This paper surveys algorithms that efficiently solve linear equations or compute eigenvalues even when the matrices involved are too large to fit in the main memory of the computer and must be stored on disks The paper focuses on scheduling techniques that result in mostly sequential data accesses and in data reuse, and on techniques for transforming algorithms that cannot be effectively scheduled The survey covers out-of-core algorithms for solving dense systems of linear equations, for the direct and iterative solution of sparse systems, for computing eigenvalues, for fast Fourier transforms, and for N-body computations The paper also discusses reasonable assumptions on memory size, approaches for the analysis of out-of-core algorithms, and relationships between out-of-core, cache-aware, and parallel algorithms

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A survey of out-of-core algorithms in numerical linear algebra

An up to date literature survey on the dynamics of underground structures is presented briefly. The dynamic response of large three-dimensional underground structures to external or internal dynamic forces or to seismic waves is numerically determined by the frequency domain boundary element method. This method is used to model both the structure and the soil medium, which are assumed to behave as linear elastic or viscoelastic bodies. The full-space dynamic fundamental solution is employed in the formulation and this requires a free soil surface discretization, confined to a finite portion around the area of interest, in addition to soil—structure interface and free structural surface discretizations. The dynamic disturbances can have a harmonic or a transient time variation. The transient case is treated with the aid of numerical Laplace transforms with respect to time. Various numerical examples involving lined cavities and long lined tunnels buried in the full- or the half-space subjected to harmonic or transient external forces or seismic waves are presented to illustrate the method and demonstrate its advantages.

Dynamic analysis of large 3-D underground structures by the bem

In this paper we derive a generalized fundamental solution for the BEM solution of problems of steady state heat conduction with arbitrarily spatially varying thermal conductivity. This is accomplished with the aid of a singular nonsymmetric generalized forcing function, D , with special sampling properties. Generalized fundamental solutions, E , are derived as locally radially symmetric responses to this nonsymmetric singular forcing function, D , at a source point ξ. Both E and D are defined in terms of the thermal conductivity of the medium. Although locally radially symmetric, E varies within the domain as the source point, ξ changes position. A boundary integral equation is formulated. Examples of generalized fundamental solutions are provided for various thermal conductivities along with the corresponding forcing function, D . Here, four numerical examples are provided. Excellent results are obtained with our formulation for variations of thermal conductivity ranging from quadratic and cubic in one dimension to exponential in two dimensions. Problems are solved in regular and irregular regions. Current work is under way investigating extensions of this general approach to further applications where nonhomogeneous property variations are an important consideration.

A generalized boundary integral equation for isotropic heat conduction with spatially varying thermal conductivity

Iterative techniques for the solution of the algebraic equations associated with the direct boundary element analysis (BEA) method are discussed. Continuum structural response analysis problems are considered, employing single- and multizone boundary element models with and without zone condensation. The impact on convergence rate and computer resource requirements associated with the sparse and blocked matrices, resulting in multizone BEA, is studied. Both conjugate gradient and generalized minimum residual preconditioned iterative solvers are applied for these problems and the performance of these algorithms is reported. Included is a quantification of the impact of the preconditioning utilized to render the boundary element matrices solvable by the respective iterative methods in a time competitive with direct methods.

Iterative solution techniques in boundary element analysis

A multi-zone boundary element analysis (BEA) capability that includes substructuring and condensation in a completely general fashion is presented. This condensation procedure is shown to be an effective way to perform blocked matrix factorizations using a reduced amount of high speed computer memory, and an approach that largely removes the effect of the boundary element zone numbering scheme on the computational resources expended due to block fill-in. In iterative problems with changing configuration, the strategy of condensing (substructuring) the unchanging portion of an overall model, in an exact fashion, and subsequently iterating on the resulting reduced model, is shown to have the potential for extending the range of such iterative problems. The approach will also allow for the simultaneous condensation and subsequent expansion of multiple boundary element zones on computers with parallel processing facility. The overall algorithm is described that allows for the assembly and solution of boundary element zones connected in a quite general way that may also be arbitrarily either condensed or maintained at their original size. The approach thus allows for both condensed and uncondensed boundary element zones to consistently coexist in the same multi-zone problem. A consistent and general formulation for the treatment of the double values of traction components at boundary element zone corners is also presented. Sample problems are described to demonstrate the efficiency and usefulness of the resulting capability.

An arbitrary condensing, noncondensing solution strategy for large scale, multi-zone boundary element analysis

An equation solver for the large unsymmetric systems of linear equations arising in the application of the boundary integral equation method to problems of linear elasticity in homogeneous or piecewise homogeneous solids is presented. The solver uses Gaussian elimination. The advantages of the solver over many existing eliminational or iterational methods are that it ignores many of the large groups of zeros that occur in piecewise homogeneous work, and also restrains growth of the number of non-zero matrix entries. Memory requirements and computation times therefore are reduced for many piecewise homogeneous problems.

J. M. Crotty

Papers

A block equation solver for large unsymmetric matrices arising in the boundary integral equation method