Showing papers in "Engineering Analysis With Boundary Elements in 1994"

Hypersingular formulation for boundary stress evaluation

Abstract: Instead of using shape function derivatives and Hooke's law, the full stress tensor is evaluated at boundary points by direct application of boundary integral identities for the displacement derivatives. It is first shown that integral equations with singular or hypersingular kernels do not give rise to unbounded terms, even when the source point is on the boundary. A general method for performing the integration is also described. Numerical results are quite interesting, since the stress components evaluated through the hypersingular integral equation method show very good accuracy even on coarse meshes.

A numerical quadrature for nearly singular boundary element integrals

Abstract: Nearly singular integrals arise in the boundary element method when analyzing thin structures and gaps and when calculating the field very near the boundary. Hayami proposed the PART method for the accurate and efficient calculation of such nearly singular integrals over general curved surface elements. In this paper, a new radial variable transformation for the method, which reduces the number of integration points for flux integrals, is proposed with theoretical error analysis using complex function theory. Also, an implementation technique of the method is proposed, which considerably reduces the number of integration points.

Solving linear diffusion equations with the dual reciprocity method in Laplace space

Abstract: The dual reciprocity method (DRM) is applied in the Laplace space to solve efficiently time-dependent diffusion problems. Since there is no discretation in time and there are no domain integrals involved in a calculation, the proposed approach seems to have provided considerable savings on computer operating costs and in data preparation, and thus leads to certain advantages over existing methods. Three numerical examples are presented, which demonstrate well the efficiency and accuracy of the new approach.

Third degree polynomial transformation for boundary element integrals: Further improvements

Abstract: The first author has previously developed a third degree polynomial coordinate transformation which improves the accuracy of numerical quadrature schemes typical of boundary element implementations. This transformation is now extended to include additional integrand variations and kernels of order 1n(1/r), 1/r, 1/r2 and 1/r3 are discussed. The optimized parameter which governs the transformation is presented in tabular form for several source to element minimum distances.

On the choice of interpolation functions used in the dual-reciprocity boundary-element method

Abstract: The dual-reciprocity boundary-element method is a very powerful technique for solving general elliptic equations of the type ∇ 2 u = b . In this method, a series of interpolation functions is used to approximate b in order to convert the associated domain integral, which it is necessary to evaluate in a traditional boundary-element analysis, into boundary integrals only. Hence the choice of interpolation functions has direct effects on the numerical results. According to Partridge and Brebbia, the adoption of a comparatively simple form of interpolation function gives the best results. Unfortunately, when b contains partial derivatives of the unknown function u ( x , y ), the adoption of such a type of interpolation function inevitably leads to the creation of singularities on all boundary and internal nodes used in a dual-reciprocity boundary-element analysis, as was pointed out by Zhu and Zhang in 1992. To avoid this problem, a functional transformation, which applies only to linear governing equations, can be employed to eliminate these derivative terms and thus to obtain better numerical results. In this paper, two new interpolation functions are proposed and examined; they are proven to be generally applicable and satisfactory.

Dual-reciprocity BEM based on global interpolation functions

Abstract: Several global shape functions are introduced to interpolate the body-force term in the dual-reciprocity boundary-element method. These global-interpolation functions, which include polynomial, trigonometric, and hyperbolic series, can be used in place of the locally based radial shape function. For the several examples presented, the global functions have demonstrated superior convergence properties.

Green's functions for heterogeneous media potential problems

Abstract: Procedures are discussed for obtaining Green's functions for potential problems in heterogeneous materials. Such fundamental solutions are required in the boundary element method. An exact funddamental solution for a linearly varying, layered two dimensional potential problem is given that does not appear to be available elsewhere.

On the convergence of the dual reciprocity boundary element method

Abstract: This paper presents a study of the convergence properties of the dual reciprocity method (DRM). DRM is one of the most popular techniques used to transform volume integrals that arise, for example, from the inhomogeneous term of Poisson's equation, into equivalent boundary integrals in the boundary element method (BEM). The transformation is carried out by expanding the inhomogeneous term into approximating functions whose particular solutions can be easily obtained. In the present paper, interpolation functions are derived from the approximating functions, and their properties are studied theoretically and numerically. The results obtained confirm that the interpolation converges to the real function.

Numerical simulation of axisymmetric viscous sintering

Abstract: In this paper we describe a numerical method to simulate particular axisymmetric viscous sintering problems. In these problems the material transport is modelled as a viscous incompressible Newtonian volume flow driven solely by surface tension. The numerical simulation is carried out by solving the governing Stokes equations for a fixed domain through a boudary element method (BEM). The resulting velocity field then determines an approximate geometry at a next time level by employing a variable step, variable order backward differences formulae (BDF) method. This numerical algorithm is demonstrated for several simply connected sintering domains, including two coalescing spheres. Furthermore, by considering the movement of a particular fluid region, some discrepancies between modelling it as a two-dimensional fluid and as an axisymmetric problem are demonstrated.

Dynamic elastoplastic analysis by BEM/FEM

Abstract: A boundary element/finite element method (BEM/FEM) hybrid scheme in the time domain is developed for the dynamic analysis of elastoplastic structures under plane strain or plane stress conditions. The FEM employs eight-noded isoparametric quadrilateral elements and discretizes the boundary as well as the interior of that portion of the structure, which is expected to become plastic. The BEM employs three-noded qudratic boundary line elements and discretizes only the boundary portion of the structure, which is expected to stay elastic during the whole time history. The FEM part can take into account Tresca, Von Mises, Drucker-Prager and Mohr-Coulomb isotropic hardening plasticity models. The BEM and FEM domains of the structure are connected at their interface through equilibrium and compatibility. The applied loads can have any transient time variation. The solution procedure follows the step-by-step implicit time integration algorithm of Newmark and employs iterations at every time step. Numerical examples are presented to illustrate the proposed scheme and assess its advantages.

Elastic analysis of thin plates with beam supports

Abstract: The results of analysis of thin plates supported on elastic beams are given for two types of models, namely, a ‘boundary condition’ type in which the plate and attached edge beam are connected continuously, and an ‘attached beam element’ type in which the connection is not fully displacement compatible. From these analyses, the authors believe that the paper provides some general guidelines on the numerical analysis of plates with attached beams. For example, the discontinuation of a beam support along a plate edge leads to a special singular condition, and the validity of a mathematical model of this will be of interest to structural designers. The numerical modelling of thin plates is based on the direct boundary element method and the procedures employed in attaching edge beams are described fully. Numerical results are given for square plates for comparison with established cases, however, the procedures are basically very general purpose and have been used in solving problems involving thin plates of general plan shape and transverse loading.

A comparative study of three boundary element approaches to transient dynamic crack problems

Abstract: Using displacement integral representations for time dependent elastic problems, three different boundary element (BE) approaches can be considered. One is based on the time domain formulation, a second one on the frequency domain formulation, and the third on the dual reciprocity formulation. In this paper, the three BE approaches are applied and compared when used for the computation of stress intensity factors (SIF) of stationary cracks under dynamic loading. In the three cases a quadratic discretization in space and a singular quarter-point (SQP) element are used. Several problems are solved and the results obtained by the three approaches compared. The dual reciprocity approach for dynamic SIF computation is presented in this paper. Conclusions and recommendations on the capabilities of the three approaches in dynamic fracture mechanics are presented.

Fundamental solutions of products of Helmholtz and polyharmonic operators

Abstract: Many problems in engineering lead to linear systems of partial differential equations. In certain cases the determinant of the system of partial differential operators reduces to a product of Helmholtz, metaharmonic (modified Helmholtz), or polyharmonic operators. Fundamental solutions of these systems are available in the mathematics literature. However, their existence is not necessarily well known to the engineering and boundary element community. This research note lists some of these solutions to facilitate their use.

Nondestructive detection of cavities by an inverse elastostatics boundary element method

Abstract: The elastostatics boundary element method is applied in an inverse problem approach to the nondestructive detection of subsurface cavities in structures. The boundary conditions at the exposed surface are overspecified: tractions are specified and displacements are used as additional data for solving the inverse problem. In the developed iterative procedure, an initial guess is made for the shape of the cavity and a grid pattern is laid out. The use of this pattern allows one of the coordinates of the interior nodes to be fixed thus reducing the number of unknowns at each cavity node to one. The initial guess will not correspond to the actual cavity, consequently, the BEM solution will yield displacements which do not agree with the reference displacements. This leads to residuals at each node. The cavity is then located by iteratively driving these residuals to zero. Newton's method and the steepest descent method are considered in this effort. Iterative updates of the cavity geometry are kept within a physically realistic feasible region. Validation cases are presented for the detection of single circular and elliptic holes located at various positions within a rectangular plate. Numerical results demonstrate the successful detection of subsurface cavities by this method, Finally, results are presented for an experiment in which the surface displacements are determined by a laser speckle photography technique. A centrally located circular hole is successfully located using these surface displacement data.

Static and transient dynamic nonlinear stress analysis by the boundary element method with implicit techniques

Abstract: This paper is concerned with the application of implicit techniques to solve static and transient dynamic elastoplastic problems. The current formulation keeps the inertial domain integral, which appears due to the use of the static fundamental solution, leading to the discretization of the entire domain. The time-domain problem is solved by a direct integration method (Houbolt scheme). Illustrative examples are presented at the end of the paper.

Dynamic analysis of elasto-plastic flexural plates by the D/BEM

Abstract: A direct domain/boundary element method is developed for the dynamic response analysis of thin elasto-plastic flexural plates of arbitrary geometry and boundary conditions subjected to any lateral loading history. The method employs the elastostatic fundamental solution of thin flexural plates in a time domain integral formulation. Thus, the plasticity effect, the inertial load and the external lateral load appear in domain integrals in the boundary integral formulation of the problem. This requires a boundary as well as an interior discretization of the plate. Quadratic isoparameteric boundary and interior elements are employed for increased accuracy. The solution is obtained by an explicit time integration scheme employed on the incremental form of the matrix equations of motion. The incremental plastic moments needed to evaluate the plasticity effect are calculated by a finite element methodology to avoid the evaluation of highly singular terms. Numerical examples are presented to illustrate the proposed method and compare it with the finite element method.

A review of error estimation and adaptivity in the boundary element method

Abstract: When using the boundary element method, the accuracy of the numerical solution depends critically on the discretization of the boundary into elements (panels). The distribution of the panels is one of the most important decisions taken when analyzing a problem, but still the vast majority of users employ empirical guidelines to distribute the panels. This paper reviews the various adaptive schemes that have been proposed for boundary elements. Numerical results are obtained for infinite fluid flow problems and free surface problems and are used to assess the reliability and effectiveness of each method.

Quasi-singular integrals in the modeling of nonlinear water waves in shallow water

Abstract: The model by Grilli et al.,5,8 based on fully nonlinear potential flow equations, is used to study propagation of water waves over arbitrary bottom topography. The model combines a higher-order boundary element method for the solution of Laplace's equation at a given time, and Lagrangian Taylor expansions for the time updating of the free surface position and potential. In this paper, both the accuracy and the efficiency of computations are improved, for wave shoaling and breaking over gentle slopes, in domains with very sharp geometry and large aspect ratio, by using quasi-singular integration techniques based on modified Telles17 and Lutz11 methods. Applications are presented that demonstrate the accuracy and the efficiency of the new approaches.

Boundary-integral equations and the boundary-element method for three-dimensional fracture mechanics

Abstract: In this paper, the displacement-discontinuity fundamental solutions for three-dimensional fracture mechanics are obtained by an integral-transform method and the boundary-integral equations are established. By using finite-part integrals, the displacement-discontinuity boundary-element method is realized. Finally, two simple numerical exaamples are given.

Modal analysis of elastic-plastic plate vibrations by integral equations

Abstract: A direct boundary element method for the vibration problems of thne elastic-plastic plates is presented. Dynamic fundamental solutions of a suitably shaped finite domain are used in modal form. The series Green's functions are separated into a quasistatic and a dynamic part. Often the series of the quasistatic part can be written in a faster converging form than the equivalent modal series. Analytical integration in the vicinity of the singularity is performed on the closed form fundamental solutions of the infinite domain, and only the non-singular differences from the actual Green's functions are represented in series form. This paper gives a general formulation of this method for Kirchhoff plates on an arbitrary elastic foundation. After integration, the resulting algebraic equations are arranged in a form most convenient for a time-stepping analysis of inelastic response. This rearrangement has to be performed only once, if the time step is kept constant. Constitutive equations are integrated by an implicit backward Euler scheme for plane stress. Applications are shown for impacted circular plates on several different foundations.

On computing boundary limiting values of boundary integrals with strongly singular and hypersingular kernels in 3D BEM for elastostatics

Abstract: A few original or recently described numerical methods are presented for direct computation of boundary limits of strongly singular and hypersingular boundary integrals appearing in three-dimensional BEM applied to elastostatics. The methods differ from each other in dealing with integrand singularity: Kutt's formulae in the radial direction, analytical integration in the radial direction of the leading homogeneous term of the integrand expansion and regularizations using Stokes' integral theorem. The advanced unified approach for their implementations is based on the polar coordinate transformation in the parameter plane of the singular element. Due to various groups of performed numerical tests for surfaces discretized by flat or curved boundary elements with linear, quadratic and cubic shape functions, it is possible to compare the methods for their stability and rate of convergence to exact values.

Boundary element analysis of anisotropic bimaterials with special Green's functions☆

Abstract: The boundary integral equations incorporating the Green's function for anisotropic solids containing planar interfaces are presented. The fundamental displacement and traction solutions are determined from the displacement Green's function of Tewary, Wagoner, and Hirth ( Journal of Materials Research , 1989, 4 , 113–123). The fundamental solutions numerically degenerate to the Kelvin solution in the isotropic limit. The boundary integral equations are formulated with the use of constant boundary elements. The constant boundary elements allow for analytical evaluation of the boundary integrals. The application of the method is demonstrated by analyzing a copper-solder system subjected to mechanical loading.

On the hyper-singular boundary-element formulation for fracture-mechanics applications

Abstract: The present paper discusses the application of the hyper-singular boundary-integral equation, the so-called traction formulation, to solve linear-elastic-fracture-mechanics (LEFM) problems. Emphasis is given to two-dimensional boundary-element implementations, combining continuous or discontinuous quadratic and quarter-point elements in order to identify the limitations and improvements for computing stress-intensity factors. The study shows the importance of considering displacement-derivative continuities into the numerical implementation and the use of quarter-point elements to improve results.

On the efficient numerical evaluation of integrals with complex singularity poles

Abstract: A simple quadrature rule is proposed for the evaluation of one-dimensional quasi-singular integrals with a complex pole. The technique is based on the concept of synthetic division of two polynomials, as a means of regularizing the quasi-singularity, followed by a quadrature using the roots of shifted Legendre polynomials as fixed abscissas. This procedure is a generalization of the technique proposed in a previous paper for singularities due to a real pole. With this technique, it is even possible to conceive of procedures for the analytical or semi-analytical evaluation of most kinds of integrals that appear in a general boundary element formulation with curved elements.

Nonlinear transient dynamic analysis of soil-pavement interaction under moving load: a coupled BEM-FEM approach

Abstract: The nonlinear transient dynamic analysis of a 3-D linear elastic pavement placed on an elastoplastic soil medium, subject to a moving load, is presented. The time domain nonlinear boundary element method for the soil medium is combined with the finite element method for the pavement. The nonlinear BEM is based on an initial stress approach. The accuracy of the algorithm is verified by comparison with available analytical solutions and published numerical results.

Boundary element analysis of thick Reissner plates in bending

Abstract: A simplified approach is presented for the derivation of boundary integral equations and fundamental solutions for thick Reissner plates in bending. Explicit expressions for fundamental solution parameters, suitable for plates with arbitrary shapes, are derived using Hankel integral transforms. Corner and singularity problems are also discussed. Case studies with different loading and boundary conditions have been analysed and the boundary element results agree very well well corresponding analytical solutions, confirming the correctness of the presented formulations.

A symmetric and positive definite variational BEM for 2-D free vibration analysis

Abstract: A general BEM model for structural dynamics is derived by using a symmetric and positive definite variational formulation. The functional employed involves domain displacements and boundary tractions and displacements. These variables are taken to be independent of one another. The boundary variables are expressed in terms of their nodal values while the domain displacement field is approximated by a linear combination of static fundamental solutions. The source point of the latter is located outside the domain. The resolving system is a linear system and for free vibration a classic linear algebraic eigenvalue problem is inferred. The stiffness and mass matrices are symmetric and positive definite and the domain integral, when associated with the inertial term, can be transformed into a boundary integral. Numerical results are presented to prove the efficiency of the method.

Dual reciprocity BEM: local versus global approximation functions for diffusion, convection and other problems

Abstract: The use of the global approximation functions (elements of Pascal's triangle, sine expansions and others) in the dual reciprocity boundary element method is compared to the better known local radial basis functions for convection, diffusion and other problems in which the volume integrals considered contain first and second derivatives of the problem variables, time derivatives and sums and products of functions, including nonlinear terms. It will be shown that whilst it is possible to obtain accurate solutions to the problems considered using the global functions, a successful solution to a given problem depends very much on the function chosen, as well as other factors.

Water-entry for a wedge in arbitrary water depth

Abstract: Numerical and experimental results are presented for the two-dimensional entry of a wedge into initially calm water of arbitrary depth. In the numerical computation, the flow field is solved by the boundary element method with nonlinear free surface boundary conditions. The case of the impact pressure in shallow water depth, which is problematical in analytical method, can be evaluated and expressed in the concept of relative water depth. The results show that the magnitude of impact pressure in shallow water is larger than that in deep water. To verify the validity of this numerical analysis, some experiments are carried out in a laboratory channel. The measured results are compared with numerical predictions, and quantitative agreements are obtained.

On fundamental solutions in time-domain boundary element analysis of linear viscoelasticity

Abstract: This paper presents an efficient method for the computation of the fundamental solutions in time-domain boundary integral equation for viscoelastic problems. The method proposed is based on the Laplace transform and the correspondence principle. The relaxation function is expanded in a sum of exponentials and the transformed fundamental solutions are inverted numerically into real time space. The proposed procedure requires a small computational effort and it is applicable in time-domain boundary element analysis of realistic viscoelastic problems. Numerical results of an example problem show the effectiveness and applicability of the proposed method.