bilum block versions of multielimination and multilevel ilu preconditioner for general sparse linear systems we introduce block versions of the multielimination incomplete lu ilum factorization preconditioning technique for solving general sparse unstructured linear systems these preconditioners have a multilevel structure and for certain types of problems may exhibit properties that are typically enjoyed by multigrid methods several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed the advantages of block ilum over point ilum include increased robustness and efficiency we compare several versions of the block ilum point ilum and the dual threshold based ilut preconditioners in particular tests with some convection diffusion problems show that it may be possible to obtain convergence that is nearly independent of the reynolds number as well as of the grid size

local radial basis function based gridfree scheme for unsteady incompressible viscous flows a local radial basic function rbf based gridfree scheme has been developed to solve unsteady incompressible navier stokes equations in primitive variables the velocity pressure decoupling is obtained by making use of a fractional step algorithm the scheme is validated over a variety of benchmark problems and found a very good agreement with the existing results comparisons with the benchmark solutions show that the developed local rbf gridfree scheme is stable and produces accurate results on domains discretized even with non uniform distribution of nodal points

burton miller type singular boundary method for acoustic radiation and scattering abstract this paper proposes the singular boundary method sbm in conjunction with burton and miller s formulation for acoustic radiation and scattering the sbm is a strong form collocation boundary discretization technique using the singular fundamental solutions which is mathematically simple easy to program meshless and introduces the concept of source intensity factors sifs to eliminate the singularities of the fundamental solutions therefore it avoids singular numerical integrals in the boundary element method bem and circumvents the troublesome placement of the fictitious boundary in the method of fundamental solutions mfs in the present method we derive the sifs of exterior helmholtz equation by means of the sifs of exterior laplace equation owing to the same order of singularities between the laplace and helmholtz fundamental solutions in conjunction with the burton miller formulation the sbm enhances the quality of the solution particularly in the vicinity of the corresponding interior eigenfrequencies numerical illustrations demonstrate efficiency and accuracy of the present scheme on some benchmark examples under 2d and 3d unbounded domains in comparison with the analytical solutions the boundary element solutions and dirichlet to neumann finite element solutions

new method for deriving eigenvalue rate with respect to support location the authors present a new method to derive the formulas of eigenvalue rate with respect to in span support location using the generalized variational principle of the rayleigh quotient following the ideas of this note the eigenvalue rate with respect to such variables as location of discrete or continuous in span supports the location of concentrated mass location of elastic support and location of a substructure can be derived without any conceptual difficulties

a meshless method for solving an inverse spacewise dependent heat source problem in this paper an effective meshless and integration free numerical scheme for solving an inverse spacewise dependent heat source problem is proposed due to the use of the fundamental solution as basis functions the method leads to a global approximation scheme in both spatial and time domains the standard tikhonov regularization technique with the generalized cross validation criterion for choosing the regularization parameter is adopted for solving the resulting ill conditioned system of linear algebraic equations the effectiveness of the algorithm is illustrated by several numerical examples

eigenvalue sensitivity analysis of planar frames with variable joint and support locations various sensitivity equations are developed in this study for eigenvalue sensitivity analysis of planar frames with variable joint and support locations the emphasis is placed on the derivation of the sensitivity equations by the continuum approach a variational form of an eigenvalue equation is first derived in which all of the quantities are expressed with respect to the local coordinate system the material derivative of this variational form is then sought to account for the changes in the member s length and orientation resulting from the perturbation of joint and support locations finally eigenvalue sensitivity equations are formulated in either domain quantities by the domain method or boundary quantities by the boundary method examples are presented to validate these derived sensitivity equations with analytical solutions as well as numerical results it is concluded that the sensitivity equation derived by the boundary method is computationally more efficient but less accurate than that of the domain method nevertheless both of them are superior to the conventional discrete analytical method in terms of computational efficiency nomenclature ait ei ij cross sectional area young s modulus and moment of inertia of member respectively b design variable j a line functional j and its total shape derivative k m global stiffness and mass matrices at m elementary stiffness and mass matrices dj pj length orientation angle and mass density of member respectively mit rij cosine and sine of member n total number of members n

solve partial differential equations by two or more radial basis functions abstract there are many partial differential equations their solution might be oscillatory or vary rapidly we will get an inaccurate result or encounter an ill conditioned problem whenever we solve it using kansa s collocation method mfs or other methods in this paper we will present a new method in which such pdes will be solved by two or more radial basis functions rbfs so these pdes can be solved efficiently and accurately

a parallel algorithm to approximate inverse factors of a matrix via sparse sparse iterations abstract in d k salkuyeh f toutounian a block version algorithm to approximate inverse factors appl math comput 162 2005 1499 1509 the authors proposed the baib algorithm to approximate inverse factors of a matrix in this paper a parallel version of the baib algorithm is presented in this method the baib algorithm is combined with computing the sparse approximate solution of a sparse linear system by sparse sparse iterations the new method does not require that the sparsity pattern be known in advance some numerical experiments on test matrices from harwell boeing collection are presented to show the efficiency of the new method and comparing to the aib algorithm

solving partial differential equations by meshless methods using radial basis functions abstract in this paper we will present some meshless methods for partial differential equations including meshless subdomain method meshless petrov galerkin method and meshless schwarz method

topology optimization of compliant mechanism and its support through a level set method abstract in the present paper the task of designing a compliant mechanism is accomplished by solving a multi objective topology optimization problem the objective function includes a target displacement at the output port and compliances of two load cases the design variables include two types of boundaries i e the traction free boundary and the dirichlet boundary the topology optimization of compliant mechanism and its support is solved by using a level set method two level set functions are used to represent a compliant mechanism and the two types of boundaries each of the two boundaries is represented by a part of the zero level set of one level set function evolution of the two boundaries is modeled by two independent hamilton jacobi equations thus the shape and topology of the two boundaries change independently the shape derivatives are derived by using a lagrangian function and the adjoint method the finite element analysis is done by modifying a fixed background mesh and the artificial weak material is not used numerical examples in two dimensions are investigated

preconditioned krylov subspace methods for solving nonsymmetric matrices from cfd applications abstract we conduct an experimental study on the behavior of several preconditioned iterative methods to solve nonsymmetric matrices arising from computational fluid dynamics cfd applications the preconditioned iterative methods consist of krylov subspace accelerators and a powerful general purpose multilevel block ilu bilum preconditioner the bilum preconditioner and an enhanced version of it are slightly modified versions of the originally proposed preconditioners they will be used in combination with different krylov subspace methods we choose to test three popular transpose free krylov subspace methods bicgstab gmres and tfqmr numerical experiments using several sets of test matrices arising from various relevant cfd applications are reported

minimum stiffness location of point support for control of fundamental natural frequency of rectangular plate by rayleigh ritz method abstract the rayleigh ritz method is employed to determine the minimum stiffness location of the elastic point support for raising the fundamental natural frequency of a rectangular plate to the second frequency of the unsupported plate which usually is the upper limit of the first frequency for a single support based on the optimal design of the support position the minimum stiffness can be obtained numerically by solving a characteristic eigenvalue sub problem in the rayleigh ritz procedure the boundary characteristic orthogonal polynomials are used for the admissible functions several typical examples of plate structures with the additional point supports are analyzed in detail and the results prove that the proposed method is very effective in the solution to the optimal design of the supports it will be shown that elastic supports can be designed like rigid ones for the purpose of increasing the fundamental natural frequency of a rectangular plate

a priori sparsity patterns for parallel sparse approximate inverse preconditioners parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm this paper demonstrates that for pde problems the patterns of powers of sparsified matrices psms can be used a priori as effective approximate inverse patterns and that the additional effort of adaptive sparsity pattern calculations may not be required psm patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the pde s green s function a parallel implementation shows that psm patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively while often giving preconditioners of comparable quality

optimization of support positions to maximize the fundamental frequency of structures in this paper the position optimization of simple supports is implemented to maximize the fundamental frequency of a beam or plate structure both elastic and rigid supports are taken into account first the frequency sensitivity with respect to the movement of a simple support is derived using the discrete method by means of the shape functions of the finite element method closed form sensitivity formulations are developed straightforwardly then a heuristic approach called evolutionary shift method is presented for optimizing support positions with a fixed grid mesh scheme based on the design sensitivity analysis the support with the highest efficiency is shifted in priority along the elementary edges with the interval step of the elementary size to facilitate the convergence of the process the interpolation technique is employed to evaluate the solution more accurately finally three numerical examples are presented to demonstrate the validity of the sensitivity analysis and the effectiveness of the optimization method copyright 2004 john wiley sons ltd

enhanced multi level block ilu preconditioning strategies for general sparse linear systems abstract this paper introduces several strategies to deal with pivot blocks in multi level block incomplete lu factorization bilum preconditioning techniques these techniques are aimed at increasing the robustness and controlling the amount of fill ins of bilum for solving large sparse linear systems when large size blocks are used to form block independent set techniques proposed in this paper include double dropping strategies approximate singular value decomposition variable size blocks and use of an arrowhead block submatrix we point out the advantages and disadvantages of these strategies and discuss their efficient implementations numerical experiments are conducted to show the usefulness of the new techniques in dealing with hard to solve problems arising from computational fluid dynamics in addition we discuss the relation between multi level ilu preconditioning methods and algebraic multi level methods

a new class of decomposition for inverting asymmetric and indefinite matrices abstract an innovative decomposition for inverting a nonsingular asymmetric and indefinite matrix a of order n n is derived in this paper the inverse of a is written as a 1 l d u where l is a lower triangular matrix d is a diagonal matrix and u is an upper triangular matrix by the method the solution of a x b may be easily and efficiently computed by matrix vector multiplications as x l d u b this technique requires a minimal amount of computer memories and can be easily transformed into parallel procedures with high efficiencies performances in inverting asymmetric and indefinite matrices and in solving systems of linear equations on an alliant fx8 computer are reported

an efficient numerical scheme for burgers equation this paper applies the multiquadric mq as a spatial approximation scheme for solving the nonlinear burgers equation for comparison purposes a low order explicit finite difference approximation of the time derivative is employed by decreasing the time step of the computation it is shown that the major numerical error is from the time integration instead of the mq spatial approximation the numerical results indicate that this mq offers an excellent approximation for all possible values of reynolds number an adaptive algorithm is also developed to adjust the mq interpolation points to the peak of the shock wave which is shown to provide an improved numerical result numerical comparisons are made with most of the existing numerical schemes for solving the burgers equation

dual reciprocity hybrid boundary node method for free vibration analysis abstract as a truly meshless method of boundary type the hybrid boundary node method hbnm has the advantages of both boundary element method bem and meshless method the main problem is that it is only suitable for the homogeneous problems now the dual reciprocity method drm is introduced into hbnm to deal with the integral for the inhomogeneous terms of the governing equations and the rigid body motion approach is employed to solve the hyper singular integrations a new meshless method named dual reciprocity hybrid boundary node method drhbnm is proposed and applied to solve free vibration problems in this method the solution composes into two parts i e the general solution and the particular solution the general solution is solved by hbnm and the particular one is obtained by drm drhbnm is a true boundary type meshless method it does not require the boundary element mesh either for the purpose of interpolation of the variables or for the integration of energy the points in the domain are only used to interpolate particular solution by the radial basis function finally the boundary variables are interpolated by the independent smooth boundary segments the q r algorithm and householder algorithm are applied to solve the eigenvalues and eigenvectors of the transformed matrix numerical examples for free vibration problems show that a good convergence with mesh refinement is achievable and the computational results for the natural circular frequencies and free vibration modes are very accurate furthermore the computation parameters have little influence on the results and can be chosen in a wide range it is shown that the present method is effective and can be widely applied in practical engineering

optimal support positions for a structure to maximize its fundamental natural frequency abstract a procedure and related theories are developed to find loci of optimal support positions for a structure to maximize its fundamental eigenvalue by increasing the support stiffness the concept of limit eigenvalue which is the upper bound of fundamental eigenvalue achieved by adding supports is introduced a condition is derived on which the fundamental eigenvalue can be reached to its limit eigenvalue a sensitivity formula of eigenvalues with respect to the change of support positions is also derived to set up an optimization problem and to obtain its optimal support positions it is found that the loci of m supports start from the maximum displacement position of the structure s first eigenfunction and end at certain positions on the nodal line of its m 1 th eigenfunction if the fundamental eigenvalue can reach its limit eigenvalue the suggested method is tested to find the loci for a beam and a plate structure

factored approximate inverse preconditioners with dynamic sparsity patterns abstract we propose two sparsity pattern selection algorithms for factored approximate inverse preconditioners to solve general sparse matrices the sparsity pattern is adaptively updated in the construction phase by using combined information of the inverse and original triangular factors of the original matrix in order to determine the sparsity pattern our first algorithm uses the norm of the inverse factors multiplied by the largest absolute value of the original factors and the second employs the norm of the inverse factors divided by the norm of the original factors experimental results show that these algorithms improve the robustness of the preconditioners to solve general sparse matrices

a sparse approximate inverse preconditioner for the conjugate gradient method a method for computing a sparse incomplete factorization of the inverse of a symmetric positive definite matrix a is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient calculations it is proved that in exact arithmetic the preconditioner is well defined if a is an h matrix the results of numerical experiments are presented

a sparsity for decomposing a symmetric matrix abstract this paper studies a sparse configuration for a new class of decomposition derived by the author based upon the sparsity the inverse of a sparse and symmetric matrix and the solution of a system of linear equations can be economically computed numerical examples are included

superconvergent gradient smoothing meshfree collocation method abstract a superconvergent meshfree collocation method with smoothed nodal gradients is presented in this method the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function while the second order smoothed gradients are computed through directly differentiating the first order smoothed gradients it is noted that the second order smoothed gradients evaluated at meshfree nodes can be conveniently expressed as two successive first order gradient smoothing operations on the meshfree shape function which facilitates a trivial numerical implementation subsequently an employment of the second order smoothed gradients in the strong form of a given problem leads to the gradient smoothing meshfree collocation method purely using nodes as the collocation points based upon a local truncation error analysis it is systematically shown that the proposed meshfree collocation method yields superconvergent solutions for odd degree basis functions a key ingredient attributed to this superconvergence property is that the second order smoothed gradients meet the consistency conditions which go one order beyond the original basis degree of meshfree approximation another interesting fact is that the present formulation enables a convergent collocation scheme when the linear basis function is used in meshfree approximation which is non feasible in the conventional collocation formulation the effectiveness of the proposed methodology is validated by numerical examples for both potential and elasticity problems numerical results well demonstrate the superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method

maximizing the natural frequency of a beam with an intermediate elastic support the minimum stiffness of a simple or point support that raises a natural frequency of a beam to its upper limit is investigated for different boundary conditions the approach produces the closed form solution for the minimum stiffness based on the derivative of a natural frequency with respect to the support position it is seen that when an intermediate elastic support is positioned properly the effect is similar to a rigid support the solution process also provides insight into the dynamics of a beam with an intermediate support for more general boundary conditions the resulting solutions can be used to guide the practical design of a support

regularized meshless method for antiplane shear problems with multiple inclusions in this paper we employ the regularized meshless method to solve antiplane shear problems with multiple inclusions the solution is represented by a distribution of double layer potentials the singularities of kernels are regularized by using a subtracting and adding back technique therefore the troublesome singularity in the method of fundamental solutions mfs is avoided and the diagonal terms of influence matrices are determined an inclusion problem is decomposed into two parts one is the exterior problem for a matrix with holes subjected to remote shear the other is the interior problem for each inclusion the two boundary densities essential and natural data along the interface between the inclusion and matrix satisfy the continuity and equilibrium conditions a linear algebraic system is obtained by matching boundary conditions and interface conditions finally numerical results demonstrate the accuracy of the present solution good agreements are obtained and compare well with analytical solutions and gong s results

boundary element free method befm and its application to two dimensional elasticity problems in this study we first discuss the moving least square approximation mls method in some cases the mls may form an ill conditioned system of equations so that the solution cannot be correctly obtained hence in this paper we propose an improved moving least square approximation imls method in the imls method the orthogonal function system with a weight function is used as the basis function the imls has higher computational efficiency and precision than the mls and will not lead to an ill conditioned system of equations combining the boundary integral equation bie method and the imls approximation method a direct meshless bie method the boundary element free method befm for two dimensional elasticity is presented compared to other meshless bie methods befm is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables and the boundary conditions can be applied easily hence it has higher computational precision for demonstration purpose selected numerical examples are given

extended concept of stair shape sparsity for the inverse of an asymmetric matrix abstract this paper extends the concept of stair shape sparsity introduced in a previous work of the author for analysis of a symmetric and sparse matrix to an asymmetric matrix the essential idea first considers an asymmetric matrix as a combination of an upper triangular matrix and a lower triangular matrix and then applies the stair shape sparsity to the upper and lower triangular matrices this enables us to avoid the operation of zero fill ins for the inverse of an asymmetric matrix the introduced sparsity provides an efficient procedure for the solution of a sparse system of linear equations

asymptotic stability of the fundamental solution method abstract in a previous paper 1988 we studied the numerical stability of the fundamental solution method applied to the dirichlet problem of laplace s equation and presented a practical scheme for examining the stability of the method using the singular value decomposition in the present paper we give an asymptotic estimate of the stability with respect to the number of collocation points

comparison of time and spatial collocation methods for the heat equation we combine a high order compact finite difference scheme to approximate the spatial derivatives and collocation techniques for the time component to numerically solve the two dimensional heat equation we use two approaches to implement the time collocation methods the first one is based on an explicit computation of the coefficients of polynomials and the second one relies on differential quadratures we also implement a spatial collocation method where differential quadratures are utilized for spatial derivatives and an implicit scheme for marching in time we compare all the three techniques by studying their merits and analyzing their numerical performance our experiments show that all of them achieve high accurate approximate solution but the time collocation method with differential quadrature offers with respect to the one with explicit polynomials less computational complexity and a better efficiency all our computations based on parallel algorithms are carried out on the cray sv1

a level set method for shape and topology optimization of both structure and support of continuum structures abstract we present a level set method for the shape and topology optimization of both structure and support two level set functions are used to implicitly represent a structure the traction free boundary and the dirichlet boundary are represented separately and are allowed to be continuously changed during the optimization the optimization problem of minimum compliance is considered the shape derivatives of both boundaries are derived by using a lagrangian function and the adjoint method the finite element analysis is done by modifying a fixed background mesh and we do not use the artificial weak material numerical examples in two dimensions are investigated

a meshless singular boundary method for three dimensional elasticity problems summary this study documents the first attempt to extend the singular boundary method sbm a novel meshless boundary collocation method for the solution of three dimensional 3d elasticity problems the sbm involves a coupling between the regularized boundary element method bem and the method of fundamental solutions mfs the main idea here is to fully inherit the dimensionality and stability advantages of the former and the meshless and integration free attributes of the later this makes it particularly attractive for problems in complex geometries and three dimensions four benchmark 3d problems in linear elasticity are well studied to demonstrate the feasibility and accuracy of the proposed method the advantages disadvantages and potential applications of the proposed method as compared with the fem bem and mfs are also examined and discussed this article is protected by copyright all rights reserved

boundary knot method based on geodesic distance for anisotropic problems the radial basis function rbf collocation techniques for the numerical solution of partial differential equation problems are increasingly popular in recent years thanks to their striking merits being inherently meshless integration free and highly accurate however the rbf based methods have markedly been limited to handle isotropic problems due to the use of the isotropic euclidean distance this paper makes the first attempt to use the geodesic distance with the rbf based boundary knot method bkm to solve 2d and 3d anisotropic helmholtz type and convection diffusion problems this approach is mathematically simple and easy to implement and spectral convergence is numerically observed for problems under complex shaped boundary numerical results show that the bkm based on the geodesic distance can produce highly accurate solutions of anisotropic problems with a relatively small number of knots this study provides a promising strategy for the rbf based methods to effectively solve anisotropic problems

approximate inverse preconditioners via sparse sparse iterations the standard incomplete lu ilu preconditioners often fail for general sparse indefinite matrices because they give rise to unstable factors l and u in such cases it may be attractive to approximate the inverse of the matrix directly this paper focuses on approximate inverse preconditioners based on minimizing i am f where am is the preconditioned matrix an iterative descent type method is used to approximate each column of the inverse for this approach to be efficient the iteration must be done in sparse mode i e with sparse matrix by sparse vector operations numerical dropping is applied to maintain sparsity compared to previous methods this is a natural way to determine the sparsity pattern of the approximate inverse this paper describes newton global and column oriented algorithms and discusses options for initial guesses self preconditioning and dropping strategies some limited theoretical results on the properties and convergence of approximate inverses are derived numerical tests on problems from the harwell boeing collection and the fidap fluid dynamics analysis package show the strengths and limitations of approximate inverses finally some ideas and experiments with practical variations and applications are presented

calculation of eigenvector derivatives for structures with repeated eigenvalues in structural optimization and system identification eigenvector derivatives provide important information for updating design model parameters when the current parameter values yield repeated eigenvalues it has not been possible previously to calculate unique eigenvector derivatives recent work has provided a method for determining unique eigenvalue derivatives for this case but methods for calculating the eigenvector sensitivities have been incomplete in this work a complete method for calculation of repeated root eigenvector derivatives is shown for the real symmetric structural eigenproblem the derivation is completed by using information from the second derivative of the eigen problem and is limited to the case of distinct eigenvalue sensitivities as an example the repeated root eigenvector sensitivities are calculated for a simple three degree of freedom beam grillage comparisons of linear approximations using these derivatives to the calculated eigenvectors support the accuracy of the formulation

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

the location of spring supports from measured vibration data detecting the position of supports within an elastic structure has many applications particularly when these supports are not fixed previous studies have presented methods for the detection of translational support locations in elastic structures based on the minimization of the difference between the measured and computed natural frequencies however these discrete supports were constrained to be at nodes of the finite element model of the elastic structure this required a fine mesh and the numerical computation of the eigenvalue derivative with respect to the support location this paper has the same purpose namely to identify the support locations but the position of the supports is now a continuous parameter when the support is located within an element the shape functions are used to produce the global stiffness matrix the advantage is that the support location now appears explicitly in the formulation of the problem and hence the analytical computation of the eigenvalue derivative is possible furthermore the mesh may be much more coarse requiring fewer degrees of freedom to detect the support locations with reduced computational effort the effect of identifying both the support stiffness and location is also discussed the proposed approach has been illustrated with simulated and experimental examples used in the earlier studies

a new approach to compute sparse approximate inverse factors of a matrix abstract the fsai algorithm is among the most effective preconditioners for solving large and sparse linear system of equations but its main disadvantage is the need to prescribe the sparsity pattern of the approximate inverse factors in advance in this paper a new method is proposed by combining the fsai preconditioner with approximating the sparse solution of a sparse linear system by sparse sparse iterations the new method does not require that the sparsity pattern be known in advance moreover it retains the inherent parallelism of the fsai algorithm some numerical experiments on test matrices from harwell boeing collection are presented

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 incomplete inverse as a preconditioner for the conjugate gradient method abstract this paper introduces a new preconditioner with a super convergence for the conjugate gradient method for solving a system of linear equations in a banded structure the idea is based upon an innovative method of the author to decompose a symmetric matrix into its inverse and the preconditioner is the result of an incomplete decomposition i e an approximation to the inverse by the presented preconditioner the rate of convergence can be significantly improved examples to demonstrate the improvement of rate of convergence and the speedup implemented on an alliant fx8 computer are included

taylor meshless method for solving non linear partial differential equations a true meshless integration free method based on taylor series named taylor meshless method tmm has been proposed recently to solve partial differential equations pdes where the shape functions are high degree polynomials and the discretization concerns only the boundary with high computational efficiency and exponential convergence the tmm has been confirmed to be very robust in linear problems including large scale cases in this paper the tmm and the automatic differentiation ad are combined with the newton method to solve non linear elliptic pdes where the ad is used to compute shape functions in a fast manner the numerical results illustrate that the proposed algorithm is very accurate and efficient in solving non linear elliptic problems

approximate inverse techniques for block partitioned matrices this paper proposes some preconditioning options when the system matrix is in block partitioned form this form may arise naturally for example from the incompressible navier stokes equations or may be imposed after a domain decomposition reordering approximate inverse techniques are used to generate sparse approximate solutions whenever these are needed in forming the preconditioner the storage requirements for these preconditioners may be much less than for incomplete lu factorization ilu preconditioners for tough large scale computational fluid dynamics cfd problems the numerical experiments show that these preconditioners can help solve difficult linear systems whose coefficient matrices are highly indefinite

a sparse approximate inverse preconditioner for parallel preconditioning of general sparse matrices a factored sparse approximate inverse is computed and used as a preconditioner for solving general sparse matrices the algorithm is derived from a matrix decomposition algorithm for inverting dense nonsymmetric matrices several strategies and special data structures are proposed to implement the algorithm efficiently sparsity patterns of the factored inverse are exploited to reduce computational cost the preconditioner possesses greater inherent parallelism than traditional preconditioners based on incomplete lu factorizations numerical experiments are used to show the effectiveness and efficiency of the proposed sparse approximate inverse preconditioner

a boundary type meshless solver for transient heat conduction analysis of slender functionally graded materials with exponential variations abstract this study presents a parallel meshless solver for transient heat conduction analysis of slender functionally graded materials fgms with exponential variations in the present parallel meshless solver a strong form boundary collocation method the boundary knot method bkm in conjunction with laplace transform is implemented to solve the heat conduction equations of slender fgms with exponential variations this method is mathematically simple easy to parallel meshless and without domain discretization however two ill posed issues the ill conditioning dense bkm matrix and numerical inverse laplace transform process may lead to incorrect numerical results here the extended precision arithmetic epa and the domain decomposition method ddm have been adopted to alleviate the effect of these two ill posed issues on numerical efficiency of the present method then the parallel algorithm has been employed to significantly reduce the computational cost and enhance the computational capacity for the fgm structures with larger length width ratio to demonstrate the effectiveness of the present parallel meshless solver for transient heat conduction analysis several benchmark examples are considered under slender fgms with exponential variations the present results are compared with the analytical solutions the conventional boundary knot method and comsol simulation

optimization of boundary conditions for maximum fundamental frequency of vibrating structures a sensitivity formula of eigenvalues with respect to the change of boundary conditions is derived using the material derivative concept based on a variational formulation the change of boundary conditions is described with the introduction of the tangential component of the design velocity field used in shape design simply supported and partially welded plates are taken as numerical examples to check the accuracy of the sensitivity formula the sensitivites of the distinct and multiple eigenvalues calculated by the formulas are compared with those calculated by finite differences optimal support locations are then determined by use of a gradient based optimization method it is shown that a crossing of eigenvalues can occur in the solution process

a meshless average source boundary node method for steady state heat conduction in general anisotropic media abstract the average source boundary node method asbnm is a recent boundary type meshless method which uses only the boundary nodes in the solution procedure without involving any element or integration notion that is truly meshless and easy to implement this paper documents the first attempt to extend the asbnm for solving the steady state heat conduction problems in general anisotropic media noteworthily for boundary type meshless meshfree methods which depend on the boundary integral equations whatever their forms are a key but difficult issue is to accurately and efficiently determine the diagonal coefficients of influence matrices in this study we develop a new scheme to evaluate the diagonal coefficients via the pure boundary node implementation based on coupling a new regularized boundary integral equation with direct unknowns of considered problems and the average source technique ast seven two and three dimensional benchmark examples are tested in comparison with some existing methods numerical results demonstrate that the present asbnm is superior in the light of overall accuracy efficiency stability and convergence rates especially for the solution of the boundary quantities

a two phase preconditioning strategy of sparse approximate inverse for indefinite matrices a two phase preconditioning strategy based on a factored sparse approximate inverse is proposed for solving sparse indefinite matrices in each phase the strategy first makes the original matrix diagonally dominant to enhance the stability by a shifting method and constructs an inverse approximation of the shifted matrix by utilizing a factored sparse approximate inverse preconditioner the two inverse approximation matrices produced from each phase are then combined to be used as a preconditioner experimental results show that the presented strategy improves the accuracy and the stability of the preconditioner on solving indefinite sparse matrices furthermore the strategy ensures that convergence rate of the preconditioned iterations of the two phase preconditioning strategy is much better than that of the standard sparse approximate inverse ones for solving indefinite matrices

a meshless integration free and boundary only rbf technique based on the radial basis function rbf nonsingular general solution and dual reciprocity method drm this paper presents an inherently meshless integration free boundary only rbf collocation technique for numerical solution of various partial differential equation systems the basic ideas behind this methodology are very mathematically simple in this study the rbfs are employed to approximate the inhomogeneous terms via the drm while nonsingular general solution leads to a boundary only rbf formulation for homogenous solution the present scheme is named as the boundary knot method bkm to differentiate it from the other numerical techniques in particular due to the use of nonsingular general solutions rather than singular fundamental solutions the bkm is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions the efficiency and utility of this new technique are validated through a number of typical numerical examples completeness concern of the bkm due to the sole use of the nonsingular part of complete fundamental solution is also discussed

a boundary face method for potential problems in three dimensions this work presents a new implementation of the boundary node method bnm for numerical solution of laplace s equation by coupling the boundary integral equations and the moving least squares mls approximation the bnm is a boundary type meshless method however it still uses the standard elements for boundary integration and approximation of the geometry thus loses the advantages of the meshless methods in our implementation here called the boundary face method the boundary integration is performed on boundary faces which are represented in parametric form exactly as the boundary representation data structure in solid modeling the integrand quantities such as the coordinates of gauss integration points jacobian and out normal are calculated directly from the faces rather than from elements in order to deal with thin structures a mixed variable interpolation scheme of 1 d mls and lagrange polynomial for long and narrow faces an adaptive integration scheme for nearly singular integrals has been developed numerical examples show that our implementation can provide much more accurate results than the bnm and keep reasonable accuracy in some extreme cases such as very irregular distribution of nodes and thin shells copyright 2009 john wiley sons ltd

novel meshless method for solving the potential problems with arbitrary domain in this article a non singular and boundary type meshless method in two dimensions is developed to solve the potential problems the solution is represented by a distribution of the kernel functions of double layer potentials by using the desingularization technique to regularize the singularity and hypersingularity of the kernel functions the source points can be located on the real boundary and therefore the diagonal terms of influence matrices are determined the main difficulty of the coincidence of the source and collocation points then disappears by employing the two point function the off diagonal coefficients of influence matrices are easily obtained the numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the results of exact solution conventional mfs and bem for the dirichlet neumann and mix type boundary conditions bcs of interior and exterior problems with simple and complicated boundaries good agreements with exact solutions are observed

an approximate algorithm for estimating treatment lags from right censored data abstract the problem of likelihood estimate of a function given limited data and a set of constraints finds many practical applications one of them is the lag function estimation for treatment effects in clinical trials however searching for the global optimum in such problems often turns out to be prohibitively complex in this paper we present an algorithm that can be effectively used to approximate such solutions this algorithm modifies standard genetic algorithms which model the natural processes of information inheritance and selective pressure the most important modifications deal with the complex strong constraints imposed on the sought solution we present few experiments with a simplified version of the problem and discuss possible future extensions that relax the restrictions

singular boundary method for solving plane strain elastostatic problems this study documents the first attempt to apply the singular boundary method sbm a novel boundary only collocation method to two dimensional 2d elasticity problems unlike the method of fundamental solutions mfs the source points coincide with the collocation points on the physical boundary by using an inverse interpolation technique to regularize the singularity of the fundamental solution of the equation governing the problems of interest three benchmark elasticity problems are tested to demonstrate the feasibility and accuracy of the proposed method through detailed comparisons with the mfs boundary element method bem and finite element method fem

local radial basis function based differential quadrature method and its application to solve two dimensional incompressible navier stokes equations local radial basis function based differential quadrature method is presented in detail in this paper the method is a natural mesh free approach like the conventional differential quadrature dq method it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots which may be distributed randomly however different from the conventional dq method the weighting coefficients in present method are determined by taking the radial basis functions rbfs instead of high order polynomials as the test functions the method works in a similar fashion as conventional finite difference schemes but with truly mesh free property in this paper we mainly concentrate on the multiquadric rbfs since they have exponential convergence the effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied and how the value of optimal c varies with the number of local support knots is also numerically demonstrated the proposed method is validated by its application to the simulation of natural convection in a square cavity excellent numerical results are obtained on an irregular knot distribution

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

an alternative approach for numerical solutions of the navier stokes equations conventional approaches for solving the navier stokes equations of incompressible fluid dynamics are the primitive variable approach and the vorticity velocity approach in this paper an alternative approach is presented in this approach pressure and one of the velocity components are eliminated from the governing equations the result is one higher order partial differential equation with one unknown for two dimensional problems or two higher order partial differential equations with two unknowns for three dimensional problems a meshless collocation method based on radial basis functions for solving the navier stokes equations using this approach is presented the proposed method is used to solve a two and a three dimensional test problem of which exact solutions are known it is found that with appropriate values of the method parameters solutions of satisfactory accuracy can be obtained copyright 2006 john wiley sons ltd

the minimum support stiffness required to raise the fundamental natural frequency of plate structures the minimum stiffness and optimum position of a flexible point support is calculated that raises the fundamental natural frequency of plate structures for a single support the maximum fundamental natural frequency of the supported structure is equal to the second natural frequency of the unsupported structure the structure is modelled using finite element analysis allowing a wide range of applications and boundary conditions if a support is positioned within a finite element then the shape functions are used to calculate the contribution of the support to the system stiffness and also the slope of the mode at the support efficient methods are used to calculate the minimum support stiffness required numerical results demonstrate that a system with flexible supports may be designed with the same fundamental natural frequency as that of a system with rigid supports examples of plate structures with one or two point supports are analysed and show that the method is efficient in these cases

parallel preconditioning with sparse approximate inverses a parallel preconditioner is presented for the solution of general sparse linear systems of equations a sparse approximate inverse is computed explicitly and then applied as a preconditioner to an iterative method the computation of the preconditioner is inherently parallel and its application only requires a matrix vector product the sparsity pattern of the approximate inverse is not imposed a priori but captured automatically this keeps the amount of work and the number of nonzero entries in the preconditioner to a minimum rigorous bounds on the clustering of the eigenvalues and the singular values are derived for the preconditioned system and the proximity of the approximate to the true inverse is estimated an extensive set of test problems from scientific and industrial applications provides convincing evidence of the effectiveness of this approach

exponential basis functions in space and time a meshless method for 2d time dependent problems in this paper we present a method based on using exponential basis functions ebfs to solve well known two dimensional time dependent engineering problems such as elasto dynamic ones the formulation has much in common with those in three dimensional problems while time is taken as the third axis the solution is first approximated by a summation of ebfs and then completed by satisfying the time dependent boundary conditions as well as the initial conditions through a collocation method this is performed by considering a series of spatial and time dependent boundary points to satisfy the boundary conditions through a mixed collocation method the solution method is presented in a time marching form which is capable of solving variety of problems such as transient heat conduction and wave propagation in solids several problems are solved to demonstrate the capabilities of the method

meshless projection iterative analysis of signorini problems using a boundary element free method this paper presents a meshless method based on coupling a boundary element free method befm with a new projection iterative scheme for signorini problems using the new iterative scheme the nonlinear complementary signorini boundary conditions are transformed into neumann boundary conditions then mixed elliptic problems are formed and are solved by the befm in this method only a nodal data structure on the boundary is required and the coefficient matrices of the discrete system equations only have to be computed once in the whole iterative process convergence of the iterative scheme is discussed theoretically numerical examples involving beach percolation and an electropaint process are given to illustrate the convergence rate and high computational efficiency of the method