A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains

Ying, Lexing; Biros, George; Zorin, Denis

doi:10.1016/j.jcp.2006.03.021

Cited by 137 publications

(181 citation statements)

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“…Bruno and Kunyansky [13] proposed a spectral integration scheme for weakly-singular integrals. Ying et al [67] extended it to arbitrary-geometry smooth surfaces. Although asymptotically optimal, this scheme is rather expensive as it requires the use of partition of unity functions, for which derivative magnitudes rapidly increase with order and as a consequence, a relatively large number of points is needed for good approximation.…”

Section: Related Workmentioning

confidence: 99%

“…Although we could use the integration scheme in [67], we have found that, for the problem sizes we are targeting, it is cheaper to use a simple upsampling-based quadrature (see discussion in Section 4.1).…”

Section: Related Workmentioning

confidence: 99%

“…However, for close interactions the accuracy of these rules deteriorates as the integrals become nearly singular. A robust, fourth-order accurate, near-singular integration scheme for smooth surfaces is presented in [67]. However, the constant in the complexity estimate of that scheme is rather high since it involves multiple evaluations and interpolations, and requires finding nearest points on the surface.…”

Section: Singular Integrationmentioning

confidence: 99%

“…We spectrally upsample 6 all vesicle surfaces by a factor of two (similar to the medium-distance part of the evaluation in [67]), then we compute the interactions at this higher resolution and then downsample to the original resolution using spectral cutoff. This computation can be done in a per-vesicle adaptive way, but in our implementation we have used uniform upsampling across all vesicles.…”

Section: Singular Integrationmentioning

confidence: 99%

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A fast algorithm for simulating vesicle flows in three dimensions

Veerapaneni

Rahimian

Biros

et al. 2011

Journal of Computational Physics

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137

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Vesicles are locally-inextensible fluid membranes that can sustain bending. In this paper, we extend "A numerical method for simulating the dynamics of 3D axisymmetric vesicles suspended in viscous flows", Veerapaneni et al. Journal of Computational Physics, 228(19), 2009 to general non-axisymmetric vesicle flows in three dimensions.Although the main components of the algorithm are similar in spirit to the axisymmetric case (spectral approximation in space, semi-implicit time-stepping scheme), important new elements need to be introduced for a full 3D method. In particular, spatial quantities are discretized using spherical harmonics, and quadrature rules for singular surface integrals need to be adapted to this case; an algorithm for surface reparameterization is neeed to ensure sufficient of the timestepping scheme, and spectral filtering is introduced to maintain reasonable accuracy while minimizing computational costs. To characterize the stability of the scheme and to construct preconditioners for the iterative linear system solvers used in the semi-implicit time-stepping scheme, we perform a spectral analysis of the evolution operator on the unit sphere.By introducing these algorithmic components, we obtain a time-stepping scheme that, in our numerical experiments, is unconditionally stable. We present results to analyze the cost and convergence rates of the overall scheme. To illustrate the applicability of the new method, we consider a few vesicle-flow interaction problems: a single vesicle in relaxation, sedimentation, shear flows, and many-vesicle flows.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Singular Integrationmentioning

confidence: 99%

Section: Singular Integrationmentioning

confidence: 99%

See 2 more Smart Citations

A fast algorithm for simulating vesicle flows in three dimensions

Veerapaneni

Rahimian

Biros

et al. 2011

Journal of Computational Physics

Self Cite

137

143

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“…To achieve high-order convergence rates, a high-order smooth nonsingular parametrization is necessary. Our manifold-based surfaces are an essential part of the boundary integral equation solver described in (Ying et al, 2006), which achieves high convergence rates in part thanks to high degree of continuity of manifold-based surfaces.…”

Section: Introductionmentioning

confidence: 99%

Manifold-based surfaces with boundaries

Tosun

Zorin

2011

Computer Aided Geometric Design

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Fast Ewald summation for Stokesian particle suspensions

Klinteberg

Tornberg

2014

Numerical Methods in Fluids

118

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We present a numerical method for suspensions of spheroids of arbitrary aspect ratio which sediment under gravity. The method is based on a periodized boundary integral formulation using the Stokes double layer potential. The resulting discrete system is solved iteratively using GMRES accelerated by the spectral Ewald (SE) method, which reduces the computational complexity to O(N log N ), where N is the number of points used to discretize the particle surfaces. We develop predictive error estimates, which can be used to optimize the choice of parameters in the Ewald summation. Numerical tests show that the method is well conditioned and provides good accuracy when validated against reference solutions.

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A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains

Cited by 137 publications

References 50 publications

A fast algorithm for simulating vesicle flows in three dimensions

A fast algorithm for simulating vesicle flows in three dimensions

Manifold-based surfaces with boundaries

Fast Ewald summation for Stokesian particle suspensions

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