Pitfalls of Discrete Adjoint Fixed-Points Based on Algorithmic Differentiation

Abstract This paper presents the development and verification of a discrete adjoint solver using algorithmic differentiation (AD). The computational cost of sensitivity evaluation using the adjoint method is largely independent of the number of design variables, making it attractive for optimization applications where the design variables are far more than objectives and constraints. To obtain the gradients of a single objective function or constraint with respect to many design variables, the nonlinear flow and the adjoint equations need to be solved once at every design cycle. This paper makes a detailed presentation of how AD is used to develop a discrete adjoint solver. The data flow diagrams of the nonlinear flow, linear and adjoint solvers are compared. Moreover, a comparison of convergence history of sensitivity, asymptotic rate of residual convergence and computational cost between the linear and adjoint solvers is also made. Two cases — the subsonic Durham turbine and transonic NASA Rotor 67 are studied in this paper. The results show that the adjoint solver has the same asymptotic rate of residual convergence and produces consistent convergence history of sensitivity as the linear solver, but the adjoint solver consumes more time and memory.

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Discrete adjoint methodology for general multiphysics problems

Structural and Multidisciplinary Optimization ◽

10.1007/s00158-021-03117-5 ◽

2022 ◽

Vol 65 (1) ◽

Author(s):

Ole Burghardt ◽

Pedro Gomes ◽

Tobias Kattmann ◽

Thomas D. Economon ◽

Nicolas R. Gauger ◽

...

Keyword(s):

Shape Sensitivity ◽

Algorithmic Differentiation ◽

Coupled Problem ◽

Reverse Mode ◽

Discrete Adjoint ◽

Numerical Solution Methods ◽

Solution Methods ◽

Design Software ◽

Adjoint Solutions ◽

Multiphysics Problems

AbstractThis article presents a methodology whereby adjoint solutions for partitioned multiphysics problems can be computed efficiently, in a way that is completely independent of the underlying physical sub-problems, the associated numerical solution methods, and the number and type of couplings between them. By applying the reverse mode of algorithmic differentiation to each discipline, and by using a specialized recording strategy, diagonal and cross terms can be evaluated individually, thereby allowing different solution methods for the generic coupled problem (for example block-Jacobi or block-Gauss-Seidel). Based on an implementation in the open-source multiphysics simulation and design software SU2, we demonstrate how the same algorithm can be applied for shape sensitivity analysis on a heat exchanger (conjugate heat transfer), a deforming wing (fluid–structure interaction), and a cooled turbine blade where both effects are simultaneously taken into account.

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