Generalized isogeometric shape sensitivity analysis in curvilinear coordinate system and shape optimization of shell structures

Abstract

A generalized sensitivity formulation described in a curvilinear coordinate system is proposed. Utilizing it, the continuum-based isogeometric shape sensitivity analysis method for the shell components is developed in the curvilinear coordinates derived from the given NURBS geometry. In isogeometric approach, the designs are embedded into the NURBS basis functions and the control points so that geometrically exact shell models can be incorporated in both response and sensitivity analyses. The precise shape sensitivities can be obtained by considering accurate and continuous normal and curvatures in the boundary integrals of the boundary resultants of the shell and their material derivatives. Through numerical examples, the developed isogeometric shape sensitivity is verified to demonstrate excellent agreements with finite difference sensitivity. Also, the importance of higher order geometric information in the sensitivity expressions is identified. For the shape optimization problem of the shell, the proposed method works well with boundary resultants accompanying severe curvature changes.

Appendix. Gradient computations for design sensitivity analysis

The unit normal vector a ₃ to the other base vectors can be calculated as

$$ {\mathbf{a}}_3=\frac{{\mathbf{a}}_1\times {\mathbf{a}}_2}{\left\Vert {\mathbf{a}}_1\times {\mathbf{a}}_2\right\Vert }. $$

(A-1)

Its gradient computation shown in (68) with respect to the shape parameter is given by

$$ \nabla {\mathbf{a}}_3\cdot \mathbf{V}=\frac{{\mathbf{V}}_{,1}\times {\mathbf{a}}_2}{\left\Vert {\mathbf{a}}_1\times {\mathbf{a}}_2\right\Vert }+\frac{{\mathbf{a}}_1\times {\mathbf{V}}_{,2}}{\left\Vert {\mathbf{a}}_1\times {\mathbf{a}}_2\right\Vert }-\frac{{\mathbf{a}}_3}{\left\Vert {\mathbf{a}}_1\times {\mathbf{a}}_2\right\Vert}\left({\mathbf{a}}_3\cdot \left({\mathbf{V}}_{,1}\times {\mathbf{a}}_2+{\mathbf{a}}_1\times {\mathbf{V}}_{,2}\right)\right) $$

(A-2)

where

$$ \nabla {\mathbf{a}}_{\alpha}\cdot \mathbf{V}=\frac{\partial }{\partial {}^n\mathbf{x}}\left(\frac{\partial {}^n\mathbf{x}}{\partial {\theta}^{\alpha }}\right)\cdot \mathbf{V}={\mathbf{V}}_{,\alpha }. $$

(A-3)

Obviously, V = V(ⁿ x) in all equations in this appendix. Also in (68), the following yields

$$ \nabla {\mathbf{a}}^{\alpha}\cdot \mathbf{V}=\nabla \left({a}^{\alpha \beta }{\mathbf{a}}_{\beta}\right)\cdot \mathbf{V}=\left(\nabla {a}^{\alpha \beta}\cdot \mathbf{V}\right){\mathbf{a}}_{\beta }+{a}^{\alpha \beta }{\mathbf{V}}_{,\beta }. $$

(A-4)

Using a ^αλ a _λμ  = δ ^α _μ , we have

$$ \nabla {a}^{\alpha \beta}\cdot \mathbf{V}=-{a}^{\alpha \lambda}\left(\nabla {a}_{\lambda \mu}\cdot \mathbf{V}\right){a}^{\mu \beta }=-{a}^{\alpha \lambda}\left({\mathbf{V}}_{,\lambda}\cdot {\mathbf{a}}_{\mu }+{\mathbf{a}}_{\lambda}\cdot {\mathbf{V}}_{,\mu}\right){a}^{\mu \beta }. $$

(A-5)

The gradient computations shown in (77) are given by

$$ \begin{array}{c}\hfill \nabla {\varGamma}_{\alpha \beta}^{\mu}\cdot \mathbf{V}=\left(\nabla {\mathbf{a}}_{\alpha, \beta}\cdot \mathbf{V}\right)\cdot {\mathbf{a}}^{\mu }+{\mathbf{a}}_{\alpha, \beta}\cdot \left(\nabla {\mathbf{a}}^{\mu}\cdot \mathbf{V}\right)\hfill \\ {}\hfill ={\mathbf{V}}_{,\alpha \beta}\cdot {\mathbf{a}}^{\mu }+{\mathbf{a}}_{\alpha, \beta}\cdot \left(\nabla {a}^{\mu \lambda}\cdot \mathbf{V}\right){\mathbf{a}}_{\lambda }+{a}^{\mu \lambda }{\mathbf{V}}_{,\lambda}\hfill \end{array} $$

(A-6)

and

$$ \begin{array}{c}\hfill \nabla {b}_{\alpha \beta}\cdot \mathbf{V}=\left(\nabla {\mathbf{a}}_3\cdot \mathbf{V}\right)\cdot {\mathbf{a}}_{\alpha, \beta }+{\mathbf{a}}_3\cdot \left(\nabla {\mathbf{a}}_{\alpha, \beta}\cdot \mathbf{V}\right)\hfill \\ {}\hfill =\left(\nabla {\mathbf{a}}_3\cdot \mathbf{V}\right)\cdot {\mathbf{a}}_{\alpha, \beta }+{\mathbf{a}}_3\cdot {\mathbf{V}}_{,\alpha \beta }.\hfill \end{array} $$

(A-7)

Also in (79), we have

$$ \nabla {b}_{\alpha}^{\beta}\cdot \mathbf{V}=\left(\nabla {b}_{\lambda \alpha}\cdot \mathbf{V}\right){a}^{\beta \lambda }+{b}_{\lambda \alpha}\left(\nabla {a}^{\beta \lambda}\cdot \mathbf{V}\right). $$

(A-8)

Moreover, the gradient computations of the material tensors in (80) can be given from (52),

$$ \begin{array}{c}\hfill \nabla {C}^{\alpha \beta \mu \lambda}\cdot \mathbf{V}=\frac{E}{2\left(1+\nu \right)}\left\{\left(\nabla {a}^{\alpha \mu}\cdot \mathbf{V}\right){a}^{\beta \lambda }+{a}^{\alpha \mu}\left(\nabla {a}^{\beta \lambda}\cdot \mathbf{V}\right)+\left(\nabla {a}^{\alpha \lambda}\cdot \mathbf{V}\right){a}^{\beta \mu }+{a}^{\alpha \lambda}\left(\nabla {a}^{\beta \mu}\cdot \mathbf{V}\right)\right\}\hfill \\ {}\hfill +\frac{2\nu E}{2\left(1+\nu \right)\left(1-\nu \right)}\left\{\left(\nabla {a}^{\alpha \beta}\cdot \mathbf{V}\right){a}^{\mu \lambda }+{a}^{\alpha \beta}\left(\nabla {a}^{\mu \lambda}\cdot \mathbf{V}\right)\right\}\hfill \end{array} $$

(A-9)

and

$$ \nabla {C}^{\alpha 3\beta 3}\cdot \mathbf{V}=\frac{E}{2\left(1+\nu \right)}\nabla {a}^{\alpha \beta}\cdot \mathbf{V}. $$

(A-10)