Two-dimensional Anisotropic Problems Research Articles

Modular implementation of magnetotelluric 2D forward modeling with general anisotropy

We present a general framework for two-dimensional finite difference modeling of magnetotelluric data in the presence of general anisotropy. Our approach is modular, allowing differential operators for a range of formulations of the governing equations, defined on several possible discrete grids, to be constructed from a basic set of first difference and averaging operators. We specifically consider two formulations of the two-dimensional anisotropic problem, one with Maxwell's equations reduced to a second order system in terms of three coupled electric components, and one in terms of coupled electric and magnetic x-components. Both formulations are discretized on a staggered grid; the second (coupled electric and magnetic) system is also implemented on a grid with fixed nodes (i.e., not staggered). The three implementations are validated and compared using a range of test models, including a half-space with general anisotropy, an infinite fault with axial anisotropy and a simple dyke model. Comparisons to analytic results (for half-space and fault models), and to results from other anisotropic codes, combined with grid-refinement convergence tests, demonstrate that our algorithms are accurate and capable of routine modeling of two-dimensional general anisotropy. These finite difference codes, demonstrating the flexibility of our numerical discretization approach, can be readily applied to other problems.

Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings

We develop a dimension-independent, Delaunay-based anisotropic mesh generation algorithm suitable for integration with adaptive numerical solvers. As such, the mesh produced by our algorithm conforms to an anisotropic metric prescribed by the solver as well as the domain geometry, given as a piecewise smooth complex. Motivated by the work of Lévy and Dassi [10-12,20], we use a discrete manifold embedding algorithm to transform the anisotropic problem to a uniform one. This work differs from previous approaches in several ways. First, the embedding algorithm is driven by a Riemannian metric field instead of the Gauss map, lending itself to general anisotropic mesh generation problems. Second we describe our method for computing restricted Voronoi diagrams in a dimension-independent manner which is used to compute constrained centroidal Voronoi tessellations. In particular, we compute restricted Voronoi simplices using exact arithmetic and use data structures based on convex polytope theory. Finally, since adaptive solvers require geometry-conforming meshes, we offer a Steiner vertex insertion algorithm for ensuring the extracted dual Delaunay triangulation is homeomorphic to the input geometries.The two major contributions of this paper are: a method for isometrically embedding arbitrary mesh-metric pairs in higher dimensional Euclidean spaces and a dimension-independent vertex insertion algorithm for producing geometry-conforming Delaunay meshes. The former is demonstrated on a two-dimensional anisotropic problem whereas the latter is demonstrated on both 3d and 4d problems.

Analysis of Aggregation-Based Multigrid

Aggregation-based multigrid with standard piecewise constant like prolongation is investigated. Unknowns are aggregated either by pairs or by quadruplets; in the latter case the grouping may be either linewise or boxwise. A Fourier analysis is developed for a model two-dimensional anisotropic problem. Most of the results are stated for an arbitrary smoother (which fits with the Fourier analysis framework). It turns out that the convergence factor of two-grid schemes can be bounded independently of the grid size. With a sensible choice of the (linewise or boxwise) coarsening, the bound is also uniform with respect to the anisotropy ratio, without requiring a specialized smoother. The bound is too large to guarantee optimal convergence properties with the V-cycle or the standard W-cycle, but a W-cycle scheme accelerated by the recursive use of the conjugate gradient method exhibits near grid independent convergence.

Numerical Convergence of a Parameterisation Method for the Solution of a Highly Anisotropic Two-Dimensional Elliptic Problem

Highly anisotropic two-dimensional elliptic problems lead to severe numerical difficulties. In this paper, starting from a simple finite volume scheme, we present a parameterisation method that allows us to obtain the solution even if the anisotropy ratio is very large. We derive a formal asymptotic limit of the two-dimensional anisotropic problem, in the case where the anisotropy ratio goes to infinity. This formal limit is used as a reference, and we show that the parameterisation method gives similar results, whereas the finite volume scheme fails to give an accurate solution. Numerical results are given, which indicate important parameters to be considered in order to obtain a good precision

An efficient boundary element method for computing accurate stresses in two-dimensional anisotropic problems

The stresses calculated by the boundary element method are accurate everywhere except in a narrow region near the boundaries. This “boundary layer effect” is due to the presence of hypersingularities in the relevant kernals. An efficient method that eliminates the boundary layer effect and yields accurate stresses everywhere in two-dimensional anisotropic material problems is presented. This method, the modified displacement gradient method, utilizes two identities—Somigliana's identity and a second identity in terms of displacement gradients and tractions. In this method, the Somigliana's identity is used as in the traditional BEM to determine the displacements and tractions at all nodes on the boundary. All the boundary data is then used to determine the displacement gradients at each of the boundary nodes. These displacement gradients and tractions are then used in the second identity to calculate the displacement gradients (and hence strains) at interior points. The stresses are then calculated using the constitutive relationships. The modified displacement gradient method is applied to several two-dimensional elasticity problems with isotropic and orthotropic materials with circular or elliptic cutouts. Numerical studies indicate that the present method gives accurate stresses even in the boundary layer region and is computationally efficient and attractive. In conjunction with the modified displacement gradient method, three approaches that use different evaluation procedures and locations for determining displacement gradients are used. In the first approach, the averaging approach, displacement gradients are averaged at nodes common to adjacent elements. In the second approach, the non-averaging approach, displacement gradients are not averaged but are stored element-wise. In the last approach, the discontinuous element approach, the gradients are evaluated at the nodes of discontinuous elements. Numerical studies indicate that all three approaches yield nearly same stress results, and hence, any one approach can be used.

An Iterative Matrix Solution Method Suitable for Anisotropic Problems

Abstract Often, the most time-consuming step in solving partial differential equations in two space partial differential equations in two space dimensions is an iterative solution of the finite-difference equations. For the closed boundary case, this solution is difficult when the equations are anisotropic, as they are normally when mesh spacing is much longer in one direction than in the other. This paper presents a solution method for use in such problems. The method is shown to be very fast in an anisotropic problem, but not as fast as other available methods in an isotropic problem. the extension to three space dimensions is outlined. Introduction In the finite-difference solution of multidimensional parabolic or elliptic equations, a set of many linear parabolic or elliptic equations, a set of many linear simultaneous algebraic equations arises. The most time-consuming part of the solution is the solving of this set of equations, normally accomplished by using some iterative method. Often the finite-difference equations are anisotropic. By anisotropic it is meant that two of the of diagonal coefficients in each equation are much larger than the other off-diagonal coefficients. When this occurs, the solution becomes more difficult for the closed boundary case. Until recently it appeared that line successive overrelaxation was the best technique to use in such a situation. Now, a method developed by Stone appears best among those published. This paper describes an alternative method for use with anisotropic systems. The method can be considered to be a specialization of a more general technique discussed by De la Vallee Poussin. It consists of a correction applied at each mesh point in a line, coupled with line successive overrelaxation. The method is shown to be very fast in a two-dimensional anisotropic problem. The extension to three dimensions is outlined, but no calculations are presented for that case. A theoretical analysis of the new method will be the subject of a later paper. paper. METHOD Consider the physical system from which the finite-difference equations are derived, as shown in Fig. 1. The system is represented by a matrix of mesh points. The finite-difference equation at each mesh point takes the form .........(1) All coefficients with the exception of di, j, are nonnegative. If the problem is elliptic (steady-slate), bi, j is zero. This equation is more commonly written as follows. .....(2) where This equation is written for each mesh point to create the set of simultaneous algebraic equations that must be solved. The matrix formed from the coefficients of these equations is often symmetric. The mesh points in Fig. 1 are arranged in a square pattern. SPEJ P. 47