Proper orthogonal decomposition and modal analysis for acceleration of transient FEM thermal analysis

A method of reducing the number of degrees of freedom and the overall computing times in finite element method (FEM) has been devised. The technique is valid for linear problems and arbitrary temporal variation of boundary conditions. At the first stage of the method standard FEM time stepping procedure is invoked. The temperature fields obtained for the first few time steps undergo statistical analysis yielding an optimal set of globally defined trial and weighting functions for the Galerkin solution of the problem at hand. Simple matrix manipulations applied to the original FEM system produce a set of ordinary differential equations of a dimensionality greatly reduced when compared with the original FEM formulation. Using the concept of modal analysis the set is then solved analytically. Treatment of non-homogeneous initial conditions, time-dependent boundary conditions and controlling the error introduced by the reduction of the degrees of freedom are discussed. Several numerical examples are included for validation of the approach