Graph Approach for Finite-Element Model of an Elastic Body in Polar Coordinates

The purpose of the article was to suggest a numerical method for analysis of the stress-strain state of elastic media based on a discrete model in form of directed graph. To analyze a deformable body using the graph approach, we partition a solid body onto elements and replace each element by its model in the form of an elementary cell. Derivation of cell equations, which is based on conversion of an element to a cell, relies on an invariant. We use the deformation energy as the invariant. The study describes a procedure to determine parameters of the elementary cell. The graph of the whole body is built following the same rule as in the elementary cell. The method is based on transforming generalized coordinates of a solid body separated into pieces to a system of generalized coordinates of the initial solid body. The specific nature of the graph model lies in the possibility to construct these matrices with no need for their numerical inversion. With the use of a unit cell having 8 degrees of freedom, the strain field is approximated by linear polynomials (with corresponds to approximated of the displacement fields by quadratic polynomials). The standard finite-element method requires 16 degrees of freedom (elements with 8 nodes) for the same purpose. The proposed graphical approach thus reduces the number of equations that describe the model. We provide numerical examples which prove the efficiency of the method.

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