On the approximation of derivatives in hexahedral 8-node finite elements

In numerical solutions of elasticity and plasticity problems, finite elements with a reduced integration are often used, especially in solving dynamic problems. In this case, for 8-node 3D elements, one point of the numerical integration is used instead of 8. In this case, it is actually assumed that the strains and stresses are constant within an element. In this case, the traditional technique of constructing the stiffness matrix for an element of a standard shape in the form of a cube with a subsequent mapping of the actual finite elements of an arbitrary shape and size onto the standard one is not necessary. Instead, the stiffness matrix can be constructed directly for a finite element of an arbitrary shape. In this case, it is expressed through the coefficients of grid operators approximating the first partial derivatives of the displacement field in the finite element. The paper considers a new approach to approximating derivatives when constructing the stiffness matrix for a 3D 8-node finite element with one integration point. The theoretical basis for this approach is the further development of the class of rare mesh FEM schemes. The obtained formulas allow one to construct incompatible FEM schemes with improved properties. The paper discusses the problems of hourglass instability, shear and volume locking. A new effective approach to solving the hourglass problem is proposed. The possibility of applying new derivative approximation formulas to finite elements of a degenerate form with a number of nodes less than eight is also discussed. It is shown that they remain applicable in a standard way in this case too. The results of the study are confirmed by the presented numerical solution results of the model static elasticity problems.