A smooth finite element method based on reproducing kernel DMS-splines. (English) Zbl 1231.65218

Summary: The element-based piecewise smooth functional approximation in the conventional finite element method (FEM) results in discontinuous first and higher order derivatives across element boundaries. Despite the significant advantages of the FEM in modelling complicated geometries, a motivation in developing mesh-free methods has been the ease with which higher order globally smooth shape functions can be derived via the reproduction of polynomials. There is thus a case for combining these advantages in a so-called hybrid scheme or a ‘smooth FEM’ that, whilst retaining the popular mesh-based discretization, obtains shape functions with uniform \(C^{p}\) (\(p\geq 1\)) continuity. One such recent attempt, a NURBS based parametric bridging method, uses polynomial reproducing, tensor-product non-uniform rational B-splines (NURBS) over a typical FE mesh and relies upon a (possibly piecewise) bijective geometric map between the physical domain and a rectangular (cuboidal) parametric domain. The present work aims at a significant extension and improvement of this concept by replacing NURBS with DMS-splines (say, of degree \(n>0\)) that are defined over triangles and provide \(C^{n-1}\) continuity across the triangle edges. This relieves the need for a geometric map that could precipitate ill-conditioning of the discretized equations. Delaunay triangulation is used to discretize the physical domain and shape functions are constructed via the polynomial reproduction condition, which quite remarkably relieves the solution of its sensitive dependence on the selected knotsets. Derivatives of shape functions are also constructed based on the principle of reproduction of derivatives of polynomials. Within the present scheme, the triangles also serve as background integration cells in weak formulations thereby overcoming non-conformability issues. Numerical examples involving the evaluation of derivatives of targeted functions up to the fourth order and applications of the method to a few boundary value problems of general interest in solid mechanics over (non-simply connected) bounded domains in 2D are presented towards the end of the paper.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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