Adjoint-based optimization for hyperbolic balance laws in the presence of discontinuities

Adjoint;Optimization;Hyperbolic;Conservation laws;Interface;Thinc

Fikl, Alexandru ; Sayadi, Taraneh ; Le Chenadec ; Vincent

关键词: Adjoint;    Optimization;    Hyperbolic;    Conservation laws;    Interface;    Thinc;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/95395/FIKL-THESIS-2016.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

In this thesis, we are interested in optimization in multiphase flows using discrete adjoint-based methods. Themain issues we will endeavor to study are the impact on the linearized andadjoint equations of discontinuous solutions, the behavior of the THINC-family of numerical schemes under linearization and the rigorous formulation of an optimization problem when the interface is represented by a Heaviside marker function.We will study the two main discontinuous wave patterns that appear in hyperbolic balance laws: contact discontinuities and shocks. These two have very different characteristics in the forward problem and we will see that they require different treatments for the adjoint formulations as well. Specifically, shocks require additional information, in the form of the shock location, to properly define first-order variations necessary for gradient methods. Another important aspect in the numerical treatment of multiphase flows is the use of an anti-diffusive numerical scheme for the interface advection equation. The scheme we will investigate is the fairly recent THINC scheme, because it is one of the few numerical schemes in the field that is differentiable (with respect to the volume fraction). However, we will need to extend the classic formulation of the THINC scheme to force it to behave correctly in the linearized and adjoint regime. Even with these extensions, we will see that the adjoint converges everywhere except at the interface, due to the discontinuity. Finally, we define a simplified optimization problem for the volume fraction, where the velocity is given by a velocity potential instead of the Navier-Stokes equations. This allows us to specifically study the formulation of the cost functional and the viability of using a Heaviside function as a representation of the interface (as opposed to e.g. level set methods). We will show that such a formulation is indeed possible and leads to well-posed optimization problems.