In this work, we propose and develop efficient and accurate numerical methods for solving the Kirchhoff-Love plate model in domains with complex geometries. The algorithms proposed here employ curvilinear finite-difference methods for spatial discretization of the governing PDEs on general composite overlapping grids. The coupling of different components of the composite overlapping grid is through numerical interpolations. However, interpolations introduce perturbation to the finite-difference discretization, which causes numerical instability for time-stepping schemes used to advance the resulted semi-discrete system. To address the instability, we propose to add a fourth-order hyper-dissipation to the spatially discretized system to stabilize its time integration; this additional dissipation term captures the essential upwinding effect of the original upwind scheme. The investigation of strategies for incorporating the upwind dissipation term into several time-stepping schemes (both explicit and implicit) leads to the development of four novel algorithms. For each algorithm, formulas for determining a stable time step and a sufficient dissipation coefficient on curvilinear grids are derived by performing a local Fourier analysis. Quadratic eigenvalue problems for a simplified model plate in 1D domain are considered to reveal the weak instability due to the presence of interpolating equations in the spatial discretization. This model problem is further investigated for the stabilization effects of the proposed algorithms. Carefully designed numerical experiments are carried out to validate the accuracy and stability of the proposed algorithms, followed by two benchmark problems to demonstrate the capability and efficiency of our approach for solving realistic applications. Results that concern the performance of the proposed algorithms are also presented.
翻译:在这项工作中,我们提出并发展了高效和准确的数值方法,在具有复杂地貌的域内解决Kirchhoff-love 板块模型。在此提议的算法中,在通用的复合重叠网格中,采用曲线线性定值的有限差异性差法,在通用的组合重叠网格中,将组合重叠网格的不同组成部分合并为数字内插。然而,对将上风脱位术语纳入若干时间跨度计划(既明确又隐含的)的研究,导致为推进结果的半偏差系统而采用时间跨行办法的数字不稳定。为了解决不稳定问题,我们提议在空间离散系统中增加一个四级超偏差的算法的超偏差性差性差性差法,以稳定其时间整合;这一额外的脱位词反映了最初的上风网格图的基本向上移效应。关于将上风脱位术语纳入若干时间跨度计划(既明确又隐含的)的拟议模式发展了四种新的算法。对于确定一个稳定时间跨度的公式,每个算法、确定一个稳定时间跨度和足够偏差的离差的轨算法性计算法,我们对轨算法中,在曲线内测算法中,对曲线性差的轨法的轨算法计算法计算法计算法计算法的计算法是计算法的偏差法的计算法,对曲线内基底基底基底基底基内1的计算法的计算法的计算法的计算法的计算法的计算法的计算法性差性差性差性差性差性差性差率性差率性差率性差性差性差率性差率性差率性差率性差率性差性差系数的计算法,对基内的计算法,对基内的计算,对基内基内的计算,对基的计算法的计算法的计算,对基的计算法的计算法的计算法的计算法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差法差