h p refinement for linearized Euler equations solved by a Galerkin discontinuous method
21
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Découvre YouScribe en t'inscrivant gratuitement
Découvre YouScribe en t'inscrivant gratuitement
21
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
- discontinuous galerkin
- software developped
- cpu memory
- physical problem
- mesh generation
- softwares used
- cartesian mesh
- finite difference
h-p Refinement for the Euler Equations
Solved by Discontinuous Galerkin Method
DSNA / PS3A
C. Peyret, M. Borrel, Ph. Delorme, O. Labbé, R. Léger, G. RahierAcknowledgments
Jacques Peter (DSNA/NUMF)
Frédéric Alauzet, Paul-Louis Georges, Loïc Maréchal (INRIA)
Pascal Frey (UPMC - Lab. J-L Lions)
Eric Quemerais (DSNA/ELCI)
2Softwares Used
yams + meshMet + tetMesh + medit (INRIA)
Mesh generation, adaptation and visualization
Space (ONERA)
Solver based on Discontinuous Galerkin Method
CAA
Flu3M (ONERA)
Solver based on Finite Volume Method
LES
Fidel (ONERA) - Software developped for research purpose
Solver based on Finite Difference Method + Cartesian Mesh
CAA
Cwipi (ONERA)
Coupling of Parallel Computations
3The Challenge
Solving a Physical Problem
Precision
Physical model
Numerical prec.
Cost Flexibility
ManPower Geometries
CPU Memory Configurations
4The Challenge
Solving a Physical Problem
Precision
Cost Flexibility
Find Concepts that give good Precision for low Cost and high Flexibility
5Physical Modeling and Mathematical formulation
Discontinuous Galerkin Method
Linearized Euler’s Equations u1
ϕ= v1
a ρ /ρSymetric Friedrich System 0 1 0
∂ ϕ+A ∂ ϕ+Bϕ=0t i i
Matrix is symetricA ∂i i
Variational formulation is diagonalizableA ni i
k k + −ϕ ∈W (ω ) |∀ψ ∈W (ω ); L(ϕ ,ψ ) = 0 A n = [A n ] +[A .n ]h hh h h h i i i i i i
L(ϕ ,ψ ) = ψ . ∂ ϕ + ψ .A ∂ ϕ + ψ .Bϕt i ih h h h h h h h
Ω Ω Ω
− o i+ ψ .[A .n ] (ϕ −ϕ )+ ψ .(Mϕ −g)− ψ .gi ih h h h h h
∂ω ∩∂Ω Ω∂ω |∂Ω hh
Fully Upwind
Scheme
6h Refinement for CAA
hp Refinement for CAA
Coupling of Methods for CAA
CFD-CAA Coupling
INRIA Mesh Tools•
yams•
Space Solver•
•Discontinuous Galerkin Method
•P1 Order
•Metric Calculator
•Linearized Euler Equationsh Refinement for CAA
Putting cells where and when it is needed to reduce mesh size
Acoustic pressure field
Unstructured meshes•
Localized energy•
Illustration of adaptative mesh
Propagation of sound generated by two impulsive monopolar sources
8h Refinement for CAA
hp Refinement for CAA
Coupling of Methods for CAA
CFD-CAA Coupling
INRIA Mesh Tools•
yams•
Space Solver•
•Discontinuous Galerkin Method
•Order Auto Adaptation
•Euler Linearized Equationshp Refinement for CAA
Introducing DG higher orders to reduce mesh size
Step 1 : Mesh adaptation on CFD data to reduce the number of cells•
Step 2 : Order adaptation to keep a uniform precision for CAA•
Step 1 Step 2
2D CFD mesh (8512 quads) CAA Order map computed for CFD mesh (P2)
P0 P1 P2 P3 P4 P5 P6
P0 P1 P2 P3 P4 P5 P6
2D Adapted mesh (3729 triangles) CAA Order map computed for adapted mesh (P1-P6)
10
