Ω∗O
δc[
c−ctar]
dΩ∗. (3.23)
Boundary Conditions
The material derivative (δ + δn∗(∂/∂ n∗) is applied to generic boundary conditions of Dirichlet (D∗φ) and Neumann (N∗φ) type. The argumentation assumes that the boundary conditions of the Old State on the Old Shape must be equal to those of the New State on the New Shape, viz. (·)|old = (·)|new. The control perturbation (δn∗ = 0) and thus the variational contributions disappear along all boundaries that do not belong to the design surface, i.e. Γ∗∩Γ∗D =∅. However, if the respective boundary is subjected to control, the variations need to be further expanded, viz.
δφ∗ = 0 or ∂(δφ∗)
∂ n∗ = 0 on Γ∗ (3.24)
δφ∗ =−δn∗∂ φ∗
∂ n∗ or ∂(δφ∗)
∂ n∗ =−δn∗ ∂
∂ n∗ [∂ φ∗
∂ n∗ ]
on Γ∗∩Γ∗D. (3.25) A variation of the primal boundary conditions from Tab. 2.2 according to (3.24)-(3.25) is listed in Tab. 3.1. Due to the concept of material derivative, local contributions disappear along almost all boundaries. In line with Eqns. (3.24)-(3.25), the following variations of primal velocity and concentration can be developed along controlled no-slip boundaries
δvi∗ =−δn∗∂ v∗i
∂ n∗ and ∂δc
∂ n∗ =−δn∗ ∂
∂ n∗ [ ∂ c
∂ n∗ ]
= 0. (3.26)
The variation of the effective pressure gradient condition ∂ p∗/∂ n∗ = ρ∗gk∗nk is further expanded
∂δp∗
∂ n∗ =−δn∗ ∂
∂ n∗ [
ρ∗gk∗nk
]
=−δn∗∂ ρ∗
∂ c
∂ c
∂ n∗ρ∗gk∗nk = 0. (3.27) and likewise vanishes due to a zero gradient condition for the concentration. The variations of the primal boundary conditions are collected in Tab. 3.1. They will be reused to derive the adjoint boundary conditions in Sec. 4.1.6.
3.5 Shape Gradient Approximation for Non-Parameterized Shapes
Shape optimization procedures in the discrete CFD surface’s design space allow access to local features and shape optima on the level of the discrete CFD resolution. However, the
”raw” adjoint shape derivatives suffer from a few well-known weaknesses, e.g. they only describe the normal deformation (cf. Sec. 3.3) but do not provide tangential information and the shape derivatives are not necessarily smooth. These deficiencies yield rough/noisy
boundary type δvi∗ δp∗ δc inlet δvi∗ = 0 ∂ δp∂ n∗∗ = 0 δc= 0 outlet ∂ δv∂ n∗∗i = 0 δp∗ = 0 ∂ n∂ δc∗ = 0 symmetry δvi∗ni= 0, ∂ δv∂ n∗i∗ti= 0 ∂ δp∂ n∗∗ = 0 ∂ n∂ δc∗ = 0 wall (slip) δvi∗ni= 0, ∂ δv∂ n∗i∗ti= 0 ∂ δp∂ n∗∗ = 0 ∂ n∂ δc∗ = 0 wall (no-slip, Γ∗ ̸⊂Γ∗D) δvi∗ = 0 ∂ δp∂ n∗∗ = 0 ∂ n∂ δc∗ = 0 wall (no-slip, Γ∗ ⊂Γ∗D) δv∗i =−δn∗∂ v∂ n∗i∗ ∂ δp∂ n∗∗ = 0 ∂ n∂ δc∗ = 0
Table 3.1: Variations of the primal boundary conditions where ti [ni] refers to the local boundary tangential [normal] and δn∗ =δu∗knk.
shape updates (cf. St¨uck and Rung [2011], Kr¨oger and Rung [2015]) and lead to distorted near-wall meshes which in turn hamper the preservation of numerical accuracy during the optimization procedure, e.g. Stavropoulou et al. [2014] and Bletzinger [2014]. As a consequence, the adjoint shape derivatives have to be regularized to obtain meaningful technical shape updates as initially proposed by Jameson and Vassberg [2000] and Vassberg and Jameson [2006a,b] in terms of an implicit, continuous smoothing operator based on an extended definition of the inner product, frequently labeled ”Sobolev-gradient”. In the following, gi∗ denotes the shape gradient. In general, the shape gradient habitat –surface-vs. volume-based– depends on the underlying surface metric. Prominent examples refer to Laplace-Beltrami (LB) or Steklov-Poincar (SP) type metrics, e.g. Schulz and Siebenborn [2016].
3.5.1 Laplace-Beltrami
Applying the smoothing operation along the design boundary Γ∗Dleads to the LB equation with a source term corresponding to the shape sensitivitys∗i =s∗ni, viz.
˜
g∗i −µ∗˜gi2∆∗Γg˜∗i =g∗i −µ∗˜gi2
[∂2g˜∗i
∂ x∗k2 −nk
∂2g˜∗i
∂ x∗k ∂ x∗lnl
]
=s∗i on Γ∗∩Γ∗D, (3.28) where the LB operator operator is introduced as the tangential Laplacian, i.e. ∆∗Γ =
∆∗−∆∗n, cf. App. B.1. For dimensional reasons, the LB Eqn. (3.28) inheres a diffusivity µ∗g˜i that corresponds to a squared length which allows for an adjustment of the gradient smoothness and influences the search direction. Note that the gradient equals the sensi-tivity derivative in the limiting case of µ∗g˜i →0. The LB Eqn. (3.28) is defined along the walled domain boundary and therefore requires a tailored two-dimensional approximation infrastructure. Classical CFD solvers may reach their limits since standardized solution routines (assembling, solving, etc.) are not necessarily available. As a consequence, Eqn.
(3.28) is often approximated explicitly, cf. Bletzinger [2014]. The explicitly filtered gradi-ent, e.g. by using consistent kernel functions (Kr¨oger and Rung [2015]), marks a first-order approximation to the implicit Sobolev-gradient, cf. St¨uck and Rung [2011]. Note that po-tentially expensive manipulations of the smoothing operation are necessary if the geometry contains, e.g., symmetry constraints. Once the boundary-based shape gradient is available along Γ∗D, a subsequent mesh deformation equation transports the surface-based gradient
information into the interior field in terms of, e.g. spring analogies or based on an addi-tional Laplacian PDE together with prescribed boundary deformations, cf. L¨ohner and Yang [1996], Crumpton and Giles [1997], Jasak and Tukovi´c [2006].
In line with the modelling of floatation, cf. Sec. 2.2.11 and Eqn. (2.111), the shape gradient ˜gi∗ enters the boundary conditions of the following Laplacian PDE
∂
∂x∗k [
µ∗gi∂gi∗
∂x∗k ]
= 0 in Ω∗ with
{g∗i = ˜gi∗ on Γ∗∩Γ∗D
g∗i = 0 on Γ∗ . (3.29) The diffusivity µ∗gi refers to the inverse (non zero) distance to the nearest wall, which suppresses distortion of the grid in the vicinity of the design wall, cf. (2.111).
3.5.2 Steklov-Poincar´ e
The SP approach refers to a novel strategy on an industrial level that employs an ellip-tic volume-based formulation where smoothed results are subsequently projected on the boundary. Unlike the previous smoothing operations along the design boundary, the SP procedure essentially tries to combine the 2D shape update in continuous space with the 3D mesh update using the discrete CFD mesh sensitivities, cf. Schulz and Siebenborn [2016], Haubner et al. [2021]. The approach exclusively operates in the fluid domain and is thus attractive due to its compatibility with a CFD solver environment. The use of standard HPC-capable solution routines (assembling, solving, etc.) supplied by the flow solver represents a major benefit of the SP procedure.
The SP-approach resembles Eqn. (3.29) based on a manipulation of the boundary con-dition along the controlled boundary Γ∗D, viz.
∂
∂x∗k [
µ∗gi∂gi∗
∂x∗k ]
= 0 in Ω∗ with
⎧⎪
⎨
⎪⎩
∂ g∗i
∂ n∗ =s∗ni on Γ∗∩Γ∗D gi∗ni = 0,∂ g∂ n∗i∗ti= 0 on Γ∗Symm
gi∗ = 0 on Γ∗
, (3.30) where the Dirichlet condition of the LB-metric translates into a Neumann condition that directly employs the sensitivity derivative, cf. Schulz and Siebenborn [2016], Haubner et al. [2021]. Accordingly, the gradient units in (3.30) differ from those in (3.29), which translates to a different employed step size within a steepest descent approach.
Note that no additional boundary-based operations are necessary and the number of equations to be solved decreases compared with the LB approach. Moreover, modifications of the boundary conditions listed in (3.30) support an intuitive introduction of additional geometric constraints. Examples included in the application chapter of this thesis refer to fixed intersection lines along a symmetry plane via g∗i = 0 on Γ∗Symm, or the realization of a mandatory flat ship transom obtained by gi∗ni = 0 along Γ∗D˜. In contrast to the LB metric, the SP approach involves only a single user-defined parameter, i.e. the diffusivity µ∗gi used in (3.30). In this thesis, the diffusivity refers to the inverse (non zero) distance to the nearest wall in line with (2.111)) and (3.29). More sophisticated SP approaches employ a nonlinear viscosity, e.g. µ∗gi = [(∂ gi∗/∂ x∗k)(∂ g∗i/∂ x∗k)](p−2)/2 in terms of a p-Laplacian approach, cf. M¨uller et al. [2021].
The SP-approach is the preferred approach of this thesis. A step in the steepest descent direction is performed once the volume-based shape gradient is available, cf. Sec. 3.7.
Prior to this, other technical constraints are incorporated into the process.