Application of the Double Approximation Method for Constructing Stiffness Matrices of Volumetric Finite Elements

Abstract

Introduction. When numerically solving problems of elasticity theory in a three-dimensional formulation by the finite element method, finite elements (FE) in the form of parallelepipeds, prisms and tetrahedra are used. Regularly, the construction of stiffness matrices of volumetric FE is based on the principle of isoparametricity, which involves the Lagrange polynomials to approximate the geometry and displacements. In computational practice, the most widespread FE are the so-called multilinear isoparametric FE with a linear law of approximation of displacements. The main disadvantage of these elements lies in the “locking” effect when modulating bending deformations. Moreover, the error of the numerical solution increases drastically in the case when the structure, in comparison to conventional deformations, undergoes significant displacements as a rigid whole. Long-term experience in solving problems of deformable solid mechanics by the finite element method has shown that existing volumetric FE have slow convergence, specifically, when modeling bending deformations of plates and shells. This study aims at constructing stiffness matrices of multilinear volumetric FE of increased accuracy allowing for rigid displacements based on the double approximation method.Materials and Methods. The mathematical apparatus of the double approximation method based on the principle of a separate representation of the distribution functions of displacements and deformations inside the element, was used to construct the stiffness matrices of volumetric FE. The storage and processing of the resulting system of equations was implemented in algorithmic terms of sparse matrices. Software development and computational experiments were carried out using the Microsoft Visual Studio 2013 64-bit computing platform and the Intel ® Parallel Studio XE 2019 compiler with the integrated Intel ® Visual Fortran Composer XE 2019 text editor. Visualization of the calculation results was performed using the descriptor graphics of the MATLAB computer mathematics package. A large eight-node SOLID185 CE of the ANSYS Mechanical software complex was used as a test sample.Results. Mathematical tool and software were developed to study the stress-strain state of massive structures under various types of external actions. The authorized application software package was verified on test examples with known analytical solutions. It has been shown that the constructed FE accurately satisfy the basic requirements for finite element modeling of spatial problems of elasticity theory.Discussion and Conclusion. The performed testing of the developed mathematical and program toolkit has shown that the finite elements constructed on the basis of the double approximation method can successfully compete with similar SOLID185 volumetric elements of the ANSYS Mechanical software complex. The proposed elements can be integrated into domestic import-substituting software systems that implement the finite element method in the form of the displacement method.