2014 CFDSC Toronto Paper Abstract

Integrated Geometry Parameterization Approach for
Aerodynamic Shape Optimization

Timothy M. Leung and David W. Zingg

Institute for Aerospace Studies, University of Toronto, 4925 Dufferin Street, Toronto, ON M3H 5T6

To be presented at the 22nd CFD Society of Canada Annual Conference, Toronto, Ontario, June 2014. The abstract was submitted on February 9, and was accepted on February 23. This abstract does not contain all the equations because of the length requirments. The PDF file for this paper (which is the version submitted to the CFDSC) is available through the link at the bottom of the page. The full paper with expanded algorithm description and results will be submitted to CFDSC on April 1.

Abstract: An integrated geometry parameterization approach that controls both geometry changes and movement of the CFD grid is presented. This approach is suitable for aerodynamic and multi-disciplinary optimization. In this approach, a B-spline free-form deformation hull is constructed over the entire CFD grid; internal B-spline control points are moved during optimization to modify the geometry. A linear-elasticity grid movement method then controls the movement of the remaining internal control points to modify the grid. A sample result on 3D case shows that the deformed grid remains high in quality even with large shape changes. The final paper will include flow solutions and optimization results.


Geometry parameterization and grid movement methods are two key components in an efficient aerodynamic shape optimization algorithm. The parameterization method defines the geometry, and a grid movement method is then applied to modify the CFD grid. A good parameterization method makes delicate trade-offs between the ability to accurately a wide range of geometry changes and a low number of design variables. A good grid movement algorithm avoids the expensive process of grid regeneration during optimization by maintaining a high-quality grid even after large shape changes.

We present an integrated geometry parameterization and grid movement approach that combines the advantages of the free-form deformation (FFD) method and the robust linear-elasticity grid movement method [6]. Our approach is unique in that the FFD defines both the geometry as well as the CFD grid. The entire CFD grid is embedded inside a FFD control grid ("hull"). During optimization, coordinates of control points near the geometry are used as design variables. Other interior control points are then solved using the linear-elasticity method. Advantages of this approach include preserving FFD's ability to seamlessly manipulate multiple unrelated grids (e.g. CFD and structural meshes in multi-disciplinary optimization), and also allowing the end-user to specify the optimizer's degrees of freedom. The linear-elasticity method allows the CFD grid to remain high in quality with large shape deformations. This approach can be used for both structured and unstructured grids.

Integrated FFD & Grid Movement

We apply the "integrated FFD method" approach to parameterize the geometry, and to control the changes in the CFD grid as the geometry changes. Our approach uses a B-spline hull constructed around the CFD grid; the nodes of the hull are therefore the control points. Note that the entire CFD volume grid is inside the boundaries of the hull. The x, y, z coordinates of every grid node can be expressed as parametric distances ξ, ζς defined by the B-spline volume. At the beginning of the optimization cycle, the parametric distances at every node are computed using a quasi-Newton method. Once the parametric distances are known, the physical position of any point inside the hull are given by:

x(ξ) = ∑∑∑ Bijk Ni,p(ξ) Nj,p(ζ) Nk,p(ς)

whereBijk are the coordinates of the control points, and N are the pth-order basis functions. The basis functions are determined recursively by the Cox-deBoor [1] relationship. During optimization, the coordinates of specific control points are used as design variables.

Once the coordinates of the control points used as design variables are modified by the optimizer, the location of the other interior points are determined by the linear elasticity grid movement algorithm [6]. In this method,
the spatial domain Ω enclosed by the hull is considered to consist of a linearly elastic material. Then the displacement of the interior points inside this elastic material is governed by the equilibrium condition:

DT σ  + f = 0  on Ω

where σ is the stress tensor, f is the vector of external forces applied at the boundary, and D is the symmetric gradient operator. The equation is subject to the prescribed deformation u at boundary Γ:

u = u   on   Γ

Note that the "boundary" also includes interior nodes that have been designated as design variables. The equations of linear elasticity are discretized on the control points of the hull using the finite-element method with trilinear elements. This leads to a linear system that is solved each time the grid is moved:

M = Ku - F = 0

where F are the elemental forces applied at the boundary nodes, and K is the global stiffness matrix. The above equation is solved using a preconditioned conjugate gradient method [5].

Aerodynamic Optimization

The aerodynamic shape optimization algorithm is based on Ref. [3]. It uses the unconstrained optimizer BFGS [4]. Gradients are computed using the discrete adjoint method, and the adjoint system is solved using a variant of GCROT [1]. The objective function used is the lift-constrained drag minimization function:

J = ωL (1-(CL/CL*)) + ωD (1-(CD/CD*))

Geometric constraints are applied as penalty terms in the objective function. Design variables include both the angle-of-attack as well as locations of the control points. The full paper will include 3D lift-constrained drag minimization results for viscous turbulent flows.

Results and Discussions

A 3D example is presented to demonstrate the method's ability to create large changes in the geometry. Starting from a planar NACA0012 wing, a large winglet is manually generated. This changes are within the norm in exploratory optimization cases. Figs. 1a and 1b show a 19 x 14 x 14 hull and the CFD grid. The control points highlighted in red are moved to generate the winglet.

view-hull-original  view-grid-original
view-hull-final  view-grid-final
Figure 1: FFD hull and CFD grid before shape changes

After generating the winglet, the grid movement method took 0.5s to solve for the new FFD control point locations, and another 0.3s to generate the new CFD grid using a single processor. Figs 1c and 1d show the deformed wing and the hull.

Fig. 2 shows a quantitative measurement of the quality of the deformed grid compared to the initial geometry.

Figure 2: Grid quality measurement

The orthogonality of the grid lines at each node is measured. The ideal nodes have an orthogonallity measure of 1.0, while very skewed grid lines will be closer to 0.0. In this case, there is only a small decrease in the quality of
the grid, despite the large deformations, demonstrating the robustness of the algorithm.


We have presented an integrated FFD and grid movement algorithm suitable for high-fidelity aerodynamic shape optimization. Grid quality tests show that the CFD grid remains high in quality after large deformations of the geometry. The full paper will further demonstrate the efficiency and effectiveness of this


  1. C. de Boor. A Practical Guide to Splines. Springer–Verlag, 1978.
  2. J. E. Hicken and D. W. Zingg. A simplified and flexible variant of GCROT for solving nonsymmetric linear systems. SIAM J on Sci Comp, 32(3):1672–1694, 2010.
  3. T. M. Leung and D. W. Zingg. Aerodynamic shape optimization of wings using a parallel Newton-Krylov approach. AIAA Journal, 50(3):540–550, 2012.
  4. J. Nocedal and S. Wright. Numerical Optimization. Springer, 1999.
  5. Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia, PA, 2nd edition, 2003.
  6. A. H. Truong, C. A. Oldfield, and D. W. Zingg. Mesh movement for a discrete-adjoint Newton-Krylov algorithm for aerodynamic optimization. AIAA Journal, 46(7):1695–1704, 2008.