Pub Date : 2025-07-14DOI: 10.1016/j.finel.2025.104405
Minqiang Xu , Lei Zhang , Boying Wu , Kai Liu
In this paper, we introduce unified Hessian recovery-based finite element methods (HRB–FEM) and finite volume methods (HRB–FVM) for 2D biharmonic equations. Within the framework of Petrov–Galerkin methods, we propose a novel formulation. Initially, we employ the Hessian recovery operator to discretize the Laplacian operator, subsequently integrating it into both the standard Lagrange finite element framework and finite volume framework. Through tailored treatments of Neumann-type boundary conditions aimed at reducing computational overhead, we extend our Hessian recovery-based FEM to address phase field equations. Numerical experiments confirm optimal order of convergence under and norms, demonstrating rates of and respectively for both proposed methods. Furthermore, a series of benchmark tests highlight the robustness of our approach and its ability to faithfully capture the physical characteristics during prolonged simulations of phase field equations.
{"title":"A novel class of Hessian recovery-based numerical methods for solving biharmonic equations and their applications in phase field modeling","authors":"Minqiang Xu , Lei Zhang , Boying Wu , Kai Liu","doi":"10.1016/j.finel.2025.104405","DOIUrl":"10.1016/j.finel.2025.104405","url":null,"abstract":"<div><div>In this paper, we introduce unified Hessian recovery-based <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> finite element methods (HRB–FEM) and finite volume methods (HRB–FVM) for 2D biharmonic equations. Within the framework of Petrov–Galerkin methods, we propose a novel <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></mrow></math></span> formulation. Initially, we employ the Hessian recovery operator to discretize the Laplacian operator, subsequently integrating it into both the standard <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msup></math></span> Lagrange finite element framework and finite volume framework. Through tailored treatments of Neumann-type boundary conditions aimed at reducing computational overhead, we extend our Hessian recovery-based FEM to address phase field equations. Numerical experiments confirm optimal order of convergence under <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> norms, demonstrating rates of <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span> respectively for both proposed methods. Furthermore, a series of benchmark tests highlight the robustness of our approach and its ability to faithfully capture the physical characteristics during prolonged simulations of phase field equations.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"251 ","pages":"Article 104405"},"PeriodicalIF":3.5,"publicationDate":"2025-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144614377","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 micropolar elasticity finite element method is widely used for analyzing advanced materials with complex microstructures, but existing implementations often suffer from computational inefficiency due to full integration schemes and shear locking in bending scenarios. This study proposes a high-performance, reduced-integration, first-order hexahedral micropolar element to address these limitations. The formulation combines standard Lagrange interpolation with uniform strain and curvature fields, ensuring patch test satisfaction and accuracy in skewed configurations. An artificial stiffness method is introduced to suppress displacement and rotational hourglass instabilities. Rigorous numerical validations, including force and displacement patch tests, cantilever beam bending, and free vibration analysis, demonstrate the superior accuracy and computational efficiency of the element. Furthermore, applications to star-shaped lattices and 3D chiral metamaterials highlight its effectiveness in capturing microstructure-dependent mechanical behaviors, such as unexpected bending deformation and tension-twist coupling. The proposed element significantly enhances computational efficiency in homogenization simulation, providing a robust and practical tool for the simulation-driven design of advanced mechanical metamaterials with complex deformation mechanisms.
{"title":"Reduced-integration hexahedral finite element for static and vibration analysis of micropolar continuum","authors":"Yu Yao , Xin Zhao , Linghao Chen , Yujie Gu , Tianqi Zhou","doi":"10.1016/j.finel.2025.104412","DOIUrl":"10.1016/j.finel.2025.104412","url":null,"abstract":"<div><div>The micropolar elasticity finite element method is widely used for analyzing advanced materials with complex microstructures, but existing implementations often suffer from computational inefficiency due to full integration schemes and shear locking in bending scenarios. This study proposes a high-performance, reduced-integration, first-order hexahedral micropolar element to address these limitations. The formulation combines standard Lagrange interpolation with uniform strain and curvature fields, ensuring patch test satisfaction and accuracy in skewed configurations. An artificial stiffness method is introduced to suppress displacement and rotational hourglass instabilities. Rigorous numerical validations, including force and displacement patch tests, cantilever beam bending, and free vibration analysis, demonstrate the superior accuracy and computational efficiency of the element. Furthermore, applications to star-shaped lattices and 3D chiral metamaterials highlight its effectiveness in capturing microstructure-dependent mechanical behaviors, such as unexpected bending deformation and tension-twist coupling. The proposed element significantly enhances computational efficiency in homogenization simulation, providing a robust and practical tool for the simulation-driven design of advanced mechanical metamaterials with complex deformation mechanisms.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"251 ","pages":"Article 104412"},"PeriodicalIF":3.5,"publicationDate":"2025-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144611919","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}
Pub Date : 2025-07-03DOI: 10.1016/j.finel.2025.104404
Andrew B. Groeneveld , Pinlei Chen
We propose a method for modeling interfacial damage and debonding under quasi-static loads using immersed meshes in 3D at finite strains. This is an extension of our previous work on an immersed variational multiscale discontinuous Galerkin (VMDG) method in 2D. The variational approach remains the same, but transitioning from 2D to 3D introduces significant complications in the computational geometry aspects. The immersed VMDG method is a stabilized interface formulation derived using variational multiscale (VMS) ideas to apply discontinuous Galerkin (DG) treatment to the interface while employing a continuous Galerkin (CG) approximation elsewhere. Key benefits of VMDG are the variationally derived stabilization terms that evolve during deformation and are free of user-defined parameters. Also, the transition from perfect bond to damage behavior at the interface is handled naturally by incorporating an interfacial gap variable governed by a yield criterion and a flow rule. To support 3D simulations, we introduce algorithms for integrating cut elements, forming interface segments, and computing the VMDG stabilization tensor. Cut-element integration is performed using voxel-based moment-fitting integration to avoid the robustness issues associated with using mesh Booleans and tetrahedral integration cells. A simplification of the stabilization tensor is also proposed to reduce the computational cost while retaining the variational character of the stabilization. Several numerical examples are presented to demonstrate the robustness, efficiency, and range of applicability of the method.
{"title":"Three-dimensional simulation of finite-strain debonding using immersed meshes","authors":"Andrew B. Groeneveld , Pinlei Chen","doi":"10.1016/j.finel.2025.104404","DOIUrl":"10.1016/j.finel.2025.104404","url":null,"abstract":"<div><div>We propose a method for modeling interfacial damage and debonding under quasi-static loads using immersed meshes in 3D at finite strains. This is an extension of our previous work on an immersed variational multiscale discontinuous Galerkin (VMDG) method in 2D. The variational approach remains the same, but transitioning from 2D to 3D introduces significant complications in the computational geometry aspects. The immersed VMDG method is a stabilized interface formulation derived using variational multiscale (VMS) ideas to apply discontinuous Galerkin (DG) treatment to the interface while employing a continuous Galerkin (CG) approximation elsewhere. Key benefits of VMDG are the variationally derived stabilization terms that evolve during deformation and are free of user-defined parameters. Also, the transition from perfect bond to damage behavior at the interface is handled naturally by incorporating an interfacial gap variable governed by a yield criterion and a flow rule. To support 3D simulations, we introduce algorithms for integrating cut elements, forming interface segments, and computing the VMDG stabilization tensor. Cut-element integration is performed using voxel-based moment-fitting integration to avoid the robustness issues associated with using mesh Booleans and tetrahedral integration cells. A simplification of the stabilization tensor is also proposed to reduce the computational cost while retaining the variational character of the stabilization. Several numerical examples are presented to demonstrate the robustness, efficiency, and range of applicability of the method.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"250 ","pages":"Article 104404"},"PeriodicalIF":3.5,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144536086","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}
Pub Date : 2025-07-01DOI: 10.1016/j.finel.2025.104398
Zhengrong Xie
The flux vector splitting (FVS) method has firstly been incorporated into the Runge–Kutta Discontinuous Galerkin (RKDG) framework for reconstructing the numerical fluxes required for the spatial semi-discrete formulation, setting it apart from the conventional RKDG approaches that typically utilize the Lax–Friedrichs flux scheme or classical Riemann solvers such as HLLC. The control equations are initially reformulated into a flux-split form. Subsequently, a variational approach is applied to this flux-split form, from which a DG spatial semi-discrete scheme based on FVS is derived. Then, FVS-RKDG is implemented in two-dimensional case by splitting the normal flux on cell interfaces instead of splitting dimension by dimension in the x and y directions Finally, the concept of “flux vector splitting based on Jacobian eigenvalue decomposition” has been applied to the conservative linear scalar transport equations and the nonlinear Burgers’ equation. This approach has led to the rederivation of the classical Lax–Friedrichs flux scheme and the provision of a Steger–Warming flux scheme for scalar cases.
{"title":"Runge–Kutta discontinuous Galerkin method based on flux vector splitting for hyperbolic conservation laws","authors":"Zhengrong Xie","doi":"10.1016/j.finel.2025.104398","DOIUrl":"10.1016/j.finel.2025.104398","url":null,"abstract":"<div><div>The flux vector splitting (FVS) method has firstly been incorporated into the Runge–Kutta Discontinuous Galerkin (RKDG) framework for reconstructing the numerical fluxes required for the spatial semi-discrete formulation, setting it apart from the conventional RKDG approaches that typically utilize the Lax–Friedrichs flux scheme or classical Riemann solvers such as HLLC. The control equations are initially reformulated into a flux-split form. Subsequently, a variational approach is applied to this flux-split form, from which a DG spatial semi-discrete scheme based on FVS is derived. Then, FVS-RKDG is implemented in two-dimensional case by splitting the normal flux on cell interfaces instead of splitting dimension by dimension in the x and y directions Finally, the concept of “flux vector splitting based on Jacobian eigenvalue decomposition” has been applied to the conservative linear scalar transport equations and the nonlinear Burgers’ equation. This approach has led to the rederivation of the classical Lax–Friedrichs flux scheme and the provision of a Steger–Warming flux scheme for scalar cases.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"250 ","pages":"Article 104398"},"PeriodicalIF":3.5,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144518096","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}
Pub Date : 2025-06-28DOI: 10.1016/j.finel.2025.104403
Chongxin Zhang, Guangzhi Du, Xinxin Sun
In this paper, two kinds of two-grid domain decomposition methods for the coupled Dual-Porosity-Navier-Stokes system are proposed and analyzed by integrating the established robin-type domain decomposition approach with a two-grid strategy. Initially, we apply the established robin-type domain decomposition approach on a coarse grid to address the coupled problem. Subsequently, on a fine grid, we employ two distinct approaches: first, to solve the matrix and microfracture subproblems, followed by the Navier–Stokes subproblem. Both approaches fundamentally approximate the interface term using the coarse-grid solution. The proposed algorithms integrate the two-grid approach with the established domain decomposition method, capitalizing on the strengths of both techniques while addressing their respective limitations. Comprehensive theoretical analysis is established, and four in-depth numerical investigations are conducted to assess the efficiency, accuracy, and robustness of the proposed algorithms by comparing them with the domain decomposition method.
{"title":"Two-grid domain decomposition methods for the coupled Dual-Porosity-Navier-Stokes system with Beavers-Joseph interface condition","authors":"Chongxin Zhang, Guangzhi Du, Xinxin Sun","doi":"10.1016/j.finel.2025.104403","DOIUrl":"10.1016/j.finel.2025.104403","url":null,"abstract":"<div><div>In this paper, two kinds of two-grid domain decomposition methods for the coupled Dual-Porosity-Navier-Stokes system are proposed and analyzed by integrating the established robin-type domain decomposition approach with a two-grid strategy. Initially, we apply the established robin-type domain decomposition approach on a coarse grid to address the coupled problem. Subsequently, on a fine grid, we employ two distinct approaches: first, to solve the matrix and microfracture subproblems, followed by the Navier–Stokes subproblem. Both approaches fundamentally approximate the interface term using the coarse-grid solution. The proposed algorithms integrate the two-grid approach with the established domain decomposition method, capitalizing on the strengths of both techniques while addressing their respective limitations. Comprehensive theoretical analysis is established, and four in-depth numerical investigations are conducted to assess the efficiency, accuracy, and robustness of the proposed algorithms by comparing them with the domain decomposition method.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"250 ","pages":"Article 104403"},"PeriodicalIF":3.5,"publicationDate":"2025-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144501860","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}
Pub Date : 2025-06-28DOI: 10.1016/j.finel.2025.104387
Somraj Sen , Satyendra Kumar Singh , Arnab Banerjee
The materials proposed for use as shock absorbers exhibit dynamic properties closely resembling those of hard connective tissues, such as human bone tissue. Moreover, cellular and porous structures, like gyroids, are increasingly preferred for implant applications due to their tailored mechanical and dynamic properties, offering superior performance compared to solid materials. This observation inspires us to investigate and evaluate the dispersion characteristics of a lightweight architectured beam inspired from gyroid cellular structures (GCS) as its unit cell, aimed at comprehending its wave propagation behavior. A simplified model of GCS is conceptualized through the assembly of prismatic space frame elements, modeled using the spectral element method (SEM) within the framework of transfer matrix formulation of the harmonic solution. The proposed architectured beam demonstrates the presence of complete attenuation bandgap regions, attributed to the coupling of various wave modes. These complete bandgaps signify that waves of all modes within the specified frequencies are attenuated. Furthermore, the bandgaps are validated through the frequency response function obtained for a beam constructed by assembling multiple unit cells. The study also explores the influence of the structural parameters, including the slenderness ratio and diameter ratio on the attenuation bandwidth, offering insights into optimizing the beam’s dynamic performance.
{"title":"Waves in a bio-inspired gyroid cellular architectured metabeam","authors":"Somraj Sen , Satyendra Kumar Singh , Arnab Banerjee","doi":"10.1016/j.finel.2025.104387","DOIUrl":"10.1016/j.finel.2025.104387","url":null,"abstract":"<div><div>The materials proposed for use as shock absorbers exhibit dynamic properties closely resembling those of hard connective tissues, such as human bone tissue. Moreover, cellular and porous structures, like gyroids, are increasingly preferred for implant applications due to their tailored mechanical and dynamic properties, offering superior performance compared to solid materials. This observation inspires us to investigate and evaluate the dispersion characteristics of a lightweight architectured beam inspired from gyroid cellular structures (GCS) as its unit cell, aimed at comprehending its wave propagation behavior. A simplified model of GCS is conceptualized through the assembly of prismatic space frame elements, modeled using the spectral element method (SEM) within the framework of transfer matrix formulation of the harmonic solution. The proposed architectured beam demonstrates the presence of complete attenuation bandgap regions, attributed to the coupling of various wave modes. These complete bandgaps signify that waves of all modes within the specified frequencies are attenuated. Furthermore, the bandgaps are validated through the frequency response function obtained for a beam constructed by assembling multiple unit cells. The study also explores the influence of the structural parameters, including the slenderness ratio and diameter ratio on the attenuation bandwidth, offering insights into optimizing the beam’s dynamic performance.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"250 ","pages":"Article 104387"},"PeriodicalIF":3.5,"publicationDate":"2025-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144501859","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}
Pub Date : 2025-06-26DOI: 10.1016/j.finel.2025.104401
Shi-An Zhou, Song Yao
In this paper, a combined Floating Projection Topology Optimization (FPTO) method is applied to stress-based topology optimization of continuum structures. To balance the need for a stable optimization process under a low P-norm and the goal of achieving a uniform stress distribution at a higher P-norm, the FPTO method here is combined with piecewise P-norm strategy. A new parameter, solid rate, is introduced, serving not only as an additional convergence criterion but also as a decision criterion for adaptively increasing the P-norm value. To validate the effectiveness of the proposed FPTO method incorporating both piecewise P-norm stabilization and the solid rate control strategy, this study conducts stiffness maximization optimization with stress constraints on various typical structures. The results show that the method effectively reduces computational oscillations in stress-based topology optimization for structures with stress concentrations, which helps to the discovery of superior designs.
{"title":"Stress-based topology optimization of continuum structures incorporating a piecewise P-norm stabilization strategy","authors":"Shi-An Zhou, Song Yao","doi":"10.1016/j.finel.2025.104401","DOIUrl":"10.1016/j.finel.2025.104401","url":null,"abstract":"<div><div>In this paper, a combined Floating Projection Topology Optimization (FPTO) method is applied to stress-based topology optimization of continuum structures. To balance the need for a stable optimization process under a low P-norm and the goal of achieving a uniform stress distribution at a higher P-norm, the FPTO method here is combined with piecewise P-norm strategy. A new parameter, solid rate, is introduced, serving not only as an additional convergence criterion but also as a decision criterion for adaptively increasing the P-norm value. To validate the effectiveness of the proposed FPTO method incorporating both piecewise P-norm stabilization and the solid rate control strategy, this study conducts stiffness maximization optimization with stress constraints on various typical structures. The results show that the method effectively reduces computational oscillations in stress-based topology optimization for structures with stress concentrations, which helps to the discovery of superior designs.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"250 ","pages":"Article 104401"},"PeriodicalIF":3.5,"publicationDate":"2025-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144480638","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}
Pub Date : 2025-06-25DOI: 10.1016/j.finel.2025.104399
Christian Austin , Sara Pollock , L. Ridgway Scott
In this paper, we discuss a finite element implementation of the SRTD algorithm described by Girault and Scott for the steady-state case of a certain 3-parameter subset of the Oldroyd models. We compare it to the well-known EVSS method, which, though originally described for the upper-convected Maxwell model, can easily accommodate the Oldroyd 3-parameter model. We obtain numerical results for both methods on two benchmark problems: the lid-driven cavity problem and the journal-bearing, or eccentric rotating cylinders, problem. We find that the resulting finite element implementation of SRTD is stable with respect to mesh refinement and is generally faster than EVSS, though is not capable of reaching as high a Weissenberg number as EVSS.
{"title":"A finite element implementation of the SRTD algorithm for an Oldroyd 3-parameter viscoelastic fluid model","authors":"Christian Austin , Sara Pollock , L. Ridgway Scott","doi":"10.1016/j.finel.2025.104399","DOIUrl":"10.1016/j.finel.2025.104399","url":null,"abstract":"<div><div>In this paper, we discuss a finite element implementation of the SRTD algorithm described by Girault and Scott for the steady-state case of a certain 3-parameter subset of the Oldroyd models. We compare it to the well-known EVSS method, which, though originally described for the upper-convected Maxwell model, can easily accommodate the Oldroyd 3-parameter model. We obtain numerical results for both methods on two benchmark problems: the lid-driven cavity problem and the journal-bearing, or eccentric rotating cylinders, problem. We find that the resulting finite element implementation of SRTD is stable with respect to mesh refinement and is generally faster than EVSS, though is not capable of reaching as high a Weissenberg number as EVSS.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"250 ","pages":"Article 104399"},"PeriodicalIF":3.5,"publicationDate":"2025-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144471163","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}
Pub Date : 2025-06-18DOI: 10.1016/j.finel.2025.104402
Sang Truong Ha , Hyeong Cheol Park , Han Young Yoon , Hyoung Gwon Choi
We developed three parallel algorithms for a meshless geometric multigrid (GMG) method proposed for the finite element discretization of elliptic partial differential equation. These methods for parallel multigrid (PMG) are based on the message passing interface (MPI) for domain decomposition and coarse matrix aggregation (CMA) algorithm for coarser levels. Using coarse matrices obtained by a parallel Galerkin condition for the present meshless GMG, we proposed a parameter by which an optimal aggregation level is determined. This parameter is defined as the ratio of total number of external interface nodes from all the subdomains before aggregation to the number of non-zero entries of gathered matrix after aggregation. Three methods —M1, M2, and M3— are classified depending on how the coarsest matrix is solved and the number of coarser levels for which CMA is applied. M1 (M2) solves the coarsest matrix via an iterative (direct) solver applying CMA only for the coarsest level, whereas M3 determines the multigrid levels with CMA based on the parameter and employs a direct solver for the coarsest matrix. We found that M3 is more efficient than the others and much more efficient in the case of complicated geometry because communication overhead is reduced compared to the other methods. Furthermore, the present PMG could achieve super-linear scalability owing to the cache effect for a large problem.
{"title":"Investigation on an optimal aggregation level for a parallel meshless multigrid method based on domain decomposition method","authors":"Sang Truong Ha , Hyeong Cheol Park , Han Young Yoon , Hyoung Gwon Choi","doi":"10.1016/j.finel.2025.104402","DOIUrl":"10.1016/j.finel.2025.104402","url":null,"abstract":"<div><div>We developed three parallel algorithms for a meshless geometric multigrid (GMG) method proposed for the finite element discretization of elliptic partial differential equation. These methods for parallel multigrid (PMG) are based on the message passing interface (MPI) for domain decomposition and coarse matrix aggregation (CMA) algorithm for coarser levels. Using coarse matrices obtained by a parallel Galerkin condition for the present meshless GMG, we proposed a parameter by which an optimal aggregation level is determined. This parameter is defined as the ratio of total number of external interface nodes from all the subdomains before aggregation to the number of non-zero entries of gathered matrix after aggregation. Three methods <strong>—M1, M2, and M3—</strong> are classified depending on how the coarsest matrix is solved and the number of coarser levels for which CMA is applied. M1 (M2) solves the coarsest matrix via an iterative (direct) solver applying CMA only for the coarsest level, whereas M3 determines the multigrid levels with CMA based on the parameter and employs a direct solver for the coarsest matrix. We found that M3 is more efficient than the others and much more efficient in the case of complicated geometry because communication overhead is reduced compared to the other methods. Furthermore, the present PMG could achieve super-linear scalability owing to the cache effect for a large problem.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"250 ","pages":"Article 104402"},"PeriodicalIF":3.5,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144307470","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}
Pub Date : 2025-06-14DOI: 10.1016/j.finel.2025.104391
Jianping Zhang, Ou Guo, Yaping Zhao, Zhijian Zuo, Mintao Chen, Haishan Lu, Shuguang Gong
The multi-material structure topology optimization method considering length scale control (LSC) using the isogeometric analysis approach is put forward. The density distribution function (DDF) is applied for improving structural smoothness, and the resulting structure has smoother and clearer boundaries compared to the conventional finite element method. The alternating active-phase algorithm and gradient algorithm are utilized for building the multi-material interpolation model and updating the design variable, respectively. The effects of different LSC scheme, the maximum length scale control (MaxLSC) domain radius , the radius of DDF influence domain and the aggregation factor on the structure performance are investigated. The results show the structure with both MaxLSC and MinLSC applied exhibits a more uniform material distribution, the structural manufacturability is effectively guaranteed. When , the jagged boundaries appear in the structure, and when , the branching structures decrease. The topological structure obtained when has the relatively uniform material distribution. When , the island phenomenon appears in the structure. When , the branching structure is reduced and thickened simultaneously, the recommended range for is . When , the topological structure exhibits more branching structure in both materials. It is proved that the effectiveness of the LSC method can still be guaranteed in the three-dimensional problem and curved edge structure.
{"title":"Multi-material structure topology optimization considering length scale control based on isogeometric analysis approach","authors":"Jianping Zhang, Ou Guo, Yaping Zhao, Zhijian Zuo, Mintao Chen, Haishan Lu, Shuguang Gong","doi":"10.1016/j.finel.2025.104391","DOIUrl":"10.1016/j.finel.2025.104391","url":null,"abstract":"<div><div>The multi-material structure topology optimization method considering length scale control (LSC) using the isogeometric analysis approach is put forward. The density distribution function (DDF) is applied for improving structural smoothness, and the resulting structure has smoother and clearer boundaries compared to the conventional finite element method. The alternating active-phase algorithm and gradient algorithm are utilized for building the multi-material interpolation model and updating the design variable, respectively. The effects of different LSC scheme, the maximum length scale control (MaxLSC) domain radius <span><math><mrow><msub><mi>R</mi><mi>max</mi></msub></mrow></math></span>, the radius of DDF influence domain <span><math><mrow><msub><mi>r</mi><mtext>fil</mtext></msub></mrow></math></span> and the aggregation factor <span><math><mrow><msub><mi>p</mi><mi>n</mi></msub></mrow></math></span> on the structure performance are investigated. The results show the structure with both MaxLSC and MinLSC applied exhibits a more uniform material distribution, the structural manufacturability is effectively guaranteed. When <span><math><mrow><msub><mi>r</mi><mtext>fil</mtext></msub><mo>≤</mo><mn>3</mn><mi>h</mi></mrow></math></span>, the jagged boundaries appear in the structure, and when <span><math><mrow><msub><mi>r</mi><mtext>fil</mtext></msub><mo>≥</mo><mn>5</mn><mi>h</mi></mrow></math></span>, the branching structures decrease. The topological structure obtained when <span><math><mrow><msub><mi>r</mi><mtext>fil</mtext></msub><mo>=</mo><mn>3.5</mn><mi>h</mi><mo>−</mo><mn>4</mn><mi>h</mi></mrow></math></span> has the relatively uniform material distribution. When <span><math><mrow><msub><mi>R</mi><mi>max</mi></msub><mo>=</mo><mn>5</mn><mi>h</mi></mrow></math></span>, the island phenomenon appears in the structure. When <span><math><mrow><msub><mi>R</mi><mi>max</mi></msub><mo>=</mo><mn>10</mn><mi>h</mi></mrow></math></span>, the branching structure is reduced and thickened simultaneously, the recommended range for <span><math><mrow><msub><mi>R</mi><mi>max</mi></msub></mrow></math></span> is <span><math><mrow><mn>6</mn><mi>h</mi><mo>−</mo><mn>8</mn><mi>h</mi></mrow></math></span>. When <span><math><mrow><msub><mi>p</mi><mi>n</mi></msub><mo>=</mo><mn>80</mn><mo>−</mo><mn>110</mn></mrow></math></span>, the topological structure exhibits more branching structure in both materials. It is proved that the effectiveness of the LSC method can still be guaranteed in the three-dimensional problem and curved edge structure.</div></div>","PeriodicalId":56133,"journal":{"name":"Finite Elements in Analysis and Design","volume":"250 ","pages":"Article 104391"},"PeriodicalIF":3.5,"publicationDate":"2025-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144280979","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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