A Nitsche-based element-free Galerkin(EFG)method for solving semilinear elliptic problems is developed and analyzed in this paper.The existence and uniqueness of the weak solution for semilinear elliptic problems are ...A Nitsche-based element-free Galerkin(EFG)method for solving semilinear elliptic problems is developed and analyzed in this paper.The existence and uniqueness of the weak solution for semilinear elliptic problems are proved based on a condition that the nonlinear term is an increasing Lipschitz continuous function of the unknown function.A simple iterative scheme is used to deal with the nonlinear integral term.We proved the existence,uniqueness and convergence of the weak solution sequence for continuous level of the simple iterative scheme.A commonly used assumption for approximate space,sometimes called inverse assumption,is proved.Optimal order error estimates in L 2 and H1 norms are proved for the linear and semilinear elliptic problems.In the actual numerical calculation,the characteristic distance h does not appear explicitly in the parameterβintroduced by the Nitsche method.The theoretical results are confirmed numerically。展开更多
This paper proposes a deep unfitted Nitsche method for solving elliptic interface problems with high contrasts in high dimensions.To capture discontinuities of the solution caused by interfaces,we reformulate the prob...This paper proposes a deep unfitted Nitsche method for solving elliptic interface problems with high contrasts in high dimensions.To capture discontinuities of the solution caused by interfaces,we reformulate the problem as an energy minimization problem involving two weakly coupled components.This enables us to train two deep neural networks to represent two components of the solution in highdimensional space.The curse of dimensionality is alleviated by using theMonte-Carlo method to discretize the unfittedNitsche energy functional.We present several numerical examples to show the performance of the proposed method.展开更多
This note deals with the existence and uniqueness of a minimiser of the following Grtzsch-type problem inf f ∈F∫∫_(Q_1)φ(K(z,f))λ(x)dxdyunder some mild conditions,where F denotes the set of all homeomorphims f wi...This note deals with the existence and uniqueness of a minimiser of the following Grtzsch-type problem inf f ∈F∫∫_(Q_1)φ(K(z,f))λ(x)dxdyunder some mild conditions,where F denotes the set of all homeomorphims f with finite linear distortion K(z,f)between two rectangles Q_1 and Q_2 taking vertices into vertices,φ is a positive,increasing and convex function,and λ is a positive weight function.A similar problem of Nitsche-type,which concerns the minimiser of some weighted functional for mappings between two annuli,is also discussed.As by-products,our discussion gives a unified approach to some known results in the literature concerning the weighted Grtzsch and Nitsche problems.展开更多
In this paper, we study Nitsche extended finite element method （XFEM） for the inter- face problem of a two dimensional diffusion equation. Specifically, we study the quadratic XFEM scheme on some shape-regular famil...In this paper, we study Nitsche extended finite element method （XFEM） for the inter- face problem of a two dimensional diffusion equation. Specifically, we study the quadratic XFEM scheme on some shape-regular family of grids and prove the optimal convergence rate of the scheme with respect to the mesh size. Main efforts are devoted onto classifying the cases of intersection between the elements and the interface and prove a weighted trace inequality for the extended finite element functions needed, and the general framework of analysing XFEM c^n be implemented then.展开更多
In the context of isogeometric analysis(IGA)of shell structures,the popularity of the solid-shell elements benefit from formulation simplicity and full 3D stress state.However some basic questions remain unresolved wh...In the context of isogeometric analysis(IGA)of shell structures,the popularity of the solid-shell elements benefit from formulation simplicity and full 3D stress state.However some basic questions remain unresolved when using solid-shell element,especially for large deformation cases with patch coupling,which is a common scene in real-life simulations.In this research,after introduction of the solid-shell nonlinear formulation and a fundamental 3D model construction method,we present a non-symmetric variant of the standardNitsche’s formulation for multi-patch coupling in associationwith an empirical formula for its stabilization parameter.An selective and reduced integration scheme is also presented to address the locking syndrome.In addition,the quasi-Newton iteration format is derived as solver,together with a step length control method.The second order derivatives are totally neglected by the adoption of the non-symmetric Nitsche’s formulation and the quasi-Newton solver.The solid-shell elements are numerically studied by a linear elastic plate example,then we demonstrate the performance of the proposed formulation in large deformation,in terms of result verification,iteration history and continuity of displacement across the coupling interface.展开更多
Topological optimization plays a guiding role in the conceptual design process.This paper conducts research on structural topology optimization algorithm within the framework of isogeometric analysis.For multi-compone...Topological optimization plays a guiding role in the conceptual design process.This paper conducts research on structural topology optimization algorithm within the framework of isogeometric analysis.For multi-component structures,the Nitsche’smethod is used to glue differentmeshes to performisogeometricmulti-patch analysis.The discrete variable topology optimization algorithm based on integer programming is adopted in order to obtain clear boundaries for topology optimization.The sensitivity filtering method based on the Helmholtz equation is employed for averaging of curved elements’sensitivities.In addition,a simple averaging method along coupling interfaces is proposed in order to ensure the material distribution across coupling areas is reasonably smooth.Finally,the performance of the algorithm is demonstrated by numerical examples,and the effectiveness of the algorithm is verified by comparing it with the results obtained by single-patch and ABAQUS cases.展开更多
基金supported by the Innovation Research Group Project in Universities of Chongqing of China(No.CXQT19018)the National Natural Science Foundation of China(Grant No.11971085)+1 种基金he Natural Science Foundation of Chongqing(Grant Nos.cstc2021jcyj-jqX0011 and cstc2020jcyj-msxm0777)an open project of Key Laboratory for Optimization and Control Ministry of Education,Chongqing Normal University(Grant No.CSSXKFKTM202006)。
文摘A Nitsche-based element-free Galerkin(EFG)method for solving semilinear elliptic problems is developed and analyzed in this paper.The existence and uniqueness of the weak solution for semilinear elliptic problems are proved based on a condition that the nonlinear term is an increasing Lipschitz continuous function of the unknown function.A simple iterative scheme is used to deal with the nonlinear integral term.We proved the existence,uniqueness and convergence of the weak solution sequence for continuous level of the simple iterative scheme.A commonly used assumption for approximate space,sometimes called inverse assumption,is proved.Optimal order error estimates in L 2 and H1 norms are proved for the linear and semilinear elliptic problems.In the actual numerical calculation,the characteristic distance h does not appear explicitly in the parameterβintroduced by the Nitsche method.The theoretical results are confirmed numerically。
基金supported by Andrew Sisson Fund of the University of MelbourneX.Y.was partially supported by the NSF grants DMS-1818592 and DMS-2109116.
文摘This paper proposes a deep unfitted Nitsche method for solving elliptic interface problems with high contrasts in high dimensions.To capture discontinuities of the solution caused by interfaces,we reformulate the problem as an energy minimization problem involving two weakly coupled components.This enables us to train two deep neural networks to represent two components of the solution in highdimensional space.The curse of dimensionality is alleviated by using theMonte-Carlo method to discretize the unfittedNitsche energy functional.We present several numerical examples to show the performance of the proposed method.
基金supported by National Natural Science Foundation of China(Grant Nos.11371268 and 11171080)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20123201110002)the Natural Science Foundation of Jiangsu Province(Grant No.BK20141189)
文摘This note deals with the existence and uniqueness of a minimiser of the following Grtzsch-type problem inf f ∈F∫∫_(Q_1)φ(K(z,f))λ(x)dxdyunder some mild conditions,where F denotes the set of all homeomorphims f with finite linear distortion K(z,f)between two rectangles Q_1 and Q_2 taking vertices into vertices,φ is a positive,increasing and convex function,and λ is a positive weight function.A similar problem of Nitsche-type,which concerns the minimiser of some weighted functional for mappings between two annuli,is also discussed.As by-products,our discussion gives a unified approach to some known results in the literature concerning the weighted Grtzsch and Nitsche problems.
基金The first author is partially supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research as part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials under Award Number DE-SC-0009249, and the Key Program of National Natural Science Foundation of China with Grant No. 91430215. The second author is supported by State Key Laboratory of Scientific and Engineering Computing （LSEC）, National Center for Mathematics and Interdisciplinary Sciences of Chinese Academy of Sciences （NCMIS）, and National Natural Science Foundation of China with Grant No. 11471026 he is thankful to the Center for Computational Mathematics and Applications, the Pennsylvania State University, where he worked on this manuscript as a visiting scholar. The authors are grateful to Professor Jinchao Xu, Dr. Yuanming Xiao and Dr. Maximilian Metti for their valuable suggestions and discussions, to Professor Haijun Wu for his valuable help on preparing the numerical example, and to the anonymous referee for the valuable comments and suggestion which lead to improvements of the paper.
文摘In this paper, we study Nitsche extended finite element method （XFEM） for the inter- face problem of a two dimensional diffusion equation. Specifically, we study the quadratic XFEM scheme on some shape-regular family of grids and prove the optimal convergence rate of the scheme with respect to the mesh size. Main efforts are devoted onto classifying the cases of intersection between the elements and the interface and prove a weighted trace inequality for the extended finite element functions needed, and the general framework of analysing XFEM c^n be implemented then.
基金supported by the Fundamental Research Funds for the Central Universities(Grant JUSRP12038)the Natural Science Foundation of Jiangsu Province(Grant BK20200611).
文摘In the context of isogeometric analysis(IGA)of shell structures,the popularity of the solid-shell elements benefit from formulation simplicity and full 3D stress state.However some basic questions remain unresolved when using solid-shell element,especially for large deformation cases with patch coupling,which is a common scene in real-life simulations.In this research,after introduction of the solid-shell nonlinear formulation and a fundamental 3D model construction method,we present a non-symmetric variant of the standardNitsche’s formulation for multi-patch coupling in associationwith an empirical formula for its stabilization parameter.An selective and reduced integration scheme is also presented to address the locking syndrome.In addition,the quasi-Newton iteration format is derived as solver,together with a step length control method.The second order derivatives are totally neglected by the adoption of the non-symmetric Nitsche’s formulation and the quasi-Newton solver.The solid-shell elements are numerically studied by a linear elastic plate example,then we demonstrate the performance of the proposed formulation in large deformation,in terms of result verification,iteration history and continuity of displacement across the coupling interface.
基金supported by the Fundamental Research Funds for the Cen-tral Universities(No.JUSRP12038)the Natural Science Foundation of Jiangsu Province(No.BK20200611)the National Natural Science Foundation of China(No.12102146).
文摘Topological optimization plays a guiding role in the conceptual design process.This paper conducts research on structural topology optimization algorithm within the framework of isogeometric analysis.For multi-component structures,the Nitsche’smethod is used to glue differentmeshes to performisogeometricmulti-patch analysis.The discrete variable topology optimization algorithm based on integer programming is adopted in order to obtain clear boundaries for topology optimization.The sensitivity filtering method based on the Helmholtz equation is employed for averaging of curved elements’sensitivities.In addition,a simple averaging method along coupling interfaces is proposed in order to ensure the material distribution across coupling areas is reasonably smooth.Finally,the performance of the algorithm is demonstrated by numerical examples,and the effectiveness of the algorithm is verified by comparing it with the results obtained by single-patch and ABAQUS cases.
