General Two-dimensional Domains Research Articles

Two-Dimensional Fourier Continuation and Applications

This paper presents a method (2D-FC) for construction of bi-periodic extensions of smooth non-periodic functions defined over general two-dimensional smooth domains. The approach can be directly generalized to domains of any given dimensionality, and even to non-smooth domains, but such generalizations are not considered here. The 2D-FC extensions are produced in a two-step procedure. In the first step the one-dimensional Continuation method is applied along a discrete set of outward boundary-normal directions to produce, along such directions, continuations that vanish outside a narrow interval beyond the boundary. Thus, the first step of the algorithm produces blending-to-zero along normals for the given function values. In the second step, the extended function values are evaluated on an underlying Cartesian grid by means of an efficient, high-order boundary-normal interpolation scheme. A Continuation expansion of the given function can then be obtained by a direct application of the two-dimensional FFT algorithm. Algorithms of arbitrarily high order of accuracy can be obtained by this method. The usefulness and performance of the proposed two-dimensional Continuation method are illustrated with applications to the Poisson equation and the time-domain wave equation within a bounded domain. As part of these examples the novel Fourier Forwarding solver is introduced which, propagating plane waves as they would in free space and relying on certain boundary corrections, can solve the time-domain wave equation and other hyperbolic partial differential equations within general domains at computing costs that grow sublinearly with the size of the spatial discretization.

A Fourier Continuation Method for the Solution of Elliptic Eigenvalue Problems in General Domains

We present a new computational method for the solution of elliptic eigenvalue problems with variable coefficients in general two-dimensional domains. The proposed approach is based on use of the novel Fourier continuation method (which enables fast and highly accurate Fourier approximation of nonperiodic functions in equispaced grids without the limitations arising from the Gibbs phenomenon) in conjunction with an overlapping patch domain decomposition strategy and Arnoldi iteration. A variety of examples demonstrate the versatility, accuracy, and generality of the proposed methodology.

Reprint of Domain decomposition multigrid methods for nonlinear reaction–diffusion problems

In this work, we propose efficient discretizations for nonlinear evolutionary reaction–diffusion problems on general two-dimensional domains. The spatial domain is discretized through an unstructured coarse triangulation, which is subsequently refined via regular triangular grids. Following the method of lines approach, we first consider a finite element spatial discretization, and then use a linearly implicit splitting time integrator related to a suitable decomposition of the triangulation nodes. Such a procedure provides a linear system per internal stage. The equations corresponding to those nodes lying strictly inside the elements of the coarse triangulation can be decoupled and solved in parallel using geometric multigrid techniques. The method is unconditionally stable and computationally efficient, since it avoids the need for Schwarz-type iteration procedures. In addition, it is formulated for triangular elements, thus yielding much flexibility in the discretization of complex geometries. To illustrate its practical utility, the algorithm is shown to reproduce the pattern-forming dynamics of the Schnakenberg model.

Domain decomposition multigrid methods for nonlinear reaction–diffusion problems

In this work, we propose efficient discretizations for nonlinear evolutionary reaction–diffusion problems on general two-dimensional domains. The spatial domain is discretized through an unstructured coarse triangulation, which is subsequently refined via regular triangular grids. Following the method of lines approach, we first consider a finite element spatial discretization, and then use a linearly implicit splitting time integrator related to a suitable decomposition of the triangulation nodes. Such a procedure provides a linear system per internal stage. The equations corresponding to those nodes lying strictly inside the elements of the coarse triangulation can be decoupled and solved in parallel using geometric multigrid techniques. The method is unconditionally stable and computationally efficient, since it avoids the need for Schwarz-type iteration procedures. In addition, it is formulated for triangular elements, thus yielding much flexibility in the discretization of complex geometries. To illustrate its practical utility, the algorithm is shown to reproduce the pattern-forming dynamics of the Schnakenberg model.

Stress intensity factor solutions for two-dimensional elastostatic problems by the hypersingular boundary integral equation

The boundary element method is a numerical method that can be advantageously used for a wide range of engineering problems, including the stress concentration problems encountered in fracture mechanics. In linear elastic fracture mechanics (LEFM), the stress intensity factor is an important parameter. Cracks, if present in the region experiencing the modes of deformation, increase the stress amplitude significantly and this high stress may lead to premature failure of the engineering components. If the value of the SIF is known, it is possible to predict whether the crack will propagate or not. As the conventional boundary integral equation (CBIE) degenerates when a mathematical crack is modelled, a previously developed dual boundary integral equation approach has been adopted in the current work. It utilizes the hypersingular boundary integral equation (HBIE) along with the CBIE. A weakly singular form of HBIE is utilized in the current work to eliminate the hypersingularity analytically. Stress intensity factors are evaluated using the crack opening displacement (COD), displacement extrapolation (DE), and the J-integral approaches. A stand-alone code has been developed for calculating the stress intensity factors of general two-dimensional domains with cracks. The code has been validated by evaluating the stress intensity factors for the standard components, for which the stress intensity factor values are available in the literature. Accurate and well-converged results are obtained proving the robustness of the code. A linear combination of the CBIE and HBIE was applied at the crack and a significant (87–97 per cent) reduction in the condition numbers for the system of equations was observed for the examples studied. Again, the results obtained are accurate and well converged.

A Second Order Accurate Embedded Boundary Method for the Wave Equation with Dirichlet Data

The accuracy of Cartesian embedded boundary methods for the second order wave equation in general two-dimensional domains subject to Dirichlet boundary conditions is analyzed. Based on the analysis, we develop a numerical method where both the solution and its gradient are second order accurate. We avoid the small-cell stiffness problem without sacrificing the second order accuracy by adding a small artificial term to the Dirichlet boundary condition. Long-time stability of the method is obtained by adding a small fourth order dissipative term. Several numerical examples are provided to demonstrate the accuracy and stability of the method. The method is also used to solve the two-dimensional TM$_z$ problem for Maxwell's equations posed as a second order wave equation for the electric field coupled to ordinary differential equations for the magnetic field.

Difference Approximations for the Second Order Wave Equation

Difference approximations are derived for the second order wave equation in one and two space dimensions, without first writing it as a first order system. Both the Dirichlet and the Neumann problems are treated for the one-dimensional case. Relations between the boundary error and the interior phase error are derived for a fully second order accurate discretization as well as a scheme that is fourth order accurate in the interior and second order accurate at the boundary. General two-dimensional domains are considered for the Dirichlet problem where the domain is embedded in a Cartesian grid and the boundary conditions are approximated by interpolation. A stable conservative scheme is derived where the time step is determined only by the interior discretization formula. Discretization cells cut by the boundary are treated implicitly, but the resulting scheme becomes explicit because the implicit dependence only is pointwise. Numerical examples are provided to verify the stability and accuracy of the proposed method.

Scalability aspects of parallel multigrid

This paper summarizes theoretical and practical investigations into the effect of parallelization by grid-partitioning on the performance of multigrid methods for the solution of partial differential equations on general two-dimensional domains. Particular emphasis will be placed on the algorithmic scalability for MIMD distributed memory systems. Experimental results for two Navier-Stokes test problems, presented in the last section of the paper, show that the theoretically predicted dependency of the combined numerical and parallel efficiencies of multigrid methods on the number of processors employed is in fact very weak. This leads to the conclusion that multigrid is an appropriate candidate for solving partial differential equations on massively parallel machines.

Parallel ELLPACK for shared memory multiprocessors

This paper describes a parallel version of ELLPACK for shared memory multiprocessors. ELLPACK is a system for numerically solving elliptic PDEs. It consists of a very high level language for defining PDE problems and selecting methods of solution, and a library of approximately 50 problem solving modules. Each of the modules performs one of the basic steps in solving an elliptic PDE: discretization, reordering of equations and unknowns, linear system solution, etc. ELLPACK may be used to solve linear elliptic PDEs posed on general two-dimensional domains or in three-dimensional boxes, nonlinear problems, time-dependent problems and systems of elliptic equations. Earlier work considered three discretization modules ( five point star, hodie and hermite collocation), two linear system solution modules ( linpack spd band and jacobi cg) and a triple module ( hodie fft), which includes both discretization and solution, all for rectangular domains and simple boundary conditions. Here we describe parallel versions of six additional modules ( hermite collocation, hodie helmholtz, five point star, band ge, sor, symmetric sor cg) for general boundary conditions and domains, and discuss modifications to the ELLPACK preprocessor, the tool that translates an ELLPACK “program” into FORTRAN. The parallelization strategy is based on kernels, with the machine-dependent and performance critical code located in a relatively small number of kernels. We report performance of these parallel ELLPACK modules on a Sequent Symmetry S81 shared memory multiprocessor with 26 processors for two simple test problems, and a problem involving a composite laminate with spatially varying fiber orientations.

Integrated topology and boundary shape optimization of 2-D solids

This study is concerned with the development of an integrated procedure for the computation of the optimal topology as well as the optimal boundary shape of a two-dimensional, linear elastic body. The topology is computed by regarding the body as a domain of the plane with a high density of material and the objective is to maximize the overall stiffness, subject to a constraint on the material volume of the body. This optimal topology is then used as the basis for a shape optimal design method that regards the body as given by boundary curves. For this case the objective is to minimize the maximum value of the Von Mises equivalent stress in the body, subject to an isoperimetric constraint on the area as well as a constraint on the stiffness. The solution procedures for the shape design are based on variational formulations for the problems and the results of a variational analysis are implemented via finite element discretizations. The discretization grids are generated automatically by an elliptical method for general two-dimensional domains. Computational results are presented for the design of a fillet, a beam and a portal frame.

Collocation software for second-order elliptic partial differential equations

We consider the collocation method for linear, second-order elliptic problems on rectangular and general two-dimensional domains. An overview of the method is given for general domains, followed by a discussion of the improved efficiencies and simplifications possible for rectangular domains. A very-high-level description is given of three specific collocation algorithms that use Hermite bicubic basic functions, (1) GENCOL (collocation on general two-dimensional domains), (2) HERMCOL (collocation on rectangular domains with general linear boundary conditions), and (3) INTCOL (collocation on rectangular domains with uncoupled boundary conditions). The linear system resulting from INTCOL has half the bandwidth of that from HERMCOL, which provides substantial benefit in solving the system. We provide some examples showing the range of applicability of the algorithms and some performance profiles illustrating their efficiency. Fortran implementations of these algorithms are given in the companion papers [10, 11].

Multi-Grid Solution of Bifurcation Problems

A continuation strategy is developed for a method of inverse iteration type to compute solutions of nonlinear eigenvalue problems. The solution curves are parametrized by a norm of the solution. Thus continuation near turning points does not represent any difficulties. A multi-grid version of the algorithm is presented and successfully applied to solve problems on general two-dimensional domains. Numerical results including a comparison with a more standard approach are also given for the continuation strategy proposed for the basic algorithm.

Numerical Computation with General Two-Dimensional Domains

Consider a bounded two dimensional domain R with boundary aR. Applications which involve processing R include approximation. numerical integration and the numer(cal solution of partial differential equations. In each case the geometric information of Rand aR must be put into a data structure (either implicitly or explicitly) as an initial phase of the problem solution. This paper presents one approach to this problem based on overlaying R by a rectangular grid. R is assumed to be defined by parameterization of pieces of an plus an orientation. Specific representations of this information are given and a dala structure presented which is convenient for using the geometric inrormation in various applications. An algorithm to produce this iniormation is given in a companion paper, AlgDrithm: A Two Dimension Domain Processor; various special techniques or the algorithm are described. The application to numerical integration is discussed in some detail. Thls 1I'ork 11'89 supported in part by NSF grant MC8-7926396 and DOE contract DE-AC026IERI099?

Numerical Methods for the First Biharmonic Equation and for the Two-Dimensional Stokes Problem

We describe in this report various methods, iterative and almost direct, for solving the first biharmonic problem on general two-dimensional domains once the continuous problem has been approximated by an appropriate mixed finite element method. Using the approach described in this report we recover some well known methods for solving the first biharmonic equation as a system of coupled harmonic equations, but some of the methods discussed here are completely new, including a conjugate gradient type algorithm. In the last part of this report we discuss the extension of the above methods to the numerical solution of the two dimensional Stokes problem in p- connected domains (p $\geq$ 1) through the stream function-vorticity formulation.