Isogeometric Approach for
Shape Optimization of
PhD Candidate: Zhenpei Wang
Supervisor: Sergio Turteltaub
Start date: 07-9-2011
Traditionally, most of the aerospace structures are
geometrically represented by Non-Uniform Rational
B-Splines (NURBS) functions and subsequently
analyzed in finite element programs that rely on
Lagrange polynomials. The conversion from NURBS
to Lagrange polynomials is significantly costly and
prone to loss of information. In contrast,
isogeometric analysis allows a seamless transition
from design to analysis based on common NURBSbased
representation provides a natural environment to
develop shape optimization. Typical applications
for aerospace structures include minimizing structural
compliance and buckling design. The design
sensitivity analysis for these objectives is facilitated
due to the fact that these are self-adjoint problems.
However, optimization problems of practical interest,
such as stress concentration reduction and passive
control, are non-self adjoint.
Figure 2: (a) Redesign and deformation process of
the body, (b) Primary structure of varying shape,
(c) Adjoint structure for stress functional
The objective of this project is the development and
numerical implementation of a shape optimization
procedure for general objective functional within the
framework of isogeometric analysis.
Several two- and three-dimensional isogeometric
shape optimization examples, all of which are nonself adjoint problems, are presented in figure3, 4 and
5. The objective in these problems is the stress
Figure 4: Two-dimensional plane stress
Design control points
Figure 5: Three-dimensional fillet design
The passive control problem for the aerospace
structures with a dynamic load will be carried out
based on the work presented above. A preliminary
result is shown in figure 6.
Figure 1: isogeometric shape design optimization
Isogeometric analysis has been applied to shape
optimization where NURBS control points have been
used as design variables to characterize the
boundary shape (see fig.1). The analytical
sensitivities of arbitrary objective functionals over
shape parameters, in many cases, are difficult to
obtain. By using the adjoint method, the sensitivity
can be expressed in terms of the variation of the
shape parameters and the corresponding primary
and adjoint fields (see fig.2). The advantage of the
isogeometric approach is that it minimizes
discretization errors present in the traditional
descriptions for design and analysis.
Figure 3: Optimal shape of an orifice in a plate
under (a) bi-axial traction and (b) shear traction
40 44 48 52 56
60 64 68 72 76
Figure 6: (b) shows a optimal hole design in a plate
under variable bi-axial traction defined in (a).
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-  A. P. Nagy. Isogeometric design optimization. Ph.D. thesis, Delft University of Technology, Delft, 2011.
-  K. Dems, Z. Mroz. Variational approach by means of adjoint systems to structural optimization and sensitivity analysis II, structure shape variation. Int. J. Solids &
Struct., 20, 527-552, 1984.
-  K. K. Choi, N. H. Kim. Structural Sensitivity Analysis and Optimization 1: Linear Systems. Springer-Verlag New York Inc. (United States), 2005.
-  S. Turteltaub. Optimal control and optimization of functionally graded materials for thermomechanical processes. Int. J. Solids & Struct., 39, 3175–3197, 2002.