Pub Date : 2024-10-16DOI: 10.1016/j.cma.2024.117460
This paper presents a novel decoupled framework for reliability-based topology optimization (RBTO) that aims to find optimal material configurations while meeting local stiffness and strength constraints. To effectively address the nonlinear displacement and stress reliability constraints, the proposed framework replaces the conventional first-order reliability method (FORM) with the more accurate Local Radial Point Interpolation Method (LRPIM). This substitution overcomes the limitations of FORM in approximating high-dimensional nonlinear problems. The framework includes the qp-relaxation criterion and a global stress aggregation technique to avoid stress singularities. For multi-constrained optimization, the adjoint vector method is used for design sensitivity analysis, followed by a gradient-based algorithm to solve the structural optimization problem. Numerical examples are presented to validate the effectiveness of the proposed RBTO methodology, demonstrating its superiority in both accuracy and reliability compared to the Sequential Optimization and Reliability Assessment (SORA) method. The comparative analysis highlights the efficiency and precision of the proposed method across different reliability approaches, making it a robust tool for addressing complex engineering challenges.
本文提出了一种新颖的基于可靠性的拓扑优化（RBTO）解耦框架，旨在找到最佳材料配置，同时满足局部刚度和强度约束。为有效解决非线性位移和应力可靠性约束，本文提出的框架用更精确的局部径向点插值法（LRPIM）取代了传统的一阶可靠性方法（FORM）。这种替代方法克服了 FORM 在逼近高维非线性问题时的局限性。该框架包括 qp 松弛准则和全局应力聚集技术，以避免应力奇点。在多约束优化方面，使用邻接向量法进行设计敏感性分析，然后使用基于梯度的算法解决结构优化问题。通过数值示例验证了所提出的 RBTO 方法的有效性，证明其与顺序优化和可靠性评估（SORA）方法相比，在准确性和可靠性方面都更胜一筹。对比分析凸显了拟议方法在不同可靠性方法中的效率和精确性，使其成为应对复杂工程挑战的强大工具。
{"title":"Reliability-based topology optimization using LRPIM surrogate model considering local stress and displacement constraints","authors":"","doi":"10.1016/j.cma.2024.117460","DOIUrl":"10.1016/j.cma.2024.117460","url":null,"abstract":"<div><div>This paper presents a novel decoupled framework for reliability-based topology optimization (RBTO) that aims to find optimal material configurations while meeting local stiffness and strength constraints. To effectively address the nonlinear displacement and stress reliability constraints, the proposed framework replaces the conventional first-order reliability method (FORM) with the more accurate Local Radial Point Interpolation Method (LRPIM). This substitution overcomes the limitations of FORM in approximating high-dimensional nonlinear problems. The framework includes the qp-relaxation criterion and a global stress aggregation technique to avoid stress singularities. For multi-constrained optimization, the adjoint vector method is used for design sensitivity analysis, followed by a gradient-based algorithm to solve the structural optimization problem. Numerical examples are presented to validate the effectiveness of the proposed RBTO methodology, demonstrating its superiority in both accuracy and reliability compared to the Sequential Optimization and Reliability Assessment (SORA) method. The comparative analysis highlights the efficiency and precision of the proposed method across different reliability approaches, making it a robust tool for addressing complex engineering challenges.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":null,"pages":null},"PeriodicalIF":6.9,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441963","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 : 2024-10-15DOI: 10.1016/j.cma.2024.117444
The accuracy of finite element solutions is closely tied to the mesh quality. In particular, geometrically nonlinear problems involving large and strongly localized deformations often result in prohibitively large element distortions. In this work, we propose a novel mesh regularization approach allowing to restore a non-distorted high-quality mesh in an adaptive manner without the need for expensive re-meshing procedures. The core idea of this approach lies in the definition of a finite element distortion potential considering contributions from different distortion modes such as skewness and aspect ratio of the elements. The regularized mesh is found by minimization of this potential. Moreover, based on the concept of spatial localization functions, the method allows to specify tailored requirements on mesh resolution and quality for regions with strongly localized mechanical deformation and mesh distortion. In addition, while existing mesh regularization schemes often keep the boundary nodes of the discretization fixed, we propose a mesh-sliding algorithm based on variationally consistent mortar methods allowing for an unrestricted tangential motion of nodes along the problem boundary. Especially for problems involving significant surface deformation (e.g., frictional contact), this approach allows for an improved mesh relaxation as compared to schemes with fixed boundary nodes. To transfer data such as tensor-valued history variables of the material model from the old (distorted) to the new (regularized) mesh, a structure-preserving invariant interpolation scheme for second-order tensors is employed, which has been proposed in our previous work and is designed to preserve important properties of tensor-valued data such as objectivity and positive definiteness. As a practically relevant application scenario, we consider the thermo-mechanical expansion of materials such as foams involving extreme volume changes by up to two orders of magnitude along with large and strongly localized strains as well as thermo-mechanical contact interaction. For this scenario, it is demonstrated that the proposed regularization approach preserves a high mesh quality at small computational costs. In contrast, simulations without mesh adaption are shown to lead to significant mesh distortion, deteriorating result quality, and, eventually, to non-convergence of the numerical solution scheme.
{"title":"A novel mesh regularization approach based on finite element distortion potentials: Application to material expansion processes with extreme volume change","authors":"","doi":"10.1016/j.cma.2024.117444","DOIUrl":"10.1016/j.cma.2024.117444","url":null,"abstract":"<div><div>The accuracy of finite element solutions is closely tied to the mesh quality. In particular, geometrically nonlinear problems involving large and strongly localized deformations often result in prohibitively large element distortions. In this work, we propose a novel mesh regularization approach allowing to restore a non-distorted high-quality mesh in an adaptive manner without the need for expensive re-meshing procedures. The core idea of this approach lies in the definition of a finite element distortion potential considering contributions from different distortion modes such as skewness and aspect ratio of the elements. The regularized mesh is found by minimization of this potential. Moreover, based on the concept of spatial localization functions, the method allows to specify tailored requirements on mesh resolution and quality for regions with strongly localized mechanical deformation and mesh distortion. In addition, while existing mesh regularization schemes often keep the boundary nodes of the discretization fixed, we propose a mesh-sliding algorithm based on variationally consistent mortar methods allowing for an unrestricted tangential motion of nodes along the problem boundary. Especially for problems involving significant surface deformation (e.g., frictional contact), this approach allows for an improved mesh relaxation as compared to schemes with fixed boundary nodes. To transfer data such as tensor-valued history variables of the material model from the old (distorted) to the new (regularized) mesh, a structure-preserving invariant interpolation scheme for second-order tensors is employed, which has been proposed in our previous work and is designed to preserve important properties of tensor-valued data such as objectivity and positive definiteness. As a practically relevant application scenario, we consider the thermo-mechanical expansion of materials such as foams involving extreme volume changes by up to two orders of magnitude along with large and strongly localized strains as well as thermo-mechanical contact interaction. For this scenario, it is demonstrated that the proposed regularization approach preserves a high mesh quality at small computational costs. In contrast, simulations without mesh adaption are shown to lead to significant mesh distortion, deteriorating result quality, and, eventually, to non-convergence of the numerical solution scheme.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":null,"pages":null},"PeriodicalIF":6.9,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441961","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 : 2024-10-15DOI: 10.1016/j.cma.2024.117404
Among several recently proposed data-driven Reduced Order Models (ROMs), the coupling of Proper Orthogonal Decompositions (POD) and deep learning-based ROMs (DL-ROMs) has proved to be a successful strategy to construct non-intrusive, highly accurate, surrogates for the real time solution of parametric nonlinear time-dependent PDEs. Inexpensive to evaluate, POD-DL-ROMs are also relatively fast to train, thanks to their limited complexity. However, POD-DL-ROMs account for the physical laws governing the problem at hand only through the training data, that are usually obtained through a full order model (FOM) relying on a high-fidelity discretization of the underlying equations. Moreover, the accuracy of POD-DL-ROMs strongly depends on the amount of available data. In this paper, we consider a major extension of POD-DL-ROMs by enforcing the fulfillment of the governing physical laws in the training process – that is, by making them physics-informed – to compensate for possible scarce and/or unavailable data and improve the overall reliability. To do that, we first complement POD-DL-ROMs with a trunk net architecture, endowing them with the ability to compute the problem’s solution at every point in the spatial domain, and ultimately enabling a seamless computation of the physics-based loss by means of the strong continuous formulation. Then, we introduce an efficient training strategy that limits the notorious computational burden entailed by a physics-informed training phase. In particular, we take advantage of the few available data to develop a low-cost pre-training procedure; then, we fine-tune the architecture in order to further improve the prediction reliability. Accuracy and efficiency of the resulting pre-trained physics-informed DL-ROMs (PTPI-DL-ROMs) are then assessed on a set of test cases ranging from non-affinely parametrized advection–diffusion–reaction equations, to nonlinear problems like the Navier–Stokes equations for fluid flows.
{"title":"PTPI-DL-ROMs: Pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEs","authors":"","doi":"10.1016/j.cma.2024.117404","DOIUrl":"10.1016/j.cma.2024.117404","url":null,"abstract":"<div><div>Among several recently proposed data-driven Reduced Order Models (ROMs), the coupling of Proper Orthogonal Decompositions (POD) and deep learning-based ROMs (DL-ROMs) has proved to be a successful strategy to construct non-intrusive, highly accurate, surrogates for the <em>real time</em> solution of parametric nonlinear time-dependent PDEs. Inexpensive to evaluate, POD-DL-ROMs are also relatively fast to train, thanks to their limited complexity. However, POD-DL-ROMs account for the physical laws governing the problem at hand only through the training data, that are usually obtained through a full order model (FOM) relying on a high-fidelity discretization of the underlying equations. Moreover, the accuracy of POD-DL-ROMs strongly depends on the amount of available data. In this paper, we consider a major extension of POD-DL-ROMs by enforcing the fulfillment of the governing physical laws in the training process – that is, by making them physics-informed – to compensate for possible scarce and/or unavailable data and improve the overall reliability. To do that, we first complement POD-DL-ROMs with a <em>trunk net</em> architecture, endowing them with the ability to compute the problem’s solution at every point in the spatial domain, and ultimately enabling a seamless computation of the physics-based loss by means of the strong continuous formulation. Then, we introduce an efficient training strategy that limits the notorious computational burden entailed by a physics-informed training phase. In particular, we take advantage of the few available data to develop a low-cost pre-training procedure; then, we fine-tune the architecture in order to further improve the prediction reliability. Accuracy and efficiency of the resulting pre-trained physics-informed DL-ROMs (PTPI-DL-ROMs) are then assessed on a set of test cases ranging from non-affinely parametrized advection–diffusion–reaction equations, to nonlinear problems like the Navier–Stokes equations for fluid flows.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":null,"pages":null},"PeriodicalIF":6.9,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142433311","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 : 2024-10-14DOI: 10.1016/j.cma.2024.117436
Computational methods, such as finite elements, are indispensable for modeling the mechanical compliance of elastic solids. However, as the size and geometric complexity of the domain increases, the cost of simulations becomes prohibitive. One example is the microstructure of a porous material, such as a piece of rock or bone sample, captured by an X-ray CT image. The solid geometry consists of numerous grains, cavities, and/or channels, with the domain large enough to allow inferring statistically converged macroscale properties. The pore-level multiscale method (PLMM) was recently proposed by the authors to reduce the associated computational cost through a divide-and-conquer strategy. The domain is decomposed into subdomains via watershed segmentation, and local basis and correction functions are built numerically, then assembled to obtain an approximate solution. However, PLMM is limited to domains corresponding to microscale porous media, incurs large errors when modeling loading conditions that generate significant bending/twisting moments locally, and it is equipped with only one mechanism to control approximation errors during a simulation. Here, we generalize PLMM into a high-order variant, called hPLMM, that removes these drawbacks. In hPLMM, appropriate mortar spaces are defined at subdomain interfaces that allow improving the boundary conditions used to solve local problems on the subdomains, thus the accuracy of the approximation. Moreover, errors can be reduced by a second mechanism wherein an upfront cost is paid prior to a simulation, useful if basis functions can be reused many times, e.g., across loading steps. Finally, the method applies not just to pore-scale, but also Darcy-scale and non-porous domains. We validate hPLMM against a range of complex 2D/3D geometries and discuss its convergence and algorithmic complexity. Implications for modeling failure and nonlinear problems are discussed.
{"title":"High-order multiscale method for elastic deformation of complex geometries","authors":"","doi":"10.1016/j.cma.2024.117436","DOIUrl":"10.1016/j.cma.2024.117436","url":null,"abstract":"<div><div>Computational methods, such as finite elements, are indispensable for modeling the mechanical compliance of elastic solids. However, as the size and geometric complexity of the domain increases, the cost of simulations becomes prohibitive. One example is the microstructure of a porous material, such as a piece of rock or bone sample, captured by an X-ray <span><math><mi>μ</mi></math></span>CT image. The solid geometry consists of numerous grains, cavities, and/or channels, with the domain large enough to allow inferring statistically converged macroscale properties. The pore-level multiscale method (PLMM) was recently proposed by the authors to reduce the associated computational cost through a divide-and-conquer strategy. The domain is decomposed into subdomains via watershed segmentation, and local basis and correction functions are built numerically, then assembled to obtain an approximate solution. However, PLMM is limited to domains corresponding to microscale porous media, incurs large errors when modeling loading conditions that generate significant bending/twisting moments locally, and it is equipped with only one mechanism to control approximation errors <em>during</em> a simulation. Here, we generalize PLMM into a high-order variant, called hPLMM, that removes these drawbacks. In hPLMM, appropriate mortar spaces are defined at subdomain interfaces that allow improving the boundary conditions used to solve local problems on the subdomains, thus the accuracy of the approximation. Moreover, errors can be reduced by a second mechanism wherein an upfront cost is paid <em>prior to</em> a simulation, useful if basis functions can be reused many times, e.g., across loading steps. Finally, the method applies not just to pore-scale, but also Darcy-scale and non-porous domains. We validate hPLMM against a range of complex 2D/3D geometries and discuss its convergence and algorithmic complexity. Implications for modeling failure and nonlinear problems are discussed.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":null,"pages":null},"PeriodicalIF":6.9,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142433309","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 : 2024-10-14DOI: 10.1016/j.cma.2024.117445
The detection and resolution of contacts among irregular-shaped particles pose significant challenges in the discrete element method (DEM) and recent advancements have introduced a machine learning-enabled approach specifically tailored for contact detection and resolution in two dimensions. Building upon this progress, this paper extends the application of machine learning-enabled DEM to encompass the more complex and realistic three-dimensional (3D) scenario. Particles are modeled using a polyhedral representation with arbitrary shapes, and contact behavior is governed by an energy-conserving contact model based on contact volumes. The efficacy of the machine learning-enabled 3D DEM is evaluated through comparative analyses of computational time and simulation results across individual contact as well as whole DEM simulations against those obtained from the conventional DEM. The findings indicate that the machine learning-enabled approach adeptly identifies and resolves contacts among 3D irregular-shaped particles while accurately reproducing the mechanical characteristics of densely contacting particle assemblies. The computational issues and challenges associated with the machine learning-enabled DEM are also discussed. The study highlights that the machine learning-enabled approach significantly enhances computational efficiency, showcasing its potential to advance complex DEM simulations in a more efficient manner.
在离散元素法（DEM）中，不规则形状颗粒间接触的检测和解析是一项重大挑战，而最近的进展则引入了一种专门针对二维接触检测和解析的机器学习方法。在这一进展的基础上，本文扩展了机器学习离散元素法的应用范围，以涵盖更复杂、更现实的三维（3D）场景。粒子采用任意形状的多面体表示法建模，接触行为由基于接触体积的能量守恒接触模型控制。通过对单个接触和整个 DEM 仿真的计算时间和仿真结果与传统 DEM 仿真结果的比较分析，评估了机器学习 3D DEM 的功效。研究结果表明，支持机器学习的方法能够巧妙地识别和解决三维不规则形状颗粒之间的接触，同时准确地再现了密集接触颗粒组件的机械特性。研究还讨论了与机器学习 DEM 相关的计算问题和挑战。研究强调，机器学习支持方法显著提高了计算效率，展示了以更高效的方式推进复杂 DEM 仿真的潜力。
{"title":"Machine-learning-enabled discrete element method: The extension to three dimensions and computational issues","authors":"","doi":"10.1016/j.cma.2024.117445","DOIUrl":"10.1016/j.cma.2024.117445","url":null,"abstract":"<div><div>The detection and resolution of contacts among irregular-shaped particles pose significant challenges in the discrete element method (DEM) and recent advancements have introduced a machine learning-enabled approach specifically tailored for contact detection and resolution in two dimensions. Building upon this progress, this paper extends the application of machine learning-enabled DEM to encompass the more complex and realistic three-dimensional (3D) scenario. Particles are modeled using a polyhedral representation with arbitrary shapes, and contact behavior is governed by an energy-conserving contact model based on contact volumes. The efficacy of the machine learning-enabled 3D DEM is evaluated through comparative analyses of computational time and simulation results across individual contact as well as whole DEM simulations against those obtained from the conventional DEM. The findings indicate that the machine learning-enabled approach adeptly identifies and resolves contacts among 3D irregular-shaped particles while accurately reproducing the mechanical characteristics of densely contacting particle assemblies. The computational issues and challenges associated with the machine learning-enabled DEM are also discussed. The study highlights that the machine learning-enabled approach significantly enhances computational efficiency, showcasing its potential to advance complex DEM simulations in a more efficient manner.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":null,"pages":null},"PeriodicalIF":6.9,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142433310","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 : 2024-10-12DOI: 10.1016/j.cma.2024.117418
Bayesian analysis enables combining prior knowledge with measurement data to learn model parameters. Commonly, one resorts to computing the maximum a posteriori (MAP) estimate, when only a point estimate of the parameters is of interest. We apply MAP estimation in the context of structural dynamic models, where the system response can be described by the frequency response function. To alleviate high computational demands from repeated expensive model calls, we utilize a rational polynomial chaos expansion (RPCE) surrogate model that expresses the system frequency response as a rational of two polynomials with complex coefficients. We propose an extension to an existing sparse Bayesian learning approach for RPCE based on Laplace’s approximation for the posterior distribution of the denominator coefficients. Furthermore, we introduce a Bayesian optimization approach, which allows to adaptively enrich the experimental design throughout the optimization process of MAP estimation. Thereby, we utilize the expected improvement acquisition function as a means to identify sample points in the input space that are possibly associated with large objective function values. The acquisition function is estimated through Monte Carlo sampling based on the posterior distribution of the expansion coefficients identified in the sparse Bayesian learning process. By combining the sparsity-inducing learning procedure with the sequential experimental design, we effectively reduce the number of model evaluations in the MAP estimation problem. We demonstrate the applicability of the presented methods on the parameter updating problem of an algebraic two-degree-of-freedom system and the finite element model of a cross-laminated timber plate.
{"title":"Maximum a posteriori estimation for linear structural dynamics models using Bayesian optimization with rational polynomial chaos expansions","authors":"","doi":"10.1016/j.cma.2024.117418","DOIUrl":"10.1016/j.cma.2024.117418","url":null,"abstract":"<div><div>Bayesian analysis enables combining prior knowledge with measurement data to learn model parameters. Commonly, one resorts to computing the maximum a posteriori (MAP) estimate, when only a point estimate of the parameters is of interest. We apply MAP estimation in the context of structural dynamic models, where the system response can be described by the frequency response function. To alleviate high computational demands from repeated expensive model calls, we utilize a rational polynomial chaos expansion (RPCE) surrogate model that expresses the system frequency response as a rational of two polynomials with complex coefficients. We propose an extension to an existing sparse Bayesian learning approach for RPCE based on Laplace’s approximation for the posterior distribution of the denominator coefficients. Furthermore, we introduce a Bayesian optimization approach, which allows to adaptively enrich the experimental design throughout the optimization process of MAP estimation. Thereby, we utilize the expected improvement acquisition function as a means to identify sample points in the input space that are possibly associated with large objective function values. The acquisition function is estimated through Monte Carlo sampling based on the posterior distribution of the expansion coefficients identified in the sparse Bayesian learning process. By combining the sparsity-inducing learning procedure with the sequential experimental design, we effectively reduce the number of model evaluations in the MAP estimation problem. We demonstrate the applicability of the presented methods on the parameter updating problem of an algebraic two-degree-of-freedom system and the finite element model of a cross-laminated timber plate.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":null,"pages":null},"PeriodicalIF":6.9,"publicationDate":"2024-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142419867","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 : 2024-10-11DOI: 10.1016/j.cma.2024.117443
This work deals with optimal control problems for constrained mechanical systems whose motion is governed by differential algebraic equations (DAEs). Both index-3 DAEs and stabilized index-2 DAEs are considered. Two alternative formulations of the optimal control problem are compared to each other. It is shown that symmetries of the optimal control problem lead to the conservation of generalized momentum maps. These generalized momentum maps are related to quadratic invariants of the optimal control problem. A direct discretization approach is newly proposed which is (i) capable to conserve the quadratic invariants, and (ii) equivalent to the indirect approach to the optimal control problem. Numerical examples are presented to access the properties of the newly developed schemes.
{"title":"Optimal control of constrained mechanical systems in redundant coordinates: Formulation and structure-preserving discretization","authors":"","doi":"10.1016/j.cma.2024.117443","DOIUrl":"10.1016/j.cma.2024.117443","url":null,"abstract":"<div><div>This work deals with optimal control problems for constrained mechanical systems whose motion is governed by differential algebraic equations (DAEs). Both index-3 DAEs and stabilized index-2 DAEs are considered. Two alternative formulations of the optimal control problem are compared to each other. It is shown that symmetries of the optimal control problem lead to the conservation of generalized momentum maps. These generalized momentum maps are related to quadratic invariants of the optimal control problem. A direct discretization approach is newly proposed which is (i) capable to conserve the quadratic invariants, and (ii) equivalent to the indirect approach to the optimal control problem. Numerical examples are presented to access the properties of the newly developed schemes.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":null,"pages":null},"PeriodicalIF":6.9,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142419877","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 : 2024-10-10DOI: 10.1016/j.cma.2024.117277
Modern adaptive finite element (FE) algorithms for solution of initial-boundary-value-problems (IBVP) employ goal-oriented error measures in order to assess the quality of computational results for the physical event under investigation. However, traditional time-stepping algorithms for solution of the corresponding dual problem run backwards-in-time, which due to additional storage requirements might become a serious drawback when an extensive number of time steps for the FEM simulation arises. In this paper, we take advantage of an end-boundary-value-problem (EBVP) associated to IBVP, with corresponding dual-problem running forwards in time. In order to obtain a unified framework for numerical approximation of primal and dual weak forms for both, IBVP and EBVP, respectively, we apply the concept of downwind and upwind approximations not only to the trial functions but also to the test functions. This results into eight different integration schemes. On this basis, as a main result of this contribution, a time-stepping algorithm is obtained, which runs forwards-in-time for the dual problem and therefore avoids the additional storage requirements of the traditional backwards-in-time stepping procedures. The presented algorithm is numerically tested and validated for a CT-specimen for elastic and elasto-plastic behavior, where the constitutive equations are written in Prandtl–Reuss type format.
{"title":"Downwind and upwind approximations for primal and dual problems of elasto-plasticity with Prandtl–Reuss type material laws","authors":"","doi":"10.1016/j.cma.2024.117277","DOIUrl":"10.1016/j.cma.2024.117277","url":null,"abstract":"<div><div>Modern adaptive finite element (FE) algorithms for solution of initial-boundary-value-problems (IBVP) employ goal-oriented error measures in order to assess the quality of computational results for the physical event under investigation. However, traditional time-stepping algorithms for solution of the corresponding dual problem run backwards-in-time, which due to additional storage requirements might become a serious drawback when an extensive number of time steps for the FEM simulation arises. In this paper, we take advantage of an end-boundary-value-problem (EBVP) associated to IBVP, with corresponding dual-problem running forwards in time. In order to obtain a unified framework for numerical approximation of primal and dual weak forms for both, IBVP and EBVP, respectively, we apply the concept of downwind and upwind approximations not only to the trial functions but also to the test functions. This results into eight different integration schemes. On this basis, as a main result of this contribution, a time-stepping algorithm is obtained, which runs forwards-in-time for the dual problem and therefore avoids the additional storage requirements of the traditional backwards-in-time stepping procedures. The presented algorithm is numerically tested and validated for a CT-specimen for elastic and elasto-plastic behavior, where the constitutive equations are written in Prandtl–Reuss type format.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":null,"pages":null},"PeriodicalIF":6.9,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142419879","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 : 2024-10-09DOI: 10.1016/j.cma.2024.117439
In dynamic vibration reduction design, the frequency-dependent material properties are crucial for the optimal configuration, especially in the problem of prescribed frequency forbidden band. In this paper, a new dynamic topology optimization method for structures with frequency-dependent material properties is proposed to achieve the vibration reduction design in the prescribed frequency forbidden band. First, a dynamic topology optimization model is established for the problem studied in this paper. This model integrates the solution method for frequency-dependent problem, dynamic isolated structures elimination method and the formulation of prescribed frequency forbidden band constraints, which are based on the research results previously developed by the authors. Additionally, different interpolation schemes are used for different number of material designs. The above optimization model is intended to consider nonlinear terms and design several frequency-dependent structures with prescribed frequency forbidden bands that are more in line with practical engineering problems, so that they can accurately avoid the operating frequency range, thus improving the service life of engineering equipment. Finally, to address common numerical problems, the "bound formulation" and "robust formulation" are employed, enhancing the applicability and robustness of the method for the application in topology optimization. The effectiveness of the developed method is supported by two types optimization problems, including single-material and bi-material examples. The cross-check results reveal that when considering frequency-dependent terms, the design results are better and closer to the practical engineering problem compared to linear structures.
{"title":"Dynamic topology optimization for structures exhibiting frequency-dependent material properties with prescribed frequency forbidden band","authors":"","doi":"10.1016/j.cma.2024.117439","DOIUrl":"10.1016/j.cma.2024.117439","url":null,"abstract":"<div><div>In dynamic vibration reduction design, the frequency-dependent material properties are crucial for the optimal configuration, especially in the problem of prescribed frequency forbidden band. In this paper, a new dynamic topology optimization method for structures with frequency-dependent material properties is proposed to achieve the vibration reduction design in the prescribed frequency forbidden band. First, a dynamic topology optimization model is established for the problem studied in this paper. This model integrates the solution method for frequency-dependent problem, dynamic isolated structures elimination method and the formulation of prescribed frequency forbidden band constraints, which are based on the research results previously developed by the authors. Additionally, different interpolation schemes are used for different number of material designs. The above optimization model is intended to consider nonlinear terms and design several frequency-dependent structures with prescribed frequency forbidden bands that are more in line with practical engineering problems, so that they can accurately avoid the operating frequency range, thus improving the service life of engineering equipment. Finally, to address common numerical problems, the \"bound formulation\" and \"robust formulation\" are employed, enhancing the applicability and robustness of the method for the application in topology optimization. The effectiveness of the developed method is supported by two types optimization problems, including single-material and bi-material examples. The cross-check results reveal that when considering frequency-dependent terms, the design results are better and closer to the practical engineering problem compared to linear structures.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":null,"pages":null},"PeriodicalIF":6.9,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142419730","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 : 2024-10-09DOI: 10.1016/j.cma.2024.117437
Simulating the flow of two fluid phases in porous media is a challenging task, especially when fractures are included in the simulation. Fractures may have highly heterogeneous properties compared to the surrounding rock matrix, significantly affecting fluid flow, and at the same time hydraulic apertures that are much smaller than any other characteristic sizes in the domain. Generally, flow simulators face difficulties with counter-current flow, generated by gravity and pressure gradients, which hinders the convergence of non-linear solvers (Newton).
In this work, we model the fracture geometry with a mixed-dimensional discrete fracture network, thus lightening the computational burden associated to an equi-dimensional representation. We address the issue of counter-current flows with appropriate spatial discretization of the advective fluid fluxes, with the aim of improving the convergence speed of the non-linear solver. In particular, the extension of the hybrid upwinding to the mixed-dimensional framework, with the use of a phase potential upstreaming at the interfaces of subdomains.
We test the method across several cases with different flow regimes and fracture network geometries. Results show robustness of the chosen discretization and a consistent improvements, in terms of Newton iterations, compared to using phase potential upstreaming everywhere.
{"title":"A hybrid upwind scheme for two-phase flow in fractured porous media","authors":"","doi":"10.1016/j.cma.2024.117437","DOIUrl":"10.1016/j.cma.2024.117437","url":null,"abstract":"<div><div>Simulating the flow of two fluid phases in porous media is a challenging task, especially when fractures are included in the simulation. Fractures may have highly heterogeneous properties compared to the surrounding rock matrix, significantly affecting fluid flow, and at the same time hydraulic apertures that are much smaller than any other characteristic sizes in the domain. Generally, flow simulators face difficulties with counter-current flow, generated by gravity and pressure gradients, which hinders the convergence of non-linear solvers (Newton).</div><div>In this work, we model the fracture geometry with a mixed-dimensional discrete fracture network, thus lightening the computational burden associated to an equi-dimensional representation. We address the issue of counter-current flows with appropriate spatial discretization of the advective fluid fluxes, with the aim of improving the convergence speed of the non-linear solver. In particular, the extension of the hybrid upwinding to the mixed-dimensional framework, with the use of a phase potential upstreaming at the interfaces of subdomains.</div><div>We test the method across several cases with different flow regimes and fracture network geometries. Results show robustness of the chosen discretization and a consistent improvements, in terms of Newton iterations, compared to using phase potential upstreaming everywhere.</div></div>","PeriodicalId":55222,"journal":{"name":"Computer Methods in Applied Mechanics and Engineering","volume":null,"pages":null},"PeriodicalIF":6.9,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142419664","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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