Conjugate heat transfer shape optimization based on the continuous adjoint method

In this paper, the continuous adjoint method for use in gradient-based op- timization methods for coupled problems including heat transfer between bodies (solids) and fluids flowing over or inside them is developed. This kind of problems are usually referred to as Conjugate Heat Transfer (CHT) problems. Emphasis is given to expand- ing the Enhanced-Surface Integral (E-SI) adjoint formulation recently published by the authors’ group for shape optimization problems in fluid mechanics only, to tackle CHT problems. This formulation ensures that the gradient of the objective function is accu- rately computed, while the computational cost is kept as low as possible

Aerodynamic Shape Optimization using the truncated Newton Method and Continuous Adjoint

This paper presents the development and application of the Truncated Newton (TN) method for shape optimization problems based on continuous adjoint. The method is presented for laminar, incompressible flows. OpenFOAM R is chosen as the CFD toolbox in which the method is developed. The Newton equations are solved using the restarted linear GMRES algorithm which requires only the product of the Hessian matrix of the objective function (with respect to the design variables) with a vector. This overcomes the cost for computing the Hessian matrix itself, which unfortunately scales with the number of design variables. The computation of Hessian-vector products is conducted via the combination of continuous adjoint and direct differentiation that gives the minimum cost. The developed method is used for the shape optimization of two 3D ducts and the speed-up gained compared to rival methods is showcased. This research was funded from the People Programme (ITN Marie Curie Actions) of the European Union’s 7 th Framework Programme (FP7/2007-2013) under REA Grant Agreement 317006 (AboutFLOW). The first author is an AboutFLOW Early Stage Researche

On the Discretization of the Continuous Adjoint to the Euler Equations in Aerodynamic Shape Optimization

In aerodynamic shape optimization, gradient-based algorithms usually rely on the adjoint method to compute gradients. Working with continuous adjoint offers a clear insight into the adjoint equations and their boundary conditions, but discretization schemes significantly affect the accuracy of gradients. On the other hand, discrete ad joint computes sensitivities consistent with the discretized flow equations, with a higher memory footprint though. This work bridges the gap between the two adjoint variants by proposing consistent discretization schemes (inspired by discrete adjoint) for the con tinuous adjoint PDEs and their boundary conditions, with a clear physical meaning. The capabilities of the new Think-Discrete-Do-Continuous adjoint are demonstrated, for in viscid flows of compressible fluids, in shape optimization in external aerodynamics

The iPGDZ+ technique for compressing primal solution time-series in unsteady adjoint - applications & assessment

Gradient-based optimization for large-scale problems governed by unsteady PDEs, in which gradients with respect to the design variables are computed using unsteady adjoint, are characterized by the backward in time integration of the adjoint equations, which require the instantaneous primal/flow fields to be available at each time-step. The most widely used technique to reduce storage requirements, at the expense of a controlled number of recomputations, is binomial check-pointing. Alternatively, one may profit of lossless and lossy compression techniques, such as iPGDZ+, this paper relies upon. iPGDZ+is a hybrid algorithm which consists of (a) an incremental variant of the Proper Generalized Decomposition (iPGD), (b) the ZFP and (c) the Zlib compression algorithms. Two different implementations of iPGDZ+are described: (a) the Compressed Full Storage (CFS ) strategy which stores the whole time-history of the flow solution using iPGDZ+and (b) the Compressed Coarse-grained Check-Pointing (3CP ) technique which combines iPGDZ+with check-pointing. Assessment in aerodynamic shape optimization problems in terms of storage saving, computational cost and representation accuracy are included along with comparisons with binomial check-pointing. The methods presented are implemented within the in-house version of the publicly available adjointOptimisation library of OpenFOAM, for solving the flow and adjoint equations and conducting the optimization

Parametric free-form shape design with PDE models and reduced basis method

We present a coupling of the reduced basis methods and free-form deformations for shape optimization and design of systems modelled by elliptic PDEs. The free-form deformations give a parameterization of the shape that is independent of the mesh, the initial geometry, and the underlying PDE model. The resulting parametric PDEs are solved by reduced basis methods. An important role in our implementation is played by the recently proposed empirical interpolation method, which allows approximating the non-affinely parameterized deformations with affinely parameterized ones. These ingredients together give rise to an efficient online computational procedure for a repeated evaluation design environment like the one for shape optimization. The proposed approach is demonstrated on an airfoil inverse design problem. © 2010 Elsevier B.V