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1.
The Brinkman model describes flow of fluid in complex porous media with
a high-contrast permeability coefficient such that the flow is dominated by Darcy in
some regions and by Stokes in others. A weak Galerkin (WG) finite element method
for solving the Brinkman equations in two or three dimensional spaces by using polynomials
is developed and analyzed. The WG method is designed by using the generalized
functions and their weak derivatives which are defined as generalized distributions.
The variational form we considered in this paper is based on two gradient operators
which is different from the usual gradient-divergence operators for Brinkman
equations. The WG method is highly flexible by allowing the use of discontinuous
functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal-order
error estimates are established for the corresponding WG finite element solutions
in various norms. Some computational results are presented to demonstrate the
robustness, reliability, accuracy, and flexibility of the WG method for the Brinkman
equations. 相似文献
2.
Lin Mu Junping Wang Xiu Ye & Shan Zhao 《Communications In Computational Physics》2014,15(5):1461-1479
A weak Galerkin (WG) method is introduced and numerically tested for the
Helmholtz equation. This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property. At the same time, the WG finite
element formulation is symmetric and parameter free. Several test scenarios are designed for a numerical investigation on the accuracy, convergence, and robustness of
the WG method in both inhomogeneous and homogeneous media over convex and
non-convex domains. Challenging problems with high wave numbers are also examined. Our numerical experiments indicate that the weak Galerkin is a finite element
technique that is easy to implement, and provides very accurate and robust numerical
solutions for the Helmholtz problem with high wave numbers. 相似文献
3.
A Parallel Domain Decomposition Algorithm for Simulating Blood Flow with Incompressible Navier-Stokes Equations with Resistive Boundary Condition 下载免费PDF全文
We introduce and study a parallel domain decomposition algorithm for
the simulation of blood flow in compliant arteries using a fully-coupled system of
nonlinear partial differential equations consisting of a linear elasticity equation and
the incompressible Navier-Stokes equations with a resistive outflow boundary condition. The system is discretized with a finite element method on unstructured moving
meshes and solved by a Newton-Krylov algorithm preconditioned with an overlapping restricted additive Schwarz method. The resistive outflow boundary condition
plays an interesting role in the accuracy of the blood flow simulation and we provide a
numerical comparison of its accuracy with the standard pressure type boundary condition. We also discuss the parallel performance of the implicit domain decomposition
method for solving the fully coupled nonlinear system on a supercomputer with a few
hundred processors. 相似文献
4.
An augmented method is proposed for solving stationary incompressible Stokes equations with a Dirichlet boundary condition along parts of the boundary. In this approach, the normal derivative of the pressure along the parts of the boundary is introduced as an additional variable and it is solved by the GMRES iterative method. The dimension of the augmented variable in discretization is the number of grid points along the boundary which is O(N). Each GMRES iteration (or one matrix-vector multiplication) requires three fast Poisson solvers for the pressure and the velocity. In our numerical experiments, only a few iterations are needed. We have also combined the augmented approach for Stokes equations involving interfaces, discontinuities, and singularities. 相似文献
5.
An Indirect-Forcing Immersed Boundary Method for Incompressible Viscous Flows with Interfaces on Irregular Domains 下载免费PDF全文
Zhijun Tan K. M. Lim B. C. Khoo & Desheng Wang 《Communications In Computational Physics》2009,6(5):997-1021
An indirect-forcing immersed boundary method for solving the incompressible
Navier-Stokes equations involving the interfaces and irregular domains is developed.
The rigid boundaries and interfaces are represented by a number of Lagrangian
control points. Stationary rigid boundaries are embedded in the Cartesian grid and
singular forces at the rigid boundaries are applied to impose the prescribed velocity
conditions. The singular forces at the interfaces and the rigid boundaries are then distributed
to the nearby Cartesian grid points using the immersed boundary method. In
the present work, the singular forces at the rigid boundaries are computed implicitly
by solving a small system of equations at each time step to ensure that the prescribed
velocity condition at the rigid boundary is satisfied exactly. For deformable interfaces,
the forces that the interface exerts on the fluid are computed from the configuration
of the elastic interface and are applied to the fluid. The Navier-Stokes equations are
discretized using finite difference method on a staggered uniform Cartesian grid by a
second order accurate projection method. The ability of the method to simulate viscous
flows with interfaces on irregular domains is demonstrated by applying to the
rotational flow problem, the relaxation of an elastic membrane and flow in a constriction
with an immersed elastic membrane. 相似文献
6.
This paper summarizes suitable material models for creep and damage of concrete which are coupled with heat and moisture transfer. The fully coupled approach or the staggered coupling is assumed. Governing equations are spatially discretized by the finite element method and the temporal discretization is done by the generalized trapezoidal method. Systems of non-linear algebraic equations are solved by the Newton method. Development of an efficient and extensible computer code based on the C++ programming language is described. Finally, successful analyses of two real engineering problems are described. 相似文献
7.
A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids 下载免费PDF全文
Weiyi Wang & Zhijun Tan 《Communications In Computational Physics》2021,30(1):227-254
In this paper, a fairly simple 3D immersed interface method based on the
CG-Uzawa type method and the level set representation of the interface is employed
for solving three-dimensional Stokes flow with singular forces along the interface. The
method is to apply the Taylor's expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes. A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy. The Stokes
equations are discretized involving the correction terms on staggered grids and then
solved by the conjugate gradient Uzawa type method. The major advantages of the
present method are the special simplicity, the ability in handling the Dirichlet boundary conditions, and no need of the pressure boundary condition. The method can
also preserve the volume conservation and the discrete divergence free condition very
well. The numerical results show that the proposed method is second order accurate
and efficient. 相似文献
8.
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
9.
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
10.
Benchmark Computations of the Phase Field Crystal and Functionalized Cahn-Hilliard Equations via Fully Implicit,Nesterov Accelerated Schemes 下载免费PDF全文
Jea-Hyun Park Abner J. Salgado & Steven M. Wise 《Communications In Computational Physics》2023,33(2):367-398
We introduce a fast solver for the phase field crystal (PFC) and functionalized Cahn-Hilliard (FCH) equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov’s accelerated gradient descent
(PAGD) method. We discretize these problems with a Fourier collocation method
in space, and employ various second-order schemes in time. We observe a significant speedup with this solver when compared to the preconditioned gradient descent
(PGD) method. With the PAGD solver, fully implicit, second-order-in-time schemes
are not only feasible to solve the PFC and FCH equations, but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are
taken into account. Benchmark computations of four different schemes for the PFC
and FCH equations are conducted and the results indicate that, for the FCH experiments, the fully implicit schemes (midpoint rule and BDF2 equipped with the PAGD
as a nonlinear time marching solver) perform better than their IMEX versions in terms
of computational cost needed to achieve a certain precision. For the PFC, the results
are not as conclusive as in the FCH experiments, which, we believe, is due to the fact
that the nonlinearity in the PFC is milder nature compared to the FCH equation. We
also discuss some practical matters in applying the PAGD. We introduce an averaged
Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good
preconditioner parameters. The sweeping-friction strategy exhibits almost as good
a performance as the case of the best manually tuned parameters. 相似文献
11.
Thomas Hagstrom Eliane Bé cache Dan Givoli & Kurt Stein 《Communications In Computational Physics》2012,11(2):610-628
Local approximate radiation boundary conditions of optimal efficiency for the convective wave equation and the linearized Euler equations in waveguide geometry are formulated, analyzed, and tested. The results extend and improve for the convective case the general formulation of high-order local radiation boundary condition sequences for anisotropic scalar equations developed in [4]. 相似文献
12.
Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schrödinger Equations with Unbounded Potentials and Nonlinearities 下载免费PDF全文
Pauline Klein Xavier Antoine Christophe Besse & Matthias Ehrhardt 《Communications In Computational Physics》2011,10(5):1280-1304
We propose a hierarchy of novel absorbing boundary conditions for the one-dimensional
stationary Schrödinger equation with general (linear and nonlinear) potential.
The accuracy of the new absorbing boundary conditions is investigated numerically
for the computation of energies and ground-states for linear and nonlinear
Schrödinger equations. It turns out that these absorbing boundary conditions and
their variants lead to a higher accuracy than the usual Dirichlet boundary condition.
Finally, we give the extension of these ABCs to N-dimensional stationary Schrödinger
equations. 相似文献
13.
Yifei Wan & Yinhua Xia 《Communications In Computational Physics》2023,33(5):1270-1331
For steady Euler equations in complex boundary domains, high-order shockcapturing schemes usually suffer not only from the difficulty of steady-state convergence but also from the problem of dealing with physical boundaries on Cartesian
grids to achieve uniform high-order accuracy. In this paper, we utilize a fifth-order
finite difference hybrid WENO scheme to simulate steady Euler equations, and the
same fifth-order WENO extrapolation methods are developed to handle the curved
boundary. The values of the ghost points outside the physical boundary can be obtained by applying WENO extrapolation near the boundary, involving normal derivatives acquired by the simplified inverse Lax-Wendroff procedure. Both equivalent expressions involving curvature and numerical differentiation are utilized to transform
the tangential derivatives along the curved solid wall boundary. This hybrid WENO
scheme is robust for steady-state convergence and maintains high-order accuracy in
the smooth region even with the solid wall boundary condition. Besides, the essentially non-oscillation property is achieved. The numerical spectral analysis also shows
that this hybrid WENO scheme has low dispersion and dissipation errors. Numerical
examples are presented to validate the high-order accuracy and robust performance of
the hybrid scheme for steady Euler equations in curved domains with Cartesian grids. 相似文献
14.
Dimension-Reduced Hyperbolic Moment Method for the Boltzmann Equation with BGK-Type Collision 下载免费PDF全文
Zhenning Cai Yuwei Fan Ruo Li & Zhonghua Qiao 《Communications In Computational Physics》2014,15(5):1368-1406
We develop the dimension-reduced hyperbolic moment method for the
Boltzmann equation, to improve solution efficiency using a numerical regularized
moment method for problems with low-dimensional macroscopic variables and high-dimensional microscopic variables. In the present work, we deduce the globally hyperbolic moment equations for the dimension-reduced Boltzmann equation based on the
Hermite expansion and a globally hyperbolic regularization. The numbers of Maxwell
boundary condition required for well-posedness are studied. The numerical scheme
is then developed and an improved projection algorithm between two different Hermite expansion spaces is developed. By solving several benchmark problems, we validate the method developed and demonstrate the significant efficiency improvement
by dimension-reduction. 相似文献
15.
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
16.
Brian C. Fabien 《Optimal control applications & methods.》2014,35(2):204-230
This paper presents an algorithm for the indirect solution of optimal control problems that contain mixed state and control variable inequality constraints. The necessary conditions for optimality lead to an inequality constrained two‐point BVP with index‐1 differential‐algebraic equations (BVP‐DAEs). These BVP‐DAEs are solved using a multiple shooting method where the DAEs are approximated using a single‐step linearly implicit Runge–Kutta (Rosenbrock–Wanner) method. An interior‐point Newton method is used to solve the residual equations associated with the multiple shooting discretization. The elements of the residual equations, and the Jacobian of the residual equations, are constructed in parallel. The search direction for the interior‐point method is computed by solving a sparse bordered almost block diagonal (BABD) linear system. Here, a parallel‐structured orthogonal factorization algorithm is used to solve the BABD system. Examples are presented to illustrate the efficiency of the parallel algorithm. It is shown that an American National Standards Institute C implementation of the parallel algorithm achieves significant speedup with the increase in the number of processors used. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
17.
A fourth-order finite difference method is proposed and studied for the
primitive equations (PEs) of large-scale atmospheric and oceanic flow based on mean
vorticity formulation. Since the vertical average of the horizontal velocity field is
divergence-free, we can introduce mean vorticity and mean stream function which are
connected by a 2-D Poisson equation. As a result, the PEs can be reformulated such that
the prognostic equation for the horizontal velocity is replaced by evolutionary equations for the mean vorticity field and the vertical derivative of the horizontal velocity.
The mean vorticity equation is approximated by a compact difference scheme due to
the difficulty of the mean vorticity boundary condition, while fourth-order long-stencil
approximations are utilized to deal with transport type equations for computational
convenience. The numerical values for the total velocity field (both horizontal and
vertical) are statically determined by a discrete realization of a differential equation at
each fixed horizontal point. The method is highly efficient and is capable of producing highly resolved solutions at a reasonable computational cost. The full fourth-order
accuracy is checked by an example of the reformulated PEs with force terms. Additionally, numerical results of a large-scale oceanic circulation are presented. 相似文献
18.
Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws 下载免费PDF全文
Andreas Hiltebrand & Siddhartha Mishra 《Communications In Computational Physics》2015,17(5):1320-1359
An entropy stable fully discrete shock capturing space-time Discontinuous
Galerkin (DG) method was proposed in a recent paper [20] to approximate hyperbolic
systems of conservation laws. This numerical scheme involves the solution of a
very large nonlinear system of algebraic equations, by a Newton-Krylov method, at
every time step. In this paper, we design efficient preconditioners for the large, nonsymmetric
linear system, that needs to be solved at every Newton step. Two sets of
preconditioners, one of the block Jacobi and another of the block Gauss-Seidel type are
designed. Fourier analysis of the preconditioners reveals their robustness and a large
number of numerical experiments are presented to illustrate the gain in efficiency that
results from preconditioning. The resulting method is employed to compute approximate
solutions of the compressible Euler equations, even for very high CFL numbers. 相似文献
19.
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
20.
Na Liu Luis Tobó n Yifa Tang & Qing Huo Liu 《Communications In Computational Physics》2015,17(2):458-486
It is well known that conventional edge elements in solving vector Maxwell's
eigenvalue equations by the finite element method will lead to the presence of spurious
zero eigenvalues. This problem has been addressed for the first order edge element
by Kikuchi by the mixed element method. Inspired by this approach, this paper
describes a higher order mixed spectral element method (mixed SEM) for the computation
of two-dimensional vector eigenvalue problem of Maxwell's equations. It
utilizes Gauss-Lobatto-Legendre (GLL) polynomials as the basis functions in the finite-element
framework with a weak divergence condition. It is shown that this method
can suppress all spurious zero and nonzero modes and has spectral accuracy. A rigorous
analysis of the convergence of the mixed SEM is presented, based on the higher
order edge element interpolation error estimates, which fully confirms the robustness
of our method. Numerical results are given for homogeneous, inhomogeneous, L-shape,
coaxial and dual-inner-conductor cavities to verify the merits of the proposed
method. 相似文献