On the second variations for the stress wave problem using the euler-lagrange and adjoint formulations

Abstract

Dynamical stress behaviors and shock transients in mechanics are important subjects to be studied. The use of the finite element method based on developed algorithms from the variational principle can give direct numerical solutions for partial derivatives of the functions in these problems. Many researchers have found it difficult to apply the finite element method to an hyperbolic type partial differential equation (PDE). The Galerkin method and the like have been used instead. The author has reported previous attempts to solve these hyperbolic type PDE's by employing variational principles in a number of papers. It is the purpose of this paper to show that under certain conditions the stationary values are strong extremals, not saddle points. This is equivalent to requiring that the second variations of the functional be positive semidefinite with discontinuities in the partials. The adjoint system can be arranged in a manner so it is a reflected mirror of the original system in time. Generalized boundary conditions employ many types of “springs” relating the various spatial partial derivatives. They are defined to satisfy the boundaries of the original and adjoint system relationship for the bilinear expression. Algorithms for use in the finite element method are simplified since the adjoint system gives exactly the same solutions as those of the original system. The second necessary condition for an extremum is satisfied by showing that the second variation of the functional is positive semi-definite.