Boundary and internal conditions for adjoint fluid-flow problems

Abstract

The ever-increasing robustness and reliability of flow-simulation methods have consolidated CFD as a major tool in virtually all branches of fluid mechanics. Traditionally, those methods have played a crucial role in the analysis of flow physics. In more recent years, though, the subject has broadened considerably, with the development of optimization and inverse design applications. Since then, the search for efficient ways to evaluate flow-sensitivity gradients has received the attention of numerous researchers. In this scenario, the adjoint method has emerged as, quite possibly, the most powerful tool for the job, which heightens the need for a clear understanding of its conceptual basis. Yet, some of its underlying aspects are still subject to debate in the literature, despite all the research that has been carried out on the method. Such is the case with the adjoint boundary and internal conditions, in particular. The present work aims to shed more light on that topic, with emphasis on the need for an internal shock condition. By following the path of previous authors, the quasi-1D Euler problem is used as a vehicle to explore those concepts. The results clearly indicate that the behavior of the adjoint solution through a shock wave ultimately depends upon the nature of the objective functional.

Characteristics‐based boundary conditions for the Euler adjoint problem

“…The rationale is analogous to that behind the principle of virtual work, in that the adjoint vector plays the part of a generalized constraint force, which should be normal to any acceptable virtual displacement of the system. The approach is shown to be fully consistent with the requirements for well-posedness of the adjoint equation, as they are determined by the Riemann problem [36][37][38].…”

“…Use was also made of the fact that, once it has been constrained to be realizable, the should also be deemed arbitrary—hence, the need for the last equation. In any case, the procedure is better illustrated with the simple example of supersonic flow .…”

“…where the symbol h o stands for the specific stagnation enthalpy. Equations (38) and (36) represent all the Cauchy data that cross the solid wall. Together, they should form the set of Riemann equations at that boundary.…”

“…Together, they should form the set of Riemann equations at that boundary. The problem with that set is the presence of all four adjoint variables in Equation (38). Had the simplified characteristic equation been rid of 1 and 4 , then one would have a closed set to solve for 2 and 3 .…”

An Extension of Adjoint Method Capabilities

“…The applications, however, also include error analysis [21,22] and grid adaptation [23,24,22,25]. More recently, Volpe and Santos [26] developed a procedure to obtain well-posed adjoint boundary conditions for the quasi 1-D Euler equations, which was later extended to the multidimensional case by Hayashi et al [28].…”