Abstract
We compare contemporary practices of global approximation using cubic B-splines in conjunction with double multiplicity of inner knots (C1-continuous) with older ideas of utilizing local Hermite interpolation of third degree. The study is conducted within the context of the Galerkin-Ritz formulation, which forms the background of the finite element structural analysis. Numerical results, concerning static and eigenvalue analysis of rectangular elastic structures in plane stress conditions, show that both interpolations lead to identical results, a finding that supports the view that they are mathematically equivalent.
Highlights
Structural analysis is usually performed using commercial codes that include finite elements of low degree, where the accuracy of the calculations increases by mesh refinement (h-version)
In order to avoid the undesired numerical oscillations caused by Lagrange polynomials of high degree, the generation of CAD-based macroelements replaced them with tensor-product B-splines [5]
The need for smoother (C1) global basis functions is encountered in second-order problems when collocation finite element methods are utilized [10, page 66]. With these situations in mind, we examine the relationship between particular Hermite elements of third degree Journal of Structures and cubic B-splines elements with multiplicity λ = 2
Summary
Structural analysis is usually performed using commercial codes that include finite elements of low (usually first or second) degree, where the accuracy of the calculations increases by mesh refinement (h-version). The aforementioned macroelements integrate the solid modelling (CAD: computeraided-design) with the analysis (CAE: computer-aided-engineering) In more detail, these macroelements use the same global approximation for both the geometry and the displacement vector. The multiplicity of λ inner knots per breakpoint in combination with a piecewise polynomial of degree p ensures Cp−λ-continuity of the variable (here: displacement components) [7, 9]. The need for smoother (C1) global basis functions is encountered in second-order problems when collocation finite element methods are utilized [10, page 66]. With these situations in mind, we examine the relationship between particular Hermite elements of third degree.
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