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Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations. (English) Zbl 1008.76041
Summary: We describe a method for optimising a pre-existing mesh of tetrahedral finite elements. 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. Then 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.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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