The immersed interface method iim 6,7,911 is an efficient finite difference approach for interface problems with discontinuities and singularities. The explicitjump immersed interface method ejiim was developed following lis fast iterative immersed interface method fiiim, recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic bvps. A finite element method is proposed for one dimensional interface problems involving discontinuities in the coefficients of the differential equations and the derivatives of the solutions. A partially penalty immersed crouzeixraviart finite element. Jan 18, 2017 we show that immersed finite element solutions inherit all desired superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. An immersed transitional interface finite element method itifem is proposed to simulate fluidstructure interaction fsi. If partial di erential equations pdes are used to model these simulations, it usually leads to the socalled interface. Let us consider an incompressible threedimensional deformable structure in. Mar 17, 2014 the nonconforming immersed finite element method ifem developed in li et al.
In the xfem, they use enrichment basis functions in addition to the standard finite element basis. The related research works can be categorized into three aspects. There are two classes of methods which belong to this type. The immersed finite element method based on a uniform cartesian mesh has been developed for the linear elasticity equations with discontinuous coefficients across an interface in this paper. Immersed finite element method for interface problems with. These methods can use a fixed mesh because ifes can handle interface jump conditions without requiring the mesh to be aligned with the interface. A consistent immersed finite element method for the. In the framework of finite element method fem, the navierstokes equations and the dynamic equation for solid are integrated using the galerkin method, and the. Pdf modeling and an immersed finite element method for. An immersed transitional interface finite element method for.
By an immersed finite element ife method, both the governing partial differential equations and the objective functional for an interface inverse problem are discretized optimally regardless of. The proposed method can be utilized on interfaceunfitted meshes such as cartesian grids consisting of cuboids. In ifem, a lagrangian solid mesh moves on top of a background eulerian fluid mesh which spans over the entire computational domain. Immersed finite element method for interface problems with algebraic multigrid solver article pdf available in communications in computational physics 154. In this paper, the immersed finite element method ifem is proposed for the solution of complex fluid and deformable structure interaction problems encountered in many physical models. Pdf solving threedimensional interface problems with. Numerical solutions of pdes involving interfaces and irregular domains, frontiers in applied mathematics, 33 2006. Immersed finite element methods for elliptic interface problems with nonhomogeneous jump conditions xiaoming he, tao lin, and yanping lin abstract.
The iim makes use of the jump conditions across the interface so that the finite difference element discretization can be accurate. Nonconforming immersed finite element methods for interface. Immersed finite element method and its applications to biological systems wing kam liu 1, yaling liu, david farrell, lucy zhang1, x. A coupled sharpinterface immersedboundaryfinite element method for flowstructure interaction with application to human phonation. The iim is a sharp interface method based on cartesian grids. Bilinear immersed finite elements for interface problems xiaoming he abstract in this dissertation we discuss bilinear immersed. In the xfem, they use enrichment basis functions in addition to. For a full discussion of existing immersed finite element methods see 1929 and the references therein. A coupled sharp interface immersed boundary finite element method for flowstructure interaction with application to human phonation. Nonconforming immersed finite element methods for interface problems xu zhang abstract in science and engineering, many simulations are carried out over domains consisting of multiple materials separated by curvessurfaces. A second order isoparametric finite element method for. Convergence of an immersed finite element method for. Immersed finite element method for interface problems with algebraic multigrid solver by wenqiang feng a thesis presented to the faculty of the graduate school of missouri university of science and technology in partial ful. Several immersed finite element methods and spaces have been proposed for solving elliptic problems over the last fifteen years.
The nonconforming immersed finite element method ifem developed in li et al. A method of lines based on immersed finite elements for. A huygens immersed finite element particleincell method for modeling plasmasurface interactions with moving interface. Anisotropic diffusion is important to many different types of common materials and media. Based on structured cartesian meshes, we develop a three. Pdf an immersed finite element method for elasticity. The flux jump condition is weakly enforced on the smooth interface. An immersed linear finite element method with interface flux. Dec 18, 2019 based on structured cartesian meshes, we develop a three.
A method of lines based on immersed finite elements for parabolic moving interface problems volume 5 issue 4 tao lin, yanping lin, xu zhang skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a. Partially penalized immersed finite element methods for. Based on an initial cartesian mesh, a mesh optimization strategy is presented by employing curved boundary elements at. Nonconforming immersed finite element methods for interface problems. Immersed finite element methods for parabolic equations with moving interface xiaoming he,1 tao lin,2 yanping lin,3,4 xu zhang2 1department of mathematics and statistics, missouri university of science and technology, rolla, missouri 65409 2department of mathematics, virginia tech, blacksburg, virginia 24061 3department of applied mathematics, hong kong polytechnic university, hung hom. This method is based on linear immersed interface finite elements iife and applies the discontinuous galerkin formulation around the interface. The immersed interface method iim has been developed in recent years particularly designed for interface problems. The immersed interface method society for industrial and. A consistent immersed finite element method for the interface. For the theoretical analysis, we restrict ourselves to the isotropic interface problems for simplicity. Fully variational implementation of immersed finite element method for fluidstructure interaction applications nitesh nama, tony jun huang, and francesco costanzo department of engineering science and mechanics the pennsylvania state university, state college, pa, usa namahuangcostanzo immersed finite element methodimmersed finite element. In particular, on interface elements, superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and interface jump conditions. Immersedinterface finiteelement methods 475 triangular immersedinterface.
Li, the immersed interface method using a finite element formulation, applied numerical mathemtics, 27 1998, 253. Request pdf a symmetric and consistent immersed finite element method for interface problems the nonconforming immersed finite element method ifem developed in li et al. However, our framework described here can be used for high order immersedinterface. For example, extended finite element methods xfem, immersed interface methods iim, and multiscale methods have been developed to solve interface problems of elliptic partial differential equations with an interfaceindependent mesh, and readers are referred to 5, 9, 19, 32, 39, 40 for more details about these methods. An immersed finite element method based on a locally anisotropic remeshing for the incompressible stokes problem f. An enriched immersed finite element method for interface. The plot on the left shows how elements are placed along an interface in a standard fe method. The interfaces do not have to be one of grid points. One is the extended finite element method xfem and the other is the immersed finite element method ifem. The proposed method can be utilized on interface unfitted meshes such as cartesian grids consisting of cuboids. Fully variational implementation of immersed finite element. In this paper, we extend the immersed finite element ife method to the stokes interface problem.
A bilinear partially penalized immersed finite element. An immersed discontinuous finite element method for stokes. The idea is to construct basis functions which satisfy the interface jump conditions. The immersed interface method using a finite element. Immersed finite element methods for parabolic equations. This article is to discuss the bilinear and linear immersed finite element ife solutions generated fromthe algebraicmultigrid solver for both stationary andmoving interface problems. We show that immersed finite element solutions inherit all desired superconvergence properties from standard finite element methods without requiring the mesh to be aligned with the interface. For the theoretical analysis, we restrict ourselves to the.
Superconvergence of immersed finite element methods for. In this paper, we develop and analyze a trilinear immersed finite element method for solving threedimensional elliptic interface problems. Basedon the ife method, the immersedfiniteelement particleincell. Superconvergence of immersed finite element methods for interface problems waixiang caoa, xu zhangb, zhimin zhanga,c abeijing computational science research center, beijing, 94, china bdepartment of mathematics and statistics, mississippi state university, mississippi state ms 39762 cdepartment of mathematics, wayne state university, detroit, mi 48202. In this paper, we develop the immersed finite element method for parabolic optimal control problems with interfaces. Communications in nonlinear science and numerical simulation, vol. A bilinear partially penalized immersed finite element method. It yields better accuracy than some existing secondorder methods, when the coefficients or the flux across the immersed curved interface is discontinuous. An immersed finite element method for elliptic interface. Immersedinterface finiteelement methods for elliptic. Convergence of an immersed finite element method for semilinear parabolic interface problems champike attanayake department of mathematics miami university 4200 e. Based on the interface conditions and the existing bilinear immersed finite element space for the interface poisson equation, we propose an immersed finite element method for the spatial. Sometimes numerical methods using a finite element formulation may be preferred for various reasons such as theoretical analysis, personal background and preferences, available resources and linear solvers, etc.
Finally, we use s h to denote the space of conforming piecewise linear polyn. The iim is a sharp interface method that has been coupled with evolution schemes such as the level set and front tracking methods and has been used in both finite difference and finite element formulations to solve several moving interface and free boundary problems. Throughout this paper, we adopt notations of standard sobolev spaces. Sheldon wang2 and others 1 department of mechanical engineering northwestern university evanston, il 60208 2 department of mathematical sciences new jersey institute of technology newark, nj 07102. Sheldon wang2 and others 1 department of mechanical engineering northwestern university evanston, il 60208 2 department of mathematical sciences new jersey institute of technology. Numerical examples show that these methods based on the bilinear ife spaces have the same optimal convergence rates as those based on the standard bilinear finite element for solutions with. The solution obtained from the iim is typically second order accurate in the infinity norm regardless of the relative. A partially penalty immersed crouzeixraviart finite.
Bilinear immersed finite elements for interface problems. An immersed transitional interface finite element method. Pdf a coupled sharpinterface immersedboundaryfinite. An ifem adopts standard finite element basis functions on elements away from interfaces but constructs. Immersed finite element method and its analysis for. A fixed mesh method with immersed finite elements for solving. A method of lines based on immersed finite elements for parabolic moving interface problems volume 5 issue 4 tao lin, yanping lin, xu zhang. An immersed finite element method based on a locally.
Pdf modeling and an immersed finite element method for an. Then the immersed spaces are applied in galerkin, finite volume element fve and discontinuous galerkin dg methods for solving interface problems. We establish the trace and inverse inequalities for trilinear ife functions for interface elements with arbitrary interface cutting. We first consider the second order elliptic interface problem with a discontinuous diffusion. Therefore, it is suitable for moving interface problems. An immersed weak galerkin method for elliptic interface problems. A second order isoparametric finite element method ipfem is proposed for elliptic interface problems. If partial di erential equations pdes are used to model these simulations, it usually leads to the socalled interface problems of pdes whose coe cients are discontinuous.
In particular, on interface elements, superconvergence occurs at roots of generalized orthogonal polynomials that satisfy both orthogonality and. A huygens immersedfiniteelement particleincell method for modeling plasmasurface interactions with moving interface. We add two penalty terms to the general iife formulation along the sides intersected with the interface. A symmetric and consistent immersed finite element method. A coupled sharpinterface immersedboundaryfinite element. A partially penalised immersed finite element method for. In this paper, on nonbody fitted triangular meshes, we have developed a partially penalty immersed crouzeixraviart finite element method for both isotropic and anisotropic elliptic interface problems. In this article, we develop and analyze a pth degree immersed nite element ife method for the second order elliptic interface.
We establish the trace and inverse inequalities for trilinear ife functions for interface elements with arbitrary interfacecutting configuration. It is well known that the standard finiteelement method can be applied to interface problems, provided that the mesh is formed according to the interface see babuska, 1970. Immersed finite element methods for parabolic equations with. A partially penalty immersed interface finite element method. Many efforts have been made to develop alternative finiteelement methods based on unfitted meshes. An immersed linear finite element method with interface. Inverse problems, interface problems, shape optimization, discontinuous coe cients, immersed nite element methods. A fixed mesh method with immersed finite elements for. An enriched immersed finite element method for interface problems with nonhomogeneous jump conditions slimane adjerid ivo babuska y ruchi guo z tao lin x abstract this article presents and analyzes a pthdegree immersed nite element ife method for elliptic interface problems with nonhomogeneous jump conditions. A symmetric and consistent immersed finite element method for. Many boundary value problems bvps or initial bvps have nonsmooth solutions, with jumps along lowerdimensional interfaces.
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