preconditioned krylov solvers for bea the performance of a number of preconditioned krylov methods is analysed for a large variety of boundary element formulations low and high order element two dimensional 2 d and three dimensional 3 d regular singular and hypersingular collocation and symmetric galerkin single and multi zone thermal and elastic continuous and discontinuous boundary formulations with and without condensation are considered preconditioned conjugate gradient cg solvers in standard form and a form effectively operating on the normal equations cgn generalized minimal residual gmres conjugate gradient squared cgs and stabilized bi conjugate gradient bi cgstab krylov solvers are employed in this study both the primitive and preconditioned matrix operators are depicted graphically to illustrate the relative amenability of the alternative formulations to solution via kryiov methods and to contrast and explain their computational performances a notable difference between 2 d and 3 d bea operators is readily visualized in this manner numerical examples are presented and the relative conditioning of the various discrete bea operators is reflected in the performance of the krylov equation solvers a preconditioning scheme which was found to be uncompetitive in the collocation bea context is shown to make iterative solution of symmetric galerkin bea problems more economical than employing direct solution techniques we conclude that the preconditioned krylov techniques are competitive with or superior to direct methods in a wide range of boundary formulated problems and that their performance can be partially correlated with certain problem characteristics

a lanczos type method for multiple starting vectors given a square matrix and single right and left starting vectors the classical nonsymmetric lanczos process generates two sequences of bior thogonal basis vectors for the right and left krylov subspaces induced by the given matrix and vectors in this paper we propose a lanczos type algorithm that extends the classical lanczos process for single starting vectors to mul tiple starting vectors given a square matrix and two blocks of right and left starting vectors the algorithm generates two sequences of biorthogonal basis vectors for the right and left block krylov subspaces induced by the given data the algorithm can handle the most general case of right and left start ing blocks of arbitrary sizes while all previously proposed extensions of the lanczos process are restricted to right and left starting blocks of identical sizes other features of our algorithm include a built in deation procedure to detect and delete linearly dependent vectors in the block krylov sequences and the option to employ look ahead to remedy the potential breakdowns that may occur in nonsymmetric lanczos type methods

the method of fundamental solutions for the numerical solution of the biharmonic equation abstract the method of fundamental solutions mfs is a relatively new technique for the numerical solution of certain elliptic boundary value problems it falls in the class of methods generally called boundary methods and like the well known boundary integral equation method is applicable when a fundamental solution of the differential equation is known in the mfs the approximate solution is a linear combination of fundamental solutions with singularities placed outside the domain of the problem the locations of the singularities are either preassigned or determined along with the coefficients of the fundamental solutions so that the approximate solution satisfies the boundary conditions as well as possible in many applications these quantities are determined by a least squares fit of the boundary conditions a nonlinear problem which is solved using standard software in this paper the mfs is formulated for biharmonic problems and is applied to a variety of standard test problems as well as to problems arising in elasticity and fluid flow

the collocation points of the fundamental solution method for the potential problem we propose an algorithm for computing harmonic functions in a two dimensional domain with prescribed dirichlet data our algorithm is a variant of what is called a fundamental solution method 1 3 this method requires us to select 2n points in the two dimensional plane n of which are called collocation points and the remaining n are called charge points representing the position of the singularities of the fundamental solution it is known that there exists a set of 2n points by which the error is exponentially small 1 3 6 however these papers are concerned mainly with existence and as far as the authors know few fast and reliable algorithms are known for good position of the points in this paper we propose a new rule for the position of the points and examine its efficiency by numerical experiments the new rule uses fft effectively

lanczos algorithm for the quadratic eigenvalue problem in engineering applications abstract a new approach of employing the lanczos two sided recursion to solve the quadratic eigenvalue problem is presented the methodology employed retains the n order quadratic problems as posed without the need to use the method of matrix augmentation traditionally used to cast the problem as a linear eigenvalue problem of order 2n pursuing the concept of finding a characteristic matrix to yield an eigenvalue square matrix the lanczos recursion has been devised to solve the quadratic eigenvalue problem appropriate proofs showing the lanczos biorthogonal transformation are included the example problems presented validate the algorithm and show the effectiveness of the method for large quadratic eigenproblems

sinc galerkin method for solving biharmonic problems there are many techniques available to numerically solve the biharmonic equation in this paper we show that the sinc galerkin method is a very effective tool in numerically solving this equation hermite interpolation is used to treat the nonhomogeneous boundary conditions our method is tested on examples and comparisons with other methods are made it is shown that the sinc galerkin method yields good results even when singularities occur at the boundaries

analytical study and numerical experiments for degenerate scale problems in the boundary element method for two dimensional elasticity for a plane elasticity problem the boundary integral equation approach has been shown to yield a non unique solution when geometry size is equal to a degenerate scale in this paper the degenerate scale problem in the boundary element method bem is analytically studied using the method of stress function for the elliptic domain problem the numerical difficulty of the degenerate scale can be solved by using the hypersingular formulation instead of using the singular formulation in the dual bem a simple example is shown to demonstrate the failure using the singular integral equations of dual bem it is found that the degenerate scale also depends on the poisson s ratio by employing the hypersingular formulation in the dual bem no degenerate scale occurs since a zero eigenvalue is not imbedded in the influence matrix for any case

solution of bem systems of equations via iterative techniques the present paper is concerned with the application of iterative techniques to solve boundary element method bem systems of equations initially a brief description of some algorithms which have been employed for symmetric positive definite matrices is given finite element method matrices for instance subsequently non symmetric versions of these algorithms are described and their performance is checked for bem matrices through various numerical experiments preconditioned algorithms were found to work quite well

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

continuum design sensitivity of transient responses using ritz and mode acceleration methods in this paper a unified continuum based sizing design sensitivity analysis method is developed for the transient dynamic response of built up structures using the direct differentiation method the variational equation of the built up structure is differentiate d with respect to the design variable to obtain a variational design sensitivity equation of the transient response the same finite element analysis model that is used to obtain an approximate solution of the variational equation of the built up structure is also used to obtain an approximate solution of the variational design sensitivity equation for large size built up structures the same superposition method is used both to reduce the dimension of the matrix equation of the built up structure and also to reduce the dimension of the design sensitivity equation the continuum based design sensitivity analysis method can be implemented outside established finite element analysis codes using postprocessing data since it does not require derivatives of the stiffness damping and mass matrices moreover the method is efficient since it does not require derivatives of basis vectors either the ritz or mode acceleration method is used to further improve accuracy and efficiency of both analysis and sensitivity results an example with damping shows that accurate design sensitivity of pointwise transient displacements is obtained using only 10 of basis vectors this example indicates that in the continuum based design sensitivity analysis method the same basis vectors that are used for analysis of the built up structure are suitable for analysis of the design sensitivity equation

coarse division transform based preconditioner for boundary element problems this paper is devoted to the development of efficient preconditioners for an iterative solution of equation sets arising in the boundary element method bem a standard collocation system of equations is transformed to a new basis associated with an auxiliary coarse division model for boundary unknowns and solved in this basis new systems have rapidly decreasing coefficients and by neglecting a large number of them it is possible to construct readily invertible sparse preconditioners for iterative procedures the specific features of the transformed matrix can be attributed to the analytical properties of integral equations although the transformation is based on an auxiliary coarse division model it does not require any additional operations with boundary elements all manipulations necessary to construct the mapping are performed on the level of algebraic equations numerical experiments included in the paper confirm a high rate of convergence of the developed iterative scheme

effective condition number for finite difference method for solving the linear algebraic equations ax b with the symmetric and positive definite matrix a from elliptic equations the traditional condition number in the 2 norm is defined by cond l n where l and n are the maximal and minimal eigenvalues of the matrix a respectively the condition number is used to provide the bounds of the relative errors from the perturbation of both a and b such a cond can only be reached by the worst situation of all rounding errors and all b for the given b the true relative errors may be smaller or even much smaller than the cond which is called the effective condition number in chan and foulser effectively well conditioned linear systems siam j sci statist comput 9 1988 963 969 and christiansen and hansen the effective condition number applied to error analysis of certain boundary collocation methods j comput appl math 54 1 1994 15 36 in this paper we propose the new computational formulas for effective condition number cond eff and define the new simplified effective condition number cond e for the latter we only need the eigenvector corresponding to the minimal eigenvalue of a which can be easily obtained by the inverse power method in this paper we also apply the effective condition number for the finite difference method for poisson s equation the difference grids are not supposed to be quasiuniform under a non orthogonality assumption the effective condition number is proven to be o 1 for the homogeneous boundary conditions such a result is extraordinary compared with the traditional cond o hmin 2 where hmin is the minimal meshspacing of the difference grids used for the non homogeneous neumann and dirichlet boundary conditions the effective condition number is proven to be o h 1 2 and o h 1 2hmin 1 respectively where h is the maximal meshspacing of the difference grids numerical experiments are carried out to verify the analysis made

computational enhancement of an unsymmetric block lanczos algorithm an unsymmetric block lanczos algorithm has been employed for the dynamic analysis of a large system which has arbitrary damping and or repeated or closely spaced eigenvalues in the algorithm development the right and left lanczos vectors are all theoretically biorthogonal to each other however these vectors may lose the biorthogonality owing to cancellation and roundoff errors for the unsymmetric case there can be a breakdown even without numerical errors this paper describes computational techniques which have led to a robust unsymmetric block lanczos algorithm

structural dynamics analysis using an unsymmetric block lanczos algorithm this paper describes the way in which an unsymmetric block lanczos algorithm can be employed for approximate eigensolutions or dynamic response solutions for large systems having arbitrary damping and or repeated or closely spaced eigenvalues to reduce a square unsymmetric matrix which may have repeated eigenvalues to block tridiagonal form an unsymmetric block lanczos algorithm is developed right and left lanczos vectors generated by the reduction process are employed to define the reduced order model based on a discrete analytical model of general linear time invariant dynamic systems a special type of starting vector is also introduced which automatically includes the static displacement this reduced order model is employed to compute approximate eigenvalues vectors and dynamic responses of the original system to verify the proposed algorithm examples of an 8 dof beam rotor system are provided for eigensolutions and dynamic responses due to step and random external forces these examples showed promising results indicating that the proposed algorithm is worthy of further study

the trefftz method using fundamental solutions for biharmonic equations in this paper the trefftz method of fundamental solution fs called the method of fundamental solution mfs is used for biharmonic equations the bounds of errors are derived for the mfs with almansi s fundamental solutions denoted as the mafs in bounded simply connected domains the exponential and polynomial convergence rates are obtained from highly and finitely smooth solutions respectively the stability analysis of the mafs is also made for circular domains numerical experiments are carried out for both smooth and singularity problems the numerical results coincide with the theoretical analysis made when the particular solutions satisfying the biharmonic equation can be found the method of particular solutions mps is always superior to the mfs and the mafs based on numerical examples however if such singular particular solutions near the singular points do not exist the local refinement of collocation nodes and the greedy adaptive techniques can be used for seeking better source points based on the computed results the mfs using the greedy adaptive techniques may provide more accurate solutions for singularity problems moreover the numerical solutions by the mafs with almansi s fs are slightly better in accuracy and stability than those by the traditional mfs hence the mafs with the afs is recommended for biharmonic equations due to its simplicity

analysis on the method of fundamental solutions for biharmonic equations abstract in this paper the error and stability analysis of the method of fundamental solution mfs is explored for biharmonic equations the bounds of errors are derived for the fundamental solutions r 2 ln r in bounded simply connected domains and the polynomial convergence rates are obtained for certain smooth solutions the bounds of condition number are also derived to show the exponential growth rates for disk domains numerical experiments are carried out to support the above analysis which is the first time to provide the rigorous analysis of the mfs using r 2 ln r for biharmonic equations

a subspace iteration algorithm for the eigensolution of large structures subject to non classical viscous damping reference lmaf article 2000 002view record in web of science record created on 2005 09 14 modified on 2017 05 10

on the equivalence of the trefftz method and method of fundamental solutions for laplace and biharmonic equations in this paper it is proved that the two approaches known in the literature as the method of fundamental solutions mfs and the trefftz method are mathematically equivalent in spite of their essentially minor and apparent differences in formulation in deriving the equivalence of the trefftz method and the mfs for the laplace and biharmonic problems it is interesting to find that the complete set in the trefftz method for the laplace and biharmonic problems are embedded in the degenerate kernels of the mfs the degenerate scale appears using the mfs when the geometrical matrix is singular the occurring mechanism of the degenerate scale in the mfs is also studied by using circulant the comparison of accuracy and efficiency of the two methods was addressed

the importance of diagonal dominance in the iterative solution of equations generated from the boundary element method the unsymmetric matrix equations generated from the boundary element method bem can be solved iteratively with convergence to the correct solution guaranteed if the boundary element system of equations can be first transformed into an equivalent diagonally dominant system a transformation is presented which selectively annihilates terms in the coefficient matrix of the system ax b until an equivalent row diagonally dominant system if available is obtained the new row diagonally dominant system is well suited for use with jacobi and gauss seidel point iterative equation solvers the diagonal dominizing transformation presented in this work is not suitable for large systems of equations but is useful as a research tool for studying the importance of diagonal dominance in the iterative solution of equations generated from the bem a simple laplacian problem is used to examine the structure of the bem equations and to introduce the diagonal dominizing transformation the importance of diagonal dominance is shown by comparing iterative convergence of positive definite symmetric positive definite and diagonally dominant systems of bem equations obtained from a plane strain elasticity problem

second kind integral equation formulation of stokes flows past a particle of arbitary shape the problem of determining the slow viscous flow of an unbounded fluid past a single solid particle is formulated exactly as a system of linear fredholm integral equations of the second kind for a

fundamental solutions method for elliptic boundary value problems we consider a procedure for solving boundary value problems for elliptic homogeneous equations known as the fundamental solutions method we prove its applicability for some second order operators as well as for fourth order ones the boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest in a specific case of the laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions numerical examples are given among which are simple signorini s problems for harmonic and lame operators

stokes flow around cylinders in a bounded two dimensional domain using multipole accelerated boundary element methods the multipole technique has recently received attention in the field of boundary element analysis as a means of reducing the order of data storage and calculation time requirements from o n2 iterative solvers or o n3 gaussian elimination to o n log n or o n where n is the number of nodes in the discretized system such a reduction in the growth of the calculation time and data storage is crucial in applications where n is large such as when modelling the macroscopic behaviour of suspensions of particles in such cases a minimum of 1000 particles is needed to obtain statistically meaningful results leading to systems with n of the order of 10 000 for the smallest problems when only boundary velocities are known the indirect boundary element formulation for stokes flow results in fredholm equations of the second kind which generally produce a well posed set of equations when discretized a necessary requirement for iterative solution methods the direct boundary element formulation on the other hand results in fredholm equations of the first kind which upon discretization produce ill conditioned systems of equations the model system here is a two dimensional wide gap couette viscometer where particles are suspended in the fluid between the cylinders this is a typical system that is efficiently modelled using boundary element method simulations the multipolar technique is applied to both direct and indirect formulations it is found that the indirect approach is sufficiently well conditioned to allow the use of fast multipole methods the direct approach results in severe ill conditioning to a point where application of the multipole method leads to non convergence of the solution iteration copyright 1999 john wiley sons ltd

the method of fundamental solutions for annular shaped domains this paper consists of two parts in the first part we combine the analysis of bogomolny a bogomolny fundamental solutions method for elliptic boundary value problems siam journal on numerical analysis 22 1985 644 669 comodi and mathon m i comodi r mathon a boundary approximation method for fourth order problems mathematical models and methods in applied sciences 1 1991 437 445 and li et al z c li r mathon p sermer boundary methods for solving elliptic equations with singularities siam journal on numerical analysis 24 1987 487 498 for the method of fundamental solutions mfs and new error bounds are derived for the bounded simply connected in a readable approach a factor of o n is removed in the error bounds of bogomolny 1985 in the second part we extend the analysis for the annular domain s a by the mfs the other fundamental solutions are needed whose source nodes may also be located uniformly on a circle inside the domain s a as in the above reference error bounds are derived in detail to display the polynomial convergence rates

an iterative svd krylov based method for model reduction of large scale dynamical systems abstract in this paper we propose a model reduction algorithm for approximation of large scale linear time invariant dynamical systems the method is a two sided projection combining features of the singular value decomposition svd based and the krylov based model reduction techniques while the svd side of the projection depends on the observability gramian the krylov side is obtained via iterative rational krylov steps the reduced model is asymptotically stable matches certain moments and solves a restricted h 2 minimization problem we present modifications to the proposed approach for employing low rank gramians in the reduction step and also for reducing discrete time systems several numerical examples from various disciplines verify the effectiveness of the proposed approach it performs significantly better than the q cover a yousouff r e skelton covariance equivalent realizations with applications to model reduction of large scale systems in c t leondes ed control and dynamic systems vol 22 academic press 1985 pp 273 348 a yousouff d a wagie r e skelton linear system approximation via covariance equivalent realizations journal of math anal and appl 196 1985 91 115 and the least squares s gugercin a c antoulas model reduction of large scale systems by least squares linear algebra appl 415 2 3 2006 290 321 methods that have a similar projection structure to the proposed method also in terms of both the h 2 and h error measures its performance is comparable to or sometimes better than that of balanced truncation moreover the method proves to be robust with respect to the perturbations due to usage of approximate gramians

a parallel multipolar indirect boundary element method for the neumann interior stokes flow problem a multipolar expansion technique is applied to the indirect formulation of the boundary element method in order to solve the two dimensional internal stokes flow second kind boundary value problems the algorithm is based on a multipolar expansion for the far field and numerical evaluation for the near field due to the nature of the algorithm it is necessary to resort to the use of an iterative solver for the resulting algebraic linear system of equations a parallel implementation is designed to take advantage of the natural domain decomposition of fast multipolar techniques and bring further improvements copyright 2000 john wiley sons ltd

boundary value problems for a compressible stokes system in bounded domains in rn some boundary value problems of dirichlet and neumann types ssociated with a compressible stokes system are studied from the point of view of the theory of hydrodynamic potentials existence and uniqueness results as well as boundary integral representations of classical solutions are given in the case of bounded domains in r n having compact but not connected boundaries of class c 1 a 0 email protected 1

the lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems many powerful methods for solving systems of equations can be regarded as projection methods most of the projection methods known for solving linear systems are orthogonal projection methods but little attention has been given to the class of nonorthogonal or oblique projection methods which is particularly attractive for large nonsymmetric systems the purpose of the paper is to present some methods in the general setting of oblique projection methods and to give some theoretical results some experiments comparing the various algorithms are reported

the almansi method of fundamental solutions for solving biharmonic problems a new fundamental solutions method for the numerical solution of two dimensional biharmonic problems is described in this method which is based on the almansi representation of a biharmonic function in the plane the approximate solution is expressed in terms of fundamental solutions of laplace s equation and is determined by a least squares fit of the boundary conditions the results of numerical experiments which demonstrate the efficacy of the method are presented

block krylov subspace methods for approximating the linear combination of ź functions arising in exponential integrators in this paper we present numerical methods for computing the matrix functions arising in exponential integrators the matrix functions are the linear combination of the form j 0 p j a b j where j is related to exponential function by reducing the corresponding systems of linear odes with polynomial inhomogeneity based on block krylov subspace methods two effective exponentials of low dimension matrix are derived to approximate the objective functions some useful a posteriori error estimates are established the algorithms can be combined with time stepping strategy to satisfy the accuracy requirements under a moderate dimension of the block krylov subspace several numerical experiments are presented to demonstrate the efficiency of the methods

error analysis of the method of fundamental solutions for linear elastostatics for linear elastostatics in 2d the trefftz methods i e the boundary methods using the particular solutions and the fundamental solutions satisfying the cauchy navier equation lead to the method of particular solutions mps and the method of fundamental solutions mfs respectively in this paper the mixed types of the displacement and the traction boundary conditions are dealt with and both the direct collocation techniques and the lagrange multiplier are used to couple the boundary conditions the former is just the mfs and the mps and the latter is also called the hybrid trefftz method htm in jirousek 1978 1992 1996 1 3 in bogomolny 1985 4 and li 2009 5 the error analysis of the mfs is given for laplace s equation and in li 2012 6 the error bounds of both mps and htm using particular solutions ps are provided for linear elastostatics in this paper our efforts are devoted to explore the error analysis of the mfs and the htm using fundamental solutions fs the key analysis is to derive the errors between fs and ps of the linear elastostatics where the expansions of the fs in li et al 2011 7 are a basic tool in analysis then the optimal convergence rates can be achieved for the mfs and the htm using fs recently the mfs has been developed with numerous reports in computation the analysis is behind the analysis of the mfs for linear elastostatics in this paper may narrow the existing gap between computation and theory of the mfs

the lanczos algorithm applied to unsymmetric generalized eigenvalue problem application of the two sided lanczos recursion to the unsymmetric generalized eigenvalue problem is presented the system matrices are real and unsymmetric therefore the recursions are performed in real arithmetic and complex arithmetic is employed in the qr algorithm used to extract the eigenvalues of the transformed tridiagonal matrix the biorthonormal transformation of the unsymmetric generalized eigenvalue problem is considered in detail with appropriate proofs presented in appendices issues relating to the computer implementation of the unsymmetric generalized eigenvalue problem are discussed the example problems solved demonstrate the working of the algorithm in extracting the complex and or real eignevalues of an unsymmetric system of matrices also the algorithm is applied to extract a few of the eigenvalues of a large fluid structure interaction problem and the results are compared with the eigenfrequencies extracted by an unsymmetric subspace iteration procedure presented in the literature

optimization of large structures subjected to dynamic loads with the multiplier method the multiplier method for optimization of large scale mechanical and structural systems subjected to dynamic loads is investigated a large scale dynamic response optimization problem is formulated and solved in several alternate ways using the first and second order forms of the equations of motion results are compared with those obtained with the sequential quadratic programming algorithm a primal method in all the cases investigated the multiplier algorithm is more efficient than the primal method therefore it is concluded that the multiplier method is more appropriate for dynamic response optimization of large scale problems

solving inverse stokes problems by modified collocation trefftz method in this paper the two dimensional inverse stokes problems governed by bi harmonic equations are stably solved by the modified collocation trefftz method mctm in some practical applications of the stokes problems part of the boundary conditions cannot be measured in advance so the mathematical descriptions of such problems are known as the inverse stokes problems when numerical simulation is adopted for solutions of the inverse stokes problems the solutions will become extremely unstable which means that small perturbations in the boundary conditions will result in large errors of the final results hence we adopted the mctm for stably and efficiently analyzing the inverse stokes problems the mctm is one kind of boundary type meshless methods so the mesh generation and the numerical quadrature can be avoided besides the numerical solution is expressed as a linear combination of t complete functions modified by a characteristic length by enforcing the satisfactions of the boundary conditions at every boundary node a system of linear algebraic equations will be yielded the unknown coefficients in the solution expression can be acquired by directly inverting the coefficient matrix the numerical solutions and their derivatives can be easily obtained by linear summation three numerical examples are provided to demonstrate the accuracy and the stability of the proposed meshless scheme for solving the two dimensional inverse stokes problems

a boundary element solution for two dimensional viscous sintering by viscous sintering is meant the process in which a granular compact is heated to a temperature at which the viscosity of the material under consideration becomes low enough for surface tension to cause the powder particles to deform and coalesce for the sake of simplicity this process is modeled in a two dimensional space the governing stokes equations describe the deformation of a two dimensional viscous liquid region under the influence of the curvature of the outer boundary however some extra conditions are needed to ensure that these equations can be solved uniquely a boundary element method is applied to solve the equations for an arbitrarily initial shaped fluid region the numerical problems that can arise in computing the curvature in particular when this is varying rapidly are discussed a number of numerical examples are shown for simply connected regions which transform themselves into circles as time increases

an efficient computational method for solving fractional biharmonic equation in this paper we first introduce the fractional biharmonic equation and an orthogonal system of basis functions for the space of continuous functions on the interval 0 l generated by the shifted chebyshev polynomials moreover we propose a computational method based on the operational matrix of fractional derivatives of these basis functions for solving the fractional biharmonic equation the main characteristic behind this approach is that it reduces the problem under consideration to solving a system of algebraic equations which greatly simplifies the problem convergence of the shifted chebyshev polynomials expansion in two dimensions is investigated also the power of this manageable method is illustrated

iterative methods in solving navier stokes equations by the boundary element method the solution of navier stokes equations of time dependent incompressible viscous fluid flow in planar geometry by the boundary domain integral method bdim is discussed the introduction of a subdomain technique to fluid flow problems is considered and improved in order to maintain the stability of bdim to avoid problems with flow kinematics computation in the sudomain mesh a segmentation technique is proposed which combines the original bdim with its subdomain variant and preserves its numerical stability in order to reduce the computational cost of bdim which greatly depends on the solution of systems of linear equations iterative methods are used conjugate gradient methods conjugate gradients squared and an improved version of the biconjugate gradient method bicgstab together with the generalized minimal residual method are used as iterative solvers different types of preconditioning from simple jacobi to incomplete lu factorization are carried out and the performance of chosen iterative methods and preconditioners are reported test examples include backward facing step flow and flow through tubular heat exchangers test computation results show that bdim is an accurate approximation technique which together with the subdomain technique and powerful iterative solvers can exhibit some significant savings in storage and cpu time requirements