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 fini...
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Editorial Office of Journal of Changjiang River Scientific Research Institute
2025-04-01
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| Series: | 长江科学院院报 |
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| Online Access: | http://ckyyb.crsri.cn/fileup/1001-5485/PDF/1735268218068-714963553.pdf |
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| author | SU Hai-dong |
| author_facet | SU Hai-dong |
| author_sort | SU Hai-dong |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-41aa36353a154ecfbb356579c400367a |
| institution | OA Journals |
| issn | 1001-5485 |
| language | zho |
| publishDate | 2025-04-01 |
| publisher | Editorial Office of Journal of Changjiang River Scientific Research Institute |
| record_format | Article |
| series | 长江科学院院报 |
| spelling | doaj-art-41aa36353a154ecfbb356579c400367a2025-08-20T02:09:09ZzhoEditorial Office of Journal of Changjiang River Scientific Research Institute长江科学院院报1001-54852025-04-0142419320110.11988/ckyyb.20240111General Formulas and Program Design for Manifold Method Based on Independent Covers Ⅰ:General FormulasSU Hai-dong01 Material and Engineering Structure Department, Changjiang River Scientific Research Institute,Wuhan 430010, China;2 Research Center of Water Engineering Safety and Disaster Prevention of Ministry of Water Resources, Wuhan 430010, ChinaManifold 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.http://ckyyb.crsri.cn/fileup/1001-5485/PDF/1735268218068-714963553.pdfpartial differential equations|series solutions|mesh division|exact geometry|independent covers|numerical manifold method |
| spellingShingle | SU Hai-dong General Formulas and Program Design for Manifold Method Based on Independent Covers Ⅰ:General Formulas 长江科学院院报 partial differential equations|series solutions|mesh division|exact geometry|independent covers|numerical manifold method |
| title | General Formulas and Program Design for Manifold Method Based on Independent Covers Ⅰ:General Formulas |
| title_full | General Formulas and Program Design for Manifold Method Based on Independent Covers Ⅰ:General Formulas |
| title_fullStr | General Formulas and Program Design for Manifold Method Based on Independent Covers Ⅰ:General Formulas |
| title_full_unstemmed | General Formulas and Program Design for Manifold Method Based on Independent Covers Ⅰ:General Formulas |
| title_short | General Formulas and Program Design for Manifold Method Based on Independent Covers Ⅰ:General Formulas |
| title_sort | general formulas and program design for manifold method based on independent covers i general formulas |
| topic | partial differential equations|series solutions|mesh division|exact geometry|independent covers|numerical manifold method |
| url | http://ckyyb.crsri.cn/fileup/1001-5485/PDF/1735268218068-714963553.pdf |
| work_keys_str_mv | AT suhaidong generalformulasandprogramdesignformanifoldmethodbasedonindependentcoversigeneralformulas |