An immersed interface method for the Navier-Stokes equations on irregular domains

An augmented method based on a Cartesian grid is proposed for the incompressible Navier Stokes equations on an irregular domain. The irregular domain is embedded into a rectangular one so that a fast Poisson solver can be used in the projection method. Unlike several methods suggested in the literature that set the force strengths as unknowns, which often results an ill-conditioned linear system, we set the jump in the normal derivative of the velocity as the augmented variable. The new approach improve the condition number of the system for the augmented variable significantly. Using the immersed interface method, we achieve second order accuracy for the velocity. Numerical results and comparisons are given to validate the new method. Some interesting new numerical experiments results are also presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new augmented immersed finite element method without using SVD interpolations

Augmented immersed interface methods have been developed recently for interface problems and problems on irregular domains including CFD applications with free boundaries and moving interfaces. In an augmented method, one or several augmented variables are introduced along the interface or boundary so that one can get efficient discretizations. The augmented variables should be chosen such that the interface or boundary conditions are satisfied. The key to the success of the augmented methods often relies on the interpolation scheme to couple the augmented variables with the governing differential equations through the interface or boundary conditions. This has been done using a least squares interpolation (under-determined) for which the singular value decomposition (SVD) is used to solve for the interpolation coefficients. In this paper, based on properties of the finite element method, a new augmented immersed finite element method (IFEM) that does not need the interpolations is proposed for elliptic interface problems that have a piecewise constant coefficient. Thus the new augmented method is more efficient and simple than the old one that uses interpolations. The method then is extended to Poisson equations on irregular domains with a Dirichlet boundary condition. Numerical experiments with arbitrary interfaces/irregular domains and large jump ratios are provided to demonstrate the accuracy and the efficiency of the new augmented methods. Numerical results also show that the number of GMRES iterations is independent of the mesh size and nearly independent of the jump in the coefficient.     

A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow

Biharmonic equations have many applications, especially in fluid and solid mechanics, but is difficult to solve due to the fourth order derivatives in the differential equation. In this paper a fast second order accurate algorithm based on a finite difference discretization and a Cartesian grid is developed for two dimensional biharmonic equations on irregular domains with essential boundary conditions. The irregular domain is embedded into a rectangular region and the biharmonic equation is decoupled to two Poisson equations. An auxiliary unknown quantity Δu along the boundary is introduced so that fast Poisson solvers on irregular domains can be used. Non-trivial numerical examples show the efficiency of the proposed method. The number of iterations of the method is independent of the mesh size. Another key to the method is a new interpolation scheme to evaluate the residual of the Schur complement system. The new biharmonic solver has been applied to solve the incompressible Stokes flow on an irregular domain.      

Fast solvers for 3D Poisson equations involving interfaces in a finite or the infinite domain

In this paper, numerical methods are proposed for Poisson equations defined in a finite or infinite domain in three dimensions. In the domain, there can exists an interface across which the source term, the flux, and therefore the solution may be discontinuous. The existence and uniqueness of the solution are also discussed. To deal with the discontinuity in the source term and in the flux, the original problem is transformed to a new one with a smooth solution. Such a transformation can be carried out easily through an extension of the jumps along the normal direction if the interface is expressed as the zero level set of a three-dimensional function. An auxiliary sphere is used to separate the infinite region into an interior and exterior domain. The Kelvin's inversion is used to map the exterior domain into an interior domain. The two Poisson equations defined in the interior and the exterior written in spherical coordinates are solved simultaneously. By choosing the mesh size carefully and exploiting the fast Fourier transform, the resulting finite difference equations can be solved efficiently. The approach in dealing with the interface has also been used with the artificial boundary condition technique which truncates the infinite domain. Numerical results demonstrate second order accuracy of our algorithms.     

Boundary collocation fast poisson solver on irregular domains

A fast Poisson solver on irregular domains, based on boundary methods, is presented. The harmonic polynomial approximation of the solution of the associated homogeneous problem provides a good practical boundary method which allows a trivial parallel processing for solution evaluation or straightforward computations of the interface values for domain decomposition/embedding.     

基于张量乘积的快速谱元算法

针对椭圆型方程的谱元离散系统构造了一种基于张量乘积的快速直接解法．分析显示，新算法的计算量仅相当于迭代方法迭代Kx＋Ky次的计算量（这里Kx，Ky分别为x，y方向的区域剖分数），特别适合那些网格不多但多项式阶数较高的谱元离散．我们还将张量乘积方法推广到具有Neumann边界条件的奇异泊松问题的求解。给出了具体的实现方法．最后，利用张量乘积构造了变形区域上椭圆型方程的预条件子，数值结果显示预条件系统的条件数与多项式阶数无关．     

A non-parameterized entropy correction for Roe's approximate Riemann solver

Summary. The main drawback with Roe's approximate Riemann solver is that non-physical expansion shocks can occur in the vicinity of sonic points. Previous work aimed at enforcing the entropy condition is based on the representation of sonic rarefaction waves. We propose a new non-parameterized approach which is based on a nonlinear Hermite interpolation of an approximate flux function and the exact resolution of non convex scalar Riemann problems. Convergence and consistency with the entropy condition are proved for scalar convex conservation laws with arbitrarily large initial data. When considering strictly hyperbolic systems of conservation laws, consistency of the resulting scheme with the entropy condition is also proved for initial data sufficiently close to a constant. Numerical results on a one-dimensional shock-tube and a two-dimensional supersonic forward facing step confirm our theoretical results. Received March 1, 1993 / Revised version received August 26, 1994     

A Cartesian‐grid collocation method based on radial‐basis‐function networks for solving PDEs in irregular domains

This paper reports a new Cartesian‐grid collocation method based on radial‐basis‐function networks (RBFNs) for numerically solving elliptic partial differential equations in irregular domains. The domain of interest is embedded in a Cartesian grid, and the governing equation is discretized by using a collocation approach. The new features here are (a) one‐dimensional integrated RBFNs are employed to represent the variable along each line of the grid, resulting in a significant improvement of computational efficiency, (b) the present method does not require complicated interpolation techniques for the treatment of Dirichlet boundary conditions in order to achieve a high level of accuracy, and (c) normal derivative boundary conditions are imposed by means of integration constants. The method is verified through the solution of second‐ and fourth‐order PDEs; accurate results and fast convergence rates are obtained. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

A uniformly well-conditioned, unfitted Nitsche method for interface problems

A finite element method for elliptic partial differential equations that allows for discontinuities along an interface not aligned with the mesh is presented. The solution on each side of the interface is separately expanded in standard continuous, piecewise-linear functions, and jump conditions at the interface are weakly enforced using a variant of Nitsche’s method. In our method, the solutions on each side of the interface are extended to the entire domain which results in a fixed number of unknowns independent of the location of the interface. A stabilization procedure is included to ensure well-defined extensions. We prove that the method provides optimal convergence order in the energy and the L ² norms and a condition number of the system matrix that is independent of the position of the interface relative to the mesh. Numerical experiments confirm the theoretical results and demonstrate optimal convergence order also for the pointwise errors.     

Debonding of an elastic inhomogeneity of arbitrary shape in anti-plane shear

We investigate the anti-plane shear problem of a curvilinear crack lying along the interface of an arbitrarily shaped elastic inhomogeneity embedded in an infinite matrix subjected to uniform stresses at infinity. Complex variable and conformal mapping techniques are used to derive an analytical solution in series form. The problem is first reduced to a non-homogeneous Riemann–Hilbert problem, the solution of which can be obtained by evaluating the associated Cauchy integral. A set of linear algebraic equations is obtained from the compatibility condition imposed on the resulting analytic function defined in the inhomogeneity and its Faber series expansion. Each of the unknown coefficients in the corresponding analytic functions can then be uniquely determined by solving the linear algebraic equations, which are written concisely in matrix form. The resulting analytical solution is then used to quantify the displacement jump across the debonded section of the interface as well as the traction distribution along the bonded section of the interface. In addition, our solution allows us to obtain mode-III stress intensity factors at the two crack tips. The solution to the anti-plane problem of a partially debonded elliptical inhomogeneity containing a confocal crack is also derived using a similar method.     

Coupling of compressible and incompressible flow regions using the multiple pressure variables approach

In many cases, multiphase flows are simulated on the basis of the incompressible Navier–Stokes equations. This assumption is valid as long as the density changes in the gas phase can be neglected. Yet, for certain technical applications such as fuel injection, this is no longer the case, and at least the gaseous phase has to be treated as a compressible fluid. In this paper, we consider the coupling of a compressible flow region to an incompressible one based on a splitting of the pressure into a thermodynamic and a hydrodynamic part. The compressible Euler equations are then connected to the Mach number zero limit equations in the other region. These limit equations can be solved analytically in one space dimension that allows to couple them to the solution of a half‐Riemann problem on the compressible side with the help of velocity and pressure jump conditions across the interface. At the interface location, the flux terms for the compressible flow solver are provided by the coupling algorithms. The coupling is demonstrated in a one‐dimensional framework by use of a discontinuous Galerkin scheme for compressible two‐phase flow with a sharp interface tracking via a ghost‐fluid type method. The coupling schemes are applied to two generic test cases. The computational results are compared with those obtained with the fully compressible two‐phase flow solver, where the Mach number zero limit is approached by a weakly compressible fluid. For all cases, we obtain a very good agreement between the coupling approaches and the fully compressible solver. Copyright © 2014 John Wiley & Sons, Ltd.     

带分数布朗Brown的随机比例方程半隐式Euler法的数值解(英文)

本文给出并分析了Poisson随机跳测度驱动的带分数Brown运动的随机比例方程半隐式Euler法的数值解,在局部Lipschitz条件下,证明了在均方意义下半隐式Euler数值解收敛到精确解.     

Convergence estimates for an optimized Schwarz method for PDEs with discontinuous coefficients

Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. When the subdomains are overlapping or nonoverlapping, these methods employ the optimal value of parameter(s) in the boundary condition along the artificial interface to accelerate its convergence. In the literature, the analysis of optimized Schwarz methods rely mostly on Fourier analysis and so the domains are restricted to be regular (rectangular). As in earlier papers, the interface operator can be expressed in terms of Poincaré–Steklov operators. This enables the derivation of an upper bound for the spectral radius of the interface operator on essentially arbitrary geometry. The problem of interest here is a PDE with a discontinuous coefficient across the artificial interface. We derive convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large. We consider two different types of Robin transmission conditions in the Schwarz iteration: the first one leads to the best estimate when h is small, whereas for the second type, we derive a convergence estimate inversely proportional to the jump in the coefficient. This latter result improves upon the rate of popular domain decomposition methods such as the Neumann–Neumann method or FETI-DP methods, which was shown to be independent of the jump. In memory of Gene Golub.     

A fast Poisson solver on disks

We present a fast/parallel Poisson solver on disks, based on efficient evaluation of the exact solution given by the Newtonian potential and the Poisson integral. Derived from an integral formulation, it is more accurate and simpler in parallel implementation and in upgrading to a higher order algorithm, than an algorithm which solves the linear system obtained from a differential formulation.     

Fast iterative solver for convection‐diffusion systems with spectral elements

We introduce a solver and preconditioning technique based on Domain Decomposition and the Fast Diagonalization Method that can be applied to tensor product based discretizations of the steady convection–diffusion equation. The method is based on a Robin–Robin interface preconditioner coupled to a fast diagonalization solver which is used to efficiently eliminate the interior degrees of freedom and perform subsidiary subdomain solves. Using a spectral element discretization, we first apply our technique to constant wind problems, and then propose a means for applying the technique as a preconditioner for variable wind problems. We demonstrate that iteration counts are mildly dependent on changes in mesh size and convection strength. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

A General Cavitation Model for the Highly Nonlinear Mie-Grüneisen Equation of State

A general one-fluid cavitation model is proposed for a family of Mie-Grüneisen equations of state (EOS), which can provide a wide application of cavitation flows, such as liquid-vapour transformation and underwater explosion. An approximate Riemann problem and its approximate solver for the general cavitation model are developed. The approximate solver, which provides the interface pressure and normal velocity by an iterative method, is applied in computing the numerical flux at the phase interface for our compressible multi-medium flow simulation on Eulerian grids. Several numerical examples, including Riemann problems and underwater explosion applications, are presented to validate the cavitation model and the corresponding approximate solver.     

基于Crouzeix-Raviart元的界面浸入有限元方法及其收敛性分析

本文对具间断系数的二阶椭圆界面问题提出一种浸入有限元方法(theimmersed finite element method), 即在界面单元上采用依赖于界面的线性多项式空间离散, 而在非界面单元上采用Crouzeix-Raviart非协调元离散. 论证表明, 该方法具有对界面问题解的最优L²-模和H¹-模收敛精度.     

轴对称Poisson方程的Trefftz有限元解法

将径向基函数应用到一类轴对称Poisson方程的数值求解中,提出了一种Trefftz有限元计算格式.非0右端项将问题的特解引入Trefftz单元域内场,致使单元刚度方程涉及区域积分.利用径向基函数对特解近似处理,可消除区域积分,从而保持Trefftz有限元法只含边界积分的优势.为获得特解,选取求解域内所有单元的节点和形心作为基本插值点,而在求解域之外构造一个虚拟边界,在其上布置一定数目的虚拟点作为额外插值点.数值算例验证了该方法的有效性和可行性.     

The Fourier-finite element method for the Poisson problem on a non-convex polyhedral cylinder

We study the Poisson problem with zero boundary datum in a (finite) polyhedral cylinder with a non-convex edge. Applying the Fourier sine series to the equation along the edge and by a corner singularity expansion for the Poisson problem with parameter, we define the edge flux coefficient and the regular part of the solution on the polyhedral cylinder. We present a numerical method for approximating the edge flux coefficient and the regular part and show the stability. We derive an error estimate and give some numerical experiments.     

A flux-limiter method for dam-break flows over erodible sediment beds

Finite volume methods for dam-break flows over erodible sediment beds require a monotone numerical flux. In the present study we present a new flux-limiter scheme based on the Lax–Wendroff method coupled with a non-homogeneous Riemann solver and a flux limiter function. The non-homogeneous Riemann solver consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The proposed method satisfy the conservation property such that the discretization of the flux gradients and the source terms are well-balanced in the numerical solution of suspended sediment models. The flux-limiter method provides accurate results avoiding numerical oscillations and numerical dissipation in the approximated solutions. Several standard test examples are considered to verify the performance and the accuracy of the proposed method.