关键词: 偏微分方程, 级数解, 网格剖分, 精确几何, 独立覆盖, 数值流形方法

Abstract:

Based on the general calculation formula of the manifold method based on independent covers presented in the previous article, we provide the flowchart of the calculation program. First, we summarize the integration methods for various geometric shapes (such as partitions, stripes, and boundary faces) that may appear in one- to three-dimensional spaces. On this basis, we develop integration programs according to simplex geometric elements of points, lines, faces, and bodies. This approach ensures the universality for any mesh shape. Next, we propose a programming strategy that separates the integration module from the integrand function module. The arbitrary combination of these two modules endows the program with extensibility and the potential to achieve universality in solving partial differential equations. Moreover, the universality of series is realized through the determination of series formulas, corresponding coordinates, coordinate transformation matrices, and series matrices. In addition, all calculation parameters can be input via formulas using user subroutines, thus achieving universality of input parameters. Ultimately, with relatively less program code, we can conduct one- to three- dimensional steady-state and transient analyses of the differential equations of motion in elasticity, conduction equations, and wave equations, including one to three types of boundary conditions.

Key words: partial differential equations, series solutions, mesh division, exact geometry, independent covers, numerical manifold method