors each describing a physical property of a component of the multiscale material object at a microscopic level and one or more morphological parameters each describing a morphology of the multiscale material object. In this second aspect, the method of use forms an inverse identification method. In such an inverse identification method, the method determines physical properties of one or more components of the multi-scale material based on given physical properties of the multi-scale material object at a macroscopic level (via the acquired tensors) and one or more morphological parameters. The determining includes minimizing the difference between the acquired tensors and candidate predicted values ​​for the acquired tensors. The candidate predicted values ​​are 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.In other words, the method (iteratively) searches for and minimizes (a set of) candidates for one or more tensors and (a set of) candidates for one or more morphological parameters, and for each group of candidates for one or more tensors and one or more morphological parameters, applies the trained neural network to obtain candidate predictions for the obtained tensors, thereby obtaining the variance to be minimized. In other words, the trained neural network is part of the objective function (i.e., the variance) of such minimization, and the one or more tensors and one or more morphological parameters are the free variables of such minimization. The variance may equivalently be called the loss function. In practice, this aspect may form part of or be part of an identification process where the macroscopic physical properties of the multi-scale material object are known (e.g., via high-fidelity numerical simulations such as finite element methods or experiments).

[0070] The method of use may include both or any of the above-mentioned aspects independently. In other words, when the neural network is trained according to the training method, it may be used for direct prediction of macroscopic physical properties of multi-scale material objects and / or for inverse identification of physical properties of components.

[0071] In either of the two aspects of the above-mentioned method of use, the inputs and outputs of the network in the online phase of use (i.e., inference or deployment) are of the same type as the respective portions of the entries in the acquired data set used in the offline stage of training according to the training method. In other words, the acquired tensors related to the macroscopic level, the acquired one or more tensors related to the microscopic level, and the acquired one or more morphological parameters are of the same type (i.e., in physical properties and / or in units) as the tensors describing the physical properties of the multi-scale material object at the macroscopic level, the one or more tensors related to the microscopic level, and the one or more morphological parameters of each entry of the provided data set in the training method, respectively.

[0072] The difference may be the norm of the difference between the obtained tensor and its predicted value. In an example, the difference may be of the following type:

[0073]

number

[0074] Here, L cal is the difference,

[0075]

number

[0076] is the obtained tensor,

[0077]

number

[0078] is the prediction of the obtained tensor, where the norm ||.|| denotes the standard Frobenius norm.

[0079] The minimization of the difference may include calculating the gradient of the candidate predictor with respect to one or more tensors of each respective candidate and calculating the gradient of the candidate predictor with respect to one or more morphological parameters of each respective candidate. In other words, the minimization is a gradient-based minimization. According to the architecture of the DMN-based framework, the method may use automatic differentiation to calculate each of the gradient of the candidate predictor with respect to one or more tensors of each respective candidate and the gradient of the candidate predictor with respect to one or more morphological parameters of each respective candidate.

[0080] A database of multi-scale materials is further provided. Each entry in such database corresponds to a respective multi-scale material. Each entry includes one or more morphological parameters each describing a morphology of the multi-scale material object. Each entry further includes a neural network trained to predict macroscopic physical properties of the multi-scale material according to a training method. Each entry may relate specifically to a multi-scale material object formed within the respective multi-scale material. Such multi-scale material objects may have standard geometries and dimensions, for example, according to a test standard.

[0081] This constitutes an improved solution for storing a database of multiscale materials. As mentioned above, multiscale materials have a microstructure that needs to be represented when stored. Known solutions in the field store multiscale materials using unstructured meshes that spatially discretize the microstructure. Storage of this microstructure for a practical range of microscopic level properties and different morphologies of components is thereby significantly memory costly. This database therefore represents an improved solution since for each multiscale material it stores only the trained network.

[0082] "Database" means any collection of data (i.e., information) organized for search and retrieval (e.g., a relational database based on a predefined structured language, e.g., SQL). A database, when stored in memory, allows for rapid search and retrieval by a computer. A database is actually structured to facilitate the storage, retrieval, modification, and deletion of data, along with various data processing operations. A database may consist of a file or set of files that may be broken down into records, each of which consists of one or more fields. A field is the basic unit of data storage. A user may retrieve data primarily through queries. Using keywords and sorting commands, a user may quickly search, reorganize, group, and select fields in many records to retrieve or create reports on a particular collection of data, according to the rules of the database management system being used.

[0083] As described above, in an example where the DMN-based neural network is composed of a first block and a second block, the first block is configured to output network parameters by using only morphological parameters without inputting one or more tensors about the object. In other words, the trained parameters of the second block (i.e., network parameters) can be obtained from the plurality of morphological parameters and the first block. In such an example, the database of the multi-scale material may store only the first block instead of the entire trained neural network (i.e., the first block and the second block). This significantly improves the required storage capacity.

[0084] These methods are computer-implemented, meaning that the steps of the method (or substantially all steps) are performed by at least one computer, or any similar system. Thus, the steps of the method are performed by the computer, possibly fully automatically, or semi-automatically. In an example, triggering of at least some of the steps of the method may be performed by user-computer interaction. The level of user-computer interaction required may depend on the level of automation foreseen and may balance the need to implement the user's wishes. In an example, this level may be user-defined and / or predefined.

[0085] A typical example of a computer implementation of the method is to execute the method by a system adapted for this purpose. The system may include a processor coupled to a memory and a graphical user interface (GUI), the memory recording a computer program including instructions for executing the method. The memory may also store a database. The memory is any hardware suitable for such storage, and may optionally include several physically separate parts (e.g., a part for the program and, optionally, a part for the database).

[0086] FIG. 1 shows an example of a system, which is a client computer system, such as a user's workstation.

[0087] The client computer of this example comprises a central processing unit (CPU) 1010 connected to an internal communication bus 1000, and a random access memory (RAM) 1070 also connected to the bus. The client computer is further provided with a graphics processing unit (GPU) 1110 associated with a video random access memory 1100 connected to the bus. The video RAM 1100 is also known in the art as a frame buffer. A mass storage device controller 1020 manages access to mass memory devices such as a hard drive 1030. Mass memory devices suitable for tangibly embodying computer program instructions and data include all forms of non-volatile memory including, by way of example, semiconductor memory devices such as EPROM, EEPROM, flash memory devices, magnetic disks such as internal hard disks and removable disks, magneto-optical disks. Any of the foregoing may be supplemented by or incorporated in specially designed ASICs (application specific integrated circuits). A network adapter 1050 manages access to a network 1060. The client computer may also include a haptic device 1090 such as a cursor control device, a keyboard, etc. A cursor control device is used in the client computer to allow a user to selectively place a cursor at any desired location on the display 1080. In addition, the cursor control device allows a user to select various commands and input control signals. The cursor control device includes some signal generating devices for input of control signals to the system. Typically, the cursor control device may be a mouse, and the buttons of the mouse are used to generate the signals. Alternatively or additionally, the client computer system may include a sensitive pad, and / or a sensitive screen.

[0088] The computer program may include instructions executable by a computer, the instructions including the means for causing the system described above to carry out the method. The program may be recordable on any data storage medium, including the memory of the system. The program may be implemented, for example, in digital electronic circuitry, or computer hardware, firmware, software, or a combination thereof. The program may be implemented, for example, as an apparatus, which is a product tangibly embodied in a machine-readable storage device for execution by a programmable processor. The method steps may be executed by a programmable processor executing a program of instructions that performs the functions of the method by operating on input data to generate output. Thus, the processor may be programmable and coupled to receive data and instructions from, and transmit data and instructions to, a data storage system, at least one input device, and at least one output device. The application program may be implemented in a high-level procedural or object-oriented programming language, or in an assembly or machine language, as appropriate. In either case, the language may be a compiled or interpreted language. The program may be a complete installation program or an update program. The application of the program on the system in either case results in instructions for carrying out the method. The computer program may alternatively be stored and executed on a server in a cloud computing environment, the server being in communication with one or more clients over a network, in which case the processing unit executes instructions constituted by the program, thereby executing the method on the cloud computing environment.

[0089] Next, an implementation of the above mentioned method will be described.

[0090] The present implementations relate to AI surrogates that incorporate physical laws that can predict the effective behavior of materials with a particular fixed microstructure. More specifically, the present implementations can relate to predicting the effective physical properties of these materials when the microstructure itself can change.

[0091] This implementation relates to the numerical simulation of the effective macroscopic mechanical behavior of heterogeneous materials containing microstructures. Heterogeneity means that multiple materials are spatially arranged within the internal structure, i.e., the microstructure. Such materials are often referred to as composites, structured materials, metamaterials, and multiscale materials. Such structured materials or metamaterials also include lattice or spinodal-like structures, where only one material is organized with the structure. The other material phase is void. These multiscale materials are frequently used in various industrial applications such as automotive, aerospace, medical, and civil engineering.

[0092] Figure 2 depicts examples of structured materials: example 210 is a unidirectional fiber composite, example 220 is a woven composite, example 230 is a lattice structure, and example 240 is a spinodal structure.

[0093] The multi-scale nature of these materials arises because their macroscopic behavior is determined by both their constituent properties (mechanical, thermal, or other physical properties) and their microstructural morphology (geometry, topology, etc.). For example, mechanical behavior is often described by a stress-strain curve, which measures the mechanical stress required to cause a certain amount of deformation of a material.

[0094] Figure 3 shows an example of a strain-stress curve. The initial slope, called Young's modulus, characterizes the elastic stiffness. For multi-scale materials, the stress-strain curve depends on the material properties and microstructural morphology of its constituents.

[0095] One such heterogeneous material is fiber-reinforced composites, where glass or carbon fibers are used as reinforcements in a polymer matrix. They can guarantee high mechanical resistance specifications while achieving a reduction in overall part weight. Figure 4 shows an example of such a material.

[0096] The microstructure of such multiscale materials can be described by several microstructural parameters that characterize their morphology.

[0097] FIG. 5 shows the volume fraction (of fibers), vf, varying from 0.2 to 0.8 for the unidirectional fiber composite.

[0098] Another geometric parameter for such unidirectional fiber composites is fiber misalignment. Due to the manufacturing process, the fibers are not perfectly aligned but exhibit some misalignment. This misalignment can be described by an initial misalignment angle.

[0099] An example of such misalignment between fibers is shown in Figure 6. Figure 6 shows three fiber composites with three different misalignment angles: (top) 0°, (middle) 1°, and (bottom) 2°. Implementations may take this configuration parameter into account if its effect on effective physical properties is studied.

[0100] Short fiber reinforced plastic composites can be described by the fiber volume fraction, fiber aspect ratio, fiber orientation, etc. The aspect ratio (ar) is a scalar defined as the ratio of length to diameter. This parameter characterizes the elongation of the fibers and typically varies from 1 (nearly spherical particles) to over 100 (highly elongated fibers). Figures 7 and 8 show examples for two values ​​of the aspect ratio. In Figure 7, ar=1 and the fibers are spherical, while in Figure 8, ar=10 and the fibers are more elongated.

[0101] The fiber orientation is a x +ay+az A vector (a x , a y , a z These three components a i Each of ,describes the statistical probability of finding a fiber in the i-th direction.,Figures 9 and 10 show examples. In Figure 9, all fibers point in the,X,direction, and the fiber orientation is described by the vector,(1,0,0). In Figure 10, the fibers are randomly oriented in 3D space, and the,fiber orientation is described by the vector,(,1,0,0).

[0102]

number

[0103] It is described by:

[0104] Such multiscale materials with associated microstructural parameters are called parametric multiscale materials. This implementation is directed to predicting the effective mechanical properties of such parametric multiscale materials. Accurate and fast prediction of effective physical behavior is required to design optimal metamaterials for industrial applications, quantify the uncertainties in material properties, and inversely identify the properties of their constituents. Efficient scale-bridging techniques between macroscopic and microscopic simulations are also needed to design and validate industrial components made of multiscale materials.

[0105] This implementation proposes a new neural network architecture for general parametric microstructures with varying volume fractions and possibly other geometric parameters. This implementation uses a DMN formulation for fixed microstructures at any given microstructural parameters and can accurately and efficiently predict valid linear and nonlinear behavior.

[0106] The implementation is based on a deep material network (DMN) formulation. Such a formulation defines the elasticity tensor of the construct

[0107]

number

[0108] is given, the effective elastic tensor of the dissimilar materials

[0109]

number

[0110] The fitting parameters of the DMN are the weights and rotations of the internal neurons, denoted by (w,θ). During the offline training procedure, these parameters are used to approximate the predicted values ​​of the DMN.

[0111]

number

[0112] is adjusted to match the value calculated by a computational homogenization technique such as the finite element method (FE-RVE) applied to a representative volume element of the microstructure. Figure 11 shows a schematic of the DMN compared to the finite element method solution.

[0113] The original DMN formulation is limited to a particular, fixed microstructure. The trained fitting parameters are specifically learned to be a compact representation of the microstructure. If the morphology of the microstructure varies with a particular microstructure parameter p, then (w,θ) can be expected to vary in some way with p as well. However, in the original DMN formulation, the fitting parameters (w,θ) are fixed.

[0114] To solve this problem, this implementation proposes to use a standard feed-forward neural network to take into account the dependence of the DMN fitting parameters (w, θ) on the fine-structure parameters p. This neural network consists of an input layer that receives the fine-structure parameters p and an output layer that computes the DMN fitting parameters (w, θ). The hidden layer consists of an affine function and a nonlinear activation function. Figure 12 shows a simple example of such a feed-forward neural network.

[0115] Thus, the neural network architecture of this implementation is two-part. An example of such a two-part neural network is shown in FIG. 13. In this example, the implementation provides a parameter vector p representing the microstructural features (volume fraction, fiber length, fiber diameter, fiber orientation, etc.) of the multi-scale material to a first neural network. The first neural network is configured to predict a compact representation (w, θ) of a particular microstructure. Using the computed compact representation, the implementation then computes the effective elasticity tensor of the microstructure using the original formulation of the DMN.

[0116] Forward functions to predict effective properties

[0117]

number

[0118] can be described by the following procedure. - providing a parameter vector p representing the microstructural features of the multiscale material, e.g., volume fraction or fiber orientation; Using this parameter vector p as input, a first neural network predicts a compact representation (w, θ) of this particular microstructure with the given parameters. - Elasticity tensor

[0119]

number

[0120] Provide the linear elastic properties (Young's modulus, shear modulus, and Poisson's ratio) of each construct, represented by: - The original DMN model is a multi-scale material macroscopic model.

[0121]

number

[0122] Take (w,θ) as input to predict

[0123] This implementation employs a generic feed-forward neural network to take into account the dependence of DMN parameters (i.e., compact representation of the microstructure) on the microstructural parameters. Therefore, it is generalizable to any parametric microstructure with multiple parameters of different nature. In particular, this implementation exploits a physics-incorporated neural network (PINN). In such a network, the architecture itself is designed to satisfy several properties justified by physics, compared to a standard feed-forward neural network. Furthermore, additional constraints based on micromechanics are included to improve the generalization ability of the neural network architecture. Thus, the resulting neural network is capable of predicting effective properties in the parametric space with high accuracy and efficiency.

[0124] The offline training process of this implementation can be performed simultaneously by using a dataset generated from multiple instances (different parametric values) of the parametric microstructure. This ensures that the proposed neural network can be efficiently trained with all the information at hand. The offline training procedure can be described as follows: - provide an experimental design (sampling points) in a parametric space describing the different microstructural parameters of the microstructure. In the example of a microstructure containing only one volume fraction parameter, the parameter space is therefore a one-dimensional interval [νf min ,νf max ] - Different input material properties

[0125]

number

[0126] Provide an experimental design (sampling points) in material space that describes - For each combined sampling point in parameter space and material space, a finite element simulation is performed. - Define a loss function that compares the predictions of the parametric DMN model with the previous dataset. - Adjust the fitting parameters of the parametric DMN model so that the DMN can closely approximate the previous dataset. This is done iteratively: for each epoch (i.e., iteration), the fitting parameters are incrementally adjusted according to the gradient of the loss function with respect to each fitting parameter.

[0127] The architecture of this implementation also provides a new compact representation of parametric microstructures with any number of parameters, where a finite number of parameters can now represent 3D meshes of an infinite number of complex microstructures, with a dramatic reduction in storage requirements.

[0128] This implementation can also be used in an inverse identification problem setting. Given the available properties of dissimilar materials that can be experimentally measured, the input constituent properties can then be efficiently calibrated or identified using this implementation. The procedure for such an inverse identification problem can be described as follows: - Train a parametric DMN to represent a given parametric microstructure. - Providing effective properties for composite materials. - Inverse identification of input construct microstructural parameters.

[0129] FIG. 14 shows an exemplary flow chart of such reverse identification according to this implementation.

[0130] DMN Architecture Next, we explain the basic architecture of the DMN formulation and its fitting parameters (w, θ).

[0131] DMNs are multi-rank laminate microstructures consisting of hierarchically nested stacks of laminates at different length scales or levels.

[0132] FIG. 15 shows an example of a DMN with three levels (i.e., level 0, level 1, and level 2). The rank n of such a multi-rank stack, also called the number of DMN layers, characterizes the number of nesting levels. The architecture corresponds to a topologically complete binary tree. Each "node" is a rank-1 stack microstructure and serves as a "mechanical building block" or "neuron" in this neural network-like architecture. FIG. 16 shows an example of a neuron of the DMN, which is described in detail later.

[0133] DMN is the stiffness tensor of its two components.

[0134]

number

[0135]

number

[0136] Based on the effective stiffness tensor of the metamaterial,

[0137]

number

[0138] At the lowest level, we calculate an approximation of

[0139]

number

[0140]

number

[0141] The inputs fed into the DMN architecture are:

[0142]

number

[0143] is calculated.

[0144] Stiffness tensor

[0145]

number

[0146]

number

[0147] are given in their respective material frames. They can be expressed in terms of several material parameters describing their elastic behavior: Young's modulus E (which describes the stiffness required per unit elongation in a particular direction), Poisson's ratio ν (which describes the relative deformation between two orthogonal directions), and shear modulus μ (which describes the shear stiffness required per unit shear deformation).

[0148] For offline training of the DMN, the implementation is as known in the art, for example according to the already cited literature (Non-Patent Document 1), as an orthotropic material.

[0149]

number

[0150]

number

[0151] In this case, the stiffness tensor is set to the 9 material parameters (E1, E2, E3, ν 12 , v 13 , v 23 , μ 12 , μ 13 , μ 23 The compliance matrix, which is the inverse of the stiffness matrix, is given by the following 6 × 6 symmetric matrix:

[0152]

number

[0153] Here, blank elements indicate zero.

[0154] Neuron Definition This implementation assigns each neuron to an effective homogenized stiffness tensor based on

[0155]

number

[0156]

number

[0157] It is defined by: - Stiffness tensors of the two constituents that make up the laminate microstructure

[0158]

number

[0159]

number

[0160] - the volume fraction f' of one of the constituents in the laminate microstructure - the rotation θ' of the current neuron relative to other neurons at the higher level

[0161] Without loss of generality, an implementation uses the volume fraction of the first constituent.

[0162] The exact formula of the Lam function depends on the spatial dimension and the physical quantities considered, i.e. mechanical, thermal, etc. Examples of formulas for mechanical behavior can be found in the already cited literature (Non-Patent Document 1). Hereinafter, an example is given for 2D mechanical behavior (i.e. plane strain or plane stress conditions) with reference to FIG. 16.

[0163] In this example, we assume that the interface between the two components is defined by a plane with its normal vector aligned in direction 2. Mechanical static equilibrium ensures that the normal direction σ 22 and σ 12 The stress component along is continuous across the interface. The strain component parallel to the interface, ε 11 These interface conditions are expressed by the strain tensor ε = (ε 11 ,ε 22 ,ε 12 )and,

[0164]

number

[0165] and

[0166]

number

[0167] and a modified stiffness tensor, denoted by:

[0168]

number

[0169] Effective strain tensor

[0170]

number

[0171]

number

[0172] Finally, the definition of the effective stress tensor

[0173]

number

[0174]

number

[0175]

number

[0176] Finally, a rotation is applied using the following rotation matrix:

[0177]

number

[0178] this is,

[0179]

number

[0180] where R T is the transpose of the rotation matrix.

[0181] In another example, the implementation can define a neuron that predicts the effective thermal conductivity of a metamaterial. In this case,

[0182]

number

[0183]

number

[0184] is the thermal conductivity tensor of the two components, which in the 2D case is a 2×2 symmetric tensor. The thermal balance equations result in the interface equations:

[0185]

number

[0186] where in this equation the strain tensor is replaced by the temperature gradient vector. Effective thermal conductivity tensor

[0187]

number

[0188] The final formula involving is formally the same as the previous type, i.e.

[0189]

number

[0190] The rotation matrix is ​​as follows:

[0191]

number

[0192] DMN fitting parameters The fitting parameters of DMN are weights w and rotations θ. The weights are defined on the leaf nodes and characterize the respective volume fractions of each material node. They are required to be non-negative. The rotations are defined on a per-node basis. They can each be described by a quaternion.

[0193] In the original DMN formulation, these fitting parameters (w, θ) are assumed to be fixed. Therefore, this is not appropriate for parametric fine-structure p, which should also vary with the fine-structure parameters. This implementation thereby takes into account the dependence of (w, θ) on p.

[0194] Neural network incorporating physical laws for fitting DMN parameters In this implementation, the functional dependence of the DMN parameters (w, θ) on the fine-structure parameters p is directly accounted for by a feedforward neural network composed of multiple layers of affine transformations and activation functions. Due to the physics-based architecture of the DMN, the architecture of the hidden layers does not significantly change the performance.

[0195] In the option depicted in Figure 17, this implementation uses a physics-based architecture where the DMN weight vector w depends only on the volume fraction parameters, and the DMN rotation vector θ varies along with all other parameters that do not change the volume fraction. Thus, the microstructural parameters are split as follows:

[0196]

number

[0197] where ν is the volume fraction of the second phase and q denotes all the other q independent parameters orthogonal to ν. Given this partition, the physics-incorporated neural network (PINN) for the DMN parameters is now given as: w(p) = w(νf) = σ(νf w1 + w0) θ(p)=θ(q)=Θ1q+θ0 where w0 and w1 are vectors of length n, and θ0 is (2 n -1) × 4 matrix, Θ1 is (2 n -1)×4×q tensors. These tensors (w0, w1, θ0, Θ1) are the fitting parameters of the newly proposed architecture. The architecture used in this option is hereafter called PINN-DMN. In the following symbols, νf (which is a scalar) can be considered as a function of the DMN weights w (which can be represented as a vector), symbol νf(w).

[0198] In another option depicted in FIG. 18, this implementation used a fully connected architecture, but the DMN fitting parameters w and θ both depend on the overall parameter vector p.

[0199] The PINN architecture compared to the fully connected architecture requires fewer fitting parameters due to the decomposition, which improves computational efficiency. Furthermore, numerical simulations suggest that the PINN architecture may provide comparable accuracy compared to the fully connected architecture and also improve generalization ability (interpolation and extrapolation) in the parametric space. This means that the proposed PINN approach maintains high accuracy in the predicted effective properties while varying the fine-structure parameters.

[0200] FIG. 19 shows an example of the application of the PINN approach in FIG. 17, where the method is used to make predictions for different values ​​of parameters p0, p1, p2.

[0201] Here, compared to the transfer learning approach, a unique offline training is required to optimize the fitting parameters of the PINN-DMN, where the neural network is trained on each p i The linear elastic behavior data in is used to evaluate the linear elastic behavior of the neural network. Furthermore, the functional dependency of the neural network naturally defines interpolation and extrapolation within or outside the training domain, and it easily extends to higher parameter dimensions.

[0202] Apart from the physics-based architecture for the functional dependency p → (w, θ), some physical constraints can also be prescribed to further improve its generalization ability, similar to physics-based machine learning approaches for nonlinear partial differential equations. This so-called physics-based approach consists of a standard feedforward neural network and a loss function that takes into account the physical equations or constraints defined by partial differential equations (PDEs). An example of such a physics-based network is shown in Figure 20, where the PDE equations are prescribed as additional terms in the loss function.

[0203] In this implementation, two additional physical constraints are prescribed on the neural network that computes p → (w, θ). The first constraint is the volume fraction constraint L νf In other words, the DMN volume fraction should be consistent with the actual volume fraction value of the microstructure for any volume fraction value. The second constraint is the orientation constraint L a which imposes that the orientation of the average material frame predicted by the DMN should also match the orientation of the actual microstructure.

[0204] Next, each of the two constraints is explained.

[0205] volume fraction constraints The DMN volume fraction can be calculated from the DMN weight vector using the following equation:

[0206]

number

[0207] In this formula, w i are the weights defined on each leaf node (material node) of the DMN architecture. They are divided into two parts: the index set

[0208]

number

[0209] As defined for component 1 by

[0210]

number

[0211] The union of these two index sets is

[0212]

number

[0213] The above formula calculates the volume fraction of phase 2 in the entire microstructure. The physics-based design of DMN allows us to calculate the actual metamaterial νf Ω The actual volume fraction of can be well approximated by the DMN volume fraction.

[0214] Then, we use the PINN-DMN framework to estimate these DMN weights w i is computed by the neural network. Thus, this implementation may prescribe the following volume fraction constraints at all volume fraction values: νf(w)=νf(σ(νf w1+w0))=νf, ∀νf∈[0,1]

[0215] This formula means that the DMN volume fraction always coincides with 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:

[0216]

number

[0217] where νf i denotes the collocation points that weakly prescribe the volume fraction constraint, and n is the number of such collocation points. Since the volume fraction is a scalar between 0 and 1, uniform sampling is adopted.

[0218] Figures 21 and 22 show examples of volume fraction constraints prescribed for PINN. Figure 21 shows the minimization of the volume fraction constraint loss function, and Figure 22 shows the predicted DMN volume fraction at the initial stage (i.e., epoch=10) and at the end of training (i.e., at convergence).

[0219] With the decrease of the loss function during the optimization iterations, the DMN volume fraction matches better with the actual volume fraction of the microstructure.

[0220] Orientation constraints The DMN does not only capture the microstructural morphology: it also learns the material orientation distribution function in the microstructure. Using the orientation tensor, this implementation proposes orientation constraints, specified in the DMN weights w and rotations θ, such that the parametric DMN generalizes such material orientation knowledge across the parametric space.

[0221] The orientation tensor concisely describes the statistics of the orientation distribution of unit vectors.

[0222]

number

[0223] is given, where

[0224]

number

[0225] denotes the two-dimensional surface of the unit sphere, and the 3×3 quadratic orientation tensor is defined by

[0226]

number

[0227] Due to the normalization constraints on e and the probability density f, a is symmetric and it can be easily shown that tr(a) = 1. Higher order orientation tensors exist, but the second order one is most frequently used to characterize the local fiber orientation due to the manufacturing process and their effects on material properties.

[0228] This implementation uses a generalization of such a (second-order) orientation tensor for the orientation distribution of rotation. In contrast to transversely isotropic fibers, where a single unit vector is sufficient to characterize its material frame, a general anisotropic material (such as an orthotropic material) requires a rotation matrix R to describe the transformation from its material frame (e1,e2,e3) to a global one. In such cases, the orientation distribution function is now defined in terms of the rotation matrix, i.e.

[0229]

number

[0230] The domain of this function SO(3) is the 3D rotation group that contains all rotation matrices in three-dimensional space that are orthogonal to the identity determinant. Each column of R, 1≦i≦3, is essentially

[0231]

number

[0232] which defines the orientation distribution of the material frame.

[0233] Figures 23-25 ​​show examples of such orientation distributions for each of the three material axes provided for the tows in a woven microstructure: Figure 23 for direction e1, Figure 24 for direction e2, and Figure 25 for direction e3.

[0234] Given this interpretation, three (second order) orientation tensors can be defined for each axis of the material frame as follows:

[0235]

number

[0236] Here, no Einstein summation is implied. The tensor a can be understood as a third-order material frame orientation tensor. For discrete probability functions defined on a mesh, the integral can be understood as a weighted average using the element volumes as weights. Note that in the case of a two-phase microstructure, it can be calculated for each phase.

[0237] In the above example, the material frame orientation tensor for the tow is given by:

[0238]

number

[0239] Thus, the tows are oriented isotropically in the XY plane for e1 and e2, and unidirectionally in the Z axis for e3. For the parent phase, its material frame coincides with the global frame. In this case, a is analogous to a unidirectional orientation tensor, and for the other phases,

[0240]

number

[0241] Such orientation tensors can also be defined for DMNs. DMN rotations θ define the rotation matrix between the local frame of the current stack and the local frame of the next nesting level. They can therefore be constructed to obtain a valid rotation matrix from the material frame (e1, e2, e3) (leaf stacks) to the global frame of the microstructure Ω (root stack). Let p(i) denote the parent of stack i in the DMN binary tree architecture. For example, in the example of a 3-layer DMN shown in Figure 15, p(4)=2 for leaf stack 4 and p 2 (4) = p(p(4)) = p(2) = 1, which is the root stack. For each leaf stack i with a DMN material node, the effective rotation matrix from the material frame to the global frame is given by:

[0242]

number

[0243] For an n-layer DMN, it is p for any leaf stack i n-1 Note that we keep (i)=1.

[0244]

number

[0245] Since there is no input rotation matrix for , material nodes that share the same leaf stack also get the same effective rotation matrix. For example, for material nodes 3 and 4 in leaf stack 5, their effective rotations are:

[0246]

number

[0247] Using these effective rotations in the leaf stack, for phase 1, the DMN material frame orientation tensor can be calculated as follows:

[0248]

number

[0249] A similar equation can be defined for phase 2. In comparison to the DMN volume fraction, the computation of the DMN orientation tensor requires both the DMN weights w and the rotations θ. Using a neural network that incorporates physics for the DMN parameters, this implementation leverages the following orientation constraints through the definition of the loss function:

[0250]

number

[0251] A similar loss function is defined for phase 2 and then summed. In this equation, p i is a collocation point in the fine structure parametric space, and a(p i ) is the actual orientation tensor of the parametric microstructure. Similar to the volume fraction constraints, the number n denotes the number of these collocation points. These collocation points can be sampled using the Latin hypercube sampling method.

[0252] Material Sampling This implementation follows the original material sampling method proposed by the already cited document (Non-Patent Document 1).

[0253]

number

[0254] are assumed to be orthotropic in their respective material frames. In total, 9+9=18 material parameters are needed to characterize its orthotropic elastic behavior, with one additional scaling parameter used to introduce the contrast in elastic moduli between the two phases. In this work, Latin hypercube sampling is used to sample this 19-dimensional space.

[0255] Total loss function These two constraints are added to a total loss function based on the mean squared error between the homogenized elasticity tensor predicted by DMN and that generated by the FE-RVE simulation.

[0256]

number

[0257] These errors between the losses at a fixed microstructural parameter value L are then p Finally, they are averaged to compute the sample

[0258]

number

[0259]

number

[0260] The total loss function is minimized using a gradient-based method. In this implementation, 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 by automatic differentiation.

[0261] Examples of specific parametric microstructures This implementation was tested on a specific parametric microstructure, i.e., a 2x2 twill woven composite with varying tow volume fraction. The objective is to evaluate the PINN-DMN for such a microstructure in terms of modulus prediction as a function of volume fraction. Inverse identification of material and microstructural parameters is also explored using the PINN-DMN as an accurate and efficient surrogate for the parametric microstructure.

[0262] 3D finite element models were constructed using TexGen with four volume fraction values ​​for the tow. Three of them (νf=0.459, νf=0.608, νf=0.729) were used to generate the training dataset, while νf=0.537 was used to test the interpolation accuracy. FE-RVE analysis was performed under Abaqus to obtain a 6×6 linear elastic stiffness tensor. Each FE-RVE model contains 75000 voxel elements and 241899 degrees of freedom. The simulation was run with 24 threads using an Intel® Xeon® Gold5220R CPU@2.20GHz and requires approximately 11 seconds.

[0263] Figures 26(a)-26(d) show FE-RVE models for woven composites with different fiber volume fractions: 26(a) for νf=0.459, 26(b) for νf=0.608, 26(c) for νf=0.729, and 26(d) for νf=0.537. The first three models are used to generate a training dataset, and the last one is used to test the interpolation accuracy.

[0264] The actual linear elastic properties of the two phases can be shown below. The matrix is ​​isotropic, while the carbon fiber tows are assumed to be transversely isotropic in the local material frame. To train the microstructures νf=0.459, νf=0.608, and νf=0.729, 400 of the 500 material samples are used as the training data set, and the remaining 100 are reserved for validation. For νf=0.537, all 500 samples are used to test the interpolation accuracy.

[0265] [Table 1]

[0266] [Table 2]

[0267] PINN-DMN was trained according to two physical constraints. The loss history was compared for 7 and 9 DMN layers. The neural network becomes more accurate with increasing layers and the loss values ​​become smaller (converge). The total loss can also be split into the FE-RVE data part and the physical constraint part. These two parts decrease monotonically. The physical constraints are used because the corresponding loss value at 9 layers is 10 -5is well within the range of 0.25–0.45, which is adequate. Results using 7 layers are reported below, as it provides sufficient accuracy. Figure 27 shows the PINN-DMN training for a woven composite. Figure 27(a) shows the loss history for 7 and 9 DMN layers, while Figure 27(b) shows the partitioning of the total loss into the FE-RVE data and physical constraints parts.

[0268] The volume fraction and material frame orientation tensor for the tow are compared to their prescribed values. The DMN volume fraction is matched to the FE-RVE data points and is in good agreement with the theoretical line, even when evaluated outside the training domain. The components of the DMN orientation tensor

[0269]

number

[0270] is shown below. Again, excellent agreement is obtained.

[0271] Figure 28 shows the validation of physical constraints for a woven composite: (a) volume fraction and (b) material frame orientation tensor for the tow.

[0272] At different volume fractions, the training, validation, and testing errors are calculated. The median error is around 2% for all four microstructures. The testing error is only slightly larger compared to the training error.

[0273] FIG. 29 shows the training, validation, and testing errors of PINN-DMN at different volume fractions represented by their respective 0.1, 0.5 (median), and 0.9 quantiles.

[0274] Homogenized moduli are calculated while varying the volume fraction with real linear elastic properties. For comparison, a fully connected architecture is also tested, using physical constraints. Recall that the fully connected architecture implies that the rotation of the DMN is also a function of the volume fraction. Both models capture well the nonlinear νf dependence of these moduli. The fully connected architecture not only increases the number of fitting parameters (here, Θ1 is added), but also does not improve the prediction accuracy compared to PINN.

[0275] Figure 30 shows the parametric DMN model. Figure 30(a) shows the in-plane Young's modulus

[0276]

number

[0277] and Fig. 30(b) shows the lateral Poisson's ratio

[0278]

number

[0279] , and Fig. 30(c) shows the in-plane shear modulus

[0280]

number

[0281] The FE-RVE training and testing data are shown.

[0282]

number

[0283] and

[0284]

number

[0285] The computational efficiency of DMN is compared with FE-RVE simulations in terms of computing the homogenized elasticity tensor given by: Due to this significant speedup, DMN can be further used for parametric analysis, uncertainty quantification, and material property calibration. The following table presents the computational speedup of DMN compared with FE-RVE in terms of predicting the homogenized elasticity tensor. For both cases, experiments were performed on an Intel® Xeon® Gold5220R CPU@2.20GHz with 24 cores.

[0286] [Table 3]

[0287] Inverse identification of input material and microstructural parameters The trained PINN-DMN can act as an accurate and computationally efficient surrogate for parametric microstructures, which can not only be used for forward prediction of effective properties under different microstructural parameters, but also be employed to identify both material and microstructural properties in inverse identification problems.

[0288] In this application, the homogenized elasticity tensor

[0289]

number

[0290] The purpose of this is to determine the elastic properties of the matrix.

[0291]

number

[0292] , Elastic properties of the tow

[0293]

number

[0294] and the volume fraction of the tow, νf.

[0295]

number

[0296] can be experimentally measured, where this implementation is based on FE-RVE simulation results for a test microstructure νf=0.537, obtained with the actual properties currently being sought.

[0297]

number

[0298] Use.

[0299]

number

[0300] This implementation is motivated by the fact that the gradient with respect to can be easily computed using automatic differentiation by the PINN-DMN architecture, and the DMN predictions and the input homogenized data

[0301]

number

[0302] We employ a gradient-based optimization approach based on a loss function that compares and with the standard Frobenius norm.

[0303]

number

[0304] Optimization iterations are unknown

[0305]

number

[0306] We need an initial guess for νf = 0.537. Therefore, the actual properties as well as the true volume fraction νf = 0.537 are randomly perturbed using a normal distribution with a coefficient of variation equal to 20%. The initial values ​​generated by the two random realizations are shown in the two tables below.

[0307] [Table 4]

[0308] [Table 5]

[0309] Given the loss history corresponding to these two initial values, the loss function converges quickly, reaching 10 within 1000 optimization iterations. -5 The input homogenized elasticity tensor

[0310]

number

[0311] is well recovered. The converged homogenized elastic tensor and the input

[0312]

number

[0313] The relative errors between the data are 0.42% and 0.26% for these two sets of initial values.

[0314] FIG. 31 shows the calibration loss history using two initial values: (a) Realization 1; (b) Realization 2.

[0315] [Table 6]

[0316] Inversely identified input parameters

[0317]

number

[0318] are reported. The material and microstructural parameters found in Realization 1 are similar to the real properties. However, in Realization 2, the identified volume fraction 0.667 is higher than the data 0.537. Meanwhile, the longitudinal Young's modulus of the tow (64.6 GPa) is also smaller than the data (78.8 GPa). This indicates the non-uniqueness of the inverse identification problem. Additional conditions can be provided by the user to further constrain the inverse problem. Some input properties can be assumed fixed, for example, the input properties of the matrix. Nevertheless, such an inverse identification problem can now be efficiently solved within a few seconds using the PINN-DMN.

[0319] [Table 7]

[0320] [Table 8]

[0321] Compressed representation of microstructures using arbitrary parameters Conventionally, parametric microstructures may be computationally described using unstructured meshes that spatially discretize the microstructure. For example, for a woven microstructure, a typical mesh such as that in Figure 32 may be provided with a fixed volume fraction value. For different parameter values, different meshes will be provided.

[0322] For each mesh, it requires storage of node coordinates and cell connectivity matrices. In this case, it requires about 4MB. Note that this is only for one specific volume fraction value. If five volume fractions are required, five fine structure meshes need to be stored. When many parameter values ​​are required, mesh-based representations can be resource intensive.

[0323] [Table 9]

[0324] The parametric DMN architecture in this implementation also serves as a new compressed representation of the parametric microstructure. In contrast to mesh-based representations, the parametric DMN approach describes the entire parametric microstructure. The 7-layer parametric DMN architecture requires only 6KB to represent the parametric microstructure with arbitrary parameter value variations.

[0325] [Table 10]

[0326] This compressed representation can be used to build a database of parametric multi-scale materials. Each entry corresponds to a particular parametric 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 shows an example of a tabular format for such entries:

[0327] [Table 11]

Claims

1. 1. A computer-implemented method for training a deep material network (DMN) based neural network configured to predict macroscopic physical properties of a multi-scale material, the multi-scale material including one or more components, the method comprising: - acquiring a dataset, each entry of said dataset corresponding to a respective multi-scale material object, said entries comprising: Tensors describing the physical properties of the multi-scale material object at a macroscopic level [0010] and, One or more tensors each describing the physical properties of components of the multi-scale material object at a microscopic level [0025] and, one or more morphological parameters (p) each describing the morphology of the multi-scale material object; and and training the neural network based on the acquired dataset to generate the one or more tensors for a multi-scale material object; [0030] and predicting a tensor describing the physical properties of the object at a macroscopic level based on the one or more morphological parameters of the object.

23. A computer-implemented method comprising:

2. The deep material network (DMN) based neural network comprises a first block and a second block, - the first block is configured to receive as input the one or more morphological parameters (p) for the object and to output values ​​of a number of network parameters ((w, θ)); and - the second block stores the values ​​of the plurality of network parameters ((w, θ)) and the one or more tensors for the object. [0045] and configured to output a prediction of the physical property of the multi-scale material object at a macroscopic level. The method of claim 1.

3. said first block is a feedforward neural network, and / or The method according to claim 2, wherein the second block comprises a DMN architecture having the plurality of network parameters ((w, θ)).

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

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

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

7. The training includes minimizing a loss function, the loss function being, for each entry, the obtained tensors describing the physical properties of the multi-scale material object at a macroscopic level; the predicted tensors describing the physical properties of the multi-scale material object at a macroscopic level; and 7. The method according to claim 1, wherein the penalty is the difference between

8. The method of claim 7 , wherein the loss function further penalizes non-compliance with volume fraction constraints of the multi-scale material object and / or non-compliance with material orientation constraints of the multi-scale material object.

9. A trainable neural network according to any one of claims 1 to 8.

10. one or more tensors each describing the physical properties of components of the multi-scale material object at a microscopic level; and one or more morphological parameters each describing the morphology of the multi-scale material object; and - obtaining a - predicting the macroscopic physical properties of the multi-scale material object by applying the neural network to the one or more tensors and the one or more parameters; and / or Tensors describing the physical properties of the multi-scale material object at a macroscopic level [0050] - obtaining a one or more tensors each describing the physical properties of components of the multi-scale material object at a microscopic level; and one or more morphological parameters each describing the morphology of the multi-scale material object; and - determining the step Including, The determining step includes minimizing a difference between the obtained tensor and a candidate predicted value of the obtained tensor, the candidate predicted value being determined by applying the neural network to one or more candidate tensors and one or more candidate morphological parameters.

10. A computer implemented method using the neural network of claim 9.

11. The step of minimizing the difference comprises: the gradient of said candidate predictor with respect to one or more tensors of each respective candidate; - the gradient of the candidate predictions with respect to one or more morphological parameters of each respective candidate; and calculating 11. The method of claim 10, comprising:

12. Each entry in the data set corresponds to a respective multi-scale material, the entries comprising: - one or more morphological parameters each describing the morphology of said multi-scale material object; - a neural network trained to predict the macroscopic physical properties of a multiscale material according to any one of claims 1 to 8; A database of multiscale materials including:

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

14. A computer readable storage medium having recorded thereon a computer program as claimed in claim 13 and / or a neural network as claimed in claim 9 and / or a database as claimed in claim 12.

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

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