高性能有限元方法研究

This work is devoted to the study of high performance finite element methods in.large scale science and engineering computing. A notable feature of the methods is that lower order schemes can attain higher order accuracy at coarse meshes so as to obtain optimal numerical.performance with the least computational cost. In recent years, the search for high performance elements represents an important area in solid and structured mechanics. We have discussed.optimization of stress modes of hybrid stress finite element methods and proposed an optimal stress mode and a hybrid element of excellent performance. Under a unified theoretical frame we have analysed the convergence of several enhanced stress/strain hybrid elements of high.performance in mechanics and engineering to disclose the mechnism of acquiring high.performance. From a geometric view point in mechanics，we have discussed the intrinsic mechanism of enhancing coarse-mesh accuracy and stability of lower order schemes for the.combined hybrid method and pointed out that the coarse-mesh accuracy of finite element schemes can be enhanced by controlling the energy error of the discrete model. By the energy-adjustable.mechanism of the combined hybrid method, we have improved the performance of the compatible isoparametric bilinear quadrilateral element. We have also established the theory frame of.combined hybrid methods for 4th-order plate bending problems. By the energy-adjustable mechanism, we have studied the improvement of plate bending elements by the combined hybrid.methods.

本项目着力于研究以大规模科学和工程计算为应用背景的高性能有限元方法。高性能有限元方法的一个显著特征是：低阶有限元格式在粗网格时可达到高数值精确度，从而以最小的计算工作量获得最佳的数值效果。近年来，探索高性能有限元方法已成为固体和结构力学中有限元研究的重要领域。我们讨论了工程中应用广泛的杂交应力有限元方法的应力模式优化问题，设计出了最佳应力模式和迄今最优的高性能的杂交元。在统一的理论框架下，对工程力学上的几个高性能丰富应力/应变杂交元给出了严格的收敛性分析，指出了其获得高性能的机理。从力学几何观点的角度，讨论了组合杂交变分原理增强低阶有限元格式粗网格精度和稳定性的内在机制，揭示了控制离散模型的能量误差可以增强有限元格式的粗网格精度的数值规律性。利用组合杂交变分原理的能量可调准机制，研究了组合杂交有限元法对协调等参双线性平板元的改进，获得粗网格高精度。建立了四阶薄板问题的组合杂交有限元方法的理论框架。利用组合杂交变分原理的能量调准机制，研究了组合杂交有限元法对薄板弯曲元的改进，获得计算简单的高精度元。