Computational mechanics based on the theory of boundary eigensolutions

The theory of boundary eigensolutions for boundary value problems is applied to the development of computational mechanics formulations. The boundary element and finite element methods that result are consistent with the mathematical theory of boundary value problems. Although the approach is quite general, this paper focuses on potential problems. For these problems, both methods employ potential and boundary flux as primary variables. Convergence characteristics of the new flux-oriented finite element method are also developed. By utilizing suitable boundary weight functions, the formulations are written exclusively in terms of bounded quantities, even for non-smooth problems involving notches, cracks and mixed boundary conditions. The results of numerical experiments indicate that the algorithms perform in concert with the underlying theory and thus provide an attractive alternative to existing approaches. Beyond this, the approach developed here provides a new perspective from which to view computational mechanics, and can be used to obtain a better understanding of boundary element and finite element methods. Comparisons with closed-form boundary eigensolutions are also presented in order to provide a means for assessing the numerical methods.