Reduction of the dimensionality of transient fem solutions using proper orthogonal decomposition

A method of reducing the number of degrees of freedom in FEM analysis has been devised. As in the case of FEM, Galerkin weighted residuals is based on weak formulation. The distinct feature of the method is the usage of a set of globally defined trial (and weighting) functions defined as a linear combination of the shape functions. The coefficients of this combination are evaluated employing the Proper Orthogonal Decomposition method ensuring optimal approximation properties of the trial functions. The resulting set of ODEs of much lower dimensionality than the standard FEM is then solved analytically using the modal analysis technique. In the included numerical examples, the number of unknowns is reduced by several orders of magnitude while the maximum error is of the order of 1%. The technique leads also to a significant reduction of the execution times. A method of controlling the error introduced by the POD is proposed.