A multi-patch nonsingular isogeometric boundary element method using trimmed elements

Abstract

One of the major goals of isogeometric analysis is direct design-to-analysis, i.e., using computer-aided design (CAD) files for analysis without the need for mesh generation. One of the primary obstacles to achieving this goal is CAD models are based on surfaces, and not volumes. The boundary element method (BEM) circumvents this difficulty by directly working with the surfaces. The standard basis functions in CAD are trimmed nonuniform rational B-spline (NURBS). NURBS patches are the tensor product of one-dimensional NURBS, making the construction of arbitrary surfaces difficult. Trimmed NURBS use curves to trim away regions of the patch to obtain the desired shape. By coupling trimmed NURBS with a nonsingular BEM, the formulation proposed here comes close achieving the goal of direct design to analysis. Example calculations demonstrate its efficiency and accuracy.

Appendix: Evaluation of boundary stresses and displacement gradients

The boundary stresses in the IGABEM may be derived from the traction-recovery method [24]. Following their development, a local Cartesian coordinate system $\overline{x}_{i}$ is defined as shown in Fig. 21 (global coordinate system is $x_{i}$), where $\overline{x}_{1}$ and $\overline{x}_{2}$ are tangential to the surface and $\overline{x}_{3}$ is directed in the outward normal direction. In order to concisely describe the formulation, the notation $(\xi _{1},\,\xi _{2})$ replaces $(\xi ,\,\eta )$ for the intrinsic coordinates.

Fig. 21

Local orthogonal system for evaluating the derivatives of displacements

The derivatives of the intrinsic coordinates are

$$\begin{aligned}&\frac{\partial {\xi _{1}}}{\partial {\overline{x}_{1}}}=\frac{1}{m_{1}},\quad \frac{\partial {\xi _{1}}}{\partial {\overline{x}_{2}}}=\frac{-\cos \theta }{|m_{1}|\sin \theta },\quad \frac{\partial {\xi _{2}}}{\partial {\overline{x}_{1}}}=0,\nonumber \\&\qquad \frac{\partial {\xi _{2}}}{\partial {\overline{x}_{2}}}=\frac{1}{|m_{2}|\sin \theta }, \end{aligned}$$

(33)

where $m_{k}$ and the angle $\theta$ are given as

$$\begin{aligned}&\left| m_{k}\right| =\sqrt{\left( \frac{\partial {x_{1}}}{\partial {\xi _{k}}}\right) ^{2}+\left( \frac{\partial {x_{2}}}{\partial {\xi _{k}}}\right) ^{2}+\left( \frac{\partial {x_{3}}}{\partial {\xi _{k}}}\right) ^{2}},\nonumber \\&\cos \theta =\frac{1}{|m_{1}||m_{2}|}\frac{\partial {x_{i}}}{\partial {\xi _{1}}}\frac{\partial {x_{i}}}{\partial {\xi _{2}}}. \end{aligned}$$

(34)

The local displacements and tractions are expressed in terms of the global displacements and tractions

$$\begin{aligned} \overline{u}_{i}=L_{ij}u_{j},\quad \overline{t}_{i}=L_{ij}t_{j}, \end{aligned}$$

(35)

in which $L_{ij}$ are entries of the rotation matrix $\varvec{L}$

$$\begin{aligned} \varvec{L}= \left[ \begin{array}{l@{\quad }l@{\quad }l} \frac{1}{|m_{1}|}\frac{\partial {x_{1}}}{\partial {\xi _{1}}} &{} \ \ \frac{1}{|m_{1}|}\frac{\partial {x_{2}}}{\partial {\xi _{1}}} &{} \ \ \frac{1}{|m_{1}|}\frac{\partial {x_{3}}}{\partial {\xi _{1}}} \\ n_{2}L_{13}-n_{3}L_{12} &{} \ \ n_{3}L_{11}-n_{1}L_{13} &{} \ \ n_{1}L_{12}-n_{2}L_{11} \\ n_{1} &{} \ \ n_{2} &{} \ \ n_{3} \end{array}\right] , \end{aligned}$$

(36)

where $n_{1},\,n_{2},$ and $n_{3}$ are the components of unit outward normal.

After some lengthy algebra using Hooke’s law, the boundary stress may be expressed in terms of the displacements and tractions of control points as

$$\begin{aligned} \sigma _{mn}=C_{mnj\alpha }u_{j}^{\alpha }+D_{mnj\alpha }t_{j}^{\alpha }, \end{aligned}$$

(37)

where the coefficients $C_{mnj\alpha }$ and $D_{mnj\alpha }$ are

$$\begin{aligned} C_{mnj\alpha }= & {} 2\mu \left\{ \frac{1}{1-\nu }\left[ L_{1m}L_{1n}\left( \frac{\partial {\xi _{k}}}{\partial {\overline{x}_{1}}}L_{1j}+\nu \frac{\partial {\xi _{k}}}{\partial {\overline{x}_{2}}}L_{2j}\right) \right. \right. \nonumber \\&+\,\left. L_{2m}L_{2n} \left( \frac{\partial {\xi _{k}}}{\partial {\overline{x}_{2}}}L_{2j}+\nu \frac{\partial {\xi _{k}}}{\partial {\overline{x}_{1}}}L_{1j} \right) \right] \nonumber \\&+\,\frac{1}{2}\left( L_{1m}L_{2n}+L_{2m}L_{1n}\right) \nonumber \\&\left. \left( \frac{\partial {\xi _{k}}}{\partial {\overline{x}_{1}}}L_{2j}+\frac{\partial {\xi _{k}}}{\partial {\overline{x}_{2}}}L_{1j} \right) \right\} \frac{\partial {N_{\alpha }}}{\partial {\xi _{k}}}, \end{aligned}$$

(38)

$$\begin{aligned} D_{mnj\alpha }= & {} \left( L_{3m}L_{1n}+L_{1m}L_{3n}\right) L_{1j}\nonumber \\&+\, \left( L_{2m}L_{3n}+L_{3m}L_{2n}\right) L_{2j} \nonumber \\&+\,\left( \frac{\nu }{1-\nu }\delta _{mn}+\frac{1-2\nu }{1-\nu }L_{3m}L_{3n} \right) L_{3j}, \end{aligned}$$

(39)

where $N_{\alpha }$ is the basis function associated with the $\alpha$th control point of the element [refer to Eqs. (16) and (17)], $\nu$ is Poisson’s ratio, $\mu$ is shear modulus, and $\delta _{ij}$ is Kronecker delta.

With the help of the local coordinate system in Fig. 21, the derivatives of global displacement can be expressed as

$$\begin{aligned} u_{m,n}=L_{rm}L_{sn}\overline{u}_{r,s}. \end{aligned}$$

(40)

When the subscript j does not equal 3, the local derivatives of displacements are

$$\begin{aligned} \overline{u}_{i,j}=\frac{\partial {\overline{u}_{i}}}{\partial {\overline{x}_{j}}}=\frac{\partial {\overline{u}_{i}}}{\partial {\xi _{k}}}\frac{\partial {\xi _{k}}}{\partial {\overline{x}_{j}}}, \end{aligned}$$

(41)

where the subscript k ranges from 1 to 2. When the subscript j equals to 3, according to Hooke’s law and $\overline{\sigma }_{i3}=\overline{t}_{i},$ the local derivatives of the displacements are

$$\begin{aligned}&\frac{\partial {\overline{u}_{1}}}{\partial {\overline{x}_{3}}}=\frac{\overline{t}_{1}}{\mu }-\left( \frac{\partial {\overline{u}_{3}}}{\partial {\xi _{k}}}\frac{\partial {\xi _{k}}}{\partial {\overline{x}_{1}}}\right) , \nonumber \\&\frac{\partial {\overline{u}_{2}}}{\partial {\overline{x}_{3}}}=\frac{\overline{t}_{2}}{\mu }-\left( \frac{\partial {\overline{u}_{3}}}{\partial {\xi _{k}}}\frac{\partial {\xi _{k}}}{\partial {\overline{x}_{2}}}\right) , \nonumber \\&\frac{\partial {\overline{u}_{3}}}{\partial {\overline{x}_{3}}}=\frac{1-2\nu }{\mu (2-2\nu )}\overline{t}_{3}-\frac{\nu }{1-\nu }\left( \frac{\partial {\overline{u}_{1}}}{\partial {\xi _{k}}}\frac{\partial {\xi _{k}}}{\partial {\overline{x}_{1}}}+\frac{\partial {\overline{u}_{2}}}{\partial {\xi _{k}}}\frac{\partial {\xi _{k}}}{\partial {\overline{x}_{2}}}\right) .\nonumber \\ \end{aligned}$$

(42)

Using Eqs. (16), (17), (35), (40)–(42), the derivatives of displacements are expressed in terms of the displacements and tractions of control points

$$\begin{aligned} u_{m,n}=A_{mnj\alpha }u_{j}^{\alpha }+B_{mnj\alpha }t_{j}^{\alpha }, \end{aligned}$$

(43)

where the coefficient $A_{mnj\alpha }$ is

$$\begin{aligned} A_{mnj\alpha }= & {} \left( L_{1m}L_{1j}+L_{2m}L_{2j}+L_{3m}L_{3j}\right) L_{1n}c_{\alpha 1} \nonumber \\&+\,\left( \frac{\nu }{\nu -1}L_{3m}+L_{1j}-L_{1m}L_{3j} \right) L_{3n}c_{\alpha 1} \nonumber \\&+\,\left( L_{1m}L_{1j}+L_{2m}L_{2j}+L_{3m}L_{3j}\right) L_{2n}c_{\alpha 2}\nonumber \\&+\,\left( \frac{\nu }{\nu -1}L_{3m}+L_{2j}-L_{2m}L_{3j} \right) L_{3n}c_{\alpha 2},\nonumber \\ \end{aligned}$$

(44)

where coefficients $c_{\alpha 1}$ and $c_{\alpha 2}$ are

$$\begin{aligned} c_{\alpha 1}= & {} \frac{\partial {\xi _{1}}}{\partial {\overline{x}_{1}}}\frac{\partial {N_{\alpha }}}{\partial {\xi _{1}}}+\frac{\partial {\xi _{2}}}{\partial {\overline{x}_{1}}}\frac{\partial {N_{\alpha }}}{\partial {\xi _{2}}}, \end{aligned}$$

(45)

$$\begin{aligned} c_{\alpha 2}= & {} \frac{\partial {\xi _{1}}}{\partial {\overline{x}_{2}}}\frac{\partial {N_{\alpha }}}{\partial {\xi _{1}}}+\frac{\partial {\xi _{2}}}{\partial {\overline{x}_{2}}}\frac{\partial {N_{\alpha }}}{\partial {\xi _{2}}}, \end{aligned}$$

(46)

and the coefficient $B_{mnj\alpha }$ is

$$\begin{aligned} B_{mnj\alpha }= & {} \frac{L_{3n}}{\mu }\left( L_{1m}L_{1j}+L_{2m}L_{2j}\right. \nonumber \\&\left. +\frac{1-2\nu }{2(1-\nu )}L_{3m}L_{3j}\right) N_{\alpha }. \end{aligned}$$

(47)