Pub Date : 2023-12-01DOI: 10.4208/cicp.oa-2023-0102
Qian Zhang,Min Zhang, Zhimin Zhang
We propose two families of nonconforming elements on cubical meshes: one for the −curl∆curl problem and the other for the Brinkman problem. The element for the −curl∆curl problem is the first nonconforming element on cubical meshes. The element for the Brinkman problem can yield a uniformly stable finite element method with respect to the viscosity coefficient $ν.$ The lowest-order elements for the −curl∆curl and the Brinkman problems have 48 and 30 DOFs on each cube, respectively. The two families of elements are subspaces of $H({rm curl};Ω)$ and $H({rm div};Ω),$ and they, as nonconforming approximation to $H({rm gradcurl};Ω)$ and $[H^1 (Ω)]^3,$ can form a discrete Stokes complex together with the serendipity finite element space and the piecewise polynomial space.
{"title":"Nonconforming Finite Elements for the −curl∆curl and Brinkman Problems on Cubical Meshes","authors":"Qian Zhang,Min Zhang, Zhimin Zhang","doi":"10.4208/cicp.oa-2023-0102","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0102","url":null,"abstract":"We propose two families of nonconforming elements on cubical meshes: one\u0000for the −curl∆curl problem and the other for the Brinkman problem. The element\u0000for the −curl∆curl problem is the first nonconforming element on cubical meshes.\u0000The element for the Brinkman problem can yield a uniformly stable finite element\u0000method with respect to the viscosity coefficient $ν.$ The lowest-order elements for the\u0000−curl∆curl and the Brinkman problems have 48 and 30 DOFs on each cube, respectively. The two families of elements are subspaces of $H({rm curl};Ω)$ and $H({rm div};Ω),$ and\u0000they, as nonconforming approximation to $H({rm gradcurl};Ω)$ and $[H^1\u0000(Ω)]^3,$ can form a\u0000discrete Stokes complex together with the serendipity finite element space and the\u0000piecewise polynomial space.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"13 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138567957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.4208/cicp.oa-2023-0142
Qiang Li,Gang Bao,Yanzhao Cao, Junshan Lin
In this work, we develop a stochastic gradient descent method for the computational optimal design of random rough surfaces in thin-film solar cells. We formulate the design problems as random PDE-constrained optimization problems and seek the optimal statistical parameters for the random surfaces. The optimizations at fixed frequency as well as at multiple frequencies and multiple incident angles are investigated. To evaluate the gradient of the objective function, we derive the shape derivatives for the interfaces and apply the adjoint state method to perform the computation. The stochastic gradient descent method evaluates the gradient of the objective function only at a few samples for each iteration, which reduces the computational cost significantly. Various numerical experiments are conducted to illustrate the efficiency of the method and significant increases of the absorptance for the optimal random structures. We also examine the convergence of the stochastic gradient descent algorithm theoretically and prove that the numerical method is convergent under certain assumptions for the random interfaces.
{"title":"A Stochastic Gradient Descent Method for Computational Design of Random Rough Surfaces in Solar Cells","authors":"Qiang Li,Gang Bao,Yanzhao Cao, Junshan Lin","doi":"10.4208/cicp.oa-2023-0142","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0142","url":null,"abstract":"In this work, we develop a stochastic gradient descent method for the computational optimal design of random rough surfaces in thin-film solar cells. We formulate the design problems as random PDE-constrained optimization problems and seek\u0000the optimal statistical parameters for the random surfaces. The optimizations at fixed\u0000frequency as well as at multiple frequencies and multiple incident angles are investigated. To evaluate the gradient of the objective function, we derive the shape derivatives for the interfaces and apply the adjoint state method to perform the computation.\u0000The stochastic gradient descent method evaluates the gradient of the objective function\u0000only at a few samples for each iteration, which reduces the computational cost significantly. Various numerical experiments are conducted to illustrate the efficiency of the\u0000method and significant increases of the absorptance for the optimal random structures.\u0000We also examine the convergence of the stochastic gradient descent algorithm theoretically and prove that the numerical method is convergent under certain assumptions\u0000for the random interfaces.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"79 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138741894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.4208/cicp.oa-2023-0112
Rihui Lan,Lili Ju,Zhu Wang, Max Gunzburger
The baroclinic-barotropic mode splitting technique is commonly employed in numerical solutions of the primitive equations for ocean modeling to deal with the multiple time scales of ocean dynamics. In this paper, a second-order implicit-explicit (IMEX) scheme is proposed to advance the baroclinic-barotropic split system. Specifically, the baroclinic mode and the layer thickness of fluid are evolved explicitly via the second-order strong stability preserving Runge-Kutta scheme, while the barotropic mode is advanced implicitly using the linearized Crank-Nicolson scheme. At each time step, the baroclinic velocity is first computed using an intermediate barotropic velocity. The barotropic velocity is then corrected by re-advancing the barotropic mode with an improved barotropic forcing. Finally, the layer thickness is updated by coupling the baroclinic and barotropic velocities together. In addition, numerical inconsistencies on the discretized sea surface height caused by the mode splitting are alleviated via a reconciliation process with carefully calculated flux deficits. Temporal truncation error is also analyzed to validate the second-order accuracy of the scheme. Finally, two benchmark tests from the MPAS-Ocean platform are conducted to numerically demonstrate the performance of the proposed IMEX scheme.
{"title":"A Second-Order Implicit-Explicit Scheme for the Baroclinic-Barotropic Split System of Primitive Equations","authors":"Rihui Lan,Lili Ju,Zhu Wang, Max Gunzburger","doi":"10.4208/cicp.oa-2023-0112","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0112","url":null,"abstract":"The baroclinic-barotropic mode splitting technique is commonly employed\u0000in numerical solutions of the primitive equations for ocean modeling to deal with the\u0000multiple time scales of ocean dynamics. In this paper, a second-order implicit-explicit\u0000(IMEX) scheme is proposed to advance the baroclinic-barotropic split system. Specifically, the baroclinic mode and the layer thickness of fluid are evolved explicitly via\u0000the second-order strong stability preserving Runge-Kutta scheme, while the barotropic\u0000mode is advanced implicitly using the linearized Crank-Nicolson scheme. At each\u0000time step, the baroclinic velocity is first computed using an intermediate barotropic velocity. The barotropic velocity is then corrected by re-advancing the barotropic mode\u0000with an improved barotropic forcing. Finally, the layer thickness is updated by coupling the baroclinic and barotropic velocities together. In addition, numerical inconsistencies on the discretized sea surface height caused by the mode splitting are alleviated\u0000via a reconciliation process with carefully calculated flux deficits. Temporal truncation\u0000error is also analyzed to validate the second-order accuracy of the scheme. Finally,\u0000two benchmark tests from the MPAS-Ocean platform are conducted to numerically\u0000demonstrate the performance of the proposed IMEX scheme.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"24 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with efficient numerical methods for the advection-diffusion equation in a heterogeneous porous medium containing fractures. A dimensionally reduced fracture model is considered, in which the fracture is represented as an interface between subdomains and is assumed to have larger permeability than the surrounding area. We develop three global-in-time domain decomposition methods coupled with operator splitting for the reduced fracture model, where the advection and the diffusion are treated separately by different numerical schemes and with different time steps. Importantly, smaller time steps can be used in the fracture-interface than in the subdomains. The first two methods are based on the physical transmission conditions, while the third one is based on the optimized Schwarz waveform relaxation approach with Ventcel-Robin transmission conditions. A discrete space-time interface system is formulated for each method and is solved iteratively and globally in time. Numerical results for two-dimensional problems with various Péclet numbers and different types of fracture are presented to illustrate and compare the convergence and accuracy in time of the proposed methods with local time stepping.
{"title":"Operator Splitting and Local Time-Stepping Methods for Transport Problems in Fractured Porous Media","authors":"Phuoc-Toan Huynh,Yanzhao Cao, Thi-Thao-Phuong Hoang","doi":"10.4208/cicp.oa-2022-0187","DOIUrl":"https://doi.org/10.4208/cicp.oa-2022-0187","url":null,"abstract":"This paper is concerned with efficient numerical methods for the advection-diffusion equation in a heterogeneous porous medium containing fractures. A dimensionally reduced fracture model is considered, in which the fracture is represented as\u0000an interface between subdomains and is assumed to have larger permeability than the\u0000surrounding area. We develop three global-in-time domain decomposition methods\u0000coupled with operator splitting for the reduced fracture model, where the advection\u0000and the diffusion are treated separately by different numerical schemes and with different time steps. Importantly, smaller time steps can be used in the fracture-interface\u0000than in the subdomains. The first two methods are based on the physical transmission conditions, while the third one is based on the optimized Schwarz waveform relaxation approach with Ventcel-Robin transmission conditions. A discrete space-time\u0000interface system is formulated for each method and is solved iteratively and globally\u0000in time. Numerical results for two-dimensional problems with various Péclet numbers and different types of fracture are presented to illustrate and compare the convergence\u0000and accuracy in time of the proposed methods with local time stepping.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"1 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138580007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.4208/cicp.oa-2023-0032
Yu Du, Haijun Wu
We extend the pure source transfer domain decomposition method (PSTDDM) to solve the perfectly matched layer approximation of Helmholtz scattering problems in heterogeneous media. We first propose some new source transfer operators, and then introduce the layer-wise and block-wise PSTDDMs based on these operators. In particular, it is proved that the solution obtained by the layer-wise PSTDDM in $mathbb{R}^2$ coincides with the exact solution to the heterogeneous Helmholtz problem in the computational domain. Second, we propose the iterative layer-wise and blockwise PSTDDMs, which are designed by simply iterating the PSTDDM alternatively over two staggered decompositions of the computational domain. Finally, extensive numerical tests in two and three dimensions show that, as the preconditioner for the GMRES method, the iterative PSTDDMs are more robust and efficient than PSTDDMs for solving heterogeneous Helmholtz problems.
{"title":"Iterative Pure Source Transfer Domain Decomposition Methods for Helmholtz Equations in Heterogeneous Media","authors":"Yu Du, Haijun Wu","doi":"10.4208/cicp.oa-2023-0032","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0032","url":null,"abstract":"We extend the pure source transfer domain decomposition method (PSTDDM) to solve the perfectly matched layer approximation of Helmholtz scattering\u0000problems in heterogeneous media. We first propose some new source transfer operators, and then introduce the layer-wise and block-wise PSTDDMs based on these\u0000operators. In particular, it is proved that the solution obtained by the layer-wise PSTDDM in $mathbb{R}^2$ coincides with the exact solution to the heterogeneous Helmholtz problem\u0000in the computational domain. Second, we propose the iterative layer-wise and blockwise PSTDDMs, which are designed by simply iterating the PSTDDM alternatively\u0000over two staggered decompositions of the computational domain. Finally, extensive\u0000numerical tests in two and three dimensions show that, as the preconditioner for the\u0000GMRES method, the iterative PSTDDMs are more robust and efficient than PSTDDMs\u0000for solving heterogeneous Helmholtz problems.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"14 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138568634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.4208/cicp.oa-2023-0027
Ling Ling Sun,Hai Bi, Yidu Yang
In this paper, based on the velocity-pressure formulation of the Stokes eigenvalue problem in $d$-dimensional case $(d=2,3),$ we propose a multigrid discretization of discontinuous Galerkin method using $mathbb{P}_k−mathbb{P}_k−1$ element $(k≥1)$ and prove its a priori error estimate. We also give the a posteriori error estimators for approximate eigenpairs, prove their reliability and efficiency for eigenfunctions, and also analyze their reliability for eigenvalues. We implement adaptive calculation, and the numerical results confirm our theoretical predictions and show that our method is efficient and can achieve the optimal convergence order $mathcal{O}(do f^{ frac{−2k}{d}} ).$
{"title":"A Multigrid Discretization of Discontinuous Galerkin Method for the Stokes Eigenvalue Problem","authors":"Ling Ling Sun,Hai Bi, Yidu Yang","doi":"10.4208/cicp.oa-2023-0027","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0027","url":null,"abstract":"In this paper, based on the velocity-pressure formulation of the Stokes eigenvalue problem in $d$-dimensional case $(d=2,3),$ we propose a multigrid discretization of\u0000discontinuous Galerkin method using $mathbb{P}_k−mathbb{P}_k−1$ element $(k≥1)$ and prove its a priori\u0000error estimate. We also give the a posteriori error estimators for approximate eigenpairs, prove their reliability and efficiency for eigenfunctions, and also analyze their\u0000reliability for eigenvalues. We implement adaptive calculation, and the numerical results confirm our theoretical predictions and show that our method is efficient and can\u0000achieve the optimal convergence order $mathcal{O}(do f^{ frac{−2k}{d}} ).$","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"105 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138716147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-01DOI: 10.4208/cicp.oa-2023-0179
J. Querales, P. Venegas
This paper deals with the numerical approximation of a pressure/displacement formulation of the elastoacoustic vibration problem in the axisymmetric case. We propose and analyze a discretization based on Lagrangian finite elements in the fluid and solid domains. We show that the scheme provides a correct approximation of the spectrum and prove quasi-optimal error estimates. We report numerical results to validate the proposed methodology for elastoacoustic vibrations.
{"title":"Numerical Approximation of an Axisymmetric Elastoacoustic Eigenvalue Problem","authors":"J. Querales, P. Venegas","doi":"10.4208/cicp.oa-2023-0179","DOIUrl":"https://doi.org/10.4208/cicp.oa-2023-0179","url":null,"abstract":"This paper deals with the numerical approximation of a pressure/displacement formulation of the elastoacoustic vibration problem in the axisymmetric case.\u0000We propose and analyze a discretization based on Lagrangian finite elements in the\u0000fluid and solid domains. We show that the scheme provides a correct approximation\u0000of the spectrum and prove quasi-optimal error estimates. We report numerical results\u0000to validate the proposed methodology for elastoacoustic vibrations.","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"30 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138716456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.4208/cicp.oa-2022-0295
Xiang Wang, Yuqing Zhang null, Zhimin Zhang
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{"title":"New Superconvergent Structures with Optional Superconvergent Points for the Finite Volume Element Method","authors":"Xiang Wang, Yuqing Zhang null, Zhimin Zhang","doi":"10.4208/cicp.oa-2022-0295","DOIUrl":"https://doi.org/10.4208/cicp.oa-2022-0295","url":null,"abstract":".","PeriodicalId":50661,"journal":{"name":"Communications in Computational Physics","volume":"1 1","pages":""},"PeriodicalIF":3.7,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70516085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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