C1-continuous spectral elements and time integration algorithms.

摘要

This thesis consists of two topics: spatial discretisation for high-order PDEs and temporal discretisation for nonlinear elastodynamics. For the spatial discretisation, we propose C1-continuous spectral elements, which overcomes the difficulty in solving high-order PDEs directly. Numerical examples involving Helmholtz and Cahn-Hilliard equations demonstrate the convergence characteristics and robust performance in two-phase system simulations. For the temporal discretisation, we propose a new class of of historical second derivative linear multistep methods that are second-order accurate, numerically simple and effective for the dynamic solution of structures involving large deformations and rotations. The testing results show that the proposed algorithms are of second-order accuracy and effective for practical nonlinear analysis.