Detailed Aerodynamic Shape Optimization Based on Adjoint Method ...

1/16The SIAM Workshop on Combinatorial Scientific ComputingDarmstadt, May 19 th -21 st , 2011.<strong>Detailed</strong> <strong>Aerodynamic</strong> <strong>Shape</strong> <strong>Optimization</strong><strong>Based</strong> on Adjoint Methodwith <strong>Shape</strong> DerivativesCaslav Ilic (DLR)Stephan Schmidt (Uni. Trier)Nicolas Gauger (RWTH Aachen)Volker Schulz (Uni. Trier)Work within the DFG SPP 1253 project

Problem Scale in <strong>Aerodynamic</strong> <strong>Optimization</strong>2/16➢Small scale● preliminary design phase● highly approximative models(algebraic, statistic)● key design parameters, oftendiscrete: O(10)● methods: EAs, simplex➢Medium scale● accurate models (CFD, CSM)● application-driven globalparametrization: O(100)● methods: simplex, gradient

Problem Scale in <strong>Aerodynamic</strong> <strong>Optimization</strong> (2)3/16➢Large scale● detailed design phase● accurate models (CFD, CSM)● “free” parametrizations: O(10 3 -10 5 ) design parameters● methods: adjoint-based gradient descent“free-form”“free-node”

Adjoint-<strong>Based</strong> Gradient Descent4/16➢Constrained gradient descent algorithms● Unconstrained (steepest gradient, BFGS...) with penalties● Sequential quadratic programming (SQP)● Interior-point (IP)➢Necessary: gradient of objective and constraints● some Hessian information benefitial➢Adjoint formulation for the gradient● computation time theoretically independent of number ofdesign parameters● must solve adjoint (dual) state per objective/constraint

Adjoint-<strong>Based</strong> Gradient Descent (2)5/16➢<strong>Optimization</strong> time independent of number of designparameters hard to achieve in practice➢Need some combination of special ingredients...● Scalable parametrizations● movable surface mesh nodes (“free-node”)● knot-based parametric surfaces (control mesh)What we did● Problem knowledge inside the optimizer (no “black-box”)● preconditioning of the gradient descent● sufficient simulation convergence (“one-shot”)● No finite differences in gradient computation● adjoining the whole tool-chain by AD● surface formulation by shape derivatives

Test Case: Transonic Shock Removal6/16➢Remove shock from a wing; fixed planform, Euler flowAoA 3 deg, Mach 0.83initial Onera M6:unstructured mesh,3 levels (by adaptation)lambda-shocklevel # points # elements # surf. nd.L1 110,000 580,000 18,000L2 290,000 1,590,000 36,000L3 820,000 4,660,000 72,000➢Objective: minimize drag➢Constraints:● preserve initial lift● preserve initial internal volume

Test Case: <strong>Optimization</strong> Performance7/16optimal drag =induced drag (planform)+ spurious drag (numerics)optimaloptimization time =~7 x simulation timeshock removed

Test Case: Section Pressure Distribution8/16

Test Case: Section <strong>Shape</strong> Distribution9/16

Surface Formulation by Adjoining11/16➢Remove the expensive term by solving the adjoint problem● PDE and b.c. linearization● adjoining shape derivative● integration by partsarbitrary scalar field in Ω● substitution of b.c.final deriv.surf. formset != 0adjoint b.c.set != 0adjoint PDE➢Details in PhD thesis by S. Schmidt, 2010.

Automatic Symbolic Derivation12/16➢Manual adjoint shape derivation tedious and error-prone, nophysical check-points for implementation➢Do it automatically on the symbolic (continuous) level● PDE, b.c., objectives, constraints... → set of s-expressions● For each phase of manual adjoining, define● expression “finality” measure (=0 final, >0 not final)● expression “cost” measure (>0, the smaller the better)● set of expression transformers (recognizer-applicator)● For each phase, determine transformation sequence whichreduces finality to zero, minimizes cost→ Combinatorial optimization problem?(Full enumeration prohibitively expensive!)● Convert final s-expression to target language (C, Fortran...)

Automatic Symbolic Derivation (2)13/16➢Implementation in Python➢Example input, 2D stationary Euler flow problem:

Automatic Symbolic Derivation (3)14/16➢Example derivation phase, linerization of energy eq.● starting: finality = 3, cost = 5.6● linearized:finality = 0, cost = 160.4

Conclusion: Practical Applicability?15/16➢Highly constrained detailed design● rough constraints (might prevent more optimal shapes)● how to transfer optimal shape back into a CAD model?➢“Advisory” optimization● free-node optimization starting from optimal shape fromapplication-fitted medium-scale parametrization● if gains useful, indicator to refine medium-scale parametrization➢Gradient by shape derivatives● if node sensitivities available, fast gradient computation onarbitrary parametrization through chain rule

The End16/16Thank You ForYour Attention!