Serge Piperno

Partitioned procedures for the transient solution of coupled aroelastic problems Part I: Model problem, theory and two-dimensional application

Abstract In order to predict the dynamic response of a flexible structure in a fluid flow, the equations of motion of the structure and the fluid must be solved simultaneously. In this paper we present several partitioned procedures for time-integrating this focus coupled problem and discuss their merits in terms of accuracy, stability, heterogeneous computing, I/O transfers, subcycling and parallel processing. All theoretical results are derived for a one-dimensional piston model problem with a compressible flow, because the complete three-dimensional aeroelastic problem is difficult to analyze mathematically. However, the insight gained from the analysis of the coupled piston problem and the conclusions drawn from its numerical investigation are confirmed with the numerical simulation of the two-dimensional transient aeroelastic response of a flexible panel in a transonic non-linear Euler flow regime.

Partitioned procedures for the transient solution of coupled aeroelastic problems. Part II: energy transfer analysis and three-dimensional applications

Abstract We consider the problem of solving large-scale nonlinear dynamic aeroelasticity problems in the time-domain using a fluid/structure partitioned procedure. We present a mathematical framework for assessing some important numerical properties of the chosen partitioned procedure, and predicting its performance for realistic applications. Our analysis framework is based on the estimation of the energy that is artificially introduced at the fluid/structure interface by the staggering process that is inherent to most partitioned solution methods. This framework also suggests alternative approaches for time-discretizing the transfer of aerodynamic data from the fluid subsystem to the structure subsystem that improves the accuracy and the stability properties of the underlying partitioned method. We apply this framework to the analysis of several partitioned procedures that have been previously proposed for the solution of nonlinear transient aeroelastic problems. Using two- and three-dimensional, transonic and supersonic, wing and panel aeroelastic applications, we validate this framework and highlight its impact on the design and selection of a staggering algorithm for the solution of coupled fluid/structure equations.

Explicit/implicit fluid/structure staggered procedures with a structural predictor and fluid subcycling for 2D inviscid aeroelastic simulations

Field time integrators with second-order-accurate numerical schemes for both the fluid and the structure are considered for unsteady Euler aeroelastic computations. We show that if these schemes are simply coupled and used straightforwardly with subcycling, then accuracy and stability properties may be lost. We present new coupling staggered procedures where momentum conservation is enforced at the interface. This is done by using a structural predictor. Continuity of structural and fluid grid displacements is not satisfied at the fluid/structure interface. However, we show on a two-degree-of-freedom aerofoil that this new type of method has many advantages, e.g. accuracy of conservation at the interface and extended stability. The supersonic flutter of a flat panel is simulated in order to numerically prove that the algorithm gives accurate results with arbitrary subcycling for the fluid in the satisfying limit of 30 time steps per period of coupled oscillation.

A Nondiffusive Finite Volume Scheme for the Three-Dimensional Maxwell's Equations on Unstructured Meshes

We prove a sufficient CFL-like condition for the L2-stability of the second-order accurate finite volume scheme proposed by Remaki for the time-domain solution of Maxwells equations in heterogeneous media with metallic and absorbing boundary conditions. We yield a very general sufficient condition valid for any finite volume partition in two and three space dimensions. Numerical tests show the potential of this original finite volume scheme in one, two, and three space dimensions for the numerical solution of Maxwells equations in the time-domain.

Discontinuous Galerkin time‐domain solution of Maxwell's equations on locally‐refined nonconforming Cartesian grids

Purpose – The use of the prominent FDTD method for the time domain solution of electromagnetic wave propagation past devices with small geometrical details can require very fine grids and can lead to very important computational time and storage. The purpose is to develop a numerical method able to handle possibly non‐conforming locally refined grids, based on portions of Cartesian grids in order to use existing pre‐ and post‐processing tools.Design/methodology/approach – A Discontinuous Galerkin method is built based on bricks and its stability, accuracy and efficiency are proved.Findings – It is found to be possible to conserve exactly the electromagnetic energy and weakly preserves the divergence of the fields (on conforming grids). For non‐conforming grids, the local sets of basis functions are enriched at subgrid interfaces in order to get rid of possible spurious wave reflections.Research limitations/implications – Although the dispersion analysis is incomplete, the numerical results are really enco...

TIME-DOMAIN PARALLEL SIMULATION OF HETEROGENEOUS WAVE PROPAGATION ON UNSTRUCTURED GRIDS USING EXPLICIT, NONDIFFUSIVE, DISCONTINUOUS GALERKIN METHODS

A general Discontinuous Galerkin framework is introduced for symmetric systems of conservations laws. It is applied to the three-dimensional electromagnetic wave propagation in heterogeneous media, and to the propagation of aeroacoustic perturbations of either uniform or nonuniform, steady solutions of the three-dimensional Euler equations. In all these linear contexts, the time evolution of some quadratic wave energy is given in a balance equation, with a volumic source term for aeroacoustics in a nonuniform flow. An explicit leap-frog time scheme along with centered numerical fluxes are used in the proposed Discontinuous Galerkin Time Domain (DGTD) method, in order to achieve a discrete equivalent of the balance equation for the wave energy. The scheme introduced is genuinely nondissipative. Numerical first-order boundary conditions are developed to bound the domain and stability is proved on arbitrary unstructured meshes and discontinuous finite elements, under some CFL-like stability condition on the time step. Numerical results obtained with a parallel implementation of the method based on mesh partitioning and message passing are presented to show the potential of the method.

Criteria for the design of limiters yielding efficient high resolution TVD schemes

Abstract High resolution, finite-volume, conservative schemes, based on the classical MUSCL (Monotonic Upwind Schemes for Conservation Laws) construction and non-linear interpolation limiters are considered. Possible criteria are established for the construction of the limiters, yielding monotonic and efficient schemes. For scalar hyperbolic conservation laws, new limiters for both upwind and centred numerical flux functions are proposed and compared with existing limiters. These limiters are also compared in the case of the computation of the two-dimensional inviscid flow around a NACA0012 airfoil with particular attention to the issues of iterative convergence to steady state and monotonicity preservation of the computed solution. We actually show that the proposed criteria do control these issues.

Design of Efficient Partitioned Procedures for the Transient Solution of Aeroelastic Problems

ABSTRACT We consider the problem of solving large-scale non-linear dynamic aeroelasticity problems in the time-domain using a fluid-structure partitioned procedure. We present a mathematical framework for assessing some important numerical properties of the chosen partitioned procedure, and predicting its performance for realistic applications. Our analysis framework is based on the estimation of the energy that is artificially introduced at the fluid-structure interface by the staggering process that is inherent to most partitioned solution methods. This framework provides a powerful means for the construction of more accurate and stable partitioned methods. Using two- and three-dimensional, transonic and supersonic, wing and panel aeroelastic applications, we validate this framework and highlight its impact on the design and selection of a staggering algorithm for the solution of coupled fluid-structure equations.

A DISSIPATION-FREE TIME-DOMAIN DISCONTINUOUS GALERKIN METHOD APPLIED TO THREE-DIMENSIONAL LINEARIZED EULER EQUATIONS AROUND A STEADY-STATE NON-UNIFORM INVISCID FLOW

We present in this paper a time-domain discontinuous Galerkin dissipation-free method for the transient solution of the three-dimensional linearized Euler equations around a steady-state solution. In the general context of a nonuniform supporting flow, we prove, using the well-known symmetrization of Euler equations, that some aeroacoustic energy satisfies a balance equation with source term at the continuous level, and that our numerical framework satisfies an equivalent balance equation at the discrete level and is genuinely dissipation-free. In the case of ℙ1 Lagrange basis functions and tetrahedral unstructured meshes, a parallel implementation of the method has been developed, based on message passing and mesh partitioning. Three-dimensional numerical results confirm the theoretical properties of the method. They include test-cases where Kelvin–Helmholtz instabilities appear.

Comparison Of A Stochastic Particle Method And A Finite Volume Deterministic Method Applied To Burgers Equation

The aim of this work is to compare two very different numerical methods, on a simple model, the Burgers equation. The first method is a stochastic particle method. In this approach, the solution of the Burgers equation is interpreted as the cumulative distribution function of the law of a stochastic process. This law satisfies a PDE of McKean-Vlasov type. The second method is based on a weak formulation of the Burgers equation considered as a conservation law satisfied on each part of the computational domain called cell or finite volume. We compare these two methods for viscous and in viscid test cases.