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

ABSTRACT This paper introduces a novel method for numerically stabilizing sequential continuous adjoint flow solvers utilizing an elliptic relaxation strategy. Unlike previous stabilization approaches, 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 smoothing features with redistribution capabilities while 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 a Reynolds number of and Froude number of , 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 and as well as a harbor ferry cruising at and 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, yielding stable and adjoint solutions of improved consistency even for complex, unstructured, two‐phase flow configurations. This demonstrates that the proposed elliptic relaxation strategy provides a practical and broadly applicable means to enhance the numerical robustness of segregated continuous adjoint solvers in industrial CFD environments.