Finite element model of a pseudominimal surface

The author of the current paper continues his study of the geometrical approach to modelling the shape of a fine elastic orthotopic material spanned on a closed contour. This regards, in particular, a reflecting surface of the reflector antenna, made of metallic mesh. The modelling is based on the class of surfaces with a constant ratio of principal curvatures. This class was introduced by the author in publications of 2016 and is called the class of pseudominimal surfaces (the class of minimal surfaces is its subclass). Pseudominimal surfaces are specified by a partial differential equation, which is very difficult to analyze. A sufficiently appropriate tool to build surfaces close to pseudominimal ones is the finite element method, applied in this article. An essential role is played by the existence theorem according to which the width of pseudominimal surfaces class is two functions of a scalar argument. An algorithm allowing one to calculate the position of a variable fifth node inside the cell for the given four nodes of the grid (the grid is not necessarily orthogonal and uniform) has been developed. This algorithm is a modification of the well-known algorithm which is effective for the finite-element modelling of minimal surfaces. The algorithm modification involves consideration of inequity of the two principal directions at the point of surface due to orthotropy. The coordinates of the fifth node are calculated from the coordinates of the four nodes using the weighting factors reflecting the ratio of the principal curvatures. The algorithm implementing the finite element method is modified to a more convenient algorithm of the stretched grids method (SGM) analogous to the modification (E.V. Popov, 1990s) for minimal surfaces modelling. The class of pseudominimal surfaces of revolution contains a family of algebraic surfaces of the fourth order. This kind of surface has been used for the algorithm testing. The author has good ground to believe that this algorithm is suitable for modelling.

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