pseudo time stepping methods for space time discontinuous galerkin discretizations of the compressible navier stokes equations the space time discontinuous galerkin discretization of the compressible navier stokes equations results in a non linear system of algebraic equations which we solve with pseudo time stepping methods we show that explicit runge kutta methods developed for the euler equations suffer from a severe stability constraint linked to the viscous part of the equations and propose an alternative to relieve this constraint while preserving locality to evaluate its effectiveness we compare with an implicit explicit runge kutta method which does not suffer from the viscous stability constraint we analyze the stability of the methods and illustrate their performance by computing the flow around a 2d airfoil and a 3d delta wing at low and moderate reynolds numbers

an operator splitting radial basis function method for the solution of transient nonlinear poisson problems abstract this paper presents an operator splitting radial basis function os rbf method as a generic solution procedure for transient nonlinear poisson problems by combining the concepts of operator splitting radial basis function interpolation particular solutions and the method of fundamental solutions the application of the operator splitting permits the isolation of the nonlinear part of the equation that is solved by explicit adams bashforth time marching for half the time step this leaves a nonhomogeneous modified helmholtz type of differential equation for the elliptic part of the operator to be solved at each time step the resulting equation is solved by an approximate particular solution and by using the method of fundamental solution for the fitting of the boundary conditions radial basis functions are used to construct approximate particular solutions and a grid free dimension independent method with high computational efficiency is obtained this method is demonstrated for some prototypical nonlinear poisson problems in heat and mass transfer and for a problem of transient convection with diffusion the results obtained by the os rbf method compare very well with those obtained by other traditional techniques that are computationally more expensive the new os rbf method is useful for both general irregular two and three dimensional geometry and provides a mesh free technique with many mathematical flexibilities and can be used in a variety of engineering applications

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

a comparative study of different sets of variables for solving compressible and incompressible flows a globally conservative galerkin least squares formulation which attains correct shock structure is developed for any choice of variables only the choice of entropy variables satisfies exactly the discrete clausius duhem inequality without any dissipative mechanisms whereas for the rest of the variables artificial diffusion is required to guarantee entropy production the limit of the formulation is well defined for entropy variables and the primitive variables p u t leading to conservative incompressible formulations the approach is stable for any continuous interpolations both for compressible and incompressible flows a comparative study of different variables is performed indicating that entropy variables and the primitive variables p u t possess the most attributes for practical problem solving

on the application of the method of fundamental solutions for the study of the stress state of a plate subjected to elastic plastic deformation abstract the plane elastoplastic problem for the stress state of a plate with a narrowing subjected to uniaxial extension is considered the behaviour of the material that hardens with plastic deformation is characterised by the ramberg osgood stress strain relation since this relation provides a smooth continuous curve over the whole elastic and plastic deformation range the same governing equation can be used for both deformation regions the paper provides a method for solving the resulting nonlinear boundary value problem the algorithm is based on meshless methods i e the method of fundamental solutions and the method of particular solutions together with a picard iteration process the approximate solution i e the stress function obtained in each iteration step is a linear combination of fundamental and particular solutions it can thus be further used to compute the values of stresses and some effective material parameters i e the young modulus and the poisson ratio at any point of the domain

numerical study of the turbulent flow past an airfoil with trailing edge separation presentation d une methode numerique par volume fini pour la resolution des equations de navier stokes bidimensionnelles incompressibles et stationnaires en coordonnees generales curvilignes application de la methode aux ecoulements turbulents sur des profils avec et sans separation au bord de sortie posterieur comparaison des calculs avec des donnees experimentales

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

multigrid acceleration for recirculating laminar and turbulent flows computed with a non orthogonal collocated finite volume scheme abstract a study has been undertaken of multigrid convergence acceleration in iterative calculations of recirculating laminar and turbulent flows the basic numerical platform is a 2d 3d non orthogonal collocated finite volume algorithm in which convection is approximated optionally by a non monotonic upstream weighted third order scheme or by a monotonie second order muscl tvd formulation and which iterates the solution towards the steady state by way of a pressure correction scheme the study investigates the performance of various multigrid implementations including the correction and full approximation multigrid variants the latter operated within a variety of cycles both with prescribed and residual driven relaxation sweeps at each grid level the multigrid implementation is documented in detail and includes a consideration of data structure issues and special algorithmic measures necessary to secure robust convergence for turbulent flows the performance characteristics of the multigrid variants are then exposed through a range of results arising from computations for laminar and turbulent conditions with turbulence being represented by the k model or full reynolds stress transport closure

the numerical solution of the biharmonic equation by conformal mapping the solution to the biharmonic equation in a simply connected region omega in the plane is computed in terms of the goursat functions the boundary conditions are conformally transplanted to the disk with a numerical conformal map a linear system is obtained for the taylor coefficients of the goursat functions the coefficient matrix of the linear system can be put in the form i k where k is the discretization of a compact operator k can be thought of as the composition of a block hankel matrix with a diagonal matrix the compactness leads to clustering of eigenvalues and the hankel structure yields a matrix vector multiplication cost of o n log n thus if the conjugate gradient method is applied to the system then superlinear convergence will be obtained numerical results are given to illustrate the spectrum clustering and superlinear convergence

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

an hp adaptive mixed discontinuous galerkin fem for nearly incompressible linear elasticity we develop the a posteriori error estimation of mixed hp version discontinuous galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions computable upper and lower bounds on the error measured in terms of a natural mesh dependent energy norm are derived the bounds are explicit in the local mesh sizes and approximation orders and are independent of the locking parameter a series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp adaptive refinement procedure

meshless local boundary integral equation method for simply supported and clamped plates resting on elastic foundation abstract simply supported and clamped thin elastic plates resting on a two parameter foundation are analyzed in the paper the governing partial differential equation of fourth order for a plate is decomposed into two coupled partial differential equations of second order one of them is poisson s equation whereas the other one is helmholtz s equation the local boundary integral equation method is used with meshless approximation for both the poisson and the helmholtz equation the moving least square method is employed as the meshless approximation independent of the boundary conditions fictitious nodal unknowns used for the approximation of bending moments and deflections are always coupled in the resulting system of algebraic equations the winkler foundation model follows from the pasternak model if the second parameter is equal to zero numerical results for a square plate with simply and or clamped edges are presented to prove the efficiency of the proposed formulation

h multigrid for space time discontinuous galerkin discretizations of the compressible navier stokes equations being implicit in time the space time discontinuous galerkin discretization of the compressible navier stokes equations requires the solution of a non linear system of algebraic equations at each time step the overall performance therefore highly depends on the efficiency of the solver in this article we solve the system of algebraic equations with a h multigrid method using explicit runge kutta relaxation two level fourier analysis of this method for the scalar advection diffusion equation shows convergence factors between 0 5 and 0 75 this motivates its application to the 3d compressible navier stokes equations where numerical experiments show that the computational effort is significantly reduced up to a factor 10 w r t single grid iterations

efficient solution algorithms for the riemann problem for real gases a procedure for constructing solutions to the riemann problem for gas dynamics with a general convex equation of state is given approximate procedures involving a local parametrization of the equation of state are introduced in order to calculate numerical fluxes in conservative finite difference schemes this leads to difference schemes which are as accurate and almost as fast as the analogous schemes for polytropic gases numerical results in one and two space variables are presented

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

algebraic turbulence modeling for unstructured and adaptive meshes an algebraic turbulence model based on the baldwin lomax model has been implemented for use on unstructured grids the implementation is based on the use of local background turbulence meshes at each time step flow variables are interpolated from the unstructured mesh onto the background structured meshes the turbulence model is executed on these meshes and the resulting eddy viscosity values are interpolated back to the unstructured mesh modifications to the algebraic model were required to enable the treatment of more compli cated flows such as confluent boundary layers and wakes the model is used in conjunction with an efficient unstructured multigrid finite element navier stokes solver in order to compute compressible turbulent flows on fully unstructured meshes solutions about single and multiple element airfoils are obtained and compared with experimental data he use of unstructured mesh techniques for computa tional fluid dynamics cfd problems has become more widespread in recent years due to the flexibility they afford in discretizing arbitrarily complex geometries and due to the possibility they offer in resolving highly localized flow phenomena through the use of adaptive meshing however research on unstructured mesh techniques for cfd has con centrated almost exclusively on the solution of the euler equa tions in two or three dimensions for viscous flow calculations about nonsimple geometries hybrid meshes have generally been employed 1 3 where a thin structured mesh is placed in the boundary layer and wake regions and an unstructured mesh is constructed in the outer inviscid region of the flowfield besides leading to an increase in coding complexity this type of compromise limits the generality of the unstruc tured mesh approach in dealing with arbitrarily complex geometries such as multiple body geometries with close toler ances where confluent boundary layers or wakes may occur and complicates the task of performing adaptive meshing in the inviscid as well as viscous regions of flow it appears that the difficulties associated with generating highly stretched unstructured meshes which are required for efficiently resolv ing viscous shear layers as well as the efficient implementa tion of a turbulence model on such meshes has generally im peded the use of fully unstructured meshes for viscous flows the use of unstructured meshes throughout the entire flowfield is advocated in the present work previous work by the author has shown how a highly stretched unstructured mesh suitable for high reynolds number viscous flow calcula tions may be constructed4 and has also discussed the develop ment of an efficient unstructured navier stokes solver for laminar flows 5 this paper is thus concerned with the effi cient implementation of a turbulence model for computing high reynolds number turbulent flows using fully unstruc tured meshes

a discontinuous galerkin finite element discretization of the euler equations for compressible and incompressible fluids using the generalized variable formulation of the euler equations of fluid dynamics we develop a numerical method that is capable of simulating the flow of fluids with widely differing thermodynamic behavior ideal and real gases can be treated with the same method as an incompressible fluid the well defined incompressible limit relies on using pressure primitive or entropy variables in particular entropy variables can provide numerical methods with attractive properties e g fulfillment of the second law of thermodynamics we show how a discontinuous galerkin finite element discretization previously used for compressible flow with an ideal gas equation of state can be extended for general fluids we also examine which components of the numerical method have to be changed or adapted especially we investigate different possibilities of solving the nonlinear algebraic system with a pseudo time iteration numerical results highlight the applicability of the method for various fluids

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows ii efficient flux quadrature a new and efficient quadrature rule for the flux integrals arising in the space time discontinuous galerkin discretization of the euler equations in a moving and deforming space time domain is presented and analyzed the quadrature rule is a factor three more efficient than the commonly applied quadrature rule and does not affect the local truncation error and stability of the numerical scheme the local truncation error of the resulting numerical discretization is determined and is shown to be the same as when product gauss quadrature rules are used details of the approximation of the dissipation in the numerical flux are presented which render the scheme consistent and stable the method is successfully applied to the simulation of a three dimensional transonic flow over a deforming wing

a complex valued first integral of navier stokes equations unsteady couette flow in a corrugated channel system for a two dimensional incompressible viscous flow a first integral of the governing equations of motion is constructed based on a reformulation of the unsteady navier stokes equations in terms of complex variables and the subsequent introduction of a complex potential field complementary solid and free surface boundary conditions are formulated the methodology is used to solve the challenging problem of unsteady couette flow between two sinusoidally varying corrugated rigid surfaces utilising two modelling approaches to highlight the versatility of the first integral in the stokes flow limit the results obtained in the case of steady flow are found to be in excellent agreement with corresponding investigations in the open literature similarly for unsteady flow the results are in accord with related investigations exploring material transfer between trapped eddies and the associated bulk flow and vice versa it is shown how the work relates to the classical complex variable method for solving the

annihilation of boundary singularities via suitable green s functions abstract a numerical integral equation method is presented for the accurate solution of viscous flow problems in which there are inherent boundary singularities where applicable this method has the advantage over existing singularity subtraction and singularity incorporation methods since natural boundary conditions can be employed without any special form of pre or post processing results are presented for the benchmark problem of stokes flow in a lid driven cavity

local mesh refinement within a multi block structured grid scheme for general flows abstract a multi level flow adaptive mesh refinement strategy has been formulated and implemented within a structured grid multiblock scheme for 2d laminar transitional and turbulent recirculating flow although the locally refined irregular grid portions are organised as block structured patches within coarse grid blocks a special data structure allows high storage economy to be achieved issues of particular interest in the study include the performance of different flow adaptation sensors based on solution properties and error and the response of the solution accuracy to inter grid interpolation practices based on bi linear bi quadratic and compact interpolation the effectiveness of the local refinement strategy especially in terms of cpu and memory resources is examined by reference to four test cases ranging from an analytically prescribed scalar field over a square domain to a transitional flow over a highly loaded compressor blade the study demonstrates that cpu and memory savings with a three level refinement for the compressor blade case are of order 70 and 90 respectively relative to the requirements of globally fine grids returning similar accuracy levels

local method of approximate particular solutions for two dimensional unsteady burgers equations the local method of approximate particular solutions lmaps is first proposed to solve burgers equations in avoiding the ill conditioning problem the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low order linear systems within the local supporting domain then the global solutions can be obtained by reformulating the local matrix in the global and sparse matrix the obtained large sparse linear systems are directly solved instead of using more complicated iterative method the numerical experiments have shown that the proposed method is suitable for solving the two dimensional unsteady burgers equations with high accuracy and efficiency

a particular solution trefftz method for non linear poisson problems in heat and mass transfer this paper proposes a computational procedure based on the trefftz method for the solution of non linear poisson problems the problem is solved by finding an approximate particular solution to the poisson equation and using boundary collocation to solve the resulting laplace equation this method results in a global collocation and hence eliminates the need for discretization of the domain in both two and three dimensions the solution procedure proposed earlier in the literature based upon the above method has been reformulated for increased computational efficiency a quasi newton iteration method along with a new heuristic for source point location is used for efficient convergence of the numerical scheme the efficacy of the new formulation has been demonstrated for two classes of problems viz the thermal explosion problem and the diffusion reaction problem in a partially wetted catalyst pellet a brief error analysis of the method as well as future research directions is presented

an improved subspace selection algorithm for meshless collocation methods choosing data points is a common problem for researchers who employ various meshless methods for solving partial differential equations on the one hand high accuracy is always desired on the other ill conditioning problems of the resultant matrices which may lead to unstable algorithms prevent some researchers from using meshless methods for example the optimal placements of source points in the method of fundamental solutions or of the centers in the radial basis functions method are always unclear intuitively such optimal locations will depend on many factors the partial differential equations the domain the trial basis used i e the employed method itself 1pt the computational precisions some user defined parameters and so on such complexity makes the hope of having an optimal centers placement unpromising in this paper we provide a data dependent algorithm that adaptively selects centers based on all the other variables copyright 2009 john wiley sons ltd

solution of low mach number aeroacoustic flows using a variational multi scale finite element formulation of the compressible navier stokes equations written in primitive variables abstract in this work we solve the compressible navier stokes equations written in primitive variables in order to simulate low mach number aeroacoustic flows we develop a variational multi scale formulation to stabilize the finite element discretization by including the orthogonal dynamic and non linear subscales together with an implicit scheme for advancing in time three additional features define the proposed numerical scheme the splitting of the pressure and temperature variables into a relative and a reference part the definition of the matrix of stabilization parameters in terms of a modified velocity that accounts for the local compressibility and the approximation of the dynamic stabilization matrix for the time dependent subscales we also include a weak imposition of implicit non reflecting boundary conditions in order to overcome the challenges that arise in the aeroacoustic simulations at low compressibility regimes the order of accuracy of the method is verified for two and three dimensional linear and quadratic elements using steady manufactured solutions several benchmark flow problems are studied including transient examples and aeroacoustic applications

an invariant method of fundamental solutions for two dimensional isotropic linear elasticity abstract in this paper we propose an invariant method of fundamental solutions for solving boundary value problems in two dimensional isotropic linear elasticity the invariant method of fundamental solutions keeps a very basic natural property the so called invariance property under trivial coordinate changes in the problem description e g dilations and or contractions the problems are solved by the tikhonov regularization method in conjunction with morozov s discrepancy principle finally five examples of boundary value problems are investigated to show the efficiency of this method the numerical convergence accuracy and stability of the proposed method with respect to the number of source points the distance between the sources and the boundary and the amount of noise added into the input data respectively are also analyzed

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows i general formulation a new space time discontinuous galerkin finite element method for the solution of the euler equations of gas dynamics in time dependent flow domains is presented the discontinuous galerkin discretization results in an efficient elementwise conservative upwind finite element method which is particularly well suited for local mesh refinement the upwind scheme uses a formulation of the hllc flux applicable to moving meshes and several formulations for the stabilization operator to ensure that monotone solutions around discontinuities are investigated the non linear equations of the space time discretization are solved using a multigrid accelerated pseudo time integration technique with an optimized runge kutta method the linear stability of the pseudo time integration method is investigated for the linear advection equation the numerical scheme is demonstrated with simulations of the flow field in a shock tube a channel with a bump and an oscillating naca 0012 airfoil these simulations show that using the data at the superconvergence points the accuracy of the numerical discretization is o h5 2 in space for smooth subsonic flows both on structured and on locally refined meshes and that the space time adaptation can significantly improve the accuracy and efficiency of the numerical method

a multilayer method of fundamental solutions for stokes flow problems the method of fundamental solutions mfs is a meshless method for the solution of boundary value problems and has recently been proposed as a simple and efficient method for the solution of stokes flow problems the mfs approximates the solution by an expansion of fundamental solutions whose singularities are located outside the flow domain typically the source points i e the singularities of the fundamental solutions are confined to a smooth source layer embracing the flow domain this monolayer implementation of the mfs monolayer mfs depends strongly on the location of the user defined source points on the one hand increasing the distance of the source points from the boundary tends to increase the convergence rate on the other hand this may limit the achievable accuracy this often results in an unfavorable compromise between the convergence rate and the achievable accuracy of the mfs the idea behind the present work is that a multilayer implementation of the mfs multilayer mfs can improve the robustness of the mfs by efficiently resolving different scales of the solution by source layers at different distances from the boundary we propose a block greedy qr algorithm bgqra which exploits this property in a multilevel fashion the proposed multilayer mfs is much more robust than the monolayer mfs and can compute stokes flows on general two and three dimensional domains it converges rapidly and yields high levels of accuracy by combining the properties of distant and close source points the block algorithm alleviates the overhead of multiple source layers and allows the multilayer mfs to outperform the monolayer mfs

on a method of atkinson for evaluating domain integrals in the boundary element method we have shown how to couple a numerical method due to atkinson for computing particular solutions to a class of elliptic differential equations with a variety of boundary element methods which alleviates the problem of domain discretization for solving inhomogeneous equations when the inhomogeneous term is of the form typically used in the dual reciprocity method atkinson s formula is such that only boundary integrals need to be approximated numerically if this method is coupled with the method of potentials it results in a computational technique which requires neither boundary nor domain discretization some numerical results are given validating our approach

a new kirchhoff plate model based on a modified couple stress theory in this paper a new kirchhoff plate model is developed for the static analysis of isotropic micro plates with arbitrary shape based on a modified couple stress theory containing only one material length scale parameter which can capture the size effect the proposed model is capable of handling plates with complex geometries and boundary conditions from a detailed variational procedure the governing equilibrium equation of the micro plate and the most general boundary conditions are derived in terms of the deflection using the principle of minimum potential energy the resulting boundary value problem is of the fourth order instead of existing gradient theories which is of the sixth order and it is solved using the method of fundamental solutions mfs which is a boundary type meshless method several plates of various shapes aspect and poisson s ratios are analyzed to illustrate the applicability of the developed micro plate model and to reveal the differences between the current model and the classical plate model moreover useful conclusions are drawn from the micron scale response of this new kirchhoff plate model

unsteady solution of incompressible navier stokes equations abstract a new time accurate finite difference numerical method for solving incompressible navier stokes equations is presented with primitive variables as the unknowns the numerical scheme is a crank nicolson implicit treatment of all terms of the equations with central differencing for space derivatives the pressure and the nonlinearities in convection terms are not lagged to obtain the incompressible solution at the advanced time level a continuous auxiliary system is introduced in pseudo time with artificial compressibility time accurate solutions are presented for two dimensional fluid flows in a square cavity with an impulsively starting lid and with an oscillating lid

the method of fundamental solutions for the solution of steady state free boundary problems abstract the method of fundamental solutions is formulated for the solution of steady state free boundary problems the method is tested on three such problems and the results compared to published results obtained with other numerical methods

the method of fundamental solutions for axisymmetric potential problems in this paper we investigate the application of the method of fundamental solutions mfs to two classes of axisymmetric potential problems in the first the boundary conditions as well as the domain of the problem are axisymmetric and in the second the boundary conditions are arbitrary in both cases the fundamental solutions of the governing equations and their normal derivatives which are required in the formulation of the mfs can be expressed in terms of complete elliptic integrals the method is tested on several axisymmetric problems from the literature and is also applied to an axisymmetric free boundary problem copyright 1999 john wiley sons ltd

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

higher order discontinuous galerkin time stepping and local projection stabilization techniques for the transient stokes problem abstract we introduce and analyze discontinuous galerkin time discretizations coupled with continuous finite element methods based on equal order interpolation in space for velocity and pressure in transient stokes problems spatial stability of the pressure is ensured by adding a stabilization term based on local projection we present error estimates for the semi discrete problem after discretization in space only and for the fully discrete problem the fully discrete pressure shows an instability in the limit of small time step length numerical tests are presented which confirm our theoretical results including the pressure instability

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

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

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 new finite element formulation for computational fluid dynamics x the compressible euler and navier stokes equations a space time element method is presented for solving the compressible euler and navier stokes equations the proposed formulation includes the variational equation predictor multi corrector algorithms and boundary conditions the variational equation is based on the time discontinuous galerkin method in which the physical entropy variables are employed a least squares operator and a discontinuity capturing operator are added resulting in a high order accurate and unconditionally stable method implicit explicit predictor multi corrector algorithms applicable to steady as well as unsteady problems are presented techniques are developed to enhance their efficiency implementation of boundary conditions is addressed in particular a technique is introduced to satisfy nonlinear essential boundary conditions and a consistent method is presented to calculate boundary fluxes numerical results are presented to demonstrate the performance of the method

a fornberg like conformal mapping method for slender regions abstract a method is presented for approximating the conformal map from the interior of an ellipse to the interior of a simply connected target region the map is represented as a truncated chebyshev series conditions that the mapping function be conformal are transplanted from the ellipse to an annulus with the joukowski map the resulting conditions on the laurent coefficients then give a system of equations for the newton update of the approximate boundary correspondence this system is a generalization of fornberg s system for maps from the disk and is solved similarly in o n log n operations by the conjugate gradient method our numerical experiments demonstrate that the maps from the ellipse to a slender target region of similar aspect ratio can be constructed with far fewer mesh points than are required for maps from the disk thus circumventing the ill conditioning due to crowding in these cases

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

a unified method for computing incompressible and compressible flows in boundary fitted coordinates a unified method for computing incompressible and compressible flows with mach uniform accuracy and efficiency is described the method is equally applicable to stationary and nonstationary flows a pressure based discretisation on a staggered grid in general boundary fitted coordinates is used for the euler equations extension to navier stokes is straightforward dimensionless variables that remain finite for all mach numbers are used mach number independent accuracy and efficiency is shown by numerical experiments

simply supported plates by the boundary integral equation method plates governed by kirchhoff s equation have been analysed by the boundary integral equation method using the fundamental solution of the biharmonic equation in the case of supported plates the boundary conditions permit the uncoupling of the field equation into two harmonic equations that originate due to the nature of the fundamental solution easier integration kernels and a simpler system of equations the calculation of bending and twisting moments and transverse shear force can be formed combining derivatives of the integral equation which defines the expression of the deflection on any point of the plate the uncoupling of the biharmonic equation into two poisson s equations involves the discretization of the domain of the studied problems nevertheless the unknown quantity of the problem does not appear in the domain integrations for which a refined discretization is unnecessary in the paper however a numerical alternative is considered to express the domain integral by means of boundary integrals in this way we need only discretize the boundary of the plate making it necessary to solve a supplementary system of equations in order to calculate the coefficients of the approximation carried out

the method of fundamental solutions for the cauchy problem in two dimensional linear elasticity in this paper the application of the method of fundamental solutions to the cauchy problem in two dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularised by employing the first order tikhonov functional while the choice of the regularisation parameter is based on the l curve method numerical results are presented for both smooth and piecewise smooth geometries as well as for constant and linear stress states the convergence and the 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

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

grid optimization for the boundary element method the quality of solution obtained using the boundary element method bem is dependent on how the boundary is discretized this is particularly true in domains of complex geometry a rule for grid optimization for the bem is derived on the bases of an asymptotic measure of the boundary element error that preserves the number of elements degrees of freedom three example problems are provided to show the advantages of grid optimization in terms of accuracy and cost

a meshless method for the numerical solution of the cauchy problem associated with three dimensional helmholtz type equations in this paper the application of the method of fundamental solutions to the cauchy problem associated with three dimensional helmholtz type equations is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the zeroth order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for under equally and over determined cauchy problems in a piecewise 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

construction of stabilization operators for galerkin least squares discretizations of compressible and incompressible flows the design and analysis of a class of stabilization operators suitable for space time galerkin least squares finite element discretizations of the symmetrized compressible navier stokes equations is discussed the obtained stabilization matrix is well defined in the incompressible limit and reduces to the matrix described in m polner j j w van der vegt and r m j van damme analysis of stabilization operators for galerkin least squares discretizations of the incompressible navier stokes equations comput meth appl mech eng 195 2006 982 1006 the stabilization matrix is investigated to give necessary and sufficient conditions for its positive definiteness under these conditions on the stabilization matrix the galerkin least squares method for the symmetrized compressible navier stokes equations satisfies the entropy condition which ensures global nonlinear stability of the discretization

a simple stabilized finite element method for solving two phase compressible incompressible interface flows abstract when computing interface flows between compressible gas and incompressible liquid fluids one faces at least to the following difficulties 1 transition from a gas density linked to the local temperature and pressure by an equation of state to a liquid density mainly constant in space 2 proper approximation of the divergence constraint in incompressible regions and 3 wave transmission at the interface the aim of the present paper is to design a global i e the same for each phase numerical method to address easily this coupling to this end the same set of primitive unknowns and equations is used everywhere in the flow but with a dynamic parameterization that changes from compressible to incompressible regions on one hand the compressible navier stokes equations are considered under weakly compressibility assumption so that a non conservative formulation can be used on the other hand the incompressible non isothermal model is retained in addition the level set transport equation is used to capture the interface position needed to identify the local characteristics of the fluid and to recover the adequate local modelling for space approximation of navier stokes equations a galerkin least squares finite element method is used two essential elements for defining this numerical scheme are the stabilization and the computation of element integral of the approximated weak form since very different concerns motivate the need for stabilization in compressible and incompressible flows the first difficulty is to design a stabilization operator suitable for both types of flows especially in mixed elements in addition some integral of discontinuous functions must be correctly computed to ensure interfacial wave transmission to overcome these two difficulties specific averages are computed especially near the interface finally the level set transport equation is computed by a quadrature free discontinuous galerkin method numerical strategies are performed and validated for 1d and 2d applications

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

space time discontinuous galerkin method for the compressible navier stokes equations a space time discontinuous galerkin finite element method for the compressible navier stokes equations is presented we explain the space time setting derive the weak formulation and discuss our choices for the numerical fluxes the resulting numerical method allows local grid adaptation as well as moving and deforming boundaries which we illustrate by computing the flow around a 3d delta wing on an adapted mesh and by simulating the dynamic stall phenomenon of a 2d airfoil in rapid pitch up maneuver

an efficient meshless technique for the solution of transversely isotropic two dimensional piezoelectricity in this paper a new meshless technique is presented for the solution of two dimensional piezoelectricity problem the technique derivation is based on the solution of the corresponding analog equation by transforming the original set of differential equations into three poisson equations with unknown right hand side terms boundary discretisation of the resulting integral equations is eliminated by the use of the method of fundamental solutions the right hand side terms are represented in the solution as particular solutions expressed in terms of radial basis functions the problem solution is then rewritten in its new form which involves complementary solution and particular solution the governing partial differential operator for piezoelectricity is applied on the obtained solution form and forced to be satisfied at a set of domain points whereas the prescribed boundary conditions are satisfied at another set of boundary points the proposed technique is implemented into computer code where several numerical examples with different boundary conditions are tested the results demonstrated excellent agreement with those obtained from analytical and fem solutions

localized mfs for the inverse cauchy problems of two dimensional laplace and biharmonic equations abstract this paper makes a first attempt to use a new localized method of fundamental solutions lmfs to accurately and stably solve the inverse cauchy problems of two dimensional laplace and biharmonic equations in complex geometries the lmfs firstly divides the whole physical domain into several small overlapping subdomains and then employs the traditional method of fundamental solutions mfs formulation in every local subdomain for calculating the unknown coefficients on the local fictitious boundary after that a sparse linear system is formed by using the governing equation for interior nodes and the nodes on under specified boundary and by using the given boundary conditions for the nodes on over specified boundary finally the numerical solutions of the inverse problems can be obtained by solving the resultant sparse system compared with the traditional mfs with the global boundary discretization the lmfs requires less computational cost which may make the lmfs suitable for solving large scale problems numerical experiments demonstrate the validity and accuracy of the proposed lmfs for the inverse cauchy problems of two dimensional laplace and biharmonic equations with noisy boundary data

the method of fundamental solutions for eigenproblems in domains with and without interior holes the main purpose of the present paper is to provide a general method of fundamental solution mfs formulation for two and three dimensional eigenproblems without spurious eigenvalues the spurious eigenvalues are avoided by utilizing the mixed potential method illustrated problems in the annular and concentric domains are studied analytically and numerically to demonstrate the issue of spurious eigenvalues by the discrete and continuous versions of the mfs with and without the mixed potential method the proposed numerical method is then verified with the exact solutions of the benchmark problems in circular and spherical domains with and without holes further studies are performed in a three dimensional peanut shaped domain in the spirit of the mfs this scheme is free from meshes singularities and numerical integrations

methods of fundamental solutions for time dependent heat conduction problems a time dependent heat conduction problem can be solved by the method of fundamental solutions using the fundamental solution to the modified helmholtz equation or the fundamental solution to the heat equation this paper presents solutions using both formulations in terms of initial and boundary conditions such formulations enable calculation of errors and variance which indicates sensitivities of solutions to uncertainties in initial and boundary conditions both errors and variance of solutions to three test problems by the two methods of fundamental solutions are used to compare performances of the methods copyright 2005 john wiley sons ltd

a general non orthogonal collocated finite volume algorithm for turbulent flow at all speeds incorporating second moment turbulence transport closure part 1 computational implementation abstract a computational procedure has been developed for predicting separated turbulent flows in complex two dimensional and three dimensional geometries the procedure is based on the fully conservative structured finite volume framework within which the volumes are non orthogonal and collocated such that all flow variables are stored at one and the same set of nodes to ease the task of discretization and to enhance the conservative property of the scheme a cartesian or datum line adapted decomposition of the velocity field has been used the solution algorithm is iterative in nature approaching the steady solution with the aid of a pressure correction scheme convection is approximated with a range of schemes among them higher order upstream weighted approximations and a tvd type muscl form the last applied principally to the transport equations governing turbulence properties effects of turbulence are represented either by two equation eddy viscosity models or by a full reynolds stress transport closure the former category includes both high and low reynolds number variants in two as well as three dimensional conditions to achieve a stable implementation of the reynolds stress equations a special interpolation practice analogous to that of rhie and chow for momentum 1 has been introduced within the general framework the procedure has been formulated so as to apply to both incompressible and compressible flows the latter may contain shocks and highly supersonic regions to achieve this range of applicability the retarded density concept has been combined with the basic pressure based algorithm to capture shock waves a full approximation multigrid scheme for convergence acceleration has been incorporated and applied in conjunction with all turbulence models including second moment closure the present first part of a twin paper focuses on numerical and turbulence modelling issues in part 2 computational results are presented for six representative applications out of fifteen recently predicted with the algorithm within an extensive validation exercise

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

unstructured cartesian refinement with sharp interface immersed boundary method for 3d unsteady incompressible flows a novel numerical method is developed for solving the 3d unsteady incompressible navier stokes equations on locally refined fully unstructured cartesian grids in domains with arbitrarily complex immersed boundaries owing to the utilization of the fractional step method on an unstructured cartesian hybrid staggered non staggered grid layout flux mismatch and pressure discontinuity issues are avoided and the divergence free constraint is inherently satisfied to machine zero auxiliary hanging nodes are used to facilitate the discretization of the governing equations the second order accuracy of the solver is ensured by using multi dimension lagrange interpolation operators and appropriate differencing schemes at the interface of regions with different levels of refinement the sharp interface immersed boundary method is augmented with local near boundary refinement to handle arbitrarily complex boundaries the discrete momentum equation is solved with the matrix free newton krylov method and the krylov subspace method is employed to solve the poisson equation the second order accuracy of the proposed method on unstructured cartesian grids is demonstrated by solving the poisson equation with a known analytical solution a number of three dimensional laminar flow simulations of increasing complexity illustrate the ability of the method to handle flows across a range of reynolds numbers and flow regimes laminar steady and unsteady flows past a sphere and the oblique vortex shedding from a circular cylinder mounted between two end walls demonstrate the accuracy the efficiency and the smooth transition of scales and coherent structures across refinement levels large eddy simulation les past a miniature wind turbine rotor parameterized using the actuator line approach indicates the ability of the fully unstructured solver to simulate complex turbulent flows finally a geometry resolving les of turbulent flow past a complete hydrokinetic turbine illustrates the potential of the method to simulate turbulent flows past geometrically complex bodies on locally refined meshes in all the cases the results are found to be in very good agreement with published data and savings in computational resources are achieved

an integral equation method for the solution of singular slow flow problems a biharmonic boundary integral equation bbie method is used to solve a two dimensional contained viscous flow problem in order to achieve a greater accuracy than is usually possible in this type of method analytic expressions are used for the piecewise integration of all the kernel functions rather than the more time consuming method of gaussian quadrature

a boundary collocation meshfree method for the treatment of poisson problems with complex morphologies a new meshfree method based on a discrete transformation of green s basis functions is introduced to simulate poisson problems with complex morphologies the proposed green s discrete transformation method gdtm uses source points that are located along a virtual boundary outside the problem domain to construct the basis functions needed to approximate the field the optimal number of green s functions source points and their relative distances with respect to the problem boundaries are evaluated to obtain the best approximation of the partition of unity condition a discrete transformation technique together with the boundary point collocation method is employed to evaluate the unknown coefficients of the solution series via satisfying the problem boundary conditions a comprehensive convergence study is presented to investigate the accuracy and convergence rate of the gdtm we will also demonstrate the application of this meshfree method for simulating the conductive heat transfer in a heterogeneous materials system and the dissolved aluminum ions concentration in the electrolyte solution formed near a passive corrosion pit

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

regularity results for the 2d critical oldroyd b model in the corotational case this paper studies the regularity results of classical solutions to the two dimensional critical oldroyd b model in the corotational case the critical case refers to the full laplacian dissipation in the velocity or the full laplacian dissipation in the non newtonian part of the stress tensor whether or not their classical solutions develop finite time singularities is a difficult problem and remains open the object of this paper is two fold firstly we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non newtonian part of the stress tensor secondly when the critical case occurs in the non newtonian part of the stress tensor we first present many interesting global a priori bounds then establish a conditional global regularity in terms of the non newtonian part of the stress tensor this criterion comes naturally from our approach to obtain a global l bound for the vorticity

laminar fluid flow and heat transfer in an internally corrugated tube by means of the method of fundamental solutions and radial basis functions abstract the paper shows application of the method of fundamental solutions in combination with the radial basis functions for analysis of fluid flow and heat transfer in an internally corrugated tube cross section of such a tube is mathematically described by a cosine function and it can potentially represent a natural duct with internal corrugations e g inside arteries the boundary value problem is described by two partial differential equations one for fluid flow problem and one for heat transfer problem and appropriate boundary conditions during solving this boundary value problem the average fluid velocity and average fluid temperature are calculated numerically in the paper the nusselt number and the product of friction factor and reynolds number are presented for some selected geometrical parameters the number and amplitude of corrugations it is shown that for a given number of corrugations a minimal value of the product of friction factor and reynolds number can be found as it was expected the nusselt number increases with increasing amplitude and number of corrugations

a general non orthogonal collocated finite volume algorithm for turbulent flow at all speeds incorporating second moment turbulence transport closure part 2 application abstract a general non orthogonal finite volume procedure documented in a companion paper presented earlier is applied to six flows one 2d laminar one 2d turbulent two 3d turbulent one inviscid transonic and one turbulent transonic the principal aim is to convey the capabilities of the method and this aim is pursued by choosing flows which feature a wide range of phenomena and processes in some cases a comparison is presented between two or more turbulence modelling practices and in these cases considerable emphasis is put on an analysis of model performance and associated physical implications

high order entropy conservative schemes and kinetic relations for van der waals fluids abstract we consider the mixed hyperbolic elliptic system of two conservation laws modeling the dynamics of van der waals fluids viscosity and capillarity effects are taken into account we introduce a new class of semidiscrete high order schemes which are entropy conservative in the sense of tadmor when the viscosity is neglected and otherwise dissipate the associated mathematical entropy our numerical schemes generalize the works by e tadmor 1987 math comput 49 91 and p g lefloch and c rohde 2000 siam j numer anal 37 2023 who proposed second order and third order entropy conservative schemes respectively following b t hayes and p g lefloch 1998 siam j numer anal 35 2169 we demonstrate numerically that balanced viscosity and capillarity terms in van der waals fluids may generate nonclassical shock waves or subsonic propagating phase transitions such waves are undercompressive and do not satisfy standard entropy criteria they must be characterized by a kinetic function which we determine numerically in this paper from vanishing viscosity and capillarity the kinetic relation is an efficient tool to discuss the interplay among the viscosity capillarity and discretization parameters in van der waals fluids

on the thermodynamics stability and hierarchy of entropy functions in fluid flow symmetric forms of systems of equations describing physical phenomena present exceptional desirable properties such as symmetric coefficient matrices well posedness of the cauchy initial value problem finite wave propagation speeds and a link to nonlinear stability principles therefore they should be used when possible to design better numerical methods in this paper such symmetric forms are reviewed for various models of fluid flow and a hierarchy of generalized entropy functions is presented standard concepts are extended to the diffusive terms

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

numerical solution of the cauchy problem for steady state heat transfer in two dimensional functionally graded materials the application of the method of fundamental solutions to the cauchy problem for steady state heat conduction in two dimensional functionally graded materials fgms is investigated the resulting system of linear algebraic equations is ill conditioned and therefore regularization is required in order to solve this system of equations in a stable manner this is achieved by employing the zeroth order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for both smooth and piecewise smooth geometries the convergence and the 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