An extrapolative approach to integration over hypersurfaces in the level set framework

Kublik, Catherine; Tsai, Richard

doi:10.1090/mcom/3282

Cited by 11 publications

(9 citation statements)

References 30 publications

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“…The functions H (φ) and δ (φ) represent regularized approximations to H(φ) and δ Γ , respectively. The design and analysis of such approximations have received significant attention in the literature during the last two decades [11,19,20,35,38,42]. Many diffuse interface methods [36,37] and level set algorithms [18,32,33] use regularized Heaviside and/or delta functions.…”

Section: Diffuse Interface Problemmentioning

confidence: 99%

“…The corresponding transformation formulas can be found, e.g., in [14]. No transformations need to be performed for the normal derivatives ∂ n U (x Γ ) to be used in (20).…”

Section: Flux Extrapolationmentioning

confidence: 99%

“…Diffuse interface methods replace surface integrals by volume integrals depending on regularized Dirac delta functions [19,20,36,37]. This family of unfitted FEM is closely related to classical immersed boundary methods [28,31] and well suited, e.g., for implementation of surface tension effects in two-phase flow models [6,18].…”

Section: Introductionmentioning

confidence: 99%

“…If tangential derivatives are needed for linear extrapolation in Step (iii), they are approximated using an interpolation stencil centered at the internal point. The way in which the interface data is projected into quadrature points distinguishes our algorithm from shifted boundary methods and extrapolation-based approaches proposed in [19,20].…”

Section: Introductionmentioning

confidence: 99%

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An unfitted finite element method using level set functions for extrapolation into deformable diffuse interfaces

Kuzmin,

Bäcker

2021

Preprint

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We explore a new way to handle flux boundary conditions imposed on level sets. The proposed approach is a diffuse interface version of the shifted boundary method (SBM) for continuous Galerkin discretizations of conservation laws in embedded domains. We impose the interface conditions weakly and approximate surface integrals by volume integrals. The discretized weak form of the governing equation has the structure of an immersed boundary finite element method. A ghost penalty term is included to extend the weak solution into the external subdomain. The calculation of interface forcing terms requires (i) construction of an approximate delta function and (ii) extrapolation of embedded boundary data into quadrature points. We accomplish these tasks using a level set function, which is given analytically or evolved numerically. A globally defined averaged gradient of this approximate signed distance function is used to construct a simple map to the closest point on the interface. The normal and tangential derivatives of the numerical solution at that point are calculated using the interface conditions and/or interpolation on uniform stencils. Similarly to SBM, extrapolation back to the quadrature points is performed using Taylor expansions. The same strategy is used to construct ghost penalty functions and extension velocities. Computations that require extrapolation are restricted to a narrow band around the interface. Numerical results are presented for elliptic, parabolic, and hyperbolic test problems, which are specifically designed to assess the error caused by the numerical treatment of interface conditions on fixed and moving boundaries in 2D.

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Section: Diffuse Interface Problemmentioning

confidence: 99%

“…The corresponding transformation formulas can be found, e.g., in [14]. No transformations need to be performed for the normal derivatives ∂ n U (x Γ ) to be used in (20).…”

Section: Flux Extrapolationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An unfitted finite element method using level set functions for extrapolation into deformable diffuse interfaces

Kuzmin,

Bäcker

2021

Preprint

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“…Such an extrapolative approach is proposed and analyzed in [13] for numerical integration on implicit surfaces that have corners and kinks.…”

Section: Definitionmentioning

confidence: 99%

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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An implicit boundary integral method for computing electric potential of macromolecules in solvent

Zhong

Ren

Tsai

2018

Journal of Computational Physics

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A numerical method using implicit surface representations is proposed to solve the linearized Poisson-Boltzmann equation that arises in mathematical models for the electrostatics of molecules in solvent. The proposed method uses an implicit boundary integral formulation to derive a linear system defined on Cartesian nodes in a narrowband surrounding the closed surface that separates the molecule and the solvent. The needed implicit surface is constructed from the given atomic description of the molecules, by a sequence of standard level set algorithms. A fast multipole method is applied to accelerate the solution of the linear system. A few numerical studies involving some standard test cases are presented and compared to other existing results.Key words. Poisson-Boltzmann equation, implicit boundary integral method, level set method, fast multipole method, electrostatics, implicit solvent model.

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An extrapolative approach to integration over hypersurfaces in the level set framework

Cited by 11 publications

References 30 publications

An unfitted finite element method using level set functions for extrapolation into deformable diffuse interfaces

An unfitted finite element method using level set functions for extrapolation into deformable diffuse interfaces

Implicit boundary integral methods for the Helmholtz equation in exterior domains

An implicit boundary integral method for computing electric potential of macromolecules in solvent

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