G. Kuhn

COMPUTATIONAL FLUID DYNAMICS BY BOUNDARY-DOMAIN INTEGRAL METHOD

Very fast development of computing enabled also the development of numerical fluid dynamics. It is numerical modelling and simulation of flow circumstances, including numerical experiments by the computer. Such procedure may have several important advantages over physical measurements on a laboratory model. It is of great importance that fluid properties (density, viscosity, compressibility, etc.) may be simply and arbitrarily changed, numerical experiment does not disturb the flow, plane flows can simply be simulated what may not be the case with laboratory experiments. The numerical experiment also has its own drawbacks and disadvantages, known to all numerical procedures, since the numerical solution represents a result of a discrete equation systems, which are not completely identical to basic physical laws of mechanics of continua. Discretisation often changes quantitatively and qualitatively the behavior of equations and thus also the solutions. Numerical simulation has also similar limitations like a laboratory experiments, since the solutions are individual discrete values only, not the functions of the flow fields.

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 Greens 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.

On the decomposition of the J-integral for 3D crack problems

The extended J-integral for 3D linear elastostatic crack problems and its application to mixed mode problems is investigated. In 3D, the decomposition of the J-integral into its parts corresponding to the symmetric mode I and both antisymmetric modes II and III is derived explicitly. The range of validity of the decomposition method is also discussed in the framework of linear elastic fracture mechanics (LEFM). It is shown analytically that in a general mixed mode case the antisymmetric part of the J-integral can be split into the parts associated with mode II and mode III only in the crack near-field.

Efficient evaluation of volume integrals in the boundary element method

Abstract Nonlinear formulations of the boundary element method, as for example elastoplastic analysis, require integration over internal cells. Due to the singularity of the integrand, this volume integration is cumbersome. The paper presents a closed concept of direct evaluation of these domain integrals by means of simple transformations and standard Gaussian quadrature rules. For regular, but nearly singular integrals, an adaptive integration scheme is presented. For the strongly singular case, a regularized formulation based on Gauss theorem is proposed. Numerical examples concerning elastoplastic analysis are included.

Primitive variable dual reciprocity boundary element method solution of incompressible Navier-Stokes equations

Abstract This paper describes the solution to transient incompressible two-dimensional Navier–Stokes equations in primitive variables by the dual reciprocity boundary element method. The coupled set of mass and momentum equations is structured by the fundamental solution of the Laplace equation. The dual reciprocity method is based on the augmented thin plate splines. All derivatives involved are calculated through integral representation formulas. Numerical example include convergence studies with different mesh size for the classical lid-driven cavity problem at Re=100 and comparison with the results obtained through calculation of the derivatives from global interpolation formulas. The accuracy of the solution is assessed by comparison with the Ghia–Ghia–Shin finite difference solution as a reference.

Comparison of the basic and the discontinuity formulation of the 3D-dual boundary element method

Abstract The dual boundary element method (Dual BEM) has established as a numerical approach for solving arbitrary 3D-crack problems in linear elastostatics. In the case of symmetrically loaded cracks — especially traction-free cracks — often the more efficient displacement discontinuity method (DDM) is used, because one obtains a reduced system of algebraic equations. In our paper we will show that the discontinuity method is just a special formulation of the basic Dual BEM and can be applied to arbitrary boundary value problems on the crack. We will present a numerical example for an unsymmetrically loaded crack and discuss which combinations of boundary conditions on the crack surfaces lead to a reduced system of algebraic equations. The savings in memory and computing time compared to the basic formulation of the Dual BEM will be quantified and illustrated by the numerical simulation of 3D crack propagation.

EVALUATION OF THE STRESS TENSOR IN 3-D ELASTOPLASTICITY BY DIRECT SOLVING OF HYPERSINGULAR INTEGRALS

A new method of direct numerical evaluation of hypersingular boundary integrals has been applied to the differentiated form of the Somigliana-identity (hypersingular identity) in 3D-elastostatics. Through this method it is possible to evaluate the stress tensor on the boundary of a complex 3D structure in a very accurate manner by employing the direct boundary element method (BEM). The geometry of the elements and their arrangements over the boundary of the structure are not subjected to any restrictions. Numerical examples illustrate the accuracy of the proposed method.

Numerical computation of hypersingular integrals and application to the boundary integral equation for the stress tensor

Abstract This paper describes a method for the numerical computation of hypersingular integrals as they appear in the evaluation of the stress tensor by hypersingular boundary integral equations in two-dimensional linear elastostatics. The method is not restricted to this type of problem however and may be easily applied to other hypersingular boundary integral equations. It allows the use of linear and circular representation of the boundary geometry and arbitrary approximation of the boundary value functions. The main advantage of the proposed method is the numerical computation of the hypersingular integrals by Kutts quadrature formulas, thus requiring little analytic pre-work. The calculated examples reveal a high level of accuracy provided the necessary continuity requirements are fulfilled, which may be achieved by Overhauser spline elements.

Dual reciprocity BEM without matrix inversion for transient heat conduction

Abstract Presence of domain integrals in the formulation of the boundary element method dramatically decreases the efficiency of this technique. Dual reciprocity boundary element method (DRBEM) is one of the most popular methods to convert domain integrals into a series of boundary integrals. This is done at the expense of generating some additional matrices and inverting one of them. The latter feature makes the DRBEM inefficient for large-scale problems. This paper describes simple means of avoiding matrix inversion for transient heat transfer problems with arbitrary set of boundary conditions. The technique is also directly applicable to other phenomena (acoustic wave propagation, elastodynamics). For the boundary conditions of Neumann and Robin type, the proposed technique produces exactly the same results as the standard approach. In the presence of Dirichlet conditions, a lower bound on the time step has been detected in the backward difference time stepping procedure. The approach has been tested on some transient heat conduction benchmark problems and accurate results have been obtained.

A field boundary element formulation for damage mechanics

Abstract In this paper, two typical and representative elastoplastic damage mechanic models are presented in the context of a field boundary element formulation. In particular, the damage models of Lemaitre and Gurson are considered. Lemaitres model is based on more phenomenological considerations, whereas Gursons model describes the failure mechanism on a micromechanic level by means of initiation, growth and coalescence of voids. The special requirements of implementing both models into a field boundary element code are investigated. The formulation is based on thermodynamics of irreversible processes and the concept of internal variables. Kinematic and isotropic hardening, isotropic damage evolution and arbitrary uniaxial stress-strain relations are considered to describe the nonlinear material behaviour at small strains. The theoretical background of both models as well as the field boundary element formulation are presented. This includes the evaluation of hypersingular boundary integrals and a consistent regularization of strongly singular domain integrals for arbitratry domain elements. The strain-space formulations of the constitutive equations are presented first in 3D and the applied to 2D examples, in order to demonstrate the validity of the proposed method.