In recent years, the rapid advancement of deep learning has significantly impacted various fields, particularly in solving partial differential equations (PDEs) in the realm of solid mechanics, benefiting greatly from the remarkable approximation capabilities of neural networks. In solving PDEs, physics‐informed neural networks (PINNs) and the deep energy method (DEM) have garnered substantial attention. The principle of minimum potential energy and complementary energy are two important variational principles in solid mechanics. However, the well‐known DEM is based on the principle of minimum potential energy, but it lacks the important form of minimum complementary energy. To bridge this gap, we propose the deep complementary energy method (DCEM) based on the principle of minimum complementary energy. The output function of DCEM is the stress function, which inherently satisfies the equilibrium equation. We present numerical results of classical linear elasticity using the Prandtl and Airy stress functions, and compare DCEM with existing PINNs and DEM algorithms when modeling representative mechanical problems. The results demonstrate that DCEM outperforms DEM in terms of stress accuracy and efficiency and has an advantage in dealing with complex displacement boundary conditions, which is supported by theoretical analyses and numerical simulations. We extend DCEM to DCEM‐Plus (DCEM‐P), adding terms that satisfy PDEs. Furthermore, we propose a deep complementary energy operator method (DCEM‐O) by combining operator learning with physical equations. Initially, we train DCEM‐O using high‐fidelity numerical results and then incorporate complementary energy. DCEM‐P and DCEM‐O further enhance the accuracy and efficiency of DCEM.
{"title":"DCEM: A deep complementary energy method for linear elasticity","authors":"Yizheng Wang, Jia Sun, Timon Rabczuk, Yinghua Liu","doi":"10.1002/nme.7585","DOIUrl":"https://doi.org/10.1002/nme.7585","url":null,"abstract":"In recent years, the rapid advancement of deep learning has significantly impacted various fields, particularly in solving partial differential equations (PDEs) in the realm of solid mechanics, benefiting greatly from the remarkable approximation capabilities of neural networks. In solving PDEs, physics‐informed neural networks (PINNs) and the deep energy method (DEM) have garnered substantial attention. The principle of minimum potential energy and complementary energy are two important variational principles in solid mechanics. However, the well‐known DEM is based on the principle of minimum potential energy, but it lacks the important form of minimum complementary energy. To bridge this gap, we propose the deep complementary energy method (DCEM) based on the principle of minimum complementary energy. The output function of DCEM is the stress function, which inherently satisfies the equilibrium equation. We present numerical results of classical linear elasticity using the Prandtl and Airy stress functions, and compare DCEM with existing PINNs and DEM algorithms when modeling representative mechanical problems. The results demonstrate that DCEM outperforms DEM in terms of stress accuracy and efficiency and has an advantage in dealing with complex displacement boundary conditions, which is supported by theoretical analyses and numerical simulations. We extend DCEM to DCEM‐Plus (DCEM‐P), adding terms that satisfy PDEs. Furthermore, we propose a deep complementary energy operator method (DCEM‐O) by combining operator learning with physical equations. Initially, we train DCEM‐O using high‐fidelity numerical results and then incorporate complementary energy. DCEM‐P and DCEM‐O further enhance the accuracy and efficiency of DCEM.","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The cover image is based on the article Stabilized Time-Series Moving Morphable Components Method for Topology Optimization by Xueyan Hu et al., https://doi.org/10.1002/nme.7562
封面图片来自 Xueyan Hu 等人撰写的文章《用于拓扑优化的稳定时序移动可变形组件法》，https://doi.org/10.1002/nme.7562。
{"title":"Featured Cover","authors":"Xueyan Hu, Zonghao Li, Ronghao Bao, Weiqiu Chen","doi":"10.1002/nme.7591","DOIUrl":"https://doi.org/10.1002/nme.7591","url":null,"abstract":"<p>The cover image is based on the article <i>Stabilized Time-Series Moving Morphable Components Method for Topology Optimization</i> by Xueyan Hu et al., https://doi.org/10.1002/nme.7562\u0000 \u0000 <figure>\u0000 <div><picture>\u0000 <source></source></picture><p></p>\u0000 </div>\u0000 </figure></p>","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/nme.7591","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142165261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Devesh Kumar, Dave Carlson, J. S. Kumar, Jianming Cao, Bruce Engelmann
Frequency response analysis often provides a great deal of information about the system response over the entire range of operation. This can be computationally expensive if time‐domain methods are used, especially for large structural models. Presence of non‐linearity in the system makes it difficult to employ standard frequency response analysis techniques, which are linear in nature. If the system contains mild‐nonlinearities and the response of the system can be assumed to be periodic, it is possible to obtain nonlinear frequency response of the system using harmonic balance techniques. This paper presents the application of the harmonic balance method for solving nonlinear structural dynamics problems. To improve robustness of the solution and capture unstable branches, the continuation procedure technique is included along with the harmonic balance method. The method developed has been implemented in MSC Nastran SOL 128.
频率响应分析通常可以提供大量有关整个工作范围内系统响应的信息。如果使用时域方法，计算成本可能会很高，尤其是对于大型结构模型。由于系统中存在非线性，因此很难采用标准的频率响应分析技术，因为这些技术本质上是线性的。如果系统包含轻度非线性，并且系统响应可以假定为周期性的，那么就有可能利用谐波平衡技术获得系统的非线性频率响应。本文介绍了谐波平衡法在非线性结构动力学问题求解中的应用。为了提高求解的鲁棒性并捕捉不稳定分支，在采用谐波平衡法的同时还采用了延续程序技术。所开发的方法已在 MSC Nastran SOL 128 中实现。
{"title":"Nonlinear frequency response analysis using MSC Nastran","authors":"Devesh Kumar, Dave Carlson, J. S. Kumar, Jianming Cao, Bruce Engelmann","doi":"10.1002/nme.7588","DOIUrl":"https://doi.org/10.1002/nme.7588","url":null,"abstract":"Frequency response analysis often provides a great deal of information about the system response over the entire range of operation. This can be computationally expensive if time‐domain methods are used, especially for large structural models. Presence of non‐linearity in the system makes it difficult to employ standard frequency response analysis techniques, which are linear in nature. If the system contains mild‐nonlinearities and the response of the system can be assumed to be periodic, it is possible to obtain nonlinear frequency response of the system using harmonic balance techniques. This paper presents the application of the harmonic balance method for solving nonlinear structural dynamics problems. To improve robustness of the solution and capture unstable branches, the continuation procedure technique is included along with the harmonic balance method. The method developed has been implemented in MSC Nastran SOL 128.","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. R. Bravo, J. A. Hernández, S. Ares de Parga, R. Rossi
This article is concerned with quadrature/cubature rules able to deal with multiple subspaces of functions, in such a way that the integration points are common for all the subspaces, yet the (nonnegative) weights are tailored to each specific subspace. These subspace‐adaptive weights cubature rules can be used to accelerate computational mechanics applications requiring efficiently evaluating spatial integrals whose integrand function dynamically switches between multiple pre‐computed subspaces. One of such applications is local hyperreduced‐order modeling (HROM), in which the solution manifold is approximately represented as a collection of basis matrices, each basis matrix corresponding to a different region in parameter space. The proposed optimization framework is discrete in terms of the location of the integration points, in the sense that such points are selected among the Gauss points of a given finite element mesh, and the target subspaces of functions are represented by orthogonal basis matrices constructed from the values of the functions at such Gauss points, using the singular value decomposition (SVD). This discrete framework allows us to treat also problems in which the integrals are approximated as a weighted sum of the contribution of each finite element, as in the energy‐conserving sampling and weighting method of C. Farhat and co‐workers. Two distinct solution strategies are examined. The first one is a greedy strategy based on an enhanced version of the empirical cubature method (ECM) developed by the authors elsewhere (we call it the subspace‐adaptive weights ECM, SAW‐ECM for short), while the second method is based on a convexification of the cubature problem so that it can be addressed by linear programming algorithms. We show in a toy problem involving integration of polynomial functions that the SAW‐ECM clearly outperforms the other method both in terms of computational cost and optimality. On the other hand, we illustrate the performance of the SAW‐ECM in the construction of a local HROMs in a highly nonlinear equilibrium problem (large strains regime). We demonstrate that, provided that the subspace‐transition errors are negligible, the error associated to hyperreduction using adaptive weights can be controlled by the truncation tolerances of the SVDs used for determining the basis matrices. We also show that the number of integration points decreases notably as the number of subspaces increases, and that, in the limiting case of using as many subspaces as snapshots, the SAW‐ECM delivers rules with a number of integration points only dependent on the intrinsic dimensionality of the solution manifold and the degree of overlapping required to avoid subspace‐transition errors. The Python source codes of the proposed SAW‐ECM are openly accessible in the public repository <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://github.com/Rbravo555/localECM">https://github.com/Rbravo555/localECM</jats:e
{"title":"A subspace‐adaptive weights cubature method with application to the local hyperreduction of parameterized finite element models","authors":"J. R. Bravo, J. A. Hernández, S. Ares de Parga, R. Rossi","doi":"10.1002/nme.7590","DOIUrl":"https://doi.org/10.1002/nme.7590","url":null,"abstract":"This article is concerned with quadrature/cubature rules able to deal with multiple subspaces of functions, in such a way that the integration points are common for all the subspaces, yet the (nonnegative) weights are tailored to each specific subspace. These subspace‐adaptive weights cubature rules can be used to accelerate computational mechanics applications requiring efficiently evaluating spatial integrals whose integrand function dynamically switches between multiple pre‐computed subspaces. One of such applications is local hyperreduced‐order modeling (HROM), in which the solution manifold is approximately represented as a collection of basis matrices, each basis matrix corresponding to a different region in parameter space. The proposed optimization framework is discrete in terms of the location of the integration points, in the sense that such points are selected among the Gauss points of a given finite element mesh, and the target subspaces of functions are represented by orthogonal basis matrices constructed from the values of the functions at such Gauss points, using the singular value decomposition (SVD). This discrete framework allows us to treat also problems in which the integrals are approximated as a weighted sum of the contribution of each finite element, as in the energy‐conserving sampling and weighting method of C. Farhat and co‐workers. Two distinct solution strategies are examined. The first one is a greedy strategy based on an enhanced version of the empirical cubature method (ECM) developed by the authors elsewhere (we call it the subspace‐adaptive weights ECM, SAW‐ECM for short), while the second method is based on a convexification of the cubature problem so that it can be addressed by linear programming algorithms. We show in a toy problem involving integration of polynomial functions that the SAW‐ECM clearly outperforms the other method both in terms of computational cost and optimality. On the other hand, we illustrate the performance of the SAW‐ECM in the construction of a local HROMs in a highly nonlinear equilibrium problem (large strains regime). We demonstrate that, provided that the subspace‐transition errors are negligible, the error associated to hyperreduction using adaptive weights can be controlled by the truncation tolerances of the SVDs used for determining the basis matrices. We also show that the number of integration points decreases notably as the number of subspaces increases, and that, in the limiting case of using as many subspaces as snapshots, the SAW‐ECM delivers rules with a number of integration points only dependent on the intrinsic dimensionality of the solution manifold and the degree of overlapping required to avoid subspace‐transition errors. The Python source codes of the proposed SAW‐ECM are openly accessible in the public repository <jats:ext-link xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"https://github.com/Rbravo555/localECM\">https://github.com/Rbravo555/localECM</jats:e","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190982","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The coupling between free and porous medium flows has received significant attention since it plays an important role in a wide range of problems from fluid‐soil interactions to biofluid dynamics. However, modeling this coupled process remains a difficult task as it often involves a domain decomposition algorithm in conjunction with a special treatment at the interface. The problem can become more challenging under non‐isothermal conditions because it requires the iterative procedure at every time step to simultaneously meet the transient mass continuity, force equilibrium, and energy balance for the entire system. This article presents a diffuse interface framework for modeling non‐isothermal Stokes‐Darcy flow and the corresponding finite element formulation that bypasses the need for explicitly splitting the domain into two, which enables the unified treatment for distinct regions with different hydrothermal flow regimes. To achieve this goal, we employ the Allen‐Cahn type phase field model to generate the diffuse geometry, where the solution field can be seen as a regularized approximation of the Heaviside indicator function, allowing us to transfer the interface conditions into a set of immersed boundary conditions. Our formulation suggests that the isothermal operator splitting strategy can be adopted without compromising accuracy if the heat and mass transfer processes are decoupled by assuming that the density and viscosity of the phase constituents are independent to the temperature. Numerical examples are also introduced to verify the implementation and to demonstrate the model capacity.
{"title":"Diffuse interface modeling of non‐isothermal Stokes‐Darcy flow with immersed transmissibility conditions","authors":"Hyoung Suk Suh","doi":"10.1002/nme.7589","DOIUrl":"https://doi.org/10.1002/nme.7589","url":null,"abstract":"The coupling between free and porous medium flows has received significant attention since it plays an important role in a wide range of problems from fluid‐soil interactions to biofluid dynamics. However, modeling this coupled process remains a difficult task as it often involves a domain decomposition algorithm in conjunction with a special treatment at the interface. The problem can become more challenging under non‐isothermal conditions because it requires the iterative procedure at every time step to simultaneously meet the transient mass continuity, force equilibrium, and energy balance for the entire system. This article presents a diffuse interface framework for modeling non‐isothermal Stokes‐Darcy flow and the corresponding finite element formulation that bypasses the need for explicitly splitting the domain into two, which enables the unified treatment for distinct regions with different hydrothermal flow regimes. To achieve this goal, we employ the Allen‐Cahn type phase field model to generate the diffuse geometry, where the solution field can be seen as a regularized approximation of the Heaviside indicator function, allowing us to transfer the interface conditions into a set of immersed boundary conditions. Our formulation suggests that the isothermal operator splitting strategy can be adopted without compromising accuracy if the heat and mass transfer processes are decoupled by assuming that the density and viscosity of the phase constituents are independent to the temperature. Numerical examples are also introduced to verify the implementation and to demonstrate the model capacity.","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190980","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marie Jeanneteau, Paul Oumaziz, Jean‐Charles Passieux, Vincent Gibiat, Jonathan Cottier
A high‐fidelity finite element model is proposed for the complete simulation of the time‐harmonic acoustic propagation in wind instruments. The challenge is to meet the extremely high accuracy required by professional musicians, in a complex domain, for all fingerings and over a wide frequency range, within an affordable computational time. Several modelling assumptions are made to limit the numerical complexity of the problem while preserving all relevant physics. A dedicated high‐performance solution strategy is also proposed, based on partitioning, condensation and model order reduction, exploiting the combinatorial nature of wind instrument fingerings. Finally, the proposed approach is applied to the simulation of an alto saxophone. An order of magnitude reduction in memory and computational cost is achieved.
{"title":"A combinatorial model reduction method for the finite element analysis of wind instruments","authors":"Marie Jeanneteau, Paul Oumaziz, Jean‐Charles Passieux, Vincent Gibiat, Jonathan Cottier","doi":"10.1002/nme.7582","DOIUrl":"https://doi.org/10.1002/nme.7582","url":null,"abstract":"A high‐fidelity finite element model is proposed for the complete simulation of the time‐harmonic acoustic propagation in wind instruments. The challenge is to meet the extremely high accuracy required by professional musicians, in a complex domain, for all fingerings and over a wide frequency range, within an affordable computational time. Several modelling assumptions are made to limit the numerical complexity of the problem while preserving all relevant physics. A dedicated high‐performance solution strategy is also proposed, based on partitioning, condensation and model order reduction, exploiting the combinatorial nature of wind instrument fingerings. Finally, the proposed approach is applied to the simulation of an alto saxophone. An order of magnitude reduction in memory and computational cost is achieved.","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142190983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Paul Hofer, Matthias Neuner, Peter Gamnitzer, Günter Hofstetter
The localization of deformations plays a crucial role in the failure of granular materials. Concerning classical continuum constitutive models, the localization of deformations is considered to be connected to the loss of ellipticity of the governing rate equilibrium equations, and entails mesh sensitivity in finite element simulations. While previous studies are often limited to strain localization analyses of individual tests, the focus of the present contribution lies on studying the localization properties in general constitutive states. For this purpose, a staggered optimization algorithm for determining the loss of ellipticity, considering both extreme values, minimum and maximum, of the determinant of the acoustic tensor, is proposed. Part of this algorithm representing a novel application of spherical Fibonacci lattices for discretizing the feasible domain of the associated optimization problem. In the presented localization study of the widely recognized modified Cam‐clay model, special attention is paid to determining the influence of the individual model parameters. Specifically, three factors favoring strain localization are found, namely (i) a low value of the ratio of the primary compression index and the recompression index, (ii) a large value of the critical state frictional constant, as well as (iii) a large value of Poisson's ratio. Moreover, a structural finite element study is performed, confirming the results of localization analyses at the constitutive level.
{"title":"Revisiting strain localization analysis for elastoplastic constitutive models in geomechanics","authors":"Paul Hofer, Matthias Neuner, Peter Gamnitzer, Günter Hofstetter","doi":"10.1002/nme.7579","DOIUrl":"https://doi.org/10.1002/nme.7579","url":null,"abstract":"The localization of deformations plays a crucial role in the failure of granular materials. Concerning classical continuum constitutive models, the localization of deformations is considered to be connected to the loss of ellipticity of the governing rate equilibrium equations, and entails mesh sensitivity in finite element simulations. While previous studies are often limited to strain localization analyses of individual tests, the focus of the present contribution lies on studying the localization properties in general constitutive states. For this purpose, a staggered optimization algorithm for determining the loss of ellipticity, considering both extreme values, minimum and maximum, of the determinant of the acoustic tensor, is proposed. Part of this algorithm representing a novel application of spherical Fibonacci lattices for discretizing the feasible domain of the associated optimization problem. In the presented localization study of the widely recognized modified Cam‐clay model, special attention is paid to determining the influence of the individual model parameters. Specifically, three factors favoring strain localization are found, namely (i) a low value of the ratio of the primary compression index and the recompression index, (ii) a large value of the critical state frictional constant, as well as (iii) a large value of Poisson's ratio. Moreover, a structural finite element study is performed, confirming the results of localization analyses at the constitutive level.","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tim van der Velden, Tim Brepols, Stefanie Reese, Hagen Holthusen
Modern inelastic material model formulations rely on the use of tensor‐valued internal variables. When inelastic phenomena include softening, simulations of the former are prone to localization. Thus, an accurate regularization of the tensor‐valued internal variables is essential to obtain physically correct results. Here, we focus on the regularization of anisotropic damage at finite strains. Thus, a flexible anisotropic damage model with isotropic, kinematic, and distortional hardening is equipped with three gradient‐extensions using a full and two reduced regularizations of the damage tensor. Theoretical and numerical comparisons of the three gradient‐extensions yield excellent agreement between the full and the reduced regularization based on a volumetric‐deviatoric regularization using only two nonlocal degrees of freedom.
{"title":"A comparative study of micromorphic gradient‐extensions for anisotropic damage at finite strains","authors":"Tim van der Velden, Tim Brepols, Stefanie Reese, Hagen Holthusen","doi":"10.1002/nme.7580","DOIUrl":"https://doi.org/10.1002/nme.7580","url":null,"abstract":"Modern inelastic material model formulations rely on the use of tensor‐valued internal variables. When inelastic phenomena include softening, simulations of the former are prone to localization. Thus, an accurate regularization of the tensor‐valued internal variables is essential to obtain physically correct results. Here, we focus on the regularization of anisotropic damage at finite strains. Thus, a flexible anisotropic damage model with isotropic, kinematic, and distortional hardening is equipped with three gradient‐extensions using a full and two reduced regularizations of the damage tensor. Theoretical and numerical comparisons of the three gradient‐extensions yield excellent agreement between the full and the reduced regularization based on a volumetric‐deviatoric regularization using only two nonlocal degrees of freedom.","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":null,"pages":null},"PeriodicalIF":2.9,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141969814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lennart Scherz, B. Kriegesmann, Claus B. W. Pedersen
The current paper introduces a new stabilization scheme for void and low‐density elements for geometrical nonlinear topology optimization. Frequently, certain localized regions in the geometrical nonlinear finite element analysis of the topology optimization have excessive artificial distortions due to the low stiffness of the void and low‐density elements. The present stabilization applies a hyperelastic constitutive material model for the numerical stabilization that is associated with the condition number of the deformation gradient and thereby, is associated with the numerical conditioning of the mapping between current configuration and reference configuration of the underlying continuum mechanics on a constitutive material model level. The stabilization method is independent upon the topology design variables during the optimization iterations. Numerical parametric studies show that the parameters for the constitutive hyperelasticity material of the new stabilization scheme are governed by the stiffness of the constitutive model of the initial physical system. The parametric studies also show that the stabilization scheme is independently upon the type of constitutive model of the physical system and the element types applied for the finite element modeling. The new stabilization scheme is numerical verified using both academic reference examples and industrial applications. The numerical examples show that the number of optimization iterations is significantly reduced compared to the stabilization approaches previously reported in the literature.
{"title":"A condition number‐based numerical stabilization method for geometrically nonlinear topology optimization","authors":"Lennart Scherz, B. Kriegesmann, Claus B. W. Pedersen","doi":"10.1002/nme.7574","DOIUrl":"https://doi.org/10.1002/nme.7574","url":null,"abstract":"The current paper introduces a new stabilization scheme for void and low‐density elements for geometrical nonlinear topology optimization. Frequently, certain localized regions in the geometrical nonlinear finite element analysis of the topology optimization have excessive artificial distortions due to the low stiffness of the void and low‐density elements. The present stabilization applies a hyperelastic constitutive material model for the numerical stabilization that is associated with the condition number of the deformation gradient and thereby, is associated with the numerical conditioning of the mapping between current configuration and reference configuration of the underlying continuum mechanics on a constitutive material model level. The stabilization method is independent upon the topology design variables during the optimization iterations. Numerical parametric studies show that the parameters for the constitutive hyperelasticity material of the new stabilization scheme are governed by the stiffness of the constitutive model of the initial physical system. The parametric studies also show that the stabilization scheme is independently upon the type of constitutive model of the physical system and the element types applied for the finite element modeling. The new stabilization scheme is numerical verified using both academic reference examples and industrial applications. The numerical examples show that the number of optimization iterations is significantly reduced compared to the stabilization approaches previously reported in the literature.","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141920799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Accurate modeling of layered composite structures often requires the use of detailed finite element models which can sufficiently resolve the kinematics and material behavior within each layer of the composite. However, individually discretizing each material layer into finite elements presents a prohibitive computational expensive given the large number of thin layers comprising some laminated composites. To address these challenges, an 8‐node layered solid hexahedral finite element is formulated with the aim of striking an appropriate balance between efficiency and fidelity. The element is discretized into an arbitrary number of distinct material layers, and employs reduced in‐plane integration within each layer. The chosen reduced integration scheme is supplemented by a novel physical stabilization approach which includes layerwise enhancements to mitigate various forms of locking phenomena. The proposed framework additionally supports the inclusion of interlaminar enhanced displacements to better represent the kinematics of general layered composite materials. The described element formulation has been implemented in the ParaDyn finite element code, and its efficacy for modeling laminated composite structures is demonstrated on a variety of verification problems.
{"title":"A layered solid finite element formulation with interlaminar enhanced displacements for the modeling of laminated composite structures","authors":"B. Giffin, Miklos J. Zoller","doi":"10.1002/nme.7581","DOIUrl":"https://doi.org/10.1002/nme.7581","url":null,"abstract":"Accurate modeling of layered composite structures often requires the use of detailed finite element models which can sufficiently resolve the kinematics and material behavior within each layer of the composite. However, individually discretizing each material layer into finite elements presents a prohibitive computational expensive given the large number of thin layers comprising some laminated composites. To address these challenges, an 8‐node layered solid hexahedral finite element is formulated with the aim of striking an appropriate balance between efficiency and fidelity. The element is discretized into an arbitrary number of distinct material layers, and employs reduced in‐plane integration within each layer. The chosen reduced integration scheme is supplemented by a novel physical stabilization approach which includes layerwise enhancements to mitigate various forms of locking phenomena. The proposed framework additionally supports the inclusion of interlaminar enhanced displacements to better represent the kinematics of general layered composite materials. The described element formulation has been implemented in the ParaDyn finite element code, and its efficacy for modeling laminated composite structures is demonstrated on a variety of verification problems.","PeriodicalId":13699,"journal":{"name":"International Journal for Numerical Methods in Engineering","volume":null,"pages":null},"PeriodicalIF":2.7,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141929702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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