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How to achieve the fast computation for voxel-based irregular structures by few finite elements?
... The results showed that the maximum inter-story drift at PGA=0.54g is 1.472. Zhang et al. [12] investigated computation for irregular building by finite element models. The smart-I FE model proposed breaks through the limitation of traditional element with regular boundary and pave a new method to implement the FE model calculation on voxel-based geometries. ...
Irregular structures have special seismic performance during earthquakes. Major irregularities in structural standards include geometric, torsional, diaphragm, out-of-plane and non-parallel. In the present paper, 5, 8 and 11-story steel structures with mass and height irregularities were modeled with SAP2000 software. The structures were subjected to 11 scaled earthquakes (near and far-fault) by nonlinear time history analysis method (direct integration). Steel structures have three types of concentrically braces including Gate, Inverted V and Knee. The results showed that the Knee brace has the best performance in the acceleration and displacement parameters of the structure. Inverted V brace has the best performance in shear force parameter. The average displacement reduction of irregular steel structures (5, 8 and 11-story) with Gate, Inverted V and Knee braces is 39.26%, 23.15%, and 49.65%, respectively. The average acceleration reduction of irregular steel structures (5, 8 and 11-story) with Gate, Inverted V and Knee braces is 28.56%, 17.23%, and 45.98%, respectively. The average reduction of column shear force in irregular steel structures (5, 8 and 11-story) with Gate, Inverted V and Knee braces is 14.11%, 24.48%, and 9.51%, respectively.
We present OBMeshfree, an Optimization-Based Meshfree solver for compactly supported nonlocal integro-differential equations (IDEs) that can describe material heterogeneity and brittle fractures. OBMeshfree is developed based on a quadrature rule calculated via an equality constrained least square problem to reproduce exact integrals for polynomials. As such, a meshfree discretization method is obtained, whose solution possesses the asymptotically compatible convergence to the corresponding local solution. Moreover, when fracture occurs, this meshfree formulation automatically provides a sharp representation of the fracture surface by breaking bonds, avoiding the loss of mass. As numerical examples, we consider the problem of modeling both homogeneous and heterogeneous materials with nonlocal diffusion and peridynamics models. Convergence to the analytical nonlocal solution and to the local theory is demonstrated. Finally, we verify the applicability of the approach to realistic problems by reproducing high-velocity impact results from the Kalthoff-Winkler experiments. Discussions on possible immediate extensions of the code to other nonlocal diffusion and peridynamics problems are provided. OBMeshfree is freely available on GitHub (Fan et al. [1]).
The discovery of fast numerical solvers prompted a clear and rapid shift towards iterative techniques in many applications, especially in computational mechanics, due to the increased necessity for solving very large linear systems. Most numerical solvers are highly dependent on the problem geometry and discretization, facing issues when any of these properties change. The newly developed Hybrid Iterative Numerical Transferable Solver (HINTS) combines a standard solver with a neural operator to achieve better performance, focusing on a single geometry at a time. In this work, we explore the “T” in HINTS, i.e., the geometry transferability properties of HINTS. We first propose to directly employ HINTS built for a specific geometry to a different but related geometry without any adjustments. In addition, we propose the integration of an operator level transfer learning with HINTS to even further improve the convergence of HINTS on new geometries and discretizations. We conduct numerical experiments for a Darcy flow problem and a plane-strain elasticity problem. The results show that both the direct application of HINTS and the transfer learning enhanced HINTS are able to accurately solve these problems on different geometries. In addition, using transfer learning, HINTS is able to converge to machine zero even faster than the direct application of HINTS.
Modeling the three-dimensional (3D) structure from a given 2D image is of great importance for analyzing and studying the physical properties of porous media. As an intractable inverse problem, many methods have been developed to address this fundamental problems over the past decades. Among many methods, the deep learning-(DL) based methods show great advantages in terms of accuracy, diversity, and efficiency. Usually, the 3D reconstruction from the 2D slice with a larger field-of-view is more conducive to simulate and analyze the physical properties of porous media accurately. However, due to the limitation of reconstruction ability, the reconstruction size of most widely used generative adversarial network-based model is constrained to 64^{3} or 128^{3}. Recently, a 3D porous media recurrent neural network based method (namely, 3D-PMRNN) (namely 3D-PMRNN) has been proposed to improve the reconstruction ability, and thus the reconstruction size is expanded to 256^{3}. Nevertheless, in order to train these models, the existed DL-based methods need to down-sample the original computed tomography (CT) image first so that the convolutional kernel can capture the morphological features of training images. Thus, the detailed information of the original CT image will be lost. Besides, the 3D reconstruction from a optical thin section is not available because of the large size of the cutting slice. In this paper, we proposed an improved recurrent generative model to further enhance the reconstruction ability (512^{3}). Benefiting from the RNN-based architecture, the proposed model requires only one 3D training sample at least and generates the 3D structures layer by layer. There are three more improvements: First, a hybrid receptive field for the kernel of convolutional neural network is adopted. Second, an attention-based module is merged into the proposed model. Finally, a useful section loss is proposed to enhance the continuity along the Z direction. Three experiments are carried out to verify the effectiveness of the proposed model. Experimental results indicate the good reconstruction ability of proposed model in terms of accuracy, diversity, and generalization. And the effectiveness of section loss is also proved from the perspective of visual inspection and statistical comparison.
Composite plates are widely used in many engineering fields such as aerospace and automotive. An accurate and efficient multiscale modeling and simulation strategy is of paramount importance to improve design and manufacture. To this end, we propose an efficient data-driven computing scheme based on the classical plate theory for the multiscale analysis of composite plates. In order to accurately describe the relationship between the macroscopic mechanical properties and the microscopic architecture, the multiscale finite element method (FE2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2$$\end{document}) is adopted to compute the generalized strain and stress fields. These data are then used to construct a database for data-driven computing. Since the database is offline populated, the data-driven computing scheme allows for a reduced computational cost when compared to the traditional multiscale method, where the concurrent coupling of different scales is still a burden. And data are obtained from a reduced structural model for computational efficiency. The proposed scheme is therefore addressed as Structural-Genome-Driven (SGD) modeling of plates. Compared to the general data-driven computational mechanics modeling of plates, SGD is found to be more efficient since the number of integration points is significantly reduced. This scheme provides a robust alternative computational tool for composite plate structures analysis.
Robotic grasping is an essential problem at both the household and industrial levels, and unstructured objects have always been difficult for grippers. Parallel-plate grippers and algorithms, focusing on partial information of objects, are one of the widely used approaches. However, most works predict single-size grasp rectangles for fixed cameras and gripper sizes. In this paper, a multi-scale grasp detector is proposed to predict grasp rectangles with different sizes on RGB-D or RGB images in real-time for hand-eye cameras and various parallel-plate grippers. The detector extracts feature maps of multiple scales and conducts predictions on each scale independently. To guarantee independence between scales and efficiency, fully matching model and background classifier are applied in the network. Based on analysis of the Cornell Grasp Dataset, the fully matching model can match all labeled grasp rectangles. Furthermore, background classification, along with angle classification and box regression, functions as hard negative mining and background predictor. The detector is trained and tested on the augmented dataset, which includes images of 320 × 320 pixels and grasp rectangles ranging from 20 to more than 320 pixels. It performs up to 98.87% accuracy on image-wise dataset and 97.83% on object-wise split dataset at a speed of more than 22 frames per second. In addition, the detector, which is trained on a single-object dataset, can predict grasps on multiple objects.
This document presents comprehensive historical accounts on the developments of finite element methods (FEM) since 1941, with a specific emphasis on developments related to solid mechanics. We present a historical overview beginning with the theoretical formulations and origins of the FEM, while discussing important developments that have enabled the FEM to become the numerical method of choice for so many problems rooted in solid mechanics.
We consider the use of extreme learning machines (ELM) for computational partial differential equations (PDE). In ELM the hidden-layer coefficients in the neural network are assigned to random values generated on [−Rm,Rm] and fixed, where Rm is a user-provided constant, and the output-layer coefficients are trained by a linear or nonlinear least squares computation. We present a method for computing the optimal or near-optimal value of Rm based on the differential evolution algorithm. The presented method enables us to illuminate the characteristics of the optimal Rm for two types of ELM configurations: (i) Single-Rm-ELM, corresponding to the conventional ELM method in which a single Rm is used for generating the random coefficients in all the hidden layers, and (ii) Multi-Rm-ELM, corresponding to a modified ELM method in which multiple Rm constants are involved with each used for generating the random coefficients of a different hidden layer. We adopt the optimal Rm from this method and also incorporate other improvements into the ELM implementation. In particular, here we compute all the differential operators involving the output fields of the last hidden layer by a forward-mode auto-differentiation, as opposed to the reverse-mode auto-differentiation in a previous work. These improvements significantly reduce the network training time and enhance the ELM performance. We systematically compare the computational performance of the current improved ELM with that of the finite element method (FEM), both the classical second-order FEM and the high-order FEM with Lagrange elements of higher degrees, for solving a number of linear and nonlinear PDEs. It is shown that the current improved ELM far outperforms the classical FEM. Its computational performance is comparable to that of the high-order FEM for smaller problem sizes, and for larger problem sizes the ELM markedly outperforms the high-order FEM.
Multiscale modeling is an effective approach for investigating multiphysics systems with largely disparate size features, where models with different resolutions or heterogeneous descriptions are coupled together for predicting the system’s response. The solver with lower fidelity (coarse) is responsible for simulating domains with homogeneous features, whereas the expensive high-fidelity (fine) model describes microscopic features with refined discretization, often making the overall cost prohibitively high, especially for time-dependent problems. In this work, we explore the idea of multiscale modeling with machine learning and employ DeepONet, a neural operator, as an efficient surrogate of the expensive solver. DeepONet is trained offline using data acquired from the fine solver for learning the underlying and possibly unknown fine-scale dynamics. It is then coupled with standard PDE solvers for predicting the multiscale systems with new boundary/initial conditions in the coupling stage. The proposed framework significantly reduces the computational cost of multiscale simulations since the DeepONet inference cost is negligible, facilitating readily the incorporation of a plurality of interface conditions and coupling schemes. We present various benchmarks to assess the accuracy and efficiency, including static and time-dependent problems. We also demonstrate the feasibility of coupling of a continuum model (finite element methods, FEM) with a neural operator, serving as a surrogate of a particle system (Smoothed Particle Hydrodynamics, SPH), for predicting mechanical responses of anisotropic and hyperelastic materials. What makes this approach unique is that a well-trained over-parametrized DeepONet can generalize well and make predictions at a negligible cost.
We propose an approach for data-driven automated discovery of material laws, which we call EUCLID (Efficient Unsupervised Constitutive Law Identification and Discovery), and we apply it here to the discovery of plasticity models, including arbitrarily shaped yield surfaces and isotropic and/or kinematic hardening laws. The approach is unsupervised, i.e., it requires no stress data but only full-field displacement and global force data; it delivers interpretable models, i.e., models that are embodied by parsimonious mathematical expressions discovered through sparse regression of a potentially large catalog of candidate functions; it is one-shot, i.e., discovery only needs one experiment. The material model library is constructed by expanding the yield function with a Fourier series, whereas isotropic and kinematic hardening is introduced by assuming a yield function dependency on internal history variables that evolve with the plastic deformation. For selecting the most relevant Fourier modes and identifying the hardening behavior, EUCLID employs physics knowledge, i.e., the optimization problem that governs the discovery enforces the equilibrium constraints in the bulk and at the loaded boundary of the domain. Sparsity promoting regularization is deployed to generate a set of solutions out of which a solution with low cost and high parsimony is automatically selected. Through virtual experiments, we demonstrate the ability of EUCLID to accurately discover several plastic yield surfaces and hardening mechanisms of different complexity.
This paper shows hidden information from the plastic deformation of metallic glasses using machine learning. Ni62Nb38 (at.%) metallic glass (MG) film and Zr64.13Cu15.75Al10Ni10.12 (at.%) BMG, as two model materials, are considered for nano-scratching and compression experiment, respectively. The interconnectedness among variables is probed using correlation analysis. The evolvement mechanism and governing system of plastic deformation are explored by combining dynamical neural networks and sparse identification. The governing system has the same basis function for different experiments, and the coefficient error is ≤ 0.14% under repeated experiments, revealing the intrinsic quality in metallic glasses. Furthermore, the governing system is conducted based on the preceding result to predict the deformation behavior. This shows that the prediction agrees well with the real value for the deformation process.
Recently, computational modeling has shifted towards the use of statistical inference, deep learning, and other data-driven modeling frameworks. Although this shift in modeling holds promise in many applications like design optimization and real-time control by lowering the computational burden, training deep learning models needs a huge amount of data. This big data is not always available for scientific problems and leads to poorly generalizable data-driven models. This gap can be furnished by leveraging information from physics-based models. Exploiting prior knowledge about the problem at hand, this study puts forth a physics-guided machine learning (PGML) approach to build more tailored, effective, and efficient surrogate models. For our analysis, without losing its generalizability and modularity, we focus on the development of predictive models for laminar and turbulent boundary layer flows. In particular, we combine the self-similarity solution and power-law velocity profile (low-fidelity models) with the noisy data obtained either from experiments or computational fluid dynamics simulations (high-fidelity models) through a concatenated neural network. We illustrate how the knowledge from these simplified models results in reducing uncertainties associated with deep learning models applied to boundary layer flow prediction problems. The proposed multi-fidelity information fusion framework produces physically consistent models that attempt to achieve better generalization than data-driven models obtained purely based on data. While we demonstrate our framework for a problem relevant to fluid mechanics, its workflow and principles can be adopted for many scientific problems where empirical, analytical, or simplified models are prevalent. In line with grand demands in novel PGML principles, this work builds a bridge between extensive physics-based theories and data-driven modeling paradigms and paves the way for using hybrid physics and machine learning modeling approaches for next-generation digital twin technologies.
A mechanics‐informed artificial neural network approach for learning constitutive laws governing complex, nonlinear, elastic materials from strain‐stress data is proposed. The approach features a robust and accurate method for training a regression‐based model capable of capturing highly nonlinear strain‐stress mappings, while preserving some fundamental principles of solid mechanics. In this sense, it is a structure‐preserving approach for constructing a data‐driven model featuring both the form‐agnostic advantage of purely phenomenological data‐driven regressions and the physical soundness of mechanistic models. The proposed methodology enforces desirable mathematical properties on the network architecture to guarantee the satisfaction of physical constraints such as objectivity, consistency (preservation of rigid body modes), dynamic stability, and material stability, which are important for successfully exploiting the resulting model in numerical simulations. Indeed, embedding such notions in a learning approach reduces a model's sensitivity to noise and promotes its robustness to inputs outside the training domain. The merits of the proposed learning approach are highlighted using several finite element analysis examples. Its potential for ensuring the computational tractability of multi‐scale applications is demonstrated with the acceleration of the nonlinear, dynamic, multi‐scale, fluid‐structure simulation of the supersonic inflation dynamics of a parachute system with a canopy made of a woven fabric.
Topology optimization by optimally distributing materials in a given domain requires non-gradient optimizers to solve highly complicated problems. However, with hundreds of design variables or more involved, solving such problems would require millions of Finite Element Method (FEM) calculations whose computational cost is huge and impractical. Here we report Self-directed Online Learning Optimization (SOLO) which integrates Deep Neural Network (DNN) with FEM calculations. A DNN learns and substitutes the objective as a function of design variables. A small number of training data is generated dynamically based on the DNN’s prediction of the optimum. The DNN adapts to the new training data and gives better prediction in the region of interest until convergence. The optimum predicted by the DNN is proved to converge to the true global optimum through iterations. Our algorithm was tested by four types of problems including compliance minimization, fluid-structure optimization, heat transfer enhancement and truss optimization. It reduced the computational time by 2 ~ 5 orders of magnitude compared with directly using heuristic methods, and outperformed all state-of-the-art algorithms tested in our experiments. This approach enables solving large multi-dimensional optimization problems.
The availability of high-quality three-dimensional (3D) microstructures is an essential prerequisite to understanding the macroscopic behaviours of heterogeneous media. Considering the high cost of 3D microscopy imaging, it is an economical way to derive large numbers of virtual 3D microstructures from limited morphological information of 2D cross-sectional images. This paper presents an efficient method that incorporates machine learning-based characterization of 2D images to statistically reconstruct 3D microstructures. The latent stochasticity of 2D images is mastered by fitting supervised machine learning models, which essentially characterize the local morphological statistics. A morphology integration scheme is developed to project the 2D morphological statistics into the 3D space, and new equivalent 3D microstructures can then be synthesized by sequentially generating voxel values from probability sampling. The new method is tested on a series of heterogeneous media with distinct morphologies, and it is also compared with two classical methods (i.e. stochastic optimization-based reconstruction and Gaussian random field transformation) in terms of reconstruction accuracy and efficiency. Besides, various microstructural descriptors are used to quantify the discrepancies between the reconstructed and target microstructures. The results confirm that the proposed method provides a highly cost-effective and widely applicable way to reproduce 3D realizations that precisely preserve the statistical characteristics, geometrical irregularities, long-distance connectivity and anisotropy that exist in 2D cross-sectional images.
Partial differential equations (PDEs) play a central role in the mathematical analysis and modeling of complex dynamic processes across all corners of science and engineering. Their solution often requires laborious analytical or computational tools, associated with a cost that is markedly amplified when different scenarios need to be investigated, for example, corresponding to different initial or boundary conditions, different inputs, etc. In this work, we introduce physics-informed DeepONets, a deep learning framework for learning the solution operator of arbitrary PDEs, even in the absence of any paired input-output training data. We illustrate the effectiveness of the proposed framework in rapidly predicting the solution of various types of parametric PDEs up to three orders of magnitude faster compared to conventional PDE solvers, setting a previously unexplored paradigm for modeling and simulation of nonlinear and nonequilibrium processes in science and engineering.
We propose a novel deep learning framework for predicting the permeability of porous media from their digital images. Unlike convolutional neural networks, instead of feeding the whole image volume as inputs to the network, we model the boundary between solid matrix and pore spaces as point clouds and feed them as inputs to a neural network based on the PointNet architecture. This approach overcomes the challenge of memory restriction of graphics processing units and its consequences on the choice of batch size and convergence. Compared to convolutional neural networks, the proposed deep learning methodology provides freedom to select larger batch sizes due to reducing significantly the size of network inputs. Specifically, we use the classification branch of PointNet and adjust it for a regression task. As a test case, two and three dimensional synthetic digital rock images are considered. We investigate the effect of different components of our neural network on its performance. We compare our deep learning strategy with a convolutional neural network from various perspectives, specifically for maximum possible batch size. We inspect the generalizability of our network by predicting the permeability of real-world rock samples as well as synthetic digital rocks that are statistically different from the samples used during training. The network predicts the permeability of digital rocks a few thousand times faster than a lattice Boltzmann solver with a high level of prediction accuracy.
We present a neural network-based method for solving linear and nonlinear partial differential equations, by combining the ideas of extreme learning machines (ELM), domain decomposition and local neural networks. The field solution on each sub-domain is represented by a local feed-forward neural network, and Ck continuity conditions are imposed on the sub-domain boundaries. Each local neural network consists of a small number of hidden layers, while its last hidden layer can be wide. The weight/bias coefficients in all the hidden layers of the local neural networks are pre-set to random values and fixed throughout the computation, and only the weight coefficients in the output layers of the local neural networks are training parameters. The overall neural network is trained by a linear or nonlinear least squares computation, not by the back-propagation type algorithms. We introduce a block time-marching scheme together with the presented method for long-time simulations of time-dependent linear/nonlinear partial differential equations. The current method exhibits a clear sense of convergence with respect to the degrees of freedom in the neural network. Its numerical errors typically decrease exponentially or nearly exponentially as the number of training parameters, or the number of training data points, or the number of sub-domains increases. Extensive numerical experiments have been performed to demonstrate the computational performance of the presented method. We also demonstrate its capability for long-time dynamic simulations with some test problems. We compare the presented method with the deep Galerkin method (DGM) and the physics-informed neural network (PINN) method in terms of the accuracy and computational cost. The current method exhibits a clear superiority, with its numerical errors and network training time considerably smaller (typically by orders of magnitude) than those of DGM and PINN. We also compare the current method with the classical finite element method (FEM). The computational performance of the current method is on par with, and often exceeds, the FEM performance in terms of the accuracy and computational cost.
This study proposes a machine learning (ML) based approach for optimizing fiber orientations of variable stiffness carbon fiber reinforced plastic (CFRP) structures, where neural networks are developed to estimate the objective function and analytical sensitivities with respect to design variables as a substitute for finite element analysis (FEA). To reduce the number of training samples and improve the regression accuracy, an active learning strategy is implemented by successively supplying effective samples along with the suboptimal process. After proper training of neural networks, a quasi-global search strategy can be applied by implementing a large number of initial designs as starting points in the optimization. In this paper, a mathematical example is first presented to show the superiority of the active learning strategy. Then a benchmark design example of a CFRP plate is scrutinized to compare the proposed ML-based with the conventional FEA-based discrete material optimization (DMO) method. Finally, topology optimization of fiber orientations is performed for design of a CFRP engine hood, in which the structural performance generated from the proposed ML-based approach achieves 12.62% improvement compared with that obtained from the conventional single-initial design method. This paper is anticipated to demonstrate a new alternative for design of fiber-reinforced composite structures.
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Fluid flow in heterogeneous porous media arises in many systems, from biological tissues to composite materials, soil, wood, and paper. With advances in instrumentations, high-resolution images of porous media can be obtained and used directly in the simulation of fluid flow. The computations are, however, highly intensive. Although machine learning (ML) algorithms have been used for predicting flow properties of porous media, they lack a rigorous, physics-based foundation and rely on correlations. We introduce an ML approach that incorporates mass conservation and the Navier–Stokes equations in its learning process. By training the algorithm to relatively limited data obtained from the solutions of the equations over a time interval, we show that the approach provides highly accurate predictions for the flow properties of porous media at all other times and spatial locations, while reducing the computation time. We also show that when the network is used for a different porous medium, it again provides very accurate predictions.
In this study, we present three methods of predicting CO2 adsorption upon displacing CH4 in dehydrated fractured shale samples. These methods include a dynamic numerical approach, a steady state numerical approach, and a machine learning (ML)-based approach. We develop a coupled formulation for including the effects of gas competitive adsorption, decompression, Knudsen diffusion, slippage, advection, and Fickian diffusion for multicomponent single-phase CO2-CH4 systems. Several stochastic micro-structures of fractured shale samples are generated and processed for pore network extraction. Extracted pore networks contain microporosities as well as mesopores and fractures. Next, we show that the dynamic numerical approach can be substituted with steady state model which is only 2.3% less accurate but 3 to 4 orders of magnitude computationally faster to execute. In order to enhance the computational limits even further, we train a deep learning model on 3982 simulated steady state cases with r-squared greater than 0.93. In addition, we define a storage quality criterion for recognizing suitable shale samples for CO2 sequestration which depends both on the rate and amount of adsorption. Using this criterion and the developed ML tool, a large number of shale pore-scale images can be ranked based on their suitability in an efficient and accurate manner. The proposed workflow is useful for selection of the proper geological locations for subsurface CO2 storage by analysing the available image datasets.
It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN with a single hidden layer can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators or complex systems from streams of scattered data. Here, we thus extend this theorem to DNNs. We design a new network with small generalization error, the deep operator network (DeepONet), which consists of a DNN for encoding the discrete input function space (branch net) and another DNN for encoding the domain of the output functions (trunk net). We demonstrate that DeepONet can learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations. We study different formulations of the input function space and its effect on the generalization error for 16 different diverse applications. Neural networks are known as universal approximators of continuous functions, but they can also approximate any mathematical operator (mapping a function to another function), which is an important capability for complex systems such as robotics control. A new deep neural network called DeepONet can lean various mathematical operators with small generalization error.
In this work, a physics-informed neural network (PINN) designed specifically for analyzing digital materials is introduced. This proposed machine learning (ML) model can be trained free of ground truth data by adopting the minimum energy criteria as its loss function. Results show that our energy-based PINN reaches similar accuracy as supervised ML models. Adding a hinge loss on the Jacobian can constrain the model to avoid erroneous deformation gradient caused by the nonlinear logarithmic strain. Lastly, we discuss how the strain energy of each material element at each numerical integration point can be calculated parallelly on a GPU. The algorithm is tested on different mesh densities to evaluate its computational efficiency which scales linearly with respect to the number of nodes in the system. This work provides a foundation for encoding physical behaviors of digital materials directly into neural networks, enabling label-free learning for the design of next-generation composites.
In this work, a physics-informed neural network (PINN) designed specifically for analyzing digital materials is introduced. This proposed machine learning (ML) model can be trained free of ground truth data by adopting the minimum energy criteria as its loss function. Results show that our energy-based PINN reaches similar accuracy as supervised ML models. Adding a hinge loss on the Jacobian can constrain the model to avoid erroneous deformation gradient caused by the nonlinear logarithmic strain. Lastly, we discuss how the strain energy of each material element at each numerical integration point can be calculated parallelly on a GPU. The algorithm is tested on different mesh densities to evaluate its computational efficiency which scales linearly with respect to the number of nodes in the system. This work provides a foundation for encoding physical behaviors of digital materials directly into neural networks, enabling label-free learning for the design of next-generation composites.
The phase-field method is a powerful and versatile computational approach for modeling the evolution of microstructures and associated properties for a wide variety of physical, chemical, and biological systems. However, existing high-fidelity phase-field models are inherently computationally expensive, requiring high-performance computing resources and sophisticated numerical integration schemes to achieve a useful degree of accuracy. In this paper, we present a computationally inexpensive, accurate, data-driven surrogate model that directly learns the microstructural evolution of targeted systems by combining phase-field and history-dependent machine-learning techniques. We integrate a statistically representative, low-dimensional description of the microstructure, obtained directly from phase-field simulations, with either a time-series multivariate adaptive regression splines autoregressive algorithm or a long short-term memory neural network. The neural-network-trained surrogate model shows the best performance and accurately predicts the nonlinear microstructure evolution of a two-phase mixture during spinodal decomposition in seconds, without the need for “on-the-fly” solutions of the phase-field equations of motion. We also show that the predictions from our machine-learned surrogate model can be fed directly as an input into a classical high-fidelity phase-field model in order to accelerate the high-fidelity phase-field simulations by leaping in time. Such machine-learned phase-field framework opens a promising path forward to use accelerated phase-field simulations for discovering, understanding, and predicting processing–microstructure–performance relationships.
We present a two-scale topology optimization framework for the design of macroscopic bodies with an optimized elastic response, which is achieved by means of a spatially-variant cellular architecture on the microscale. The chosen spinodoid topology for the cellular network on the microscale (which is inspired by natural microstructures forming during spinodal decomposition) admits a seamless spatial grading as well as tunable elastic anisotropy, and it is parametrized by a small set of design parameters associated with the underlying Gaussian random field. The macroscale boundary value problem is discretized by finite elements, which in addition to the displacement field continuously interpolate the microscale design parameters. By assuming a separation of scales, the local constitutive behavior on the macroscale is identified as the homogenized elastic response of the microstructure based on the local design parameters. As a departure from classical FE$^2$-type approaches, we replace the costly microscale homogenization by a data-driven surrogate model, using deep neural networks, which accurately and efficiently maps design parameters onto the effective elasticity tensor. The model is trained on homogenized stiffness data obtained from numerical homogenization by finite elements. As an added benefit, the machine learning setup admits automatic differentiation, so that sensitivities (required for the optimization problem) can be computed exactly and without the need for numerical derivatives - a strategy that holds promise far beyond the elastic stiffness. Therefore, this framework presents a new opportunity for multiscale topology optimization based on data-driven surrogate models.
The hierarchical deep-learning neural network (HiDeNN) is systematically developed through the construction of structured deep neural networks (DNNs) in a hierarchical manner, and a special case of HiDeNN for representing Finite Element Method (or HiDeNN-FEM in short) is established. In HiDeNN-FEM, weights and biases are functions of the nodal positions, hence the training process in HiDeNN-FEM includes the optimization of the nodal coordinates. This is the spirit of r-adaptivity, and it increases both the local and global accuracy of the interpolants. By fixing the number of hidden layers and increasing the number of neurons by training the DNNs, rh-adaptivity can be achieved, which leads to further improvement of the accuracy for the solutions. The generalization of rational functions is achieved by the development of three fundamental building blocks of constructing deep hierarchical neural networks. The three building blocks are linear functions, multiplication, and inversion. With these building blocks, the class of deep learning interpolation functions are demonstrated for interpolation theories such as Lagrange polynomials, NURBS, isogeometric, reproducing kernel particle method, and others. In HiDeNN-FEM, enrichment functions through the multiplication of neurons is equivalent to the enrichment in standard finite element methods, that is, generalized, extended, and partition of unity finite element methods. Numerical examples performed by HiDeNN-FEM exhibit reduced approximation error compared with the standard FEM. Finally, an outlook for the generalized HiDeNN to high-order continuity for multiple dimensions and topology optimizations are illustrated through the hierarchy of the proposed DNNs.
A new method combining computational homogenization and the Artificial Neural Network (ANN) is proposed to construct elastoplastic composites database efficiently for data-driven computational mechanics (DDCM). The numerical calculations are performed on the representative volume element (RVE) of elastoplastic composites to collect a small set of high-fidelity data containing stress and strain tensors, which is then enriched using ANN to provide sufficient data for DDCM. To justify the validity and efficiency of the proposed method, two composite plates made of matrix-round inclusion material and fiber reinforced material, respectively, are considered. The former is composed of nonlinear materials, while the latter has nonlinearity caused by fiber buckling and plasticity of matrix in the RVE. ANN shows good ability to learn nonlinear relationships between stress and strain from the data collected by computational homogenization, and to generate new data efficiently. This work is expected to uncover the possibility of applying the data generated by artificial intelligence to DDCM in solving mechanical problems of composites. Comparisons demonstrate that the proposed method not only reduces the computational cost for database construction, but also shows excellent accuracy of DDCM even for three-dimensional problems.
Solving topology optimization problem is very computationally demanding especially when high-resolution results are sought for. In the present work, a problem-independent machine learning (PIML) technique is proposed to reduce the computational time associated with finite element analysis (FEA) which constitutes the main bottleneck of the solution process. The key idea is to construct the structural analysis procedure under the extended multi-scale finite element method (EMsFEM) framework, and establish an implicit mapping between the shape functions of EMsFEM and element-wise material densities of a coarse-resolution element through machine learning (ML). Compared with existing works, the proposed mechanistic-based ML technique is truly problem-independent and can be used to solve any kind of topology optimization problems without any modification once the easy-to-implement off-line training is completed. It is demonstrated that the proposed approach can reduce the FEA time significantly. In particular, with the use of the proposed approach, a topology optimization problem with 200 million of design variables can be solved on a personal workstation with an average of only two minutes for FEA per iteration step.
Neural operators [1], [2], [3], [4], [5] have recently become popular tools for designing solution maps between function spaces in the form of neural networks. Differently from classical scientific machine learning approaches that learn parameters of a known partial differential equation (PDE) for a single instance of the input parameters at a fixed resolution, neural operators approximate the solution map of a family of PDEs [6], [7]. Despite their success, the uses of neural operators are so far restricted to relatively shallow neural networks and confined to learning hidden governing laws. In this work, we propose a novel nonlocal neural operator, which we refer to as nonlocal kernel network (NKN), that is resolution independent, characterized by deep neural networks, and capable of handling a variety of tasks such as learning governing equations and classifying images. Our NKN stems from the interpretation of the neural network as a discrete nonlocal diffusion reaction equation that, in the limit of infinite layers, is equivalent to a parabolic nonlocal equation, whose stability is analyzed via nonlocal vector calculus. The resemblance with integral forms of neural operators allows NKNs to capture long-range dependencies in the feature space, while the continuous treatment of node-to-node interactions makes NKNs resolution independent. The resemblance with neural ODEs, reinterpreted in a nonlocal sense, and the stable network dynamics between layers allow for generalization of NKN's optimal parameters from shallow to deep networks. This fact enables the use of shallow-to-deep initialization techniques [8]. Our tests show that NKNs outperform baseline methods in both learning governing equations and image classification tasks and generalize well to different resolutions and depths.
The accurate and efficient reconstruction of the porous media is the fundamental link to revealing its structural features and physical properties. In this work, we propose a deep learning (DL)-based algorithm to reconstruct large-scale three-dimensional (3D) porous media that can be treated as the representative element volumes (REVs), based on generative adversarial networks (GAN) and convolutional neural networks (CNN), named LGCNN. The proposed framework consists of a machine learning based (ML-based) reconstruction method for small-scale porous media and an adjustable splicing algorithm to achieve the REVs reconstruction. On this basis, four special neural networks are established to reconstruct the porous media and ensure the connectivity between the adjacent porous media during the splicing process. Subsequently, the detailed validation of LGCNN against traditional reconstruction methods and other deep learning algorithms is performed. The results show that the reconstruction speed (600³ voxels) of LGCNN (10 min) is much faster than traditional numerical reconstruction methods including QSGS (642 min), CCSIM (5973 min), and SD (33,628 min) with higher accuracy on structural parameters (e.g., porosity and pore size distribution), when compared with real porous media. In particular, the size of constructed porous media is far larger than previous ML-based reconstruction algorithms as much as 3–4 orders of magnitude, indicating the puissant ability of LGCNN to be used for high-resolution or multi-scale reconstruction.
This paper presents a model-free data-driven strategy for linear and non-linear finite element computations of open-cell foam. Employing sets of material data, the data-driven problem is formulated as the minimization of a distance function subjected to the physical constraints from the kinematics and the balance laws of the mechanical problem.
The material data sets of the foam are deduced here from representative microscopic material volumes. These volume elements capture the stochastic characteristics of the open-cell microstructure and the properties of the polyurethane material. Their computation provides the required stress–strain data used in the macroscopic finite element computations without postulating a specific constitutive model. The paper shows how to derive suitable material data sets for the different cases of (non-)linear and (an-)isotropic material behavior efficiently.
Exemplarily, we compare data-driven finite element computations with linearized and finite deformations and show that in a data-driven computation the linear kinematic is sufficiently accurate to capture the material’s non-linearity up to 50% of straining. The numerical example of a rubber sealing illustrates possible areas of application, the expenditure, and the proposed strategy’s versatility.
Splitting method is a powerful method to handle application problems by splitting physics, scales, domain, and so on. Many splitting algorithms have been designed for efficient temporal discretization. In this paper, our goal is to use temporal splitting concepts in designing machine learning algorithms and, at the same time, help splitting algorithms by incorporating data and speeding them up. We propose a machine learning assisted splitting scheme which improves the efficiency of the scheme meanwhile preserves the accuracy. We consider a recently introduced multiscale splitting algorithms, where the multiscale problem is solved on a coarse grid. To approximate the dynamics, only a few degrees of freedom are solved implicitly, while others explicitly. This splitting concept allows identifying degrees of freedom that need implicit treatment. In this paper, we use this splitting concept in machine learning and propose several strategies. First, the implicit part of the solution can be learned as it is more difficult to solve, while the explicit part can be computed. This provides a speed-up and data incorporation for splitting approaches. Secondly, one can design a hybrid neural network architecture because handling explicit parts requires much fewer communications among neurons and can be done efficiently. Thirdly, one can solve the coarse grid component via PDEs or other approximation methods and construct simpler neural networks for the explicit part of the solutions. We discuss these options and implement one of them by interpreting it as a machine translation task. This interpretation of the splitting scheme successfully enables us using the Transformer since it can perform model reduction for multiple time series and learn the connection between them. We also find that the splitting scheme is a great platform to predict the coarse solution with insufficient information of the target model: the target problem is partially given and we need to solve it through a known problem which approximates the target. Our machine learning model can incorporate and encode the given information from two different problems and then solve the target problems. We conduct four numerical examples and the results show that our method is stable and accurate.
Understanding the gas adsorption behavior in shale nanopores is essential for the reservoir estimation and performance prediction of shale gas, which is still unclear considering the complexity of geological environment and nanoporous structure of shale. In general, traditional methods based on experiments and molecular dynamics (MD) simulations are always expensive and time consuming. In this work, a machine learning (ML) framework to predict the methane adsorption behavior in shale nanopores is constructed from the microscopic and kinetic theory perspectives, where three novel parameters related to potential energy distribution (PED) are proposed to represent the methane adsorption characteristics of shale slit nanopores. Machine learning algorithm based on the uniformly constructed dataset is introduced to realize fast and accurate prediction of methane adsorption behavior, which is well validated by typical inorganic and organic models. Moreover, the application of the proposed ML model to predict the adsorption behavior of methane in different geological conditions (e.g., pressure and temperature) is performed, indicating its feasibility to predict the gas adsorption behavior in shale nanopores with ultra-fast computation speed, that would be beneficial for the exploitation and development of shale gas reservoirs. The insights gained in this work is also instructive for the prediction of nano-confined fluid behavior under complex environmental factors.
Failure trajectories, probable failure zones, and damage indices are some of the key quantities of relevance in brittle fracture mechanics. High-fidelity numerical solvers that reliably estimate these relevant quantities exist but they are computationally demanding requiring a high resolution of the crack. Moreover, independent simulations need to be carried out even for a small change in domain parameters and/or material properties. Therefore, fast and generalizable surrogate models are needed to alleviate the computational burden but the discontinuous and complex nature of fracture mechanics presents a major challenge to developing such models. We propose a physics-informed variational formulation of DeepONet (V-DeepONet) for brittle fracture analysis. V-DeepONet is trained to map the initial configuration of the defect to the relevant fields of interests (e.g., damage and displacements). Once the network is trained, the entire global solution can be rapidly obtained for any initial crack configuration and loading steps on that domain. While the original DeepONet is solely data-driven, we take a different path to train the V-DeepONet by imposing the governing equations in a variational form with some labeled data. We demonstrate the effectiveness of V-DeepOnet through two benchmarks of brittle fracture and verify its accuracy using results from high-fidelity solvers. Encoding the physical laws to the model with data enhancement in training renders the surrogate model capable of accurately performing both interpolation and extrapolation tasks. Considering that fracture modeling is very sensitive to fluctuations, the proposed V-DeepONet with a hybrid training strategy is able to predict the quantities of interests with good accuracy, which can be easily extended to a wide array of dynamical systems with complex responses.
This work proposes a data-driven approach, G-MAP123, using discrete data directly for nonlinear elastic materials to solve boundary value problems, avoiding analytic-function based constitutive models. G-MAP123 is formulated in the current configuration in which the Cauchy stress and the left Cauchy–Green strain are adopted as the stress–strain measures of the data. Data generated under both uniaxial tension and equibiaxial tension experiments is used. A data search employing stress triaxiality as the index is here proposed for the stress update. Furthermore, including additional data from other loading paths is also rendered possible. Comparison with reference analytic-function based models such as Arruda-Boyce, Yeoh, Mooney–Rivlin and Van der Waals is carried out. Results show that the predictions from G-MAP123 are in agreement with all those of the reference models. Moreover, the classic experimental data of rubber from Treloar is here used to demonstrate the capability of the proposed G-MAP123 in the practical setting. This approach opens a new avenue to modeling soft materials accurately and conveniently at large deformation, directly from the data.
This work proposes a simple but robust 4-node 24-DOF facet shell element for static analysis of small-scale thin shell structures. To accommodate the size effects, the modified couple stress theory is employed as the theoretical basis. The element is constructed via two main innovations. First, the trial functions that can a priori satisfy related governing differential equations are adopted as the basic functions for formulating the element interpolations. Second, the generalized conforming theory and the penalty function method are employed to meet the C¹ continuity requirement in weak sense for ensuring the computation convergence property. Several benchmarks of shells with different geometries are tested to assess the new facet shell element's capability. The numerical results reveal that the element can effectively simulate the size-dependent mechanical behaviors of small-scale thin shells, exhibiting satisfactory numerical accuracy and low susceptibility to mesh distortion. Moreover, as the shell element uses only six degrees of freedom per node, it can be incorporated into the commonly available finite element programs very readily.
Knitting is an effective technique for producing complex three‐dimensional surfaces owing to the inherent flexibility of interlooped yarns and recent advances in manufacturing providing better control of local stitch patterns. Fully yarn‐level modelling of large‐scale knitted membranes is not feasible. Therefore, we use a two‐scale homogenisation approach and model the membrane as a Kirchhoff‐Love shell on the macroscale and as Euler‐Bernoulli rods on the microscale. The governing equations for both the shell and the rod are discretised with cubic B‐spline basis functions. For homogenisation we consider only the in‐plane response of the membrane. The solution of the nonlinear microscale problem requires a significant amount of time due to the large deformations and the enforcement of contact constraints, rendering conventional online computational homogenisation approaches infeasible. To sidestep this problem, we use a pre‐trained statistical Gaussian Process Regression (GPR) model to map the macroscale deformations to macroscale stresses. During the offline learning phase, the GPR model is trained by solving the microscale problem for a sufficiently rich set of deformation states obtained by either uniform or Sobol sampling. The trained GPR model encodes the nonlinearities and anisotropies present in the microscale and serves as a material model for the membrane response of the macroscale shell. The bending response can be chosen in dependence of the mesh size to penalise the fine out‐of‐plane wrinkling of the membrane. After verifying and validating the different components of the proposed approach, we introduce several examples involving membranes subjected to tension and shear to demonstrate its versatility and good performance.
Permeability prediction of porous media from numerical approaches is an important supplement for experimental measurements with the benefits of being more economical and efficient. However, the accuracy and reliability of traditional numerical approaches are strongly dependent on the high-resolution images of porous media, which greatly limits their implementation for engineering applications. Herein, a semi-supervised machine learning approach is proposed to predict the permeability of porous media from low-resolution images. This approach consists of an autoencoder (AE) module trained with unlabeled data to assist the backbone convolutional neural network (CNN) in the prediction by providing a mapping of the low-resolution porous media to high-resolution features. The low-resolution information from CNN trained with small amount of labeled data and high-resolution information from AE trained with larger amount of unlabeled data are comprehensively considered in this approach. The prediction performance of AE-CNN from low-resolution images is examined against the results from traditional approaches of CNN and lattice Boltzmann method (LBM) by the mean-square errors (MSE) and R-Squared (R2) calculations. Using 5-fold cross-validation method, the average value of R2 is 0.896 on the test dataset by AE-CNN, compared to 0.869 for the traditional CNN without the AE. The MSEs for AE-CNN are 0.022 and 0.064 on the training and test datasets respectively in the best-performance fold, while without AE, the MSEs for only CNN are 0.034 and 0.083 on the training and test datasets respectively in the best-performance fold, implying that AE modules can substantially improve the prediction performance from low-resolution images of porous media. As for the simulation results of LBM approach, its prediction reliability (average R2: 0.42; MSE: 0.37 and 0.36 in the best-performance fold) is extremely lower than that of CNN-based machine learning algorithms owing to huge numerical error at the blurred boundaries of low-resolution images.
Deep learning and the collocation method are merged and used to solve partial differential equations describing structures' deformation. We have considered different types of materials: linear elasticity, hyperelasticity (neo‐Hookean) with large deformation, and von Mises plasticity with isotropic and kinematic hardening. The performance of this deep collocation method (DCM) depends on the architecture of the neural network and the corresponding hyperparameters. The presented DCM is meshfree and avoids any spatial discretization, which is usually needed for the finite element method (FEM). We show that the DCM can capture the response qualitatively and quantitatively, without the need for any data generation using other numerical methods such as the FEM. Data generation usually is the main bottleneck in most data‐driven models. The deep learning model is trained to learn the model's parameters yielding accurate approximate solutions. Once the model is properly trained, solutions can be obtained almost instantly at any point in the domain, given its spatial coordinates. Therefore, the deep collocation method is potentially a promising standalone technique to solve partial differential equations involved in the deformation of materials and structural systems as well as other physical phenomena.
We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs). The approach allows to train neural networks while respecting the PDEs as a strong constraint in the optimisation as apposed to making them part of the loss function. The resulting models are discretised in space by the finite element method (FEM). The method applies to both stationary and transient as well as linear/nonlinear PDEs. We describe implementation of the approach as an extension of the existing FEM framework FEniCS and its algorithmic differentiation tool dolfin-adjoint. Through series of examples we demonstrate capabilities of the approach to recover coefficients and missing PDE operators from observations. Further, the proposed method is compared with alternative methodologies, namely, physics informed neural networks and standard PDE-constrained optimisation. Finally, we demonstrate the method on a complex cardiac cell model problem using deep neural networks.
Predicting the production performance of multistage fractured horizontal wells is essential for developing unconventional resources such as shale gas and oil. Accurate predictions of the production performance of wells that have not been put into production are necessary to optimize hydraulic fracture parameters prior to operation. However, traditional analytic methods are made inefficient by their strong dependency on historical production data and their huge computational expense. To conquer this issue, we developed deep belief network (DBN) models to predict the production performance of unconventional wells effectively and accurately. We ran 815 numerical simulation cases to construct a database for model training and optimized the hyperparameters of our network model using the Bayesian optimization algorithm. DBN models exhibit greater prediction accuracy and generalization ability than traditional machine-learning techniques such as back-propagation (BP) neural networks, and support vector regression (SVR). We also used the trained DBN model as a proxy to optimize the fracturing design and obtained outstanding results. Our proposed model could predict the production performance of an unconventional well instantaneously with considerable accuracy and shows excellent reusability, making it a powerful tool in optimizing fracturing designs. Our work lays a solid basis for anticipating the production performance of unconventional reservoirs and sheds light on the construction of data-driven models in the areas of energy conversion and utilization.
For many systems in science and engineering, the governing differential equation is either not known or known in an approximate sense. Analyses and design of such systems are governed by data collected from the field and/or laboratory experiments. This challenging scenario is further worsened when data-collection is expensive and time-consuming. To address this issue, this paper presents a novel multi-fidelity physics informed deep neural network (MF-PIDNN). The framework proposed is particularly suitable when the physics of the problem is known in an approximate sense (low-fidelity physics) and only a few high-fidelity data are available. MF-PIDNN blends physics informed and data-driven deep learning techniques by using the concept of transfer learning. The approximate governing equation is first used to train a low-fidelity physics informed deep neural network. This is followed by transfer learning where the low-fidelity model is updated by using the available high-fidelity data. MF-PIDNN is able to encode useful information on the physics of the problem from the approximate governing differential equation and hence, provides accurate prediction even in zones with no data. Additionally, no low-fidelity data is required for training this model. Two examples involving function approximations with linear and nonlinear correlation are presented to illustrate the effectiveness of transfer learning in solving multi-fidelity problems. Applicability and utility of MF-PIDNN are illustrated in solving four benchmark reliability analysis problems. Case studies presented illustrate interesting features of the proposed approach.
Coupling pore network and finite element methods for rapid modelling of deformation - Volume 897 - Samuel Fagbemi, Pejman Tahmasebi