Boundary Element Methods for the Dynamic Analysis of Elastic, Viscoelastic, and Piezoelectric Solids
Authors: Gaul L. , Kögl M. , Moser F. , Schanz M. 
Published: January 2005

As an alternative to domain discretization methods, the boundary element method (BEM) provides a powerful tool for the solution of single‐field and multifield initial value/boundary value problems. These are encountered, for example, in the analysis of elastic solids and compressible fluids and their interaction at contacting boundaries. As a basis for a boundary element analysis of these problems, one can employ singular boundary integral equations as well as multifield variational formulations. To eliminate the domain integrals from these formulations, different types of fundamental solutions are employed as weighting and approximation functions.

When using fundamental solutions that fulfill thedynamicfield equations, one obtains singular boundary integral equations and all domain integrals are eliminated. Such a formulation in time domain is presented for the calculation of solids with generalized viscoelastic constitutive behavior. The time convolution is numerically approximated by an efficient convolution quadrature that uses the Laplace transformed fundamental solution. Infinite domains can be easily treated because the fundamental solution fulfills the Sommerfeld radiation condition.

If the fundamental solutions that are employed in the boundary element formulation fulfill only thestaticpart of the field equation, the representation formula contains both boundary and domain integrals. Several techniques are available for transforming these domain integrals to the boundary, two of which will be presented here. The first technique is the Dual Reciprocity Method, which uses particular solutions to achieve an approximate transformation of the domain integral to the boundary. This technique is very versatile and will be applied to anisotropic elastodynamics and the coupled field problem of dynamic piezoelectricity. The second technique provides an analytical transformation to the boundary by constructing a so‐called fundamental solution of higher order and will be applied – in the framework of the hybrid boundary element method – to elastodynamics.

Both techniques yield mass matrices for the domain integral, similar to those obtained with the finite element method (FEM). Since static fundamental solutions are employed, the system matrices do not depend on frequency or time, which makes the formulations especially suited for the calculation of the eigenvalue problem, transient and forced vibration analysis.

Direct boundary element formulations based upon the collocation method lead to nonsymmetric matrices, which is a disadvantage when the matrices are coupled with symmetric finite element matrices. A symmetric system of equations can be obtained by starting from a variational principle. This will be demonstrated by means of elastodynamics, for which a boundary element formulation is derived from a multifield version of Hamilton's principle. A superposition of static fundamental solutions is used to approximate the field in the domain, and a fundamental solution of higher order is employed for transforming the domain integral over the inertia term to the boundary. The resulting mass matrix is symmetric and positive‐definite. The formulation involves the calculation of singular boundary integrals if the load points of the fundamental solutions are placed on the nodes of the boundary discretization. To avoid the occurrence of singular integrals, the load points are placed outside the domain. This way, only regular integrals have to be evaluated numerically, and the resulting boundary element formulation becomes nonsingular.

The boundary element formulations described here, along with their implementation, are the result of recent progress in boundary elements research. The numerical examples indicate that for linear dynamic field problems, the presented methods can be seen as alternatives to well‐established domain discretization methods, in particular the finite element method. In the interest of brevity, aspects of numerically efficient implementations, error estimation, and adaptivity have not been addressed.