elastoplastic boundary problems in pies comparing to bem and fem the main aim of the paper is the application of a global way of defining the domain and global integration over the domain to the process of numerical solving of elastoplastic problems by the parametric integral equations system pies the paper presents mathematical formalism of pies for elastoplastic boundary value problems and its numerical implementation two crucial elements of the numerical algorithm are parametric surfaces used to define yield regions and 2d series applied to approximate solutions characteristics of pies are presented in comparison with fem and bem the paper consists of practical examples solved by pies whose results are confronted with other numerical experimental and analytical solutions

experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

compressed graphs and the minimum degree algorithm the minimum degree algorithm is well known and widely used to find a permutation matrix p such that a sparse symmetric positive definite matrix a can be factored as pap t ll t to yield a lower triangular matrix l with relatively few nonzero entries in this paper we show that by compressing the graph of a when possible the run time of the minimum degree algorithm can be significantly reduced some empirical results for a collection of test matrices are presented

high performance pcg solvers for fem structural analysis the preconditioned conjugate gradient algorithm is a well known and powerful method used to solve large sparse symmetric positive definite linear systems such system are generated by the finite element discretisation in structural analysis but users of finite elements in this contest generally still rely on direct methods it is our purpose in the present work to highlight the improvement brought forward by some new preconditioning techniques and show that the preconditioned conjugate gradient method performs better than efficient direct methods

algebraic multigrid amg experiences and comparisons algebraic multigrid methods are designed for the solution of sparse linear systems of equations using multigrid principles in contrast to standard multigrid methods amg does not take advantage of the origin of a particular system of equations at hand nor does it exploit any underlying geometrical situation fully automatically and based solely on algebraic information contained in the given matrix amg constructs a sequence of grids and corresponding operators a special amg algorithm will be presented for a wide range of problems including certain problems which do not have a continuous background this algorithm yields an iterative method which exhibits a convergence behavior typical for multigrid methods

on a class of preconditioning methods for dense linear systems from boundary elements in this paper we discuss several sparse matrix based preconditioners suitable for preconditioning dense linear systems of boundary element equations all preconditioners involve only o n nonzeros we provide a framework for constructing operator splitting based preconditioners and use it to analyze a class of sparse preconditioners for singular integral equations a more efficient preconditioner is proposed that has a band 2 structure

weight adjusted discontinuous galerkin methods wave propagation in heterogeneous media time domain discontinuous galerkin dg methods for wave propagation require accounting for the inversion of dense elemental mass matrices where each mass matrix is computed with respect to a parameter weighted l 2 inner product in applications where the wavespeed varies spatially at a subelement scale these matrices are distinct over each element necessitating additional storage in this work we propose a weight adjusted dg wadg method that reduces storage costs by replacing the weighted l 2 inner product with a weight adjusted inner product this equivalent inner product results in an energy stable method but does not increase storage costs for locally varying weights a priori error estimates are derived and numerical examples are given illustrating the application of this method to the acoustic wave equation with heterogeneous wavespeed

incomplete cholesky factorizations with limited memory we propose an incomplete cholesky factorization for the solution of large scale trust region subproblems and positive definite systems of linear equations this factorization depends on a parameter p that specifies the amount of additional memory in multiples of n the dimension of the problem that is available there is no need to specify a drop tolerance our numerical results show that the number of conjugate gradient iterations and the computing time are reduced dramatically for small values of p we also show that in contrast with drop tolerance strategies the new approach is more stable in terms of number of iterations and memory requirements

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

evaluation of regular and singular domain integrals with boundary only discretization theory and fortran code in this paper a set of boundary integrals are derived based on a radial integration technique to accurately evaluate two dimensional 2d and three dimensional 3d regular and singular domain integrals a self contained fortran code is listed and described for numerical implementation of these boundary integrals the main feature of the theory is that only the boundary of the integration domain needs to be discretized into elements this feature cannot only save considerable efforts in discretizing the integration domain into internal cells as in the conventional method but also make computational results for singular domain integrals more accurate since the integrals have been regularized some examples are provided to verify the correctness of the presented formulations and the included code

numerical evaluation of two dimensional singular boundary integrals theory and fortran code in this paper an approach is presented for the numerical evaluation of weakly strongly hyper and supersingular boundary integrals which exist in the cauchy principal value sense in two dimensional problems in this approach the singularities involved in integration kernels are analytically removed by expressing the nonsingular parts of the integration kernels as polynomials of the distance r a self contained fortran code is listed and described for implementation of the proposed approach the attached code is also able to evaluate general regular integrals using gaussian quadrature which enables the code to evaluate any two dimensional boundary integral some examples are provided to verify the correctness of the presented formulations and the included code

regular incomplete factorizations of real positive definite matrices abstract we study the classes of matrices which admit a regular incomplete factorization with respect to any graph set we extend the construction to the class of real positive definite matrices the convergence of the basic iteration associated with the splittings we have is discussed

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

preconditioning highly indefinite and nonsymmetric matrices standard preconditioners like incomplete factorizations perform well when the coefficient matrix is diagonally dominant but often fail on general sparse matrices we experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices the permutations and scalings are those developed by olschowka and neumaier linear algebra appl 240 1996 pp 131 151 and by duff and koster siam j matrix anal appl 20 1999 pp 889 901 tech report ral tr 99 030 rutherford appleton laboratory chilton uk 1999 we target highly indefinite nonsymmetric problems that cause difficulties for preconditioned iterative solvers our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing

robust approximate inverse preconditioning for the conjugate gradient method we present a variant of the ainv factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix the new preconditioner is breakdown free and when used in conjunction with the conjugate gradient method results in a reliable solver for highly ill conditioned linear systems we also investigate an alternative approach to a stable approximate inverse algorithm based on the idea of diagonally compensated reduction of matrix entries the results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented

analysis of a monte carlo method for nonlinear radiative transfer it has recently been proved that solutions of nonlinear radiative transfer problems satisfy a maximum principle andreev kozmanov and rachilov u s s r comput math math phys 23 104 1983 mercier siam j math anal in press in this article it is shown that monte carlo solutions of such problems obtained using the method of fleck and cummings j comput phys 8 313 1971 must satisfy this maximum principle for sufficiently small time steps but can violate it for sufficiently large time steps analyses of the frequency dependent and grey cases are given and a numerical solution violating the maximum principle is discussed

robust approaches to handling complex geometries with galerkin difference methods abstract the galerkin difference gd basis is a set of continuous piecewise polynomials defined using a finite difference like grid of degrees of freedom the one dimensional gd basis functions are naturally extended to multiple dimensions using a tensor product construction on quadrilateral elements the gd basis can be used to define the solution space for a discontinuous galerkin finite element discretization of partial differential equations in this work we propose two approaches to handling complex geometries within this setting 1 using nonconforming curvilinear gd elements and 2 coupling affine gd elements with curvilinear simplicial elements in both cases the semidiscrete discontinuous galerkin method is provably energy stable even when variational crimes are committed additionally for both element types a weight adjusted mass matrix is used which ensures that only the reference mass matrix must be inverted we also present sufficient conditions on the treatment of metric terms for the curvilinear nonconforming gd elements to ensure that the scheme is both constant preserving and conservative numerical experiments confirm the stability results and demonstrate the accuracy of the coupled schemes

orderings for factorized sparse approximate inverse preconditioners the influence of reorderings on the performance of factorized sparse approximate inverse preconditioners is considered some theoretical results on the effect of orderings on the fill in and decay behavior of the inverse factors of a sparse matrix are presented it is shown experimentally that certain reorderings like minimum degree and nested dissection can be very beneficial the benefit consists of a reduction in the storage and time required for constructing the preconditioner and of faster convergence of the preconditioned iteration in many cases of practical interest

finding exact and approximate block structures for ilu preconditioning sparse matrices which arise in many applications often possess a block structure that can be exploited in iterative and direct solution methods these block matrices have as their entries small dense blocks with constant or variable dimensions block versions of incomplete lu factorizations which have been developed to take advantage of such structures give rise to a class of preconditioners that are among the most effective available this paper presents general techniques for automatically determining block structures in sparse matrices a standard graph compression algorithm used in direct sparse matrix methods is considered along with two other algorithms which are also capable of unraveling approximate block structures

the incorporation of the semi implicit linear equations into newton s method to solve radiation transfer equations abstract for large time steps the nonlinear equations of radiation transfer may not be solved adequately by the semi implicit linear approximation to yield physical solutions this deficiency is rectified in three steps the equations of the semi implicit linear method are modified the modified equations are incorporated into newton s method to solve nonlinear equations and the transfer equations are solved by the resulting method the new method also uses the photon free method to search for the solution in a lower dimensional space than the space of the underlying transfer equations two algorithms are developed from the new method they solve the modified semi implicit linear equations by different approaches the first is a physics approach it solves the linear equations approximately by the grey transport approximation the second is a mathematical approach it solves the linear equations exactly by the sherman morrison woodbury formula of linear algebra however both algorithms yield the solution to the nonlinear system derived by the photon free method therefore their solutions are equal to within the specified tolerance of the nonlinear solver moreover both methods can take advantage of the unconditional stability which comes with the implicit differencing of the time derivative the time step which both methods can take is much larger than the time step in which time discretization error is discernible we shall relate the mathematical approach to the photon free method numerical results for three test problems are presented

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

some history of the conjugate gradient and lanczos methods this paper gives some of the history of the conjugate gradient and lanczos algorithms and an annotated bibliography for the period 1948 1976

galerkin differences for acoustic and elastic wave equations in two space dimensions abstract the galerkin difference method a finite element method built using standard galerkin projection techniques but employing nonstandard basis functions was originally developed for one space dimension in 1 here the method is extended to two space dimensions using a tensor product construction theoretical and computational evidence shows the method behaves as expected for the acoustic wave equation for the elastic wave equation the approximations are found to be at least as accurate as predicted but with a free surface the scheme may exhibit an unexpected superconvergence extension to curvilinear mapped grids is also considered for acoustics in all cases the use of a tensor product construction allows for efficient solution of the linear system involving the mass matrix which implies optimal linear time solutions with respect to the number of degrees of freedom

boundary element analysis of uncoupled transient thermo elastic problems with time and space dependent heat sources abstract a boundary element method bem for the analysis of two and three dimensional uncoupled transient thermo elastic problems involving time and space dependent heat sources is presented the domain integrals are efficiently treated using the cartesian transformation and the radial integration methods without considering any internal cells similar to the dual reciprocity method drm some internal points without any connectivity are considered however in contrast to the drm any arbitrary mesh free interpolation method can be used in the present formulation there is no need to find any particular solutions and the shape functions in the mesh free interpolation method can be arbitrary and sufficiently complicated unlike the drm the generated system of equations contains the unknowns only on the boundary after finding the primary unknowns on the boundary the temperature displacement and stress components at all internal points can directly be found without solving any system of equations three examples with different forms of heat sources are presented to demonstrate the efficiency and accuracy of the proposed method although the proposed bem is mathematically more complicated than domain methods such as the finite element method fem it is more efficient from a modelling viewpoint since only the surface mesh has to be generated in the presented method

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance

a skew symmetric discontinuous galerkin spectral element discretization and its relation to sbp sat finite difference methods this paper shows that the discontinuous galerkin collocation spectral element method with gauss lobatto points dgsem gl satisfies the discrete summation by parts sbp property and can thus be classified as an sbp sat simultaneous approximation term scheme with a diagonal norm operator in the same way sbp sat finite difference schemes can be interpreted as discontinuous galerkin type methods with a corresponding weak formulation based on an inner product formulation common in the finite element community this relation allows the use of matrix vector notation common in the sbp sat finite difference community to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for dgsem gl and diagonal norm sbp sat based on this result a skew symmetric energy stable discretely conservative dgsem gl formulation applicable to general diagonal norm sbp sat schemes for the nonlinear burgers equation is constructed

a petrov galerkin finite element method for solving the neutron transport equation abstract a finite element method using different trial and test spaces is introduced for solving the neutron transport equation in spherical geometry it is shown that the widely used discrete ordinates method can also be thought of as such a finite element technique in which integrals appearing in the difference equations are replaced by one point gauss quadrature formulas midpoint rule comparison of accuracy between the new method and the discrete ordinates method is discussed and numerical examples are given to illustrate the greater accuracy of the new technique

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

a low storage curvilinear discontinuous galerkin method for wave problems the low storage curvilinear discontinuous galerkin lsc dg method reduces the storage requirements for solving symmetric and linear linear symmetric conservation laws in curvilinear domains when compared to the standard discontinuous galerkin method we perform a semidiscrete a priori convergence analysis of lsc dg and determine sufficient conditions on sequences of meshes to guarantee the same rate of convergence as the original upwind dg method on curvilinear domains computational results confirm the optimal order convergence on sample mesh sequences that satisfy the sufficiency conditions additional results show that the sufficient conditions for optimal order convergence of lsc dg may also be necessary conditions

asymptotic analysis of a computational method for time and frequency dependent radiative transfer we consider a time dependent energy dependent nonlinear radiative transfer problem in which opacities are large o 1 and interior sources are small o an asymptotic analysis of this problem as 0 leads to the equilibrium diffusion equation in the interior of the system along with boundary conditions and initial conditions for this equation we apply the same asymptotic analysis to a discrete version of the problem in which the frequency variable is discretized by the multigroup method the direction variable by the discrete ordinates method the time variable by the fully implicit method and the spatial variable by a subcell balance method we find that as 0 the discrete solution satisfies a robust discretized version of the correct equilibrium diffusion equation with boundary conditions and initial conditions that are remarkably accurate the analysis thus predicts that if a spatial grid is chosen that resolves interior temperature gradients then the numerical method obtains an accurate solution in the interior of the system even though the optical thickness of the spatial cells tends to and boundary layers in the transport solution are not resolved we go a step further to analyze problems that are optically thin at some photon frequencies but thick at others and show that once again the discrete solution is remarkably accurate we present numerical results that verify these and other predictions of the analyses

an effective boundary element algorithm for 2d and 3d elastoplastic problems abstract novel methods are described for removing the strong singularities arising in the domain integrals of elastoplasticity and for solving the non linear equation set the former employs a new transformation from domain integrals to cell boundary integrals the number of system equations is minimised by using the plastic multiplier as the primary unknown and an incremental variable stiffness iterative algorithm is developed for solving these equations excellent convergence is achieved and some numerical examples demonstrate the algorithm s effectiveness

numerical evaluation of various levels of singular integrals arising in bem and its application in hydrofoil analysis the sound implementation of the boundary element method bem is highly dependent on an accurate numerical integration of singular integrals in this paper a set of various types of singular domain integrals with three dimensional boundary element discretization is evaluated based on a transformation integration technique in the bem the integration domain body surface needs to be discretized into small elements for each element the integral i x p x is calculated on the domain ds several types of integrals ib a and ic a are numerically and analytically computed and compared with the relative error the method is extended to evaluate singular integrals which arise in the solution of the three dimensional laplace s equation an example of the elliptic hydrofoil is performed to study the physical accuracy the results obtained using both numerical and analytical methods are shown in good agreement with the experimental data

modification of the boundary integral equation for the navier lame equations in 2d elastoplastic problems and its numerical solving summary the paper presents the generalization of the modification of classical boundary integral equation and obtaining parametric integral equation system for 2d elastoplastic problems the modification was made to obtain such equations for which numerical solving does not require application of finite or boundary elements this was achieved through the use of curves and surfaces for modeling introduced at the stage of analytical modification of the classic boundary integral equation for approximation of plastic strains the lagrange polynomials with various number and arrangement of interpolation nodes were used reliability of the modification was verified on examples with analytical solutions copyright 2016 john wiley sons ltd

an effective method for numerical evaluation of general 2d and 3d high order singular boundary integrals in this paper a robust method is presented for numerical evaluation of weakly strongly hyper and super singular boundary integrals which exist in the cauchy principal value sense in two and three dimensional problems in this method the singularities involved in integration kernels are analytically removed by expressing the non singular parts of the integration kernels as power series in the local distance of the intrinsic coordinate system for three dimensional boundary integrals the radial integration method 1 is applied to transform the surface integral into a line integral over the contour of the surface and to remove various orders of singularities within the radial integrals some examples are provided to verify the correctness and robustness of the presented method

weight adjusted discontinuous galerkin methods curvilinear meshes traditional time domain discontinuous galerkin dg methods result in large storage costs at high orders of approximation due to the storage of dense elemental matrices in this companion paper to j chan r j hewett and t warburton weight adjusted discontinuous galerkin methods wave propagation in heterogeneous media preprint https arxiv org abs 1608 01944 2016 we propose weight adjusted dg wadg methods for curvilinear meshes which reduce storage costs while retaining energy stability a priori error estimates show that high order accuracy is preserved under sufficient conditions on the mesh which are illustrated through convergence tests with different sequences of meshes numerical and computational experiments verify the accuracy and performance of wadg for a model problem on curved domains

new analytical expressions in radial integration bem for stress computation with several kinds of variable coefficients abstract this paper presents a set of new analytical expressions for evaluating radial integrals appearing in the stress computation of several kinds of variable coefficient elastic problems using the radial integration boundary element method ribem the strong singularity involved in the stress integral equation is explicitly removed from the derivation of the analytical expressions this approach can improve the computational efficiency considerably and can overcome the time consuming deficiency of ribem in computing involved radial integrals in addition because it can solve many kinds of variable coefficient elastic problems this approach has a very wide applicability the fourth order spline radial basis function rbf is employed to approximate the unknowns appearing in domain integrals caused by the variation of the shear modulus the radial integration method is utilized to convert domain integrals to the boundary which results in a pure boundary discretization algorithm numerical examples are given to demonstrate the efficiency of the presented formulations

a linear algebraic development of diffusion synthetic acceleration for three dimensional transport equations linear algebraic formulations of discretized mono energetic steady state linear boltzmann transport equations bte in three dimensions are presented the discretizations consist of a discrete ordinates collocation in angle and a petrov galerkin finite element method in space a matrix development of diffusion synthetic acceleration dsa is given for three dimensional 3 d rectangular geometry it is shown that the dsa consistently differenced diffusion approximation to the bte is actually singular in three dimensions although the dsa preconditioner itself is nonsingular numerical results are presented that demonstrate the effectiveness of the derived dsa preconditioner in the thick and thin limits for problems with nonconstant coefficients and nonuniform spatial zoning posed on finite domains with an incident flux prescribed at the boundaries

semi analytic treatment of the three dimensional poisson equation via a galerkin bie method a systematic treatment of the three dimensional poisson equation via singular and hypersingular boundary integral equation techniques is investigated in the context of a galerkin approximation developed to conveniently deal with domain integrals without a volume fitted mesh the proposed method initially converts domain integrals featuring the newton potential and its gradient into equivalent surface integrals then the resulting boundary integrals are evaluated by means of well established cubature methods in this transformation weakly singular domain integrals defined over simply or multiply connected domains with lipschitz boundaries are rigorously converted into weakly singular surface integrals combined with the semi analytic integration approach developed for potential problems to accurately calculate singular and hypersingular galerkin surface integrals this technique can be employed to effectively deal with mixed boundary value problems without the need to partition the underlying domain into volume cells sample problems are included to validate the proposed approach

discretely conservative finite difference formulations for nonlinear conservation laws in split form theory and boundary conditions the lax wendroff theorem stipulates that a discretely conservative operator is necessary to accurately capture discontinuities the discrete operator however need not be derived from the divergence form of the continuous equations indeed conservation law equations that are split into linear combinations of the divergence and product rule form and then discretized using any diagonal norm skew symmetric summation by parts sbp spatial operator yield discrete operators that are conservative furthermore split form discretely conservation operators can be derived for periodic or finite domain sbp spatial operators of any order examples are presented of a fourth order sbp finite difference operator with second order boundary closures sixth and eighth order constructions are derived and are supplied in an accompanying text file

a meshless bem for isotropic heat conduction problems with heat generation and spatially varying conductivity in this paper a new and simple boundary domain integral equation is presented for heat conduction problems with heat generation and non homogeneous thermal conductivity since a normalized temperature is introduced to formulate the integral equation temperature gradients are not involved in the domain integrals the green s function for the laplace equation is used and therefore the derived integral equation has a unified form for different heat generations and thermal conductivities the arising domain integrals are converted into equivalent boundary integrals using the radial integration method rim by expressing the normalized temperature using a series of basis functions and polynomials in global co ordinates numerical examples are given to demonstrate the robustness of the presented method copyright 2005 john wiley sons ltd

a linear algebraic analysis of diffusion synthetic acceleration for the boltzmann transport equation a linear algebraic formulation of discretized mono energetic steady state linear boltzmann transport equations in slab geometry is presented the discretization consists of a discrete ordinates collocation in angle and a diamond difference method in space by expressing diffusion synthetic acceleration in this formalism asymptotic results are obtained that prove the effectiveness of the associated preconditioner in various asymptotic regimes including the asymptotic diffusion limit these results hold for problems with nonconstant coefficients and nonuniform spatial zoning posed on finite domains with an incident flux and or reflection prescribed at the boundaries numerical results are also presented which demonstrate these results

sparse approximate inverse preconditioners using norm minimization techniques we investigate the use of sparse approximate inverse preconditioners for the iterative solution of unsymmetric linear systems of equations we consider the approximations proposed by cosgrove diaz and griewank internat j comput math 44 1992 pp 91 110 and huckle and grote a new approach to parallel preconditioning with sparse approximate inverses tech report sccm 94 03 stanford university 1994 which are based on norm minimization techniques such methods are of particular interest because of the considerable scope for parallelization we propose a number of enhancements which may improve their performance when run in a sequential environment these methods can perform unfavorably when compared with other techniques however they can be successful when other methods fail and simulations indicate that they can be competitive when considered in a parallel environment

robust preconditioners for linear elasticity fem analyses summary this paper deals with two forms of preconditioner which can be easily used with a conjugate gradient solver to replace a direct solution subroutine in a traditional engineering nite element package they are tested in such a package final over a range of 2 d and 3 d elasticity problems from geotechnical engineering quadratic basis functions are used a number of modications to the basic incomplete choleski ic 0 factorization preconditioner are considered an algorithm to reduce positive o diagonal entries is shown in numerical experiments to ensure stability but at the expense of slow convergence an alternative algorithm of jennings and malik is more successful and a relaxation parameter is introduced which can make a further signicant improvement in performance while maintaining stability a heuristic for determining a near optimal value of is proposed a second form of preconditioning symmetrically scaled element by element due to bartelt is also shown to perform robustly over a range of problems it does not require assembly of the global stiness matrix and has great potential for parallelization 1997 by john wiley sons ltd

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

an energy stable discontinuous galerkin spectral element discretization for variable coefficient advection problems we develop a stable split form discontinuous galerkin spectral element method for the solution of variable coefficient linear hyperbolic systems of conservation laws in our presentation we start with the simplest problem and introduce complexity as we progress we begin with the approximation of scalar conservation laws in one space dimension we then extend the derivation to hyperbolic systems finally we approximate the two dimensional problem numerical experiments are performed to compare the approximations

nurbs enhanced line integration boundary element method for 2d elasticity problems with body forces abstract a nurbs enhanced boundary element method for 2d elasticity problems with body forces is proposed in this paper the non uniform rational b spline nurbs basis functions are applied to construct the geometry and the model can be reproduced exactly at all stages since the refinement will not change the shape of the boundary both open curves and closed curves are considered the fields are approximated by the traditional lagrangian basis functions in parameter space rather than by the same nurbs basis functions for geometry approximation the parametric boundary elements and collocation nodes are defined from the knot vector of the curve and the refinement of the nurbs curve is easy boundary conditions can be imposed directly since the lagrangian basis functions have the property of delta function in addition most methods for the treatment of singular integrals in traditional boundary element method can be applied in the proposed method to overcome the difficulty for evaluation of the domain integrals in problems with body forces a line integration method is further applied in this paper to compute the domain integrals without additional volume discretizations numerical examples have shown the accuracy of the proposed method

a deterministic photon free method to solve radiation transfer equations a new method to solve radiation transfer equations is presented in the absence of scattering material motion and heat conduction the photon variables can be eliminated from the fully implicit multi group discrete ordinate finite difference finite element equations of continuum radiation transfer to yield a smaller set of equations which depends only on temperature the solution to this smaller set of equations is used to generate the solution to the original set of equations from which the reduced set is derived the reduced system simplifies to a nonlinear heat equation in the regime of strong absorption and strong emission we solve the reduced set of equations by the newton gmres method in which the jacobian update is preconditioned by a linearization of this nonlinear heat equation the performances of this new method and of the semi implicit linear method which is preconditioned by grey transport acceleration combined with diffusion synthetic acceleration are compared on two test problems the test results indicate that the new method can take larger time steps requires less memory is more accurate and is competitive in speed with the semi implicit linear method

a linear discontinuous spatial differencing scheme forsnradiative transfer calculations various types of linear discontinuous spatial differencing schemes have been developed for thesn discrete ordinates equations approximating the linear boltzmann transport equation it has been shown through an asymptotic analysis that the 1d slab geometry lumped linear discontinuous scheme not only goes over to a convergent and robust differencing of the diffusion equation in the monoenergetic thick diffusion limit but it also yields the correct interior solution even when boundary layers are left unresolved by the spatial mesh in the present work we generalize this scheme to obtain a 1d slab geometry lumped linear discontinuous scheme for the nonlinear radiative transfer equation and the associated material temperature equation we present a full nonlinear energy dependent asymptotic analysis of the behavior of this scheme in the thick equilibrium diffusion limit we find that this scheme goes over to a convergent and robust differencing of the equilibrium diffusion equation on the interior of the mesh but it does not yield the exact interior solution when boundary layers are left unresolved by the spatial mesh nevertheless the interior solution obtained with spatially unresolved boundary layers is always well behaved and fairly accurate computational results are presented which test the predictions of our asymptotic analysis and demonstrate the efficiency of our solution technique

polynomial preconditioners for conjugate gradient calculations dubois greenbaum and rodrigue proposed using a truncated neumann series as an approximation to the inverse of a matrix a for the purpose of preconditioning conjugate gradient iterative approximations to ax b if we assume that a has been symmetrically scaled to have unit diagonal and is thus of the form i g then the neumann series is a power series in g with unit coefficients the incomplete inverse was thought of as a replacement of the incomplete cholesky decomposition suggested by meijerink and van der vorst in the family of methods iccg n the motivation for the replacement was the desire to have a preconditioned conjugate gradient method which only involved vector operations and which utilized long vectors we here suggest parameterizing the incomplete inverse to form a preconditioning matrix whose inverse is a polynomial in g we then show how to select the parameters to minimize the condition number of the product of the polynomial and i g theoretically the resulting algorithm

multilevel ilu decomposition in this paper the multilevel ilu mlilu decomposition is introduced during an incomplete gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels on these levels some smoothing steps are computed the mlilu decomposition exists and the corresponding iterative scheme converges for all symmetric and positive definite matrices convergence rates independent of the number of unknowns are shown numerically for several examples many numerical experiments including unsymmetric and anisotropic problems problems with jumping coefficients as well as realistic problems are presented they indicate a very robust convergence behavior of the mlilu method

a new preconditioning technique for solving large sparse linear systems abstract a new sparse approximate triangular factorization technique for solving large sparse linear system by iterative methods is proposed the method is based on the description of the triangular factorization of a matrix as a product of elementary matrices and provides a general scheme for constructing incomplete preconditioners for the given matrix in particular the familiar incomplete choleski decomposition can be incorporated into this scheme the algorithm based on choice by value compares favorably with the incomplete choleski preconditioner and equipped with a user controlled parameter is able to tackle extremely ill conditioned problems arising in structural analysis semiconductor simulation oil reservoir modelling and other applications when applied to a positive definite symmetric matrix the algorithm produces a preconditioning matrix preserving that property

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

multi patch discontinuous galerkin isogeometric analysis for wave propagation explicit time stepping and efficient mass matrix inversion abstract we present a class of spline finite element methods for time domain wave propagation which are particularly amenable to explicit time stepping the proposed methods utilize a discontinuous galerkin discretization to enforce continuity of the solution field across geometric patches in a multi patch setting which yields a mass matrix with convenient block diagonal structure over each patch we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight adjusted approximation of the mass matrix inverse this approximation restores a tensor product structure while retaining provable high order accuracy and semi discrete energy stability we also estimate the maximum stable timestep for spline based finite elements and show that the use of spline spaces results in less stringent cfl restrictions than equivalent c 0 or discontinuous finite element spaces finally we explore the use of optimal knot vectors based on l 2 n widths we show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep and present a simple heuristic method for approximating optimal knot positions numerical experiments confirm the accuracy and stability of the proposed methods

fast approximate solution of large scale sparse linear systems abstract in this paper we apply the fast approximate inversion algorithm of frederickson to solve a large scale sparse linear system arising from the triangular finite element solution to the second order elliptic problems the main advantage of this algorithm is that the rate of convergence is independent of n for equations in the class considered several approximate inversion techniques for the algorithm are proposed and numerical results for these techniques are also provided

boundary element analysis in thermoelasticity with and without internal cells in this paper a set of internal stress integral equations is derived for solving thermoelastic problems a jump term and a strongly singular domain integral associated with the temperature of the material are produced in these equations the strongly singular domain integral is then regularized using a semi analytical technique to avoid the requirement of discretizing the domain into internal cells domain integrals included in both displacement and internal stress integral equations are transformed into equivalent boundary integrals using the radial integration method rim two numerical examples for 2d and 3d respectively are presented to verify the derived formulations copyright 2003 john wiley sons ltd

using approximate inverses in algebraic multilevel methods this paper deals with the iterative solution of large sparse symmetric positive definite systems we investigate preconditioning techniques of the two level type that are based on a block factorization of the system matrix whereas the basic scheme assumes an exact inversion of the submatrix related to the first block of unknowns we analyze the effect of using an approximate inverse instead we derive condition number estimates that are valid for any type of approximation of the schur complement and that do not assume the use of the hierarchical basis they show that the two level methods are stable when using approximate inverses based on modified ilu techniques or explicit inverses that meet some row sum criterion on the other hand we bring to the light that the use of standard approximate inverses based on convergent splittings can have a dramatic effect on the convergence rate these conclusions are numerically illustrated on some examples

on galerkin difference methods energy stable difference methods for hyperbolic initial boundary value problems are constructed using a galerkin framework the underlying basis functions are lagrange functions associated with continuous piecewise polynomial approximation on a computational grid both theoretical and computational evidence shows that the resulting methods possess excellent dispersion properties in the absence of boundaries the spectral radii of the operators for the first and second derivative matrices are bounded independent of discretization order with boundaries the spectral radius of the first order derivative matrix appears to be bounded independent of discretization order and grows only slowly with discretization order for problems in second order form

choosing the forcing terms in an inexact newton method an inexact newton method is a generalization of newton s method for solving f x 0 f mathbb r n to mathbb r n in which at the kth iteration the step s k from the current approximate solution x k is required to satisfy a condition f x k f x k s k leqslant eta k f x k for a forcing term eta k in 0 1 in typical applications the choice of the forcing terms is critical to the efficiency of the method and can affect robustness as well promising choices of the forcing terms are given their local convergence properties are analyzed and their practical performance is shown on a representative set of test problems