A Spectral Boundary Element Method

A new computational model by integrating the boundary element method and the compactly supported radial basis functions (CSRBF) is developed for three-dimensional (3D) linear elasticity with the presence of body forces. The corresponding displacement and stress particular solution kernels across the supported radius in the CSRBF are obtained for inhomogeneous term interpolation. Subsequently, the classical dual reciprocity boundary element method, in which the domain integrals due to the presence of body forces are transferred into equivalent boundary integrals, is formulated by introducing locally supported displacement and stress particular solution kernels for solving the inhomogeneous 3D linear elastic system. Finally, several examples are presented to demonstrate the accuracy and efficiency of the present method.

Radial basis function (RBF) interpolation algorithm for data transfer on coupling interface is derived from the shape function, and the global D-Lanczos (GLDL) method is introduced to solve the displacement transfer matrix. In order to improve the computational efficiency, a local compactly supported RBF (CSRBF) interpolation algorithm is proposed based on the partition acceleration treatment, and a comparison of the algorithmic complexity is made among the global Multi-Quadrics (MQ), CSRBF and the local CSRBF. Numerical test shows the significant shortening of CPU time by using the local CSRBF interpolation algorithm.

A dual reciprocity boundary element method based on global shapes functions belonging to a complete set to transform domain integrals into boundary ones is presented. Governing equations of Poisson and Helmholtz type are tested. It is demonstrated that the complete set shape functions are more efficient and accurate in approximating domain integrals than those based on incomplete set. INTRODUCTION The application of Boundary Element Method to linear partial differential operators, or to nonlinear and time-dependent ones, often leads to domain integrals that are cumbersome to handle. The dual reciprocity method pioneered by Nardini and Brebbia* is a powerful numerical technique which converts domain integrals into boundary ones using a set of global shape functions to approximate the integrand. It has been successfully applied in many problems involving dynamic, diffusion and nonlinear type operators. See Partridge, et al. for a comprehensive bibliographical review. The success of the dual reciprocity hinges on the ability to express the global shape functions into the Laplacian of a set of known functions such that the divergence theorem can be applied to resolve the domain integral into a boundary one. Such a constraint has so far limited the selection of shape functions to a set of radial functions of the form 1 + ar -f br^ -f • • •• Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 344 Boundary Elements These radial functions do not form a complete set hence may lack convergence properties. Although practical evidences have indicated that those shape functions work quite satisfactorily, it is still desirable to use shape functions from a set that is complete in the mathematical sense. This paper present a dual reciprocity BEM based on global shape functions derived from a polynomial series. Perhaps due to its mathematical completeness, we are able to demonstrate that the numerical results are more efficient and accurate than the traditional radial set. DUAL RECIPROCITY For many non-standard, transient, or nonlinear partial differential equations it is possible to extract a standard, time-independent, linear partial differential operator C and lump the remainder to the right-hand-side: jC% = 6(z, ?/, Z, f , %, . . . , %,., %„ , %,, %,, %= . • •) (1) For the purpose of BEM implementation, these non-standard portions can be formally regarded as body forces. By the application of generalized Green's theorem = / Jr (2) where v is a conjugate variable, £* is the adjoint operator of £, B» and B* are operators resulting from integration by parts, which may be viewed as generalized normal derivatives, F is the solution boundary, and fi the solution domain, a boundary integral equation can be formed as: cu = / (u*q uq*) j bu* dSl (3) where quantities denoted by asterisk are free-space Green's functions satisfying £*u* = —6, with 6 a Dirac delta function, c is a constant dependent on the Cauchy principal value integration of the singularity at the base point, q and q* are generalized normal derivatives of u and u*. To circumvent the domain integration, the function b can be approximated by an expansion based on a set of global shape functions /, such that &*fX£ (4) j=l in which a, are coefficients determined by collocation, and n^ is the number of collocation nodes. The shape functions are chosen such that particular solutions of the set of inhomogeneous equations Ciij = /,• (5) Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 345 are explicitly known. The domain integral in (3) can be transformed as [ / j^bu da = i ŷ /ju (6) j=l L ^1 J The last expression was a consequence of the application of (2). Substitution of the above into (3) yields / (u*q uq} dY = Y a, \cu, I (%*& titf) dT (7) •/r fr(. L 7r J cu We hence in principal have achieved the elimination of the domain integral. However, the detail of the implementation is somewhat dependent on the type of body force terms. We shall illustrate below the implementation by examples. EXAMPLES Operator of V^u = b(x, y) type For the governing equation of Poisson type, q in (7) is du/dn, and u* is — lnr/2%for 2-D or l/4%r for 3-D geometries. For simplicity, only two-dimensional problems will be studied herein. If the right-hand-side 6(x, y) is a mathematical expression whose particular solution is known as Up, the solution of (8) can be expressed as % = %& + %? (9) where u^ is the homogeneous solution and V2%& = 0 (10) It is apparent that Uh = u — Up satisfies the boundary integral equation c(u «„) = / [u(q q,} -(uu,)q'] (11) If we approximate the right-hand-side b(x,y) as (4), and find the particular solution term by term as V = /,(12) Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 346 Boundary Elements it is clear that j=l By the above interpretation, we have provided an alternative way for deriving (7), which is rewritten below with explicit reference to coordinates as c(x)u(x)f(u*(x',x)q(x')-u(x')q-(x',x)] dT(x') j r

A plasticity model with yield limit degradation is implemented in a boundary element program to study the fracture behavior of quasi-brittle materials. A special integration method is applied to deal with the singular integrations encountered in the volume integrals over internal cells. Strain-softening damage localizations are investigated. It is found that the damage tends to localize into a zone of one cell wide which leads to an incorrect result when different cell meshes are used in the analysis. Some sort of localization limiter has to be incorporated into the analysis in order to achieve meaningful results. INTRODUCTION In the simulation of fracture behavior of quasi-brittle materials such as concrete or ceramics, it is essential to take into account the evolution of a relative large fracture process zone forms in front of the crack tip. The process zone consists of hundreds of microcracks and its overall behavior can generally be simulated either by discrete-cracking models such as the fictitious crack model [4, 5] or smeared-cracking models such as strain-softening damage models [2, 3]. In this paper the fracture process zone is simulated by a plasticity model with yield limit degradation. When the stresses near the crack tip reach the yield surface, the yield surface will contract rather than expand. This results in a decrease in stresses during an increase in strains, i.e., a strain softening. The decrease in stresses simulates a material damage caused by mainly the formation of microcracks. Difficulty has been encountered in the finite element analysis involving strain-softening behavior. As different meshes are used in an analysis, Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 304 Boundary Elements different results are obtained. The results are not objective with respect to finite element meshes. Also, the energy dissipation of a structure at failure decreases with the refinement of the finite element mesh. It implies that the structure will fail at near zero energy dissipation when a very fine mesh is used. This is in contradiction with the reality. All these problems are considered due to damage localization [1,2,3]. In this paper, the plasticity model with yield limit degradation is incorporated into a boundary element program to study the damage localization problems. Many publications on the boundary element analysis of elastoplastic problems [7-14] have been seen since the first paper on the problem published in 1971 by Swedlow and Cruse [6]. However, the literature on the study of damage localization problems using boundary element approach is rarely seen. The initial stress boundary element method is applied to deal with the nonlinear damage problems, For localization problems, damage tends to localize into a small region of the body, and thus the internal cells required for integration purposes are needed only in a small region rather than the entire body. A special semi-analytical integration method for triangular cell is used to carry out the volume integration over the cell that contains a singular node. A rectangular panel with three different cell meshes are analyzed. It is shown that the results are not objective to the cell meshes even though all other conditions are kept the same. BOUNDARY ELEMENT FORMULATION FOR ELASTOPLASTIC ANALYSIS For elastoplastic problems based on small strain theory, the basic equations in incremental form are as follows: The governing equations are da^ + db* = 0 (1) with boundary conditions du;(x) = (x) x e I (2) cR (x) x e I I + T^T and total strain increment de^ = de^ 4de^= (du + dUjJ/2 (3) where de^ and de^ are elastic and plastic strain increments. T is the boundary of body 0 under consideration, da^ ,db; ,du;, dr; are the stress, body force (per unit volume), displacement, and traction increment respectively. du ± and df j are prescribed displacements and tractions on boundary I and r%. Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 305 The yield criterion is F(a%,k)= f(a$ <Kk) = 0 (4) where F and f are yield functions, $ is a work hardening function, and k is the work hardening parameter. The constitutive equations are de'u) = d<^ do, = DV^U (5) where D^= &pvl(l-2v)] 6fo + M<Wji + W

Compactly supported radial basis functions for shallow water equations

The boundary element method (BEM) is a numerical method for solving boundary-value or initial-value problems formulated by use of boundary integral equations. In the BEM, only the boundaries - that is, surfaces for three-dimensional problems or curves for two dimensional (2D) problems - of a problem domain need to be discretized. However the boundary element method (BEM) has been limited to solving problems with a few thousand degrees of freedom (DOFs) on a personal computer. This is because the conventional BEM, in general, produces dense and nonsymmetric matrices. The main idea of the fast multipole (FM) BEM is to employ iterative solvers to solve the BEM system of equations. Using this method we can solve models with more than one million equations on a laptop computer. In this paper, the governing equations for elasticity problems are reviewed first. Numerical examples are provided to demonstrate the accuracy and efficiencies of fast multipole method (FMM) for solving 2D elasticity problems.

The interaction of acoustic waves with submerged structures remains one of the most difficult and challenging problems in underwater acoustics. Many techniques such as coupled Boundary Element (BE)/Finite Element (FE) or coupled Infinite Element (IE)/Finite Element approximations have evolved. In this paper, we focus on the steady-state formulation only, and study a general coupled hp-adaptive BE/FE method. The work is devoted to a study on viscoelastic scattering problems in linear acoustic medium by means of hp boundary and finite element approximations. A particular emphasis is placed on an a-posteriori error estimation for the viscoelastic scattering problems.

Radial Basis Functions (RBF) are popular for interpolating scattered data. In this context, the solution of the system of linear algebraic equations (SLAE) is the most time-consuming operation. Techniques fail with large point sets consisting of more than several thousands of points when direct methods and global support are used. In this paper we demostrate that the solution of the SLAE in the wavelet domain is suitable for the problem of image interpolation by means of Compactly-Supported Radial Basis Functions (CSRBF). The iterative solution of SLAE with highly irregular matrices cannot be accelerated by wavelet transformation and subsequent sparcification if the transformed matrix is still highly irregular. To solve the SLAE in the wavelet domain, the ordering of the samples defines the spacial relationship and the energy of the coefficients in the low frequency domain. Two sorting algorithms for the wavelet domain solution are tested and compared with the spacial solution of the SLAE. Examples of image interpolation by means of CSRBF demostrate the superiority of the solution in the wavelet domain using GMRES iterative method.

Solution of the dam-reservoir interaction problem using a combination of FEM, BEM with particular integrals, modal analysis and substructuring

A boundary element method for analysing non-linear phenomena on gas-liquid two-phase free surface at middle and higher Reynolds numbers has been developed. Employing this method we analyze two kinds of non-linear phenomena on gas-liquid two-phase free surface (l) Solitary wave on free falling thin liquid film. (2) Surface tension drives convective flows in open-boat type crystal-growth techniques in microgravity environment. INTRODUCTION Although some new schemes of discretizing convective terms are developed as QUICK and QUICK ER in finite difference methods[FDM] and finite element methods[FEM], but it is very difficult to avoid numerical diffusion introduced by convective discretizations, especially, at high Reynolds number. In [1], they developed a boundary element iterative scheme(BEIS) to calculate convective terms and compared the advantages of using upwind or central difference schemes and BEIS. As shown as numerical results, BEIS of the convective terms was the most accurate of the two approaches. However, computation was complex. The main objective of this research is to develop a boundary element method for analysing some non-linear phenomena on gas-liquid two-phase free surface at middle and higher Reynolds numbers. On the basis of the boundary element formulation with the pressure penalty function, the divergence theorem is applied to the non-linear convective volume integral of the boundary element formulation. Consequently, velocity gradients are eliminated, the complete formulation is written in terms of velocity. This avoids the difficulty of convective discretizations, and provides considerable reducing storage and computational requirement while improving accuracy. In order to calculate free surface, we construct shooting iterative method of free surface on the basis of normal stress balance condition Project supported by the National Natural Science Foundation of China Transactions on Modelling and Simulation vol 6, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 168 Free and Moving Boundary Problems similar to [2] [3]. The method is applied to analyze two kinds of free surface problems. (1) Solitary wave on free falling thin liquid film. This problem was researched by Wasden and Kheshgi etc. with finite difference method [4] and finite element method [5]. However, in finite difference method[4], one applied an adjustment procedure for wave velocity in order to obtain reasonable flow fields, in finite element methods[5] numerical solution was limited to low flow rates. Moreover, the upwind schemes of discretizing convective momentum terms common to FDM and FEM introduce numerical diffusion or oscillatory behavior of the wave peak region, more importantly, the upwind scheme lacks sensitivity to cross stream diffusion and source terms which are especially important in the case of a thin film. Therefore it is valuable to research a non-traditional numerical method such as BEM. (2) Surface tension drives convective flows in open-boat type crystal growth techniques in microgravity environment. The analytical or numerical study of free surface problem in crystal-growth techniques is very difficult. We shall restrict ourselves to model a problem of open-boat type crystal growth technology, and we only study the melt-gas free boundary, assume the melt-crystal interface shape is flat. The free surface problem of open-boat type is simplified as the free surface problem in open rectangular cavity. The left vertical boundary of the cavity can be interpreted as the crucible wall. The right vertical boundary of the cavity can be interpreted as the melt-crystal interface. The temperatures equal #/ and 0,., respectively. Some important numerical results are presented in paper. BASIC EQUATIONS We consider a system of steady incompressible viscous liquid in a open rectangular cavity with differentially heated lateral walls (the temperature difference is A0*). There is a gradient of surface tension on the free surface since the distribution of temperature is non-homogeneous. The gradient of surface tension drives the convection of liquid. On the basis of the basic law of fluid mechanics and Boussinesq approximation, the governing equations of these problems can be described by a tensor notation and a non-dimensional form as follow: continuity equation «,-„• = o (i) momentum equations energy equation V,-^,=0 (3) where, #,„% is the Kronecker delta symbol, m = x for solitary wave, m = y for thermocapillary convection. The lengths, velocities are scaled with respect to H, Transactions on Modelling and Simulation vol 6, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X

Kussmaul's (1969) formulation of acoustic scattering problems involves the superposition of a double-layer distribution and a simple-layer distribution on the boundary.This ensured the existence of a unique solution for all wave numbers of the incoming wave. However,the boundary integral equation involves a highly singular kernel which proves difficult to treat numerically. An alternative approach is to place the double-layer distribution on an interior boundary similar (similarly situated) to the given boundary.This avoids the singularity problems and enables excellent numerical results to be achieved.Some numerical results are presented. INTRODUCTION Problems of hard acoustic scattering may be formulated in terms of boundary integral equations.Direct formulations are based upon the Helmholtz formula;indirect formulations utilise layer potentials.This is a Neumann boundary-value problem which has a unique solution subject to regular behaviour at infinity.However the relevant classical integral equations,whether direct or indirect,fail at wave numbers corresponding with the eigenfrequencies of the interior Dirichlet problem.In this paper we show that a relatively simple, nonsingular,supplement to the classical indirect integral equation ensures a a unique solution at all wave numbers. An immediate consequence is that meaningful numerical solutions can be achieved,for a wide range of wavenumbers,by computationally effective boundary-element methods (BEMs). Some results for scattering by a hard sphere are provided.For completeness we mention some recent papers presenting alternative resolutions of the difficulty. Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 282 Boundary Elements Rego Silva et al [1] Numerical Implementation of a Hypersingular BEM formulation for Acoustic Radiation Problems,following the approach of Burton and Miller [2].Here the hypersingular integral is evaluated numerically by an algorithm of Guiggiani et al [3] building essentially upon Hadamard's finite-part integration. Liu and Rizzo [4] A weakly-singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems,again following the approach of Burton and Miller. However here the hypersingular integral is weakened by a regularisation technique,which enables it to be evaluated numerically by straightforward methods. Wu et al [8] Vectorization and parallelization of the acoustic boundary element code BEMAP on the IBM ES/3090 VF,exploits the supercomputing facilities to improve the CHIEF method originated by Schenck [15]. Kirkup and Henwood [5] Computational Solution of Acoustic Radiation Problems by Kussmaul's Boundary Element Method,adopting Kussmaul's representation [6] of the wave function as the superposition of a simple-layer and a double-layer potential generated from sources on the boundary.This formulation also involves a hypersingular integral,which is evaluated numerically by the method of Terai [7],an early exploitation of Hadamard's finite part integration. Kussmaul's approach may be modified by locating the double-layer sources on an interior boundary,similar and similarly situated to the given boundary.This eliminates the hypersingular integral and at the same time provides a unique solution at all wave numbers. Brief details are given in the following section. KUSSMAUL FORMULATION Any solution, ,of the Helmholtz equation existing on the infinite domain B exterior to a closed (simply connected) surface dB (fig. 1) of order 0(r~') as /•co can generally be represented by a simple-layer potential dq ; q € dB , p € B+dB (1) dB where o(q) is the source density at q and dq is the area element at q ; also Transactions on Modelling and Simulation vol 1, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 283 /-',-'» ; r=|f-g| (2) where k is the wave number associated with , defined by

ABSTRACTA local radial basis function meshless (LRBFM) method is developed to solve coupled radiative and conductive heat transfer problems in multidimensional participating media, in which compact support radial basis functions (RBFs) augmented on a polynomial basis are employed to construct the trial function, and the radiative transfer equation (RTE) and energy conservation equation are discretized directly at nodes by the collocation method. LRBFM belongs to a class of truly meshless methods which require no mesh or grid, and can be readily implemented in a set of uniform or irregular node distributions with no node connectivity. Performances of the LRBFM is compared to numerical results reported in the literature via a variety of coupled radiative and conductive heat transfer problems in 1D and 2D geometries. It is demonstrated that the local radial basis function meshless method provides high accuracy and great efficiency to solve coupled radiative and conductive heat transfer problems in multidimensional participating media with uniform and irregular node distribution, especially for coupled heat transfer problems in irregular geometry with Cartesian coordinates. In addition, it is extremely simple to implement.

A Monte Carlo solution method for linear elasticity

A novel approach, the spline boundary element method (SBEM), is proposed to analyze the quasi-static characteristics of the coplanar waveguide. The scalar potential and the Green function of the boundary integral equation in the SBEM are expanded in terms of the third order B-spline function. The total matrices in the proposed method become the singular and symmetric matrices, rather than the nonsingular and nonsymmetric matrices in the boundary element method (BEM). The number of discrete boundary knots is much smaller than that of the BEM due to the flexibility of the B-spline interpolation. Test results show that the new method can considerably reduce CPU time and memory, and is simpler and more efficient than the BEM. >

The compactly supported radial basis function (CS-RBF) is improved and used to design a new response surface model. The model is incorporated into stochastic global optimal methods to develop a fast and efficient global optimal design strategy with the main target to reduce the number of function calls that involve computationally heavy procedures such as, for example, the repetitive usage of finite element analysis which is generally required in solving inverse problems. In order to employ a multistep method to automatically adjust the support of the CS-RBF to realize the "best" tradeoff between computational efficiency and accuracy, a cluster algorithm is proposed to decompose the sample points into a nested sequence of subsets. To validate the proposed algorithm, typical numerical results on two different examples are reported.