Shape Optimization using grid free solver and evolutionary algorithm

S. M. Deshpande1, G. N. Shashi Kumar2, A. K. Mahendra2


1 Engineering Mechanics Unit, Jawaharlal Nehru Center for Advanced Scientific

Research, Jakkur, Bangalore 500064, INDIA. email: smd@jncasr.ac.in

2 Machine Dynamics Division, BARC, Mumbai 400085, INDIA


Grid free solvers are very attractive numerical methods of obtaining solution of PDE of fluid dynamics. The kinetic grid free solver called LSKUM (Least Square Kinetic Upwind Method) has been very successfully applied to a large number of practical flow problems [1], [2], [3] and [4]. LSKUM requires a cloud of points around a body (airfoil, wing, flight vehicle etc.) and connectivity N(Po) for each point Po in the cloud. The connectivity N(Po) is a set of neighbors of Po and is generally obtained by connectivity preprocessor. The cloud can be generated by using any method for example, structured grid generator leads to a simple cloud of points, several overlapping grids leads to a chimera cloud. The LSKUM operating on a cloud of points is a very powerful tool of obtaining numerical solution of inviscid [2] as well as viscous flows [4] for geometrically complex configurations. Combining LSKUM with an evolutionary algorithm (such as GA for example) is a very attractive possibility in aerodynamic shape optimization. Sashi Kumar, Mahendra and Deshpande [5] were the first to apply LSKUM to a shape optimization problem.


Salient features of LSKUM combined with GA are:

  1. The computational domain around airfoil has two clouds. One is the background cloud which could be a Cartesian grid or points obtained from an elliptic PDE based grid generator. The other cloud called inner cloud is a relatively high density cloud around the airfoil. In the present test case it is obtained by elliptic PDE based method. This inner cloud has a hole which lies completely inside the airfoil.

  2. Points in the computational domain are flagged, separate flags for points inside, points on the

airfoil and interior points. Ray tracing is used to blank the points inside the aerofoil. Connectivity is generated for all the points except for those which are blanked.

  1. Points on the airfoil act as control points during optimization. Only these points are allowed to change as optimization proceeds.

  2. The given population of shapes contains a number of airfoil configurations (chromosomes). The population is generated by allowing the y-coordinate of body points within an allowed band. The variation is around NACA 0012 airfoil. LSKUM – NS code is used to compute viscous flow around each airfoil . Several CFD calls are required during optimization.

  3. A suitably chosen objective function (OF) is taken as a fitness function which is used for GA search and selection procedure. The high fitness values are used to select shapes as parents to produce off springs. For example parents are chosen based on Roulette wheel method [6] where the probability of a parent of being chosen is proportional to its fitness value.

  4. Each of these parents produce two off springs by cross over and a simple one point cross over scheme is applied. The probability of crossover is set at 50% and then mutation is applied to off spring. Mutation is carried out by randomly selecting a gene (control point) and then changing its coordinates by an arbitrary amount within a prescribed band. A new population is thus produced.

  5. The technique of elitism is used. The best and the second best members in each generation are assigned to the next generation without cross over or mutation [7].


Figs 1, 2, 3, 4 show typical results where pressure contours for a shape in one generation, variation of OF, variation of average lift and drag coefficients with generation and best shape in each generation (up to 20 generations) are shown.

Figure 1 Pressure contours for a typical airfoil shape in the 1st generation

Figure 2 Variation of objective function (OF) with generation



Generations

Figure 3:Variation of average lift and average drag coefficients with generation

Figure 4:Best shape in each generation

(1st, 5th, 10th,15th and 20th generation)



References

[1] Ghosh A. K. and Deshpande S. M. “Least Squares Kinetic Upwind Method for

Inviscid Compressible Flows” AIAA paper no 95-1735, 1995.

[2] Deshpande S. M. , Anandha narayanan K, Praveen C. and Ramesh V. “Theory

and applications of 3D LSKUM based on entropy variables” Intl J. for Numerical

methods in Fluids vol 40, pp 47-62, 2002.

[3] Ramesh V. and Deshpande S. M. “Least Squares Kinetic Upwind Method on

moving grids for unsteady Euler Computations” Computers and Fluids Journal,

vol 30/5, pp 621-641, may 2001.

[4] Mahendra A. K. “Application of Least Squares Kinetic Upwind Method to

strongly rotating viscous flows” MSc thesis, Dept. of Aerospace Engg, IISc,

Bangalore, Feb 2003.

[5] Shashi Kumar G. N., Mahendra A. K., Deshpande S. M. “Optimization of airfoil shape using GA and grid free solver” 7th Aeronautical Society of India Intl Conference on CFD. 11-12 August 2004, Bangalore, India.

[6] David E. Goldberg “Genetic Algorithms in search, optimization and machine learning” Addison – Wesley Publishing Co. Inc, Reading, 1989.

[7] B. Vanden Braembussche RA, Manna M. “Inverse design and optimization methods” Lecture series 1997-05, Von Karman Institute for Fluid Dynamics, Chausse de Waterloo 72, B-1640, Rhode Saint Gevese, Belgium, Aprial 1997.