Computational Mechanics(1987) 2, 185-196 Computational Mechanics © Springer-Verlag 1987 Dynamic analysis of 3-D structures by a transformed boundary element method S. Ahmad and G. D. Manolis Department of Civil Engineering,State Universityof New York, Buffalo, NY 14260, USA Abstract. In this work, an advancedimplementation of the direct boundary elementmethod applicableto periodic(steady- state) and transientdynamic problemsinvolving three-dimensional structuresof arbitrary shapeand connectivity is presented. Interior, exteriorand halfspacetype of problems can all be solvedby the presentmethod. The discussionfirst focuseson the formulation of the method, followedby material pertaining to the fundamentalsingular solutions and to the isoparametric boundary elementsused for discretizingthe surfaceof the problem. Subsequently, numerical integrationtechniquesand the solutionalgorithmare introduced.Thismethodology has beenincorporated in a versatile,general purposecomputer program. Finally, the stabilityand highaccuracy of this dynamic analysis techniqueare establishedthrough comparisons with available analytical and numericalresults. 1 Introduction The dynamic analysis of three-dimensional structures by retaining the continuous distribution of mass, is a challenging and difficult proposition. This is especially true if problems involving media of infinite or semi-infinite extent are under consideration. Classical analytical techniques (Graff 1974) are applicable to a small class of highly idealized problems. Experiments are expensive to perform and it is very difficult to scale gravity correctly. Thus, recourse must be made to numerical methods of solution. There are currently two major categories of numerical techniques for dynamic analyses, namely approximate continuum and discrete (lumped parameter) models. The most widely used approximate continuum method at present is the finite element method (FEM). In principle, it is a very versatile technique because it can handle complex structure geometry, medium inhomogeneities and complicated material behavior in both two and three dimensions. The major deficiency of the FEM is in modelling media of infinite or semi-infinite extent by a mesh of finite size. This situation is remedied by use of transmitting boundaries (Kausel 1975) and of hybrid techniques (Gupta 1982). The finite difference method (FDM) has been used less frequently than the FEM, primarily because of the difficulties associated with handling complicated geometries. The key idea behind a lumped parameter approach is the determination of values for the mass, stiffness and damping coefficients that essentially represent the problem. Thus, a typical multi degree- of-freedom representation is achieved. These discrete models are particularly useful for soil-structure interaction problems: There, use of frequency dependent coefficients (impedance functions), effectively uncouples the structure from the medium with consideration of the interaction phenomenon. Thus, an efficient analysis of the structure alone is possible. Exact expressions for impedance functions can be obtained for very few cases only (Arnold 1955). In most of the examples, analytic expressions for these functions are obtained under relaxed boundary conditions, i.e., by assuming uncoupling (Veletsos 1971). In recent years, the boundary element method (BEM) has emerged as a strong candidate for the numerical solution of complicated dynamic problems (Banerjee 1981). This is so because a substantial reduction in the size of the problem can be achieved and the radiation damping condition is automatically satisfied. The following approaches have been used for the general two-dimensional