General Formulas and Program Design for Manifold Method Based on Independent Covers Ⅰ:General Formulas

Manifold method based on independent covers is a novel approach for numerically solving partial differential equations. By constructing approximate functions, it generates a “partitioned series solution” for partial differential equations. This method not only achieves the main functions of the finite element method (FEM) and other numerical techniques but also outperforms them in certain aspects, such as mesh generation flexibility and computational stability. However this also means that its calculation formulas and program design are different from existing methods. This paper reviews the major research outcomes in solid computation in recent years, and summarizes a set of simple and general calculation formulas in which the shape function of the local approximation function is expressed as the product of the Partition of Unity (PU) function, coordinate transformation matrix, and series matrix. The shape function and its derivatives under various scenarios are discussed in details. Different matrices and the time integration method are also given. These formulas can be applied to solve the differential equations of motion in elasticity, conduction equations, and wave equations, covering one-to-three-dimensional steady-state and transient analyses, along with three types of boundary conditions. They offer features such as high-order series, arbitrary mesh shapes, accurate boundary geometric simulation, precise application of essential boundary conditions, and local analytical series near the crack tip. Utilizing these formulas, a general program for the new method can be developed.