Johan Helsing

On the numerical evaluation of elastostatic fields in locally isotropic two-dimensional composites

We present a fast algorithm for the calculation of elastostatic fields in locally isotropic composites. The method uses an integral equation approach due to Sherman, combined with the fast multipole method and an adaptive quadrature technique. Accurate solutions can be obtained with inclusions of arbitrary shape at a cost roughly proportional to the number of points needed to resolve the interface. Large-scale problems, with hundreds of thousands of interface points can be solved using modest computational resources.

Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning

We take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points. Use of non-standard integral equations, graded meshes, interpolatory quadrature, and compressed inverse preconditioning are techniques that are explored, developed, mixed, and tested on some familiar problems in materials science. The recursive compressed inverse preconditioning, the major novelty of the paper, turns out to be particularly powerful and, when it applies, eliminates the need for mesh grading completely. In an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of 10^-^1^2. In another example, five times as large but with a conductivity ratio of only a hundred, we achieve an estimated relative error of 10^-^1^4.

On the evaluation of layer potentials close to their sources

When solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel. Standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources. Close to the sources this is not so. Cancellation and nearly singular kernels may cause serious degradation. This paper presents a new scheme based on a mix of composite polynomial quadrature, layer density interpolation, kernel approximation, rational quadrature, high polynomial order corrected interpolation and differentiation, temporary panel mergers and splits, and a particular implementation of the GMRES solver. Criteria for which mix is fastest and most accurate in various situations are also supplied. The paper focuses on the solution of the Dirichlet problem for Laplaces equation in the plane. In a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close-to-touching contours. Densities are computed and potentials are evaluated, rapidly and accurate to almost machine precision, at points that lie arbitrarily close to the boundaries.

Effective conductivity of aggregates of anisotropic grains

General expressions for the effective conductivity σeff of overall isotropic aggregates of grains are derived within the average field approximation. The grains are modeled by ellipsoids with possibly anisotropic conductivity tensors. Explicit expressions for σeff are stated for a polycrystal, which corrects an earlier result of Bolotin and Moskalenko [V. V. Bolotin and V. N. Moskalenko, J. Appl. Mech. Tech. Phys. 8, 3 (1967)], and for systems resembling the structure of nodular and grey cast iron. Percolation of randomly distributed two‐dimensional disks in a three‐dimensional continuous medium is studied. The percolation threshold for the number density of disks of radius unity is numerically determined to be 0.233±0.002 per unit volume.

An integral equation method for elastostatics of periodic composites

Abstract An interface integral equation is presented for the elastostatic problem in a two-dimensional isotropic composite. The displacement is represented by a single layer force density on the component interfaces. In a simple numerical example involving hexagonal arrays of disks the force density is expanded in a Fourier series. This leads to an algorithm with superalgebraic convergence. The integral equation is solved to double precision accuracy with a modest computational effort. Effective moduli are extracted both for dilute arrays where previously three digit accurate results were available, and for dense arrays where previously no results were available.

Solving integral equations on piecewise smooth boundaries using the RCIP method: a tutorial

Recursively compressed inverse preconditioning (RCIP) is a numerical method for obtaining highly accurate solutions to integral equations on piecewise smooth surfaces. The method originated in 2008 as a technique within a scheme for solving Laplace’s equation in two-dimensional domains with corners. In a series of subsequent papers, the technique was then refined and extended as to apply to integral equation formulations of a broad range of boundary value problems in physics and engineering. The purpose of the present paper is threefold: first, to review the RCIP method in a simple setting; second, to show how easily the method can be implemented in MATLAB; third, to present new applications of RCIP to integral equations of scattering theory on planar curves with corners.

Integral equation methods for elliptic problems with boundary conditions of mixed type

Laplaces equation with mixed boundary conditions, that is, Dirichlet conditions on parts of the boundary and Neumann conditions on the remaining contiguous parts, is solved on an interior planar domain using an integral equation method. Rapid execution and high accuracy is obtained by combining equations which are of Fredholms second kind with compact operators on almost the entire boundary with a recursive compressed inverse preconditioning technique. Then an elastic problem with mixed boundary conditions is formulated and solved in an analogous manner and with similar results. This opens up for the rapid and accurate solution of several elliptic problems of mixed type.

Integral equation methods and numerical solutions of crack and inclusion problems in planar elastostatics

We present algorithms for crack and inclusion problems in planar linear elastostatics. The algorithms are based on new integral equations. For the pure crack problem the integral equations are of Fredholms second kind. Our algorithms show great stability and allow for solutions to problems more complex than have previously been possible. Our results are orders of magnitudes more accurate than those of previous investigators, which rely on integral equations of Fredholms first kind.

Stress intensity factors for a crack in front of an inclusion

A stable numerical algorithm is presented for an elastostatic problem involving a crack close to and in front of an inclusion interface. The algorithm is adaptive and based on an integral equation of Fredholm’s second kind. This enables accurate analysis also of rather difficult situations. Comparison with stress intensity factors of cracks close to straight infinite bimaterial interfaces are made.

Thermal conductivity of cast iron: Models and analysis of experiments

Cast iron can be viewed as a composite material. We use effective medium and other theories for the overall conductivity of a composite, expressed in the conductivities, the volume fractions, and the morphology of the constituent phases, to model the thermal conductivity of grey and white cast iron and some iron alloys. The electronic and the vibrational contributions to the conductivities of the microconstituents (alloyed ferrite, cementite, pearlite, graphite) are discussed, with consideration of the various scattering mechanisms. Our model gives a good account of measured thermal conductivities at 300 K. It is easily extended to describe the thermal conductivity of other materials characterized by having several constituent phases of varying chemical composition.