Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations

A method for optimising a pre-existing mesh of tetrahedral finite elements is described. It is based on a series of mesh connectivity and node position searches of the landscape defining mesh quality. A Riemannian metric, reflecting the a posteriori error measure, is used to calculate element size and shape. A functional is defined which embodies both shape and size quality of an element with respect to the metric, and is used to gauge mesh quality. A heuristic-based local search strategy is adopted – local in the sense that it has no hill-climbing abilities. The paper presents applications of the method to complex, steady-state and time-dependent problems which highlight its anisotropic, feature-capturing abilities. Numerical evidence is provided which suggests that the computational complexity (time) of the proposed algorithm varies linearly with the number of elements or nodes of the finite element mesh.