The adaptive immersed interface finite element method for elliptic and Maxwell interface problems

Chen, Zhiming; Xiao, Yuanming; Zhang, Linbo

doi:10.1016/j.jcp.2009.03.044

Cited by 51 publications

(30 citation statements)

References 16 publications

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“…Moreover, M h will be refined to produce the new mesh in the adaptive computations. As in [21], we can see that the mesh quality will not deteriorate during the adaptive iterations if the newest vertex bisection algorithm is used to refine the meshes. …”

Section: 3 For Case (Ii) M H Is Then a Body-fitted Mesh Letmentioning

confidence: 76%

“…For interface problems, an exact solution can have strong singularity, so it is essential to develop an adaptive immersed interface finite element method (AIIFEM). In [21], the AIIFEM is developed for solving elliptic and Maxwell interface problems with singularity. They introduce new finite element basis functions for the elements having nonempty intersection with the interface and obtain an AIIFEM which is quasi-optimal in the sense that the energy error decays as CN − 1 3 , where N is the number of degrees of freedom.…”

Section: Introductionmentioning

confidence: 99%

“…They introduce new finite element basis functions for the elements having nonempty intersection with the interface and obtain an AIIFEM which is quasi-optimal in the sense that the energy error decays as CN − 1 3 , where N is the number of degrees of freedom. Motivated by [21], we deal with our elasticity interface problems using the AIIFEM.…”

Section: Introductionmentioning

confidence: 99%

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The Adaptive Immersed Interface Finite Element Method for Elasticity Interface Problems

Chang¹

2012

JCM

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In this paper, we propose adaptive finite element methods with error control for solving elasticity problems with discontinuous coefficients. The meshes in the methods do not need to fit the interfaces. We establish a residual-based a posteriori error estimate which is λindependent multiplicative constants; the Lamé constant λ steers the incompressibility. The error estimators are then implemented and tested with promising numerical results which will show the competitive behavior of the adaptive algorithm.Mathematics subject classification: 49J20, 65N30.

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Section: 3 For Case (Ii) M H Is Then a Body-fitted Mesh Letmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Adaptive Immersed Interface Finite Element Method for Elasticity Interface Problems

Chang¹

2012

JCM

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“…PHG provides a tool, phgSurfaceCut, to create a new conforming tetrahedral mesh with respect to a given set of cut points by locally subdividing all tetrahedra containing cut points into smaller tetrahedra. That tool was used in [8] and it worked well for the test cases with simple surfaces presented in [8], but is not robust enough and may crash for complicated surfaces as those addressed in this paper. Thus we have revised and improved the underlying algorithm used in the phgSurfaceCut function and it's now robust and can reliably generate a new mesh with any given surface.…”

Section: Fit the Surface (Surface Cut)mentioning

confidence: 99%

Automated Parallel and Body-Fitted Mesh Generation in Finite Element Simulation of Macromolecular Systems

Xie

Liu

et al. 2016

Commun. Comput. Phys.

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Mesh generation is a bottleneck for finite element simulations of biomolecules. A robust and efficient approach, based on the immersed boundary method proposed in [8], has been developed and implemented to generate large-scale mesh body-fitted to molecular shape for general parallel finite element simulations. The molecular Gaussian surface is adopted to represent the molecular surface, and is finally approximated by piecewise planes via the tool phgSurfaceCut in PHG [43], which is improved and can reliably handle complicated molecular surfaces, through mesh refinement steps. A coarse background mesh is imported first and then is distributed into each process using a mesh partitioning algorithm such as space filling curve [5] or METIS [22]. A bisection method is used for the mesh refinements according to the molecular PDB or PQR file which describes the biomolecular region. After mesh refinements, the mesh is optionally repartitioned and redistributed for load balancing. For finite element simulations, the modification of region mark and boundary types is done in parallel. Our parallel mesh generation method has been successfully applied to a sphere cavity model, a DNA fragment, a gramicidin A channel and a huge Dengue virus system. The results of numerical experiments show good parallel efficiency. Computations of electrostatic potential and solvation energy also validate the method. Moreover, the meshing process and adaptive finite element computation can be integrated as one PHG project to avoid the mesh importing and exporting costs, and improve the convenience of application as well.

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“…Interface problems occur in many physical applications [1–4] such as composite materials [5, 6], fluid mechanics [7, 8], cell and bubble formation, crystal growth [9–11], biochemical processing, mining, material sciences [12], biological systems and sciences [13, 14], heat conduction [15], and heat and mass transfer [16–20].…”

Section: Introductionmentioning

confidence: 99%

Some numerical techniques for solving elliptic interface problems

Kumar

Joshi

2010

Numerical Methods Partial

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Many physical phenomena can be modeled by partial differential equations with singularities and interfaces. The standard finite difference and finite element methods may not be successful in giving satisfactory numerical results for such problems. Hence, many new methods have been developed. Some of them are developed with the modifications in the standard methods, so that they can deal with the discontinuities and the singularities. In this article, a survey has been done on some recent efficient techniques to solve elliptic interface problems.

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The adaptive immersed interface finite element method for elliptic and Maxwell interface problems

Cited by 51 publications

References 16 publications

The Adaptive Immersed Interface Finite Element Method for Elasticity Interface Problems

The Adaptive Immersed Interface Finite Element Method for Elasticity Interface Problems

Automated Parallel and Body-Fitted Mesh Generation in Finite Element Simulation of Macromolecular Systems

Some numerical techniques for solving elliptic interface problems

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