# Interpolation Using B-Splines and Relevant Macroelements

• Christopher G. Provatidis
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 256)

## Abstract

In this chapter, we present the theoretical background and discuss the numerical performance of CAD-macroelements in which the solution is approximated with B-splines. Univariate as well as two-dimensional tensor-product interpolation will be considered. Not only the usual Curry–Schoenberg B-splines normalized to provide a partition of unity, but also “reduced cardinal B-splines” are studied (to fully explain some older papers of the CAD/CAE group at NTUA since 1989). Numerical experiments of this chapter restrict to the Galerkin–Ritz formulation and refer to domains or structures of simple 2D primitive shapes (rectangles, circles, and ellipses). The numerical analysis is performed using a single macroelement only, without domain decomposition. The results are also compared with the FEM solution of the same mesh density.

## Keywords

B-splines Truncated power series Reduced cardinal B-splines De Boor formulation MATLAB Tensor product Solved problems

## Nomenclature

$$\xi$$

Normalized coordinate $$\left( {\xi = x/L} \right)$$, with $$\xi \in \left[ {0,1} \right]$$ and $$x \in \left[ {0,L} \right]$$

$$n$$

Number of breakpoint spans in the interval $$\xi \in \left[ {0,1} \right]$$

$$(n + 1)$$

Number of breakpoints

$$p$$

Degree of polynomial

$$(n + p)$$

Number of coefficients in truncated power series formulation [for $$C^{(p-1)\_}$$continuity]

$$\phi_{i} ,U_{i}$$

Shape functions and nodal displacements, respectively, based on $$(n + 3)$$ test points

$$\bar{\phi }_{i} ,\bar{U}_{i}$$

Reduced shape functions and nodal displacements, respectively, associated to the breakpoints only. Based on the first derivative (slope) at the two ends

$$\hat{\phi }_{i} ,\hat{U}_{i}$$

Reduced shape functions and nodal displacements, respectively, associated to the breakpoints only. Based on the second derivative (curvature) at the two ends

$$(n_{p} + 1)$$

Number of control points $$\left( {P_{0} ,P_{1} , \ldots ,P_{{n_{p} }} } \right)$$

$$N_{i,p} (\xi )$$

De Boor basis functions, $$i = 0, \ldots ,n + p - 1$$

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