A Time-Finite-Difference DRBEM for Transient Heat Conduction in Anisotropic Solids (Study on Two-Dimensional Problems)

Modal analysis of plates using the dual reciprocity boundary element method

IG-DRBEM of three-dimensional transient heat conduction problems

Boundary Element Analysis of Cracks in Shear Deformable Plates and Shells. Topics in Engineering, Vol 43

The dual reciprocity method (DRM) and the radial integration method (RIM) are techniques used in BEM to transform domain integrals into boundary integrals. They are suitable for boundary element formulations where a complete fundamental solution is either unavailable or very complex, because in these cases one or more terms can remain as domain integrals in order to use a simpler fundamental solution. Thus, a large number of problems can be solved with the knowledge of a few number of fundamental solutions and additional terms as inertia or non-linear effects, can be treated as body forces and taken to the boundary. In both methods, the remaining terms are approximated through a finite series expansion involving proposed approximating functions and coefficients to be determined. This expansion is substituted in the generated domain integrals that are, subsequently, tr ansformed into boundary integrals. In this paper, a boundary element formulation for thin shallow shells with no domain discretization is presented. The domain integrals due to the curvature of the shells are tran sformed into boundary integrals using the radial integration method. Two approximation functions are used in this work. The first is the radial ba sis function that has been used extensively in the DRM given by:

A subregion DRBEM formulation for the dynamic analysis of two-dimensional cracks

Isogeometric dual reciprocity BEM for solving non-Fourier transient heat transfer problems in FGMs with uncertainty analysis

On the convergence of the dual reciprocity boundary element method

Comparison between continuous and discontinuous boundary elements in the multidomain dual reciprocity method for the solution of the two-dimensional Navier–Stokes equations

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

The Kriging integration method applied to the boundary element analysis of Poisson problems

The two‐dimensional transient elastodynamic problems are solved numerically by using the coupling of the dual reciprocity boundary element method (DRBEM) in spatial domain with the differential quadrature method (DQM) in time domain. The DRBEM with the fundamental solution of the Laplace equation transforms the domain integrals into the boundary integrals that contain the first‐ and the second‐order time derivative terms. Thus, the application of DRBEM to elastodynamic problems results in a system of second‐order ordinary differential equations in time. This system is then discretized by the polynomial‐based DQM with respect to time, which gives a system of linear algebraic equations after the imposition of both the boundary and the initial conditions. Therefore, the solution is obtained at any required time level at one stroke without the use of an iterative scheme and without the need of very small step size in time direction. The numerical results are visualized in terms of graphics. Copyright © 2008 John Wiley & Sons, Ltd.

Triple reciprocity method for unknown function's domain integral in boundary integral equation

It is known that the pH level of the extracellular tumour environment directly effects the progression of the tumour. In this study, the mathematical model for the acid-mediated tumour cell invasion consisting of a system of nonlinear reaction diffusion equations describing the interaction between the density of the tumour cells, normal cells and the concentration of protons produced by the tumour cells is solved numerically using the combined application of dual reciprocity boundary element method (DRBEM) and finite difference method. The space derivatives in the model are discretised by DRBEM using the fundamental solution of Laplace equation considering the time derivative and the nonlinearities as the nonhomogenity. The resulting systems of ordinary differential equations after the application of DRBEM are then discretised using forward difference. Because of the highly nonlinear character of the model, there arises difficulties in solving the model especially for two-dimensions and the boundary-only nature of DRBEM discretisation gives the advantage of having solutions with a lower computational cost. The proposed method is tested with different kinds of carrying capacities which also depend on time. The results of the numerical simulations are compared among each case and seen to confirm the expected behaviour of the model.

PurposeThe purpose of this paper is to describe further developments on a novel formulation of the boundary element method (BEM) for inelastic problems using the dual reciprocity method (DRM) but using object-oriented programming (OOP). As the BEM formulation generates a domain integral due to the inelastic stresses, the DRM is employed in a modified form using polyharmonic spline approximating functions with polynomial augmentation. These approximating functions produced accurate results in BEM applications for a range of problems tested, and have been shown to converge linearly as the order of the function increases.Design/methodology/approachA programming class named DRMOOP, written in C++ language and based on OOP, was developed in this research. With such programming, general matrix equations can be easily established and applied to different inelastic problems. A vector that accounts for the influence of the inelastic strains on the displacements and boundary forces is obtained.FindingsThe C++ DRMOOP class has been implemented and tested with the BEM formulation applied to classical elastoplastic problem and the results are reported at the end of the paper.Originality/valueAn object-oriented technology and the C++ DRMOOP class applied to elastoplastic problems.

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.