Showing papers in "Computing and Visualization in Science in 1998"

Geometric surface mesh optimization

Abstract: This paper presents a surface mesh optimization method suitable to obtain a geometric finite element mesh, given an initial arbitrary surface triangulation. The first step consists of constructing a geometric support, $G^1$ continuous, associated with the initial surface triangulation, which represents an adequate approximation of the underlying surface geometry. The initial triangulation is then optimized with respect to this geometry as well as to the element shape quality. A specific application of this technique to the geometric mesh simplification is then outlined, which aims at reducing the number of mesh entities while preserving the geometric approximation of the surface. Several examples of surface meshes intended for different application areas emphasize the efficiency of the proposed approach.

Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes

Abstract: The present paper deals with an anisotropic mesh adaptation (AMA) of triangulation which can be employed for the numerical solution various problems of physics. AMA tries to construct an optimal triangulation of the domain of computation in the sense that an “error” of the solution of the problem considered is uniformly distributed over the whole triangulation. First, we describe the main idea of AMA. We define an optimal triangle and an optimal triangulation. Then we describe the process of optimization of the triangulation and the complete multilevel computational process. We apply AMA to a problem of CFD, namely to inviscid compressible flow. The computational results for a channel flow are presented.

Efficient calculation of intrinsic low-dimensional manifolds for the simplification of chemical kinetics

Abstract: During the last years the interest in the numerical simulation of reacting flows has grown considerably and numerical methods are available, which allow to couple chemical kinetics with flow and molecular transport. The use of detailed physical and chemical models, involving several hundred species, is restricted to very simple flow configurations like one-dimensional systems or two-dimensional systems with very simple geometries, and models are required, which simplify chemistry without sacrificing accuracy. One method to simplify the chemical kinetics is based on Intrinsic Low-Dimensional Manifolds (ILDM). They represent attractors for the chemical kinetics, i.e. fast chemical processes relax towards them, and slow chemical processes represent movements within the manifolds. Thus the identification of the ILDMs allows a decoupling of the fast time scales. The concept has been verified by many different reacting flow calculations. However, one remaining problem of the method is the efficient calculation of the low-dimensional manifolds. This problem is addressed in this paper. We present an efficient, robust method, which allows to calculate intrinsic low-dimensional manifolds of chemical reaction systems. It is based on a multi-dimensional continuation process. Examples are shown for a typical combustion system. The method is not restricted to this problem class, but can be applied to other reacting flows or dynamic systems provided that a large number of decaying components can be eliminated from the system.

An adaptive subdivision technique for the approximation of attractors and invariant measures

Abstract: Recently subdivision techniques have been introduced in the numerical investigation of the temporal behavior of dynamical systems. In this article we intertwine the subdivision process with the computation of invariant measures and propose an adaptive scheme for the box refinement which is based on the combination of these methods. Using this new algorithm the numerical effort for the computation of box coverings is in general significantly reduced, and we illustrate this fact by several numerical examples.

Solving discretized optimization problems by partially reduced SQP methods

Abstract: The solution of discretized optimization problems is a major task in many application areas from engineering and science. These optimization problems present various challenges which result from the high number of variables involved as well as from the properties of the underlying process to be optimized. They also provide several strucures which have to be exploited by efficient numerical solution approaches. In this paper we focus on partially reduced SQP methods which are shown to be particularly well suited for this problem class. In practical applications the efficiency of this approach is demonstrated for optimization problems resulting from discretized DAE as well as from discretized PDE. The practically important issues of inexact solution of linearized subproblems and of working range validation are tackled as well.

HP90: A general and flexible Fortran 90 hp -FE code

Abstract: A general 2D-hp-adaptive Finite Element (FE) implementation in Fortran 90 is described. The implementation is based on an abstract data structure, which allows to incorporate the full hp-adaptivity of triangular and quadrilateral finite elements. The h-refinement strategies are based on h2-refinement of quadrilaterals and h4-refinement of triangles. For p-refinement we allow the approximation order to vary within any element. The mesh refinement algorithms are restricted to 1-irregular meshes. Anisotropic and geometric refinement of quadrilateral meshes is made possible by additionally allowing double constrained nodes in rectangles. The capabilities of this hp-adaptive FE package are demonstrated on various test problems.

The Finite Volume Scharfetter-Gummel method for steady convection diffusion equations

Abstract: A priori error estimates are given for the Finite Volume Scharfetter-Gummel (FVSG) discretization of the steady convection diffusion equation by showing that the FVSG method gives the same discretization as the Edge Averaged Finite Element method of Markowich and Zlamal [9] and Xu and Zikatanov [14]. The analysis also suggests a class of modifications for triangulations containing obtuse angles. Numerical results comparing the FVSG method and a modified FVSG method to other discretizations are included.

Numerical solution of parabolic equations related to level set formulation of mean curvature flow

Abstract: Numerical approximation of a nonlinear diffusion equation of mean curvature flow type is discussed Computational results related to image processing are presented

An adaptive Rothe method for the wave equation

Abstract: The adaptive Rothe method approaches a time-dependent PDE as an ODE in function space. This ODE is solved virtually using an adaptive state-of-the-art integrator. The actual realization of each time-step requires the numerical solution of an elliptic boundary value problem, thus perturbing the virtual function space method. The admissible size of that perturbation can be computed a priori and is prescribed as a tolerance to an adaptive multilevel finite element code, which provides each time-step with an individually adapted spatial mesh. In this way, the method avoids the well-known difficulties of the method of lines in higher space dimensions. During the last few years the adaptive Rothe method has been applied successfully to various problems with infinite speed of propagation of information. The present study concerns the adaptive Rothe method for hyperbolic equations in the model situation of the wave equation. All steps of the construction are given in detail and a numerical example (diffraction at a corner) is provided for the 2D wave equation. This example clearly indicates that the adaptive Rothe method is appropriate for problems which can generally benefit from mesh adaptation. This should be even more pronounced in the 3D case because of the strong Huygens' principle.

Real time numerical simulation and visualization of electrochemical drilling

Abstract: Gas turbines have to be provided with holes in order to provide cooling; these holes are made using an electrochemical drilling technique. Since this process is tedious and expensive, computer simulations are very useful. Such a model needs to incorporate the relevant physical processes. A simulation system including real time user interaction and visualization together with efficient numerical techniques has been developed using an object oriented design.

Numerical methods for diffusion-reaction-transport processes in unsaturated porous media

Abstract: This paper introduces a new grid refinement and coarsening technique for the approximation of partial differential equations including a first order time derivative. This hierarchical movement algorithm is based on the nested iteration method. The combination of this algorithm, a quasi Newton method and the Schur-complement multi-grid method leads to an efficient method for the solution of partial differential equations describing complex real life problems. As a test case, diffusion-reaction-transport processes in heterogeneous unsaturated porous media are considered. Some simulation results are presented.