Numerical simulation of tridimensional electromagnetic shaping of liquid metals

Summary We describe a numerical method to compute free surfaces in electromagnetic shaping and levitation of liquid metals. We use an energetic variational formulation and optimization techniques to compute, a critical point. The surfaces are represented by piecewise linear finite elements. Each step of the algorithm requires solving an elliptic boundary value problem in the exterior of the intermediate surfaces. This is done by using an integral representation on these surfaces.     

MILU-CG方法和绕旋转圆柱流动的数值研究

本文采用以修正的不完全LU分解作预处理器的共轭梯度法(MILU-CG),结合高阶隐式差分格式,改进了作者(1992)提出的基于区域分解、有限差分法与涡法杂交的数值方法(HDV).系统地研究了雷诺数Re=1000,200,旋转速度比α∈(0.5,3.25)范围内,绕旋转圆柱从突然起动到充分发展,长时间内尾流旋涡结构和阻力、升力系数的变化规律.计算所得流线与实验流场显示相比,完全吻合.首次揭示了临界状态时的旋涡结构特性,并指出最佳升阻比就在该状态附近得到.     

A cascadic multigrid method for a kind of semilinear elliptic problem

In this paper, we analyze a cascadic multigrid method for semilinear elliptic problems in which the derivative of the semilinear term is Hölder continuous. We first investigate the standard finite element error estimates of this kind of problem. We then solve the corresponding discrete problems using the cascadic multigrid method. We prove that the algorithm has an optimal order of convergence in energy norm and quasi-optimal computational complexity. We also report some numerical results to support the theory.     

Convergence of an adaptive Ka?anov FEM for quasi-linear problems

We design an adaptive finite element method to approximate the solutions of quasi-linear elliptic problems. The algorithm is based on a Ka?anov iteration and a mesh adaptation step is performed after each linear solve. The method is thus inexact because we do not solve the discrete nonlinear problems exactly, but rather perform one iteration of a fixed point method (Ka?anov), using the approximation of the previous mesh as an initial guess. The convergence of the method is proved for any reasonable marking strategy and starting from any initial mesh. We conclude with some numerical experiments that illustrate the theory.     

Constructing Order Two Superconvergent WG Finite Elements on Rectangular Meshes

In this paper, we introduce a stabilizer free weak Galerkin (SFWG) finite element method for second order elliptic problems on rectangular meshes. With a special weak Gradient space, an order two superconvergence for the SFWG finite element solution is obtained, in both $L^2$ and $H^1$ norms. A local post-process lifts such a $P_k$ weak Galerkin solution to an optimal order $P_{k+2}$ solution. The numerical results confirm the theory.     

有限元分层快速高精度算法

1.引言 多重网格法和区域分解法实质是有限元空间的分解,在子空间上实行逐次校正迭代或并行校正迭代。[1]对一维有限元空间,利用正交化过程,消去单元内结点,修改单元角点的基函数,提出了所谓快速高精度算法。实例表明,这一算法十分有效。本文对一般区域Ω R~d(d=1,2,3)上有限元空间进行分层正交分解,提出所谓分层快速高精度算法。     

外推瀑布多网格法(EXCMG)——-大规模求解椭圆问题的新算法

基于有限元的渐近展开式,导出了新的外推公式,它们更精确地逼近密网上的有限元解(而不是微分方程的解).提出了新的外推瀑布型多网格法(EXCMG),采用新外推公式及其二次插值提供密网上的好初值.数值实验表明,新方法有很高的精度和效率.最后在PC机上求解了大规模二维椭圆问题.     

Variable-Elliptic-Vortex Method for Incompressible Flow Simulation

A variable-elliptic-vortex method, which is a generalization of the elliptic-vortex method proposed by the author in [1], is presented for the numerical simulation of incompressible flows. The most attractive feature of the new method is that the numerical vortex blobs used in this model like actual vortex blobs can be translated, rotated and deformed in elliptic shape. The new method provides a more reasonable and more accurate approach for flow simulation than the fixed-vortex methods. Numerical examples are presented to demonstrate the performance of the new method.     

Stressed state of elastic plane with cavities

An algorithm for the numerical solution of the problem of defining the stressed state of a loaded plane with cavities of arbitrary configuration is given. A system of singular integral equations is given, which is obtained by using displacement potentials, is quantizied by the method of discrete singularities and finite differences. Existence and uniqueness conditions for solutions are formulated. The algorithm is debugged on a test example and is applied to a study of interrelation of elliptic cavities when the distance between them decreases.Translated from Dinamicheskie Sistemy, No. 7, pp. 8–13, 1988.     

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations

We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell's equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their "immediate" neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.     

On the mixed finite element method with Lagrange multipliers

In this note we analyze a modified mixed finite element method for second‐order elliptic equations in divergence form. As a model we consider the Poisson problem with mixed boundary conditions in a polygonal domain of R ². The Neumann (essential) condition is imposed here in a weak sense, which yields the introduction of a Lagrange multiplier given by the trace of the solution on the corresponding boundary. This approach allows to handle nonhomogeneous Neumann boundary conditions, theoretically and computationally, in an alternative and usually easier way. Then we utilize the classical Babu?ka‐Brezzi theory to show that the resulting mixed variational formulation is well posed. In addition, we use Raviart‐Thomas spaces to define the associated finite element method and, applying some elliptic regularity results, we prove the stability, unique solvability, and convergence of this discrete scheme, under appropriate assumptions on the mesh sizes. Finally, we provide numerical results illustrating the performance of the algorithm for smooth and singular problems. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 192–210, 2003     

Passerelles volumes finis — éléments finis

The aim of this work is to introduce a new algorithm for the dicretization of second order elliptic operators in the context of finite volume schemes. The technique consists in matching to a finite volume discretization based on a given mesh, a finite element volume representation on the same given mesh. An inverse operator is also built. The results of numerical experiments concerning a system of two-dimensional, nonlinear partial differential equations on a unstructured mesh are presented.     

Finite element approximation of a problem with a nonlinear Newton boundary condition

The paper is concerned with the study of an elliptic boundary value problem with a nonlinear Newton boundary condition. The existence and uniqueness of the solution of the continuous problem is established with the aid of the monotone operator theory. The main attention is paid to the investigation of the finite element approximation using numerical integration for the computation of nonlinear boundary integrals. The solvability of the discrete finite element problem is proved and the convergence of the approximate solutions to the exact one is analysed. Received April 15, 1996 / Revised version received November 22, 1996     

Front tracking for two-phase flow in reservoir simulation by adaptive mesh

In this article, an algorithm for the numerical approximation of two-phase flow in porous media by adaptive mesh is presented. A convergent and conservative finite volume scheme for an elliptic equation is proposed, together with the finite difference schemes, upwind and MUSCL, for a hyperbolic equation on grids with local refinement. Hence, an IMPES method is applied in an adaptive composite grid to track the front of a moving solution. An object-oriented programmation technique is used. The computational results for different examples illustrate the efficiency of the proposed algorithm. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 673–697, 1997     

Adaptive Numerical Treatment of Elliptic Systems on Manifolds

Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2- and 3-manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2- and 3-manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unity-based method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4-component covariant elliptic system on a Riemannian 3-manifold which arises in general relativity. A number of operator properties and solvability results recently established in [55] are first summarized, making possible two quasi-optimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented; more detailed examples using MC for this application may be found in [26].     

Generation of large-scale vortex dislocations in a three-dimensional wake-type flow

Numerical study of three-dimensional evolution of wake-type flow and vortex dislocations is performed by using a compact finite diffenence-Fourier spectral method to solve 3-D incompressible Navier-Stokes equations. A local spanwise nonuniformity in momentum defect is imposed on the incoming wake-type flow. The present numerical results have shown that the flow instability leads to three-dimensional vortex streets, whose frequency, phase as well as the strength vary with the span caused by the local nonuniformity. The vortex dislocations are generated in the nonuniform region and the large-scale chain-like vortex linkage structures in the dislocations are shown. The generation and the characteristics of the vortex dislocations are described in detail.     

Local and parallel finite element algorithms based on two-grid discretizations

A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. The theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for finite element solutions on general shape-regular grids. Some numerical experiments are also presented to support the theory.

     

Adaptive finite element methods for elliptic equations over hierarchical T-meshes

Isogeometric analysis using NURBS (Non-Uniform Rational B-Splines) as basis functions gives accurate representation of the geometry and the solution but it is not well suited for local refinement. In this paper, we use the polynomial splines over hierarchical T-meshes (PHT-splines) to construct basis functions which not only share the nice smoothness properties as the B-splines, but also allow us to effectively refine meshes locally. We develop a residual-based a posteriori error estimate for the finite element discretization of elliptic equations using PHT-splines basis functions and study their approximation properties. In addition, we conduct numerical experiments to verify the theory and to demonstrate the effectiveness of the error estimate and the high order approximations provided by the numerical solution.     

Finite Element Approximation of Hemivariational Inequalities

The paper deals with a finite element approximation of elliptic and parabolic variational inequalities. Elliptic hemivariational inequalities are equivalently expressed as a system consisting of one equation and one inclusion for a couple of unknowns, namely a primal variable u and an element belonging to a multivalued mapping at u. Both components of the solution are approximated independently each other by a finite element method. Parabolic inequalities are transformed into a system of elliptic ones by using an appropriate time discretization. A numerical experiment is realized by using non-smooth optimization methods.