GPPS-NA-2018-133 CAD-BASED ADJOINT MULTIDISCIPLINARY OPTIMIZATION OF A RADIAL TURBINE UNDER STRUCTURAL CONSTRAINTS

Most adjoint-based optimization frameworks only consider aerodynamic performance and constraints, leading to designs that need to pass through revisions by structural requirements. Only in recent years, adjoint optimization frameworks have been extended to include structural constraints. These frameworks also make use of CAD-based parametrizations to maintain a connection to the master CAD geometry and to serve as the connection between the fluid and solid domains. In this work, a CAD-based adjoint multidisciplinary optimization framework for turbomachinery components is presented. A CAD-based parametrization is used for defining the shape freedom, from which the fluid and solid grids are generated, and a Reynolds-Averaged Navier-Stokes solver is used to compute the efficiency. The maximum von Mises stress is computed using a linear stress solver based on the Finite Element Method. The CFD and stress solvers each have adjoint capabilities, permitting an efficient computation of gradients at a cost independent of the size of the design space. An adjoint optimization of a radial turbine is performed with the objective of maximizing the aerodynamic efficiency while adhering to the structural constraints. Results show that within a reduced design time an aerodynamic optimal design can be achieved whilst keeping the mechanical stresses within range of the prescribed tolerance. INTRODUCTION In the field of turbomachinery, multidisciplinary optimizations (MDO’s) have been widely applied using gradient-free optimization methods. These methods are straightforward to implement due to their non-intrusive nature of not requiring any source code access. However, gradient-free methods require a high number of iterations to converge towards an optimum and the design space is limited by the curse of dimensionality, i.e., the computational effort increases exponentially with respect to the number of design parameters. Alternatively, gradient-based optimization methods use gradient information to converge towards a local optimum, typically with less iterations while allowing larger degrees of freedom. However, this requires computing the gradient of the objective with respect to the design parameters. The required gradient can traditionally be computed using a non-invasive approach such as finite differences (FD). However, the cost of using FD is also proportional to the number of design parameters : evaluations for 1st n n + 1 order FD and evaluations for 2nd order FD. The adjoint n 2 method (Pironneau, 1974; Jameson, 1988) allows a gradient calculation at a cost proportional to the number of objectives, rather than the number of design parameters. Since the number of objectives is generally much less than the number of design parameters, this significantly speeds up the gradient computation, and as a result, the optimization. State of the art adjoint optimizations in turbomachinery focus on aerodynamic cost functions and constraints (Wang and He, 2010; Walther and Nadarajah, 2013; Luo et al., 2014). Only recently have adjoint MDO’s been extended to include structural constraints (Verstraete et al., 2017). Including these constraints within a CAD-based adjoint MDO framework enables the efficient design of geometries that are not only aerodynamically optimal, but also structurally feasible. In this work, the adjoint MDO framework CADO (Verstraete, 2010) of the von Karman Institute for Fluid Dynamics is used to optimize the efficiency of a radial turbine under structural constraints. First, the adjoint MDO framework will be briefly presented. This will be followed by a discussion of the numerical setup, including the flow and structural solvers and the optimization method. Finally, the optimization results are discussed. ADJOINT MULTIDISCIPLINARY OPTIMIZATION FRAMEWORK The optimization begins with CAD parameters α ∈ R that are used to describe the geometry (figure 1). The number of CAD parameters is defined by . A CAD-based approach n is chosen to give users the possibility of imposing geometrical constraints for an optimization. Geometrical constraints can be used to define e.g. constraints on a shape’s curvature for manufacturing purposes. Additionally, the CAD surface serves as the interface between the structured mesh of the fluid domain and the unstructured mesh of the solid domain. The CAD parameters are used as inputs to α the CAD kernel which generates the geometry. Based on the CAD geometry, a structured mesh is generated for the CFD calculation and an unstructured mesh is generated for the structural solver. Following the mesh generation, CFD and CSM analyses are carried out to compute the performance parameters of interest These . y include the efficiency and the maximum von Mises stress η for the fluid and structural disciplines, respectively. σmax The performance parameters are then used to define an objective which is to be minimized. , J Discrete adjoint implementations of the CFD and CSM solvers, combined with forward differentiated implementations of the CAD kernel and mesh generation, allows an efficient calculation of the sensitivities of Jα ∈ Rn the objective function with respect to the CAD design parameters. A more detailed discussion of this framework can be found in (Verstraete et al., 2017). Previous work used the open-source structural solver Calculix (Dhondt and Wittig, 1998) and an inverse distance interpolation for the structural grid generation (Verstraete et al., 2017). In this paper, an in-house adjoint structural solver is used, which is also used to morph the unstructured grid using a linear elastic analogy. Compared to the inverse distance method, the linear elastic analogy has shown to require less remeshing for this radial turbine geometry. Figure 1 Flowchart of Multidisciplinary Framework in CADO (CAD-based Optimization), in-house optimization code of the von Karman Institute for Fluid Dynamics Numerical Setup The numerical setup for this optimization is based on the same setup as in (Verstraete et al., 2017), which will be briefly summarized in this section. An adjoint MDO of a radial turbine is performed, using CAD design parameters to modify the geometry. In total, design parameters are defined for the optimization. 4 n = 2 11 design parameters are used to define the shape of the meridional passage (figure 2), 12 parameters define the blade angle distribution from leading to trailing edge of the hub and shroud (figure 3), and one parameter is used for the trailing edge cut back definition (figure 4).