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1.
Consider the electromagnetic scattering of a time-harmonic plane wave by
an open cavity which is embedded in a perfectly electrically conducting infinite ground
plane. This paper is concerned with the numerical solutions of the transverse electric
and magnetic polarizations of the open cavity scattering problems. In each polarization, the scattering problem is reduced equivalently into a boundary value problem of
the two-dimensional Helmholtz equation in a bounded domain by using the transparent boundary condition (TBC). An a posteriori estimate based adaptive finite element
method with the perfectly matched layer (PML) technique is developed to solve the
reduced problem. The estimate takes account ofboththe finite element approximation
error and the PML truncation error, where the latter is shown to decay exponentially
with respect to the PML medium parameter and the thickness of the PML layer. Numerical experiments are presented and compared with the adaptive finite element TBC
method for both polarizations to illustrate the competitive behavior of the proposed
method. 相似文献
2.
An Edge-Based Smoothed Finite Element Method with TBC for the Elastic Wave Scattering by an Obstacle 下载免费PDF全文
Ze Wu Junhong Yue Ming Li Ruiping Niu & Yufei Zhang 《Communications In Computational Physics》2021,30(3):709-748
Elastic wave scattering has received ever-increasing attention in military and medical fields due to its high-precision solution. In this paper, an edge-based smoothed finite element method (ES-FEM) combined with the transparent boundary condition (TBC) is proposed to solve the elastic wave scattering problem by a rigid obstacle with smooth surface, which is embedded in an isotropic and homogeneous elastic medium in two dimensions. The elastic wave scattering problem satisfies Helmholtz equations with coupled boundary conditions obtained by Helmholtz decomposition. Firstly, the TBC of the elastic wave scattering is constructed by using the analytical solution to Helmholtz equations, which can truncate the boundary value problem (BVP) in an unbounded domain into the BVP in a bounded domain. Then the formulations of ES-FEM with the TBC are derived for Helmholtz equations with coupled boundary conditions. Finally, several numerical examples illustrate that the proposed ES-FEM with the TBC (ES-FEM-TBC) can work effectively and obtain more stable and accurate solution than the standard FEM with the TBC (FEM-TBC) for the elastic wave scattering problem. 相似文献
3.
Construction and Analysis of an Adapted Spectral Finite Element Method to Convective Acoustic Equations 下载免费PDF全文
Andreas Hü ppe Gary Cohen Sé bastien Imperiale & Manfred Kaltenbacher 《Communications In Computational Physics》2016,20(1):1-22
The paper addresses the construction of a non spurious mixed spectral finite
element (FE) method to problems in the field of computational aeroacoustics. Based
on a computational scheme for the conservation equations of linear acoustics, the extension
towards convected wave propagation is investigated. In aeroacoustic applications,
the mean flow effects can have a significant impact on the generated sound
field even for smaller Mach numbers. For those convective terms, the initial spectral
FE discretization leads to non-physical, spurious solutions. Therefore, a regularization
procedure is proposed and qualitatively investigated by means of discrete eigenvalues
analysis of the discrete operator in space. A study of convergence and an application
of the proposed scheme to simulate the flow induced sound generation in the process
of human phonation underlines stability and validity. 相似文献
4.
A Moving-Mesh Finite Element Method and Its Application to the Numerical Solution of Phase-Change Problems 下载免费PDF全文
M. J. Baines M. E. Hubbard P. K. Jimack & R. Mahmood 《Communications In Computational Physics》2009,6(3):595-624
A distributed Lagrangian moving-mesh finite element method is applied to
problems involving changes of phase. The algorithm uses a distributed conservation
principle to determine nodal mesh velocities, which are then used to move the nodes.
The nodal values are obtained from an ALE (Arbitrary Lagrangian-Eulerian) equation,
which represents a generalization of the original algorithm presented in Applied
Numerical Mathematics, 54:450–469 (2005). Having described the details of the generalized
algorithm it is validated on two test cases from the original paper and is then
applied to one-phase and, for the first time, two-phase Stefan problems in one and two
space dimensions, paying particular attention to the implementation of the interface
boundary conditions. Results are presented to demonstrate the accuracy and the effectiveness
of the method, including comparisons against analytical solutions where
available. 相似文献
5.
Convergent Adaptive Finite Element Method Based on Centroidal Voronoi Tessellations and Superconvergence 下载免费PDF全文
Yunqing Huang Hengfeng Qin Desheng Wang & Qiang Du 《Communications In Computational Physics》2011,10(2):339-370
We present a novel adaptive finite element method (AFEM) for elliptic equations
which is based upon the Centroidal Voronoi Tessellation (CVT) and superconvergent
gradient recovery. The constructions of CVT and its dual Centroidal Voronoi
Delaunay Triangulation (CVDT) are facilitated by a localized Lloyd iteration to produce
almost equilateral two dimensional meshes. Working with finite element solutions
on such high quality triangulations, superconvergent recovery methods become
particularly effective so that asymptotically exact a posteriori error estimations can be
obtained. Through a seamless integration of these techniques, a convergent adaptive
procedure is developed. As demonstrated by the numerical examples, the new AFEM
is capable of solving a variety of model problems and has great potential in practical
applications. 相似文献
6.
Two-Grid Method for Miscible Displacement Problem by Mixed Finite Element Methods and Mixed Finite Element Method of Characteristics 下载免费PDF全文
The miscible displacement of one incompressible fluid by another in a porous
medium is governed by a system of two equations. One is elliptic form equation for
the pressure and the other is parabolic form equation for the concentration of one of
the fluids. Since only the velocity and not the pressure appears explicitly in the concentration
equation, we use a mixed finite element method for the approximation of
the pressure equation and mixed finite element method with characteristics for the
concentration equation. To linearize the mixed-method equations, we use a two-grid
algorithm based on the Newton iteration method for this full discrete scheme problems.
First, we solve the original nonlinear equations on the coarse grid, then, we
solve the linearized problem on the fine grid used Newton iteration once. It is shown
that the coarse grid can be much coarser than the fine grid and achieve asymptotically
optimal approximation as long as the mesh sizes satisfy $h=H^2$ in this paper. Finally,
numerical experiment indicates that two-grid algorithm is very effective. 相似文献
7.
An r-Adaptive Finite Element Method for the Solution of the Two-Dimensional Phase-Field Equations 下载免费PDF全文
G. Beckett J. A. Mackenzie & M. L. Robertson 《Communications In Computational Physics》2006,1(5):805-826
An adaptive moving mesh method is developed for the numerical solution
of two-dimensional phase change problems modelled by the phase-field equations. The
numerical algorithm is relatively simple and is shown to be more efficient than fixed grid
methods. The phase-field equations are discretized by a Galerkin finite element method.
An adaptivity criterion is used that ensures that the mesh spacing at the phase front
scales with the diffuse interface thickness. 相似文献
8.
An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems 下载免费PDF全文
Ye Li 《Communications In Computational Physics》2016,19(2):442-472
In this paper, we propose a uniformly convergent adaptive finite element
method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed
nonlinear eigenvalue problems under constraints with applications in Bose-Einstein
condensation (BEC) and quantum chemistry. We begin with the time-independent
Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed
nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations
for the problem are reviewed to confirm the asymptotic behaviors of the solutions
in the boundary/interior layer regions. By using the normalized gradient flow, we
propose an adaptive finite element with hybrid basis to solve the singularly perturbed
nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively
to the small parameter ε. Extensive numerical results are reported to show the
uniform convergence property of our method. We also apply the AFEM-HB to compute
the ground and excited states of BEC with box/harmonic/optical lattice potential
in the semiclassical regime (0<ε≪1). In addition, we give a detailed error analysis of
our AFEM-HB to a simpler singularly perturbed two point boundary value problem,
show that our method has a minimum uniform convergence order $\mathcal{O}$(1/$(NlnN)^\frac{2}{3}$). 相似文献
9.
M. Holst J. A. McCammon Z. Yu Y. C. Zhou & Y. Zhu 《Communications In Computational Physics》2012,11(1):179-214
We consider the design of an effective and reliable adaptive finite element
method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first
complete solution and approximation theory for the Poisson-Boltzmann equation, the
first provably convergent discretization and also allowed for the development of a
provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this
regularization technique which can be shown to be less susceptible to such instability.
We establish a priori estimates and other basic results for the continuous regularized
problem, as well as for Galerkin finite element approximations. We show that the new
approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme
for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is
one of the first results of this type for nonlinear elliptic problems, is based on usingcontinuous and discrete a priori L∞ estimates. To provide a high-quality geometric
model as input to the AFEM algorithm, we also describe a class of feature-preserving
adaptive mesh generation algorithms designed specifically for constructing meshes of
biomolecular structures, based on the intrinsic local structure tensor of the molecular
surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages
of the new regularization scheme are demonstrated with FETK through comparisons
with the original regularization approach for a model problem. The convergence and
accuracy of the overall AFEM algorithm is also illustrated by numerical approximation
of electrostatic solvation energy for an insulin protein. 相似文献
10.
An Adaptive Finite Element Method with a Modified Perfectly Matched Layer Formulation for Diffraction Gratings 下载免费PDF全文
For numerical simulation of one-dimensional diffraction gratings both in TE
and TM polarization, an enhanced adaptive finite element method is proposed in this
paper. A modified perfectly matched layer (PML) formulation is proposed for the truncation
of the unbounded domain, which results in a homogeneous Dirichlet boundary
condition and the corresponding error estimate is greatly simplified. The a posteriori
error estimates for the adaptive finite element method are provided. Moreover, a lower
bound is obtained to demonstrate that the error estimates obtained are sharp. 相似文献
11.
In this work, we propose an efficient multi-mesh adaptive finite element
method for simulating the dendritic growth in two- and three-dimensions. The governing equations used are the phase field model, where the regularity behaviors of the
relevant dependent variables, namely the thermal field function and the phase field
function, can be very different. To enhance the computational efficiency, we approximate these variables on different h-adaptive meshes. The coupled terms in the system
are calculated based on the implementation of the multi-mesh h-adaptive algorithm
proposed by Li (J. Sci. Comput., pp. 321-341, 24 (2005)). It is illustrated numerically
that the multi-mesh technique is useful in solving phase field models and can save
storage and the CPU time significantly. 相似文献
12.
Analysis and Numerical Solution of Transient Electromagnetic Scattering from Overfilled Cavities 下载免费PDF全文
A hybrid finite element (FEM) and Fourier transform method is implemented to analyze the time domain scattering of a plane wave incident on a 2-D overfilled cavity embedded in the infinite ground plane. The algorithm first removes the
time variable by Fourier transform, through which a frequency domain problem is obtained. An artificial boundary condition is then introduced on a hemisphere enclosing
the cavity that couples the fields from the infinite exterior domain to those inside. The
exterior problem is solved analytically via Fourier series solutions, while the interior
region is solved using finite element method. In the end, the image functions in frequency domain are numerically inverted into the time domain. The perfect link over
the artificial boundary between the FEM approximation in the interior and analytical
solution in the exterior indicates the reliability of the method. A convergence analysis
is also performed. 相似文献
13.
A Decoupled Energy Stable Adaptive Finite Element Method for Cahn–Hilliard–Navier–Stokes Equations 下载免费PDF全文
In this paper, we propose, analyze, and numerically validate an adaptive
finite element method for the Cahn–Hilliard–Navier–Stokes equations. The adaptive
method is based on a linear, decoupled scheme introduced by Shen and Yang [30].
An unconditionally energy stable discrete law for the modified energy is shown for
the fully discrete scheme. A superconvergent cluster recovery based a posteriori error
estimations are constructed for both the phase field variable and velocity field function,
respectively. Based on the proposed space and time discretization error estimators, a
time-space adaptive algorithm is designed for numerical approximation of the Cahn–Hilliard–Navier–Stokes equations. Numerical experiments are presented to illustrate
the reliability and efficiency of the proposed error estimators and the corresponding
adaptive algorithm. 相似文献
14.
Acoustic Scattering Problems with Convolution Quadrature and the Method of Fundamental Solutions 下载免费PDF全文
Labarca Ignacio & Hiptmair Ralf 《Communications In Computational Physics》2021,30(4):985-1008
Time-domain acoustic scattering problems in two dimensions are studied.
The numerical scheme relies on the use of the Convolution Quadrature (CQ) method
to reduce the time-domain problem to the solution of frequency-domain Helmholtz
equations with complex wavenumbers. These equations are solved with the method
of fundamental solutions (MFS), which approximates the solution by a linear combination of fundamental solutions defined at source points inside (outside) the scatterer for
exterior (interior) problems. Numerical results show that the coupling of both methods
works efficiently and accurately for multistep and multistage based CQ. 相似文献
15.
A Hybrid Finite Element-Laplace Transform Method for the Analysis of Transient Electromagnetic Scattering by an Over-Filled Cavity in the Ground Plane 下载免费PDF全文
Junqi Huang Aihua W. Wood & Michael J. Havrilla 《Communications In Computational Physics》2009,5(1):126-141
A hybrid finite element-Laplace transform method is implemented to analyze the time domain electromagnetic scattering induced by a 2-D overfilled cavity
embedded in the infinite ground plane. The algorithm divides the whole scattering
domain into two, interior and exterior, sub-domains. In the interior sub-domain which
covers the cavity, the problem is solved via the finite element method. The problem is
solved analytically in the exterior sub-domain which slightly overlaps the interior subdomain and extends to the rest of the upper half plane. The use of the Laplace transform leads to an analytical link condition between the overlapping sub-domains. The
analytical link guides the selection of the overlapping zone and eliminates the need
to use the conventional Schwartz iteration. This dramatically improves the efficiency
for solving transient scattering problems. Numerical solutions are tested favorably
against analytical ones for a canonical geometry. The perfect link over the artificial
boundary between the finite element approximation in the interior and analytical solution in the exterior further indicates the reliability of the method. An error analysis
is also performed. 相似文献
16.
An Efficient Finite Element Method with Exponential Mesh Refinement for the Solution of the Allen-Cahn Equation in Non-Convex Polygons 下载免费PDF全文
Emine Celiker & Ping Lin 《Communications In Computational Physics》2020,28(4):1536-1560
In this paper we consider the numerical solution of the Allen-Cahn type
diffuse interface model in a polygonal domain. The intersection of the interface with
the re-entrant corners of the polygon causes strong corner singularities in the solution.
To overcome the effect of these singularities on the accuracy of the approximate solution, for the spatial discretization we develop an efficient finite element method with
exponential mesh refinement in the vicinity of the singular corners, that is based on
($k$−1)-th order Lagrange elements, $k$≥2 an integer. The problem is fully discretized by
employing a first-order, semi-implicit time stepping scheme with the Invariant Energy
Quadratization approach in time, which is an unconditionally energy stable method.
It is shown that for the error between the exact and the approximate solution, an accuracy of $\mathcal{O}$($h^k$+$τ$) is attained in the $L^2$-norm for the number of $\mathcal{O}$($h^{−2}$ln$h^{−1}$) spatial
elements, where $h$ and $τ$ are the mesh and time steps, respectively. The numerical
results obtained support the analysis made. 相似文献
17.
Yongyue Jiang Ping Lin Zhenlin Guo & Shuangling Dong 《Communications In Computational Physics》2015,18(1):180-202
In this paper, we compute a phase field (diffuse interface) model of Cahn-Hilliard
type for moving contact line problems governing the motion of isothermal
multiphase incompressible fluids. The generalized Navier boundary condition proposed
by Qian et al. [1] is adopted here. We discretize model equations using a continuous
finite element method in space and a modified midpoint scheme in time. We
apply a penalty formulation to the continuity equation which may increase the stability
in the pressure variable. Two kinds of immiscible fluids in a pipe and droplet displacement
with a moving contact line under the effect of pressure driven shear flow
are studied using a relatively coarse grid. We also derive the discrete energy law for
the droplet displacement case, which is slightly different due to the boundary conditions.
The accuracy and stability of the scheme are validated by examples, results and
estimate order. 相似文献
18.
An Adaptive Modal Discontinuous Galerkin Finite Element Parallel Method Using Unsplit Multi-Axial Perfectly Matched Layer for Seismic Wave Modeling 下载免费PDF全文
Yang Xu Xiaofei Chen Wei Zhang & Xiao Pan 《Communications In Computational Physics》2022,31(4):1083-1113
The discontinuous Galerkin finite element method (DG-FEM) is a high-precision numerical simulation method widely used in various disciplines. In this paper, we derive the auxiliary ordinary differential equation complex frequency-shifted
multi-axial perfectly matched layer (AODE CFS-MPML) in an unsplit format and combine it with any high-order adaptive DG-FEM based on an unstructured mesh to simulate seismic wave propagation. To improve the computational efficiency, we implement
Message Passing Interface (MPI) parallelization for the simulation. Comparisons of
the numerical simulation results with the analytical solutions verify the accuracy and
effectiveness of our method. The results of numerical experiments also confirm the
stability and effectiveness of the AODE CFS-MPML. 相似文献
19.
Retrieving Topological Information of Implicitly Represented Diffuse Interfaces with Adaptive Finite Element Discretization 下载免费PDF全文
We consider the finite element based computation of topological quantities of implicitly represented surfaces within a diffuse interface framework. Utilizing an adaptive finite element implementation with effective gradient recovery techniques, we discuss how the Euler number can be accurately computed directly from the numerically solved phase field functions or order parameters. Numerical examples and applications to the topological analysis of point clouds are also presented. 相似文献
20.
A High Frequency Boundary Element Method for Scattering by Convex Polygons with Impedance Boundary Conditions 下载免费PDF全文
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