Elliptic Relaxation Strategies to Support Numerical Stability of Segregated Continuous Adjoint Flow Solvers

TL;DR

The paper addresses numerical instabilities in sequential continuous adjoint CFD solvers by introducing an elliptic relaxation PDE that smooths explicit adjoint sources with a single tunable filter width, interpretable as a Gaussian kernel. This stabilization preserves integral properties while redistributing troublesome terms such as ATC, enabling robust adjoint solves on unstructured industrial grids and across two-phase flows. Validation on a laminar cylinder test confirms accuracy and conservativity, and applications to ship-hull shape optimization demonstrate substantial drag reductions when using inhomogeneous, grid-local filter widths. The work provides practical guidance that inhomogeneous relaxation offers the best stability-accuracy balance for industrial-scale adjoint problems, with significant implications for reliable, high-fidelity optimization in marine engineering.

Abstract

This paper introduces a novel method for numerically stabilizing sequential continuous adjoint flow solvers utilizing an elliptic relaxation strategy. The proposed approach is formulated as a Partial Differential Equation (PDE) containing a single user-defined parameter, which analytical investigations reveal to represent the filter width of a probabilistic density function or Gaussian kernel. Key properties of the approach include (a) smoothing features with redistribution capabilities while (b) preserving integral properties. The technique targets explicit adjoint cross-coupling terms, such as the Adjoint Transpose Convection (ATC) term, which frequently causes numerical instabilities, especially on unstructured grids common in industrial applications. A trade-off is made by sacrificing sensitivity consistency to achieve enhanced numerical robustness. The method is validated on a two-phase, laminar, two-dimensional cylinder flow test case at Re=20 and Fn=0.75, focusing on minimizing resistance or maximizing lift. A range of homogeneous and inhomogeneous filter widths is evaluated. Subsequently, the relaxation method is employed to stabilize adjoint simulations during shape optimizations that aim at drag reduction of ship hulls. Two case studies are considered: A model-scale bulk carrier traveling at Re=7.246E+06 and Fn=0.142 as well as a harbor ferry cruising at Re=2.43E+08 and Fn=0.4 in full-scale conditions. Both cases, characterized by unstructured grids prone to adjoint divergence, demonstrate the effectiveness of the proposed method in overcoming stability challenges. The resulting optimizations achieve superior outcomes compared to approaches that omit problematic coupling terms.

Figures (29)

• Figure 1: Schematic representation of a Finite-Volume arrangement (a) in the field and (b) along the boundary.
• Figure 2: Submerged cylinder case ($\mathrm{Re}_\mathrm{D} = 20$, $\mathrm{Fn}=0.75$): (a) Schematic of the initial configuration showing the controlled cylinder shape with $u_\mathrm{i}$ (red), and (b) a segment of the structured numerical grid surrounding the cylinder.
• Figure 3: Submerged cylinder case ($\mathrm{Re}_\mathrm{D} = 20$, $\mathrm{Fn}=0.75$): Normalized shape sensitivity of a drag functional along the normalized upper cylinder contour for seven different elliptic relaxation strategies with varying filter width ($\mu_\varphi/\Delta x = [0, 10^{-1}, 1, 10^1]$, $\mu_\varphi/D = [10^{-1}, 1, 10^1]$) applied to (left) the gravitational, (center) momentum convection, and (right) momentum diffusion contribution to the right-hand side of the adjoint concentration balance.
• Figure 4: Submerged cylinder case ($\mathrm{Re}_\mathrm{D} = 20$, $\mathrm{Fn}=0.75$): Normalized shape sensitivity of a drag functional along the normalized upper cylinder contour for seven different elliptic relaxation strategies with varying filter width ($\mu_\varphi/\Delta x = [0, 10^{-1}, 1, 10^1]$, $\mu_\varphi/D = [10^{-1}, 1, 10^1]$) applied to (left) the momentum and (center) volume fraction convection contribution to the right-hand side of the adjoint momentum equations as well as (right) all explicit sources to the adjoint momentum and concentration equations.
• Figure 5: Submerged cylinder case ($\mathrm{Re}_\mathrm{D} = 20$, $\mathrm{Fn}=0.75$): Converged explicit volume fraction convection contribution to the $x_2$ adjoint momentum equation for an inhomogeneous filter width of $\mu_\varphi / \Delta x^2 = [10^{-1}, 1, 10]$ from top to bottom, respectively.