Optimization Based on a Continuous Adjoint-RANS Simulation Approach (FORM-PRO)


Sensitivity information, when available, can substantially facilitate many CFD-related applications. The adjoint technique provides a very cost-effective way to calculate sensitivities (or derivatives), particularly in the case of many control parameters and costly objective evaluations, such as in viscous CFD. Typical applications are

  • shape and topology optimisation with all discrete mesh points being considered as DOF
  • targeted error estimation and grid analysis/adaptation
  • flow control

In order to obtain the cost function sensitivities, the variation of the cost function of interest is extended by the variation of the RANS-constraints in a Lagrangian manner, weighted by the adjoint (or Lagrangian) multipliers. Integration by parts yields the adjoint (or dual) RANS equations, which are solved for the adjoint multipliers to eliminate the parameter-dependent flow variations. Hence, the computational effort for the sensitivity analysis becomes independent of the number of control (or shape) parameters. In multi-objective applications, the costs for evaluating the Jacobian matrix scales with the number of objectives.



Example of superimposed resistance sensitivities (red) for an initial (blue) tanker hull (left=stern, right=bow).

Computational Approach

A 2D continuous-adjoint solver has been derived for incompressible flow on the basis of a segregated adjoint pressure-correction methodology using a finite-volume discretisation. The respective focus is on

  • appropriate (robust) discretisiation schemes for the adjoint problem
  • formulation of adjoint boundary conditions corresponding to the objective functions
  • verification and validation of flow sensitivities against analytical solutions and results of a linearised solver

The continuous adjoint solver has been integrated into a flexible 3D CFD-environment. Attention is given to the FreSCo+ software, which is designed to simulate general ship-hydrodynamic problems. In particular, the following aspects are in the centre of interest for future research

  • improved turbulence treatment for high Reynolds-number flows
  • extension of the continuous adjoint method to cost functions related to ship-hull design and propulsion efficiency
  • grid adaptation and objective oriented error analysis


The adjoint method provides the gradient distribution over a complete surface at the cost of one primal solution. The shape sensitivities are indicated by black vectors on the design surface:


Simple Validation examples included refer to an adjoint sensitivity analysis for viscous flows; Left: Drag force sensitivities w.r.t. surface-normal shape perturbations for a NACA0020 profile - shape variations following these sensitivities lead to a drag increase. Right: Sensitivities of the (negative) pressure loss for a channel flow. The corresponding boundary modifications lead to a reduced pressure drop between inlet and outlet. 

First hydrodynamic applications devoted to the shape sensitivity of a tanker to drag forces are illustrated below (Left: bow; Right: stern).         



The project is funded by the German Ministry of Economics and Technology under the aegis of the BMWi-project FORM-PRO within the framework program "Schiffahrt und Meerestechnik für das 21. Jahrhundert". The work is performed in colaboration with HSVA and Friendship Systems.

Project Duration



Dipl.-Ing Arthur Stück

Dipl.-Ing. Jörn Kröger

Prof. Dr.-Ing. Thomas Rung