General Formulas and Program Design for Manifold Method Based on Independent Covers Ⅲ: Example Verification

Based on the general calculation formulas and programming methods proposed in the previous two articles, we conduct a comprehensive validation across one- to three-dimensional problems. These involve the displacement field, temperature field, seepage field, sound field, electrostatic field, and potential flow field by solving the differential equations of motion in elasticity, the conduction equation, and the wave equation (covering both steady-state and transient analyses). The provided examples demonstrate the distinctive characteristics of the manifold method based on independent covers. These features include the ability to handle meshes of any shape and connection, the accurate simulation of geometric boundaries, the precise application of essential boundary conditions, high-order series approximation, and the use of analytical series near crack tips. Finally, we summarize the entire article and propose a new term “series manifold element (series element)”.