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.     

On the method of functional equations and the performance of desingularized boundary element methods

A correspondence is made between the reciprocal relation for linear elliptic partial differential equations and the Riesz integral representation. The former relates the boundary distributions and appropriate normal fluxes of two arbitrary solutions, and the latter expresses a continuous linear functional in terms of an integral involving a representing function. When sufficient regularity conditions are met, the representing function is identified with the unknown boundary distribution. In principle, the representing function may be expressed in terms of the images of a complete set of orthonormal basis functions with known normal fluxes, as suggested by Kupradze [Kupradze VD. On the approximate solution of problems in mathematical physics. Russ Math Surv 1967; 22: 59–107]; in practice, the representing function is computed by solving integral equations using boundary element methods. The basic procedure involves expressing the representing function in terms of finite-element or other basis functions, and requiring the satisfaction of the reciprocal relationship with a suitable set of test functions such as Green's functions and their dipoles. When the singular points are placed at the boundary, we obtain the standard boundary integral equation method. When the singular points are placed outside the domain of solution, we obtain a system of functional equations and associated class of desingularized boundary integral methods. When sufficient regularity conditions are met and the test functions comprise a complete set, then in the limit of infinite discretization the numerical solution converges to the unknown boundary distribution. An overview of formulations is presented with reference to Laplace's equation in two dimensions. Numerical experimentation shows that, in general, the solution obtained by desingularized methods becomes increasingly less accurate as the singular points of Green's functions move farther away from the boundary, but the loss of accuracy is significant only when the exact solution shows pronounced variations. Exceptions occur when the integral equation does not have a unique solution. In contrast, and in agreement with previous findings, the condition number of the linear system increases rapidly with the distance of the singular points from the boundary, to the extent that a dependable solution cannot be obtained when the singularities are located even a moderate distance away from the boundary. The desingularized formulation based on Green's function dipoles is superior in accuracy and reliability to the one that uses Green functions. The implementation of the method to the equations of elastostatics and Stokes flow are also discussed.     

Iterative solution of systems of equations in the dual reciprocity boundary element method for the diffusion equation

In this paper the diffusion equation is solved in two-dimensional geometry by the dual reciprocity boundary element method (DRBEM). It is structured by fully implicit discretization over time and by weighting with the fundamental solution of the Laplace equation. The resulting domain integral of the diffusive term is transformed into two boundary integrals by using Green's second identity, and the domain integral of the transience term is converted into a finite series of boundary integrals by using dual reciprocity interpolation based on scaled augmented thin plate spline global approximation functions. Straight line geometry and constant field shape functions for boundary discretization are employed. The described procedure results in systems of equations with fully populated unsymmetric matrices. In the case of solving large problems, the solution of these systems by direct methods may be very time consuming. The present study investigates the possibility of using iterative methods for solving these systems of equations. It was demonstrated that Krylov-type methods like CGS and GMRES with simple Jacobi preconditioning appeared to be efficient and robust with respect to the problem size and time step magnitude. This paper can be considered as a logical starting point for research of iterative solutions to DRBEM systems of equations. © 1998 John Wiley & Sons, Ltd.     

Improved element-free Galerkin method for two-dimensional potential problems

Potential difficulties arise in connection with various physical and engineering problems in which the functions satisfy a given partial differential equation and particular boundary conditions. These problems are independent of time and involve only space coordinates, as in Poisson's equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, they usually cannot be solved with analytical solutions. The element-free Galerkin (EFG) method is a meshless method for solving partial differential equations on which the trial and test functions employed in the discretization process result from moving least-squares (MLS) interpolants. In this paper, by using the weighted orthogonal basis function to construct the MLS interpolants, we derive the formulae of an improved EFG (IEFG) method for two-dimensional potential problems. There are fewer coefficients in the improved MLS (IMLS) approximation than in the MLS approximation, and in the IEFG method fewer nodes are selected in the entire domain than in the conventional EFG method. Hence, the IEFG method should result in a higher computing speed.     

A semi‐analytic collocation technique for steady‐state strongly nonlinear advection‐diffusion‐reaction equations with variable coefficients

The aim of this paper is to present a new semi‐analytic numerical method for strongly nonlinear steady‐state advection‐diffusion‐reaction equation (ADRE) in arbitrary 2‐D domains. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. Each basis function used in the algorithm is a sum of an analytic basis function and a special correcting function which is chosen to satisfy the homogeneous boundary conditions of the problem. The polynomials, trigonometric functions, conical radial basis functions, and the multiquadric radial basis functions are used in approximation of the ADRE. This allows us to seek an approximate solution in the analytic form which satisfies the boundary conditions of the initial problem with any choice of free parameters. As a result, we separate the approximation of the boundary conditions and the approximation of the ADRE inside the solution domain. The numerical examples confirm the high accuracy and efficiency of the proposed method in solving strongly nonlinear equations in an arbitrary domain.     

Solving high‐order partial differential equations with indirect radial basis function networks

This paper reports a new numerical method based on radial basis function networks (RBFNs) for solving high‐order partial differential equations (PDEs). The variables and their derivatives in the governing equations are represented by integrated RBFNs. The use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high‐order derivatives. The proposed RBFN method is verified successfully through the solution of thin‐plate bending and viscous flow problems which are governed by biharmonic equations. For thermally driven cavity flows, the solutions are obtained up to a high Rayleigh number of 10⁷. Copyright © 2005 John Wiley & Sons, Ltd.     

Radial integration BEM for transient heat conduction problems

In this paper, a new boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method. The normalized temperature is introduced to formulate integral equations, which makes the representation very simple and having no temperature gradients involved. The Green's function for the Laplace equation is adopted in deriving basic integral equations for time-dependent problems with varying heat conductivities and, as a result, domain integrals are involved in the derived integral equations. The radial integration method is employed to convert the domain integrals into equivalent boundary integrals. Based on the central finite difference technique, an implicit time marching solution scheme is developed for solving the time-dependent system of equations. Numerical examples are given to demonstrate the correctness of the presented approach.     

Meshless boundary integral equations with equilibrium satisfaction

A known feature of any mixed interpolation boundary integral equations (BIE)-based methods is that equilibrium is not generally guaranteed in the numerical solution. Here, a complete meshless technique, based on the boundary element-free method (BEFM) with complete equilibrium satisfaction for 2D elastostatic analysis is proposed. The BEFM adopted is a meshless method based on boundary integral equations such as local boundary integral equation (LBIE) method and boundary node method (BNM), differing from them with respect to the integration domain and the approximation scheme.     

Time-domain transient elastodynamic analysis of 3-D solids by BEM

The numerical implementation of the Direct Boundary Element formulation for time-domain transient analysis of three-dimensional solids is presented in a most general and complete manner. The present formulation employs the space and time dependent fundamental solution (Stokes' solution) and Graffi's dynamic reciprocal theorem to derive the boundary integral equations in the time domain. A time-stepping scheme is then used to solve the boundary initial value problem by marching forward in time. Higher order shape functions are used to approximate the field quantities in space as well as in time, and a combination of analytical (time-integration) and numerical (spatial-integration) integration is carried out to form a system of linear equations. At the end of each time step, these equations are solved to obtain the unknown field quantities at that time. Finally, the accuracy and reliability of this algorithm is demonstrated by solving a number of example problems and comparing the results against the available analytical and numerical solution.     

Trefftz Methods for Time Dependent Partial Differential Equations

In this paper we present a mesh-free approach to numerically solving a class of second order time dependent partial differential equations which include equations of parabolic, hyperbolic and parabolic-hyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to standard reaction-diffusion and wave equation forms. To solve initial boundary value problems for these equations, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a sequence of boundary value problems for inhomogeneous modified Helmholtz equations. These boundary value problems are then solved by a combination of the method of particular solutions and Trefftz methods. To do this, a variety of techniques is proposed for numerically computing a particular solution for the inhomogeneous modified Helmholtz equation. Here, we focus on the Dual Reciprocity Method where the source term is approximated by radial basis functions, polynomial or trigonometric functions. Analytic particular solutions are presented for each of these approximations. The Trefftz method is then used to solve the resulting homogenous equation obtained after the approximate particular solution is subtracted off. Two types of Trefftz bases are considered, F-Trefftz bases based on the fundamental solution of the modified Helmholtz equation, and T-Trefftz bases based on separation of variables solutions. Various techniques for satisfying the boundary conditions are considered, and a discussion is given of techniques for mitigating the ill-conditioning of the resulting linear systems. Finally, some numerical results are presented illustrating the accuracy and efficacy of this methodology.     

Numerical operational methods for time-dependent linear problems

A general and systematic discussion on the use of the operational method of Laplace transform for numerically solving complex time-dependent linear problems is presented. Application of Laplace transform with respect to time on the governing differential equations as well as the boundary and initial conditions of the problem reduces it to one independent of time, which is solved in the transform domain by any convenient numerical technique, such as the finite element method, the finite difference method or the boundary integral equation method. Finally, the time domain solution is obtained by a numerical inversion of the transformed solution. Eight existing methods of numerical inversion of the Laplace transform are systematically discussed with respect to their use, range of applicability, accuracy and computational efficiency on the basis of some framework vibration problems. Other applications of the Laplace transform method in conjunction with the finite element method or the boundary integral equation method in the areas of earthquake dynamic response of frameworks, thermaliy induced beam vibrations, forced vibrations of cylindrical shells, dynamic stress concentrations around holes in plates and viscoelastic stress analysis are also briefly described to demonstrate the generality and advantages of the method against other known methods.     

Sensitivity analysis and optimal design in charge transport problems

The focus of this paper is the accurate and efficient determination of sensitivities of electrostatic potential and charge density in charge transport problems, and the use of these sensitivities to carry out optimal design. Direct differentiations of the boundary integral formulation of Poisson's equation for charge conservation and of the non-linear partial differential equation for current continuity are carried out to obtain equations satisfied by the sensitivities. Methods for solving the sensitivity equations are discussed. A numerical implementation of the methods is validated through several examples. It is demonstrated that the Design Sensitivity Coefficients (DSCs) of the quantities of interest in charge transport are obtained accurately and that optimal design problems can be solved efficiently using these DSCs.     

Solution of Poisson's equation by iterative DRBEM using compactly supported,positive definite radial basis function

In the numerical solution of three-dimensional boundary value problems, the matrix size can be so large that it is beyond a computer's capacity to solve it. To overcome this difficulty, an iterative dual reciprocity boundary element method (DRBEM) is developed to solve Poisson's equation without the need of assembling a matrix. The DRBEM procedure requires that the right hand side of Poisson's equation be approximated by a radial basis function interpolation. In the iterative solution, it is found that only compactly supported, positive definite radial basis functions lead to converged results.     

Integral equation method for soulution of boundary value problems of structural mechanics,Part I. Ordinary differential equations

This paper presents a method of numerical solution of boundary value problems governed by a set of ordinary differential equations. The highest derivative is chosen as the unknown function. Governing equations are transformed into a set of integral equations. The kernels of integral equations turn out to be influence functions for deflection and/or bending moments of a corresponding beam, and can, therefore, be computed using well-known methods of structural analysis. Finally, the unknown function (highest derivative) is approximated in a defined manner and the solution is obtained through numerical integration.     

The method of moments and its optimization

The method of moments is a semidiscrete numerical method for solving partial differential equations. The method approximates the solution of a partial differential equation by a finite sum of products of two functions. One function in the product is an unknown function of a single variable and the other function (moment function) is a prescribed function in the remaining variables. Using variational technique we obtain a finite system of boundary value problems of ordinary differential equations for the unknown functions. The main goal of this paper is the study of the theoretical background and numerical effectiveness of the method of moments for solving linear partial differential equations on rectangular-like domains. The mathematical formulation of the method together with error estimates and the theory of optimal moment functions are given. If for the one-dimensional moment functions piecewise polynomials of degree K are used then finite element type error bounds are obtained for the approximate solution in two dimensions. We also consider the numerical implementation of the method through the factorization method and efficient initial value methods. Several numerical examples showing the efficiency of the method are presented.     

On the use of radial basis functions in the solution of elliptic boundary value problems

A numerical technique for solving elliptic boundary value problems is developed. The method expands the solution in terms of radial basis functions and chooses expansion coefficients such that the governing equations and boundary conditions are approximately satisfied. The method is demonstrated through its application to some non-trivial problems involving the Laplace and biharmonic equations.     

An advanced boundary element method for solving 2D and 3D static problems in Mindlin's strain‐gradient theory of elasticity

An advanced boundary element method (BEM) for solving two‐ (2D) and three‐dimensional (3D) problems in materials with microstructural effects is presented. The analysis is performed in the context of Mindlin's Form‐II gradient elastic theory. The fundamental solution of the equilibrium partial differential equation is explicitly derived. The integral representation of the problem, consisting of two boundary integral equations, one for displacements and the other for its normal derivative, is developed. The global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2D and 3D problems, respectively. Representative 2D and 3D numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2D problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Inverse fracture problems in piezoelectric solids by local integral equation method

The meshless local Petrov–Galerkin (MLPG) method is used to solve the inverse fracture problems in two-dimensional (2D) piezoelectric body. Electrical boundary conditions on the crack surfaces are not specified due to unknown dielectric permittivity of the medium inside the crack. Both stationary and transient dynamic boundary conditions are considered here. The analyzed domain is covered by small circular subdomains surrounding nodes spread randomly over the analyzed domain. A unit step function is chosen as test function in deriving the local integral equations (LIE) on the boundaries of the chosen subdomains. The Laplace-transform technique is applied to eliminate the time variation in the governing equation. The local integral equations are nonsingular and take a very simple form. The spatial variation of the Laplace transforms of displacements and electrical potential are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. The singular value decomposition (SVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after MLS approximation. The Stehfest algorithm is applied for the numerical Laplace inversion to retrieve the time-dependent solutions.     

Radial integration boundary integral and integro-differential equation methods for two-dimensional heat conduction problems with variable coefficients

This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional heat conduction problems with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Numerical solution of two-dimensional mixed problems with variable coefficients by the boundary-domain integral and integro-differential equation methods

This paper presents a formulation of the boundary-domain integral equation (BDIE) and the boundary-domain integro-differential equation (BDIDE) methods for the numerical solution of two-dimensional mixed boundary-value problems (BVP) for a second-order linear elliptic partial differential equation (PDE) with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the BVP to a BDIE or BDIDE. The numerical formulation of the BDIDE employs an approximation for the boundary fluxes in terms of the potential function within the domain cells; therefore, the solution is fully described in terms of the variation of the potential function along the boundary and domain. Linear basis functions localised on triangular elements and standard quadrature rules are used for the calculation of boundary integrals. For the domain integrals, we have implemented Gaussian quadrature rules for two dimensions with Duffy transformation, by mapping the triangles into squares and eliminating the weak singularity in the process. Numerical examples are presented for several simple problems with square and circular domains, for which exact solutions are available. It is shown that the present method produces accurate results even with coarse meshes. The numerical results also show that high rates of convergence are obtained with mesh refinement.