Examples of Applications of Low-Complexity Shape Optimization in ...

Examples of Applications of Low-Complexity Shape Optimization in Car Industry Bijan Mohammadi, Stephane Moreau, Mugurel Stanciu Univ. Montpellier & Valeo Abstract We show how to reduce the complexity of numerical shape design using incomplete sensitivity concepts. Introduction Optimization is a permanent and major demand. However, the complexity has always forced to consider in optimization lower level models and rougher approximations than for a pure simulation. Our aim here is to overcome this bottleneck for shape optimization problems developing optimization methods only requiring functional evaluations and incomplete sensitivity analysis. We see that this sometimes leads to cost functional reformulations made possible after a better understanding of the physic of the problem. Hence, looking for reduced complexity approaches forces the user to better understand its problem. The ingredients of the talk are illustrated on aerodynamic shape optimization problems provided by Valeo-TM for blade design in car cooling devices. These configurations require a reformulation of the functional and constraint to fit the validity domain of incomplete sensitivity. We also take advantage of these examples to show that the impact of the parameterization is essential for the design to succeed. Global Optimization and Solution of Stefan Problems A fundamental remark on classical gradient based minimization algorithms, having a continuous representation as a Cauchy problem for a first order dynamic system is that they can find the global minimum if the initial condition belongs to the attraction basin of the infimum (global minimum) and that otherwise the minimizing sequence they build is in principle captured by a local minimum. In that sense, the problem of global minimization with a gradient-based algorithm becomes the prescription of an initial condition for the mentioned Cauchy problem in the suitable attraction basin. On the other hand, one notice that minimization algorithms, including non-deterministic ones such as genetic algorithms, are discrete forms of first or second order ODE (or system of ODEs) [Mohammadi-Saiac, 2003]. A particular class of problems consists of when the infimum Jm of the functional is known: its value but not where it is reached. Hence, global optimization can be seen as a boundary value problem for an equation for the control parameter x: F(x’’,x’,x,u(x))=0, x(0)=x0, J(x(L))=Jm . This is similar to the solution of the interface between water and ice. This frontier is unknown. The free boundary is known as iso-zero temperature line. The case Jm is unknown can be handled looking for the solution of a min-min problem. It has been shown that this problem has a solution if F = x’’+ x’ + dJ/dx [Dufour et al., 2005]. The word solution should be understood as: one can find a point enough close to the infimum.

We use numerical methods for BVP (instead of initial value problem) to solve this problem. If one would like to keep an existing minimization platform, this can be done using a shooting approach introducing a new unknown x’(0)=v and minimizing h(v)= |J(xv(L))-Jm|. Mathematical background and applications of this approach to academic and industrial problems are given in [Mohammadi-Saiac 2003, Dufour et al., 2005, Ivorra 2006]. Sensitivity and Incomplete Sensitivity Sensitivity analysis is a powerful tool in constrained optimization even if a non-gradient based method is used for optimization. Indeed, this information permits to discriminate between points of the same Pareto front introducing robustness issues during optimization: the best design is the one around which constraints have low sensitivity. Hence, two points on a Pareto front can be compared if one considers the sensitivity of the functional with respect to the independent variables not being control parameter. However, sensitivity evaluation for large dimension minimization problems is not an easy task. The most efficient approach is to use an adjoint variable with the difficulty that it requires the development of dedicated software [Pironneau 1973 & 1984, Jameson 1988]. Automatic differentiation brings some simplification, but does not avoid the main difficulty of intermediate states storage, even though check-pointing techniques bring some relief [Griewank, 1995]. We reduce the cost of sensitivity evaluation introducing the concept of incomplete sensitivity [Pironneau-Mohammadi, 2001& 2005]. Consider a general simulation loop, leading from the shape parameterization to the cost functional (here for simplicity we consider all independent variables being control parameters): J(x) : x

Q(x)

U(Q(x))

J(x,Q(x),U(q(x)))

The Jacobian of J: dJ/dx = J,x + J,Q Q,x + J,U U,Q Q,x The last term involves state equation linearization and is the most expensive. Our definition of incomplete sensitivity neglects this contribution. This is only valid if the cost function is, or can be reformulated, to have the following characteristics: • J and x are both defined on the shape (or a same part of it), • J is of the form J(x) = f(x, Q(x)) g(U), which means that it involves a product of geometrical and state based functions. • The shape curvature is not too high (this has to be quantified). In practice we rather approximate or partially evaluate the last term in the exact gradient either using coarser mesh for sensitivity analysis or using a low complexity model. This is similar to what is usually done in domain decomposition where different level of complexity is used for modelling when possible. More precisely, the loop (x to Q(x) to U(Q(x)) above is replaced by (x to q(x) to u(q(x)) U(Q(x)) / u(q(x)) ) where u denotes a low complexity model and q a limited set of geometric entities involved in the full computation loop. In particular, we are interested by low complexity models only involving wall based geometric information. The scaling term in the second loop is important for incomplete sensitivity accuracy. This coefficient is frozen during linearization. The sensitivity we consider is given by: dJ/dx = J,x + J,Q Q,x + J,U u,q q,x ( U(Q(x)) / u(q(x)) ) Wall functions are natural candidates for u. This approach has been widely used in shape optimization for flows ranging from sub to hypersonic [Mohammadi, 2004].

Validation & Verification A remark when simulating or optimizing complex systems is that information on the uncertainties on the results is probably more important that the results themselves obtained, for instance, for a given functioning point of the system. For instance, it is essential to be able to identify dominant independent variables in a system. Indeed, these parameters will be those to be monitored more accurately and for which accurate measurement should be provided. We recover the robustness issue above. This concept should therefore also include different weight for the independent variables following their impact. Also it is important to be able to quantify the impact on the results of a given modelling, or its modification, and also the way a model is implemented. It is important to be able to provide such information in an incremental way, following the evolution of the software. This means that we need sensitivity of the result with respect to the different independent variables for the discrete form of the model and also that we need to be able to do that without re-deriving the continuous model. Hence, sensitivity concept is central in validation and Verification issues which refer to all of the activities that are aimed at making sure that the software will function as required. Indeed, it is important to include robustness issues into the specifications using sensitivity analysis. Application to fan optimization The fan system is the key element that manages the air flow in the car engine cooling module. It supplies a minimum flow rate for the heat exchangers in order to dissipate the necessary heat and create a pressure rise to overcome the air restrictions. The optimization is necessary to improve the fan performances while maintaining strong constraints such as low axial packaging and low sound level emitted. The fan efficiency is defined as the ratio of the produce of the flow rate by the pressure rise to the torque by the rotation rate. Pressure rise does not enter the validity domain of incomplete sensitivity defined above. However, from the momentum equation, it can be linked to a linear combination of lift and drag coefficients involving inflow incidence, using periodicity of the calculation [Mohammadi-Moreau-Stanciu, 2003]. The initial geometry has been provided by VALEO Motors and Actuators and is an intuitively optimized fan. Flow solution is by TASCflow on a structured multi-block O-type elliptic mesh. CAD-based and CAD-free parameterizations have been used to represent shape deformations. The CAD-based parameterization concerns the sweep and the stagger of the blade while the CAD-free parameterization aims to search for an optimum in a very large space made of all nodes on the shape in the computational mesh. The number of control parameters in the CAD-based parameterization is around 10 and in the CAD-free parameterization around 5000. This former parameterization is an intermediate choice between level set type approaches [Sethian, 2004] and CAD –based approaches. Its interest is clear when one does not know the type of geometry one should look for. It is therefore not very useful in this example. What we would like to show is that a low level parameterization might be more efficient if it is wisely chosen. Indeed, the results in the CAD-based parameterization are superior for this case.

This result confirms experimental data showing that the blade has a maximal efficiency for a forward sweep. After sweep optimization, the new blade is used as initial design for stagger optimization from which a CAD-free design is performed. The new design has been experimentally tested for several operating points. The optimization process is relatively fast (30 to 70 iterations for each step) and requires affordable computational resources (the full optimization costs is twice the direct simulation), hence it is well adapted to industrial needs.

Stator-rotor configuration in a typical cooling device showing typical geometric constraint to account for during fan blade optimization.

Typical computational domain.

The mesh is monitored around the body using an elliptic mesh generation algorithm. This is enforced during mesh deformation in the optimization procedure. This is highly important when turbulence modelling is used. Incomplete sensitivity permits highly fine meshes during optimization and wall-based sensitivity analysis. This makes monitoring mesh quality easier during optimization.

Initial and final shapes for a primal efficiency optimization for a sweep parameterization.

Convergence histories for a primal efficiency optimization for a sweep parameterization. Efficiency gains 6 percents.

Initial and finale shapes for a second level efficiency optimization for a stagger parameterization.

Convergence histories for a second level efficiency optimization for a stagger parameterization. Efficiency gains another 6 percents.

Initial and final shapes for the final level optimization using a CAD-free parameterization.

Convergence histories for a CAD-free parameterization. This is a final stage optimization around the previous optimal shape. Despite the parameter space is large, most of the gain has been obtained previously. The previous low dimension parameterization seems to be enough for this problem. Efficiency only gains 2 percents.

Efficiency vs. fan regime (rpm) Experimental validation of the optimized blade which quite overperforms the original blade. Concluding remarks A global optimization approach based on the solution of boundary value problems has been presented. Incomplete sensitivity analysis has been used to reduce the complexity of the optimization procedure using only wall based information in the analysis. References [Pironneau, 1973] Pironneau, O. On optimal shapes for Stokes flow, J. Fluid Mech. Vol. 70(2), pp. 331-340. [Pironneau, 1984] Pironneau, O. Optimal shape design for elliptic systems, Springer-Verlag. [Jameson, 1988] Jameson, A. Aerodynamic Design via Control Theory, J. of Scientific Computing, Vol. 3, pp. 233-260. [Mohammadi-Saiac, 2003] Saiac, J.H. & Mohammadi, B. Pratique de la simulation numérique, Dunod. [Mohammadi-Moreau-Stanciu, 2003] Mohammadi, B. S. Moreau, S. Stanciu, M. Low complexity models to improve incomplete sensitivity for shape optimization, IJCFD, Vol.11, Number 2, pp. 245-266. [Debiane et al, 2004] Debiane, L. Ivorra, B. Mohammadi, B. Nicoud, F. Poinsot, Th. Ern, A. Pitsch, H. Temperature and pollution control in flames, Center for Turbulence Research Summer Program Briefs. [Mohammadi-Pironneau, 2004] Mohammadi, B. Pironneau, O. Shape optimization in fluid mechanics, Annual Review of Fluid Mechanics, Vol. 36: 255-279. [Mohammadi-Pironneau, 2002] Mohammadi, B. & Pironneau, O. Applied shape optimization for fluids, Oxford University Press. [Ivorra, 2006] Ivorra, B. Global optimization and boundary value problems, PhD. University of Montpellier. [Dufour et al., 2005] Dufour, JP. Ivorra, B. Mohammadi, B. Redont, P. Dynamical systems and global optimization in finite time, , Optimal control: Applications and Methods, to appear.

[Griewank, 1995] Griewank, A. Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation, Optimization Methods and Software, Vol. 1, pp. 35-54. [Sethian, 2004] Sethian, J.A. Fast Marching Methods and Level Set Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Sciences, Cambridge University Press. [Mohammadi, 2004] Mohammadi, B. Optimization of aerodynamic and acoustic performances of supersonic civil transports, International J. Numerical Methods for Heat & Fluid Flow, Vol. 14. num 7.