An Object-oriented Framework for Rapid Discrete Adjoint Development using OpenFOAM

Abstract

The adjoint method is an efficient approach for computing derivatives because its computational cost is independent of the number of design variables. Using the derivatives computed from the adjoint method, a gradient-based optimization can handle complex design problems such as full-scale aircraft. Despite the above advantages, implementing the adjoint method for a partial differential equation based primal solver is a time-consuming task. To lower the barrier for adjoint implementations, we propose an object-oriented framework to rapidly develop the discrete adjoint method based on OpenFOAM; an open-source, multiphysics package that contains more than 80 primal solvers involving a wide range of disciplines such as aerodynamics, hydrodynamics, heat transfer, structures, combustion, and multiphase flow. The proposed framework provides high-level interfaces that allow us to implement the adjoint method for any existing steady-state OpenFOAM primal solvers by adding or modifying only O(100) lines of source code. In this paper we introduce the overall structure of the proposed adjoint framework and detail the adjoint development process by starting with a simple scalar transport equation and then extending the development to the Navier-Stokes equations. So far, we have implemented adjoint methods for five primal solvers and four turbulence models. We observe excellent adjoint scalability with up to O(10) million cells and O(1000) CPU cores, and the maximal error in the adjoint derivatives is less than 0.1%. To further demonstrate the benefit of having the flexibility to rapidly develop the adjoint method for different solvers and turbulence models, we showcase three successful aerodynamic shape optimizations that cover incompressible, compressible, full turbulence, and transitional turbulence conditions. Our proposed adjoint framework has the potential of becoming a useful tool to handle high-fidelity multidisciplinary design optimization problems for general engineering systems such as aircraft, cars, ships, and turbomachinery.

Comparing Matrix-based and Matrix-free Discrete Adjoint Approaches to the Euler Equations

“…Compared to previous matrix-based adjoint implementations [14,15], the approach presented in this paper is a hybrid methodology where only the compact terms of the Jacobian have been stored; the dissipation terms, with a much larger stencil, were differentiated by hand and evaluated in a matrix-free manner when solving the adjoint equations. The resulting adjoint product is very computationally efficient when compared to a purely matrix-free adjoint product at the cost of a moderate increase in memory usage.…”

“…An exception to this is if the Jacobian is sparse with some known structure, in which case storage of the Jacobian may be less than O (10) times that of the state vector. In this work, a wrap-around approach to the discrete adjoint is presented using a sparse Jacobian constructed by the complex step method and accelerated by graph colouring as performed in [14] and [15]. In [14], Lyu et al use forward-mode automatic differentiation with graph colouring to construct the sparse block Jacobian for the three-dimension Reynolds Averaged Navier Stokes equations.…”

“…In [14], Lyu et al use forward-mode automatic differentiation with graph colouring to construct the sparse block Jacobian for the three-dimension Reynolds Averaged Navier Stokes equations. In [15] He et al instead use finite differences to provide a more generic 'black-box' framework within openFOAM. The advantage of this method is that it avoids the complexity and development time associated with reverse mode automatic differentiation, while providing a very efficient adjoint product code.…”

Free-Form Deformation Parameterization on the Aerodynamic Optimization of Morphing Trailing Edge

“…Many optimization problems that use the FFD technique for the optimization of the whole wing and aircraft are available. He et al [24] applied the FFD parameterization technique on UAV and Common Research Model (CRM) wings, using 127 and 201 design variables, respectively. In a subsequent study [25], they applied the FFD parameterization method in the optimization of an Ahmed body and of a full model car.…”

Aerostructural wing shape optimization assisted by algorithmic differentiation

“…He et al 27 presented an open-source, object-oriented MDO framework which provides a high-level interface to implement discrete adjoint method for existing or new steadystate primal solvers of OpenFOAM 26 with minimum impact on the source code. To do so, partial derivatives were calculated by finite differences accelerated by an advanced graph coloring algorithm.…”