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In any optimization framework, a robust and reliable mesh morpher is necessary to undertake the adaptation of the CFD mesh to the updated boundaries at each optimization cycle. Morphing has its share of challenges, namely to main- tain high mesh quality (avoid distorted elements and tangles) even during extreme deformations. In this work, the Finite Transformation Rigid Motion Mesh Morpher (FT–R3M) is presented, an improved version of the Rigid Motion Mesh Morpher [5], that eliminates the need for sub-cycling, making it more efficient in terms of CPU time. FT–R3M, which bears some similarities to [4], is a mesh–less mesh mor- phing tool, since it does not require any inertial quantities, that gracefully propagates the movement of the boundaries (surface mesh) to the internal nodes of the mesh (volume mesh), by keeping the motion of its parts (referred to as stencils) as–rigid– as–possible. It is an optimization–based method, which means that the interior nodes of the computational mesh are displaced to minimize a distortion metric, namely the deformation energy. Since FT–R3M is minimizing the deformation energy between the initial and the final configuration, as opposed to R3M, in which the deformation energy is minimized from each sub-cycle to another, there is a significant gain in terms of the quality of the resulting mesh. The efficiency of the morpher proposed in this article will be demonstrated in small and medium–size cases.

Uncertainties, in the form of either non–predictable shape imperfections (manufacturing) or flow conditions which are not absolutely fixed (environmental) are involved in all aerodynamic shape optimization problems. In this paper, a work- flow for performing aerodynamic shape optimization under uncertainties, by taking manufacturing uncertainties into account is proposed. The uncertainty quantification (UQ) for the objective function is carried out based on the non–intrusive Polynomial Chaos Expansion (niPCE) method which relies upon the CFD software as a black– box tool. PCE is combined with an evolutionary algorithm optimization platform. CAD–free techniques are used to control the shape and simultaneously generate shape imperfections; next to this, a morphing/smoothing tool adapts the CFD mesh to any new shape. In the cases presented in this paper, all CFD evaluations are per- formed in the OpenFOAM environment.

In adjoint based shape optimization problems, after the sensitivities have been computed, there are two ways of dealing with the necessary changes to the mesh. The first one is re-meshing based on the new shape, while the second one is adapting the existing mesh (by moving the nodes) to fit the new shape. Re-meshing may be time consuming, as it is introduced as a separate step inside the optimization loop, and tedious as it may require by-hand manipulation. It also introduces inconsistencies in the process as the sensitivities have been computed at isoconnectivity. Morphing also has its share of challenges, namely maintaining the mesh quality (avoiding distorted and negative cells) while deforming it. Within this context, various mesh morphing techniques have been developed. The Spring Analogy [1] is simple but may suffer robustness issues. Laplacian smoothing [11] is suitable for translation but does not account for rotation. The linear elasticity approach [3] does not account for mesh anisotropy and is difficult to implement for general meshes because finite elements are used to solve the equations. Finally, the Radial Basis Functions [7] are promising but computationally heavy as the matrices involved are full, restricting the mesh size and complicating the implementation. The Rigid Motion Mesh Morpher approach, proposed in the present study, aims at overcoming the above-mentioned limitations, being more flexible and essentially mesh-less, since it does not require any inertial quantities or cell connectivities related to the mesh. Firstly, the set of boundary nodes (nodes defining the shape) of the mesh is identified. The prescribed motion of these nodes (their velocities) is known. Then, all nodes are grouped into “stencils” which are required to deform in an as-rigid-as-possible way. Hence, every stencil has a rotation velocity and a translation velocity. Those, along with the velocities of all internal nodes, form the set of unknowns. The as-close-to-rigid-as-possible motion is ensured by attempting to minimize a metric representing the difference of the actual deformation from a perfectly rigid motion (a translation plus a rotation). It will be shown that this quantity is related to the anisotropic deformation energy. Next, the resulting system of equations is solved, having the prescribed motion of the boundary nodes as boundary conditions.

Uncertainties, in the form of either non–predictable shape imperfections (manufacturing) or flow conditions which are not absolutely fixed (environmental) are involved in all aerodynamic shape optimization problems. In this paper, a work- flow for performing aerodynamic shape optimization under uncertainties, by taking manufacturing uncertainties into account is proposed. The uncertainty quantification (UQ) for the objective function is carried out based on the non–intrusive Polynomial Chaos Expansion (niPCE) method which relies upon the CFD software as a black– box tool. PCE is combined with an evolutionary algorithm optimization platform. CAD–free techniques are used to control the shape and simultaneously generate shape imperfections; next to this, a morphing/smoothing tool adapts the CFD mesh to any new shape. In the cases presented in this paper, all CFD evaluations are per- formed in the OpenFOAM environment.

Adjoint-based shape optimization at isoconnectivity through robust mesh deformation Rigid Motion Mesh Morpher (R3M)

This paper presents a general framework to derive an unsteady discrete adjoint method as part of a gradient-based aerodynamic shape optimization process. The method calculates the gradient of an objective function w.r.t. the design variables of the aerodynamic shape. The formulation of the adjoint equations is introduced within a generic time-dependent optimal design problem. Results are shown that demonstrate the solution of the unsteady adjoint equations for a representative row of nozzle guide vanes of a high pressure turbine.

Adjoint methods allow to compute the gradient of a cost function at an expense that is independent of the number of design variables, thus representing an appealing tool to those who deal with large-scale gradient-based optimization processes. However, researchers in the field are currently facing difficulties related to poor convergence and high storage memory requirements. Both continuous and discrete adjoints require a fully converged solution to the primal problem. It is argued that, specifically in Computational Fluid Dynamics, the presence of numerical adjustments (e.g. non-orthogonal correctors) and segregated solution algorithms yields a solution that, while acceptable as a mere aerodynamics study, is not accurate enough to produce a robust adjoint system. In the past decade some new PDE discretization schemes have emerged aiming to remove some constraints imposed by classical finite volumes. The main goal is to allow for more freedom in both the physical model (e.g. strong anisotropy in a material property) and its corresponding numerical model (e.g. the possibility to use polyhedral meshes with strongly non-orthogonal and/or non-convex cells), while improving solution accuracy and convergence properties at the same time. One of such new schemes is known as Mimetic Finite Differences; the present work introduces a specific implementation of it and extends the scheme to cater for convection-diffusion reaction problems. Preliminary numerical results are also included.