Publications

Modular machine learning-based elastoplasticity: Generalization in the context of limited data

Published in Computer Methods in Applied Mechanics and Engineering, 2023

The development of highly accurate constitutive models for materials that undergo path-dependent processes continues to be a complex challenge in computational solid mechanics. Challenges arise both in considering the appropriate model assumptions and from the viewpoint of data availability, verification, and validation. Recently, data-driven modeling approaches have been proposed that aim to establish stress-evolution laws that avoid user-chosen functional forms by relying on machine learning representations and algorithms. However, these approaches not only require a significant amount of data but also need data that probes the full stress space with a variety of complex loading paths. Furthermore, they rarely enforce all necessary thermodynamic principles as hard constraints. Hence, they are in particular not suitable for low-data or limited-data regimes, where the first arises from the cost of obtaining the data and the latter from the experimental limitations of obtaining labeled data, which is commonly the case in engineering applications. In this work, we discuss a hybrid framework that can work on a variable amount of data by relying on the modularity of the elastoplasticity formulation where each component of the model can be chosen to be either a classical phenomenological or a data-driven model depending on the amount of available information and the complexity of the response. The method is tested on synthetic uniaxial data coming from simulations as well as cyclic experimental data for structural materials. The discovered material models are found to not only interpolate well but also allow for accurate extrapolation in a thermodynamically consistent manner far outside the domain of the training data. This ability to extrapolate from limited data was the main reason for the early and continued success of phenomenological models and the main shortcoming in machine learning-enabled constitutive modeling approaches. Training aspects and details of the implementation of these models into Finite Element simulations are discussed and analyzed. © 2023 Elsevier B.V.

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Enhancing phenomenological yield functions with data: Challenges and opportunities

Published in European Journal of Mechanics, A/Solids, 2023

The formulation of history-dependent material laws has been a significant research and industrial activity in solid mechanics for over a century. A large variety of models has been developed, tailored for the description of different families of materials. However, model selection for a specific problem is a delicate issue and there still remain open problems. In fact, the catalog of yield models is continuously being enriched by experts to meet new needs and tailor models to new experimental evidence. We propose here an alternative approach, that is a flexible model-data-driven plasticity formulation, and analyze its challenges and opportunities. A phenomenological yield model is locally improved by a data-driven correction term. The data-based correction component is described by a surrogate model built from machine-learning techniques. In this regard, the framework is versatile, as shown by the fact that similar performances are obtained with Support Vector Regression, Gaussian Process Regression, and Neural Networks. The convexity of model-data yield functions is guaranteed by employing convex extensions of the adopted machine-learning techniques. The proposed approach is substantiated by reproducing a highly anisotropic yield response with tension/compression asymmetries from a simple phenomenological model enhanced with a limited number of synthetic data points of yield onset. It is shown that a crucial role is played by both model and data, the former allowing to use a limited number of data and the latter significantly enhancing model performance. © 2023 Elsevier Masson SAS

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The mixed Deep Energy Method for resolving concentration features in finite strain hyperelasticity

Published in Journal of Computational Physics, 2022

The introduction of Physics-informed Neural Networks (PINNs) has led to an increased interest in deep neural networks as universal approximators of PDEs in the solid mechanics community. Recently, the Deep Energy Method (DEM) has been proposed. DEM is based on energy minimization principles, contrary to PINN which is based on the residual of the PDEs. A significant advantage of DEM, is that it requires the approximation of lower order derivatives compared to formulations that are based on strong form residuals. However both DEM and classical PINN formulations struggle to resolve fine features of the stress and displacement fields, for example concentration features in solid mechanics applications. We propose an extension to the Deep Energy Method (DEM) to resolve these features for finite strain hyperelasticity. The developed framework termed mixed Deep Energy Method (mDEM) introduces stress measures as an additional output of the NN to the recently introduced pure displacement formulation. Using this approach, Neumann boundary conditions are approximated more accurately and the accuracy around spatial features which are typically responsible for high concentrations is increased. In order to make the proposed approach more versatile, we introduce a numerical integration scheme based on Delaunay integration, which enables the mDEM framework to be used for random training point position sets commonly needed for computational domains with stress concentrations, i.e. domains with holes, notches, etc. We highlight the advantages of the proposed approach while showing the shortcomings of classical PINN and DEM formulations. The method is offering comparable results to Finite-Element Method (FEM) on the forward calculation of challenging computational experiments involving domains with fine geometric features and concentrated loads, but additionally offers unique capabilities for the solution of inverse problems and parameter estimation in the context of hyperelasticity. © 2021 Elsevier Inc.

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On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling

Published in Computer Methods in Applied Mechanics and Engineering, 2022

Data-driven constitutive modeling is an emerging field in computational solid mechanics with the prospect of significantly relieving the computational costs of hierarchical computational methods. Additionally, this data-driven paradigm could enable a seamless connection of experimental data probing material responses with numerical simulations at the structural level. Traditionally, these surrogates have just been trained using datasets which map strain inputs to stress outputs for elastic and inelastic materials directly. Recently, artificial neural networks (ANNs) have instead been trained to additionally incorporate the underlying physical laws in the construction of these models. However, ANNs do not offer convergence guarantees from an engineering point of view and are majorly reliant on user-specified parameters. In contrast to ANNs, Gaussian process regression (GPR) is based on nonparametric modeling principles as well as on fundamental statistical knowledge and hence allows for strict convergence guarantees. Motivated by the recent work by Frankel et al. (2021) which is based on rewriting the stress output as a linear combination of an irreducible integrity basis, in this work we present a physics-informed and data-driven constitutive modeling approach for isotropic and anisotropic hyperelastic materials at finite strain. The trained surrogates are able to respect physical principles such as material frame indifference, material symmetry, thermodynamic consistency, stress-free undeformed configuration, and the local balance of angular momentum. Our approach is based on probabilistic machine learning and uniquely can be used in the big data context while maintaining the benefits of GPR. As sampling in the mixed invariant space poses a unique challenge, we additionally present the first sampling approach that directly generates space-filling points in the invariant space corresponding to a bounded domain of the deformation gradient tensor. The sampling technique is based on simulated annealing and provides more efficient and reliable physics-informed constitutive models. Overall, the presented approach is tested on synthetic data from isotropic and anisotropic constitutive laws and shows surprising accuracy even far beyond the limits of the training domain, indicating that the resulting surrogates can efficiently generalize as they incorporate knowledge about the underlying physics. © 2022 Elsevier B.V.

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Machine-learning convex and texture-dependent macroscopic yield from crystal plasticity simulations

Published in Materialia, 2022

The influence of the microstructure of a polycrystalline material on its macroscopic deformation response is still one of the major problems in materials engineering. For materials characterized by elastic-plastic deformation responses, predictive computational models to characterize crystal-plasticity (CP) have been developed. However, due to their large demand of computational resources, CP simulations cannot be straightforwardly implemented in hierarchical computational models such as FE2. This bottleneck intensifies the need for the development of macroscopic simulation tools that can be directly informed by microstructural quantities. Using a 3D finite element solver for CP, we generate a macroscopic yield function database based on general loading conditions and crystallographic texture. Leveraging the advancement in statistical modeling we describe and apply a machine learning framework for predicting plane stress macroscopic yield as a function of crystallographic texture. The convexity of the data-driven yield function is guaranteed by using partially input convex neural networks as the predictive tool. Furthermore, in order to allow for the predicted yield function to be directly incorporated in time-integration schemes, as needed for the finite element method, the yield surfaces are interpreted as the boundaries of signed distance function level sets. Results generated for an example cube texture are discussed. © 2022 Acta Materialia Inc.

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Local approximate Gaussian process regression for data-driven constitutive models: development and comparison with neural networks

Published in Computer Methods in Applied Mechanics and Engineering, 2022

Hierarchical computational methods for multiscale mechanics such as the FE2 and FE-FFT methods are generally accompanied by high computational costs. Data-driven approaches are able to speed the process up significantly by enabling to incorporate the effective micromechanical response in macroscale simulations without the need of performing additional computations at each Gauss point explicitly. Traditionally artificial neural networks (ANNs) have been the surrogate modeling technique of choice in the solid mechanics community. However they suffer from severe drawbacks due to their parametric nature and suboptimal training and inference properties for the investigated datasets in a three dimensional setting. These problems can be avoided using local approximate Gaussian process regression (laGPR). This method can allow the prediction of stress outputs at particular strain space locations by training local regression models based on Gaussian processes, using only a subset of the data for each local model, offering better and more reliable accuracy than ANNs. A modified Newton–Raphson approach specific to laGPR is proposed to accommodate for the local nature of the laGPR approximation when solving the global structural problem in a FE setting. Hence, the presented work offers a complete and general framework enabling multiscale calculations combining a data-driven constitutive prediction using laGPR, and macroscopic calculations using an FE scheme that we test for finite-strain three-dimensional hyperelastic problems. © 2021 Elsevier B.V.

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Learning hyperelastic anisotropy from data via a tensor basis neural network

Published in Journal of the Mechanics and Physics of Solids, 2022

Anisotropy in the mechanical response of materials with microstructure is common and yet is difficult to assess and model. To construct accurate response models given only stress–strain data, we employ classical representation theory, novel neural network layers, and L1 regularization. The proposed tensor-basis neural network can discover both the type and orientation of the anisotropy and provide an accurate model of the stress response. The method is demonstrated with data from hyperelastic materials with off-axis transverse isotropy and orthotropy, as well as materials with less well-defined symmetries induced by fibers or spherical inclusions. Both plain feed-forward neural networks and input-convex neural network formulations are developed and tested. Using the latter, a polyconvex potential can be established, which, by satisfying the necessary growth condition can guarantee the existence of boundary value problem solutions. © 2022

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Interval and fuzzy physics-informed neural networks for uncertain fields

Published in Probabilistic Engineering Mechanics, 2022

Temporally and spatially dependent uncertain parameters are regularly encountered in engineering applications. Commonly these uncertainties are accounted for using random fields and processes, which require knowledge about the appearing probability distributions functions that is not readily available. In these cases non-probabilistic approaches such as interval analysis and fuzzy set theory are helpful to analyze uncertainty. Partial differential equations involving fuzzy and interval fields are traditionally solved using the finite element method where the input fields are sampled using some basis function expansion methods. This approach however relies on information about the spatial correlation of the fields, which is not always obtainable. In this work we utilize physics-informed neural networks (PINNs) to solve interval and fuzzy partial differential equations. The resulting network structures termed interval physics-informed neural networks (iPINNs) and fuzzy physics-informed neural networks (fPINNs) show promising results for obtaining bounded solutions of equations involving spatially and/or temporally uncertain parameter fields. In contrast to finite element approaches, no correlation length specification of the input fields as well as no Monte-Carlo simulations are necessary. In fact, information about the input interval fields is obtained directly as a byproduct of the presented solution scheme. Furthermore, all major advantages of PINNs are retained, i.e. meshfree nature of the scheme, and ease of inverse problem set-up. © 2022 Elsevier Ltd

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Enhancing high-fidelity nonlinear solver with reduced order model

Published in Scientific Reports, 2022

We propose the use of reduced order modeling (ROM) to reduce the computational cost and improve the convergence rate of nonlinear solvers of full order models (FOM) for solving partial differential equations. In this study, a novel ROM-assisted approach is developed to improve the computational efficiency of FOM nonlinear solvers by using ROM’s prediction as an initial guess. We hypothesize that the nonlinear solver will take fewer steps to the converged solutions with an initial guess that is closer to the real solutions. To evaluate our approach, four physical problems with varying degrees of nonlinearity in flow and mechanics have been tested: Richards’ equation of water flow in heterogeneous porous media, a contact problem in a hyperelastic material, two-phase flow in layered porous media, and fracture propagation in a homogeneous material. Overall, our approach maintains the FOM’s accuracy while speeding up nonlinear solver by 18–73% (through suitable ROM-assisted FOMs). More importantly, the proximity of ROM’s prediction to the solution space leads to the improved convergence of FOMs that would have otherwise diverged with default initial guesses. We demonstrate that the ROM’s accuracy can impact the computational efficiency with more accurate ROM solutions, resulting in a better cost reduction. We also illustrate that this approach could be used in many FOM discretizations (e.g., finite volume, finite element, or a combination of those). Since our ROMs are data-driven and non-intrusive, the proposed procedure can easily lend itself to any nonlinear physics-based problem. © 2022, The Author(s).

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A classification-pursuing adaptive approach for Gaussian process regression on unlabeled data

Published in Mechanical Systems and Signal Processing, 2022

Some areas of mechanical and system engineering such as dynamic systems commonly exhibit highly fluctuating responses over given parametric domains. Therefore, classifying some quantities of interest over the parametric domain for designing new systems turns out to be a highly challenging task. In this context, an innovative adaptive sampling algorithm named Monte Carlo-intersite Voronoi (MiVor) is proposed for design applications based on the classification of one or more continuous quantities of interest useful for parametric studies. In contrast to reliability analysis problems, no probabilistic setting and information is needed. The proposed technique is able to efficiently detect two or more classes of highly imbalanced decision regions and to accurately describe the boundary between these regions in a robust manner. To the best of the authors knowledge it is the first adaptive scheme for classification-pursuing parametric studies that combines information from (potentially) multiple class label outputs and the accompanying continuous values for efficient sampling involving (possibly) multiple class outputs. The resulting surrogates utilize only a small number of observations which are obtained in an active manner. The capabilities of the presented algorithm to provide accurate classification are demonstrated on three dynamic applications with various dimensionality and under consideration of a combination of different first-passage failure scenarios. Comparisons with two regression-based adaptive schemes show that the proposed algorithm outperforms existing methods. For instance, in the case of a quarter-car problem, more than 99% of points are correctly classified using the proposed approach at convergence, whereas less than 80% of reference samples are correctly classified with standard approaches. Similar performances (> 95%) are also obtained with MiVor for a non-linear oscillator of Duffing’s type and a three-degrees-of-freedom mass-spring system with three and six-dimensional parametric spaces respectively. © 2021 Elsevier Ltd

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State-of-the-Art and Comparative Review of Adaptive Sampling Methods for Kriging

Published in Archives of Computational Methods in Engineering, 2021

Metamodels aim to approximate characteristics of functions or systems from the knowledge extracted on only a finite number of samples. In recent years kriging has emerged as a widely applied metamodeling technique for resource-intensive computational experiments. However its prediction quality is highly dependent on the size and distribution of the given training points. Hence, in order to build proficient kriging models with as few samples as possible adaptive sampling strategies have gained considerable attention. These techniques aim to find pertinent points in an iterative manner based on information extracted from the current metamodel. A review of adaptive schemes for kriging proposed in the literature is presented in this article. The objective is to provide the reader with an overview of the main principles of adaptive techniques, and insightful details to pertinently employ available tools depending on the application at hand. In this context commonly applied strategies are compared with regards to their characteristics and approximation capabilities. In light of these experiments, it is found that the success of a scheme depends on the features of a specific problem and the goal of the analysis. In order to facilitate the entry into adaptive sampling a guide is provided. All experiments described herein are replicable using a provided open source toolbox. © 2020, The Author(s).

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PI/PID controller stabilizing sets of uncertain nonlinear systems: an efficient surrogate model-based approach

Published in Nonlinear Dynamics, 2021

Closed forms of stabilizing sets are generally only available for linearized systems. An innovative numerical strategy to estimate stabilizing sets of PI or PID controllers tackling (uncertain) nonlinear systems is proposed. The stability of the closed-loop system is characterized by the sign of the largest Lyapunov exponent (LLE). In this framework, the bottleneck is the computational cost associated with the solution of the system, particularly including uncertainties. To overcome this issue, an adaptive surrogate algorithm, the Monte Carlo intersite Voronoi (MiVor) scheme, is adopted to pertinently explore the domain of the controller parameters and classify it into stable/unstable regions from a low number of nonlinear estimations. The result of the random analysis is a stochastic set providing probability information regarding the capabilities of PI or PID controllers to stabilize the nonlinear system and the risk of instabilities. The minimum of the LLE is proposed as tuning rule of the controller parameters. It is expected that using a tuning rule like this results in PID controllers producing the highest closed-loop convergence rate, thus being robust against model parametric uncertainties and capable of avoiding large fluctuating behavior. The capabilities of the innovative approach are demonstrated by estimating robust stabilizing sets for the blood glucose regulation problem in type 1 diabetes patients. © 2021, The Author(s).

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Model-data-driven constitutive responses: Application to a multiscale computational framework

Published in International Journal of Engineering Science, 2021

Computational multiscale methods for analyzing and deriving constitutive responses have been used as a tool in engineering problems because of their ability to combine information at different length scales. However, their application in a nonlinear framework can be limited by high computational costs, numerical difficulties, and/or inaccuracies. In this paper, a hybrid methodology is presented which combines classical constitutive laws (model-based), a data-driven correction component, and computational multiscale approaches. A model-based material representation is locally improved with data from lower scales obtained by means of a nonlinear numerical homogenization procedure, leading to a model-data-driven approach. Therefore, macroscale simulations explicitly incorporate the true microscale response, maintaining the same level of accuracy that would be obtained with online micro-macro simulations but with a computational cost comparable to classical model-driven approaches. In the proposed approach, both model and data play a fundamental role allowing for the synergistic integration between a physics-based response and a machine learning black-box. Numerical applications are implemented in two dimensions for different tests investigating both material and structural responses in large deformations. Overall, the presented model-data-driven methodology proves to be more versatile and accurate than methods based on classical model-driven, as well as pure data-driven techniques. In particular, a lower number of training samples is required and robustness is higher than for simulations which solely rely on data. © 2021 Elsevier Ltd

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A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial networks

Published in Nature Computational Science, 2021

Here we employ and adapt the image-to-image translation concept based on conditional generative adversarial networks (cGAN) for learning a forward and an inverse solution operator of partial differential equations (PDEs). We focus on steady-state solutions of coupled hydromechanical processes in heterogeneous porous media and present the parameterization of the spatially heterogeneous coefficients, which is exceedingly difficult using standard reduced-order modeling techniques. We show that our framework provides a speed-up of at least 2,000 times compared to a finite-element solver and achieves a relative root-mean-square error (r.m.s.e.) of less than 2% for forward modeling. For inverse modeling, the framework estimates the heterogeneous coefficients, given an input of pressure and/or displacement fields, with a relative r.m.s.e. of less than 7%, even for cases where the input data are incomplete and contaminated by noise. The framework also provides a speed-up of 120,000 times compared to a Gaussian prior-based inverse modeling approach while also delivering more accurate results. © 2021, The Author(s), under exclusive licence to Springer Nature America, Inc.

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A machine learning based plasticity model using proper orthogonal decomposition

Published in Computer Methods in Applied Mechanics and Engineering, 2020

Data-driven material models have many advantages over classical numerical approaches, such as the direct utilization of experimental data and the possibility to improve performance of predictions when additional data is available. One approach to develop a data-driven material model is to use machine learning tools. These can be trained offline to fit an observed material behaviour and then be applied in online applications. However, learning and predicting history dependent material models, such as plasticity, is still challenging. In this work, a machine learning based material modelling framework is proposed for both elasticity and plasticity. The machine learning based hyperelasticity model is developed with the Feed forward Neural Network (FNN) directly whereas the machine learning based plasticity model is developed by using of a novel method called Proper Orthogonal Decomposition Feed forward Neural Network (PODFNN). In order to account for the loading history, the accumulated absolute strain is proposed to be the history variable of the plasticity model. Additionally, the strain–stress sequence data for plasticity is collected from different loading–unloading paths based on the concept of sequence for plasticity. By means of the POD, the multi-dimensional stress sequence is decoupled leading to independent one dimensional coefficient sequences. In this case, the neural network with multiple output is replaced by multiple independent neural networks each possessing a one-dimensional output, which leads to less training time and better training performance. To apply the machine learning based material model in finite element analysis, the tangent matrix is derived by the automatic symbolic differentiation tool AceGen. The effectiveness and generalization of the presented models are investigated by a series of numerical examples using both 2D and 3D finite element analysis. © 2020 Elsevier B.V.

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Surrogate model approach for investigating the stability of a friction-induced oscillator of Duffing’s type

Published in Nonlinear Dynamics, 2019

Parametric studies are required to detect instability regimes of dynamic systems. This prediction can be computationally demanding as it requires a fine exploration of large parametric space due to the disrupted mechanical behavior. In this paper, an efficient surrogate strategy is proposed to investigate the behavior of an oscillator of Duffing’s type in combination with an elasto-plastic friction force model. Relevant quantities of interest are discussed. Sticking time is considered using a machine learning technique based on Gaussian processes called kriging. The largest Lyapunov exponent is considered as an efficient indicator of chaotic motion. This indicator is estimated using a perturbation method. A dedicated adaptive kriging strategy for classification called MiVor is utilized and appears to be highly proficient in order to detect instabilities over the parametric space and can furthermore be used for complex response surfaces in multi-dimensional parametric domains. © 2019, Springer Nature B.V.

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Jan N. Fuhg

Publications