An implicit interface boundary integral method for Poisson’s equation on arbitrary domains

Kublik, Catherine; Tanushev, Nicolay M.; Tsai, Richard

doi:10.1016/j.jcp.2013.03.049

Cited by 33 publications

(55 citation statements)

References 41 publications

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“…In this section, we first describe the implicit boundary integral formulation of [11] as it lays out the foundation of the proposed algorithm. The main contribution of this paper is presented in Sects.…”

Section: The Implicit Boundary Integral Methodsmentioning

confidence: 99%

“…The Jacobian J Γ required by this formulation can be easily evaluated by computing the curvatures H and G directly using standard centered finite differencing applied to the distance function, as in [11], or it could be very easily computed by singular value decomposition applied to a difference approximation of P Γ , as in [12]. The exact integral formulation we use in this work was first proposed in [11] and extended in [12].…”

Section: Exact Integral Formulations Using Signed Distance Functionsmentioning

confidence: 99%

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Implicit boundary integral methods for the Helmholtz equation in exterior domains

Chen

Tsai

2017

Res Math Sci

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We propose a new algorithm for solving Helmholtz equations in exterior domains with implicitly represented boundaries. The algorithm not only combines the advantages of implicit surface representation and the boundary integral method, but also provides a new way to compute a class of the so-called hypersingular integrals. The keys to the proposed algorithm are the derivation of the volume integrals which are equivalent to any given integrals on smooth closed hypersurfaces, and the ability to approximate the natural limit of the singular integrals via seamless extrapolation. We present numerical results for both two-and three-dimensional scattering problems at near resonant frequencies as well as with non-convex scattering surfaces. Keywords: Helmholtz equation, Hypersingular integrals, Level set methods, Closest point projection, Boundary integrals BackgroundLet Γ be a closed and compact C 2 hypersurface that separates R m , m = 2, 3, into a simply connected and bounded open region Ω and its complement. We consider the solution of the following Neumann boundary value problem for the Helmholtz equation:(1.1)For wave scattering by a sound hard boundary ∂Ω, a total wave field u tot (x) is a function satisfying(1.2) This total wave field is then written artificially as the sum u tot = u inc + u sc , where u inc is a known function that is used to model the far-field condition of u tot and u sc is the solution of (1

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Section: The Implicit Boundary Integral Methodsmentioning

confidence: 99%

Section: Exact Integral Formulations Using Signed Distance Functionsmentioning

confidence: 99%

Implicit boundary integral methods for the Helmholtz equation in exterior domains

Chen

Tsai

2017

Res Math Sci

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“…The idea for the present work originated in [10] where the authors proposed a formulation for computing integrals of the form…”

Section: Introductionmentioning

confidence: 99%

“…In [10], with the choice of ϕ = d ∂ being a signed distance function to ∂ , the integral (1) is expressed as an average of integrals over nearby level sets of d ∂ , where these nearby level sets continuously sweep a thin tubular neighborhood around the boundary ∂ of radius . Consequently, (1) is equivalent to the volume integral shown on the right hand side below:…”

Section: Introductionmentioning

confidence: 99%

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Integration over curves and surfaces defined by the closest point mapping

Kublik

Tsai

2016

Res Math Sci

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We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wishes to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of the Jacobian matrix of the closest point mapping. Additionally, we extend the formulation-initially derived to integrate over manifolds of codimension one-to include integration along curves in three dimensions. Some numerical examples using very simple discretizations are presented to demonstrate the efficacy of the formulation.

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Projection‐based model reduction for the immersed boundary method

Luo

Hao

2021

Numer Methods Biomed Eng

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Fluid-structure interactions are central to many biomolecular processes, and they impose a great challenge for computational and modeling methods. In this paper, we consider the immersed boundary method (IBM) for biofluid systems, and to alleviate the computational cost, we apply reduced-order techniques to eliminate the degrees of freedom associated with the large number of fluid variables. We show how reduced models can be derived using Petrov-Galerkin projection and subspaces that maintain the incompressibility condition. More importantly, the reduced-order model (ROM) is shown to preserve the Lyapunov stability. We also address the practical issue of computing coefficient matrices in the ROM using an interpolation technique. The efficiency and robustness of the proposed formulation are examined with test examples from various applications.

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An implicit interface boundary integral method for Poisson’s equation on arbitrary domains

Cited by 33 publications

References 41 publications

Implicit boundary integral methods for the Helmholtz equation in exterior domains

Implicit boundary integral methods for the Helmholtz equation in exterior domains

Integration over curves and surfaces defined by the closest point mapping

Projection‐based model reduction for the immersed boundary method

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