Fluid Dynamics Colloquium Grid-free Methods for Compressible Flows Praveen. C ∗ ARDB CFD Center Dept. of Aerospace Engg., IISc Bangalore-12 Research supervisor: Prof. S M Deshpande Dr. A K Ghosh (ADA) ∗ CTFD Division, NAL, Bangalore-17 email: cpravn@gmail.com November 9, 2005 ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
Outline of presentation 1. Why meshless methods ? 2. Kinetic schemes 3. FAME mesh and connectivity generation 4. Tests for connectivity 5. 3-D results with q-LSKUM on FAME mesh 6. Kinetic Meshless Method 7. 2-D results with KMM ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
� �✁ ✁ Why meshless methods ? • Grid generation - complex and time-consuming • 3-D configurations - several hundred man-hours • Large deformations - moving bodies, extrusion, fragmentation, moving interfaces • Meshless methods - do not require a mesh - only point distributions and local connectivity ✆✄✆✄✆✄✆✄✆✄✆ ✂✄✂ ✝✄✝✄✝✄✝✄✝✄✝ ☎✄☎ ✝✄✝✄✝✄✝✄✝✄✝ ✆✄✆✄✆✄✆✄✆✄✆ ☎✄☎ ✂✄✂ P ✝✄✝✄✝✄✝✄✝✄✝ ✆✄✆✄✆✄✆✄✆✄✆ ✂✄✂ ☎✄☎ o ✆✄✆✄✆✄✆✄✆✄✆ ✝✄✝✄✝✄✝✄✝✄✝ ☎✄☎ ✂✄✂ ✝✄✝✄✝✄✝✄✝✄✝ ✆✄✆✄✆✄✆✄✆✄✆ ☎✄☎ ✂✄✂ ✝✄✝✄✝✄✝✄✝✄✝ ✆✄✆✄✆✄✆✄✆✄✆ ☎✄☎ ✂✄✂ ☎✄☎ ✂✄✂ ☎✄☎ ✂✄✂ Connectivity ☎✄☎ ✂✄✂ ☎✄☎ ✂✄✂ C o Point distribution Connectivity ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
Why meshless methods ? • Grid generation - complex and time-consuming • 3-D configurations - several hundred man-hours • Large deformations - moving bodies, extrusion, fragmentation • Meshless methods - do not require a mesh - only point distributions and local connectivity 1. Any type of grid or combination of grids (FAME) can be used. Different topologies can be combined (Ramesh, Anandhanarayanan) - chimera clouds - few 10s of man-hours (Anandhanarayanan) 2. Easier to generate point distributions - fast and automatic point generation - L¨ ohner et al. - advancing front point generation - Varma et al. - quadtree/octree point generation 3. Simple data structure - ease of adaptation - add points wherever required 4. Include a priori knowledge about local behaviour of solution 5. Less sensitive to quality of point distributions 6. Well suited for moving boundary problems ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
Meshless methods Collocation methods Integral methods LSKUM - Ghosh and Deshpande DEM - Nayroles Batina EFG - Belytschko et al. Morinishi PUFEM - Duarte and Oden LSFDU - Balakrishnan et al. hp-clouds - Babuska and Melenk FPM - L¨ ohner et al. FVPM - Heitel, Junk et al. Simple and easy to implement Requires numerical quadrature Truly meshless Background mesh required Fluid problems Structural mechanics ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
Meshless methods Collocation methods Integral methods LSKUM - Ghosh and Deshpande DEM - Nayroles Batina EFG - Belytschko et al. Morinishi PUFEM - Duarte and Oden LSFDU - Balakrishnan et al. hp-clouds - Babuska and Melenk FPM - L¨ ohner et al. FVPM - Heitel, Junk et al. Simple and easy to implement Requires numerical quadrature Truly meshless Background mesh required Fluid problems Structural mechanics LSKUM - Least Squares Kinetic Upwind Method (1989) • meshless, kinetic, upwind scheme • subsonic, transonic, supersonic, hypersonic flows • higher order scheme (Ghosh and Deshpande, Dauhoo et al.) • 2-D moving body (Ramesh, Sachin, Chandrashekar) • 3-D complex configurations (Ramesh, Anandhanarayanan, Mahendra et al.) • 2-D laminar viscous flows (Mahendra et al., Anandhanarayanan) ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
More applications of meshless methods • Structural mechanics – Extrusion and molding - large mesh deformation – Casting - propagation of interfaces between solids and liquids – Failure processes - propagation of cracks • Airbag simulation (Kuhnert et al.) - fluid-structure interaction • Simulation of high explosive detonation, blast propagation, and shock wave diffraction for determining and assessing target vulnerability and weapon lethality (Lohner et al.) – Hybrid approach - Cartesian + gridless – Overall speed-up of 8, and – memory savings of one order compared to unstructured grid method – Accuracy comparable to unstructured grid method ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
Basic Idea • Assume a function variation f h = f o + a ( x − x o ) + b ( y − y o ) • Fit ( a, b ) using least squares • Use f h as a shape function, or • Estimate derivatives ∂f ∂x ≈ ∂f h ∂f ∂y ≈ ∂f h ∂x = a, ∂y = b ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
Full stencil ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
Split stencil ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
Euler equations • System of hyperbolic conservation laws - mass, momentum and energy ∂U ∂t + ∂ X + ∂ ∂yGY + ∂ ∂xG ∂zG Z = 0 • Vector of conserved variables ρ ρu x ρ = Density U = ρu y where u x , u y , u z = Fluid velocity ρu z E = Energy per unit volume E • Flux vectors ρu x ρu y ρu z p + ρu 2 ρu x u y ρu x u z x p + ρu 2 G X = ρu x u y , GY = , G Z = ρu y u z y p + ρu 2 ρu x u z ρu y u z z ( E + p ) u x ( E + p ) u y ( E + p ) u z ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
Least Squares Kinetic Upwind Method ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
Kinetic schemes • Exploit connection between Boltzmann and Euler/Navier-Stokes equations ∂F ∂t + v∂F ∂x = 0 ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
Kinetic schemes • Exploit connection between Boltzmann and Euler/Navier-Stokes equations ∂F ∂t + v∂F ∂x = 0 = ⇒ F ( x, t + ∆ t ) = F ( x − v ∆ t, t ) ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
Kinetic schemes • Exploit connection between Boltzmann and Euler/Navier-Stokes equations ∂F ∂t + v∂F ∂x = 0 = ⇒ F ( x, t + ∆ t ) = F ( x − v ∆ t, t ) • Stencil for LSKUM • Semi-discrete LSKUM approximation � ∂F � � ∂F � d F o d t + v + | v | + v − | v | = 0 2 ∂x 2 ∂x C 1 C 2 � �� � � �� � v ≥ 0 v ≤ 0 ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
• Moments of discretized BE � ∂G + � � ∂G − � d U o d t + + = 0 ∂x ∂x C 1 C 2 • In 3-D we need six split stencils (two per axis) � ∂ � ∂ � � d U o X + X − + ∂xG + ∂xG d t ∆ x ≤ 0 ∆ x ≥ 0 � ∂ � ∂ � � ∂yGY + ∂yGY − + + ∆ y ≤ 0 ∆ y ≥ 0 � ∂ � ∂ � � Z + Z − + ∂zG + ∂zG = 0 ∆ z ≤ 0 ∆ z ≥ 0 • Integrate the ODE system in time using a Runge-Kutta scheme • Robust scheme – Entropy consistent – Positivity preservation ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
FAME mesh and connectivity generation ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
FAME mesh • FAME - Feature Associated Mesh Embedding (DERA, UK, now called QinetiQ) • Multiple overlapping meshes - body-fitted mesh around simple parts and + background Cartesian mesh • Cartesian mesh adapted based on local features - geometry and body-fitted mesh • Local body-fitted mesh - 4 to 8 layers in the wall normal direction • Extra meshes to resolve sharp geometrical features - wing-fuselage intersection, trailing-edge, wing-tip ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
LSKUM + FAME • Store separation problem - multiple bodies in relative motion • Moving grids: Refine, coarsen and blank • Interpolation in overlapping regions • Grid-free method better suited for this application - No interpolation required - consistent update at all points • Collaborative project between QinetiQ, ADA and CFD Center • Phase I - demonstrate LSKUM on stationary FAME mesh ARDB CFD Center, Dept of Aerospace Engg., IISc First Prev Next Last Go Back Full Screen Close Quit
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