Pub Date : 2025-01-29DOI: 10.1016/j.cma.2025.117782
Rami Cassia, Rich Kerswell
Machine learning techniques are being used as an alternative to traditional numerical discretization methods for solving hyperbolic partial differential equations (PDEs) relevant to fluid flow. Whilst numerical methods are higher fidelity, they are computationally expensive. Machine learning methods on the other hand are lower fidelity but can provide significant speed-ups. The emergence of physics-informed neural networks (PINNs) in fluid dynamics has allowed scientists to directly use PDEs for evaluating loss functions. The downfall of this approach is that the differential form of systems is invalid at regions of shock inherent in hyperbolic PDEs such as the compressible Euler equations. To circumvent this problem we propose the Godunov loss function: a loss based on the finite volume method (FVM) that crucially incorporates the flux of Godunov-type methods. These Godunov-type methods are also known as approximate Riemann solvers and evaluate intercell fluxes in an entropy-satisfying and non-oscillatory manner, yielding more physically accurate shocks. Our approach leads to superior performance compared to standard PINNs that use regularized PDE-based losses as well as FVM-based losses, as tested on the 2D Riemann problem in the context of time-stepping and super-resolution reconstruction.
{"title":"Godunov loss functions for modelling of hyperbolic conservation laws","authors":"Rami Cassia, Rich Kerswell","doi":"10.1016/j.cma.2025.117782","DOIUrl":"10.1016/j.cma.2025.117782","url":null,"abstract":"<div><div>Machine learning techniques are being used as an alternative to traditional numerical discretization methods for solving hyperbolic partial differential equations (PDEs) relevant to fluid flow. Whilst numerical methods are higher fidelity, they are computationally expensive. Machine learning methods on the other hand are lower fidelity but can provide significant speed-ups. The emergence of physics-informed neural networks (PINNs) in fluid dynamics has allowed scientists to directly use PDEs for evaluating loss functions. The downfall of this approach is that the differential form of systems is invalid at regions of shock inherent in hyperbolic PDEs such as the compressible Euler equations. To circumvent this problem we propose the Godunov loss function: a loss based on the finite volume method (FVM) that crucially incorporates the flux of Godunov-type methods. These Godunov-type methods are also known as approximate Riemann solvers and evaluate intercell fluxes in an entropy-satisfying and non-oscillatory manner, yielding more physically accurate shocks. Our approach leads to superior performance compared to standard PINNs that use regularized PDE-based losses as well as FVM-based losses, as tested on the 2D Riemann problem in the context of time-stepping and super-resolution reconstruction.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117782"},"PeriodicalIF":6.9,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-29DOI: 10.1016/j.cma.2025.117779
Stephan Wulfinghoff
In computational homogenization, the microscopic problem is regularly solved via Galerkin-projection methods to speed up the computation. By evaluating the involved integrals by hyper-reduction techniques, a very high efficiency can be achieved. Here, a novel hyper-reduction method is proposed and applied to magnetostatics. The method combines the ideas of microstructural clustering with the empirical identification/correction of a reduced set of integration points, not being taken from the set of finite element integration points. The results show that the macroscopic response (2D) is hardly distinguishable from the finite element results already for 12 integration points at a phase contrast of 1000 for a porous microstructure. The online costs (but also the offline costs) are thus found to be particularly low. Further, a two-scale example is discussed and the code is made available online.
{"title":"Empirically corrected cluster cubature (E3C)","authors":"Stephan Wulfinghoff","doi":"10.1016/j.cma.2025.117779","DOIUrl":"10.1016/j.cma.2025.117779","url":null,"abstract":"<div><div>In computational homogenization, the microscopic problem is regularly solved via Galerkin-projection methods to speed up the computation. By evaluating the involved integrals by hyper-reduction techniques, a very high efficiency can be achieved. Here, a novel hyper-reduction method is proposed and applied to magnetostatics. The method combines the ideas of microstructural clustering with the empirical identification/correction of a reduced set of integration points, <em>not</em> being taken from the set of finite element integration points. The results show that the macroscopic response (2D) is hardly distinguishable from the finite element results already for 12 integration points at a phase contrast of <span><math><mo>∼</mo></math></span>1000 for a porous microstructure. The online costs (but also the offline costs) are thus found to be particularly low. Further, a two-scale example is discussed and the code is made available online.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117779"},"PeriodicalIF":6.9,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-29DOI: 10.1016/j.cma.2025.117761
Shengjun Liu, Hanchao Liu, Ting Zhang, Xinru Liu
Geometric partial differential equations (geometric PDEs) are defined on manifolds in Riemannian space, specifically tailored for modeling the temporal evolution of surfaces in natural sciences and engineering. For varying initial surfaces (initial conditions), traditional numerical methods require re-solving the equation even for the same geometric PDE, which significantly hinders the efficiency of simulations. The efficient predictive capabilities of neural networks (NNs) makes them a powerful tool for solving differential equations. The solution of geometric PDEs governs the continuous evolution of the surface over time, making it challenging for most NNs to handle solution prediction for geometric PDEs with varying initial surfaces. We propose a novel neural operator-based framework for solving geometric PDEs. Once trained, our model can predict the solution of the same geometric PDE under arbitrary initial conditions (initial surfaces). To the best of our knowledge, this is the first attempt to solve geometric PDEs using the neural operator. Firstly, we employ a learned continuous Signed Distance Function (SDF) representation method (DeepSDF) to convert the initial mesh surface into an implicit level-set representation, thereby avoiding the difficulties associated with solving explicit geometric PDEs. Subsequently, by integrating the multi-scale module, we design a Multi-Scale Implicit U-Net enhanced Factorized Fourier Neural Operator (MS-IUFFNO) for solving implicit geometric PDEs. The innovative structure of the neural operator substantially improves the prediction accuracy and long-term stability for solving geometric PDEs with reduced computational complexity. In addition, we construct datasets to train neural operators to solve the mean curvature flow and Willmore flow, which are representative of geometric PDEs. Finally, a numerical benchmark is conducted to compare MS-IUFFNO to several classical neural operator models for solving the mean curvature flow and Willmore flow, where results show that our model exhibits superior performance in terms of prediction accuracy, extrapolation capability, and stability.
{"title":"MS-IUFFNO: Multi-scale implicit U-net enhanced factorized fourier neural operator for solving geometric PDEs","authors":"Shengjun Liu, Hanchao Liu, Ting Zhang, Xinru Liu","doi":"10.1016/j.cma.2025.117761","DOIUrl":"10.1016/j.cma.2025.117761","url":null,"abstract":"<div><div>Geometric partial differential equations (geometric PDEs) are defined on manifolds in Riemannian space, specifically tailored for modeling the temporal evolution of surfaces in natural sciences and engineering. For varying initial surfaces (initial conditions), traditional numerical methods require re-solving the equation even for the same geometric PDE, which significantly hinders the efficiency of simulations. The efficient predictive capabilities of neural networks (NNs) makes them a powerful tool for solving differential equations. The solution of geometric PDEs governs the continuous evolution of the surface over time, making it challenging for most NNs to handle solution prediction for geometric PDEs with varying initial surfaces. We propose a novel neural operator-based framework for solving geometric PDEs. Once trained, our model can predict the solution of the same geometric PDE under arbitrary initial conditions (initial surfaces). To the best of our knowledge, this is the first attempt to solve geometric PDEs using the neural operator. Firstly, we employ a learned continuous Signed Distance Function (SDF) representation method (DeepSDF) to convert the initial mesh surface into an implicit level-set representation, thereby avoiding the difficulties associated with solving explicit geometric PDEs. Subsequently, by integrating the multi-scale module, we design a Multi-Scale Implicit U-Net enhanced Factorized Fourier Neural Operator (MS-IUFFNO) for solving implicit geometric PDEs. The innovative structure of the neural operator substantially improves the prediction accuracy and long-term stability for solving geometric PDEs with reduced computational complexity. In addition, we construct datasets to train neural operators to solve the mean curvature flow and Willmore flow, which are representative of geometric PDEs. Finally, a numerical benchmark is conducted to compare MS-IUFFNO to several classical neural operator models for solving the mean curvature flow and Willmore flow, where results show that our model exhibits superior performance in terms of prediction accuracy, extrapolation capability, and stability.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117761"},"PeriodicalIF":6.9,"publicationDate":"2025-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-28DOI: 10.1016/j.cma.2025.117753
Xianghao Meng , James L. Beck , Yong Huang , Hui Li
In the last few decades, Markov chain Monte Carlo (MCMC) methods have been widely applied to Bayesian updating of structural dynamic models in the field of structural health monitoring. Recently, several MCMC algorithms have been developed that incorporate neural networks to enhance their performance for specific Bayesian model updating problems. However, a common challenge with these approaches lies in the fact that the embedded neural networks often necessitate retraining when faced with new tasks, a process that is time-consuming and significantly undermines the competitiveness of these methods. This paper introduces a newly developed adaptive meta-learning stochastic gradient Hamiltonian Monte Carlo (AM-SGHMC) algorithm. The idea behind AM-SGHMC is to optimize the sampling strategy by training adaptive neural networks, and due to the adaptive design of the network inputs and outputs, the trained sampler can be directly applied to various Bayesian updating problems of the same type of structure without further training, thereby achieving meta-learning. Additionally, practical issues for the feasibility of the AM-SGHMC algorithm for structural dynamic model updating are addressed, and two examples involving Bayesian updating of multi-story building models with different model fidelity are used to demonstrate the effectiveness and generalization ability of the proposed method.
{"title":"Adaptive meta-learning stochastic gradient Hamiltonian Monte Carlo simulation for Bayesian updating of structural dynamic models","authors":"Xianghao Meng , James L. Beck , Yong Huang , Hui Li","doi":"10.1016/j.cma.2025.117753","DOIUrl":"10.1016/j.cma.2025.117753","url":null,"abstract":"<div><div>In the last few decades, Markov chain Monte Carlo (MCMC) methods have been widely applied to Bayesian updating of structural dynamic models in the field of structural health monitoring. Recently, several MCMC algorithms have been developed that incorporate neural networks to enhance their performance for specific Bayesian model updating problems. However, a common challenge with these approaches lies in the fact that the embedded neural networks often necessitate retraining when faced with new tasks, a process that is time-consuming and significantly undermines the competitiveness of these methods. This paper introduces a newly developed adaptive meta-learning stochastic gradient Hamiltonian Monte Carlo (AM-SGHMC) algorithm. The idea behind AM-SGHMC is to optimize the sampling strategy by training adaptive neural networks, and due to the adaptive design of the network inputs and outputs, the trained sampler can be directly applied to various Bayesian updating problems of the same type of structure without further training, thereby achieving meta-learning. Additionally, practical issues for the feasibility of the AM-SGHMC algorithm for structural dynamic model updating are addressed, and two examples involving Bayesian updating of multi-story building models with different model fidelity are used to demonstrate the effectiveness and generalization ability of the proposed method.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117753"},"PeriodicalIF":6.9,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-28DOI: 10.1016/j.cma.2025.117777
Han Dong , Hongjiang Wang , Jiahao Zhong , Chaohui Huang , Weizhe Wang , Yingzheng Liu
A reduced-order peridynamic (PD) model is developed to accelerate fracture simulations of composite materials. This reduced-order PD model is constructed based on a set of projection basis functions extracted from the flexibility matrix corresponding to the initial configuration, rather than from snapshots. Thus, this approach eliminates dependence on datasets with prior knowledge, resulting in superior generalization. During the calculation, the projection basis functions are adaptively updated with the damage evolution. Several two- and three-dimensional numerical examples involving the fracture of composites are investigated to validate the numerical accuracy and computational efficiency of the model. The proposed model accurately captures various fracture characteristics while significantly improving the computational efficiency. This work presents a feasible approach for accelerating fracture simulations, which is of great significance for shortening the design cycle of composite materials and enhancing the efficiency of failure analysis.
{"title":"A snapshot-free reduced-order peridynamic model for accelerating fracture analysis of composites","authors":"Han Dong , Hongjiang Wang , Jiahao Zhong , Chaohui Huang , Weizhe Wang , Yingzheng Liu","doi":"10.1016/j.cma.2025.117777","DOIUrl":"10.1016/j.cma.2025.117777","url":null,"abstract":"<div><div>A reduced-order peridynamic (PD) model is developed to accelerate fracture simulations of composite materials. This reduced-order PD model is constructed based on a set of projection basis functions extracted from the flexibility matrix corresponding to the initial configuration, rather than from snapshots. Thus, this approach eliminates dependence on datasets with prior knowledge, resulting in superior generalization. During the calculation, the projection basis functions are adaptively updated with the damage evolution. Several two- and three-dimensional numerical examples involving the fracture of composites are investigated to validate the numerical accuracy and computational efficiency of the model. The proposed model accurately captures various fracture characteristics while significantly improving the computational efficiency. This work presents a feasible approach for accelerating fracture simulations, which is of great significance for shortening the design cycle of composite materials and enhancing the efficiency of failure analysis.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117777"},"PeriodicalIF":6.9,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143072093","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-27DOI: 10.1016/j.cma.2025.117736
Riccardo Pellegrini , Zhaoyuan Wang , Frederick Stern , Matteo Diez
This paper introduces a nonlinear structural reduced order model (ROM) specifically developed for fluid–structure interaction (FSI) simulations involving high impact loads and large deflections, such as those arising in water slamming of flexible structures. The model is based on a nonlinear modal expansion trained offline using prestressed eigenfrequency analyses performed by nonlinear full-order computational structural dynamics based on finite elements. The training uses the eigenfrequencies as a function of the deflection and is non-intrusive, which means that the knowledge of the system’s full-order matrices is not required. Eigenfrequencies and deflections are evaluated under a prescribed set of static loads, which are derived from fully transient computational fluid dynamics (CFD) simulations. The resulting ROM is coupled with CFD using partitioned one- and two-way FSI schemes. Focusing on the impact of an elastic aluminum plate onto still water, the research investigates scenarios with varied horizontal and vertical velocities in three distinct experimental conditions, which cover moderate to strong hydroelastic interactions. Namely, the proposed nonlinear ROM and its linear counterpart are assessed against two FSI benchmark sets. The first set consists in comparing the ROM versus the full-order model (FOM) under prescribed external load, via one-way FSI coupling. The second set consists in comparing the ROM versus experimental data, via two-way tightly-coupled FSI. Comparisons of the nonlinear ROM versus the FOM under prescribed loads achieve an average error equal to 2.7%. Comparisons of the nonlinear ROM under two-way tightly-coupled FSI versus experiments show an average error equal to 4.5%. Comparisons of nonlinear versus linear ROM highlight the need for nonlinear models to accurately capture peak values and trends, especially in scenarios with large deflections.
{"title":"A non-intrusive nonlinear structural ROM for partitioned two-way fluid–structure interaction computations","authors":"Riccardo Pellegrini , Zhaoyuan Wang , Frederick Stern , Matteo Diez","doi":"10.1016/j.cma.2025.117736","DOIUrl":"10.1016/j.cma.2025.117736","url":null,"abstract":"<div><div>This paper introduces a nonlinear structural reduced order model (ROM) specifically developed for fluid–structure interaction (FSI) simulations involving high impact loads and large deflections, such as those arising in water slamming of flexible structures. The model is based on a nonlinear modal expansion trained offline using prestressed eigenfrequency analyses performed by nonlinear full-order computational structural dynamics based on finite elements. The training uses the eigenfrequencies as a function of the deflection and is non-intrusive, which means that the knowledge of the system’s full-order matrices is not required. Eigenfrequencies and deflections are evaluated under a prescribed set of static loads, which are derived from fully transient computational fluid dynamics (CFD) simulations. The resulting ROM is coupled with CFD using partitioned one- and two-way FSI schemes. Focusing on the impact of an elastic aluminum plate onto still water, the research investigates scenarios with varied horizontal and vertical velocities in three distinct experimental conditions, which cover moderate to strong hydroelastic interactions. Namely, the proposed nonlinear ROM and its linear counterpart are assessed against two FSI benchmark sets. The first set consists in comparing the ROM versus the full-order model (FOM) under prescribed external load, via one-way FSI coupling. The second set consists in comparing the ROM versus experimental data, via two-way tightly-coupled FSI. Comparisons of the nonlinear ROM versus the FOM under prescribed loads achieve an average error equal to 2.7%. Comparisons of the nonlinear ROM under two-way tightly-coupled FSI versus experiments show an average error equal to 4.5%. Comparisons of nonlinear versus linear ROM highlight the need for nonlinear models to accurately capture peak values and trends, especially in scenarios with large deflections.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117736"},"PeriodicalIF":6.9,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-27DOI: 10.1016/j.cma.2024.117698
Amin Yousefpour, Shirin Hosseinmardi, Carlos Mora, Ramin Bostanabad
Topology optimization (TO) provides a principled mathematical approach for optimizing the performance of a structure by designing its material spatial distribution in a pre-defined domain and subject to a set of constraints. The majority of existing TO approaches have a nested nature, and leverage numerical solvers for design evaluations during the optimization and hence rely on discretizing the design and state variables. Contrary to these approaches, herein we develop a new class of TO methods based on the framework of Gaussian processes (GPs) whose mean functions are parameterized via deep neural networks. Specifically, we place GP priors on all design and state variables to represent them via parameterized continuous functions. These GPs share a deep neural network as their mean function but have as many independent kernels as there are state and design variables. We estimate all the parameters of our model in a single optimization loop that optimizes a penalized version of the performance metric where the penalty terms correspond to the state equations and design constraints. Attractive features of our approach include having a built-in continuation nature since the performance metric is optimized at the same time that the state equations are solved, and being discretization-invariant and accommodating complex domains and topologies. To test our method against conventional TO approaches implemented in commercial software, we evaluate it on canonical problems involving the minimization of dissipated power in Stokes flow. The results indicate that our approach does not need filtering techniques, has consistent computational costs, and is highly robust against random initializations and problem setup.
{"title":"Simultaneous and meshfree topology optimization with physics-informed Gaussian processes","authors":"Amin Yousefpour, Shirin Hosseinmardi, Carlos Mora, Ramin Bostanabad","doi":"10.1016/j.cma.2024.117698","DOIUrl":"10.1016/j.cma.2024.117698","url":null,"abstract":"<div><div>Topology optimization (TO) provides a principled mathematical approach for optimizing the performance of a structure by designing its material spatial distribution in a pre-defined domain and subject to a set of constraints. The majority of existing TO approaches have <span><math><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></math></span> a nested nature, and <span><math><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></math></span> leverage numerical solvers for design evaluations during the optimization and hence rely on discretizing the design and state variables. Contrary to these approaches, herein we develop a new class of TO methods based on the framework of Gaussian processes (GPs) whose mean functions are parameterized via deep neural networks. Specifically, we place GP priors on all design and state variables to represent them via parameterized continuous functions. These GPs share a deep neural network as their mean function but have as many independent kernels as there are state and design variables. We estimate all the parameters of our model in a single optimization loop that optimizes a penalized version of the performance metric where the penalty terms correspond to the state equations and design constraints. Attractive features of our approach include <span><math><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></math></span> having a built-in continuation nature since the performance metric is optimized at the same time that the state equations are solved, and <span><math><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></math></span> being discretization-invariant and accommodating complex domains and topologies. To test our method against conventional TO approaches implemented in commercial software, we evaluate it on canonical problems involving the minimization of dissipated power in Stokes flow. The results indicate that our approach does not need filtering techniques, has consistent computational costs, and is highly robust against random initializations and problem setup.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117698"},"PeriodicalIF":6.9,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-27DOI: 10.1016/j.cma.2025.117741
R. Ortigosa , J. Martínez-Frutos , A. Pérez-Escolar , I. Castañar , N. Ellmer , A.J. Gil
This manuscript introduces a novel neural network-based computational framework for constitutive modeling of thermo-electro-mechanically coupled materials at finite strains, with four key innovations: (i) It supports calibration of neural network models with various input forms, such as , , , or , with representing the deformation gradient tensor, and the electric field and electric displacement field, respectively and finally, and , the temperature and entropy fields. These models comply with physical laws and material symmetries by utilizing isotropic or anisotropic invariants corresponding to the material’s symmetry group. (ii) A calibration approach is developed for the case of experimental data, where entropy is typically unmeasurable. (iii) The framework accommodates models like , specially convenient for the imposition of polyconvexity across the three physics involved. A detailed calibration study is conducted evaluating various neural network architectures and considering a large variety of ground truth thermo-electro-mechanical constitutive models. The results demonstrate excellent predictive performance on larger datasets, validated through complex finite element simulations using both ground truth and neural network-based models. Crucially, the framework can be straightforwardly extended to scenarios involving other physics.
{"title":"A generalized theory for physics-augmented neural networks in finite strain thermo-electro-mechanics","authors":"R. Ortigosa , J. Martínez-Frutos , A. Pérez-Escolar , I. Castañar , N. Ellmer , A.J. Gil","doi":"10.1016/j.cma.2025.117741","DOIUrl":"10.1016/j.cma.2025.117741","url":null,"abstract":"<div><div>This manuscript introduces a novel neural network-based computational framework for constitutive modeling of thermo-electro-mechanically coupled materials at finite strains, with four key innovations: (i) It supports calibration of neural network models with various input forms, such as <span><math><mrow><msub><mrow><mi>Ψ</mi></mrow><mrow><mi>n</mi><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>θ</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>n</mi><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>η</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>Υ</mi></mrow><mrow><mi>n</mi><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>,</mo><msub><mrow><mi>E</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>η</mi><mo>)</mo></mrow></mrow></math></span>, or <span><math><mrow><msub><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>θ</mi><mo>)</mo></mrow></mrow></math></span>, with <span><math><mi>F</mi></math></span> representing the deformation gradient tensor, <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> the electric field and electric displacement field, respectively and finally, <span><math><mi>θ</mi></math></span> and <span><math><mi>η</mi></math></span>, the temperature and entropy fields. These models comply with physical laws and material symmetries by utilizing isotropic or anisotropic invariants corresponding to the material’s symmetry group. (ii) A calibration approach is developed for the case of experimental data, where entropy <span><math><mi>η</mi></math></span> is typically unmeasurable. (iii) The framework accommodates models like <span><math><mrow><msub><mrow><mi>e</mi></mrow><mrow><mi>n</mi><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>F</mi><mo>,</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mi>η</mi><mo>)</mo></mrow></mrow></math></span>, specially convenient for the imposition of polyconvexity across the three physics involved. A detailed calibration study is conducted evaluating various neural network architectures and considering a large variety of ground truth thermo-electro-mechanical constitutive models. The results demonstrate excellent predictive performance on larger datasets, validated through complex finite element simulations using both ground truth and neural network-based models. Crucially, the framework can be straightforwardly extended to scenarios involving other physics.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117741"},"PeriodicalIF":6.9,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143071663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-27DOI: 10.1016/j.cma.2025.117757
Shima Baharlouei , Jamie M. Taylor , Carlos Uriarte , David Pardo
Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It utilizes a separated representation form for the parametric PDE solution via a deep neural network and a least-squares solver. In this approach, the output of the deep neural network consists of a vector-valued function, interpreted as basis functions for the parametric solution space, and the least-squares solver determines the optimal solution within the constructed solution space for each given parameter. The LS-Net method requires a quadratic loss function for the least-squares solver to find optimal solutions given the set of basis functions. In this study, we consider loss functions derived from the Deep Fourier Residual and Physics-Informed Neural Networks approaches. We also provide theoretical results similar to the Universal Approximation Theorem, stating that there exists a sufficiently large neural network that can theoretically approximate solutions of parametric PDEs with the desired accuracy. We illustrate the LS-net method by solving one- and two-dimensional problems. Numerical results clearly demonstrate the method’s ability to approximate parametric solutions.
{"title":"A Least-Squares-Based Neural Network (LS-Net) for Solving Linear Parametric PDEs","authors":"Shima Baharlouei , Jamie M. Taylor , Carlos Uriarte , David Pardo","doi":"10.1016/j.cma.2025.117757","DOIUrl":"10.1016/j.cma.2025.117757","url":null,"abstract":"<div><div>Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It utilizes a separated representation form for the parametric PDE solution via a deep neural network and a least-squares solver. In this approach, the output of the deep neural network consists of a vector-valued function, interpreted as basis functions for the parametric solution space, and the least-squares solver determines the optimal solution within the constructed solution space for each given parameter. The LS-Net method requires a quadratic loss function for the least-squares solver to find optimal solutions given the set of basis functions. In this study, we consider loss functions derived from the Deep Fourier Residual and Physics-Informed Neural Networks approaches. We also provide theoretical results similar to the Universal Approximation Theorem, stating that there exists a sufficiently large neural network that can theoretically approximate solutions of parametric PDEs with the desired accuracy. We illustrate the LS-net method by solving one- and two-dimensional problems. Numerical results clearly demonstrate the method’s ability to approximate parametric solutions.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117757"},"PeriodicalIF":6.9,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143072094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-01-27DOI: 10.1016/j.cma.2025.117745
Teo Lara , Ken Kamrin
We introduce a general simulation approach to model fluid-submerged solid contact of highly deformable objects within the Eulerian Incompressible Reference Map Technique (RMT) for fluid-solid interaction. Our approach allows solid bodies to undergo finite deformations, contact, and, importantly, self-contact while immersed in a fluid satisfying the Navier–Stokes equations. All solid boundaries are modeled using a single levelset field, which is used to produce a key secondary field that identifies when opposing surfaces approach. This secondary field can then be used to create the appropriate contact penalty forces allowing treatment of both multi-body and self-contact under the same algorithm. The method also provides modularity, being directly integrable within the current RMT framework, conserving the desirable properties of the RMT. This technique is demonstrated in multiple cases. We simulate submerged contact of two elastic disks represented with a single levelset, showing that the method approaches the analytical Hertzian prediction for contact between cylinders along their parallel axes. A grid resolution study confirms the convergence of the method. We also model a more intricate example, with several highly deformable hyperelastic objects undergoing simultaneous self- and multi-body contact as they settle within a fluid due to gravity.
{"title":"Unified Eulerian method for fluid-immersed self- and multi-body solid contact","authors":"Teo Lara , Ken Kamrin","doi":"10.1016/j.cma.2025.117745","DOIUrl":"10.1016/j.cma.2025.117745","url":null,"abstract":"<div><div>We introduce a general simulation approach to model fluid-submerged solid contact of highly deformable objects within the Eulerian Incompressible Reference Map Technique (RMT) for fluid-solid interaction. Our approach allows solid bodies to undergo finite deformations, contact, and, importantly, <em>self-contact</em> while immersed in a fluid satisfying the Navier–Stokes equations. All solid boundaries are modeled using a single levelset field, which is used to produce a key secondary field that identifies when opposing surfaces approach. This secondary field can then be used to create the appropriate contact penalty forces allowing treatment of both multi-body and self-contact under the same algorithm. The method also provides modularity, being directly integrable within the current RMT framework, conserving the desirable properties of the RMT. This technique is demonstrated in multiple cases. We simulate submerged contact of two elastic disks represented with a single levelset, showing that the method approaches the analytical Hertzian prediction for contact between cylinders along their parallel axes. A grid resolution study confirms the convergence of the method. We also model a more intricate example, with several highly deformable hyperelastic objects undergoing simultaneous self- and multi-body contact as they settle within a fluid due to gravity.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"437 ","pages":"Article 117745"},"PeriodicalIF":6.9,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143072294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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