Isogeometric configuration design optimization of three-dimensional curved beam structures for maximal fundamental frequency

Abstract

This paper presents an isogeometric configuration design optimization of curved beam structures for maximizing fundamental eigenfrequency. A shear-deformable beam model is used in the response analyses of structural vibrations within an isogeometric framework using the NURBS basis functions. An analytical design sensitivity of repeated eigenvalues is used. A special attention is paid to the computation of design velocity field and optimal design of beam structures constrained on a curved surface, where both designs of the embedded beams and the curved surface are simultaneously varied during the optimal design process. The developed design optimization method is demonstrated through several illustrative examples.

Appendix . Nonlinear mapping of embedded curves

In this appendix, we analytically illustrate the FFD process may yield a nonlinear mapping of embedded curves even if the target parent domain is a linear B-spline surface. We assume both of the square and trapezoidal domains respectively for initial and target parent domains, shown in Fig. 11, are composed of a single linear B-spline surface element, and the basis functions are given by:

$$ \left. \begin{array}{lcl} {\tilde{W}_{1}} &=& (1 - {{\tilde \xi }^{1}})(1 - {{\tilde \xi }^{2}})\\ {\tilde{W}_{2}} &=& {{\tilde \xi }^{1}}(1 - {{\tilde \xi }^{2}})\\ {\tilde{W}_{3}} &=& (1 - {{\tilde \xi }^{1}}){{\tilde \xi }^{2}}\\ {\tilde{W}_{4}} &=& {{\tilde \xi }^{1}}{{\tilde \xi }^{2}} \end{array}\right\}, $$

(79)

where the corresponding control point positions are ${\tilde {\mathbf {B}}}_{1}^{0}(0,0)$, ${\tilde {\mathbf {B}}}_{2}^{0}(1,0)$, ${\tilde {\mathbf {B}}}_{3}^{0}(0,1)$, and ${\tilde {\mathbf {B}}}_{4}^{0}(1,1)$ for the initial parent domain, and ${{\tilde {\mathbf {B}}}_{1}}(0,0)$, ${\tilde {\mathbf {B}}}_{2}^{0}(2,0)$, ${\tilde {\mathbf {B}}}_{3}^{0}(0.5,2)$, and ${\tilde {\mathbf {B}}}_{4}^{0}(1.5,2)$ for the target parent domain. The target parent domain is expressed by a linear combination of the basis functions of (79) and the control point positions ${{\tilde {\mathbf {B}}}_{i}}$ (i = 1,..., 4) through (39), as:

$$ {\tilde{\mathbf{X}}}({\tilde \xi^{1}},{\tilde \xi^{2}}) = \left[{\begin{array}{*{20}{c}} {2{{\tilde \xi }^{1}} + 0.5{{\tilde \xi }^{2}} + 0.3{{\tilde \xi }^{1}}{{\tilde \xi }^{2}}}\\ {2{{\tilde \xi }^{2}}} \end{array}}\right]. $$

(80)

Using (41) and b₁ = b₂ = 1, the surface parametric coordinates corresponding to the curve parametric position is obtained by:

$$ \left. \begin{array}{c} {{\tilde \xi }^{1}} = {X_{1}^{0}}\\ {{\tilde \xi }^{2}} = {X_{2}^{0}} \end{array}\right\}. $$

(81)

Consequently, substituting (81) into (80) yields:

$$ {\tilde{\mathbf{X}}}({X_{1}^{0}},{X_{2}^{0}}) = \left[{\begin{array}{*{20}{c}} {2{X_{1}^{0}} + 0.5{X_{2}^{0}} + 0.3{X_{1}^{0}}{X_{2}^{0}}}\\ {2{X_{2}^{0}}} \end{array}}\right], $$

(82)

which shows that the linear curve in the initial parent domain is mapped into a quadratic curve in the target parent domain.