Three-dimensional finite-volume theory for elastic stress analysis in solid mechanics

The finite-volume theory is a powerful numerical technique for structural analysis in solid mechanics and has emerged as an alternative to the finite-element method. The finite-volume theory is an equilibrium-based approach that employs surface-averaged tractions and displacements acting on the faces of a subvolume. In addition, this theory employs the equilibrium equations at the subvolume level and continuity conditions between adjacent subvolumes along subvolume faces. The finite-volume theory has been successfully used for two-dimensional structural analyses. However, in the context of three-dimensional solid mechanics analysis, this numerical technique has encountered instability problems related to the interpretation between adjacent subvolumes' faces. These problems result in the singularity of the global stiffness matrix, which has delayed the publication of a three-dimensional version of the finite-volume theory for continuum elastic structures. These numerical instability problems can be solved by employing a modified stiffness matrix, where the Tikhonov regularization method is used to artificially add minimal stiffness in the main diagonal of the global stiffness matrix. This contribution proposes the stress analysis of three-dimensional continuum elastic structures by the finite-volume theory, where problems with analytical solutions are employed for verification.