Numerical Approximation of Port-Hamiltonian Systems for Hyperbolic or Parabolic PDEs with Boundary Control

We consider the design of structure-preserving discretization methods for the solution of systems of boundary controlled Partial Differential Equations (PDEs) thanks to the port-Hamiltonian formalism. We first provide a novel general structure of infinite-dimensional port-Hamiltonian systems (pHs) for which the Partitioned Finite Element Method (PFEM) straightforwardly applies. The proposed strategy is applied to abstract multidimensional linear hyperbolic and parabolic systems of PDEs. Then we show that instructional model problems based on the wave equation, Mindlin equation and heat equation fit within this unified framework. Secondly, we introduce the ongoing project SCRIMP (Simulation and Control of Interactions in Multi-Physics) developed for the numerical simulation of infinite-dimensional pHs. SCRIMP notably relies on the FEniCS open-source computing platform for the finite element spatial discretization. Finally, we illustrate how to solve the considered model problems within this framework by carefully explaining the methodology. As additional support, companion interactive Jupyter notebooks are available.

Numerical Analysis of the Tip Leakage Flow Field in a Transonic Axial Compressor with Circumferential Casing Treatment

This paper reports on numerical investigations aimed at understanding the influence of circumferential casing grooves on the tip leakage flow and its resulting vortical structures.The results and conclusions are based on steady state 3D numerical simulations of the well-known transonic axial compressor NASA Rotor 37 near stall operating conditions.The calculations carried out on the casing treatment configuration reveal an important modification of the vortex topology at the rotor tip clearance.Circumferential grooves limit the expansion of the tip leakage vortex in the direction perpendicular to the blade chord,but generate a set of secondary tip leakage vortices due to the interaction with the leakage mass flow.Finally,a deeper investigation of the tip leakage flow is proposed.

Modal Decomposition for the Analysis of the Rotor-stator Interactions in Multi-stage Compressors

A modal analysis method of the rotor-stator interactions in multistage compressors has been developed by LMFA. This method, based on a double modal decomposition of the flow over space and time, has been applied to nu- merical and experimental results of the high-speed 3Y2-stage compressor CREATE based at LMFA, Lyon-France. It reveals the presence of a very strong rotor-stator interaction which completely drives the flow at casing behind all the rotors. This modal analysis method applied to an unsteady RANS simulation permits to calculate the en- ergy of the rotor-stator interactions and to plot energetic meridian maps to explain experimental results and to analyze the interaction in the whole machine.

SECOND-ORDER CONVERGENCE PROPERTIES OF TRUST-REGION METHODS USING INCOMPLETE CURVATURE INFORMATION, WITH AN APPLICATION TO MULTIGRID OPTIMIZATION

Convergence properties of trust-region methods for unconstrained nonconvex optimization is considered in the case where information on the objective function＇s local curvature is incomplete, in the sense that it may be restricted to a fixed set of ＂test directions＂ and may not be available at every iteration. It is shown that convergence to local ＂weak＂ minimizers can still be obtained under some additional but algorithmically realistic conditions. These theoretical results are then applied to recursive multigrid trust-region methods, which suggests a new class of algorithms with guaranteed second-order convergence properties.

Time-Harmonic Acoustic Scattering in a Complex Flow:A Full Coupling Between Acoustics and Hydrodynamics

For the numerical simulation of time harmonic acoustic scattering in a complex geometry,in presence of an arbitrary mean flow,the main difficulty is the coexistence and the coupling of two very different phenomena:acoustic propagation and convection of vortices.We consider a linearized formulation coupling an augmented Galbrun equation(for the perturbation of displacement)with a time harmonic convection equation(for the vortices).We first establish the well-posedness of this time harmonic convection equation in the appropriatemathematical framework.Then the complete problem,with Perfectly Matched Layers at the artificial boundaries,is proved to be coercive+compact,and a hybrid numerical method for the solution is proposed,coupling finite elements for the Galbrun equation and a Discontinuous Galerkin scheme for the convection equation.Finally a 2D numerical result shows the efficiency of the method.

An Unpreconditioned Boundary-Integral for Iterative Solution of Scattering Problems with Non-Constant Leontovitch Impedance Boundary Conditions

This paper concerns the electromagnetic scattering by arbitrary shaped three dimensional imperfectly conducting objects modeled with non-constant Leontovitch impedance boundary condition.It has two objectives.Firstly,the intrinsically wellconditioned integral equation(noted GCSIE)proposed in[30]is described focusing on its discretization.Secondly,we highlight the potential of this method by comparison with two other methods,the first being a two currents formulation in which the impedance condition is implicitly imposed and whose the convergence is quasioptimal for Lipschitz polyhedron,the second being a CFIE-like formulation[14].In particular,we prove that the new approach is less costly in term of CPU time and gives a more accurate solution than that obtained from the CFIE formulation.Finally,as expected,It is demonstrated that no preconditioner is needed for this formulation.

Matched Asymptotic Expansions of the Eigenvalues of a 3-D Boundary-Value Problem Relative to Two Cavities Linked by a Hole of Small Size

In this article,we consider a domain consisting of two cavities linked by a hole of small size.We derive a numerical method to compute an approximation of the eigenvalues of an elliptic operator without refining in the neighborhood of the hole.Several convergence rates are obtained and illustrated by numerical simulations.