A variational framework for flow optimization using semi-norm constraints

When considering a general system of equations describing the space-time evolution (flow) of one or several variables, the problem of the optimization over a finite period of time of a measure of the state variable at the final time is a problem of great interest in many fields. Methods already exist in order to solve this kind of optimization problem, but sometimes fail when the constraint bounding the state vector at the initial time is not a norm, meaning that some part of the state vector remains unbounded and might cause the optimization procedure to diverge. In order to regularize this problem, we propose a general method which extends the existing optimization framework in a self-consistent manner. We first derive this framework extension, and then apply it to a problem of interest. Our demonstration problem considers the transient stability properties of a one-dimensional (in space) averaged turbulent model with a space- and time-dependent model "turbulent viscosity". We believe this work has a lot of potential applications in the fluid dynamics domain for problems in which we want to control the influence of separate components of the state vector in the optimization process.Comment: 30 page

Progressive construction of a parametric reduced-order model for PDE-constrained optimization

An adaptive approach to using reduced-order models as surrogates in PDE-constrained optimization is introduced that breaks the traditional offline-online framework of model order reduction. A sequence of optimization problems constrained by a given Reduced-Order Model (ROM) is defined with the goal of converging to the solution of a given PDE-constrained optimization problem. For each reduced optimization problem, the constraining ROM is trained from sampling the High-Dimensional Model (HDM) at the solution of some of the previous problems in the sequence. The reduced optimization problems are equipped with a nonlinear trust-region based on a residual error indicator to keep the optimization trajectory in a region of the parameter space where the ROM is accurate. A technique for incorporating sensitivities into a Reduced-Order Basis (ROB) is also presented, along with a methodology for computing sensitivities of the reduced-order model that minimizes the distance to the corresponding HDM sensitivity, in a suitable norm. The proposed reduced optimization framework is applied to subsonic aerodynamic shape optimization and shown to reduce the number of queries to the HDM by a factor of 4-5, compared to the optimization problem solved using only the HDM, with errors in the optimal solution far less than 0.1%

Hamiltonian discontinuous Galerkin FEM for linear, rotating incompressible Euler equations: inertial waves

A discontinuous Galerkin finite element method (DGFEM) has been developed and tested for linear, three-dimensional, rotating incompressible Euler equations. These equations admit complicated wave solutions. The numerical challenges concern: (i) discretisation of a divergence-free velocity field; (ii) discretisation of geostrophic boundary conditions combined with no-normal flow at solid walls; (iii) discretisation of the conserved, Hamiltonian dynamics of the inertial-waves; and, (iv) large-scale computational demands owing to the three-dimensional nature of inertial-wave dynamics and possibly its narrow zones of chaotic attraction. These issues have been resolved: (i) by employing Dirac’s method of constrained Hamiltonian dynamics to our DGFEM for linear, compressible flows, thus enforcing the incompressibility constraints; (ii) by enforcing no-normal flow at solid walls in a weak form and geostrophic tangential flow —along the wall; (iii) by applying a symplectic time discretisation; and, (iv) by combining PETSc’s linear algebra routines with our high-level software. We compared our simulations with exact solutions of three-dimensional compressible and incompressible flows, in (non)rotating periodic and partly periodic cuboids (Poincar´e waves). Additional verifications concerned semi-analytical eigenmode solutions in rotating cuboids with solid walls

Kinematics of fluid particles on the sea surface. Hamiltonian theory

We derive the John-Sclavounos equations describing the motion of a fluid particle on the sea surface from first principles using Lagrangian and Hamiltonian formalisms applied to the motion of a frictionless particle constrained on an unsteady surface. The main result is that vorticity generated on a stress-free surface vanishes at a wave crest when the horizontal particle velocity equals the crest propagation speed, which is the kinematic criterion for wave breaking. If this holds for the largest crest, then the symplectic two-form associated with the Hamiltonian dynamics reduces instantaneously to that associated with the motion of a particle in free flight, as if the surface did not exist. Further, exploiting the conservation of the Hamiltonian function for steady surfaces and traveling waves we show that particle velocities remain bounded at all times, ruling out the possibility of the finite-time blowup of solutions