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In this paper, a meshless discrete scheme based on the generalized finite difference method (GFDM) is proposed to solve the biharmonic interface problem. This scheme turns the interface problem to be two non-interface subproblems coupled with the interface conditions. The interface conditions act as the boundary conditions of the subproblems because the interface plays the role of their boundary. Therefore, the proposed method has the advantage of dealing with problems with complex geometrical interfaces. Due to the GFDM approximates the derivatives of the unknown variables by using a linear combination of the values for nearby nodes, the proposed method also has the advantage in dealing with the interface condition with the jump of derivatives. Four numerical examples are provided to verify the accuracy and the stability of the GFDM for biharmonic interface problems. The numerical results show that the H1 errors can reach almost the same convergence rate of the L∞ and L2 errors. The convergence rates are almost and even higher than fourth-order when the fourth-order Taylor Series expansion is adopted in the GFDM.

The second moment of inertia of a continuous axisymmetric object with an arbitrary shape made of a nonhomogeneous material is usually calculated by dividing it into small domains. However, it is a burdensome process to specify the density of the small domains. In this paper, a technique of easily calculating the second moment of inertia of an axisymmetric nonhomogeneous material using boundary integral equations is proposed. The calculations of the mass, primary moment, and center of mass of an arbitrarily shaped object made of a nonhomogeneous material are also shown. A formulization of the boundary element method is utilized, and a technique for the direct numerical integration of the axisymmetric domain using an axisymmetric interpolation method without the need to carry out domain division is proposed. Heterogeneous materials include laminated material composites, in which the density distribution is discontinuous. Axisymmetric interpolation using harmonic and biharmonic functions has a weak point that can be easily overcome using a scale method.

In this study, we develop an approximate formulation for a generalized form of the biharmonic problem based on pseudospectral meshless radial point interpolation (PSMRPI). The boundary conditions are considered as simply supported or clamped, with application to the theory of static analysis of thin-plates. The rigorous steps to analyze such problem are defining the high order derivatives, implementing multiple boundary conditions especially when the geometry of the domain of the problem is complex. In PSMRPI method the nodal points do not need to be regularly distributed and can even be quite arbitrary. It is easy to have high order derivatives of unknowns in terms of the values at nodal points by constructing operational matrices. Furthermore, it is observed that the multiple boundary conditions can be imposed by applying PSMRPI on nodal points near the boundaries of the domain. The main results on the generalized biharmonic problem are demonstrated by some examples to show the validity and trustworthiness of PSMRPI technique. Also, a comparison with the previously standard studied method for the biharmonic problem is done.

In this paper, three meshless methods are proposed to solve boundary value problems. The special purpose Trefftz function method, the method of fundamental solutions and the symmetry method of fundamental solutions are compared. We considered six numerical examples for harmonic and biharmonic problems as a flow through the cylindrical fibers arranged in a regular square array, deflection of a uniformly loaded square plate and torsion of a bar. The boundary value problems defined for a repeating element of the domain are solved by use of the collocation technique. The accuracy and stability of compared methods are investigated. Root mean square error on the boundary and condition number of the matrix are presented as functions of a number of collocation points.

In this paper, we propose an improved Generalize Finite Integration Method with Volterra operator to solve multi-dimensional biharmonic equations. The novelty of the proposed method is that its mth order integration matrix will be exactly formulated by using Volterra operator. This leads to a significant improvement in the accuracy of the approximation. Furthermore, the original strict requirement of uniformly distributed collocation points is waived in the proposed method due to the use of piecewise polynomials. This contributes to another distinct advantage in placing collocation points for problems with geometrically complex domains. Four numerical examples are constructed to demonstrate these advantages. Comparisons on the accuracies obtained from quadrature formulas with and without Volterra operator are given. Numerical results indicate that the proposed method is capable for solving multi-dimensional biharmonic equations with superior efficiency, higher accuracy and fewer memory usage.

Solving fourth or higher order differential equations using localized numerical methods as the finite element, the finite difference and the localized radial basis functions (LRBFs), show difficulties to get accurate results. So many authors adopted iterative methods to deal with biharmonic equation by decoupling it into two Poisson’s problems. In this paper we investigate the formulation of a mixed fourth order boundary value problem as an optimal control one. Then, we establish a new iterative method by coupling an optimization iterative scheme and localized radial basis functions meshless collocation method to deal with the numerical solution of such problem. To transform the problem into an optimal control one, we firstly construct the constraints functions by splitting the biharmonic equation into two coupled Laplace equations. The Neumann boundary condition is used as energy-like error functional to be minimized. Theoretical analysis of the existence and uniqueness of the solution of such formulated optimization problem and its equivalence to the initial biharmonic problem are also demonstrated. Finally we show the effectiveness of the proposed method by solving problems in both convex and non-convex regular and irregular domain.

The localized method of fundamental solutions (LMFS) is proposed in this paper for solving two-dimensional boundary value problems, governed by Laplace and biharmonic equations, in complicated domains. Traditionally, the method of fundamental solutions (MFS) is a global method and the resultant matrix is dense and ill-conditioned. In this paper, it is the first time that the LMFS, the localized version of the MFS, is proposed. In the LMFS, the solutions at every interior node are expressed as linear combinations of solutions at some nearby nodes, while the numerical procedures of MFS are implemented within every local subdomain. The satisfactions of governing equation at interior nodes and boundary conditions at boundary nodes can yield a sparse system of linear algebraic equations, so the numerical solutions can be efficiently acquired by solving the resultant sparse system. Six numerical examples are given to demonstrate the effectiveness of the proposed LMFS.

The problem of viscous incompressible flow in a periodic cell with a porous body is solved. The Stokes flow model is adopted to describe the flow outside the body and the Brinkman equation is applied to find the filtration velocity field inside the porous domain. The conditions on the boundary between the free fluid and the porous medium for the porous body of arbitrary shape are obtained. The boundary value problem for the joint solution of the biharmonic and Brinkman equations for the stream functions outside and inside the porous body are then solved using a boundary element method. Good agreement of the numerical and analytical models for the Kuwabara circular cell model is shown for the fluid flow through a porous circular cylinder. The fluid flow past a circular, square, triangular cylinders and a circular body of uneven surface (an idealized model of a viral capsid) in a rectangular periodic cell are calculated. Comparison of the results obtained with the numerical solution from a CFD ANSYS/FLUENT model shows good accuracy of the developed mathematical model.

In this paper, a boundary element method is applied to buckling analysis of thin plates with arbitrary shapes under various load types. The governing differential equation is converted into equivalent integral equation using the static fundamental solutions of the biharmonic equation. The arising domain integrals due to in-plane stresses are evaluated applying the radial integration method. The in-plane stresses are implemented through Gaussian integration points along radial direction in the convex domains. For the concave domains, an idea has been introduced to compute radial integration along new direction from an auxiliary point to the field point. This method can be easily applied to buckling analysis of thin plates with arbitrarily distributed and concentrated edge loading. Six sample problems are presented to illustrate the effectiveness and accuracy of the proposed method.

In this paper, the geometrically nonlinear bending analysis of nanoplates with general shapes and boundary conditions is highlighted. The governing equations are derived based on the classical plate theory using nonlocal differential constitutive relation of Eringen and von Karman's nonlinear strains. The boundary-only method is developed by using the principle of the analog equation (PAE). According to the PAE, the original governing differential equations are replaced by three uncoupled equations with fictitious sources under the same boundary conditions, namely two Poisson equations and one biharmonic equation. Subsequently, the fictitious sources are established using a technique based on the boundary element method and approximated by using the radial basis functions. The solution of the actual problem is attained from the known integral representations of the potential and plate problems. Therefore, the kernels of the boundary integral equations are conveniently established and readily calculated that the complex nanoplates can be easily analyzed. The accuracy of the proposed methodology is evaluated by comparing the obtained results with available solutions. Moreover, the influences of nonlocal parameter on the various characteristics of effective distributed loads are elucidated. Finally, the effects of nonlocal parameter, von Karman's nonlinearity and aspect ratio on nonlinear bending responses are studied.

The method of fundamental solutions (MFS) has been known as an effective boundary meshless method for solving homogeneous differential equations with smooth boundary conditions and boundary shapes. Despite many attractive features of the MFS, the determination of the source location and the boundaries with sharp corners still pose a certain degree of challenges. In this paper, we revisit another powerful boundary method, the method of approximate fundamental solutions (MAFS), which approximates the fundamental solution using trigonometric functions. In the MAFS, the fundamental solutions for various governed equations can be easily constructed. The placement of the source points is also simple. In this paper, we will apply the MAFS for solving the Laplace equation with non-harmonic boundary conditions and the biharmonic equation with non-biharmonic boundary conditions with highly irregular or non-smooth domains. We will compare the performance of the MAFS and the MFS in these types of problems.

A higher-order boundary element procedure is presented for the free vibration analysis of flexible base plates of rigid fluid storage tanks resting on elastic foundation. The main principles of the procedure are replacing the biharmonic operator of the thin plate vibration problem by two successive harmonic operators, and representing the forcing terms (plate inertia, fluid loading and foundation influence) by applying the dual reciprocity boundary element formulation. The fluid effect on the plate dynamics is incorporated into the analysis by invoking another boundary element solution, which expresses the fluid pressure over the plate surface in terms of plate deflection. The performance of the method is thoroughly investigated and the nature of dynamic plate–foundation–fluid interaction is studied from several perspectives. The method provides excellent predictions, within the limits of Kirchhoff plate, potential flow, and Pasternak foundation models.

In this paper, the meshless singular boundary method (SBM) is first presented for the numerical solution of biharmonic problems. The SBM uses the source intensity factor to isolate the singularity of fundamental solutions and an inverse interpolation technique (IIT) to compute the source intensity factor, and thus avoids the fictitious boundary required in the method of fundamental solutions. However, the IIT may worsen the performance of the SBM. Then, a further development of the improved SBM (ISBM) is also presented for boundary-only analysis of biharmonic problems. In the ISBM, a new regularization technique is developed to compute the source intensity factor, which leads to higher computational precision and better numerical stability. Some numerical examples illustrate the efficiency and accuracy of both the SBM and the ISBM.

In the present paper, the inverse Cauchy problems of Laplace equation and biharmonic equation are transformed, by using the method of fundamental solutions (MFS) and the Trefftz method (TM), to the systems of linear equations for determining the expansion coefficients. Then, we propose three different conditioners together with the conjugate gradient method (CGM) to solve the resultant ill-posed linear systems. They are the post-conditioning CGM and the pre-conditioning CGM based on the idea of equilibrated norm for the conditioned matrices, as well as a minimum-distance conditioner. These algorithms are convergent fast and accurate by solving the inverse Cauchy problems under random noise.

In this paper we numerically solve both the direct and the inverse Cauchy problems of biharmonic equation by using a multiple-scale Trefftz method (TM). The approximate solution is expressed to be a linear combination of T-complete bases, and the unknown coefficients are determined to satisfy the boundary conditions, by solving a resultant linear equations system. We introduce a better multiple-scale in the T-complete bases by using the concept of equilibrated norm of the coefficient matrix, such that the explicit formulas of these multiple scales can be derived. The condition number of the coefficient matrix can be significantly reduced upon using these better scales; hence, the present multiple-scale Trefftz method (MSTM) can effectively solve the inverse Cauchy problem without needing of the overspecified data, which is an incomplete Cauchy problem . Numerical examples reveal the efficiency that the new method can provide a highly accurate numerical solution even the problem domain might have a corner singularity, and the given boundary data are subjected to a large random noise.

A fast multipole boundary element method (BEM) for solving large-scale thin plate bending problems is presented in this paper. The method is based on the Kirchhoff thin plate bending theory and the biharmonic equation governing the deflection of the plate. First, the direct boundary integral equations and the conventional BEM for thin plate bending problems are reviewed. Second, the complex notation of the kernel functions, expansions and translations in the fast multipole BEM are presented. Finally, a few numerical examples are presented to show the accuracy and efficiency of the fast multipole BEM in solving thin plate bending problems. The bending rigidity of a perforated plate is evaluated using the developed code. It is shown that the fast multipole BEM can be applied to solve plate bending problems with good accuracy. Possible improvements in the efficiency of the method are discussed.

We develop and compare a number of mesh-free formulations for solution of plate bending problems using either the Moving Least Square-approximation or Point Interpolation Method-approximation. For solution of the original biharmonic problem, we develop only the local weak formulation to avoid the 4th order derivatives of deflections. The decomposition of the biharmonic operator leads to lower order derivatives of field variables in both the developed strong and local weak formulations which are applicable to arbitrary boundary value problems for thin plate bending. The modified differentiation technique is proposed, in order to increase the accuracy of higher order derivatives of field variables. The accuracy, convergence of accuracy and computational efficiency are studied in simple boundary value problems for circular plate. The discussed methods give reasonable numerical results when applied to decomposed problem, while the methods applied to original biharmonic problem either fail or give unreliable results.

A method is proposed for the construction of Green's functions for the Sophie Germain equation in regions of irregular shape with mixed boundary conditions imposed. The method is based on the boundary integral equation approach where a kernel vector function B satisfies the biharmonic equation inside the region. This leads to a regular boundary integral equation where the compensating loads and moments are applied to the boundary. Green's function is consequently expressed in terms of the kernel vector function B , the fundamental solution function of the biharmonic equation, and kernel functions of the inverse regular integral operators. To compute moments and forces, the kernel functions are differentiated under the integral sign. The proposed method appears highly effective in computing both displacements and stress components.

The regular hybrid boundary node method (RHBNM) is a new technique for the numerical solutions of the boundary value problems. By coupling the moving least squares (MLS) approximation with a modified functional, the RHBNM retains the meshless attribute and the reduced dimensionality advantage. Besides, since the source points of the fundamental solutions are located outside the domain, ‘boundary layer effect’ is also avoided. However, an initial restriction of the present method is that it is only suitable for the problems which the governing differential equation is in second order. Now, a new variational formulation for the RHBNM is presented further to solve the biharmonic problems, in which the governing differential equation is in fourth order. The modified variational functional is applied to form the discrete equations of the RHBNM. The MLS is employed to approximate the boundary variables, while the domain variables are interpolated by a linear combination of fundamental solutions of both the biharmonic equation and Laplace’s equation. Numerical examples for some biharmonic problems show that the high accuracy with a small node number is achievable. Furthermore, the computation parameters have been studied. They can be chosen in a wide range and have little influence on the results. It is shown that the present method is effective and can be widely applied in practical engineering.

We propose a geometric modeling method in R3 based on the so-called potential field (PF) modeling technique. The method is a new technique for surface reconstruction from a data set of scattered points taken on a surface. In this method, the construction of a geometric model is based on the solution of an elliptic boundary value problem determined using the method of fundamental solutions (MFS). We consider two such problems, Laplace's equation subject to Dirichlet boundary conditions and the biharmonic Dirichlet problem. We present the results of numerical experiments which demonstrate the efficacy of these approaches. We also identify differences and potential difficulties in the application of these approaches and propose ways of addressing them. © 2009 Elsevier Ltd. All rights reserved. 33 12 1403 1410 Cited By :3

A Galerkin boundary node method (GBNM) is developed in this paper for solving biharmonic problems. The GBNM combines an equivalent variational form of boundary integral formulations for governing equations with the moving least-squares approximations for construction of the trial and test functions. In this approach, only a nodal data structure on the boundary of a domain is required. In addition, boundary conditions can be implemented accurately and the system matrices are symmetric. The convergence of this method and numerical examples are given to show the efficiency.

Differential quadrature Trefftz method (DQTM) is developed to deal with plate problems defined in irregular domains. DQTM divides the solution into two parts, a particular solution for inhomogeneous biharmonic equation and the general solution for homogeneous biharmonic equation. For the former, differential quadrature method based on the interpolation of the highest derivative (DQIHD) is involved. For the latter, polynomial basis functions are adopted instead of fundamental solutions. We will show that DQTM not only keeps the advantages of traditional differential quadrature method (DQM), high efficiency and accuracy, but also has no difficulties to deal with geometrically irregular domains.

A boundary element method (BEM)-based variational method is presented for the solution of elliptic PDEs describing the mechanical response of general inhomogeneous anisotropic bodies of arbitrary geometry. The equations, which in general have variable coefficients, may be linear or nonlinear. Using the concept of the analog equation of Katsikadelis the original equation is converted into a linear membrane (Poisson) or a linear plate (biharmonic) equation, depending on the order of the PDE under a fictitious load, which is approximated with radial basis function series of multiquadric (MQ) type. The integral representation of the solution of the substitute equation yields shape functions, which are global and satisfy both essential and natural boundary conditions, hence the name generalized Ritz method . The minimization of the functional that produces the PDE as the associated Euler–Lagrange equation yields not only the Ritz coefficients but also permits the evaluation of optimal values for the shape parameters of the MQs as well as optimal position of their centers, minimizing thus the error. If a functional does not exists or cannot be constructed as it is the usual case of nonlinear PDEs, the Galerkin principle can be applied. Since the arising domain integrals are converted into boundary line integrals, the method is boundary-only and, therefore, it maintains all the advantages of the pure BEM. Example problems are studied, which illustrate the method and demonstrate its efficiency and great accuracy.

In this paper, we apply the effective condition number in Huang and Li [Effective condition number and superconvergence of Trefftz method coupled with high order FEM for singularity problems, Eng Anal Bound Elem 2006;30:270–83] to the penalty Trefftz methods (TMs) with high order elements such as Adini's elements for Poisson's equations with singularities. The best superconvergence rates O ( h 3.5 ) in H 1 norm can be reached, where h is the maximal boundary length of elements. For the penalty and the penalty plus hybrid Trefftz methods, the bounds of effective condition numbers retain the same. However, for the penalty TM combination, not only are the algorithms simple, but also the numerical solutions are more accurate. Hence we prefer the penalty TM combination to the penalty plus hybrid TM combination. Such a recommendation is based on the effective condition number, but not on the traditional condition number, based on which the penalty TM combination was declined in Li and Yan [Global superconvergence for blending surfaces by boundary penalty plus hybrid FEMs for biharmonic equations. Appl Numer Math 2001;39:61–85]. The new stability analysis in this paper by means of effective condition number may re-evaluate the existing numerical methods; this paper reports the results of Motz's problem. Note that in Huang and Li [Effective condition number and superconvergence of Trefftz method coupled with high order FEM for singularity problems, Eng Anal Bound Elem 2006;30:270–83], no stability analysis was provided.

The solution of the equations governing the steady incompressible slow viscous fluid flow is analysed using a novel technique based on a Laplacian decomposition instead of the more traditional approaches based on the biharmonic streamfunction formulation or the velocity-pressure formulation. This results in the need to solve the Laplace equations for the pressure and other auxiliary harmonic functions which arise from the ideas of Almansi's decomposition. These equations, which become coupled through the boundary conditions, are numerically solved using the boundary element method (BEM). Results both on the boundary and inside the solution domain are presented and discussed for a simple benchmark test example and a few applications in smooth and non-smooth geometries in order to illustrate that the Laplacian decomposition in combination with BEM provides an efficient technique, in terms of accuracy and convergence, to investigate numerically a Stokes flow.

This paper reports a new boundary-integral-equation method (BIEM) for numerically solving biharmonic problems with Dirichlet boundary conditions. For the solution of these problems in convex polygons, it was found that the accuracy of the conventional BIEM is significantly reduced, and spurious oscillatory behaviour is often observed in the boundary solutions especially for areas near corners (Mai-Duy N, Tanner RI. An effective high order interpolation scheme in BIEM for biharmonic boundary value problems. Eng Anal Bound Elem 2005;29:210–23). In this study, a new treatment for these difficulties is proposed. The unknown functions in boundary integrals are approximated using a domain-type interpolation scheme rather than traditional boundary-type interpolation schemes. Two test problems are considered to validate the formulation and to demonstrate the attractiveness of the proposed method.

This paper reports a new indirect radial-basis-function (RBF) collocation method for numerically solving biharmonic boundary-value problems. The RBF approximations are constructed through an integration process. The new feature here is that integration constants are eliminated pointwise using the prescribed boundary conditions rather than employed as network weights. This treatment of integration constants is particularly well suited for the case of discretizing the governing differential equations on a set of scattered points. The proposed method is verified through the solution of benchmark test problems. Accurate results and high convergence rates are obtained.

This paper presents an effective high order boundary integral equation method (BIEM) for the solution of biharmonic equations. All boundary values including geometries are approximated by high order radial basis function networks (RBFNs) rather than the conventional low order Lagrange interpolation schemes. For a better quality of approximation, the networks representing the boundary values and their derivatives are constructed by using integration processes. Prior conversions of network weights into nodal variable values are employed in order to form a square system of equations. Numerical results show that the proposed BIEM attains a great improvement in solution accuracy, convergence rate and computational efficiency over the linear- and quadratic-BIEMs.

The purpose of this paper is to extend the boundary approximation method proposed by Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], i.e. the collocation Trefftz method called in this paper, for biharmonic equations with singularities. First, this paper derives the Green formulas for biharmonic equations on bounded domains with a non-smooth boundary, and corner terms are developed. The Green formulas are important to provide all the exterior and interior boundary conditions which will be used in the collocation Trefftz method. Second, this paper proposes three crack models (called Models I, II and III), and the collocation Trefftz method provides their most accurate solutions. In fact, Models I and II resemble Motz's problem in Li et al. [SIAM J. Numer. Anal. 24 (1987) 487], and Model III with all the clamped boundary conditions originated from Schiff et al. [The mathematics of finite elements and applications III, 1979]. Moreover, effects on d 1 of different boundary conditions are investigated, and a brief analysis of error bounds for the collocation Trefftz method is made. Since accuracy of the solutions obtained in this paper is very high, they can be used as the typical models in testing numerical methods. The computed results show that as the singularity models, Models I and II are superior to Model III, because more accurate solutions can be obtained by the collocation Trefftz method.

An adaptive scheme for the boundary element method (BEM), based on an error estimate using a sample point error analysis, is applied to a bending problem for a thin elastic plate. Both the boundary element refinement and relocation, i.e. the h- and r-version adaptive schemes, are implemented by employing the non-dimensional error indicator proposed in the present paper. Effectiveness of the proposed adaptive strategy for the boundary element plate bending analysis is demonstrated through some numerical applications of plate bending.

The scattered data interpolation problem is investigated. Instead of the direct use of radial basis functions, the interpolation function is sought as a solution of a higher order partial differential equation supplied with the interpolation equations as special boundary conditions. In this paper the methods based on the biharmonic and the bi-Helmholtz equations are analysed. The interpolation problem is reformulated in variational forms. Existence and uniqueness theorems are proved in Sobolev spaces. The approximation properties of this interpolation are also investigated. A representation theorem is proved which shows the similarity to the method of radial basis functions based on the fundamental solution of the applied partial differential operator. To solve the appearing biharmonic/bi-Helmholtz equation, a multi-level method is presented which is based on a quadtree/octtree cell system generated by the interpolation points. It is shown that the overall computational cost of the presented method is much less than that of the traditional method of radial basis functions. The method makes it possible to avoid the solution of large, fully populated and often ill-conditioned systems of linear equations. Finally, some applications to solving partial differential equations are outlined. The biharmonic/bi-Helmholtz interpolation technique immediately defines a grid-free method, but can be combined with the boundary element method as well. A possible application in the dual reciprocity method is also presented.

Particular solutions of higher order radial basis functions of conical and spline types are obtained for the Laplacian, Helmholtz-type, and polyharmonic operators. These particular solutions are needed in the implementation of the Dual Reciprocity Boundary Element Method.

Advanced multiple reciprocity boundary element method formulations are developed for solution of certain plate bending problems when either the fundamental solutions are not available or these are rather complex. The present formulation preserves a pure boundary character and the derived boundary integral equations are nonsingular. The fundamental solutions as well as all the integral kernels involved have been derived and are available.

The problem of thin plates according to Kirchhoff's theory is formulated by means of two coupled Poisson equations, which are expressed in integral form using the second theorem of Green in the classical way. The domain integrals are evaluated approximating the integrands by a series of simple domain functions whose coefficients are calculated by a collocation procedure at points placed along the boundary and domain, which originates a certain extra number of unknowns. The approximated domain integrals are then expressed by equivalent boundary integrals, the extra unknowns requiring the use of integral equations associated to the internal points used to approximate the domain integrals. This formulation avoids the problems associated with the singular character of the fundamental solution of the biharmonic equation.

An adaptive boundary element method based on an error estimate using sample point error analysis is applied to a bending problem for a thin elastic plate. The errors on each boundary element are estimated on the basis of the magnitude of the solution discrepancy except at the boundary nodes employed for the analysis. A boundary element refinement, the h -version adaptive scheme, is implemented by employing the nondimensional error indicator composed of two different kinds of indicators (extended error indicators), of which the dimensions are of deflection and of slope. The properties of the nondimensional error indicators are examined through some simple numerical examples. The effectiveness of the proposed adaptive solution procedure is demonstrated through analyses of square plates with various edge conditions.

A boundary element method is developed for the analysis of thick plates. The behaviour of the plate is described by Reissner's equations which permit the satisfaction of three boundary conditions. The solution is expressed in terms of two potentials, one biharmonic and one Bessel's. The potentials are established from their integral representations after the solution of six boundary equations, three integral and three differential. Numerical results are presented for various edge conditions and are compared against analytical or finite element solutions to illustrate the accuracy and the effectiveness of the proposed method.

The problem of flow separation at a corner which results in a free surface being formed has been investigated using the boundary element method (BEM). The free surface is approximated by a series of analytical functions and solutions are obtained over a wide range of values of the surface tension and fluid flow rates. The contact angle is determined and a comparison with values which have either been assumed, or determined, by previous authors is made and discussed.