468 research outputs found
A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces
We present a simple, accurate method for computing singular or nearly
singular integrals on a smooth, closed surface, such as layer potentials for
harmonic functions evaluated at points on or near the surface. The integral is
computed with a regularized kernel and corrections are added for regularization
and discretization, which are found from analysis near the singular point. The
surface integrals are computed from a new quadrature rule using surface points
which project onto grid points in coordinate planes. The method does not
require coordinate charts on the surface or special treatment of the
singularity other than the corrections. The accuracy is about , where
is the spacing in the background grid, uniformly with respect to the point
of evaluation, on or near the surface. Improved accuracy is obtained for points
on the surface. The treecode of Duan and Krasny for Ewald summation is used to
perform sums. Numerical examples are presented with a variety of surfaces.Comment: to appear in Commun. Comput. Phy
A Cartesian grid-based boundary integral method for moving interface problems
This paper proposes a Cartesian grid-based boundary integral method for
efficiently and stably solving two representative moving interface problems,
the Hele-Shaw flow and the Stefan problem. Elliptic and parabolic partial
differential equations (PDEs) are reformulated into boundary integral equations
and are then solved with the matrix-free generalized minimal residual (GMRES)
method. The evaluation of boundary integrals is performed by solving equivalent
and simple interface problems with finite difference methods, allowing the use
of fast PDE solvers, such as fast Fourier transform (FFT) and geometric
multigrid methods. The interface curve is evolved utilizing the
variables instead of the more commonly used variables. This choice
simplifies the preservation of mesh quality during the interface evolution. In
addition, the approach enables the design of efficient and stable
time-stepping schemes to remove the stiffness that arises from the curvature
term. Ample numerical examples, including simulations of complex viscous
fingering and dendritic solidification problems, are presented to showcase the
capability of the proposed method to handle challenging moving interface
problems
Target-Tailored Source-Transformation for Scene Graph Generation
Scene graph generation aims to provide a semantic and structural description
of an image, denoting the objects (with nodes) and their relationships (with
edges). The best performing works to date are based on exploiting the context
surrounding objects or relations,e.g., by passing information among objects. In
these approaches, to transform the representation of source objects is a
critical process for extracting information for the use by target objects. In
this work, we argue that a source object should give what tar-get object needs
and give different objects different information rather than contributing
common information to all targets. To achieve this goal, we propose a
Target-TailoredSource-Transformation (TTST) method to efficiently propagate
information among object proposals and relations. Particularly, for a source
object proposal which will contribute information to other target objects, we
transform the source object feature to the target object feature domain by
simultaneously taking both the source and target into account. We further
explore more powerful representations by integrating language prior with the
visual context in the transformation for the scene graph generation. By doing
so the target object is able to extract target-specific information from the
source object and source relation accordingly to refine its representation. Our
framework is validated on the Visual Genome bench-mark and demonstrated its
state-of-the-art performance for the scene graph generation. The experimental
results show that the performance of object detection and visual relation-ship
detection are promoted mutually by our method
ADI schemes for heat equations with irregular boundaries and interfaces in 3D with applications
In this paper, efficient alternating direction implicit (ADI) schemes are
proposed to solve three-dimensional heat equations with irregular boundaries
and interfaces. Starting from the well-known Douglas-Gunn ADI scheme, a
modified ADI scheme is constructed to mitigate the issue of accuracy loss in
solving problems with time-dependent boundary conditions. The unconditional
stability of the new ADI scheme is also rigorously proven with the Fourier
analysis. Then, by combining the ADI schemes with a 1D kernel-free boundary
integral (KFBI) method, KFBI-ADI schemes are developed to solve the heat
equation with irregular boundaries. In 1D sub-problems of the KFBI-ADI schemes,
the KFBI discretization takes advantage of the Cartesian grid and preserves the
structure of the coefficient matrix so that the fast Thomas algorithm can be
applied to solve the linear system efficiently. Second-order accuracy and
unconditional stability of the KFBI-ADI schemes are verified through several
numerical tests for both the heat equation and a reaction-diffusion equation.
For the Stefan problem, which is a free boundary problem of the heat equation,
a level set method is incorporated into the ADI method to capture the
time-dependent interface. Numerical examples for simulating 3D dendritic
solidification phenomenons are also presented
Kernel Free Boundary Integral Method for 3D Stokes and Navier Equations on Irregular Domains
A second-order accurate kernel-free boundary integral method is presented for
Stokes and Navier boundary value problems on three-dimensional irregular
domains. It solves equations in the framework of boundary integral equations,
whose corresponding discrete forms are well-conditioned and solved by the GMRES
method. A notable feature of this approach is that the boundary or volume
integrals encountered in BIEs are indirectly evaluated by a Cartesian
grid-based method, which includes discretizing corresponding simple interface
problems with a MAC scheme, correcting discrete linear systems to reduce large
local truncation errors near the interface, solving the modified system by a CG
method together with an FFT-based Poisson solver. No extra work or special
quadratures are required to deal with singular or hyper-singular boundary
integrals and the dependence on the analytical expressions of Green's functions
for the integral kernels is completely eliminated. Numerical results are given
to demonstrate the efficiency and accuracy of the Cartesian grid-based method
Kernel-free boundary integral method for two-phase Stokes equations with discontinuous viscosity on staggered grids
A discontinuous viscosity coefficient makes the jump conditions of the
velocity and normal stress coupled together, which brings great challenges to
some commonly used numerical methods to obtain accurate solutions. To overcome
the difficulties, a kernel free boundary integral (KFBI) method combined with a
modified marker-and-cell (MAC) scheme is developed to solve the two-phase
Stokes problems with discontinuous viscosity. The main idea is to reformulate
the two-phase Stokes problem into a single-fluid Stokes problem by using
boundary integral equations and then evaluate the boundary integrals indirectly
through a Cartesian grid-based method. Since the jump conditions of the
single-fluid Stokes problems can be easily decoupled, the modified MAC scheme
is adopted here and the existing fast solver can be applicable for the
resulting linear saddle system. The computed numerical solutions are second
order accurate in discrete -norm for velocity and pressure as well as
the gradient of velocity, and also second order accurate in maximum norm for
both velocity and its gradient, even in the case of high contrast viscosity
coefficient, which is demonstrated in numerical tests
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