Symmetric Second-Order Systems

Abstract

For the approximation problem it is practical to use the general first-order formulation. However, for the coupled system it is desirable to have a symmetric second-order formulation because this is the form of the finite element model. This allows us to use the standard finite element matrix formulation for the whole system. The symmetrization of the matrices is not absolutely necessary. The non-symmetric matrices could be treated as an explicit element in a implicit-explicit formulation [MFH89]. However, this solution may lead to a very small time step based on the stability requirement for the explicit scheme. The small time step dominates the whole analysis unless a more complicated strategy using different time steps for different parts of the model is employed. Thus the symmetrization of the matrices is a very desirable feature.