INDUSTRIAL APPLICATION OF CONTINUOUS ADJOINT FLOW SOLVERS FOR THE - PDF document

CFD & OPTIMIZATION 2011 - 069 An ECCOMAS Thematic Conference 23-25 May 2011, Antalya TURKEY INDUSTRIAL APPLICATION OF CONTINUOUS ADJOINT FLOW SOLVERS FOR THE OPTIMIZATION OF AUTOMOTIVE EXHAUST SYSTEMS C. Hinterberger ∗ , M. Olesen ∗ ∗ Faurecia Emissions Control Technologies, Germany GmbH e-mail: Christof.Hinterberger, Mark.Olesen@faurecia.com Key words: Geometry Optimization, Adjoint Flow Solver, Exhaust Systems Abstract. A continuous adjoint geometry optimization tool (CAGO), has been developed at Faurecia Emissions Control Technologies [1], which finds suitable shapes for catalyst inlet cones directly from the package space. CAGO produces a design proposal, which is used as a reference surface in the CAD process. Subsequently, the CAD design is validated in a fully compressible flow analysis. For further optimization sensitivities are computed by solving the adjoint flow fields with a frozen density and frozen viscosity assumption. This pseudo-compressible continuous adjoint method is described in the paper in detail in a general form including scalar transport. 1

C. Hinterberger, M. Olesen 1 OPTIMIZATION OF EXHAUST SYSTEMS Meeting backpressure and flow uniformity requirements within severe packaging con- straints presents a particular challenge in the layout of catalyst inlet cones. Figure 1 illustrates how the flow uniformity of a closed coupled catalyst can be influenced by the design of the inlet cone. Figure 1: Close coupled catalyst and a 3-in-1 manifold of a 12 cylinder engine. Especially for complex package spaces it can prove difficult to find designs that fulfill the uniformity targets. To address this problem, a continuous adjoint geometry optimization tool (CAGO) has been developed at Faurecia Emissions Control Technologies that finds suitable shapes for catalyst inlet cones directly from the package space[1]. The tool is based on the continuous adjoint formulation derived and implemented by Othmer et al. [2, 3]. The implementation uses the open source CFD toolbox OpenFOAM 1 [5]. As shown in Figure 2, CAGO begins from the provided package space and produces a design proposal that can be used as a reference surface (in IGES format) during the CAD process. Once a corresponding CAD geometry has been established — including consideration of manufacturing and durability constraints — a fully compressible flow analysis is pre- formed, in which the catalysts are modelled as anisotropic porosities. In this analysis, the surface sensitivities for the optimization of the flow uniformity and the pressure drop (see Figure 2b) are also computed by solving the adjoint flow fields with a frozen density and frozen viscosity assumption. This pseudo-compressible continuous adjoint method is described in the methodology section of the paper. For systems that are equipped with a fuel vaporizer — which is an exhaust system component for introducing additional fuel to support DPF (diesel particulate filter) regeneration — the surface sensitivities for the uniformity of the fuel vapour distribution in front of a DOC (diesel oxidation catalyst) can 1 OpenFOAM R � is a registered trademark of OpenCFD Ltd. 2

C. Hinterberger, M. Olesen be also calculated. The resulting surface sensitivities can be used to assess the effect of geometry modifications and to identify areas that are critical with respect to manufactur- ing tolerances. In the current optimization workflow, geometry modifications at this later stage are either applied manually in the CAD model or by morphing the CFD surface mesh directly. Since the number of additional constraints ( e.g., manufacturing) increases at these later design stages, further automation is difficult to define or implement and some degree of manual intervention must be accepted. Nevertheless, by enhancing the usability of the geometry manipulation tools, further improvement of the design process is still possible. Figure 2: (a, b) Workflow of design and optimization process. (b) surface sensitivities for pressure drop and flow uniformity, showing areas which have to be expanded (green) or shrunk (red) to improve results. Figure 3: Schematic of the optimization with CAGO: (a) package space and (b) computational mesh and boundary conditions. 3

C. Hinterberger, M. Olesen Figure 4: Automatic geometry optimization with CAGO; (a, c) volumetric sensitivities for energy dissi- pation, (b, d) volumetric sensitivities for flow uniformity. 4

C. Hinterberger, M. Olesen 2 CONTINUOUS ADJOINT METHOD Our optimization process uses two different adjoint solvers. The first is the solver used in our automatic geometry optimization tool CAGO, which is based on an incompressible continuous adjoint solver originally developed by Othmer et al. [2], and which is described in detail in[1]. CAGO operates on a fixed (non-moving) mesh for the package space and uses a level set method to describe the geometry (Figure 3). The geometry is adjusted automatically according to computed sensitivity fields. This is shown in Figure 4 where the development of the cone geometry over the solver iterations can be seen. CAGO is used to generate a design proposal for the CAD designer. The CAD design is then validated using a fully compressible CFD simulation. For the further optimization of this model, sensitivities are calculated using a pseudo-compressible continuous adjoint method, which uses a frozen density and frozen turbulence assump- tion. This method is described in this section in a general form including scalar transport, so that it can be easily transferred to other applications ( e.g. optimization of heat ex- changers, as seen in [4]). The derivation of the equations follows the theoretical paper of Othmer[3]. 2.1 Concept of the adjoint CFD method At the beginning of an optimization problem, a cost function J = J ( c , ξ ), which is to be minimized, is defined. The cost function depends on the geometry, which is specified via a design vector c and it depends on the flow field ξ . The total variation of the cost function with respect to a design change is thus given as follows � � = ∂J ∂ c · δ c + ∂J δJ = δ c J + δ ξ J ∂ξ · δξ (1) ���� ���� geometry flow When the design changes by an amount δ c , the flow field changes accordingly by δξ and this affects the cost function J , causing the variation δ ξ J . Additionally, there could be a direct (geometrical) dependence of J on the geometry, causing the variation δ c J . The variation can also be written as δ c J = ∂J ∂ c · δ c = ∂ c J · δ c . For geometry optimization, it is quite convenient to define a sensitivity field in the following form: δJ = ∂ c L · δ c (2) ���� ���� design variation sensitivity in which the sensitivity field ( ∂ c L ) relates how the cost function J is affected by an arbitrary, small, and reasonable smooth, design variation δ c . Since the sensitivities ∂ c L have to account for how design variations influence the flow field ξ , an augmented cost function � L = J + Ψ · R (3) Ω 5

C. Hinterberger, M. Olesen introduces the state equations R ( ξ ) = 0 , which are the Navier-Stokes equations written in residual form, as an constraint into the optimization problem, with the adjoint flow field Ψ acting as Lagrange multiplier. The augmented cost function L depends on the design c and on the flow field ξ , and therefore its total variation is δL = δ c L + δ ξ L = ∂ c L · δ c + ∂ ξ L · δξ ���� ���� (4) geometry flow By requiring δ ξ L ≡ 0 (5) the total variation (4) becomes δL ≡ δ c L = ∂ c L · δ c , which includes the sensitivity ∂ c L defined in (11). This requirement defines the adjoint flow field Ψ. With δ (Ψ R ) = Ψ δ R + R δ Ψ and R = 0 , the variation of the augmented cost function (3) is given as � δL = δJ + Ψ · δ R (6) Ω and (5) expands to � δ ξ L = δ ξ J + Ψ · δ ξ R ≡ 0 (7) Ω From this, an equation system A (Ψ) = 0 can be derived for the adjoint flow field Ψ, in which the boundary conditions and source terms depend on J (see section 2.5.2). The adjoint equations A (Ψ) = 0 are very similar to the flow equations R ( ξ ) = 0 , and can be solved with a similar numerical cost. Once the adjoint equations are solved, the variation of the cost function with respect to an arbitrary design change, can be computed directly from the primal flow ξ and adjoint flow field Ψ, as follows � δ c L = δ c J + Ψ · δ c R (8) Ω without the need of an extra CFD–solution of the state equations. This can be easily applied for the computation of volumetric sensitivities, as it will be shown later, but for surface sensitivities it is difficult to compute δ c R . However, since the total variation of the state equation is zero, δ R = δ c R + δ ξ R = 0 (9) the computation of shape sensitivities can be defined alternatively: � δ c L = δ c J − Ψ · δ ξ R (10) Ω 6