on the numerical evaluation of electrostatic fields in dense random dispersions of cylinders we consider the interface boundary value problem which arises in the evaluation of electrostatic fields in composite materials consisting of dense random dispersions of cylinders in a uniform background this is a well studied problem from the viewpoint of homogenization and effective medium theory but one for which accurate numerical simulations have been difficult to obtain size effects in particular have been neglected due to the expense of solving the field equation in the presence of large numbers of close to touching inclusions such features require very fine discretizations even with the use of adaptive gridding and cause the linear systems which arise to be highly ill conditioned in this paper we present a new integral equation method for the solution of the interface problem which uses a recently developed method of images to resolve the close to touching interactions and the fast multipole method to compute far field interactions only minutes of workstation time are needed to solve the field equation with thousands of inclusions allowing us to carry out large scale statistical studies of the effective conductivity of random two phase materials at a variety of volume fractions and contrast ratios

a fast and robust solver for the scattering from a layered periodic structure containing multi particle inclusions we present a solver for plane wave scattering from a periodic dielectric grating with a large number m of inclusions lying in each period of its middle layer such composite material geometries have a growing role in modern photonic devices and solar cells the high order scheme is based on boundary integral equations and achieves many digits of accuracy with ease the usual way to periodize the integral equation via the quasi periodic green s function fails at wood s anomalies we instead use the free space green s kernel for the near field add auxiliary basis functions for the far field and enforce periodicity in an expanded linear system this is robust for all parameters inverting the periodic and layer unknowns we are left with a square linear system involving only the inclusion scattering coefficients preconditioning by the single inclusion scattering matrix this is solved iteratively in o m time using a fast matrix vector product numerical experiments show that a diffraction grating containing m 1000 inclusions per period can be solved to 9 digit accuracy in under 5 minutes on a laptop

a higher order scheme for two dimensional quasi static crack growth simulations an efficient scheme for the simulation of quasi static crack growth in two dimensional linearly elastic isotropic specimens is presented the crack growth is simulated in a stepwise manner where an extension to the already existing crack is added in each step in a local coordinate system each such extension is represented as a polynomial of some user specified degree n the coefficients of the polynomial describing an extension are found by requiring that the mode ii stress intensity factor is equal to zero at certain points of the extension if a crack grows from a pre existing crack so that a kink develops the leading term describing the crack shape close to the kink will in a local coordinate system be proportional to x 3 2 we therefore allow the crack extensions to contain such a term in addition to the monomial terms the discontinuity in the crack growth direction at a kink the kink angle is determined by requiring that the mode ii stress intensity factor should be equal to zero for an infinitesimal extension of the existing crack to implement the scheme accurate values of the stress intensity factors and t stress are needed in each step of the simulation these fracture parameters are computed using a previously developed integral equation of the second kind c 2007 elsevier b v all rights reserved less

solving the quintic by iteration equations that can be solved using iterated rational maps are characterized an equation is computable if and only if its galois group is within a5 of solvable we give explicitly a new solution to the quintic polynomial in which the transcendental inversion of the icosahedral map due to hermite and kronecker is replaced by a purely iterative algorithm the algorithm requires a rational map with icosahedral symmetries we show all rational maps with given symmetries can be described using the classical theory of invariant polynomials

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 fredholm s 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 fredholm s first kind

gauss jacobi quadratures for weakly strongly hyper and nearly singular integrals in boundary integral equation methods for domains with sharp edges and corners abstract we present gauss jacobi quadrature rules in terms of hypergeometric functions for the discretization of weakly singular strongly singular hypersingular and nearly singular integrals that arise in integral equation formulations of potential problems for domains with sharp edges and corners the rules are tailored to weight functions with algebraic endpoint singularities of a fairly general form thus allowing one to easily incorporate a wide class of domains into the analysis numerical examples illustrate the accuracy and stability of the proposed algorithms it is shown that the same level of high accuracy can be achieved for any choice of the external variable the usefulness of the method is exemplified by application to the solution of a singular integral equation that arises in time harmonic electromagnetic scattering by either closed or open perfectly conducting cylindrical objects with edges and corners such as polygon cylinders and bent strips some practical aspects concerning the role of nearby singularities in achieving a highly accurate solution of singular integral equations are also discussed

relation between effective conductivity and susceptibility of two component rhombic checkerboard the heterogeneity of composite leads to extra charge concentration at the boundaries of different phases which results in essentially nonzero effective electric susceptibility the relation between tensors of effective electric susceptibility ef and effective conductivity ef of the infinite two dimensional two component regular composite with rhombic cell structure has been established the degrees of electric field singularity at corner points of cells are found by constructing the integral equation for the effective conductivity problem the limits of weak and strong contrast of partial conductivities 1 2 are considered the results are valid for thin films and cylindrical samples

electrostatics and heat conduction in high contrast composite materials we present a robust integral equation method for the calculation of the electrostatic and thermal properties of systems made of piecewise homogeneous high contrast materials by high contrast we mean that the electrical or thermal conductivity ratios are high our approach involves a modification of the standard integral representation using in electrostatic terms the polarization charge as an unknown it is related to the perturbation approach proposed by tausch and white j tausch j white capacitance extraction of 3 d conductor systems in dielectric media with high permittivity ratios ieee trans microwave theory tech 47 1999 18 26 but based on a simple form of projection rather than on the solution of auxiliary integral equations numerical implementation is straightforward in the multiple dielectric case

the effective conductivity of random checkerboards an algorithm is presented for the fast and accurate solution of the electrostatic equation on multi component random checkerboards it relies on a particular choice of integral equation extended as to separate ill conditioning due to singular fields in corners from ill conditioning due to interaction of clusters of well conducting squares at large distances two separate preconditioners take care of the two separate phenomena in a series of numerical examples effective conductivities are computed for random checkerboards containing up to 104 squares with conductivity ratios of up to 106 the achievable relative precision in these examples is on the order of 10 11

checkerboard composites with separated phases recently the authors have developed a method siam j appl math 61 1839 1556 2001 capable of solving in closed form boundary value problems for four phase doubly periodic checkerboard composites with continuity between the different phases the method is based upon a novel conformal mapping that preserves the doubly periodic nature of the physical problem the aim of the current article is to explore generalizations of that approach where we now replace continuity between some phases by nonconducting or perfectly conducting strips thereby modeling debonding or electrodes in electrochemical devices the specific objective is to determine effective resistivities and related parameters for these four phase objects in a concise and explicit form

on the multiplicity of an arbitrary aα eigenvalue of a connected graph abstract the a matrix of a graph g is defined by nikiforov as a g d g 1 a g where 0 1 a g and d g respectively denotes the adjacency matrix and the degree diagonal matrix of g the eigenvalues of a g are called the a eigenvalues of g in this paper we prove that for a connected graph g of order n and with maximum degree 2 and for any 0 1 the multiplicity of an arbitrary a eigenvalue of g which is denoted as m g is bounded by m g 2 n 2 1 the equality holds if and only if g and satisfy one of the following conditions i g k n with n 3 and n 1 ii g c n with even order and 2 2 1 cos 2 j n j 1 2 n 2 2 iii g c n with odd order and 2 2 1 cos 2 j n j 1 2 n 1 2 iv g k n 2 n 2 with n 4 and n 2 applying this result we solve an open conjecture posed by zhou et al in 32 one year ago

an efficient algorithm for modelling progressive damage accumulation in disordered materials this paper presents an efficient algorithm for the simulation of progressive fracture in disordered quasi brittle materials using discrete lattice networks the main computational bottleneck involved in modelling the fracture simulations using large discrete lattice networks stems from the fact that a new large set of linear equations needs to be solved every time a lattice bond is broken using the present algorithm the computational complexity of solving the new set of linear equations after breaking a bond reduces to a simple triangular solves forward elimination and backward substitution using the already cholesky factored matrix this algorithm using the direct sparse solver is faster than the fourier accelerated iterative solvers such as the preconditioned conjugate gradient pcg solvers and eliminates the critical slowing down associated with the iterative solvers that is especially severe close to the percolation critical points numerical results using random resistor networks for modelling the fracture and damage evolution in disordered materials substantiate the efficiency of the present algorithm in particular the proposed algorithm is especially advantageous for fracture simulations wherein ensemble averaging of numerical results is necessary to obtain a realistic lattice system response

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

an accurate and fast analysis for strongly interacting multiple crack configurations including kinked v and branched y cracks in this study multiple interacting cracks in an infinite plate are analyzed to determine the overall stress field as well as stress intensity factors for crack tips and singular wedges at crack kinks the problem is formulated using integral equations expressed in terms of unknown edge dislocation distributions along crack lines these distributions derive from an accurate representation of the crack opening displacements using power series basis terms obtained through wedge eigenvalue analysis which leads to both polynomial and non polynomial power series the process is to choose terms of the series and their exponents such that the tractions on the crack faces are virtually zero compared to the far field loading applying the method leads to a set of linear algebraic equations to solve for the unknown weighting coefficients for the power series basis terms since no numerical integration is required unlike in other methods in most cases solution takes just a few seconds on a pc the accuracy and efficiency of the method are first demonstrated with a simple example of three aligned cracks with small ligaments between their tips under tensile loading the results are compared to exact results as well as to those of other numerical methods including recent fie fem and bem approaches said to have fast computation times thereafter some new and challenging crack interaction problems including branched y cracks two kinked v cracks are solved from a parametric study of the various crack configurations stress intensity factors are graphed and tabulated to demonstrate subtleties in the magnitudes of the crack interactions

an adaptive fast multipole accelerated poisson solver for complex geometries abstract we present a fast direct and adaptive poisson solver for complex two dimensional geometries based on potential theory and fast multipole acceleration more precisely the solver relies on the standard decomposition of the solution as the sum of a volume integral to account for the source distribution and a layer potential to enforce the desired boundary condition the volume integral is computed by applying the fmm on a square box that encloses the domain of interest for the sake of efficiency and convergence acceleration we first extend the source distribution the right hand side in the poisson equation to the enclosing box as a c 0 function using a fast boundary integral based method we demonstrate on multiply connected domains with irregular boundaries that this continuous extension leads to high accuracy without excessive adaptive refinement near the boundary and as a result to an extremely efficient black box fast solver

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 laplace s 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

a high order 3d boundary integral equation solver for elliptic pdes in smooth domains we present a high order boundary integral equation solver for 3d elliptic boundary value problems on domains with smooth boundaries we use nystrom s method for discretization and combine it with special quadrature rules for the singular kernels that appear in the boundary integrals the overall asymptotic complexity of our method is o n3 2 where n is the number of discretization points on the boundary of the domain and corresponds to linear complexity in the number of uniformly sampled evaluation points a kernel independent fast summation algorithm is used to accelerate the evaluation of the discretized integral operators we describe a high order accurate method for evaluating the solution at arbitrary points inside the domain including points close to the domain boundary we demonstrate how our solver combined with a regular grid spectral solver can be applied to problems with distributed sources we present numerical results for the stokes navier and poisson problems

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

harmonic density interpolation methods for high order evaluation of laplace layer potentials in 2d and 3d abstract we present an effective harmonic density interpolation method for the numerical evaluation of singular and nearly singular laplace boundary integral operators and layer potentials in two and three spatial dimensions the method relies on the use of green s third identity and local taylor like interpolations of density functions in terms of harmonic polynomials the proposed technique effectively regularizes the singularities present in boundary integral operators and layer potentials and recasts the latter in terms of integrands that are bounded or even more regular depending on the order of the density interpolation the resulting boundary integrals can then be easily accurately and inexpensively evaluated by means of standard quadrature rules a variety of numerical examples demonstrate the effectiveness of the technique when used in conjunction with the classical trapezoidal rule to integrate over smooth curves in two dimensions and with a chebyshev type quadrature rule to integrate over surfaces given as unions of non overlapping quadrilateral patches in three dimensions

nonlinear optimization quadrature and interpolation we present a nonlinear optimization procedure for the design of generalized gaussian quadratures for a fairly broad class of functions while some of the components of the algorithm have been published previously we introduce an improved procedure for the determination of an acceptable initial point for the continuation scheme that stabilizes the newton type process used to find the quadratures the resulting procedure never failed when applied to chebyshev systems for which the existence and uniqueness of generalized gaussian quadratures are well known it also worked for many non chebyshev systems for which the generalized gaussian quadratures are not guaranteed to exist the performance of the algorithm is illustrated with several numerical examples some of the presented quadratures integrate efficiently large classes of singular functions

fourth order accurate evaluation of integrals in potential theory on exterior 3d regions we present two methods for the rapid high order accurate evaluation of integrals in potential theory on general unbounded 3d regions our methods allow for direct calculation of derivatives of the integrals as well one of the methods uses a fourth order compact stencil and the other uses a nonstandard variant of richardson extrapolation both methods involve calculation of discontinuities in high order derivatives of the integrals across the boundary of the integration region the extrapolation method in addition involves correction for the discontinuities in truncation error the number of operations required for the methods is essentially equal to twice the number of operations needed to solve poisson s equation on a regular grid both methods avoid problems associated with using quadrature methods to evaluate integrals with singular kernels numerical results are presented for experiments on a variety of geometries in free space

a numerical model of stress driven grain boundary diffusion the stress driven grain boundary diffusion problem is a continuum model of mass transport phenomena in microelectronic circuits due to high current densities electromigration and gradients in normal stress along grain boundaries the model involves coupling many different equations and phenomena and difficulties such as nonlocality stiffness complex geometry and singularities in the stress tensor near corners and junctions make the problem difficult to analyze rigorously and simulate numerically we present a new numerical approach to this problem using techniques from semigroup theory to represent the solution the generator of this semigroup is the composition of a type of dirichlet to neumann map on the grain boundary network with the laplace operator on the network to compute the former we solve the equations of linear elasticity several times once for each basis function on the grain boundary we resolve singularities in the stress field near corners and junctions by adjoining special singular basis functions to both finite element spaces 2d for elasticity 1d for grain boundary functions we develop data structures to handle jump discontinuities in displacement across grain boundaries singularities in the stress field complicated boundary conditions at junctions and interfaces and the lack of a natural ordering for the nodes on a branching grain boundary network the method is used to study grain boundary diffusion for several geometries

an accurate boundary value problem solver applied to scattering from cylinders with corners in this paper we consider the classic problem of scattering of waves from perfectly conducting cylinders with piecewise smooth boundaries the scattering problems are formulated as integral equations and solved using a nystrom scheme where the corners of the cylinders are efficiently handled by a method referred to as recursively compressed inverse preconditioning rcip this method has been very successful in treating static problems in nonsmooth domains and the present paper shows that it works equally well for the helmholtz equation in the numerical examples we focus on scattering of e and h waves from a cylinder with one corner even at a size kd 1000 where k is the wavenumber and d the diameter the scheme produces at least 13 digits of accuracy in the electric and magnetic fields everywhere outside the cylinder

a fast direct solver for the integral equations of scattering theory on planar curves with corners we describe an approach to the numerical solution of the integral equations of scattering theory on planar curves with corners it is rather comprehensive in that it applies to a wide variety of boundary value problems here we treat the neumann and dirichlet problems as well as the boundary value problem arising from acoustic scattering at the interface of two fluids it achieves high accuracy is applicable to large scale problems and perhaps most importantly does not require asymptotic estimates for solutions instead the singularities of solutions are resolved numerically the approach is efficient however only in the low and mid frequency regimes once the scatterer becomes more than several hundred wavelengths in size the performance of the algorithm of this paper deteriorates significantly we illustrate our method with several numerical experiments including the solution of a neumann problem for the helmholtz equation given on a domain with nearly 10000 corner points

direct solvers performance on h adapted grids we analyse the performance of direct solvers when applied to a system of linear equations arising from an h adapted c 0 finite element space theoretical estimates are derived for typical h refinement patterns arising as a result of a point edge or face singularity as well as boundary layers they are based on the elimination trees constructed specifically for the considered grids theoretical estimates are compared with experiments performed with mumps using the nested dissection algorithm for construction of the elimination tree from metis library the numerical experiments provide the same performance for the cases where our trees are identical with those constructed by the nested dissection algorithm and worse performance for some cases where our trees are different we also present numerical experiments for the cases with mixed singularities where how to construct optimal elimination trees is unknown in all analysed cases the use of h adaptive grids significantly reduces the cost of the direct solver algorithm per unknown as compared to uniform grids the theoretical estimates predict and the experimental data confirm that the computational complexity is linear for various refinement patterns in most cases the cost of the direct solver per unknown is lower when employing anisotropic refinements as opposed to isotropic ones

on the computation of stress fields on polygonal domains with v notches the interior stress problem is solved numerically for a single edge notched specimen under uniaxial load the algorithm is based on a modification of a fredholm second kind integral equation with compact operators due to muskhelishvili several singular basis functions for each of the seven corners in the geometry enable high uniform resolution of the stress field with a modest number of discretization points as a consequence notch stress intensity factors can be computed directly from the solution this is an improvement over other procedures where the stress field is not resolved in the corners and where notch stress intensity factors are computed in a roundabout way via a path independent integral numerical examples illustrate the superior stability and economy of the new scheme less

a fast and stable solver for singular integral equations on piecewise smooth curves a scheme for the numerical solution of singular integral equations on piecewise smooth curves is presented it relies on several techniques reduction nystrom discretization composite quadrature recursive compressed inverse preconditioning and multipole acceleration the scheme is fast and stable its computational cost grows roughly logarithmically with the precision sought and linearly with overall system size when the integral equation models a boundary value problem the achievable accuracy may be close to the condition number of that problem times machine epsilon this is illustrated by application to elastostatic problems involving zigzag shaped cracks with up to twenty thousand corners and branched cracks with hundreds of triple junctions

a finite element method for domains with corners a new finite element method is devised for the numerical solution of elliptic boundary value problems with geometrical singularities in it the singularity is eliminated form the computational domain in an exact fashion this is in contrast to other common methods such as those which use a refined mesh in the singularity region or those which use special singular finite elements in them the singularity is treated as a part of the numerical scheme the new method is illustrated on an elliptic differential equation in a domain containing a re entrant corner numerical experiments show that the new method yields result which are generally much more accurate than those obtained by using the standard finite element method with mesh refinement in the singularity region both methods require about the same computing time

elastostatic computations on aggregates of grains with sharp interfaces corners and triple junctions we present a fast algorithm for the calculation of elastostatic fields in two dimensional assemblies of elastic grains separated by sharp grain boundaries the algorithm uses an integral equation approach combined with the fast multipole method and recursive compression to resolve stress concentrations also very close to grain boundary junctions singular basis functions on analytic form are not required accurate results can be obtained at a cost roughly proportional to the number of grains in the assembly large scale problems with thousands of grains are solved using modest computational resources