Automated process chains for optimal fluid dynamics design are gaining
importance in automotive applications. So far, industrial implementations of optimal design processes in fluid dynamics are mostly restricted to shape optimization with bionic algorithms. However, on the basis of the recent progress in fluid dynamic topology optimization, as well as in industrial applications of adjoint-based shape design, we propose an adjoint-based design process chain comprising both topology and shape optimization. Starting from the available installation space, this approach delivers a tailored optimal geometry, making use of the whole potential of the available design space at each stage. The central element of this design process is the computation of sensitivity maps – for both topological and shape sensitivities.
1 CFD optimization: current status and limitations The classical way of part design for industrial applications comprises an iteration between the design engineer and the measurement or computation department. Implementing automatic optimization methods into the development process has the potential of shortening this iteration significantly. Such methods are based on a parametric geometry description and automatically yield a design that is optimal with respect to the desired properties and to the imposed constraints.
The techniques that are currently being employed for CFD optimization tasks in the
automotive development process are mainly robust workhorses of the bionic type, i.e.
genetic algorithms and evolutionary strategies [1]-[4]. Typical applications range from air ducts for cabin ventilation over intake ports to engine cooling jackets and exhaust systems.
Although successfully applied in the series development process and being by far more
effective than the conventional design process of trial and error, the limits of these methods are quickly reached: (1) Their computational cost puts a hard constraint on the affordable number of design parameters, i. e. on the explorable design space, and (2) an efficient CAD parametrization with, on the one hand, a maximum of design freedom and on the other, a complete controllability in terms of given design domain restrictions is not a trivial task at all.
Thse two aspects are the actual challenges of current CFD optimization methodologies.
Possible remedies are (1) the use of adjoint methods, where the cost of a sensitivity
computation is independent of the number of design variables, and (2) to perform the
optimization with a CAD-free geometry description, i. e. a surface mesh for shape optimization and a volume mesh for topology optimization, respectively [5, 6]. What this looks like for typical automotive optimization tasks will be demonstrated in the remainder of this paper. We first introduce the idea of adjoint-based computation of sensitivities, apply the methdology to the Navier-Stokes equations and employ it to compute topological sensitivities of air ducts. Recent results of the application of a commercial adjoint solver to drag reduction of a complete passenger car are shown, before the paper closes by pointing out the open issues on the road towards an adjoint-based CFD optimization process chain comprising both topology and shape optimization.
2 Computation of sensitivities with adjoint methods
While already established in the aerospace sector, adjoint methods have been recognized by the automotive industry as an efficient optimization tool only recently [6, 7]. The adjoint approach to optimal design consists of (a) computing the sensitivities of the cost function via an adjoint state, and (b) feeding these sensitivities into a gradient-based optimization algorithm [8]-[11]. Gradient-based algorithms can of course be employed without making use of adjoint states. In that case, however, the conventional computation
of sensitivities of the cost function J wrt. the design parameters = ( 1, . . . , n)T
via finite differences dJ d = J( + k) − J( ) k
for each k, incurs a computational cost of n + 1 solver calls. On the other hand, using an adjoint state, the whole sensitivity can be obtained via one call to the CFD solver (“primal”) and one call to the adjoint solver (“dual”), i. e. via two solver calls - independent of the number of design variables n. Thus, especially for problems with a huge number of design parameters, this method pays off.
It is this characteristic of the adjoint method – its cost independence of the number of design variables – that opens up unparalleled possibilities for design optimization. One oes not have to concern oneself anymore with a complicated CAD parametrization, but – for the sake of maximum design freedom – can draw on plentiful resources: surface mesh or volume mesh representations. The design parameters are then the normal displacements of each of the surface nodes or the porosity value of each volume cell, respectively. Thus, one is dealing with a number of design variables of the order of the order of 105 or even more which
cannot be handled by conventional methods anymore.
3 Topological sensitivities
In structure mechanics, topology optimization is a well-established concept for design optimization with respect to tension or stiffness [12]. Its transfer to computational fluid dynamics, however, just began three years ago with the pioneering work of Borrvall and Petersson [13]. Since then, this topic has received quite some interest both in academia and in the industry [14]-[21]. The starting point for fluid dynamic topology optimization is a volume mesh of the entire installation space. Based on a computation of the flow solution inside this domain, a suitable local criterion is applied to decide whether a fluid cell is “good” or “bad” for the flow in terms of the chosen cost function. In order to iteratively remove the identified
bad cells from the fluid domain, they are either punished via a momentum loss term, or holes are inserted into the flow domain, with their positions being determined from an evaluation of the so-called topological asymptotic.
In the former case, the momentum loss term is usually realized via a finite cell porosity, i. e. the whole design domain is treated as a porous medium: Each cell is assigned an individual porosity i, which is modeled via Darcy’s law. The value of i determines if the cell is fluid-like (low porosity values) or has a rather solid character (high values of i). In other words, the porosity field controls the geometry, and the i are the actual design variables.
With such a setting, an adjoint method can be applied to elegantly compute the sensitivities of the chosen cost function wrt. the porosity of each cell. The obtained sensitivities can then be fed into a gradient-based optimization algorithm – possibly with some penalization of intermediate porosity values in order to enforce a “digital” porosity distribution, and after several iterations, the desired optimum topology is finally extracted as an isosurface of the obtained porosity distribution or simply as the collection of all non-porous cells.
In a recent study, Othmer et al. [21] were able to verify the applicability of this methodology to typical automotive objective functions, including dissipated power, equal mass flow through different outlets, flow uniformity and angular momentum of the flow in the outlet plane. In that proof-of-concept study, Automatic Differentiation techniques were applied to an academic CFD code in order to obtain a discrete adjoint solver. For industrial-sized problems, however, this code is not suitable. Therefore, we implemented the methodology via a continuous adjoint into the professional CFD environment Open- FOAM [22]. The underlying equations of this implementation will be shown in the following section, before we show examples of topological sensitivity maps for an air duct segment.
to be continiue
importance in automotive applications. So far, industrial implementations of optimal design processes in fluid dynamics are mostly restricted to shape optimization with bionic algorithms. However, on the basis of the recent progress in fluid dynamic topology optimization, as well as in industrial applications of adjoint-based shape design, we propose an adjoint-based design process chain comprising both topology and shape optimization. Starting from the available installation space, this approach delivers a tailored optimal geometry, making use of the whole potential of the available design space at each stage. The central element of this design process is the computation of sensitivity maps – for both topological and shape sensitivities.
1 CFD optimization: current status and limitations The classical way of part design for industrial applications comprises an iteration between the design engineer and the measurement or computation department. Implementing automatic optimization methods into the development process has the potential of shortening this iteration significantly. Such methods are based on a parametric geometry description and automatically yield a design that is optimal with respect to the desired properties and to the imposed constraints.
The techniques that are currently being employed for CFD optimization tasks in the
automotive development process are mainly robust workhorses of the bionic type, i.e.
genetic algorithms and evolutionary strategies [1]-[4]. Typical applications range from air ducts for cabin ventilation over intake ports to engine cooling jackets and exhaust systems.
Although successfully applied in the series development process and being by far more
effective than the conventional design process of trial and error, the limits of these methods are quickly reached: (1) Their computational cost puts a hard constraint on the affordable number of design parameters, i. e. on the explorable design space, and (2) an efficient CAD parametrization with, on the one hand, a maximum of design freedom and on the other, a complete controllability in terms of given design domain restrictions is not a trivial task at all.
Thse two aspects are the actual challenges of current CFD optimization methodologies.
Possible remedies are (1) the use of adjoint methods, where the cost of a sensitivity
computation is independent of the number of design variables, and (2) to perform the
optimization with a CAD-free geometry description, i. e. a surface mesh for shape optimization and a volume mesh for topology optimization, respectively [5, 6]. What this looks like for typical automotive optimization tasks will be demonstrated in the remainder of this paper. We first introduce the idea of adjoint-based computation of sensitivities, apply the methdology to the Navier-Stokes equations and employ it to compute topological sensitivities of air ducts. Recent results of the application of a commercial adjoint solver to drag reduction of a complete passenger car are shown, before the paper closes by pointing out the open issues on the road towards an adjoint-based CFD optimization process chain comprising both topology and shape optimization.
2 Computation of sensitivities with adjoint methods
While already established in the aerospace sector, adjoint methods have been recognized by the automotive industry as an efficient optimization tool only recently [6, 7]. The adjoint approach to optimal design consists of (a) computing the sensitivities of the cost function via an adjoint state, and (b) feeding these sensitivities into a gradient-based optimization algorithm [8]-[11]. Gradient-based algorithms can of course be employed without making use of adjoint states. In that case, however, the conventional computation
of sensitivities of the cost function J wrt. the design parameters = ( 1, . . . , n)T
via finite differences dJ d = J( + k) − J( ) k
for each k, incurs a computational cost of n + 1 solver calls. On the other hand, using an adjoint state, the whole sensitivity can be obtained via one call to the CFD solver (“primal”) and one call to the adjoint solver (“dual”), i. e. via two solver calls - independent of the number of design variables n. Thus, especially for problems with a huge number of design parameters, this method pays off.
It is this characteristic of the adjoint method – its cost independence of the number of design variables – that opens up unparalleled possibilities for design optimization. One oes not have to concern oneself anymore with a complicated CAD parametrization, but – for the sake of maximum design freedom – can draw on plentiful resources: surface mesh or volume mesh representations. The design parameters are then the normal displacements of each of the surface nodes or the porosity value of each volume cell, respectively. Thus, one is dealing with a number of design variables of the order of the order of 105 or even more which
cannot be handled by conventional methods anymore.
3 Topological sensitivities
In structure mechanics, topology optimization is a well-established concept for design optimization with respect to tension or stiffness [12]. Its transfer to computational fluid dynamics, however, just began three years ago with the pioneering work of Borrvall and Petersson [13]. Since then, this topic has received quite some interest both in academia and in the industry [14]-[21]. The starting point for fluid dynamic topology optimization is a volume mesh of the entire installation space. Based on a computation of the flow solution inside this domain, a suitable local criterion is applied to decide whether a fluid cell is “good” or “bad” for the flow in terms of the chosen cost function. In order to iteratively remove the identified
bad cells from the fluid domain, they are either punished via a momentum loss term, or holes are inserted into the flow domain, with their positions being determined from an evaluation of the so-called topological asymptotic.
In the former case, the momentum loss term is usually realized via a finite cell porosity, i. e. the whole design domain is treated as a porous medium: Each cell is assigned an individual porosity i, which is modeled via Darcy’s law. The value of i determines if the cell is fluid-like (low porosity values) or has a rather solid character (high values of i). In other words, the porosity field controls the geometry, and the i are the actual design variables.
With such a setting, an adjoint method can be applied to elegantly compute the sensitivities of the chosen cost function wrt. the porosity of each cell. The obtained sensitivities can then be fed into a gradient-based optimization algorithm – possibly with some penalization of intermediate porosity values in order to enforce a “digital” porosity distribution, and after several iterations, the desired optimum topology is finally extracted as an isosurface of the obtained porosity distribution or simply as the collection of all non-porous cells.
In a recent study, Othmer et al. [21] were able to verify the applicability of this methodology to typical automotive objective functions, including dissipated power, equal mass flow through different outlets, flow uniformity and angular momentum of the flow in the outlet plane. In that proof-of-concept study, Automatic Differentiation techniques were applied to an academic CFD code in order to obtain a discrete adjoint solver. For industrial-sized problems, however, this code is not suitable. Therefore, we implemented the methodology via a continuous adjoint into the professional CFD environment Open- FOAM [22]. The underlying equations of this implementation will be shown in the following section, before we show examples of topological sensitivity maps for an air duct segment.
to be continiue
