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The FDM in arbitrary curvilinear co‐ordinates—formulation, numerical approach and applications
Author(s) -
Tworzydlo Wlodzimierz W.
Publication year - 1989
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620280203
Subject(s) - curvilinear coordinates , covariant transformation , curvature , numerical analysis , finite element method , computation , mathematics , manifold (fluid mechanics) , algorithm , computer science , mathematical analysis , geometry , mechanical engineering , engineering , structural engineering
Abstract The basic concept, formulation and numerical application of a fully automatic version of the finite difference method (FDM) on a two‐dimensional manifold embedded in three‐dimensional Euclidean space are presented. This version of the FDM was developed in order to enable automatic solution of problems formulated in arbitrary curvilinear co‐ordinate systems in terms of covariant derivatives (e.g. shell equations). In the classical version of the FDM all operations in the curvilinear co‐ordinates (the derivation of base vectors, curvature tensors and so on) have to be performed manually. The goal of the present work was to transfer this job to a computer, in order to minimize the user's effort during the numerical formulation of the physical problem. The relevant version of the fully automatized program FIDAMF, based on the FDM on arbitrary irregular grids, 1,2 has been worked out. This code performs automatically all stages of the numerical analysis, starting from the mesh generation and approximation of the shape of the manifold, through computation of necessary objects on this manifold, to the solution of linear or non‐linear problems formulated in terms of covariant derivatives. The method has been extended to the analysis of problems in which the co‐ordinate system changes during the computation (e.g. the analysis of large deformations in the convectional description). This version of the curvilinear FDM was applied to the analysis of large deformations of hyperelastic membrane shells. The approach and the numerical routines, although used here with the FDM, can be combined with any other approximation method, in particular the finite element method.