Finite Element Analysis of Structures Using -Continuous Cubic B-Splines or Equivalent Hermite Elements

We compare contemporary practices of global approximation using cubic B-splines in conjunction with double multiplicity of inner knots (-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. 1. Introduction 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 (-version). Alternatively, keeping the number and the position of the nodal points unaltered, the numerical solution improves using polynomials of higher degree (-version) [1]. As an extension of the above -version “philosophy,” tensor-product Lagrange polynomials as well as CAD-based (Gordon-Coons) macroelements—based on several interpolations—have been used [2–4]. The aforementioned macroelements integrate the solid modelling (CAD: computer-aided-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. In order to avoid the undesired numerical oscillations caused by Lagrange polynomials of high degree, the next generation of CAD-based macroelements replaced them with tensor-product B-splines [5]. Since 2005, the nonuniform-rational-B-splines (NURBS) interpolation has started to prevail [6]. A careful study of literature reveals that most of recent papers referring to the so-called isogeometric analysis (IGA) start with some essentials on the definition of B-splines and relevant recursive formulas due to de Boor [7]. It should be recalled that NURBS is an extension of B-splines (nonuniform and rational) modified on the basis of weighting coefficients, thus producing fully controlled sculptured surfaces [8, 9]. In a B-splines expansion, the multiplicity of the inner knots plays a significant role in the continuity of the variables. In general, the multiplicity of inner knots per breakpoint in combination with a piecewise polynomial of degree ensures -continuity of the variable (here: displacement components) [7, 9]. Thus considering cubic B-splines () in conjunction with double inner knots (), -continuity