The Buffered Block Forward Backward technique for solving electromagnetic wave scattering problems

This work focuses on efficient numerical techniques for solving electromagnetic wave scattering problems. The research is focused on three main areas: scattering from perfect electric conductors, 2D dielectric scatterers and 3D dielectric scattering objects. The problem of fields scattered from perfect electric conductors is formulated using the Electric Field Integral Equation. The Coupled Field Integral Equation is used when a 2D homogeneous dielectric object is considered. The Combined Field Integral Equation describes the problem of scattering from 3D homogeneous dielectric objects. Discretising the Integral Equation Formulation using the Method of Moments creates the matrix equation that is to be solved. Due to the large number of discretisations necessary the resulting matrices are of significant size and therefore the matrix equations cannot be solved by direct inversion and iterative methods are employed instead. Various iterative techniques for solving the matrix equation are presented including stationary methods such as the ”forwardbackward” technique, as well its matrix-block version. A novel iterative solver referred to as Buffered Block Forward Backward (BBFB) method is then described and investigated. It is shown that the incorporation of buffer regions dampens spurious diffraction effects and increases the computational efficiency of the algorithm. The BBFB is applied to both perfect electric conductors and homogeneous dielectric objects. The convergence of the BBFB method is compared to that of other techniques and it is shown that, depending on the grouping and buffering used, it can be more effective than classical methods based on Krylov subspaces for example. A possible application of the BBFB, namely the design of 2D dielectric photonic band-gap TeraHertz waveguides is investigated. i

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

Convergence and stability of finite difference schemes for some elliptic equations

The problem of convergence and stability of finite difference schemes used for solving boundary value problems for some elliptic partial differential equations has been studied in this thesis. Generally a boundary value problem is first replaced by a discretized problem whose solution is then found by numerical computation. The difference between the solution of the discretized problem and the exact solution of the boundary value problem is called the discretization error. This error is a measure of the accuracy of the numerical solution, provided the roundoff error is negligible. Estimates of the discretization error are obtained for a class of elliptic partial differential equations of order 2m (M ≥ 1) with constant coefficients in a general n-dimensional domain. This result can be used to define finite difference approximations with an arbitrary order of accuracy. The numerical solution of a discretized problem is usually obtained by solving the resulting system of algebraic equations by some iterative procedure. Such a procedure must be stable in order to yield a numerical solution. The stability of such an iteration scheme is studied in a general setting and several sufficient con­ditions of stability are obtained. When a higher order differential equation is solved numeri­cally, roundoff error can accumulate during the computations. In order to reduce this error the differential equation is sometimes replaced by several lower order differential equations. The method of splitting is analyzed for the two-dimensional biharmonic equation and the convergence of the discrete solution to the exact solution is discussed. An iterative procedure is presented for obtaining the numerical solution. It is shown that this method is also applicable to non-rectangular domains. The accuracy of numerical solutions of a nonselfadjoint elliptic differential equation is discussed when it is solved with a finite non-zero mesh size. This equation contains a parameter which may take large values. Some extensions to the two-dimensional Navier-Stokes equations are also presented

Improving Performance of Iterative Methods by Lossy Checkponting

Iterative methods are commonly used approaches to solve large, sparse linear systems, which are fundamental operations for many modern scientific simulations. When the large-scale iterative methods are running with a large number of ranks in parallel, they have to checkpoint the dynamic variables periodically in case of unavoidable fail-stop errors, requiring fast I/O systems and large storage space. To this end, significantly reducing the checkpointing overhead is critical to improving the overall performance of iterative methods. Our contribution is fourfold. (1) We propose a novel lossy checkpointing scheme that can significantly improve the checkpointing performance of iterative methods by leveraging lossy compressors. (2) We formulate a lossy checkpointing performance model and derive theoretically an upper bound for the extra number of iterations caused by the distortion of data in lossy checkpoints, in order to guarantee the performance improvement under the lossy checkpointing scheme. (3) We analyze the impact of lossy checkpointing (i.e., extra number of iterations caused by lossy checkpointing files) for multiple types of iterative methods. (4)We evaluate the lossy checkpointing scheme with optimal checkpointing intervals on a high-performance computing environment with 2,048 cores, using a well-known scientific computation package PETSc and a state-of-the-art checkpoint/restart toolkit. Experiments show that our optimized lossy checkpointing scheme can significantly reduce the fault tolerance overhead for iterative methods by 23%~70% compared with traditional checkpointing and 20%~58% compared with lossless-compressed checkpointing, in the presence of system failures.Comment: 14 pages, 10 figures, HPDC'1

The Sixth Copper Mountain Conference on Multigrid Methods, part 2

The Sixth Copper Mountain Conference on Multigrid Methods was held on April 4-9, 1993, at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Application of viscous-inviscid interaction methods to transonic turbulent flows

Two different viscous-inviscid interaction schemes have been developed for the analysis of steady transonic flows. The viscous and inviscid solutions are coupled through the displacement concept using a transpiration velocity. In the semi-inverse interaction method, the viscous and inviscid equations are solved in an explicitly separate manner and the displacement thickness is iteratively updated by a simple coupling algorithm. In the simultaneous interaction method, local solutions of viscous and inviscid equations are treated simultaneously, and the displacement thickness is treated as an unknown and obtained as a part of the solution through a global iteration procedure;The inviscid flow region is described by a direct finite-difference solution of a velocity potential equation in conservative form. The potential equation is solved on a numerically generated mesh by an approximate factorization (AF2) scheme in the semi-inverse method and by a successive line overrelaxation (SLOR) scheme in the simultaneous method. The boundary-layer equations are used for the viscous flow region. The continuity and momentum equations are solved inversely in a coupled manner using a fully implicit finite-difference scheme. The energy equation is solved uncoupled. The FLARE approximation is used in the reversed flow region and its effectiveness is studied by using a windward differencing scheme;The algebraic and one-half equation turbulence models are utilized to describe the Reynolds shear stress in turbulent flow calculations. Parameters affecting the convergence rate of the interaction procedure are discussed. The calculation schemes are evaluated by studying (1) an incompressible laminar flow over a flat plate with a trough, (2) a turbulent transonic flow over an axisymmetric boattail with a cylindrical plume simulator, (3) a turbulent transonic flow over an axisymmetric bump attached to a circular cylinder. The predictions are compared with experimental data and other available numerical results. The simultaneous interaction method becomes more efficient and reliable than the semi-inverse method as the separation size grows. The prediction obtained by the one-half equation turbulence model is generally in good agreement with the measurements, but disagreement is noticeable after the reattachment point

Application of viscous-inviscid interaction methods to transonic turbulent flows

Two different viscous-inviscid interaction schemes were developed for the analysis of steady, turbulent, transonic, separated flows over axisymmetric bodies. The viscous and inviscid solutions are coupled through the displacement concept using a transpiration velocity approach. In the semi-inverse interaction scheme, the viscous and inviscid equations are solved in an explicitly separate manner and the displacement thickness distribution is iteratively updated by a simple coupling algorithm. In the simultaneous interaction method, local solutions of viscous and inviscid equations are treated simultaneously, and the displacement thickness is treated as an unknown and is obtained as a part of the solution through a global iteration procedure. The inviscid flow region is described by a direct finite-difference solution of a velocity potential equation in conservative form. The potential equation is solved on a numerically generated mesh by an approximate factorization (AF2) scheme in the semi-inverse interaction method and by a successive line overrelaxation (SLOR) scheme in the simultaneous interaction method. The boundary-layer equations are used for the viscous flow region. The continuity and momentum equations are solved inversely in a coupled manner using a fully implicit finite-difference scheme