Adaptive pseudospectral solution of a diffuse interface model

A diffuse interface type model, using an energy-based variational formulation with a free energy that is a function of the density and its gradients is presented. All of the boundary terms are retained and related to external surface forces, which can be of particular interest when considering the fluid–fluid–solid region. The numerical solution of these types of problems can be troublesome if a thin transition layer is desired. Here, Chebyshev pseudospectral methods with mesh adaptation for the solution of diffuse interface type problems are studied. A mesh adaptation algorithm based in the equidistribution principle following a continuation process is derived. In order to achieve high precision for problems exhibiting thin transition layers, a modified version of the arc-length monitor function is proposed which yields a sufficiently smooth coordinate transformation. At every step of the continuation process, a fixed number of iterations is implemented, so that the equidistribution equations are not solved completely at each step, which saves a considerable amount of computational effort. Numerical results for the static phase field model exhibiting thin transition layers are presented.     

Hybrid Surface Mesh Adaptation for Climate Modeling

Solution-driven mesh adaptation is becoming quite popular for spatial error control in the numerical simulation of complex computational physics applications, such as climate modeling. Typically, spatial adaptation is achieved by element subdivision （h adaptation） with a primary goal of resolving the local length scales of interest. A sec- ond, less-popular method of spatial adaptivity is called ＂mesh motion＂ （r adaptation）; the smooth repositioning of mesh node points aimed at resizing existing elements to capture the local length scales. This paper proposes an adaptation method based on a combination of both element subdivision and node point repositioning （rh adaptation）. By combining these two methods using the notion of a mobility function, the proposed approach seeks to increase the flexibility and extensibility of mesh motion algorithms while providing a somewhat smoother transition between refined regions than is pro- duced by element subdivision alone. Further, in an attempt to support the requirements of a very general class of climate simulation applications, the proposed method is designed to accommodate unstructured, polygonal mesh topologies in addition to the most popular mesh types.     

Optimal adaptive grids of least-squares finite element methods in two spatial dimensions

This article concerns a procedure to generate optimal adaptive grids for convection dominated problems in two spatial dimensions based on least-squares finite element approximations. The procedure extends a one dimensional equidistribution principle which minimizes the interpolation error in some norms. The idea is to select two directions which can reflect the physics of the problems and then apply the one dimensional equidistribution principle to the chosen directions. Model problems considered are the two dimensional convection-diffusion problems where boundary and interior layers occur. Numerical results of model problems illustrating the efficiency of the proposed scheme are presented. In addition, to avoid skewed mesh in the optimal grids generated by the algorithm, an unstructured local mesh smoothing will be considered in the least-squares approximations. Comparisons with the Gakerkin finite element method will also be provided.     

A dynamic mesh adaptation method for magnetohydrodynamics problems

A dynamic adaptation method is used to numerically solve the MHD equations. The basic idea behind the method is to use an arbitrary nonstationary coordinate system for which the numerical procedure and the mesh refinement mechanism are formulated as a unified differential model. Numerical examples of multidimensional MHD flows on dynamic adaptive meshes are given to illustrate the method.     

Anisotropic mesh adaptation for numerical solution of boundary value problems

We present an efficient mesh adaptation algorithm that can be successfully applied to numerical solutions of a wide range of 2D problems of physics and engineering described by partial differential equations. We are interested in the numerical solution of a general boundary value problem discretized on triangular grids. We formulate a necessary condition for properties of the triangulation on which the discretization error is below the prescribed tolerance and control this necessary condition by the interpolation error. For a sufficiently smooth function, we recall the strategy how to construct the mesh on which the interpolation error is below the prescribed tolerance. Solving the boundary value problem we apply this strategy to the smoothed approximate solution. The novelty of the method lies in the smoothing procedure that, followed by the anisotropic mesh adaptation (AMA) algorithm, leads to the significant improvement of numerical results. We apply AMA to the numerical solution of an elliptic equation where the exact solution is known and demonstrate practical aspects of the adaptation procedure: how to control the ratio between the longest and the shortest edge of the triangulation and how to control the transition of the coarsest part of the mesh to the finest one if the two length scales of all the triangles are clearly different. An example of the use of AMA for the physically relevant numerical simulation of a geometrically challenging industrial problem (inviscid transonic flow around NACA0012 profile) is presented. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

An iterative method for the creation of structured hexahedral meshes over complex orography

In this paper we propose a technique for measuring the quality of hexahedral Cartesian meshes used to model meso-scale atmospheric circulation in 3D. It is used to verify the progress of a novel method for satisfying the Delaunay criterion for structured hexahedral meshes over complex orography with high gradients and wide gradient variability. Based on a simile with potential energy, the iterative method of mesh smoothing is shown to improve mesh quality with logarithmic convergence. The method is evaluated in a practical application in a specific geographic location.     

Model Adaptation Enriched with an Anisotropic Mesh Spacing for Nonlinear Equations: Application to Environmental and CFD Problems

Goal of this paper is to suitably combine a model with an anisotropic mesh adaptation for the numerical simulation of nonlinear advection-diffusion-reaction systems and incompressible flows in ecological and environmental applications. Using the reduced-basis method terminology, the proposed approach leads to a noticeable computational saving of the online phase with respect to the resolution of the reference model on nonadapted grids. The search of a suitable adapted model/mesh pair is to be meant, instead, in an offline fashion.     

Goal-oriented mesh adaptation for flux-limited approximations to steady hyperbolic problems

The development of adaptive numerical schemes for steady transport equations is addressed. A goal-oriented error estimator is presented and used as a refinement criterion for conforming mesh adaptation. The error in the value of a linear target functional is measured in terms of weighted residuals that depend on the solutions to the primal and dual problems. The Galerkin orthogonality error is taken into account and found to be important whenever flux or slope limiters are activated to enforce monotonicity constraints. The localization of global errors is performed using a natural decomposition of the involved weights into nodal contributions. A nodal generation function is employed in a hierarchical mesh adaptation procedure which makes each refinement step readily reversible. The developed simulation tools are applied to a linear convection problem in two space dimensions.     

A variational approach to optimal meshes

Summary. A variational approach for the optimization of triangular or tetrahedral meshes is presented. Starting from some very basic assumptions we will rigorously demonstrate that the functional controlling optimality is of a certain type related to energy functionals in non linear elasticity. It will be proved that these functionals attain their minima over admissible sets of mesh deformations which respect boundary conditions. In addition the injectivity of the deformed mesh is discussed. Thereby it is possible to construct suitable meshes for various numerical applications. Received March 14, 1994 / Revised version received August 8, 1994     

A posteriori error control for finite element approximations of elliptic eigenvalue problems

We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.     

Heuristic algorithm for generating multilevel Cartesian meshes inmultidimensional regions with moving boundaries

Universal algorithm for fully automated generation of multilevel Cartesian mesh in arbitrary planar regions with moving boundaries is developed and discussed in details. It uses multilevel tree technology and heuristic enabling flexible mesh adaptation to changes of boundaries. Efficient mixed coding for mesh storing is presented. Suggested algorithm generates meshes with properties most desirable for computing various field distribution problems. Main parts of elaborated algorithm are generalized for multidimensional meshes.     

Bakhvalov-Shishkin网格上求解边界层问题的差分进化算法

该文在Bakhvalov-Shishkin网格上求解具有左边界层或右边界层的对流扩散方程,并采用差分进化算法对Bakhvalov-Shishkin网格中的参数进行优化,获得了该网格上具有最优精度的数值解.对三个算例进行了数值模拟,数值结果表明:采用差分进化算法求解具有较高的计算精度和收敛性,特别是边界层的数值解精度明显...     

扩展型动网格的Chebyshev有限谱方法

给出了基于非均匀网格的Chebyshev有限谱方法．提出了可生成两种类型扩展型动网格的均布格式．一种类型的网格被用来提高波面附近的分辨率,另一种类型则用在梯度较大的流动区域．由于采用Chebyshev多项式作为基函数,该方法具有高阶精度．从上个时间步到当前时间步,两套不均匀网格间的物理量采用Chebyshev多项式插值．为使方法在时间离散方面保持高精度,采用了Adams-Bashforth预报格式和Adams-Moulton校正格式．为了避免由Korteweg-deVries(KdV)方程的弥散项引起的数值振荡,给出了一种非均匀网格下的数值稳定器．给出的方法与具有分析解的Burgers方程的非线性对流扩散问题和KdV方程的单孤独波和双孤独波传播问题进行了比较,结果非常吻合．     

Simulation of planar ionization wave front propagation on an unstructured adaptive grid

The paper deals with the numerical solution of a basic 2D model of the propagation of an ionization wave. The system of equations describing this propagation consists of a coupled set of reaction–diffusion-convection equations and a Poissons equation. The transport equations are solved by a finite volume method on an unstructured triangular adaptive grid. The upwind scheme and the diamond scheme are used for the discretization of the convection and diffusion fluxes, respectively. The Poisson equation is also discretized by the diamond scheme. Numerical results are presented. We deal in more detail with numerical tests of the grid adaptation technique and its influence on the numerical results. An original behavior is observed. The grid refinement is not sufficient to obtain accurate results for this particular phenomenon. Using a second order scheme for convection is necessary.     

Simulation of flows with strong shocks with an adaptive conservative scheme

In the present work, numerical simulations of unsteady flows with moving shocks are presented. An unsteady mesh adaptation method, based on error equidistribution criteria, is adopted to capture the most important flow features. The modifications to the topology of the grid are locally interpreted in terms of continuous deformation of the finite volumes built around the nodes. The arbitrary Lagrangian–Eulerian formulation of the Euler equations is then applied to compute the flow variable over the new grid without resorting to any explicit interpolation step. The numerical results show an increase in the accuracy of the solution, together with a strong reduction of the computational costs, with respect to computations with a uniform grid using a larger number of nodes.     

Nash Point Equilibria in the Calculus of Variations

In this paper, the theory of Nash point equilibria for variational functionals including the following topics: existence in convex and non-convex cases, the applications to P. D. E., and the partial regularity, is studied. In the non-convex case, for a class of functionals, it is shown that the non-trivial solutions of the related systems of Euler equations are exactly the local Nash point equilibria and the trivial solution can not be a Nash point equilibrium.     

Grid equidistribution for reaction–diffusion problems in one dimension

The numerical solution of a linear singularly-perturbed reaction–diffusion two-point boundary value problem is considered. The method used is adaptive movement of a fixed number of mesh points by monitor-function equidistribution. A partly heuristic argument based on truncation error analysis leads to several suitable monitor functions, but also shows that the standard arc-length monitor function is unsuitable for this problem. Numerical results are provided to demonstrate the effectiveness of our preferred monitor function. AMS subject classification Primary: 65L50; secondary: 65L10, 65L12, 65L70 Research supported by the Boole Centre for Research in Informatics, National University of Ireland, Cork, Ireland. Natalia Kopteva: This paper was written while the first author was visiting the Department of Mathematics, National University of Ireland, Cork, Ireland.     

A Simple and Effective Self-adaptive Moving Mesh for Enthalpy Formulations of Phase Change Problems

A self-adaptive moving mesh for enthalpy formulations of diffusiondriven phase change problems is described. The PDE is approximatedby a finite element method on a moving, irregular mesh. Themesh is determined by a novel equidistribution principle andis concentrated in the phase transition regions and uniformin regions of pure phase.     

Adaptive mesh refinement for the control of cost and quality in finite element analysis

Adaptive strategies are a necessary tool to make finite element analysis applicable to engineering practice. In this paper, attention is restricted to mesh adaptivity. Traditionally, the most common mesh adaptive strategies for linear problems are used to reach a prescribed accuracy. This goal is best met with an h-adaptive scheme in combination with an error estimator. In an industrial context, the aim of the mechanical simulations in engineering design is not only to obtain greatest quality but more often a compromise between the desired quality and the computation cost (CPU time, storage, software, competence, human cost, computer used). In this paper, we propose the use of alternative mesh refinement criteria with an h-adaptive procedure for 3D elastic problems. The alternative mesh refinement criteria (MR) are based on: prescribed number of elements with maximum accuracy, prescribed CPU time with maximum accuracy and prescribed memory size with maximum accuracy. These adaptive strategies are based on a technique of error in constitutive relation (the process could be used with other error estimators) and an efficient adaptive technique which automatically takes into account the steep gradient areas. This work proposes a 3D method of adaptivity with the latest version of the INRIA automatic mesh generator GAMHIC3D.     

Adaptive simplification of huge sets of terrain grid data for geosciences applications

We propose and discuss a new Lepp-surface method able to produce a small triangular approximation of huge sets of terrain grid data by using a two-goal strategy that assures both small approximation error and well-shaped 3D triangles. This is a refinement method which starts with a coarse initial triangulation of the input data, and incrementally selects and adds data points into the mesh as follows: for the edge e having the highest error in the mesh, one or two points close to (one or two) terminal edges associated with e are inserted in the mesh. The edge error is computed by adding the triangle approximation errors of the two triangles that share e, while each L₂-norm triangle error is computed by using a curvature tensor (a good approximation of the surface) at a representative point associated with both triangles. The method produces triangular approximations that capture well the relevant features of the terrain surface by naturally producing well-shaped triangles. We compare our method with a pure L₂-norm optimization method.