European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberhardsteiner et.al. (eds.) Vienna, Austria, September 10-14, 2012 TOPOLOGY OPTIMIZATION IN FLUID DYNAMICS USING ADJOINT-BASED TRUNCATED NEWTON Dimitrios I. Papadimitriou, Evangelos M. Papoutsis-Kiachagias and Kyriakos C. Giannakoglou National Technical University of Athens, School of Mechanical Engineering, Lab. Of Thermal Turbomachines, Parallel CFD & Optimization Unit Zografou, Athens, 15710, Greece e-mails: [email protected], [email protected], [email protected] Keywords: Continuous adjoint approach, topology optimization, second-order sensitivities, Truncated Newton, loss minimization, duct flows. Abstract. This paper proposes a truncated Newton algorithm for efficiently solving topology optimization problems in fluid mechanics, such as the design of ducts with optimal performance. In topology optimization problems, where the number of design variables i.e. the porosity func- tion values at each grid cell (in cell-centered methods) or node (vertex-centered), are too many, the adjoint approach is, by far, the most efficient way to compute the gradient required by any descent algorithm, since the CPU cost per gradient computation is independent of the number of the design variables. Although the Newton method requires only a few cycles to locate the optimal solution, the computation of the exact Hessian matrix is prohibitively expensive since its cost scales with the number of design variables. The proposed truncated Newton solves the Newton equation iteratively, without computing the exact Hessian matrix itself. Instead, Hessian-vector products are efficiently computed using second-order sensitivity analysis based on the adjoint approach and direct differentiation. Just a few conjugate gradient iterations for the solution of the Newton equations are enough to satisfactorily accelerate the convergence rate of the objective function value. Thus far, the truncated Newton was applied to shape opti- mization problems according to geometrical parameterization schemes which define the design variables. It is the first time such an algorithm is presented for the solution of topology opti- mization problems. The method is applied to the topology optimization of ducted laminar flows, by minimizing the total pressure losses between the given inlet and outlet boundaries of the flow domain. Dimitrios I. Papadimitriou, Evangelos M. Papoutsis-Kiachagias and Kyriakos C. Giannakoglou 1 INTRODUCTION Topology optimization is quite a new area of interest in fluid dynamics. In structure mechan- ics, the first relevant paper dates back to 1988, [1], where the scope was to optimize structures for maximum stiffness, by controlling the material layout within a predefined space. For this purpose, a local (i.e. at each grid node or cell, depending on the formulation) criterion that defines whether this should be treated as solid or not was used. In structural mechanics, other applications of topology optimization can be found in [2] and [3] for structures with high defor- mations. Topology optimization methods have also been developed for acoustic problems, [4], and the design of micromechanisms, [5]. In CFD-based optimization, the first topology optimization methods were developed for creeping flows, where viscous effects dominate [6, 7]. Setting up a topology optimization method in fluid mechanics requires some of the flow equations to be replaced by CFE + α · CSS = 0 (1) over the computational domain. In eq. 1, CFE denotes the flow equations in their conventional form (i.e. those to be satisfied at any point within the flow domain, according to the selected flow model), α is the so-called porosity field and CSS stands for the conditions at the surrounding solid part of the domain. Topology optimization aims at computing the optimal field of α which minimizes an objective function F , such as the viscous losses between the pre-specified inlet(s) to and outlet(s) from the flow domain to be determined. Other objective functions could be used instead; for instance, a topology optimization problem might aim at designing a flow channel that maximizes the heat exchange between the flowing fluid and the surrounding solid, etc. From the practical point of view, particularly when manifolds with more than one exits must be designed, the problem is usually constrained, since the optimal solution must also fulfill other requirements, such as a desirable mass flow rate per exit, etc. Based on the above formulation, the topology optimization problem has as many unknowns (design variables, i.e. porosity α values) as the number of grid nodes (in vertex-centered schemes) or that of grid cells (in cell- centered schemes). As such, all gradient-based solution methods, must be supported by tools for computing the gradient of F at a cost that doesn’t scale with the number of design variables; this is why, the adjoint method used in this paper is the perfect choice. Upon completion of the optimization problem, the computed α values determine the shape of the sought flow channel. Areas with zero α (practically, α ≤ ; is an infinitesimal positive quantity) correspond to parts of the domain where fluid flows since, there CFE = 0. In contrast, areas with non-zero α (practically, α ≥ ) correspond to the surrounding solid since, there, CSS =0. The interface between the two distinct areas is the boundary (line in 2D or surface in 3D) of the channel to be designed. In contrast to structural mechanics, the literature of topology optimization in fluid mechanics is not that rich. In [8], the laminar Navier-Stokes equations were used as the flow model. In [9], topology optimization for turbulent flows is demonstrated, by making, however, the frozen turbulence assumption. In the same paper, the adjoint approach is used to compute the gradient of the total pressure losses function with respect to the porosity control variables. Since its first appearance in [10], the adjoint approach has been efficiently used for the shape design and optimization of various configurations, [11, 12, 13], by providing the gradient of the objective function with respect to the design parameters at a cost independent of the number of these parameters, to efficiently drive a gradient-based optimization method towards the optimal solution. Extensions of the adjoint approach to the computation of second-order sensitivity derivatives, for use in the Newton method, can also be found, [14, 15]. In these studies, a 2 Dimitrios I. Papadimitriou, Evangelos M. Papoutsis-Kiachagias and Kyriakos C. Giannakoglou method for the computation of the exact Hessian matrix was proposed and its use within the Newton optimization algorithm accelerated the convergence of F to its minimum. However, since the cost for computing the Hessian matrix still scales with the number of design variables, its use is restricted to problems with reasonably few of them. Should the problem size increases, an efficient alternative is the exactly initialized quasi-Newton method, where the exact Hessian matrix is computed only at the first cycle and, then, is approximately updated as in standard quasi-Newton methods; however, in the case of topology optimization with so many design variables, the exact Hessian matrix cannot be computed even once. Alternatively, the Newton system of equations may be solved iteratively (truncation of the Newton equations, [16, 17]) requiring only the computation of Hessian-vector products instead of the complete Hessian matrix. This algorithm applied to shape optimization problems was presented by the authors in [18]. This method proved to perform well in cases with a relatively high number of design variables (an order of 50 design variables was used). The truncated Newton based on the continuous adjoint approach and direct differentiation has also been pre- sented in variational data assimilation problems in meteorology, [19, 20, 21, 22] and, along with Automatic Differentiation techniques, in [23]. In this paper, the truncated Newton method is applied to topology optimization of laminar flows. The design variables are many more than those used in shape optimization problems. Two applications are shown concerning the design of 2D ducts. The first one, including 38400 design variables (or grid cells) in total, has one inlet and one outlet with a squared blockage in the center and the second one contains 73600 design variables and has two inlets, two out- lets and three squared blockages. The truncated Newton algorithm is found to accelerate the convergence rate, minimizing the objective function value faster and to a lower value than a conventional gradient-based algorithm. 2 PROBLEM FORMULATION AND OPTIMIZATION METHOD Let us assume that a flow channel, connecting inlet SI and outlet SO boundaries which are specified by the designer, must be designed, so as to give a flow with minimum total pressure losses. The flow is considered to be laminar and the fluid is incompressible. This is a typical op- timization problem in internal aerodynamics, associated with an objective function expressing the mass-averaged total pressure losses, by Z Z 1 2 1 2 F = − p + v vinidS − p + v vinidS (2) SI 2 SO 2 Starting point for the formulation of the topology optimization problem, apart from the objective function of eq. 2, is the system of flow equations which define the so-called state or primal problem. After artificially introducing the porosity field α into the conventional flow equations for the laminar flow of an incompressible fluid, these become, [9], @v Rp = j =0 @xj @v @p @ @v @v v i − i j Ri = vj + ν + + αvi =0 ; i = 1; 2 (3) @xj @xi @xj @xj @xi Based on the notation of eq. 1, the last terms in the momentum equations are those previously abbreviated to CSS in eq. 1. In areas where α ≥ , eqs. 3 are satisfied only if vi = 0 and these define the solid surrounding of the channel, see also [9]. In eqs. 3, p is the static pressure 3 Dimitrios I. Papadimitriou, Evangelos M. Papoutsis-Kiachagias and Kyriakos C.