A continuous adjoint formulation for the computation of topological and surface sensitivities of ducted flows

Abstract

SUMMARYTopology optimization of fluid dynamic systems is a comparatively young optimal design technique. Its central ingredient is the computation of topological sensitivity maps. Whereas, for finite element solvers, implementations of such sensitivity maps have been accomplished in the past, this study focuses on providing this functionality within a professional finite volume computational fluid dynamics solver. On the basis of a continuous adjoint formulation, we derive the adjoint equations and the boundary conditions for typical cost functions of ducted flows and present first results for two-and three-dimensional geometries. Emphasis is placed on the versatility of our approach with respect to changes in the objective function. We further demonstrate that surface sensitivity maps can also be computed with the implemented functionality and establish their connection with topological sensitivities.

Topology optimization of inertia driven dosing units

“…The optimization method has recently been extended to transient and dynamic flow problems (Kreissl et al, 2011;Deng et al, 2011) though still limited to laminar flows. Traditionally the finite element method has been used for the modeling of topology optimization problems; however, fluid flow problems have also been optimized using the finite volume method (Othmer, 2008), the Lattice-Boltzmann method (Pingen et al, 2007) and kinetic gas theory (Evgrafov et al, 2008). A few works on flow components that are able to pump a fluid and have been designed by topology optimization have been promoted.…”

Convergence of Cell Based Finite Volume Discretizations for Problems of Control in the Conduction Coefficients

“…Presently, FVMs represent a standard choice of discretization within engineering communities dealing with computational fluid dynamics, transport, and convection-reaction problems. This is particularly important as control in the coefficients is being applied in these application domains, see for example [2,6,12,[16][17][18][19][20]27,28,30,31,[33][34][35][36][37][40][41][42]46]. Among various flavours of FVMs, cell based approaches, where all variables are associated only with cell centers, are particularly attractive.…”

“…The few exceptions include [30,33,36,37] in the context of topology optimization within computational fluid dynamics, and [26] where a control of a steady state heat conduction boundary value problem (BVP) is treated numerically. Despite these recent efforts, we have not even scratched the surface as far as our understanding of the interplay between the control in the coefficients and FVMs is concerned.…”

“…We provide characterizations of the limit points of both globally optimal solutions and two kinds of stationary points to the discretized problems, see Theorems 4.7, 5.3, and 5.5. Such an analysis is lacking in the works utilizing FVM discretizations of control in the coefficients problems [26,30,33,36,37]. For conforming FEM approximations a related study, which however deals only with globally optimal solutions, is [39].…”

Linking Parametric Cad With Adjoint Surface Sensitivities

“…As an alternative and much more efficient way of computing gradient information, adjoint methods have been the subject of considerable research in recent years [1,2,3,4,5]. The advantage of an adjoint method arises from the fact that the gradient computation becomes independent of the number of design variables.…”