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1.
This paper presents a fourth-order Cartesian grid based boundary integral
method (BIM) for heterogeneous interface problems in two and three dimensional
space, where the problem interfaces are irregular and can be explicitly given by parametric curves or implicitly defined by level set functions. The method reformulates the
governing equation with interface conditions into boundary integral equations (BIEs)
and reinterprets the involved integrals as solutions to some simple interface problems
in an extended regular region. Solution of the simple equivalent interface problems for
integral evaluation relies on a fourth-order finite difference method with an FFT-based
fast elliptic solver. The structure of the coefficient matrix is preserved even with the
existence of the interface. In the whole calculation process, analytical expressions of
Green’s functions are never determined, formulated or computed. This is the novelty
of the proposed kernel-free boundary integral (KFBI) method. Numerical experiments
in both two and three dimensions are shown to demonstrate the algorithm efficiency
and solution accuracy even for problems with a large diffusion coefficient ratio. 相似文献
2.
This work proposes a generalized boundary integral method for variable coefficients
elliptic partial differential equations (PDEs), including both boundary value
and interface problems. The method is kernel-free in the sense that there is no need
to know analytical expressions for kernels of the boundary and volume integrals in
the solution of boundary integral equations. Evaluation of a boundary or volume integral
is replaced with interpolation of a Cartesian grid based solution, which satisfies
an equivalent discrete interface problem, while the interface problem is solved by a
fast solver in the Cartesian grid. The computational work involved with the generalized
boundary integral method is essentially linearly proportional to the number
of grid nodes in the domain. This paper gives implementation details for a second-order
version of the kernel-free boundary integral method in two space dimensions
and presents numerical experiments to demonstrate the efficiency and accuracy of
the method for both boundary value and interface problems. The interface problems
demonstrated include those with piecewise constant and large-ratio coefficients and
the heterogeneous interface problem, where the elliptic PDEs on two sides of the interface
are of different types. 相似文献
3.
Shaoqiang Tang & Xi Zhu 《Communications In Computational Physics》2021,29(2):399-419
In this paper, we propose accurate numerical boundary conditions for atomic
simulations of twin boundary. The heterogeneity of the lattice structure induces physical reflection across the twin boundary. When numerical boundary and the twin
boundary coincide, the goal is to reproduce the correct amount of physical reflection.
In particular, we consider waves periodic in the direction parallel to the twin boundary and reduce the problem into a complex-valued chain motion. Using Laplace transform, we design time history kernel (THK) treatment. We further design matching
boundary conditions (MBC) by reproducing physical reflection at long wave limit and
a specific wave number. Reflection analysis and numerical tests demonstrate the effectiveness of the proposed THK and MBC treatments. 相似文献
4.
A. Rathsfeld G. Schmidt & B. H. Kleemann 《Communications In Computational Physics》2006,1(6):984-1009
The integral equation method for the simulation of the diffraction by optical gratings is an efficient numerical tool if profile gratings determined by simple cross-section curves are considered. This method in its recent version is capable to tackle
profile curves with corners, gratings with thin coated layers, and diffraction scenarios
with unfavorably large ratio period over wavelength. We discuss special implementational issues including the efficient evaluation of the quasi-periodic Green kernels,
the quadrature algorithm, and the iterative solution of the arising systems of linear
equations. Finally, as an example we present the simulation of echelle gratings which
demonstrates the efficiency of our approach. 相似文献
5.
Fast Multipole Accelerated Boundary Integral Equation Method for Evaluating the Stress Field Associated with Dislocations in a Finite Medium
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Degang Zhao Jingfang Huang & Yang Xiang 《Communications In Computational Physics》2012,12(1):226-246
In this paper, we develop an efficient numerical method based on the boundary integral equation formulation and new version of fast multipole method to solve
the boundary value problem for the stress field associated with dislocations in a finite
medium. Numerical examples are presented to examine the influence from material
boundaries on dislocations. 相似文献
6.
A Well-Conditioned Hypersingular Boundary Element Method for Electrostatic Potentials in the Presence of Inhomogeneities within Layered Media
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In this paper, we will present a high-order, well-conditioned boundary element
method (BEM) based on Müller's hypersingular second kind integral equation
formulation to accurately compute electrostatic potentials in the presence of inhomogeneity
embedded within layered media. We consider two types of inhomogeneities:
the first one is a simple model of an ion channel which consists of a finite height cylindrical
cavity embedded in a layered electrolytes/membrane environment, and the second
one is a Janus particle made of two different semi-spherical dielectric materials.
Both types of inhomogeneities have relevant applications in biology and colloidal material,
respectively. The proposed BEM gives$\mathcal{O}$(1) condition numbers, allowing fast
convergence of iterative solvers compared to previous work using first kind of integral
equations. We also show that the second order basis converges faster and is more
accurate than the first order basis for the BEM. 相似文献
7.
In a two-dimensional (2D) photonic crystal (PhC) composed of circular cylinders
(dielectric rods or air holes) on a square or triangular lattice, various PhC devices
can be created by removing or modifying some cylinders. Most existing numerical
methods for PhC devices give rise to large sparse or smaller but dense linear systems,
all of which are expensive to solve if the device is large. In a previous work [Z. Hu et al.,
Optics Express, 16 (2008), 17383-17399], an efficient Dirichlet-to-Neumann (DtN) map
method was developed for general 2D PhC devices with an infinite background PhC
to take full advantage of the underlying lattice structure. The DtN map of a unit cell is
an operator that maps the wave field to its normal derivative on the cell boundary and
it allows one to avoid computing the wave field in the interior of the unit cell. In this
paper, we extend the DtN map method to PhC devices with a finite background PhC.
Since there is no bandgap effect to confine the light in a finite PhC, a different technique
for truncating the domain is needed. We enclose the finite structure with a layer
of empty boundary and corner unit cells, and approximate the DtN maps of these cells
based on expanding the scattered wave in outgoing plane waves. Our method gives
rise to a relatively small and sparse linear systems that are particularly easy to solve. 相似文献
8.
Fast Spectral Collocation Method for Surface Integral Equations of Potential Problems in a Spheroid
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This paper proposes a new technique to speed up the computation of the
matrix of spectral collocation discretizations of surface single and double layer operators
over a spheroid. The layer densities are approximated by a spectral expansion
of spherical harmonics and the spectral collocation method is then used to solve surface
integral equations of potential problems in a spheroid. With the proposed technique,
the computation cost of collocation matrix entries is reduced from O(M2N4) to
O(MN4), where N2is the number of spherical harmonics (i.e., size of the matrix) and
M is the number of one-dimensional integration quadrature points. Numerical results
demonstrate the spectral accuracy of the method. 相似文献
9.
A Highly Scalable Boundary Integral Equation and Walk-on-Spheres (BIE-WOS) Method for the Laplace Equation with Dirichlet Data
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In this paper, we study a highly scalable communication-free parallel domain boundary decomposition algorithm for the Laplace equation based on a hybrid method combining boundary integral equations and walk-on-spheres (BIE-WOS)
method, which provides a numerical approximation of the Dirichlet-to-Neumann
(DtN) mapping for the Laplace equation. The BIE-WOS is a local method on the
boundary of the domain and does not require a structured mesh, and only needs a
covering of the domain boundary by patches and a local mesh for each patch for a local BIE. A new version of the BIE-WOS method with second kind integral equations is
introduced for better error controls. The effect of errors from the Feynman-Kac formula
based path integral WOS method on the overall accuracy of the BIE-WOS method is
analyzed for the BIEs, especially in the calculation of the right hand sides of the BIEs.
For the special case of flat patches, it is shown that the second kind integral equation
of BIE-WOS method can be simplified where the local BIE solutions can be given in
closed forms. A key advantage of the parallel BIE-WOS method is the absence of communications during the computation of the DtN mapping on individual patches of
the boundary, resulting in a complete independent computation using a large number
of cluster nodes. In addition, the BIE-WOS has an intrinsic capability of fault tolerance for exascale computations. The nearly linear scalability of the parallel BIE-WOS
method on a large-scale cluster with 6400 CPU cores is verified for computing the DtN
mapping of exterior Laplace problems with Dirichlet data for several domains. 相似文献
10.
Rongfang Gong Xiaoliang Cheng & Weimin Han 《Communications In Computational Physics》2016,19(1):226-250
In this paper, we introduce and study a new method for solving inverse
source problems, through a working model that arises in bioluminescence tomography
(BLT). In the BLT problem, one constructs quantitatively the bioluminescence source
distribution inside a small animal from optical signals detected on the animal's body
surface. The BLT problem possesses strong ill-posedness and often the Tikhonov regularization
is used to obtain stable approximate solutions. In conventional Tikhonov
regularization, it is crucial to choose a proper regularization parameter for trade off
between the accuracy and stability of approximate solutions. The new method is
based on a combination of the boundary condition and the boundary measurement
in a parameter-dependent single complex Robin boundary condition, followed by the
Tikhonov regularization. By properly adjusting the parameter in the Robin boundary
condition, we achieve two important properties for our new method: first, the regularized
solutions are uniformly stable with respect to the regularization parameter
so that the regularization parameter can be chosen based solely on the consideration
of the solution accuracy; second, the convergence order of the regularized solutions
reaches one with respect to the noise level. Then, the finite element method is used to
compute numerical solutions and a new finite element error estimate is derived for discrete
solutions. These results improve related results found in the existing literature.
Several numerical examples are provided to illustrate the theoretical results. 相似文献
11.
Victor Isakov Shingyu Leung & Jianliang Qian 《Communications In Computational Physics》2011,10(4):1044-1070
We propose a fast local level set method for the inverse problem of gravimetry.
The theoretical foundation for our approach is based on the following uniqueness
result: if an open set D is star-shaped or x3-convex with respect to its center of gravity,
then its exterior potential uniquely determines the open set D. To achieve this
purpose constructively, the first challenge is how to parametrize this open set D as its
boundary may have a variety of possible shapes. To describe those different shapes we
propose to use a level-set function to parametrize the unknown boundary of this open
set. The second challenge is how to deal with the issue of partial data as gravimetric
measurements are only made on a part of a given reference domain Ω. To overcome
this difficulty, we propose a linear numerical continuation approach based on the single
layer representation to find potentials on the boundary of some artificial domain
containing the unknown set D. The third challenge is how to speed up the level set inversion
process. Based on some features of the underlying inverse gravimetry problem
such as the potential density being constant inside the unknown domain, we propose a
novel numerical approach which is able to take advantage of these features so that the
computational speed is accelerated by an order of magnitude. We carry out numerical
experiments for both two- and three-dimensional cases to demonstrate the effectiveness
of the new algorithm. 相似文献
12.
A High Frequency Boundary Element Method for Scattering by Convex Polygons with Impedance Boundary Conditions
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S. N. Chandler-Wilde S. Langdon & M. Mokgolele 《Communications In Computational Physics》2012,11(2):573-593
We consider scattering of a time harmonic incident plane wave by a convex polygon with piecewise constant impedance boundary conditions. Standard finite or boundary element methods require the number of degrees of freedom to grow at least linearly with respect to the frequency of the incident wave in order to maintain accuracy. Extending earlier work by Chandler-Wilde and Langdon for the sound soft problem, we propose a novel Galerkin boundary element method, with the approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh with smaller elements closer to the corners of the polygon. Theoretical analysis and numerical results suggest that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency of the incident wave. 相似文献
13.
Background Eccrineangiomatoushamartoma(EAH)is a rare benign disease that comprises vascular and eccrine malformations.Pain is one of the most common symptoms and surgical resection is generally required.Objective To report a novel surgical technique that can alleviate severe pain caused by a large lesion in an EAH patient.Methods On magnetic resonance imaging(MRI),a subcutaneous abnormal signal was detected in the right hip and groin area of an 18-year-old male.The lesion was carefully elevated from the normal tissue above the plane of the deep fascia and was sutured in situ.Additionally,a small sample of the central lesion tissue was collected for histological examination.Results The visual analog scale score of pain decreased from 9 points to 2 points at 1week and to 0 points at 2 weeks after the surgery.The only complication observed was local numbness,which recovered spontaneously.No recurrence of pain was observed after 1 year of surgery.Histological findings demonstrated that the numbers of eccrine sweat glands and abnormal vessels decreased post-surgery.Conclusion The surgical procedure used in our case may represent a novel,easy,safe,and effective option for the treatment of a large,severely painful EAH. 相似文献
14.
A Lattice Boltzmann Method for the Advection-Diffusion Equation with Neumann Boundary Conditions
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In this paper, we study a lattice Boltzmann method for the advection-diffusion equation with Neumann boundary conditions on general boundaries. A
novel mass conservative scheme is introduced for implementing such boundary conditions, and is analyzed both theoretically and numerically.Second order convergence is predicted by the theoretical analysis, and numerical
investigations show that the convergence is at or close to the predicted rate. The numerical investigations include time-dependent problems and a steady-state diffusion
problem for computation of effective diffusion coefficients. 相似文献
15.
Explicit time stepping schemes for the immersed boundary method require
very small time steps in order to maintain stability. Solving the equations that arise
from an implicit discretization is difficult. Recently, several different approaches have
been proposed, but a complete understanding of this problem is still emerging. A
multigrid method is developed and explored for solving the equations in an implicit-time discretization of a model of the immersed boundary equations. The model problem consists of a scalar Poisson equation with conformation-dependent singular forces
on an immersed boundary. This model does not include the inertial terms or the incompressibility constraint. The method is more efficient than an explicit method, but
the efficiency gain is limited. The multigrid method alone may not be an effective
solver, but when used as a preconditioner for Krylov methods, the speed-up over the
explicit-time method is substantial. For example, depending on the constitutive law
for the boundary force, with a time step 100 times larger than the explicit method,
the implicit method is about 15-100 times more efficient than the explicit method. A
very attractive feature of this method is that the efficiency of the multigrid preconditioned Krylov solver is shown to be independent of the number of immersed boundary
points. 相似文献
16.
Morteza Meftah Akhilesh Yadav Anthony C. Wong Amar S. Ranawat Chitranjan S. Ranawat 《The Journal of arthroplasty》2013
Cup positioning is an important variable for short and long term function, stability, and durability of total hip arthroplasty (THA). This novel method utilizes internal and external bony landmarks, and the transverse acetabular ligament for positioning the acetabular component. The cup is placed parallel and superior to the transverse ligament and inside the anterior wall notch of the true acetabulum, then adjusted for femoral version and pelvic tilt and obliquity based on weight bearing radiographs. In 78 consecutive THAs, the mean functional anteversion and abduction angles were 17.9 ° ± 4.7 ° and 41.7 ° ± 3.8 °, respectively. 96% of the functional anteversion measurements and 100% of the functional abduction angles were within the safe zone. This technique is an easy, reproducible, and accurate method for functional cup placement. 相似文献
17.
Yunyun Ma & Jiguang Sun 《Communications In Computational Physics》2022,31(5):1546-1560
We propose a numerical method for a non-selfadjoint Steklov eigenvalue
problem of the Helmholtz equation. The problem is formulated using boundary integrals. The Nyström method is employed to discretize the integral operators, which
leads to a non-Hermitian generalized matrix eigenvalue problems. The spectral indicator method (SIM) is then applied to calculate the (complex) eigenvalues. The convergence is proved using the spectral approximation theory for (non-selfadjoint) compact
operators. Numerical examples are presented for validation. 相似文献
18.
运用醋酸锂处理酵母细胞,建立并优化了酵母完整细胞的高效质粒转化体系,转化率达104μg-1.通过对影响转化的诸因子的研究,发现载运DNA是影响酵母完整细胞转化效率的最主要因素,热休克处理及处理时间、聚乙二醇相对分子质量及浓度等对转化率也有显著影响。 相似文献
19.
Ming-Jyh Chern Dedy Zulhidayat Noor Ching-Biao Liao & Tzyy-Leng Horng 《Communications In Computational Physics》2015,18(4):1072-1094
A direct-forcing immersed boundary method (DFIB) with both virtual force
and heat source is developed here to solve Navier-Stokes and the associated energy
transport equations to study some thermal flow problems caused by a moving rigid
solid object within. The key point of this novel numerical method is that the solid object,
stationary or moving, is first treated as fluid governed by Navier-Stokes equations
for velocity and pressure, and by energy transport equation for temperature in every
time step. An additional virtual force term is then introduced on the right hand side
of momentum equations in the solid object region to make it act exactly as if it were
a solid rigid body immersed in the fluid. Likewise, an additional virtual heat source
term is applied to the right hand side of energy equation at the solid object region
to maintain the solid object at the prescribed temperature all the time. The current
method was validated by some benchmark forced and natural convection problems
such as a uniform flow past a heated circular cylinder, and a heated circular cylinder
inside a square enclosure. We further demonstrated this method by studying a mixed
convection problem involving a heated circular cylinder moving inside a square enclosure.
Our current method avoids the otherwise requested dynamic grid generation in
traditional method and shows great efficiency in the computation of thermal and flow
fields caused by fluid-structure interaction. 相似文献
20.
M. Bergmann J. Hovnanian & A. Iollo 《Communications In Computational Physics》2014,15(5):1266-1290
An accurate cartesian method is devised to simulate incompressible viscous
flows past an arbitrary moving body. The Navier-Stokes equations are spatially discretized onto a fixed Cartesian mesh. The body is taken into account via the ghost-cell
method and the so-called penalty method, resulting in second-order accuracy in velocity. The accuracy and the efficiency of the solver are tested through two-dimensional
reference simulations. To show the versatility of this scheme we simulate a three-dimensional self propelled jellyfish prototype. 相似文献