3,860 research outputs found
Accurate computation of Galerkin double surface integrals in the 3-D boundary element method
Many boundary element integral equation kernels are based on the Green's
functions of the Laplace and Helmholtz equations in three dimensions. These
include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell's
equations. Integral equation formulations lead to more compact, but dense
linear systems. These dense systems are often solved iteratively via Krylov
subspace methods, which may be accelerated via the fast multipole method. There
are advantages to Galerkin formulations for such integral equations, as they
treat problems associated with kernel singularity, and lead to symmetric and
better conditioned matrices. However, the Galerkin method requires each entry
in the system matrix to be created via the computation of a double surface
integral over one or more pairs of triangles. There are a number of
semi-analytical methods to treat these integrals, which all have some issues,
and are discussed in this paper. We present novel methods to compute all the
integrals that arise in Galerkin formulations involving kernels based on the
Laplace and Helmholtz Green's functions to any specified accuracy. Integrals
involving completely geometrically separated triangles are non-singular and are
computed using a technique based on spherical harmonics and multipole
expansions and translations, which results in the integration of polynomial
functions over the triangles. Integrals involving cases where the triangles
have common vertices, edges, or are coincident are treated via scaling and
symmetry arguments, combined with automatic recursive geometric decomposition
of the integrals. Example results are presented, and the developed software is
available as open source
Planewave density interpolation methods for 3D Helmholtz boundary integral equations
This paper introduces planewave density interpolation methods for the
regularization of weakly singular, strongly singular, hypersingular and nearly
singular integral kernels present in 3D Helmholtz surface layer potentials and
associated integral operators. Relying on Green's third identity and pointwise
interpolation of density functions in the form of planewaves, these methods
allow layer potentials and integral operators to be expressed in terms of
integrand functions that remain smooth (at least bounded) regardless the
location of the target point relative to the surface sources. Common
challenging integrals that arise in both Nystr\"om and boundary element
discretization of boundary integral equation, can then be numerically evaluated
by standard quadrature rules that are irrespective of the kernel singularity.
Closed-form and purely numerical planewave density interpolation procedures are
presented in this paper, which are used in conjunction with Chebyshev-based
Nystr\"om and Galerkin boundary element methods. A variety of numerical
examples---including problems of acoustic scattering involving multiple
touching and even intersecting obstacles, demonstrate the capabilities of the
proposed technique
Boundary integral methods in high frequency scattering
In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources
A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow
We present an efficient discontinuous Galerkin scheme for simulation of the
incompressible Navier-Stokes equations including laminar and turbulent flow. We
consider a semi-explicit high-order velocity-correction method for time
integration as well as nodal equal-order discretizations for velocity and
pressure. The non-linear convective term is treated explicitly while a linear
system is solved for the pressure Poisson equation and the viscous term. The
key feature of our solver is a consistent penalty term reducing the local
divergence error in order to overcome recently reported instabilities in
spatially under-resolved high-Reynolds-number flows as well as small time
steps. This penalty method is similar to the grad-div stabilization widely used
in continuous finite elements. We further review and compare our method to
several other techniques recently proposed in literature to stabilize the
method for such flow configurations. The solver is specifically designed for
large-scale computations through matrix-free linear solvers including efficient
preconditioning strategies and tensor-product elements, which have allowed us
to scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores.
We validate our code and demonstrate optimal convergence rates with laminar
flows present in a vortex problem and flow past a cylinder and show
applicability of our solver to direct numerical simulation as well as implicit
large-eddy simulation of turbulent channel flow at as well as
.Comment: 28 pages, in preparation for submission to Journal of Computational
Physic
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