The Immersed Interface/Multigrid Methods for Interface Problems

Adams, Loyce M.; Li, Zhilin

doi:10.1137/s1064827501389849

Cited by 99 publications

(126 citation statements)

References 7 publications

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“…In the following, we assume that φ : Ω → IR is a level-set function resulting from a segmentation process based on the image u 0 . Here, φ is usually considered as the asymptotic limit of the solution of some level-set propagation (1). We emphasize that this is not a restriction, because starting from a variety of other segmentation results (parametric surfaces, characteristic functions, phase-field etc.…”

Section: A Virtual Gridmentioning

confidence: 99%

“…We refer to the local virtual nodes by the local indices of their constraining parent nodes. A virtual node located at 1 2 (x j k +x j l ) is referred to as (j k , j l ). This implicitly gives us the set of constraining parent nodes D j from Section 2.3.…”

Section: Hashing Topologymentioning

confidence: 99%

“…The "Immersed Interface Method" (IIM) [5,27,50,45] uses adaptive finite difference stencils near the interface. Efficient solvers are presented in [28,1]. A combination with finite volume methods is discussed in [8,7] and an extension to "Immersed Finite Elements" in 1D and 2D is proposed in [29,30].…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore we fix the notational convention that the first index always refers to the regular node inside the domain Ω − . E. g. the tetrahedron T 0 in line 21 has the indices {j 0 j 0 , j 0 j 1 , j 0 j 2 , j 0 j 3 } which means that it contains one regular node x j0 ∈ N − and three virtual nodes 1 2 (x j0 + x j1 ), 1 2 (x j0 + x j2 ), 1 2 (x j0 + x j3 ). After the lookup table has been created, the set of virtual nodes N v can be easily generated from the sets of local virtual nodes N loc v .…”

mentioning

confidence: 99%

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Composite finite elements for 3D image based computing

Liehr

Preußer

Rumpf

et al. 2008

Comput. Visual Sci.

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Abstract. We present an algorithmical concept for modeling and simulation with partial differential equations (PDEs) in image based computing where the computational geometry is defined through previously segmented image data. Such problems occur in applications from biology and medicine where the underlying image data has been acquired through e. g. computed tomography (CT), magnetic resonance imaging (MRI) or electron microscopy (EM). Based on a level-set description of the computational domain, our approach is capable of automatically providing suitable composite finite element functions that resolve the complicated shapes in the medical/biological data set. It is efficient in the sense that the traversal of the grid (and thus assembling matrices for finite element computations) inherits the efficiency of uniform grids away from complicated structures. The method's efficiency heavily depends on precomputed lookup tables in the vicinity of the domain boundary or interface. A suitable multigrid method is used for an efficient solution of the systems of equations resulting from the composite finite element discretization. The paper focuses on both algorithmical and implementational details. Scalar and vector valued model problems as well as real applications underline the usability of our approach.

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Section: A Virtual Gridmentioning

confidence: 99%

Section: Hashing Topologymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Composite finite elements for 3D image based computing

Liehr

Preußer

Rumpf

et al. 2008

Comput. Visual Sci.

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“…This way of treating irregular points works for our problem, where the boundary condition is in the form of (23). For problems with jump conditions, this are somewhat different (see [4]). …”

Section: Substituting Into Eqn(38) Leads Tomentioning

confidence: 99%

Solving partial differential equations on irregular domains with moving interfaces, with applications to superconformal electrodeposition in semiconductor manufacturing

Sethian¹,

Shan²

2008

Journal of Computational Physics

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We present a numerical algorithm for solving partial differential equations on irregular domains with moving interfaces. Instead of the typical approach of solving in a larger rectangular domain, our approach performs most calculations only in the desired domain. To do so efficiently, we have developed a one-sided multigrid method to solve the corresponding large sparse linear systems.Our focus is on the simulation of the electrodeposition process in semiconductor manufacturing in both two and three dimensions. Our goal is to track the position of the interface between the metal and the electrolyte as the features are filled and to determine which initial configurations and physical parameters lead to superfilling.We begin by motivating the set of equations which model the electrodeposition process. Building on existing models for superconformal electrodeposition, we develop a model which naturally arises from a conservation law form of surface additive evolution. We then introduce several numerical algorithms, including a conservative material transport level set method and our multigrid method for one-sided diffusion equations. We then analyze the accuracy of our numerical methods. Finally, we compare our result with experiment over a wide range of physical parameters.

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Computational methods for optical molecular imaging

Chen

Wei

Cong

et al. 2008

Commun. Numer. Meth. Engng.

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SummaryA new computational technique, the matched interface and boundary (MIB) method, is presented to model the photon propagation in biological tissue for the optical molecular imaging. Optical properties have significant differences in different organs of small animals, resulting in discontinuous coefficients in the diffusion equation model. Complex organ shape of small animal induces singularities of the geometric model as well. The MIB method is designed as a dimension splitting approach to decompose a multidimensional interface problem into one-dimensional ones. The methodology simplifies the topological relation near an interface and is able to handle discontinuous coefficients and complex interfaces with geometric singularities. In the present MIB method, both the interface jump condition and the photon flux jump conditions are rigorously enforced at the interface location by using only the lowest-order jump conditions. This solution near the interface is smoothly extended across the interface so that central finite difference schemes can be employed without the loss of accuracy. A wide range of numerical experiments are carried out to validate the proposed MIB method. The second-order convergence is maintained in all benchmark problems. The fourth-order convergence is also demonstrated for some three-dimensional problems. The robustness of the proposed method over the variable strength of the linear term of the diffusion equation is also examined. The performance of the present approach is compared with that of the standard finite element method. The numerical study indicates that the proposed method is a potentially efficient and robust approach for the optical molecular imaging.

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The Immersed Interface/Multigrid Methods for Interface Problems

Cited by 99 publications

References 7 publications

Composite finite elements for 3D image based computing

Composite finite elements for 3D image based computing

Solving partial differential equations on irregular domains with moving interfaces, with applications to superconformal electrodeposition in semiconductor manufacturing

Computational methods for optical molecular imaging

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