Algorithm for Studying Harmonic Oscillations in an Ideal Fluid with Absolutely Solid Inclusions Using the Finite Element Method

The current work is the first part of constructing a universal algorithm for solving the problem of sound diffraction on a system of elastic inhomogeneous anisotropic bodies. In the current work, a variation of this problem is posed in the first approximation - this is the problem of determining the pressure in a liquid region for a given pressure distribution on its outer boundary. The fluid that fills the area is considered ideal. It is understood that the pressure both inside the liquid and at its boundary performs harmonic oscillations (steady-state). An arbitrary number of solid simply connected bodies are located inside the region. It is required to determine the pressure of the liquid, taking into account the influence of solids. To solve the problem, the finite element method is used, the application algorithm of which is described in detail. The liquid region is divided into tetrahedral elements, inside which the unknown pressure is approximated using the shape functions and the introduced local coordinate system. For each tetrahedral element, a local matrix is constructed based on the transformed homogeneous Helmholtz equation, which is satisfied by the pressure inside the liquid due to the fact that the oscillation field is steady. Local matrices of elements make it possible to form a sparse global matrix for a system of equations, the solution of which determines the required pressure values at the grid nodes. The paper describes in detail the calculation of the elements of local matrices, taking into account liquid and absolutely solid boundaries of tetrahedral elements; taking into account pinnings (due to a given pressure fluctuation at the boundary of an ideal liquid region), including the transformation of all elements of local matrices and the right parts of systems of linear algebraic equations when taking into account pinnings; the union of local matrices and the corresponding right-hand sides of equations into a global sparse matrix. The described steps of the algorithm are universal for a wide class of problems. Also, a method for further generalization of the problem to the case of inclusion of elastic bodies with a complex internal structure into the liquid region is described.