the generalized singular value decomposition and the method of particular solutions a powerful method for solving planar eigenvalue problems is the method of particular solutions mps which is also well known under the name point matching method the implementation of this method usually depends on the solution of one of three types of linear algebra problems singular value decomposition generalized eigenvalue decomposition or generalized singular value decomposition we compare and give geometric interpretations of these different variants of the mps it turns out that the most stable and accurate of them is based on the generalized singular value decomposition we present results to this effect and demonstrate the behavior of the generalized singular value decomposition in the presence of a highly ill conditioned basis of particular solutions

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 numerical solution of laplace s equation in three dimensions we consider the dirichlet problem for laplace s equation on a simply connected three dimensional region with a smooth boundary this problem is easily converted to the solution of a fredholm integral equation of the second kind based on representing the harmonic solution as a double layer potential function we solve this integral equation formulation by using galerkin s method with spherical harmonics as the basis functions this approach leads to small linear systems and once the galerkin coefficients for the region have been calculated the computation time is small for any particular boundary function the major disadvantage of the method is the calculation of the galerkin coefficients each of which is a four fold integral with a singular integrand theoretical and computational details of the method are presented

stabilized multiscale finite element method for the stationary navier stokes equations abstract in the paper a stabilized multiscale finite element method for the stationary incompressible navier stokes equations is considered the method is a petrov galerkin approach based on the multiscale enrichment of the standard polynomial space enriched with the unusual bubble functions which no longer vanish on every element boundary for the velocity space the stability of the p 1 p 0 triangular element or the q 1 p 0 quadrilateral element is established and the optimal error estimates of the stabilized multiscale finite element method for the stationary navier stokes equations are obtained

integration of thiele continued fractions and the method of fundamental solutions for solving unconfined seepage problems unconfined seepage through an earth dam or a levee is recognized as a challenging problem this complexity is mainly due to the fact that determination of the phreatic line through the dam levee body is not straightforward for simulation purposes mesh generation as well as accurate and smooth alteration of the phreatic line at the junction with the downstream slope referred to as exit point for an unconfined seepage problem with complex geometry generally makes cumbersome numerical solutions this study presents an innovative boundary type meshfree method to determine the phreatic line location in unconfined seepage problems the current study explicitly addresses the problem that alternative methods commonly face to deal with the exit point the method is developed based upon integrating the method of fundamental solutions mfs particle swarm optimization pso algorithm and thiele continued fractions tcf to accurately estimate the phreatic line location the proposed framework uses mfs to solve the flow continuity equation tcf to generate the phreatic line and pso to optimize the phreatic line location generated by tcf as a boundary method mfs only deals with the boundaries of the domain and consequently it only takes the exact position of phreatic line as a variable boundary the proposed approach employs tcf to guarantee that the phreatic line is tangent to the downstream slope at the exit point a characteristics which is important especially for the cases where abrupt changes occur in the phreatic line near the exit point for comparison and validation purposes the phreatic lines determined by the proposed approach for two unconfined seepage problems are compared and verified against those obtained from alternative analytical and numerical methods as well as a set of experimental results an excellent agreement is demonstrated upon comparison of the proposed method to the results attained from the analytical solutions and experimental tests

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

multi quadric quasi interpolation method coupled with fdm for the degasperis procesi equation in this paper the degasperis procesi dp equation which contains nonlinear high order derivatives and discontinuous solutions is recast to its equivalent form u p based on the equivalent formulation of dp equation a new numerical method is presented the multi quadric mq quasi interpolation method coupled with finite difference method is applied to approximate the spatial derivatives and the third order tvd method is used to approximate the time derivative several examples are presented to demonstrate the effectiveness of the proposed method

sequential optimal truss generator for frequency ranges abstract the concept of an ordered set of optimum designs is introduced here and all the design variables and behavioral variables belonging to the optimum designs are represented by successive piecewise taylor series a most natural direct and efficient way of generating or sweeping out all the optimum designs sequentially in the set is devised the procedure is started with the eigenvalue analysis on the truss defined by the set of all the minimum cross sectional areas neither any further eigenvalue analysis nor application of any optimization technique is required in the proposed procedure it is demonstrated that the proposed method is efficient not only for an ordered set of optimum designs of a large truss associated with the single lowest eigenvector but also for that associated with multiple lowest eigenvectors

developability conditions for prestress optimization of a curved surface developability conditions are presented for prestressed curved surfaces to be reduced to plane sheets by relaxing the stresses those conditions are applied to an optimization problem for minimizing the deviation of stresses from the target values two formulations based on the displacements defined by the local and global coordinates of finite elements are presented it is shown that the numbers of constraints by those formulations are same if triangular elements are used and there is an upper bound in the number of the elements in a cutting sheet performances of the two formulations are compared in the numerical examples and the local formulation is shown to lead to more accurate estimation of stresses that are actually generated by connecting and stretching the plane sheets

approximation by solutions of elliptic equations in semilocal spaces in the present work we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain by solutions of the same equations in a larger domain we construct an abstract framework which allows us to deal with such density questions simultaneously for various norms more specifically we study approximations with respect to the norms of semilocal banach spaces of distributions these spaces are required to satisfy certain postulates we establish density results for elliptic operators with constant coefficients which unify and extend previous results in our density results may possess holes and it is required to satisfy the segment condition we observe that analogous density results do not hold in spaces where the infinitely smooth functions are not dense finally we provide applications related to the method of fundamental solutions

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

applicability and applications of the method of fundamental solutions in the present work we investigate the applicability of the method of fundamental solutions for the solution of boundary value problems of elliptic partial differential equations and elliptic systems more specifically we study whether linear combinations of fundamental solutions can approximate the solutions of the boundary value problems under consideration in our study the singularities of the fundamental solutions lie on a prescribed pseudo boundary the boundary of a domain which embraces the domain of the problem under consideration we extend previous density results of kupradze and aleksidze and of bogomolny to more general domains and partial differential operators and with respect to more appropriate norms our domains may possess holes and their boundaries are only required to satisfy a rather weak boundary requirement namely the segment condition our density results are with respect to the norms of the spaces c l analogous density results are obtainable with respect to holder norms we have studied approximation by fundamental solutions of the laplacian biharmonic and m harmonic and modified helmholtz and poly helmholtz operators in the case of elliptic systems we obtain analogous density results for the cauchy navier operator as well as for an operator which arises in the linear theory of thermo elasticity we also study alternative formulations of the method of fundamental solutions in cases when linear combinations of fundamental solutions of the equations under consideration are not dense in the solution space finally we show that linear combinations of fundamental solutions of operators of order m 4 with singularities lying on a prescribed pseudo boundary are not in general dense in the corresponding solution space

convergence of the method of fundamental solutions for solving the boundary value problem of modified helmholtz equation in this paper we derive the rate of convergence for the method of fundamental methods in solving the boundary value problem of a homogeneous modified helmholtz equation by using the fourier sine expansion to find particular solutions we establish the convergence results for solving the boundary value problem of a non homogeneous equation

a meshless method for solving the cauchy problem in three dimensional elastostatics the application of the method of fundamental solutions to the cauchy problem in three dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the first order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for both under and equally determined cauchy problems in a piece wise smooth geometry the convergence accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

a fast fourier galerkin method solving a boundary integral equation for the biharmonic equation we develop a fast fourier galerkin method for solving a boundary integral equation which is a reformulation of the dirichlet problem of the biharmonic equation the proposed method is based on a splitting of the resulting boundary integral operator that is we write the operator as a sum of two integral operators one having the fourier basis functions as eigenfunctions and the other whose representation matrix in the fourier basis can be compressed to a sparse matrix having only mathcal o n log n number of nonzero entries where n is the order of the fourier basis functions used in the method we then project the solution of the boundary integral equation onto the space spanned by the fourier basis functions this leads to a system of linear equations a fast solver for the system is based on a compression of its coefficient matrix we show that the method has the optimal convergence order mathcal o n q where q denotes the degree of regularity of the exact solution of the boundary int

sensitivity of optimum solutions of problem parameters solution of the optimum sensitivity problem yields the values of derivatives of the optimal objective function and design variables with respect to those physical quantities which were kept constant as problem parameters during optimization examples of these sensitivity derivatives might include derivatives of cross section al area and structural mass with respect to allowable stress and derivatives of fuel consumed and wing aspect ratio with respect to aircraft range derivation of the sensitivity equations that yield the sensitivity derivatives directly which avoids the costly and inaccurate perturb and reoptimize approach is discussed and solvability of the equations is examined the equations apply to optimum solutions obtained by direct search methods as well as those generated by procedures of the sequential unconstrained minimization technique sumt class applications are discussed for the use of the sensitivity derivatives in extrapolation of the optimal objective function and design variable values for incremented parameters optimization with multiple objectives and decomposition of large optimization problems several aspects of these applications and verification of the sensitivity equations are presented through numerical examples

stress analysis of flat plates with shear using explicit stiffness matrix an explicit expression for the stiffness matrix is worked out for a triangular plate bending element considering the effect of transverse shear deformation the element has twelve nodes on the sides and four nodes internal to it the formulation is displacement type and the use of area co ordinates makes it possible to obtain the shape functions explicitly separate polynomials are assumed for transverse displacement and rotations to obtain the element stiffness matrix no matrix inversion or numerical integration need be carried out and only a few matrix multiplications of low order are necessary the element which is initially of thirty five degrees of freedom can be reduced to a thirty degrees of freedom one by condensation of the internal nodes an interesting feature of the element developed is that the values of nodal moments computed at a node point considering different elements surrounding the node do not vary significantly thus the nodal moments can be obtained directly at node points also the element does not give rise to any inconvenience like locking even for very thin plates the straightforward approach in formation of the element stiffness will cut down the storage space considerably and will also call for less cpu time thus making the use of the element well suited to low capacity computers a number of plate bending problems have been worked out using the present element for different thickness to side ratios and a comparison has been made with the available results good accuracy has been observed in all cases even for a small number of elements

efficient calculation of optimum design sensitivity two methods for obtaining the sensitivity of an optimized design to changes in problem parameters are presented the first order method uses gradient information to estimate the required sensitivity the second order method uses second derivatives of the design objective and constraint functions to provide a quadratic approximation to the new design which results from changing the specified parameter it is shown that the sensitivity of the optimized design may be discontinuous with respect to the new parameter this discontinuity is accounted for in the present methods these methods are compared to a previous approach based on the kuhn tucker necessary conditions for optimality it is shown that the present methods provide the appropriate sensitivity information in an efficient manner the methods are demonstrated by a structural synthesis example

an exponentially convergent nonpolynomial finite element method for time harmonic scattering from polygons in recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems as with their close cousin the method of particular solutions high efficiency comes from using solutions to the helmholtz equation as basis functions we present and analyze such a method for the scattering of two dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions the solution is obtained by minimizing a least squares functional which we discretize in such a way that a matrix least squares problem is obtained we give computable exponential bounds on the rate of convergence of the least squares functional that are in very good agreement with the observed numerical convergence challenging numerical examples including a nonconvex polygon with several corner singularities and a cavity domain are solved to around 10 digits of accuracy with a few seconds of cpu time the examples are implemented concisely with mpspack a matlab toolbox for wave computations with nonpolynomial basis functions developed by the authors a code example is included

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

diffusion problems using transient discrete source superposition discrete source superposition techniques seek sets of discrete singular source strengths which satisfy the boundary conditions of a particular problem and necessarily the governing differential equation the sum of the fields produced by such sources thus provides an approximation to the required solution in many respects these techniques are a special case of conventional boundary integral equation methods where density distributions of smeared sources are sought in this paper an attempt is made to extend the discrete source superposition technique to transient problems the example addressed is the transient diffusion equation the method developed does seem to have some promise it is found that even coarse spatial discretization can yield quite accurate results and that surprisingly large timesteps can be employed and accuracy maintained further although no direct comparisons are made the absence of spatial integrations indicates that the computational effort per timestep is likely to be less than for conventional bie

multiquadric method for the numerical solution of a biphasic mixture model a computational algorithm based on the multiquadric method has been devised to solve the biphasic mixture model the model includes a set of constitutive equations for the fluid flows through the solid phase a set of momentum equations for stress strain equilibrium and a continuity equation for the solid phase and the fluid phase the numerical method does not require the generation of mesh as in the finite element method and hence gives high flexibility in applying the method to irregular geometry numerical examples are made to compute the solution of the confined compression problem which approximates the nonlinear response of soft hydrated tissues under external loading the numerical results are compared with spilker s penalty finite element method

the method of fundamental solutions for inverse boundary value problems associated with the steady state heat conduction in anisotropic media in this paper the method of fundamental solutions is applied to solve some inverse boundary value problems associated with the steady state heat conduction in an anisotropic medium since the resulting matrix equation is severely ill conditioned a regularized solution is obtained by employing the truncated singular value decomposition while the optimal regularization parameter is chosen according to the l curve criterion numerical results are presented for both two and three dimensional problems as well as exact and noisy data the convergence and stability of the proposed numerical scheme with respect to increasing the number of source points and the distance between the fictitious and physical boundaries and decreasing the amount of noise added into the input data respectively are analysed a sensitivity analysis with respect to the measure of the accessible part of the boundary and the distance between the internal measurement points and the boundary is also performed the numerical results obtained show that the proposed numerical method is accurate convergent stable and computationally efficient and hence it could be considered as a competitive alternative to existing methods for solving inverse problems in anisotropic steady state heat conduction copyright 2005 john wiley sons ltd

source nodes on elliptic pseudo boundaries in the method of fundamental solutions for laplace s equation selection of pseudo boundaries abstract in the method of fundamental solutions mfs source nodes on circles outside the solution domains s has been widely applied in numerical computation in this paper source nodes on an ellipse are proposed for solving laplace s equation using the mfs and a robust error and stability analysis is established bounds on errors and condition numbers are derived for bounded simply connected domains polynomial convergence rates can be achieved but the exponential growth of the condition number is also obtained in previous stability analysis of the mfs circulant matrices have been always employed this is the first time the stability analysis is explored for non circular pseudo boundaries based on new techniques without using circulant matrices the criteria for evaluating numerical techniques are provided and some strategies for choosing pseudo boundaries are suggested

an enhanced cell based smoothed finite element method for the analysis of reissner mindlin plate bending problems involving distorted mesh summary in this work an enhanced cell based smoothed finite element method fem is presented for the reissner mindlin plate bending analysis the smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed fem is reformulated and the relationship between the original approach and the present method in curvature smoothing is established to improve the accuracy of shear strain in a distorted mesh we span the shear strain space over the adjacent element this is performed by employing an edge based smoothing technique through a simple area weighted smoothing procedure on mitc4 assumed shear strain field a three field variational principle is utilized to develop the mixed formulation the resultant element formulation is further reduced to a displacement based formulation via an assumed strain method defined by the edge smoothing technique as the result a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement rotation fields and smoothing operators can be shown to improve the solution accuracy in cell based smoothed fem for reissner mindlin plate bending analysis several numerical examples are presented to demonstrate the accuracy of the proposed formulation copyright 2013 john wiley sons ltd

two projection methods for the solution of signorini problems abstract two iterative methods based on projection and boundary element methods are considered for signorini problems the regularized problem with the projection boundary condition is first deduced from the signorini problem by using the equivalence between the signorini boundary condition and the projection fixed point problem our methods formulate the signorini boundary condition into a sequence of robin boundary conditions in the new boundary condition there is a penalty parameter the convergence speed of the first method is greatly influenced by the value of which is difficult to choose for individual problems to improve the performance of this method we present a self adaptive projection method which adjusts the parameter automatically per iteration based on the iterative data the main result of this work is to provide the convergence of the methods under mild assumptions as the iteration process is given by the potential and its derivative on the boundary of the domain the unknowns of the problem are computed explicitly by using the boundary element method both theoretical results and numerical experiments indicate efficiency of the methods proposed

numerical analysis of thick plates bending behaviour of a rectangular plate is analysed with the help of a refined higher order theory the theory is based on a higher order displacement model and the three dimensional hooke s laws for plate material giving rise to a more realistic quadratic variation of the transverse shearing strains and linear variation of the transverse normal strain through the plate thickness it is shown that the segmentation method for the numerical analysis of plates simply supported on two opposite edges can be considered to be the most competitive method in terms of efficiency economy reliability and accuracy in such applications

a method for solving an inverse biharmonic problem this paper deals with the problem of determining of an unknown coefficient in an inverse boundary value problem using a nonconstant overspecified data it has been shown that the solution to this inverse problem exists and is unique

comparable upper and lower bounds for boundary values of neumann eigenfunctions and tight inclusion of eigenvalues for smooth bounded domains in rn we prove upper and lower l2 bounds on the boundary data of neumann eigenfunctions and we prove quasiorthogonality of this boundary data in a spectral window the bounds are tight in the sense that both are independent of the eigenvalues this is achieved by working with an appropriate norm for boundary functions which includes a spectral weight that is a function of the boundary laplacian this spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery type eigenfunctions these bounds are closely related to wave equation estimates due to tataru using this we bound the distance from an arbitrary helmholtz parameter e 0 to the nearest neumann eigenvalue in terms of boundary normal derivative data of a trial function u solving the helmholtz equation e u 0 this inclusion bound improves over previously known bounds by a factor of e5 6 analogously to a recently improved inclusion bound in the dirichlet case due to the first two authors finally we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of neumann eigenpairs on smooth planar domains we show that the new inclusion bound improves the relative accuracy in a computed neumann eigenvalue around the 42000th from nine to fourteen digits with negligible extra numerical effort

linked interpolation for reissner mindlin plate elements part i a simple quadrilateral in most plate elements using the reissner mindlin assumptions the interpolations used for the lateral displacements w and the rotation involve the independent representation of each variable by its nodal values usually with identical interpolations to ensure a higher order of expansion for displacement w its representation is linked in the present paper with both sets of nodal variables conditions necessary for the use of such expansions are established here and the paper shows the development of a linear quadrilateral element q4bl whose performance and robustness are good although it possesses one singularity if only three degrees of freedom are prescribed in part ii we apply the identical formulation to develop a triangular element t3bl which performs equally well and is fully robust

critical sets in parametric optimization we deal with one parameter families of optimization problems in finite dimensions the constraints are both of equality and inequality type the concept of a generalized critical point g c point is introduced in particular every local minimum kuhn tucker point and point of fritz john type is a g c point under fairly weak even generic conditions we study the set consisting of all g c points due to the parameter the set is pieced together from one dimensional manifolds the points of can be divided into five characteristic types the subset of nondegenerate critical points first type is open and dense in nondegenerate means strict complementarity nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints a nondegenerate critical point is completely characterized by means of four indices the change of these indices along is presented finally the kuhn tucker subset of is studied in more detail in particular in connection with the failure of the mangasarian fromowitz constraint qualification

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

the effective condition number applied to error analysis of certain boundary collocation methods abstract the boundary collocation method bcm is widely used in the engineering community for the numerical solution of linear boundary value problems it is commonly stated that the results computed by this method are affected by large rounding errors this conclusion is based on a simple worst case analysis using the conventional condition number for linear systems in this paper we present a refined error analysis for a class of boundary value problems using the concept of effectively well conditioned systems this analysis leads to much smaller a priori bounds for the rounding error in the computed bcm solution these smaller bounds imply that the numerically computed solution expresses much more accurately than previously supposed properties of the ideal bcm solution computed with infinite precision therefore the intrinsic properties of the bcm can be investigated through numerical computations to a far larger extent than previously supposed thus allowing for a numerical diagnostic investigation of the bcm for example when the bcm is applied to problems with certain geometries it is observed that the approximation does not converge using our new smaller error bounds we conclude that the lack of convergence of the bcm solution for these geometries is due to the bcm itself and not to rounding errors

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

mathematical and numerical studies on meshless methods for exterior unbounded domain problems the method of fundamental solutions mfs is an efficient meshless method for solving boundary value problems in an exterior unbounded domain the numerical solution obtained by the mfs is accurate while the corresponding matrix equation is ill conditioned a modified mfs mmfs with the proper basis functions is proposed by the introduction of the modified trefftz method mtm the concrete expressions of the corresponding condition numbers are given in mathematical forms and the solvability by these methods is mathematically proven thereby the optimal parameter minimizing the condition number is also mathematically given numerical experiments show that the condition numbers of the matrices corresponding to the mtm and the mmfs are reduced and that the numerical solution by the mmfs is more accurate than the one by the conventional method

an integral equations method combined minimum norm solution for 3d elastostatics cauchy problem abstract in this paper we establish new density results for the equilibrium equations based on the denseness result of the elastic potential functions the cauchy problem for the equilibrium equations is investigated for this ill posed problem we construct a regularizing solution using the single layer potential function the well posedness of the regularizing solution as well as the convergence property is rigorously analyzed the advantage of the proposed scheme is that the regularizing solution is of the explicit analytic solution and therefore is easy to be implemented the method combines minimum norm solution with morozov discrepancy principle to solve an inverse problem convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method the numerical convergence accuracy and stability of the method with respect to the discretisation about the integral equations on pseudo boundary and the distance between the pseudo boundary and the real boundary of the solution domain and decreasing the amount of noise added into the input data respectively are also analysed with some examples

free vibration analysis of isotropic rectangular plates using a high order triangular finite element with shear in this paper free vibration analysis of isotropic rectangular plates with different thickness ratios different boundary conditions and different aspect ratios has been investigated using a high order triangular element the first order shear deformation theory fosdt is used to include the effect of transverse shear deformation the element has 18 nodes on the sides and six internal nodes the geometry of the element is expressed by three linear shape functions of area coordinates the formulation is displacement type and the use of area coordinates makes the shape functions for field variables to be expressed explicitly no numerical integration is required to get the element stiffness and consistent mass matrices the element has 51 degrees of freedom which can be reduced to 39 degrees of freedom by guyan reduction mass condensation or eigenvalue economization scheme for the degrees of freedom associated with the internal nodes rotary inertia has been included in the consistent mass matrix numerical examples are presented to show the accuracy and convergence characteristics of the element

analysis of the method of fundamental solutions for the modified helmholtz equation the method of fundamental solutions mfs was first proposed by kupradze in 1963 since then the mfs has been extensively applied for solving various kinds of problems in science and engineering however few theoretical works have been reported in the literature in this paper we devote our work to the error analysis and stability of the mfs for the case of the modified helmholtz equation for disk domains a convergence analysis of the mfs was provided by li j comput appl math 159 113125 2004 for solving the modified helmholtz equation in this paper the error bounds of the mfs for bounded and simply connected domains are derived for smooth solutions of the modified helmholtz equation the exponential convergence rates can be achieved for analytic solutions the bounds of condition numbers of the mfs are derived for both disk domains and the bounded and simply connected domains to give the exponential growth via the number of fundamental solutions used numerical experiments are carried out to support the theoretical analysis moreover the analysis of this paper is applied to parabolic equations and some reviews of proof techniques and the analytical characteristics of the mfs are addressed

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

multilevel compact radial functions based computational schemes for some elliptic problems abstract compactly supported radial basis functions cs rbfs have been recently introduced in the context of the dual reciprocity method as a possible cure of dense matrices and ill conditioning problems when using the classical radial basis functions however the support scaling factor and slow convergence rate of the cs rbfs have also raised issues on the effectiveness of the cs rbfs in this paper two multilevel schemes have been proposed to alleviate these problems

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

on solving boundary value problems of modified helmholtz equations by plane wave functions plane wave functions e x w in r2 where 0 x x y w cos sin and x w x cos y sin are used as basis functions to solve boundary value problems of modified helmholtz equations u x 2u x 0 x u x h x x where is the laplace operator and a bounded and simply connected domain in r2 approximations of the exact solution of the above problem by plane wave functions are explicitly constructed for the case that is a disc and the order of approximations is derived a computational algorithm by collocation methods based on a simple singular decomposition of circular matrices is proposed and numerical examples are shown to demonstrate the efficiency of the methods

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

an augmented rbf method for solving fractional sturm liouville eigenvalue problems in this work we construct numerical schemes using radial basis functions rbfs to solve sturm liouville eigenvalue problems in which the so called fractional derivatives are involved we take into account the fact that the solutions to such problems might be nonsmooth and consequently we augment the space generated by rbf base functions with some fractional polynomials to overcome this problem we show through some examples the effectiveness of our method

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

a meshless method for some inverse problems associated with the helmholtz equation in this paper a new numerical scheme based on the method of fundamental solutions is proposed for the numerical solution of some inverse boundary value problems associated with the helmholtz equation including the cauchy problem since the resulting matrix equation is badly ill conditioned a regularized solution is obtained by employing truncated singular value decomposition while the regularization parameter for the regularization method is provided by the l curve method numerical results are presented for problems on smooth and piecewise smooth domains with both exact and noisy data and the convergence and stability of the scheme are investigated these results show that the proposed scheme is highly accurate computationally efficient stable with respect to the noise in the data and convergent with respect to decreasing the amount of data noise and increasing the distance between the physical and fictitious boundaries and could be considered as a competitive alternative to existing methods for these problems

a direct application of higher order parametric programming techniques to structural optimization computational methods based on a sequence of parametric programming problems are presented for solving constrained optimization problems cop without any parameter an auxiliary parametric programming problem appp is formulated in order to solve cop the procedure is started with an arbitrary initial solution which is the trivial solution of appp corresponding to the initial value of the parameter then the optimal solution for the final value of the parameter which is the optimal solution of cop is estimated by taylor s expansion with respect to the parameter where higher order terms are incorporated it is shown that the incorporation of the higher order terms indeed leads to a faster convergence of the solution as an extension of the method a general algorithm is presented for optimum design problems with state variable constraints which are implicit functions of the design variables logarithmic penalty functions are incorporated and the weight coefficients for the penalty terms are updated continuously the derivatives of the state variables with respect to the parameter and their sensitivity coefficients are expressed explicitly in terms of those of the design variables finally a method of simultaneous analysis and optimization is developed for trusses with geometrical non linearity

study on connections of the mfs trefftz method indirect biem and invariant mfs in the three dimensional laplace problems containing spherical boundaries following the success of a study on the method of fundamental solutions using an image concept 13 we extend to solve the three dimensional laplace problems containing spherical boundaries by using the three approaches the case of eccentric sphere for the laplace problem is considered the optimal locations for the source distribution to include the foci in the mfs are also examined by using the image concept in the 3d problems whether a free constant is required or not in the mfs is also studied the error distribution is discussed after comparing with the analytical solution derived by using the bispherical coordinates besides the relationship between the trefftz bases and the singularity in the mfs for the three dimensional laplace problems is also addressed it is found that one source of the mfs contains several interior and exterior trefftz sets through a degenerate kernel on the contrary one single trefftz base can be superimposed by some lumped sources in the mfs through an indirect biem based on this finding the relationship between the fictitious boundary densities of the indirect biem and the singularity strength in the mfs can be constructed due to the fact that the mfs is a lumped version of an indirect biem

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

source field superposition analysis of two dimensional potential problems this method for problems governed by laplace s equation involves representing the potential field in a given solution domain as a superposition of fields generated by a number of sousces

the method of fundamental solutions for solving options pricing models abstract this paper provides a foundation of the method of fundamental solutions mfs for the options pricing models governed by the black scholes equation in which both the european option and american options are considered in the solution procedure no artificial boundary conditions are imposed for both datum and infinite sides of the stock prices in the cases of the european options no time marching procedures are required and numerical results are verified with the exact solutions since the free boundary conditions are considered for the american options boundary update procedure is thus applied at the same time numerical results are compared with the results in the literatures these numerical results indicate the mfs is an effective and robust meshless numerical solution for solving the options pricing models

bi expansion approximation of the bessel functions applied to analytical integration based bem for reissner plates the paper deals with the boundary element analysis of plates modeled by the reissner theory the integral equations are obtained in standard fashion by boundary collocation the fundamental solution is expressed in terms of modified bessel functions using different series expansions for small and large arguments the transition from one to the other form is regulated by an error based criterion for selecting the number of terms which have to be used in the series expansion attention is paid to the computational aspects influencing the accuracy of the numerical model the boundary variables are described by using a quadratic b spline interpolation and the evaluation of the boundary integrals is developed almost entirely in analytical form some numerical experiments test and compare the performances of the numerical model in particular the solution within the boundary layer is studied

the method of fundamental solutions applied to boundary value problems on the surface of a sphere abstract in this work we propose using the method of fundamental solutions mfs to solve boundary value problems for the helmholtz beltrami equation on a sphere we prove density and convergence results that justify the proposed mfs approximation several numerical examples are considered to illustrate the good performance of the method

the mfs for the solution of harmonic boundary value problems with non harmonic boundary conditions we investigate applications of the method of fundamental solutions mfs for the numerical solution of two dimensional boundary value problems in complex geometries governed by the laplace equation and subject to dirichlet boundary conditions which are not harmonic such problems can be very challenging because of the appearance of boundary singularities we consider several ways of choosing the boundary collocation points as well as the source points in the mfs we show that with an appropriate such choice the mfs yields highly accurate results

shape stress trade off design of membrane structures for specified sequence of boundary shapes a new method is presented for parametrically generating optimal shapes and stresses of a membrane structure supported by a frame the problem considered here is a so called trade off design problem where the weighted sum of the deviations of the initial stress and the shape from the desired distribution is to be minimized the membrane is discretized by using the finite element method and optimal shapes are expanded with respect to a parameter defining the shape of the boundary frame an approximate method is presented without exact calculation of the hessian of the stresses with respect to the nodal coordinates the explicit developability conditions derived by the authors are used to guarantee the curved surface to be reduced to a set of plane cutting patterns after removing the pretensioning force in the examples trade off designs are found for an hp type membrane structure and the efficiency of the proposed method is discussed

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

a triangular element based on reissner mindlin plate theory a new triangular plate blending element based on the reissner mindlin theory is developed through a mixed formulation emanating from the hu washizu variational principle a main feature of the formulation is the use of a linear transverse shear interpolation scheme with discrete constraint conditions on the edges

numerical analysis of acoustic modes using the linear least squares method of fundamental solutions the method of fundamental solutions mfs has been proved to be an accurate and efficient meshless numerical method to solve acoustic eigenproblems traditionally the technique of the singular value decomposition svd is employed to obtain the corresponding contours of acoustical modes after the eigenvalues are solved however it is found that the mode shapes are sensitive to the source locations of the mfs in this paper we try to derive a robust meshless numerical scheme to obtain the contours of acoustical modes based on the linear least squares method of fundamental solutions lsmfs by specifying an additional normalized dual boundary condition the failure for determining the mode shapes by specifying a normalized data at boundary locations near or on the nodes are examined moreover it is demonstrated that the mode shapes of degenerate eigenmodes can be distinguished by specifying the boundary data at different boundary points furthermore a normalization procedure is introduced for degenerate eigenmodes three numerical experiments with regular and irregular boundaries are carried out to validate the proposed method mode shapes obtained by the linear lsmfs are in good agreement with the analytical solutions and also the results obtained by the finite element method in addition the robustness and accuracy of the eigenvalues obtained with respect to different locations of source points by the linear lsmfs in conjunction with direct determinant search method are also revisited

sensitivity analysis of discrete structural systems methods for calculating sensitivity derivatives for discrete structural systems are surveyed primarily covering literature published during the past two decades methods are described for calculating derivatives of static displacements and stresses eigenvalues and eigenvectors transient structural response and derivatives of optimum structural designs with respect to problem parameters the survey is focused on publications developed in nonstructural fields such as electronics controls and physical chemistry which are directly applicable to structural problems most notable among the nonstructural based methods are the adjoint variable technique from control theory and the green s function and fast methods from physical chemistry

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 method of fundamental solutions for helmholtz type equations in composite materials in this paper we introduce and develop the method of fundamental solutions mfs for solving helmholtz type elliptic partial differential equations in composite materials this study builds upon the previous developments and applications of the mfs to linear and nonlinear heat conduction elasticity and functionally graded composite layered materials numerical results are presented and discussed for four examples involving both the modified helmholtz and the helmholtz equations in two dimensional or three dimensional bounded or unbounded smooth or non smooth composite domains it was found that the method produces numerical results which are in good agreement with the analytical solutions where available

a new preconditioner for indefinite and asymmetric matrices we present a novel preconditioner for numerical solutions of large sparse linear systems with indefinite and asymmetric matrices this new preconditioner named as product preconditioner ps is constructed by two fairly simple preconditioners the distribution of eigenvalues and the form of the eigenvectors of the preconditioned matrix are analyzed moreover an upper bound on the degree of the minimal polynomial is also studied numerical experiments with several examples show that the proposed ps performs better than block diagonal preconditioner bd and block triangular preconditioner bt as well as the constraint preconditioner sc in terms of the number of iteration and computational time

an engineers defence of the patch test this convergence criterion is proved for any legalizable finite element formulation i e any model for which a conforming displacement version is possible using the same nodal variables engineers may therefore take control of and responsibility for convergence for most of the elements in service today stummel s counter example is explained as a technical misunderstanding due to inadequate documentation of the patch test it is suggested without proof that the test properly applied is universal provided that it is always combined with an adequate test of stability

the method of fundamental solutions in solving coupled boundary value problems for m eeg the estimation of neuronal activity in the human brain from electroencephalography eeg and magnetoencephalography meg signals is a typical inverse problem whose solution process requires an accurate and fast forward solver in this paper the method of fundamental solutions is for the first time proposed as a meshfree boundary type and easy to implement alternative to the boundary element method bem for solving the m eeg forward problem the solution of the forward problem is obtained by numerically solving a set of coupled boundary value problems for the three dimensional laplace equation numerical accuracy convergence and computational load are investigated the proposed method is shown to be a competitive alternative to the state of the art bem for m eeg forward solving

an adaptive method of fundamental solutions for solving the laplace equation abstract in this paper we propose a residual type adaptive method of fundamental solutions amfs for solving the two dimensional laplace equation an error estimator is defined only on the boundary of the domain initial distributions of source points and collocation points are determined by using approaches proposed in chen et al 2006 the adding removing and stopping strategies are designed so that the required accuracy can be satisfied within finite steps numerical experiments reveal that amfs improves the accuracy of the mfs approximation obtained from uniformly distributed sources and collocation points which makes the mfs more practical for non harmonic and non smooth boundary conditions moreover it is shown that the error estimator becomes equidistributed after an adaptive iteration a detailed comparison between amfs and mfs using uniformly distributed points is also presented for each numerical example

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

the method of fundamental solutions and quasi monte carlo method for diffusion equations the laplace transform is applied to remove the time dependent variable in the diffusion equation for non harmonic initial conditions this gives rise to a non homogeneous modified helmholtz equation which we solve by the method of fundamental solutions to do this a particular solution must be obtained which we find through a method suggested by atkinson to avoid costly gaussian quadratures we approximate the particular solution using quasi monte carlo integration which has the advantage of ignoring the singularity in the integrand the approximate transformed solution is then inverted numerically using stehfest s algorithm two numerical examples are given to illustrate the simplicity and effectiveness of our approach to solving diffusion equations in 2 d and 3 d 1998 john wiley sons ltd

fmm accelerated source model technique for many scatterer problems motivated by the lack of scalability of the source model technique smt to many scatterer problems a new smt based method for 3 d many scatterer problems is developed the large time complexity of the smt is reduced by deriving an implicit compact representation of the scattering problem via a fast multipole formulation which divides the interactions into far and near ones while grouping and approximating the far interactions this results in significant acceleration of classical smt which is demonstrated in a variety of 3 d many body scattering problems the improvement is notable already at the minimal number of unknowns necessary and grows with the size of the problem

mathematical foundation of the mfs for certain elliptic systems in linear elasticity the method of fundamental solutions mfs is a trefftz type technique in which the solution of an elliptic boundary value problem is approximated by a linear combination of translates of fundamental solutions with singularities placed on a pseudo boundary i e a surface embracing the domain of the problem under consideration in this work we develop a mathematical framework for the numerical implementation of the mfs in elliptic systems we obtain density results with respect to the c l norms which establish the applicability of the method in certain systems arising from the theory of elastostatics and thermo elastostatics the domains in our density results may possess holes and they satisfy the segment condition

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

numerical investigation on convergence of boundary knot method in the analysis of homogeneous helmholtz modified helmholtz and convection diffusion problems abstract this paper concerns a numerical study of convergence properties of the boundary knot method bkm applied to the solution of 2d and 3d homogeneous helmholtz modified helmholtz and convection diffusion problems the bkm is a new boundary type meshfree radial function basis collocation technique the method differentiates from the method of fundamental solutions mfs in that it does not need the controversial artificial boundary outside physical domain due to the use of non singular general solutions instead of the singular fundamental solutions the bkm is also generally applicable to a variety of inhomogeneous problems in conjunction with the dual reciprocity method drm therefore when applied to inhomogeneous problems the error of the drm confounds the bkm accuracy in approximation of homogeneous solution while the latter essentially distinguishes the bkm mfs and boundary element method in order to avoid the interference of the drm this study focuses on the investigation of the convergence property of the bkm for homogeneous problems the given numerical experiments reveal rapid convergence high accuracy and efficiency mathematical simplicity of the bkm