Bernstein–Galerkin approach in elastostatics

Abstract

Since 1994, two main meshless methods have been developed and widely used: these are the element free Galerkin method and the meshless local Petrov-Galerkin method. Both methods solve partial differential equations by posing a numerical approximation to the solution using the moving least squares technique. Using Bernstein polynomials as the shape functions of Galerkin weak form-based methods improves the numerical approximation achieved at boundaries without losing accuracy inside the domain.

A coupled FEM–Bernstein approach for computing the J k integrals

“…6 Inner evaluation points in Bernstein domain. a n r = 3, n θ = 6, b n r = 7, n θ = 13 be reached for relatively low-order approximations [40], numerical dissipation may increase remarkably with the p-refinement due to evaluation of binomial terms and powers of very high order [37,41]. This problem can be relieved by applying the binomial multiplicative formula, which allows binomial terms to be computed more efficiently [39,50]:…”

“…Let u(x) be a function defined in a one-dimensional domain Ω, which is intended to be approximated by a numerical function u h (x) resulting from the linear combination: (12) where N is the number of evaluation points in the domain, φ k is the shape function associated to point k and a k is the kth evaluation point parameter, which does not coincide with the actual value of u h at x k , since Bernstein expansion is a non-interpolating scheme and therefore suffers from the lack of Delta Kronecker property, a k = u h (x k ). The shape function φ k is then written in the form [41]:…”

“…Interface value problems were reviewed by Liu et al [40], while Bernstein-Galerkin and Bernstein-Petrov-Galerkin formulations were applied to high even-order differential equations by Doha et al [36]. Numerical behaviour for free vibration analysis was checked by Valencia et al [37], and comparison of computational performances between Bernstein and MLS approximations in elastostatics benchmarks was performed by Garijo et al [41].…”