Efficient implementation of the Fourier dual reciprocity boundary element method using two-dimensional fast Fourier transforms

The dual reciprocity boundary element method, since its birth about two decades ago, has been successfully applied to solve many engineering problems. The application of this method to study the diffraction and refraction of water waves is one of many successful examples. However, most of the previous work has been concentrated on the diffraction and refraction of linear waves as they are relatively easier to deal with in comparison with the nonlinear waves. In this paper, a review of the application of boundary element method and the dual reciprocity boundary element method in this area is given first. Then, some recent progress in modeling the diffraction and refraction of weakly nonlinear waves will be reported. For a handful of papers in the literature, in which the nonlinearity was considered, the technique of marching with time was commonly adopted, which made the computation very expensive. Based on the Boussinesq equation, a model is proposed for modeling the diffraction and refraction process of monochromatic incident waves. The model is based on the assumption that the ratio of wave amplitude to the wavelength is large enough that the nonlinear dispersion effects must be taken into consideration and yet small enough so that a perturbation expansion can be carried out to cast a nonlinear problem into a series of linear problems in order to boost numerical efficiency. At each order, the dual reciprocity boundary element method is adopted to further enhance the numerical efficiency. Model validation has been conducted against some previously published experimental data and a significant improvement comparing to the wave runups predicted by linear models has been observed. Our preliminary results suggest that the model can be used for a wide range of diffraction and refraction problems in coastal and ocean engineering. Transactions on Modelling and Simulation vol 27, © 2001 WIT Press, www.witpress.com, ISSN 1743-355X

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

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

This paper describes a computational model based on the dual reciprocity boundary element method (DRBEM) for the solution of two-dimensional incompressible viscous flow problems. The model is based on the Navier-Stokes equations in velocity-vorticity variables. The model includes the solution of vorticity transport equation for vorticity whose solenoidal vorticity components are obtained by solving Poisson equations involving the velocity and vorticity components. Both the Poisson equations and the vorticity transport equations are solved iteratively using DRBEM and combined to determine the velocity and vorticity vectors. In DRBEM, all source terms, advective terms and time dependent terms are converted into boundary integrals and hence the computational domain of the problem reduces by one. Internal points are considered wherever solution is required. The model has been applied to simulate two-dimensional incompressible viscous flow problems with low Reynolds (Re) number in a typical square cavity. Results are obtained and compared with other models. The DRBEM model has been found to be reasonable and satisfactory.

On the convergence of the dual reciprocity boundary element method

In this paper, we investigate the fully developed, laminar, forced convection flow of an electrically non-conducting, viscous, biomagnetic fluid in the 2D cross-section (cavity) of a long impermeable pipe. The fluid is under the influence of a point magnetic source placed below the cavity. The dual reciprocity boundary element method (DRBEM) with constant and linear elements is used for solving the governing equations resulting from the Navier–Stokes and energy equations together with magnetization and buoyancy forces. The fundamental solution of Laplace equation is made use of converting differential equations to boundary integral equations by taking all the terms other than Laplacian as inhomogeneity in the Poisson’s equations for the velocity components, pressure and the temperature of the fluid. The unknown pressure boundary conditions are approximated through momentum equations by using finite difference approximation for the pressure gradients and DRBEM coordinate matrix for the other terms. All the space derivatives are also calculated by DRBEM coordinate matrix which is one of the main advantages of DRBEM. Pipe axis velocity is also computed. The effects of magnetization and the buoyancy force on the fluid with or without viscous dissipation term in the energy equation are investigated in square and lid-driven cavities for several values of magnetic (Mn) and Rayleigh (Ra) numbers. It is observed that the flow and heat transfer are significantly affected with increasing values of Mn and Ra. DRBEM gives small sized linear systems due to its boundary only nature at a considerably low computational expense.

BEM and FEM based numerical simulations for biomagnetic fluid flow

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

Axisymmetric multiquadrics

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.

Hw/Sw Co-Design technique for 2D fast fourier transform algorithm on Zynq SoC

Dual reciprocity boundary element method for convective-diffusive solid-liquid phase change problems, Part 1. Formulation

Simulation of the thermal wave propagation in biological tissues by the dual reciprocity boundary element method

Groundwater optimization and parameter estimation by genetic algorithm and dual reciprocity boundary element method