Adjoint node-based shape optimization of free-floating vessels

Abstract

The paper is concerned with a node-based, gradient-driven, continuous adjoint two-phase flow procedure to optimize the shapes of free-floating vessels and discusses three topics. First, we aim to convey that elements of a Cahn–Hilliard formulation should augment the frequently employed Volume-of-Fluid two-phase flow model to maintain dual consistency. It is seen that such consistency serves as the basis for a robust primal/adjoint coupling in practical applications at huge Reynolds and Froude numbers. The second topic covers different adjoint coupling strategies. A central aspect of the application is the floating position, particularly the trim and the sinkage, that interact with a variation of hydrodynamic loads induced by the shape updates. Other topics addressed refer to the required level of density coupling and a more straightforward—yet non-frozen—adjoint treatment of turbulence. The third part discusses the computation of a descent direction within a node-based environment. We will illustrate means to deform both the volume mesh and the hull shape simultaneously and at the same time obey technical constraints on the vessel’s displacement and its extensions. The Hilbert-space approach provides smooth shape updates using the established coding infrastructure of a computational fluid dynamics algorithm and provides access to managing additional technical constraints. Verification and validation follow from a submerged 2D cylinder case. The application includes a full-scale offshore supply vessel at $$\mathrm{Re} = 3 \times 10^8$$ Re = 3 × 10 8 and $$\mathrm{Fn} = 0.37$$ Fn = 0.37 . Results illustrate that the fully parallel procedure can automatically reduce the drag of an already pre-optimized shape by 9–13% within $$\approx\,{\mathcal{O}}$$ ≈ O (10,000-30,000) CPUh depending on the considered couplings and floatation aspects.

Shape Transformation Approaches for Fluid Dynamic Optimization

“…The numerical procedure for the solution of the primal (16) and adjoint system ( 19) is based upon the finite volume method (FVM) [9,34]. The implicit numerical approximation is second-order accurate in space and time and supports arbitrary polyhedral cells as well as local grid refinement.…”

“…The segregated algorithm uses a cell-centered, co-located storage arrangement for all transport properties. A detailed derivation of this hybrid adjoint approach can be found in [6,9,11]. The primal and adjoint pressure-velocity coupling utilizes a pressure-correction scheme and parallelization is realized by means of a domain decomposition approach [35,36].…”

“…In a second step the adjoint state ŷ is computed which is associated with the Lagrange multipliers and given by the solution of the adjoint equations in (19). For the shape derivative j (Ω)u we consider the boundary formulation (9) from which the descent direction u is obtained by applying one of the three different strategies. The process to determine the modified shape deformation with p-Laplace extension is summarized in Algorithm 2.…”

“…This paper is concerned with continuous adjoint-and gradient-based shape optimization methods. Adjoint-based techniques have successfully been applied in various areas of industrial applications [6][7][8][9] in order to compute shape sensitivities (shape derivatives) required by a gradient-based optimization method. The appeal of adjoint methods is that the computation of the derivatives is independent of the number of design or control variables of the problem, as this method does not require the derivative of the state variable with respect to the control.…”

Parameter-free shape optimization: various shape updates for engineering applications

“…θ max = 0.02 D, based on the experience of the authors on this particular case, cf. [53]. Further investigations in combination with the line search method are presented in Appendix A.…”