A stabilized multi-level method for non-singular finite volume solutions of the stationary 3D Navier–Stokes equations

This paper proposes and analyzes a stabilized multi-level finite volume method (FVM) for solving the stationary 3D Navier?CStokes equations by using the lowest equal-order finite element pair without relying on any solution uniqueness condition. This multi-level stabilized FVM consists of solving the nonlinear problem on the coarsest mesh and then performing one Newton correction step on each subsequent mesh, thus only solving a large linear system. An optimal convergence rate for the finite volume approximations of nonsingular solutions is first obtained with the same order as that for the usual finite element solution by using a relationship between the stabilized FVM and a stabilized finite element method. Then the multi-level finite volume approximate solution is shown to have a convergence rate of the same order as that of the stabilized finite volume solution of the stationary Navier?CStokes equations on a fine mesh with an appropriate choice of the mesh size: ${ h_{j} ~ h_{j-1}^{2}, j = 1,\ldots, J}$ . Finally, numerical results presented validate our theoretical findings.     

Use of geometry in finite element thermal radiation combined with ray tracing

In heat transfer for space applications, the exchanges of energy by radiation play a significant role. In this paper, we present a method which combines the geometrical definition of the model with a finite element mesh. The geometrical representation is advantageous for the radiative component of the thermal problem while the finite element mesh is more adapted to the conductive part. Our method naturally combines these two representations of the model. The geometrical primitives are decomposed into cells. The finite element mesh is then projected onto these cells. This results in a ray tracing acceleration technique. Moreover, the ray tracing can be performed on the exact geometry, which is necessary if specular reflectors are present in the model. We explain how the geometrical method can be used with a finite element formulation in order to solve thermal situation including conduction and radiation. We illustrate the method with the model of a satellite.     

A Lyapunov function for a two-chemical species version of the chemotaxis model

Since the accuracy of finite element solutions of partial differential equations is generally mesh dependent, especially when solutions have singularities and discontinuities, a proper mesh generation is often important and sometimes crucial for an accurate numerical approximation of such problems. In this paper, the mesh transformation method is applied to the boundary value problems of elliptic partial differential equations, and it is proved that the method leads to the optimal finite element solutions. AMS subject classification (2000) 73C50, 65K10, 65N12, 65N30     

Numerical relaxation of nonconvex functionals in elastoplasticity

We consider an exemplary problem of finite elastoplasticity which is formulated on the basis of an incremental variational principle. For a specific choice of material parameters the potential becomes non(quasi‐)convex. This gives rise to the occurrence of microstructures and the convergence of standard finite element approximations is not guaranteed, because the results become highly mesh‐dependent. This phenomena can be avoided by means of a relaxed potential calculated by partial rank‐one convexification.     

Computing non-smooth minimizers with the mesh transformation method

^** Corresponding author. Email: lizp{at}math.pku.edu.cn A regularized mesh transformation method is applied to solvea variational problem allowing a non-smooth minimizer. Sincethe mesh lines can be made to match the discontinuity set ofthe minimizer, the method efficiently improves the approximatingproperty of the numerical solution. Error bounds dominated bythe error of the energy approximation have been derived, whichverify that the numerical solution obtained by the mesh transformationmethod is the optimal finite element solution in the sense thatthe corresponding error norm is minimized among all admissiblemesh distributions. Numerical experiments are given to showthe efficiency of the method.     

A constrained optimization approach to finite element mesh smoothing

The quality of a finite element solution has been shown to be affected by the quality of the underlying mesh. A poor mesh may lead to unstable and/or inaccurate finite element approximations. Mesh quality is often characterized by the “smoothness” or “shape” of the elements (triangles in 2-D or tetrahedra in 3-D). Most automatic mesh generators produce an initial mesh where the aspect ratio of the elements are unacceptably high. In this paper, a new approach to produce acceptable quality meshes from a topologically valid initial mesh is presented. Given an initial mesh (nodal coordinates and element connectivity), a “smooth” final mesh is obtained by solving a constrained optimization problem. The variables for the iterative optimization procedure are the nodal coordinates (excluding, the boundary nodes) of the finite element mesh, and appropriate bounds are imposed on these to prevent an unacceptable finite element mesh. Examples are given of the application of the above method for 2- and 3-D meshes generated using automatic mesh generators. Results indicate that the new method not only yields better quality elements when compared with the traditional Laplacian smoothing, but also guarantees a valid mesh unlike the Laplacian method.     

Numerical computation of stress induced microstructure

The mesh transformation method is applied on a two-dimensional elastic crystal model to study the formation of laminated microstructure in austenite-martensite phase transition when certain external loads are applied. Numerical experiments show that simple laminated microstructures with various volume fractions and twin width can be obtained by varying the loads. Numerical experiments also show that second order laminated microstructure with branched needle-like laminates can also be obtained by certain loads.     

Numerical computation of stress induced microstructure

The mesh transformation method is applied on a two-dimensional elastic crystal model to study the formation of laminated microstructure in austenite-martensite phase transition when certain external loads are applied. Numerical experiments show that simple laminated microstructures with various volume fractions and twin width can be obtained by varying the loads. Numerical experiments also show that second order laminated microstructure with branched needle-like laminates can also be obtained by certain loads.     

Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems

In this paper we propose a method for improving the convergence rate of the mixed finite element approximations for the Stokes eigenvalue problem. It is based on a postprocessing strategy that consists of solving an additional Stokes source problem on an augmented mixed finite element space which can be constructed either by refining the mesh or by using the same mesh but increasing the order of the mixed finite element space. Dedicated to Ivan Hlaváček on the occasion of his 75th birthday     

不可压缩核废料污染问题的变网格特征有限元法

1 引言 多孔介质中的核废料污染问题是环境保护领域的重要课题。对于不可压缩二维模型,它是地层中迁移型耦合抛物型方程组的初边值问题:     

解Stokes特征值问题的一种两水平稳定化有限元方法

基于局部Gauss积分,研究了解Stokes特征值问题的一种两水平稳定化有限元方法．该方法涉及在网格步长为H的粗网格上解一个Stokes特征值问题,在网格步长为h=O(H2)的细网格上解一个Stokes问题．这样使其能够仍旧保持最优的逼近精度,求得的解和一般的稳定化有限元解具有相同的收敛阶,即直接在网格步长为h的细网格上解一个Stokes特征值问题．因此,该方法能够节省大量的计算时间．数值试验验证了理论结果．     

Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes

We consider an algorithm called FEMWARP for warping triangular and tetrahedral finite element meshes that computes the warping using the finite element method itself. The algorithm takes as input a two- or three-dimensional domain defined by a boundary mesh (segments in one dimension or triangles in two dimensions) that has a volume mesh (triangles in two dimensions or tetrahedra in three dimensions) in its interior. It also takes as input a prescribed movement of the boundary mesh. It computes as output updated positions of the vertices of the volume mesh. The first step of the algorithm is to determine from the initial mesh a set of local weights for each interior vertex that describes each interior vertex in terms of the positions of its neighbors. These weights are computed using a finite element stiffness matrix. After a boundary transformation is applied, a linear system of equations based upon the weights is solved to determine the final positions of the interior vertices. The FEMWARP algorithm has been considered in the previous literature (e.g., in a 2001 paper by Baker). FEMWARP has been successful in computing deformed meshes for certain applications. However, sometimes FEMWARP reverses elements; this is our main concern in this paper. We analyze the causes for this undesirable behavior and propose several techniques to make the method more robust against reversals. The most successful of the proposed methods includes combining FEMWARP with an optimization-based untangler.     

A two-grid finite element method for the Allen-Cahn equation with the logarithmic potential

In this paper, we present a two-grid finite element method for the Allen-Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen-Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two-grid finite element method. The optimal convergence order of the two-grid finite element method in H¹ norm is achieved when the mesh sizes satisfy h = O(H²). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two-grid method can save the CPU time while keeping the same convergence rate.     

晶体微观结构的数值计算

晶体微观结构是晶体材料在特定物理条件下其多个能量极小平衔态在空间形成的某种微尺度的规则分布．几何非线性的连续介质力学理论可以用能量极小化原理来解释晶体微观结构的形成，并用Young测度来刻画平衡态各变体在空间的概率分布．定性的理解与定量地分析和计算晶体材料的微观结构对于发展和改进高级晶体功能材料，如形状记忆合金、铁电体、磁至伸缩材料等，有重要的意义．本文回顾了近年来晶体微观结构数值计算方面的最新进展．介绍了计算晶体微观结构的几种数值方法及有关的数值分析结果。     

Application of a Krylov subspace iterative method in a multi-level adaptive technique to solve the mild-slope equation in nearshore regions

A multi-level adaptive numerical technique is applied to a nonlinear formulation of the mild-slope equation, to obtain the nearshore wave field, where the dominant processes of wave transformation are shoaling, refraction and diffraction. The advantage of this formulation over the traditional elliptic, parabolic and hyperbolic formulations is to require a lower minimum number of grid nodes per wavelength, thus, its capacity to predict the wave field for larger coastal areas. The efficiency of the interactions between the grid mesh levels, where two robust Krylov subspace iterative methods, the Bi-CGSTAB and the GMRES, are applied to solve the governing equation, is tested, for several hierarchies of grid mesh levels. The results show that the multi-level adaptive technique is efficient only if the GMRES iterative method is applied, and that for six grid mesh levels good results can be achieved for a residual as low as 10⁻³ for the finest grid.     

A robust hierarchical basis preconditioner on general meshes

Summary. In this paper, we introduce a multi-level direct sum space decomposition of general, possibly locally refined linear or multi-linear finite element spaces. The resulting additive Schwarz preconditioner is optimal for symmetric second order elliptic problems. Moreover, it turns out to be robust with respect to coefficient jumps over edges in the coarsest mesh, perturbations with positive zeroth order terms, and, after a further decomposition of the spaces, also with respect to anisotropy along the grid lines. Important for an efficient implementation is that stable bases of the subspaces defining our decomposition, consisting of functions having small supports can be easily constructed. Received September 8, 1995 / Revised version received October 31, 1996     

Configurational–Force–Based Adaptive Methods in Nonlocal Gradient-Type Inelastic Solids

We propose a configurational-force-based framework for h-adaptive finite element discretizations of solids with nonlocal, gradient-type constitutive response. Typical applications are related to gradient-type damage mechanics, strain gradient plasticity and regularized brittle fracture. On the theoretical side, we outline a general incremental variational framework for the multifield problem of gradient-type dissipative solids, where generalized internal variable fields account for the current state of evolving microstructures. The Euler equations of the multifield variational principle define the macroscopic balance of momentum along with balance-type evolution equations for the generalized internal variables in the physical space as well as the balance of configurational forces in the material space. We propose a staggered computational scheme for satisfying those balances in both the physical as well as the material space. The coupled micro- and macro-structural balances of momentum and internal variables provide a solution in the physical space for a given finite element mesh. The balance in the material space is then used to provide an indicator for the quality of the finite element mesh and accounts for a subsequent h-type mesh refinement. Such a configurational-force-based approach provides in a natural and unified format mesh refinement indicators for a broad class of complex nonlocal problems. This framework is applied to damage-type regularized brittle fracture. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A dynamic data structure suitable for adaptive mesh refinement in finite element method

This paper describes a dynamic data structure and its implementation, used for an optimum mesh generator. The implementation of this mesh generator was a part of a software package implemented to solve electromagnetic field problems using the finite element method. This mesh generator takes advantage of the Delaunay algorithm, which maximizes the summation of the smallest angles in all triangles and thus creates a mesh that is proved to be an optimum mesh for use in the finite element method. The dynamic data structure is explained and the source code is reviewed. The programs have been written in Pascal programming language.     

Postprocessing and higher order convergence for the mixed finite element approximations of the eigenvalue problem

In this paper, we propose a method to improve the convergence rate of the lowest order Raviart-Thomas mixed finite element approximations for the second order elliptic eigenvalue problem. Here, we prove a supercloseness result for the eigenfunction approximations and use a type of finite element postprocessing operator to construct an auxiliary source problem. Then solving the auxiliary additional source problem on an augmented mixed finite element space constructed by refining the mesh or by using the same mesh but increasing the order of corresponding mixed finite element space, we can increase the convergence order of the eigenpair approximation. This postprocessing method costs less computation than solving the eigenvalue problem on the finer mesh directly. Some numerical results are used to confirm the theoretical analysis.     

The composite mini element: a new mixed FEM for the Stokes equations on complicated domains

We introduce a new finite element method, the composite mini element, for the mixed discretization of the Stokes equations on two and three-dimensional domains that may contain a huge number of geometric details. Instead of a geometric resolution of the domain and the boundary condition by the finite element mesh the shape of the finite element functions is adapted to the geometric details. This approach allows low-dimensional approximations even for problems with complicated geometric details such as holes or rough boundaries. It turns out that the method can be viewed as a coarse scale generalization of the classical mini element approach, i.e. it reduces the computational effort while the approximation quality depends linearly on the (coarse) mesh size in the usual way. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)