Numerical Solution of the Biharmonic Problem by the High-Order Finite Element Method on Triangular Meshes

Mathematical modeling of boundary value problems for elliptic partial differential equations of the fourth order is a fundamental problem of computationalmechanics, since equations with a biharmonic operator describe the bending of thin elastic plates and the dynamics of slow viscous fluid flowsin hydrodynamics problems. Traditional grid approaches often experience significant difficulties in approximating the conditions of rigid clamping onthe contour, which necessitates the development of methods for approximate solution of such problems. In this paper, a computational scheme of thefinite element method for solving a two-dimensional biharmonic equation in a square domain is proposed and investigated. To ensure the requiredclass of smoothness at the interelement boundaries, polynomial splines of the fifth degree on triangles are used. The discretization of the computationaldomain is implemented using the Union Jack pattern - one of the classic triangulation patterns. Verification and cross-validation of the obtained computationalresults were performed by comparing them with the classical analytical solution of S. P. Tymoshenko, as well as with the geometric meshfree method of R  functions of Academician of the National Academy of Sciences of Ukraine V. L. Rvachev. A series of computational experiments were conducted for the initial levels of grid discretization, on the basis of which an analysis of the effectiveness of the selected approach was performed. It was established that the use of a fully isotropic Union Jack pattern provides excellent indicators of the conditionality of the global stiffness matrix and minimizes the geometric error of discretization due to the ideal mirror symmetry relative to the center of the cell. The numerical results obtained using the proposed algorithm showed high accuracy, which is mathematically comparable to the quasi-analytical solutions of the R - functionsmethod. At the current stage of the algorithm, the integration of the shape functions was carried out numerically, which determined the high computational cost of the procedure. The scientific novelty of the work lies in the comprehensive study of the interaction of fifth-degree splines with a regular topology of a triangular mesh of the Union Jack type when solving boundary value problems with a biharmonic operator. The results obtained and theformulated geometric regularities create a reliable computational basis for scaling the method to areas of more complex geometry. Based on the analysis,a promising direction for further research is outlined, which consists in constructing explicit analytical expressions for the elements of stiffness matrices.The derivation of such closed formulas in the future will allow us to completely abandon the procedure of numerical integration, which will radically increase the speed of formation of global matrices and eliminate rounding errors.