Investigation on Imposing Essential Boundary Conditions in Higher Order Particle Discretization Scheme

Abstract

In this paper, we address the issue of specifying Dirichlet boundary conditions (BCs) in higher order Particle Discretization Scheme FEM (HO-PDS-FEM). HO-PDS has its roots in particle-based methods and formulated using variational principle. It approximates functions as the union of local function expansions over the elements of a chosen spatial tessellation and the derivatives over the conjugate tessellation. Unlike ordinary FEM, approximation scheme of HO-PDS-FEM do not possess Kronecker delta property. Therefore, when solving a boundary value problem (BVP) in N-dimension, the given N -1 dimensional information along the boundary, ∂Ω, is insufficient to evaluate all the coefficients, which include higher order gradients in N dimension, required for approximating the given BCs with HO-PDS. In previous studies, a curious solution was found to resolve this issue; the coefficients which cannot be evaluated from the N - 1 dimensional boundary value information can be left as unknowns, if the order of basis functions in set Q, which is used for approximating derivatives, is higher than that of the set P, which are used for approximating the unknown function (i.e. |Q| > |P|). In this paper, we shed some light on why this settings, which seems to disagree with calculus, resolves the issue of specifying Dirichlet BCs. Solving BVPs with strong singularities, it is shown that the setting |Q| > |P| produces solutions of expected degree of accuracy and convergence rate. Further, we explore the use of trigonometric base functions in set Q to improve the accuracy of crack tip stress fields. BVPs with complex cracking patterns such as branching and bendng are also analyzed. Results confirm the applicability of the setting |Q| = |P|+1 and establish the superiority of HO-PDS-FEM over original PDS-FEM. © 2023 American Institute of Physics Inc.. All rights reserved.