Isogeometric Analysis of Solids in Boundary Representation

Abstract

In this chapter, we present boundary-oriented numerical methods to analyze three-dimensional solid structures. For the analysis, the original geometry of the solid is employed according to the isogeometric paradigm. For the parametrization of the domain, the idea of the scaled boundary finite element method is adopted. Hence, the boundary of the solid is sufficient to describe the entire domain. The presented approaches employ analytical and numerical solution methods such as the Galerkin and collocation methods. To illustrate the applicability in the analysis procedure, three formulations are elaborated and demonstrated by means of numerical examples. The advantages compared to standard numerical methods are discussed thoroughly.

The financial support of the German Research Foundation (DFG) under Grant No. KL1345/10-1 is gratefully acknowledged.

Appendix

The normal vectors $\varvec{n}^\xi$, $\varvec{n}^\eta$ and $\varvec{n}^\zeta$ are perpendicular to the surface described by the parameters ($\eta$,  $\zeta$), ($\zeta$,  $\xi$) and ($\xi$,  $\eta$), respectively, see Fig. 3. They are summarized as

$$\begin{aligned} \begin{aligned} \varvec{n}^\xi&= [n_{\hat{x}}^\xi ,\, n_{\hat{y}}^\xi , \,n_{\hat{z}}^\xi ]^T = {\dfrac{\hat{\varvec{x}}_s,_{\eta } \times \hat{\varvec{x}}_s,_{\zeta }}{\Vert \hat{\varvec{x}}_s,_{\eta } \times \hat{\varvec{x}}_s,_{\zeta }\Vert }} = \dfrac{\varvec{x}_s,_{\eta } \times \varvec{x}_s,_{\zeta }}{\Vert \varvec{x}_s,_{\eta } \times \varvec{x}_s,_{\zeta }\Vert } \\&= \frac{1}{g^\xi }\begin{bmatrix} {y}_s,_{\eta }{z}_s,_{\zeta }-{z}_s,_{\eta }{y}_s,_{\zeta }\\ {z}_s,_{\eta }{x}_s,_{\zeta }-{x}_s,_{\eta }{z}_s,_{\zeta }\\ {x}_s,_{\eta }{y}_s,_{\zeta }-{y}_s,_{\eta }{x}_s,_{\zeta } \end{bmatrix}, \end{aligned} \end{aligned}$$

(A.1)

$$\begin{aligned} \begin{aligned} \varvec{n}^\eta&= [n_{\hat{x}}^\eta ,\, n_{\hat{y}}^\eta , \,n_{\hat{z}}^\eta ]^T = {\dfrac{\hat{\varvec{x}}_s,_{\zeta } \times \hat{\varvec{x}}_s,_{\xi }}{\Vert \hat{\varvec{x}}_s,_{\zeta } \times \hat{\varvec{x}}_s,_{\xi }\Vert }} = \dfrac{\varvec{x}_s,_{\zeta } \times (\varvec{x}_s-\hat{\varvec{x}}_0)}{\Vert \varvec{x}_s,_{\zeta } \times (\varvec{x}_s-\hat{\varvec{x}}_0)\Vert }\\&=\frac{1}{g^\eta }\begin{bmatrix} (z_s-\hat{z}_0){y}_s,_{\zeta }-({y}_s-\hat{y}_0){z}_s,_{\zeta }\\ (x_s-\hat{x}_0){z}_s,_{\zeta }-({z}_s-\hat{z}_0){x}_s,_{\zeta }\\ ({y}_s-\hat{y}_0){x}_s,_{\zeta }-({x}_s-\hat{x}_0){y}_s,_{\zeta } \end{bmatrix}, \end{aligned} \end{aligned}$$

(A.2)

$$\begin{aligned} \begin{aligned} \varvec{n}^\zeta&= [n_{\hat{x}}^\zeta ,\, n_{\hat{y}}^\zeta , \,n_{\hat{z}}^\zeta ]^T = {\dfrac{\hat{\varvec{x}}_s,_{\xi } \times \hat{\varvec{x}}_s,_{\eta }}{\Vert \hat{\varvec{x}}_s,_{\xi } \times \hat{\varvec{x}}_s,_{\eta }\Vert }} = \dfrac{(\varvec{x}_s-\hat{\varvec{x}}_0) \times \varvec{x}_s,_{\eta }}{\Vert (\varvec{x}_s-\hat{\varvec{x}}_0) \times \varvec{x}_s,_{\eta }\Vert }\\&= \frac{1}{g^\zeta }\begin{bmatrix} (y_s-\hat{y}_0){z}_s,_{\eta }-({z}_s-\hat{z}_0){y}_s,_{\eta }\\ (z_s-\hat{z}_0){x}_s,_{\eta }-({x}_s-\hat{x}_0){z}_s,_{\eta }\\ ({x}_s-\hat{x}_0){y}_s,_{\eta }-({y}_s-\hat{y}_0){x}_s,_{\eta } \end{bmatrix} \end{aligned} \end{aligned}$$

(A.3)

where $g^\xi$, $g^\eta$ and $g^\eta$ are considered according to Chen et al. (2015, 2016). The transformation of an infinitesimal surface element ${\text {d}}S$ is derived by employing Eqs. (1) and (A.1)–(A.3) as

$$\begin{aligned} {\text {d}}S^\xi&=|\hat{\varvec{x}}_s,_{\eta } \times \hat{\varvec{x}}_s,_{\zeta }|{\text {d}}\eta {\text {d}}\zeta =|\xi \varvec{x}_s,_{\eta } \times \xi \varvec{x}_s,_{\zeta }|{\text {d}}\eta {\text {d}}\zeta =\xi ^2g^\xi {\text {d}}\eta {\text {d}}\zeta , \end{aligned}$$

(A.4)

$$\begin{aligned} {\text {d}}S^\eta&=|\hat{\varvec{x}}_s,_{\zeta } \times \hat{\varvec{x}}_s,_{\xi }|{\text {d}}\zeta {\text {d}}\xi =|\xi \varvec{x}_s,_{\zeta } \times (\varvec{x}_s-\hat{\varvec{x}}_0)|{\text {d}}\zeta {\text {d}}\xi =\xi g^\eta {\text {d}}\zeta {\text {d}}\xi , \end{aligned}$$

(A.5)

$$\begin{aligned} {\text {d}}S^\zeta&=|\hat{\varvec{x}}_s,_{\xi } \times \hat{\varvec{x}}_s,_{\eta }|{\text {d}}\xi {\text {d}}\eta =|(\varvec{x}_s-\hat{\varvec{x}}_0) \times \xi \varvec{x}_s,_{\eta }|{\text {d}}\xi {\text {d}}\eta =\xi g^\zeta {\text {d}}\xi {\text {d}}\eta . \end{aligned}$$

(A.6)