Conformal mapping for the efficient MFS solution of Dirichlet boundary value problems

In this work, we use conformal mapping to transform harmonic Dirichlet problems of Laplace’s equation which are defined in simply-connected domains into harmonic Dirichlet problems that are defined in the unit disk. We then solve the resulting harmonic Dirichlet problems efficiently using the method of fundamental solutions (MFS) in conjunction with fast fourier transforms (FFTs). This technique is extended to harmonic Dirichlet problems in doubly-connected domains which are now mapped onto annular domains. The solution of the resulting harmonic Dirichlet problems can be carried out equally efficiently using the MFS with FFTs. Several numerical examples are presented.      

Hybrid dynamic iterations for the solution of initial value problems

Many scientific problems are posed as Ordinary Differential Equations (ODEs). A large subset of these are initial value problems, which are typically solved numerically. The solution starts by using a known state space of the ODE system to determine the state at a subsequent point in time. This process is repeated several times. When the computational demand is high due to large state space, parallel computers can be used efficiently to reduce the time to solution. Conventional parallelization strategies distribute the state space of the problem amongst cores and distribute the task of computing for a single time step amongst the cores. They are not effective when the computational problems have fine granularity, for example, when the state space is relatively small and the computational effort arises largely from the long time span of the initial value problem. We propose a hybrid dynamic iterations method¹ which combines conventional sequential ODE solvers with dynamic iterations to parallelize the time domain. Empirical results demonstrate a factor of two to four improvement in performance of the hybrid dynamic iterations method over a conventional ODE solver on an 8 core processor. Compared to Picard iterations (also parallelized in the time domain), the proposed method shows better convergence and speedup results when high accuracy is required.     

An efficient numerical technique for the solution of nonlinear singular boundary value problems

In this work, a new technique based on Green’s function and the Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems (SBVPs) is proposed. The technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on the ADM, the proposed technique avoids solving a sequence of transcendental equations for the undetermined coefficients. It approximates the solution in the form of a series with easily computable components. Additionally, the convergence analysis and the error estimate of the proposed method are supplemented. The reliability and efficiency of the proposed method are demonstrated by several numerical examples. The numerical results reveal that the proposed method is very efficient and accurate.     

The hp-version of the boundary element method for Helmholtz screen problems

We study the boundary element method for weakly singular and hypersingular integral equations of the first kind on screens resulting from the Dirichlet and Neumann problems for the Helmholtz equation. It is shown that the hp-version with geometrical refined meshes converges exponentially fast in both cases. We underline our theoretical results by numerical experiments for the pure h-, p-versions, the graded mesh and the hp-version with geometrically refined mesh.     

An efficient solution algorithm for boundary element equations

Boundary element techniques result in the solution of a linear system of equations of the type HU = GQ + B, which can be transformed into a system of equations of the type AX = F. The coefficient matrix A requires the storage of a full matrix on the computer. This storage requirement, of the order of n^*n memory positions (n = number of equations), for a very large n is often considered negative for the boundary element method. Here, two algorithms are presented where the memory requirements to solve the system are only n^*(n - 1)/2 and n^*n/4 respectively. The algorithms do not necessitate any external storage devices nor do they increase the computational efforts.     

A fast and accurate computational technique for efficient numerical solution of nonlinear singular boundary value problems

This paper is concerned with design and implementation of a computational technique for the efficient solution of a class of singular boundary value problems. The method is based on a modified homotopy analysis method. The method is illustrated by six examples, two of which arise in chemical engineering: the first problem arises in the study of thermal explosions, while the second problem arises in the study of heat and mass transfer within the porous catalyst particles. Numerical results reveal that our method provides better results as compared to some existing methods. Furthermore, it is a powerful tool for dealing with different types of problems with strong nonlinearity.     

Explicit monotone iterations providing upper and lower bounds for finite element solution with nonlinear radiation boundary conditions

In this paper we consider explicit monotone iterations of the finite element solutions for the radiation cooling problem with the nonlinear boundary conditions. These iterations provide upper and lower bounds, and convergence proofs are given. Finally, we give some numerical examples to demonstrate the effectiveness.     

Asymptotic boundary conditions with immersed finite elements for interface magnetostatic/electrostatic field problems with open boundary

Many of the magnetostatic/electrostatic field problems encountered in aerospace engineering, such as plasma sheath simulation and ion neutralization process in space, are not confined to finite domain and non-interface problems, but characterized as open boundary and interface problems. Asymptotic boundary conditions (ABC) and immersed finite elements (IFE) are relatively new tools to handle open boundaries and interface problems respectively. Compared with the traditional truncation approach, asymptotic boundary conditions need a much smaller domain to achieve the same accuracy. When regular finite element methods are applied to an interface problem, it is necessary to use a body-fitting mesh in order to obtain the optimal convergence rate. However, immersed finite elements possess the same optimal convergence rate on a Cartesian mesh, which is critical to many applications. This paper applies immersed finite element methods and asymptotic boundary conditions to solve an interface problem arising from electric field simulation in composite materials with open boundary. Numerical examples are provided to demonstrate the high global accuracy of the IFE method with ABC based on Cartesian meshes, especially around both interface and boundary. This algorithm uses a much smaller domain than the truncation approach in order to achieve the same accuracy.     

On boundary conditions for the numerical solution of fluid dynamic problems

The stability and accuracy of various boundary treatments are analyzed for a finite difference scheme proposed by the author for the numerical solution of problems in fluid dynamics. The theory of Gustafsson, Kreiss and Sundstrom is used to establish stability and the theory of Skollermo is used to compare the accuracy of the various methods. The accuracy preductions are compared with computed results.     

An efficient gradient technique for the solution of optimal control problems

An iterative gradient-like technique — referred to as the double operator gradient method — is described. The method is shown to be generally much more efficient than either the fixed step or steepest descent type gradient methods. In addition, the method is applied to a well known singular optimal control problem which has recently been efficiently solved using a control-averaging technique. The double operator gradient method is shown to give a good suboptimal solution in relatively small computational time, being considerably more efficient than the controlaveraging technique for the extensive range of cases considered.     

Numerical solution of two-point boundary value problems

A new algorithm is presented for the numerical solution of linear and non-linear two-point boundary value problems with explicit type of boundary conditions. It is a systematic iterative method which reduces the problem to a corresponding initial value problem. The method has been tested to work for a set of 36 simultaneous differential equations 4 of which were non-linear.

Extension of this method for the more complicated case of implicit boundary conditions will be discussed in a separate paper.     

Adaptive meshing for two-dimensional thermoelastic problems using Hermite boundary elements

In the present work, error indicators for the potential and elastostatic problems are used in a combined fashion to implement an adaptive meshing scheme for the solution of two-dimensional steady-state thermoelastic problems using the Boundary Element Method. These error indicators exploit in their formulation the possibility of generating two different numerical solutions from just one analysis using Hermite elements. The first solution is the standard one obtained from an analysis using Hermite elements. The second is a “reduced” solution obtained representing the field variables inside an element using some of the degrees of freedom of the Hermite element together with Lagrangian shape functions. The basic idea behind the computation of the error indicator is to compare these two solutions, on an element by element basis, to obtain an estimate of the magnitude of the error in the numerical solution corresponding to the Hermite elements. In this sense, it is assumed that the bigger the difference between these two solutions, the bigger the error in the original solution with Hermite elements. Since the thermoelastic problem in its uncoupled fashion is considered, the former approach is applied to both problems, heat conduction and thermoelastic. Since both numerical solutions for each one of these problems are obtained from just one analysis, the computational cost of the proposed error indicators is very low.     

Diffuse conditions for efficient solution of multislab radiation transport problems

In this article, we make use of recently developed spectral nodal methods for anisotropically scattering media and we derive mathematical conditions for the diffuse reflection and transmission of radiation in the discrete ordinates formulation of particle transport theory for plane-parallel applications. The conditions arise from a suitable reformulation of spatially discretized equations defined on the boundary layers of a multislab domain. As a result, the boundary layers can be removed from the radiation transport calculations and replaced with exact and numerically stable equivalent conditions. In order to illustrate the applicability and computational merit of our discrete ordinates conditions for diffuse reflection and transmission in radiation transport calculations, we perform numerical experiments with atmospheric radiative transfer and nuclear reactor core models.     

Contribution to free boundary problems using boundary elements: Trefftz approach

There are two main approaches to the formulation of boundary methods, these are boundary integral equations and approximations by complete systems of solutions (Trefftz method). The latter has been the subject of extensive studies by one of the authors oriented to clarifying the foundations of the method and increasing its versatility. The present paper is devoted to explain the application of this procedure to free boundary problems such as Signorini and friction problems in elasticity.     

Normalized implicit methods for the solution of non-linear elliptic boundary value problems

New normalized implicit methods are presented for the solution of self-adjoint elliptic P.D.E.'s in two space dimensions. These methods are used in inner—outer iterative procedures in conjunction with Picard and Newton methods leading to improved composite iterative schemes for the solution of nonlinear elliptic boundary value problems. Applications of the derived methods include a nonlinear 2D magnetohydrodynamic problem and the 2D-Troesch's problem.     

An efficient solution to multi-objective control problems with LMI objectives

We revisit a technique for solving multi-objective control problems through affinely parameterizing the closed-loop system with the Youla parameterization and confining the search of the Youla parameter to finite-dimensional subspaces. It is pretty well-known how to solve such problems if the closed-loop specifications are formulated in terms of the solvability of linear matrix inequalities. However, all approaches proposed so far suffer from a substantial inflation of size of the resulting optimization problems if improving the approximation accuracy. On the basis of a novel state-space approach to solving static output feedback control problems by convex optimization for a specific class of plants, we reveal how the growth of the size of the optimization problems can be considerably reduced to arrive at more efficient algorithms. As an additional ingredient we discuss how to use the so-called mixed controller as a starting point for a genuine multi-objective design in order to improve the algorithms.