Momentary finite element for elasticity 3D problems

Abstract

A description of a new 8-node finite element in the form of a hexahedron is given for solving elasticity 3D problems. This finite element has the following features. This is a linear approximation of functions in the element, one point of integration and taking into account the moments of forces in the element. The finite element is based on “rare mesh” FEM schemes—finite element schemes in the form of n-dimensional cubes (square, cube, etc.) with templates in the form of inscribed simplexes (triangle, tetrahedron, etc.). Among the rare mesh schemes, schemes in 3-dimensional and 7-dimensional spaces are successful, in which the simplex can be arranged symmetrically with respect to the center of the n-dimensional cube. The rare mesh FEM schemes have not the hourglass instability due to the fact that the template of the finite element operator has the form of a simplex. Compared to traditional linear finite elements in the form of a simplex, rare mesh schemes are more economical and converge better, since they do not have the effect of overestimated shear stiffness. Moment FEM schemes are constructed by rare mesh schemes higher dimensional projection, respectively, on a two-dimensional or three-dimensional finite element mesh. The resulting finite elements are close to the known polylinear elements and surpass them in efficiency. The schemes contain parameters that allow you to control the convergence of numerical solutions. The possibility of applying this approach to the construction of numerical schemes for solving other problems of mathematical physics is discussed.