Pub Date : 2026-01-29DOI: 10.1016/j.cma.2026.118784
Wen-Chia Yang , Deborah L. Sulsky
This study investigates the numerical stability of the Material Point Method (MPM) by analyzing the energy behavior and spectral properties of single-step updates. An energy-based analysis is first performed to quantify the energy variation introduced during each time step, followed by a spectral analysis that identifies critical time step constraints through the amplification matrix. Closed-form expressions for the critical integration parameter and the non-dimensional critical time step are derived, highlighting their dependence on mass parameters and particle distribution. The stability analyses identify key factors affecting the stability limits in MPM, including mass matrix selection, velocity projection, partially occupied grid cells, and integration errors in the particle-based formulation. Numerical experiments validate the analytical predictions and reveal the influence of particle-based integration errors on stability. A simple stabilization coefficient is proposed, which modifies shape function gradients in partially filled edge cells, significantly extending the stable time step range without increasing computational cost. The proposed framework offers practical guidelines for selecting stable time steps and enhancing the robustness of MPM simulations.
{"title":"Stability analyses and instability mitigation for the material point method","authors":"Wen-Chia Yang , Deborah L. Sulsky","doi":"10.1016/j.cma.2026.118784","DOIUrl":"10.1016/j.cma.2026.118784","url":null,"abstract":"<div><div>This study investigates the numerical stability of the Material Point Method (MPM) by analyzing the energy behavior and spectral properties of single-step updates. An energy-based analysis is first performed to quantify the energy variation introduced during each time step, followed by a spectral analysis that identifies critical time step constraints through the amplification matrix. Closed-form expressions for the critical integration parameter and the non-dimensional critical time step are derived, highlighting their dependence on mass parameters and particle distribution. The stability analyses identify key factors affecting the stability limits in MPM, including mass matrix selection, velocity projection, partially occupied grid cells, and integration errors in the particle-based formulation. Numerical experiments validate the analytical predictions and reveal the influence of particle-based integration errors on stability. A simple stabilization coefficient is proposed, which modifies shape function gradients in partially filled edge cells, significantly extending the stable time step range without increasing computational cost. The proposed framework offers practical guidelines for selecting stable time steps and enhancing the robustness of MPM simulations.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"452 ","pages":"Article 118784"},"PeriodicalIF":7.3,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146072452","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 : 2026-01-29DOI: 10.1016/j.cma.2026.118775
Catherine Choquet, Théo Coiffard
This paper presents a unified numerical modeling framework for simulating fluid flow across heterogeneous media and multiple flow regimes, from very low-velocity porous flows to free-fluid Navier-Stokes regimes. The proposed approach builds upon the Lattice-Boltzmann (LB) method, exploiting its kinetic formulation and inherent multiscale character. Unlike conventional continuum models that rely on distinct partial differential equations (Darcy, Brinkman, Forchheimer, or Navier-Stokes) and require complex coupling strategies at interfaces, the present scheme introduces a scaling parameter (with ϵ the Knudsen number and ) to incorporate the effects of both microscopic structure and observation scale within a single LB formulation. We show that adjusting α, even abruptly, enables simulations in highly heterogeneous media without invoking separate PDE models and interface conditions, or introducing ad hoc force terms. Theoretical analysis based on Chapman-Enskog expansions demonstrates that the proposed LB scheme recovers well-known continuum (PDE) limits under appropriate scaling. Numerical benchmarks validate its accuracy and stability across Darcy, Brinkman, Forchheimer, and Stokes regimes, as well as intermediate transitions, confirming the potential of the method as a fully kinetic and genuinely multiscale alternative to traditional PDE-based approaches.
{"title":"A multiscale lattice Boltzmann model for simulating Stokes to pre-Darcy flow","authors":"Catherine Choquet, Théo Coiffard","doi":"10.1016/j.cma.2026.118775","DOIUrl":"10.1016/j.cma.2026.118775","url":null,"abstract":"<div><div>This paper presents a unified numerical modeling framework for simulating fluid flow across heterogeneous media and multiple flow regimes, from very low-velocity porous flows to free-fluid Navier-Stokes regimes. The proposed approach builds upon the Lattice-Boltzmann (LB) method, exploiting its kinetic formulation and inherent multiscale character. Unlike conventional continuum models that rely on distinct partial differential equations (Darcy, Brinkman, Forchheimer, or Navier-Stokes) and require complex coupling strategies at interfaces, the present scheme introduces a scaling parameter <span><math><mrow><mi>θ</mi><mo>=</mo><msup><mi>ϵ</mi><mi>α</mi></msup></mrow></math></span> (with ϵ the Knudsen number and <span><math><mrow><mi>α</mi><mo>∈</mo><msub><mi>R</mi><mo>+</mo></msub></mrow></math></span>) to incorporate the effects of both microscopic structure and observation scale within a single LB formulation. We show that adjusting <em>α</em>, even abruptly, enables simulations in highly heterogeneous media without invoking separate PDE models and interface conditions, or introducing <em>ad hoc</em> force terms. Theoretical analysis based on Chapman-Enskog expansions demonstrates that the proposed LB scheme recovers well-known continuum (PDE) limits under appropriate scaling. Numerical benchmarks validate its accuracy and stability across Darcy, Brinkman, Forchheimer, and Stokes regimes, as well as intermediate transitions, confirming the potential of the method as a fully kinetic and genuinely multiscale alternative to traditional PDE-based approaches.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"452 ","pages":"Article 118775"},"PeriodicalIF":7.3,"publicationDate":"2026-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146071613","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 : 2026-01-28DOI: 10.1016/j.cma.2026.118742
Chun Hean Lee , Antonio J. Gil , Tadas Jaugielavičius , Thomas Richardson , Sébastien Boyaval , Damien Violeau , Javier Bonet
This paper presents a new first-order hyperbolic framework with relaxation (or dissipation) terms for large strain viscoelastic solids. The framework is based on a compressible Maxwell-type viscoelastic model and integrates linear momentum conservation, geometric conservation laws, and evolution equations for internal variables. First, we propose a polyconvex strain energy function that is jointly convex with respect to the deformation measures and internal variables. Second, we introduce a generalised convex entropy function to symmetrise the hyperbolic system in terms of dual conjugate (entropy) variables. Third, we demonstrate that the system is hyperbolic (i.e., real wave speeds) under all deformation states, and that the relaxation terms correctly capture viscoelastic dissipation. Fourth, we present an upwinding Smoothed Particle Hydrodynamics (SPH) [1–3] scheme that enforces the second law of thermodynamics semi-discretely and uses the time rate of the generalised convex entropy to monitor internal dissipation and stabilise the simulation. Finally, the proposed framework is validated through numerical examples and benchmarked against the in-house Updated Reference Lagragian SPH [2,3] and vertex-centred finite volume [4–7] algorithms, demonstrating stability, accuracy, and consistent energy dissipation.
{"title":"Symmetrisation and hyperbolicity of first-order conservation laws in large strain compressible viscoelasticity using the smoothed particle hydrodynamics method","authors":"Chun Hean Lee , Antonio J. Gil , Tadas Jaugielavičius , Thomas Richardson , Sébastien Boyaval , Damien Violeau , Javier Bonet","doi":"10.1016/j.cma.2026.118742","DOIUrl":"10.1016/j.cma.2026.118742","url":null,"abstract":"<div><div>This paper presents a new first-order hyperbolic framework with relaxation (or dissipation) terms for large strain viscoelastic solids. The framework is based on a compressible Maxwell-type viscoelastic model and integrates linear momentum conservation, geometric conservation laws, and evolution equations for internal variables. First, we propose a polyconvex strain energy function that is jointly convex with respect to the deformation measures and internal variables. Second, we introduce a generalised convex entropy function to symmetrise the hyperbolic system in terms of dual conjugate (entropy) variables. Third, we demonstrate that the system is hyperbolic (i.e., real wave speeds) under all deformation states, and that the relaxation terms correctly capture viscoelastic dissipation. Fourth, we present an upwinding Smoothed Particle Hydrodynamics (SPH) [1–3] scheme that enforces the second law of thermodynamics semi-discretely and uses the time rate of the generalised convex entropy to monitor internal dissipation and stabilise the simulation. Finally, the proposed framework is validated through numerical examples and benchmarked against the in-house Updated Reference Lagragian SPH [2,3] and vertex-centred finite volume [4–7] algorithms, demonstrating stability, accuracy, and consistent energy dissipation.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"452 ","pages":"Article 118742"},"PeriodicalIF":7.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146071712","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 : 2026-01-28DOI: 10.1016/j.cma.2026.118774
Zakaria Chafia , Julien Yvonnet , Jérémy Bleyer
The prediction of the mechanical response of strongly heterogeneous structures containing defects critically depends on accurately capturing crack nucleation at micro scale. Fully resolved (high-fidelity) models are costly, whereas homogenized approaches may fail to represent initiation near heterogeneities. An efficient multiscale method is proposed in this work to simulate crack nucleation and propagation by bridging a high-fidelity micro-subdomain, dedicated to initiation, with a homogenized macro-subdomain used for propagation. The two subdomains overlap, may be discretized with nonconforming meshes, and are coupled through an energy-based formulation. The main contribution lies in the use, at the macro scale, of a surrogate anisotropic damage model constructed offline within the DDHAD (Data-Driven Harmonic Analysis of Damage) framework. This model reproduces direction-dependent crack propagation, while nucleation is resolved at the micro scale by the high-fidelity model. Significant computational speed-ups are achieved as compared to high-resolution simulations of the entire structure, and by accurately capturing initiation of the cracks in the microstructure. Examples on heterogeneous media exhibiting strong preferred crack orientations are presented to illustrate the potential of the approach.
{"title":"A bridging-domain approach for multiscale modeling of anisotropic fracture in large-scale heterogeneous structures","authors":"Zakaria Chafia , Julien Yvonnet , Jérémy Bleyer","doi":"10.1016/j.cma.2026.118774","DOIUrl":"10.1016/j.cma.2026.118774","url":null,"abstract":"<div><div>The prediction of the mechanical response of strongly heterogeneous structures containing defects critically depends on accurately capturing crack nucleation at micro scale. Fully resolved (high-fidelity) models are costly, whereas homogenized approaches may fail to represent initiation near heterogeneities. An efficient multiscale method is proposed in this work to simulate crack nucleation and propagation by bridging a high-fidelity micro-subdomain, dedicated to initiation, with a homogenized macro-subdomain used for propagation. The two subdomains overlap, may be discretized with nonconforming meshes, and are coupled through an energy-based formulation. The main contribution lies in the use, at the macro scale, of a surrogate anisotropic damage model constructed offline within the DDHAD (Data-Driven Harmonic Analysis of Damage) framework. This model reproduces direction-dependent crack propagation, while nucleation is resolved at the micro scale by the high-fidelity model. Significant computational speed-ups are achieved as compared to high-resolution simulations of the entire structure, and by accurately capturing initiation of the cracks in the microstructure. Examples on heterogeneous media exhibiting strong preferred crack orientations are presented to illustrate the potential of the approach.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"452 ","pages":"Article 118774"},"PeriodicalIF":7.3,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146072456","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 : 2026-01-27DOI: 10.1016/j.cma.2026.118769
Mario Kapl , Aljaž Kosmač , Vito Vitrih
We present a novel method for solving high-order partial differential equations (PDEs) over planar multi-patch geometries with possibly extraordinary vertices demonstrated on the basis of the polyharmonic equation of order m, m ≥ 1, which is a particular linear elliptic PDE of order 2m. Our approach is based on the concept of Isogeometric Tearing and Interconnecting (IETI) and allows to couple the numerical solution of the PDE with Cs-smoothness, , across the edges of the multi-patch geometry. The proposed technique relies on the use of a particular class of multi-patch geometries, called bilinear-like Gs multi-patch parameterizations, to represent the multi-patch domain. The coupling between the neighboring patches is done via the use of Lagrange multipliers and leads to a saddle point problem, which can be solved first by a small dual problem for a subset of the Lagrange multipliers followed by local, parallelizable problems on the single patches for the coefficients of the numerical solution. Several numerical examples for the polyharmonic equation of order , and , i.e. for the Poisson’s, the biharmonic and the triharmonic equation, respectively, are shown to demonstrate the potential of our IETI method for solving high-order problems over planar multi-patch geometries with possibly extraordinary vertices.
{"title":"An Isogeometric Tearing and Interconnecting (IETI) method for solving high order partial differential equations over planar multi-patch geometries","authors":"Mario Kapl , Aljaž Kosmač , Vito Vitrih","doi":"10.1016/j.cma.2026.118769","DOIUrl":"10.1016/j.cma.2026.118769","url":null,"abstract":"<div><div>We present a novel method for solving high-order partial differential equations (PDEs) over planar multi-patch geometries with possibly extraordinary vertices demonstrated on the basis of the polyharmonic equation of order <em>m, m</em> ≥ 1, which is a particular linear elliptic PDE of order 2<em>m</em>. Our approach is based on the concept of Isogeometric Tearing and Interconnecting (IETI) and allows to couple the numerical solution of the PDE with <em>C<sup>s</sup></em>-smoothness, <span><math><mrow><mi>s</mi><mo>≥</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></math></span>, across the edges of the multi-patch geometry. The proposed technique relies on the use of a particular class of multi-patch geometries, called bilinear-like <em>G<sup>s</sup></em> multi-patch parameterizations, to represent the multi-patch domain. The coupling between the neighboring patches is done via the use of Lagrange multipliers and leads to a saddle point problem, which can be solved first by a small dual problem for a subset of the Lagrange multipliers followed by local, parallelizable problems on the single patches for the coefficients of the numerical solution. Several numerical examples for the polyharmonic equation of order <span><math><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow></math></span>, <span><math><mrow><mi>m</mi><mo>=</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><mi>m</mi><mo>=</mo><mn>3</mn></mrow></math></span>, i.e. for the Poisson’s, the biharmonic and the triharmonic equation, respectively, are shown to demonstrate the potential of our IETI method for solving high-order problems over planar multi-patch geometries with possibly extraordinary vertices.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"452 ","pages":"Article 118769"},"PeriodicalIF":7.3,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146072458","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 : 2026-01-25DOI: 10.1016/j.cma.2026.118759
Celine Lauff , Matti Schneider , Thomas Böhlke
Continuum damage mechanics is characterized by mesh-dependent results unless specific countermeasures are taken. The most popular remedies involve introducing either nonlocality via filtering or a gradient extension for the damage variable(s). Such approaches have their limitations, e.g., they are hard to integrate into conventional finite-element codes, involve parameters that are non-trivial to determine experimentally and are incompatible with a scale transition that is both physically and mathematically sensible. The work at hand considers an alternative route to obtain mesh-independent damage models, namely via convex relaxation. Such convex damage models were considered before, but they are usually not capable of representing softening behavior. Schwarz et al. (Continuum Mech. Thermodyn., 33, pp. 69–95, 2021) proposed such a strategy by considering the convex envelope of a rate-limited simple damage model, i.e., an isotropic damage model without tension-compression anisotropy at small strains. However, they were not able to compute the envelope explicitly and provided an approximation only. In the work at hand, we introduce a number of conditions on the damage-degradation function which permit us to compute the convex envelope analytically for a large class of damage-degradation functions used in small-strain isotropic damage models. Interestingly, the obtained models involve a one-dimensional damaged microstructure, i.e., damage distributions emerge naturally. The resulting model is structurally simple and purely local, i.e., gradient-free, thermodynamically consistent and readily integrated into standard finite-element codes via traditional user subroutines. We discuss the computational and solid mechanical aspects of the ensuing model and demonstrate its numerical robustness via dedicated computational experiments. We also show that the model permits to be homogenized by considering a representative volume element study for an industrial-scale fiber-reinforced composite.
连续损伤力学的特点是网格依赖的结果，除非采取具体的对策。最流行的补救措施包括通过滤波或对损伤变量进行梯度扩展来引入非定域性。这种方法有其局限性，例如，它们很难集成到传统的有限元代码中，涉及的参数在实验中是不平凡的，并且与物理和数学上都合理的尺度转换不兼容。手头的工作考虑了另一种途径来获得网格无关的损伤模型，即通过凸松弛。以前考虑过这种凸损伤模型，但它们通常不能代表软化行为。Schwarz等人（连续介质力学）。Thermodyn。， 33, pp. 69-95, 2021)通过考虑速率受限简单损伤模型（即小应变下无拉压各向异性的各向同性损伤模型）的凸包线提出了这种策略。然而，他们不能明确地计算包络线，只能提供一个近似值。在手头的工作中，我们引入了一些关于损伤退化函数的条件，这些条件允许我们解析地计算用于小应变各向同性损伤模型的一类损伤退化函数的凸包络。有趣的是，所获得的模型涉及一维损伤微观结构，即损伤分布自然出现。所得模型结构简单，纯局部，即无梯度，热力学一致，并易于通过传统用户子程序集成到标准有限元代码中。我们讨论了随后模型的计算和实体力学方面，并通过专门的计算实验证明了其数值鲁棒性。我们还表明，通过考虑工业规模的纤维增强复合材料的代表性体积单元研究，该模型允许均质化。
{"title":"Numerically robust local continuum damage models with softening response via convex relaxation","authors":"Celine Lauff , Matti Schneider , Thomas Böhlke","doi":"10.1016/j.cma.2026.118759","DOIUrl":"10.1016/j.cma.2026.118759","url":null,"abstract":"<div><div>Continuum damage mechanics is characterized by mesh-dependent results unless specific countermeasures are taken. The most popular remedies involve introducing either nonlocality via filtering or a gradient extension for the damage variable(s). Such approaches have their limitations, e.g., they are hard to integrate into conventional finite-element codes, involve parameters that are non-trivial to determine experimentally and are incompatible with a scale transition that is both physically and mathematically sensible. The work at hand considers an alternative route to obtain mesh-independent damage models, namely via <em>convex relaxation</em>. Such convex damage models were considered before, but they are usually not capable of representing <em>softening behavior</em>. Schwarz et al. (Continuum Mech. Thermodyn., 33, pp. 69–95, 2021) proposed such a strategy by considering the convex envelope of a rate-limited simple damage model, i.e., an isotropic damage model without tension-compression anisotropy at small strains. However, they were not able to compute the envelope explicitly and provided an approximation only. In the work at hand, we introduce a number of conditions on the damage-degradation function which permit us to compute the convex envelope analytically for a large class of damage-degradation functions used in small-strain isotropic damage models. Interestingly, the obtained models involve a one-dimensional damaged microstructure, i.e., <em>damage distributions</em> emerge naturally. The resulting model is structurally simple and purely local, i.e., gradient-free, thermodynamically consistent and readily integrated into standard finite-element codes via traditional user subroutines. We discuss the computational and solid mechanical aspects of the ensuing model and demonstrate its numerical robustness via dedicated computational experiments. We also show that the model permits to be homogenized by considering a representative volume element study for an industrial-scale fiber-reinforced composite.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"452 ","pages":"Article 118759"},"PeriodicalIF":7.3,"publicationDate":"2026-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146048293","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 : 2026-01-24DOI: 10.1016/j.cma.2026.118768
Duc-Vinh Nguyen , Mohamed Jebahi , Francisco Chinesta
Recent research has highlighted the potential of physics-informed neural networks (PINNs) as an efficient methodology for approximating solutions of boundary value problems in solid mechanics. Nevertheless, their ability to accurately capture highly heterogeneous solutions of complex problems remains limited and requires further investigation. The present paper explores new strategies to address this challenge. In line with existing approaches based on local refinement of collocation (training) points, a weighted version of the loss function is first proposed to better balance the physical residuals across the entire computational domain. Although this modification improves overall performance, the approximation accuracy remains unsatisfactory. To overcome this limitation, an enriched version of PINN is developed to more effectively capture locally heterogeneous distributions of state variables. Specifically, wavelet-based enrichment functions are designed to approximate local high-frequency components of the full-field solution, thereby simplifying the task of the neural network, which is then required only to approximate the global smooth component of the solution. This approach achieves satisfactory accuracy even with relatively simple neural network architectures and few collocation points, as demonstrated through several benchmark problems. Therefore, the proposed enrichment concept represents a promising direction for further improving the performance of PINNs as solvers in computational mechanics, paving the way for their application to more complex problems.
{"title":"Wavelet-based enrichment for physics informed neural networks to approximate localized and heterogeneous solutions in solid mechanics","authors":"Duc-Vinh Nguyen , Mohamed Jebahi , Francisco Chinesta","doi":"10.1016/j.cma.2026.118768","DOIUrl":"10.1016/j.cma.2026.118768","url":null,"abstract":"<div><div>Recent research has highlighted the potential of physics-informed neural networks (PINNs) as an efficient methodology for approximating solutions of boundary value problems in solid mechanics. Nevertheless, their ability to accurately capture highly heterogeneous solutions of complex problems remains limited and requires further investigation. The present paper explores new strategies to address this challenge. In line with existing approaches based on local refinement of collocation (training) points, a weighted version of the loss function is first proposed to better balance the physical residuals across the entire computational domain. Although this modification improves overall performance, the approximation accuracy remains unsatisfactory. To overcome this limitation, an enriched version of PINN is developed to more effectively capture locally heterogeneous distributions of state variables. Specifically, wavelet-based enrichment functions are designed to approximate local high-frequency components of the full-field solution, thereby simplifying the task of the neural network, which is then required only to approximate the global smooth component of the solution. This approach achieves satisfactory accuracy even with relatively simple neural network architectures and few collocation points, as demonstrated through several benchmark problems. Therefore, the proposed enrichment concept represents a promising direction for further improving the performance of PINNs as solvers in computational mechanics, paving the way for their application to more complex problems.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"452 ","pages":"Article 118768"},"PeriodicalIF":7.3,"publicationDate":"2026-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023763","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 : 2026-01-24DOI: 10.1016/j.cma.2026.118764
Aswanth Thani , Adrian Buganza Tepole
We present a physics-informed surface autoencoder (PISA) framework for Kirchhoff-Love thin shell analysis. The method constructs global C1 surface parameterizations directly from unstructured point clouds for both single-patch surfaces homeomorphic to disks, and multi-patch parameterizations for closed genus-zero surfaces. In the multi-patch case, a classification network assigns probabilistic labels to points, and the autoencoder learns overlapping charts with smooth transitions, ensuring global C1 continuity. With the learned parameterizations, we introduce a decoder for the displacement field and compute differential geometric quantities such as the metric and second fundamental form in the reference and deformed surfaces. Then, we enforce equilibrium by minimizing the total potential energy. The approach is validated on classical shell benchmarks, including the Scordelis-Lo roof, pinched cylinder, and hemisphere under pressure. We showcase the flexibility of the framework with complex geometries such as the Stanford Bunny and dura mater. Compared with traditional spline-based parameterizations and existing machine learning approaches, PISA offers a pipeline for generating smooth surface maps for complex geometries and integrates the surface representation into the physics-informed solver. Importantly, the thin shell analysis pipeline proposed works directly with unstructured point cloud data. Thus, this PISA framework’s potential applications range from engineering structures to biological membranes such as heart valves, skin, and dura mater.
{"title":"Physics informed surface autoencoders for thin shell analysis","authors":"Aswanth Thani , Adrian Buganza Tepole","doi":"10.1016/j.cma.2026.118764","DOIUrl":"10.1016/j.cma.2026.118764","url":null,"abstract":"<div><div>We present a physics-informed surface autoencoder (PISA) framework for Kirchhoff-Love thin shell analysis. The method constructs global <em>C</em><sup>1</sup> surface parameterizations directly from unstructured point clouds for both single-patch surfaces homeomorphic to disks, and multi-patch parameterizations for closed genus-zero surfaces. In the multi-patch case, a classification network assigns probabilistic labels to points, and the autoencoder learns overlapping charts with smooth transitions, ensuring global <em>C</em><sup>1</sup> continuity. With the learned parameterizations, we introduce a decoder for the displacement field and compute differential geometric quantities such as the metric and second fundamental form in the reference and deformed surfaces. Then, we enforce equilibrium by minimizing the total potential energy. The approach is validated on classical shell benchmarks, including the Scordelis-Lo roof, pinched cylinder, and hemisphere under pressure. We showcase the flexibility of the framework with complex geometries such as the Stanford Bunny and dura mater. Compared with traditional spline-based parameterizations and existing machine learning approaches, PISA offers a pipeline for generating smooth surface maps for complex geometries and integrates the surface representation into the physics-informed solver. Importantly, the thin shell analysis pipeline proposed works directly with unstructured point cloud data. Thus, this PISA framework’s potential applications range from engineering structures to biological membranes such as heart valves, skin, and dura mater.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"452 ","pages":"Article 118764"},"PeriodicalIF":7.3,"publicationDate":"2026-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023859","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 : 2026-01-24DOI: 10.1016/j.cma.2026.118755
Wasim Akram, Manil T. Mohan
We investigate an optimal control problem governed by a stationary convective Brinkman-Forchheimer extended Darcy (CBFeD) model formulated as a hemivariational inequality in both two- and three-dimensional settings. This framework captures complex incompressible fluid flow through porous media by simultaneously accounting for convection, viscous damping, and nonlinear resistance effects, while naturally incorporating non-smooth frictional interactions through a subdifferential boundary condition. A key contribution of this work is a rigorous stability analysis of the CBFeD hemivariational inequality with respect to perturbations in both the external force density and the associated superpotential. Building on this analysis, we establish the existence of optimal controls when the external force density is treated as the control variable under admissible constraints. This result extends existing optimal control theories to a broader class of nonsmooth, nonlinear flow models in porous media. From a computational perspective, we propose a fully implementable numerical scheme for the resulting optimal control problem and prove its convergence. The method is based on finite element discretization and is applicable in both two and three dimensions, making it suitable for practical simulations. Numerical experiments are presented to illustrate the effectiveness of the proposed approach and to confirm the theoretical findings.
{"title":"Optimal control of a hemivariational inequality of stationary convective Brinkman-Forchheimer extended Darcy equations with numerical approximation","authors":"Wasim Akram, Manil T. Mohan","doi":"10.1016/j.cma.2026.118755","DOIUrl":"10.1016/j.cma.2026.118755","url":null,"abstract":"<div><div>We investigate an optimal control problem governed by a stationary convective Brinkman-Forchheimer extended Darcy (CBFeD) model formulated as a hemivariational inequality in both two- and three-dimensional settings. This framework captures complex incompressible fluid flow through porous media by simultaneously accounting for convection, viscous damping, and nonlinear resistance effects, while naturally incorporating non-smooth frictional interactions through a subdifferential boundary condition. A key contribution of this work is a rigorous stability analysis of the CBFeD hemivariational inequality with respect to perturbations in both the external force density and the associated superpotential. Building on this analysis, we establish the existence of optimal controls when the external force density is treated as the control variable under admissible constraints. This result extends existing optimal control theories to a broader class of nonsmooth, nonlinear flow models in porous media. From a computational perspective, we propose a fully implementable numerical scheme for the resulting optimal control problem and prove its convergence. The method is based on finite element discretization and is applicable in both two and three dimensions, making it suitable for practical simulations. Numerical experiments are presented to illustrate the effectiveness of the proposed approach and to confirm the theoretical findings.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"452 ","pages":"Article 118755"},"PeriodicalIF":7.3,"publicationDate":"2026-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146048313","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 : 2026-01-24DOI: 10.1016/j.cma.2026.118732
Dean J. Maxam, Pranav Chengala Madhusoodana, Kumar K. Tamma
This article details the analysis and synthesis of a novel family of explicit single-step integration methods for the second-order dynamics of structural and multibody systems. The family is derived from the Generalized Single-Step Single-Solve framework, pertaining to the class of linear multistep methods. The new explicit methods achieve second-order accuracy with optimal starting error, controllable numerical dissipation, and an option for explicit or implicit treatment of damping; the latter yields a stability limit which scales optimally with modal damping ratio, unlike prior methods with only incidental gains. The new family is compared with existing explicit methods on the basis of numerical accuracy and stability. Its superior performance for linear and nonlinear systems is demonstrated by numerical examples.
{"title":"A new family of explicit generalized single-step single-stage integration methods for structural/multibody dynamics with improved stability for viscous damping","authors":"Dean J. Maxam, Pranav Chengala Madhusoodana, Kumar K. Tamma","doi":"10.1016/j.cma.2026.118732","DOIUrl":"10.1016/j.cma.2026.118732","url":null,"abstract":"<div><div>This article details the analysis and synthesis of a novel family of explicit single-step integration methods for the second-order dynamics of structural and multibody systems. The family is derived from the Generalized Single-Step Single-Solve framework, pertaining to the class of linear multistep methods. The new explicit methods achieve second-order accuracy with optimal starting error, controllable numerical dissipation, and an option for explicit or implicit treatment of damping; the latter yields a stability limit which scales optimally with modal damping ratio, unlike prior methods with only incidental gains. The new family is compared with existing explicit methods on the basis of numerical accuracy and stability. Its superior performance for linear and nonlinear systems is demonstrated by numerical examples.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":"452 ","pages":"Article 118732"},"PeriodicalIF":7.3,"publicationDate":"2026-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146048404","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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