Lagrangian Coordinates ∂ X ∂ t ( x , t ) = u ( X ( x , t ) , t ) , X ( x , t = 0) = x D t a = ( ∂ ˜ a /∂ t ) X , ( ∂ a /∂ x ) t = ( � ρ | t =0 / � ρ )( ∂ ˜ a /∂ X ) t The system reads (for regular solutions) Euler Coord. Lagrange Coord. a ( x , t ) ← → � a ( X , t ) D t z = 0 , D t y = 0 z t = 0 , � � y t = 0 1 /ρ − u + − � ρ D t u P = 0 u 1 / � ρ ( L ) � e Pu � ρ t =0 � u + P =0 x � e � P � u D t · = ∂ t · + u ∂ x · t X KOKH (CEA) Int MultiMat Fl. Guidel May 2011 20 / 110
Lagrange-Remap Process We only consider a single timestep: t n → t n +1 . Final Euler Variable: Map the Euler variable Update the Lagrange Resample the solution onto the Lagrange Variable over the original mesh variable. a n +1 � ∼ � a j a n a n a n +1 j = � j j j a n +1 a n +1 a n +1 j − 1 j +1 j Remapping Step: Compute the Euler variable at time t n +1 � � � a j − 1 a j a j +1 Lagrange Step: ∆ t solve system (L) a n a n a n t n → t n +1 j − 1 j j +1 ∆ x KOKH (CEA) Int MultiMat Fl. Guidel May 2011 21 / 110
Outline The Numerical Scheme 3 Lagrange Step Remapping Step: General Structure Remapping Step: Optimizing the Numerical Diffusion KOKH (CEA) Int MultiMat Fl. Guidel May 2011 22 / 110
Scheme for the Lagrange Step Acoustic Scheme (Despr´ es) / Suliciu-Type Relaxation Scheme ρ j − 1 /ρ n 1 / � − u j +1 / 2 + u j − 1 / 2 j + λ = 0 u j − u n � P j +1 / 2 − P j − 1 / 2 j ρ n e j − e n � j P j +1 / 2 u j +1 / 2 − P j − 1 / 2 u j − 1 / 2 j z j = z n y j = y n � � j , j , λ = ∆ t / ∆ x Flux Formulas This step � preserves max[ ρ n j ( c n j ) 2 , ρ n j +1 ( c n j +1 ) 2 ] min( ρ n j , ρ n ( ρ c ) ∗ j +1 / 2 = j +1 ) ( P , u )- P j +1 / 2 = 1 j +1 ) − 1 2( P n j + P n j +1 / 2 ( u n j +1 − u n constant 2( ρ c ) ∗ j ) profiles. u j +1 / 2 = 1 j +1 ) − 1 1 2( u n j + u n ( P n j +1 − P n j ) 2 ( ρ c ) ∗ j +1 / 2 KOKH (CEA) Int MultiMat Fl. Guidel May 2011 23 / 110
CFL Condition In the sequel we shall suppose that λ = ∆ t / ∆ x verifies � � | u j +1 / 2 | , ( ρ c ) ∗ j +1 / 2 / min( ρ n j , ρ n λ × max j +1 ) ≤ C ( ⋆ ) j ∈ Z where usually C ≃ 0 . 8. (numerical of the present talks performed with C = 0 . 999) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 24 / 110
Scheme for the Remapping Step General Form ρ W = ( ρ y , ρ, ρ u , ρ e ) T � � u j +1 / 2 � ( ρ W ) j +1 / 2 − u j − 1 / 2 � ρ n +1 W n +1 − � ρ j W j + λ ( ρ W ) j − 1 / 2 j j − λ � ( ρ W ) j ( u j +1 / 2 − u j − 1 / 2 ) = 0 ( z n +1 − z n z j − 1 / 2 ) − λ z n j ) + λ ( u j +1 / 2 � z j +1 / 2 − u j − 1 / 2 � j ( u j +1 / 2 − u j − 1 / 2 ) = 0 j Building the scheme boils down to specify the following terms � y j +1 / 2 , � ρ j +1 / 2 , u j +1 / 2 , � ε j +1 / 2 , � � z j +1 / 2 KOKH (CEA) Int MultiMat Fl. Guidel May 2011 25 / 110
Defining the Fluxes Enforce consistency for z j +1 / 2 =? � y j +1 / 2 , � � ρ j +1 / 2 , � ε j +1 / 2 . u j +1 / 2 =? � Upwind choice ( j + 1 / 2 = upw ) � y j +1 / 2 =? according to the sign of u j +1 / 2 for ρ j +1 / 2 =? � phasic quantities ρ 0 , ρ 1 and ρ 0 ε 0 , ρ 1 ε 1 . � ( ρε ) j +1 / 2 =? Upwind choice too for u KOKH (CEA) Int MultiMat Fl. Guidel May 2011 26 / 110
Defining the Fluxes Enforce consistency for z j +1 / 2 =? � y j +1 / 2 , � � ρ j +1 / 2 , ε j +1 / 2 . � � u j +1 / 2 =? Upwind choice ( j + 1 / 2 = upw ) z j +1 / 2 � � y j +1 / 2 = � ( ρ 1 ) j +1 / 2 / � ρ j +1 / 2 according to the sign of u j +1 / 2 for z j +1 / 2 � z j +1 / 2 ) � ρ j +1 / 2 = � � phasic quantities ( ρ 1 ) j +1 / 2 + (1 − � ( ρ 0 ) j +1 / 2 ρ 0 , ρ 1 and ρ 0 ε 0 , ρ 1 ε 1 . � z j +1 / 2 � z j +1 / 2 ) � ( ρε ) j +1 / 2 = � ( ρ 1 ε 1 ) j +1 / 2 + (1 − � ( ρ 0 ε 0 ) j +1 / 2 Upwind choice too for u KOKH (CEA) Int MultiMat Fl. Guidel May 2011 26 / 110
Defining the Fluxes Enforce consistency for z j +1 / 2 =? � y j +1 / 2 , � � ρ j +1 / 2 , ε j +1 / 2 . � � u j +1 / 2 =? Upwind choice ( j + 1 / 2 = upw ) z j +1 / 2 � � y j +1 / 2 = � ( ρ 1 ) upw / � ρ j +1 / 2 according to the sign of u j +1 / 2 for z j +1 / 2 � z j +1 / 2 ) � ρ j +1 / 2 = � � phasic quantities ( ρ 1 ) upw + (1 − � ( ρ 0 ) upw ρ 0 , ρ 1 and ρ 0 ε 0 , ρ 1 ε 1 . � z j +1 / 2 � z j +1 / 2 ) � ( ρε ) j +1 / 2 = � ( ρ 1 ε 1 ) upw + (1 − � ( ρ 0 ε 0 ) upw Upwind choice too for u KOKH (CEA) Int MultiMat Fl. Guidel May 2011 26 / 110
Defining the Fluxes Enforce consistency for z j +1 / 2 =? � � y j +1 / 2 , � ρ j +1 / 2 , � ε j +1 / 2 . u j +1 / 2 = � � u upw Upwind choice ( j + 1 / 2 = upw ) z j +1 / 2 � � y j +1 / 2 = � ( ρ 1 ) upw / � ρ j +1 / 2 according to the sign of u j +1 / 2 for z j +1 / 2 � z j +1 / 2 ) � ρ j +1 / 2 = � � phasic quantities ( ρ 1 ) upw + (1 − � ( ρ 0 ) upw ρ 0 , ρ 1 and ρ 0 ε 0 , ρ 1 ε 1 . � z j +1 / 2 � z j +1 / 2 ) � ( ρε ) j +1 / 2 = � ( ρ 1 ε 1 ) upw + (1 − � ( ρ 0 ε 0 ) upw Upwind choice too for u KOKH (CEA) Int MultiMat Fl. Guidel May 2011 26 / 110
Defining the Fluxes Enforce consistency for z j +1 / 2 =? � � y j +1 / 2 , � ρ j +1 / 2 , � ε j +1 / 2 . u j +1 / 2 = � � u upw Upwind choice ( j + 1 / 2 = upw ) z j +1 / 2 � � y j +1 / 2 = � ( ρ 1 ) upw / � ρ j +1 / 2 according to the sign of u j +1 / 2 for z j +1 / 2 � z j +1 / 2 ) � ρ j +1 / 2 = � � phasic quantities ( ρ 1 ) upw + (1 − � ( ρ 0 ) upw ρ 0 , ρ 1 and ρ 0 ε 0 , ρ 1 ε 1 . � z j +1 / 2 � z j +1 / 2 ) � ( ρε ) j +1 / 2 = � ( ρ 1 ε 1 ) upw + (1 − � ( ρ 0 ε 0 ) upw Upwind choice too for u KOKH (CEA) Int MultiMat Fl. Guidel May 2011 26 / 110
Constraints for the flux � z j +1 / 2 Suppose u j − 1 / 2 > 0 and u j +1 / 2 > 0. Under CFL condition ( ⋆ ), we have a “trust interval” for � z j +1 / 2 that ensures stability for both y and z , and consistency for both fluxes � z j +1 / 2 and � y j +1 / 2 . stability for z n +1 stability for y n +1 j j consistency for � z j +1 / 2 consistency for � y j +1 / 2 � � � � � � � � � � � z j +1 / 2 ∈ � m j − 1 / 2 , M j − 1 / 2 a j , A j b j , B j d j +1 / 2 , D j +1 / 2 � = ∅ � �� � ∈ z n j KOKH (CEA) Int MultiMat Fl. Guidel May 2011 27 / 110
Constraints for the flux � z j +1 / 2 Suppose u j − 1 / 2 > 0 and u j +1 / 2 > 0. Under CFL condition ( ⋆ ), we have a “trust interval” for � z j +1 / 2 that ensures stability for both y and z , and consistency for both fluxes � z j +1 / 2 and � y j +1 / 2 . stability for z n +1 stability for y n +1 j j consistency for � z j +1 / 2 consistency for � y j +1 / 2 � � � � � � � � � � � z j +1 / 2 ∈ � m j − 1 / 2 , M j − 1 / 2 a j , A j b j , B j d j +1 / 2 , D j +1 / 2 � = ∅ � �� � ∈ z n j KOKH (CEA) Int MultiMat Fl. Guidel May 2011 27 / 110
Constraints for the flux � z j +1 / 2 Suppose u j − 1 / 2 > 0 and u j +1 / 2 > 0. Under CFL condition ( ⋆ ), we have a “trust interval” for � z j +1 / 2 that ensures stability for both y and z , and consistency for both fluxes � z j +1 / 2 and � y j +1 / 2 . stability for z n +1 stability for y n +1 j j consistency for � z j +1 / 2 consistency for � y j +1 / 2 � � � � � � � � � � � � z j +1 / 2 ∈ m j − 1 / 2 , M j − 1 / 2 a j , A j b j , B j d j +1 / 2 , D j +1 / 2 � = ∅ � �� � ∈ z n j How should we choose � z j +1 / 2 ? KOKH (CEA) Int MultiMat Fl. Guidel May 2011 27 / 110
Choosing the z -flux � z j +1 / 2 (1) Suppose u j − 1 / 2 > 0 and u j +1 / 2 > 0. Despr´ es-Lagouti` ere Strategy: Limited Downwind Strategy We choose the most downwinded possible value for � z j +1 / 2 such that � ��� ��� ��� � � z j +1 / 2 ∈ m j − 1 / 2 , M j − 1 / 2 a j , A j b j , B j d j +1 / 2 , D j +1 / 2 � z j +1 / 2 values axis z n j +1 Choosing the � z j +1 / 2 value is z j +1 / 2 � an explicit step: no CPU cost z n j +1 is needed, nor any recursive procedure. � z j +1 / 2 z n j +1 z j +1 / 2 � KOKH (CEA) Int MultiMat Fl. Guidel May 2011 28 / 110
Choosing the flux � z j +1 / 2 (2) What if u j − 1 / 2 < 0 and u j +1 / 2 > 0 ? For sake of “security” we opt for the upwind choice, i.e. z j +1 / 2 = z n � j Other cases The cases u j − 1 / 2 > 0, u j +1 / 2 < 0 and u j − 1 / 2 < 0, u j +1 / 2 < 0 can be examined following the same lines and provide similar formulas for the flux � z j +1 / 2 . KOKH (CEA) Int MultiMat Fl. Guidel May 2011 29 / 110
Outline Numerical Results 4 1D Interface Advection Sod-Type Shock Tube 2D Air-R22 Shock-Interface Interaction Kelvin-Helmholtz Instability KOKH (CEA) Int MultiMat Fl. Guidel May 2011 30 / 110
1D Interface Advection(1) Test Description Advection of a 1D “bubble” (pulse) involving two fluids Inner bubble state Outter bubble State Analytical EOS Tabulated EOS ρ = 10 3 , P = 10 5 , u = 10 3 P = 10 5 , u = 10 3 ρ = 50 , 1 m long domain discretized over a 100-cell mesh Periodic boundary conditions t = 3 . 0 s (1 524 000 time steps) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 31 / 110
1D Interface Advection(2) EOS for the Inner Bubble Fluid (Stiffened Gas Equation) π 1 = 6 × 10 8 Pa P = ( γ 1 − 1) ρε − γ 1 π 1 , γ 1 = 4 . 4 , EOS for the Outter Bubble Fluid (Tabulated van der Waals Fluid) � γ 0 − 1 � ( ρε + a 0 ρ 2 ) − a 0 ρ 2 , γ 0 = 1 . 4 , b 0 = 10 − 3 , a 0 = 5 . P = 1 − b 0 ρ Discretization of the ( ρ, P ) ∈ [0 , 990] × [10 4 , 10 9 ] over a uniform 10 3 × 10 3 grid. ( ρ, P ) �→ ρε is provided thanks to a Q 1 interpolation ( ρ, ε ) �→ P computed with a Newton method P 1 = P 0 resolved with a Dichotomy Algorithm KOKH (CEA) Int MultiMat Fl. Guidel May 2011 32 / 110
1D Interface Advection(3): Initial State color function (discretized) color function (exact) 1 0.8 Color Function 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 x(m) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 33 / 110
Interface Advection (4): Color Function at t = 0 . 01 s (step 5080) upwind anti-diffusive exact 1 0.8 Color Function 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 x(m) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 34 / 110
Interface Advection (5): Color Function at t = 0 . 03 s (step 15240) upwind anti-diffusive exact 1 0.8 Color Function 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 x(m) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 35 / 110
Interface Advection (6): Color Function at t = 0 . 1 s (step 50800) upwind anti-diffusive exact 1 0.8 Color Function 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 x(m) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 36 / 110
Interface Advection (7): Color Function at t = 3 . 0 s (step 1524000) upwind anti-diffusive exact 1 0.8 Color Function 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 x(m) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 37 / 110
Interface Advection (8): Pressure & Velocity at t = 3 . 0 s Pressure Velocity 101000 1001 upwind upwind anti-diffusive anti-diffusive exact exact 100500 1000.5 Pressure (Pa) Velocity (m/s) 100000 1000 99500 999.5 99000 999 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x(m) x(m) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 38 / 110
Interface Advection (9): Numerical Diffusion 100 upwind anti-diffusive 80 diffusion cells (%) 60 40 20 0 1 10 100 1000 10000 100000 1e+06 time steps KOKH (CEA) Int MultiMat Fl. Guidel May 2011 39 / 110
Shock Tube 1 (1) Test Description Riemann Problem with two perfect gases, adapted from the Sod Shock Tube Test Left State Right State γ = 1 . 4 γ = 2 . 4 ρ = 1 . 0 ρ = 0 . 125 P = 1 . 0 P = 0 . 1 u = 0 . 0 u = 0 . 0 The domain is discretized over a 300-cell mesh. t = 0 . 14 s KOKH (CEA) Int MultiMat Fl. Guidel May 2011 40 / 110
Shock Tube 1 (2): Velocity 0.9 upwind anti-diffusive exact 0.8 0.7 0.6 Velocity (m/s) 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 41 / 110
Shock Tube 1 (3): Pressure 1 upwind anti-diffusive exact 0.9 0.8 0.7 Pressure (Pa) 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 42 / 110
Shock Tube 1 (4): Color Function 1 upwind anti-diffusive exact 0.8 0.6 Color Function 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 43 / 110
Shock Tube 1 (5): Mass Fraction 1 upwind anti-diffusive exact 0.8 0.6 Mass Fraction 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 44 / 110
Shock Tube 1 (6): Density 1 upwind anti-diffusive exact 0.9 0.8 0.7 Density (kg/m 3 ) 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x(m) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 45 / 110
Shock Tube 1 (7): Numerical Diffusion of The Color Function 25 upwind anti-diffusive percent of cell number with numerical diffusion 20 15 10 5 0 0 2 4 6 8 10 time steps KOKH (CEA) Int MultiMat Fl. Guidel May 2011 46 / 110
Shock Tube 1 (8): Velocity (50 000 cells) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 47 / 110
Shock Tube 1 (9): Pressure (50 000 cells) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 48 / 110
Shock Tube 1 (10): Color Function (50 000 cells) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 49 / 110
Shock Tube 1 (11): Mass Fraction (50 000 cells) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 50 / 110
Shock Tube 1 (12): Density (50 000 cells) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 51 / 110
2D Air-R22 Shock-Interface Interaction (1) Test Description A planar shock hits a bubble initially at rest. Domain discretized over a 5000 × 1000 mesh Two perfect gases Fluid 0 Fluid 0 Pre-Shock State Post-Shock State Fluid 1 Pre-Shock State Ref : Haas&Sturtevant(Experiment), Quirk&Karni, Shyue, . . . KOKH (CEA) Int MultiMat Fl. Guidel May 2011 52 / 110
2D Air-R22 Shock-Interface Interaction (2) EOS Parameters & Initial Values location density pressure u 1 u 2 γ ( kg . m − 3 ) ( m . s − 1 ) ( m . s − 1 ) ( Pa ) 1 . 59 × 10 5 air (post-shock) 1 . 686 − 113 . 5 0 1 . 4 1 . 01325 × 10 5 air (pre-shock) 1 . 225 0 0 1 . 4 1 . 01325 × 10 5 R22 3 . 863 0 0 1 . 249 KOKH (CEA) Int MultiMat Fl. Guidel May 2011 53 / 110
2D Air-R22 Shock-Interface Interaction (3) Color Function t = 50 , µ s (anti-diffusive) t = 50 µ s (upwind) t = 239 µ s (anti-diffusive) t = 239 µ s (upwind)
2D Air-R22 Shock-Interface Interaction (4) Color Function t = 430 , µ s (anti-diffusive) t = 430 µ s (upwind) t = 540 µ s (anti-diffusive) t = 540 µ s (upwind)
2D Air-R22 Shock-Interface Interaction (5) Color Function t = 1020 , µ s (anti-diffusive) t = 1020 µ s (upwind)
2D Air-R22 Shock-Interface Interaction (6) Anti-Diffusive Solver Experiment
2D Air-R22 Shock-Interface Interaction (7) Anti-Diffusive Solver Experiment
2D Air-R22 Shock-Interface Interaction (8) Anti-Diffusive Solver Experiment
2D Air-R22 Shock-Interface Interaction (9) Anti-Diffusive Solver Experiment
2D Air-R22 Shock-Interface Interaction (10) Anti-Diffusive Solver Upwind Solver Anti-diffusive Solver Upwind Solver 250 250 200 200 150 150 t ( µ s) t ( µ s) 100 100 V s V s 50 50 V u V u V d V d V r V r V t V t V t V t 0 0 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 x (mm) x (mm)
2D Air-R22 Shock-Interface Interaction (11) Velocity ( m / s ) V s V R V T V ui V uf V di V df Haas & Sturtevant (Exp.) 415 240 540 73 90 78 78 Quirk & Karni 420 254 560 74 90 116 82 Shyue (tracking) 411 243 538 64 87 82 60 Shyue (capturing) 411 244 534 65 86 98 76 Upwind Solver 411 243 524 66 86 83 62 Anti-Diffusive Solver 411 243 525 65 86 85 64
2D Air-R22 Shock-Interface Interaction (12) Anti-Diffusive Solver vs Results of obtained by Shyue. Anti-diffusive Solver 250 200 150 t ( µ s) 100 V s 50 V u V d V r V t V t 0 0 10 20 30 40 50 60 70 x (mm) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 63 / 110
Kelvin-Helmholtz Instability (1) The domain is discretized over a 1000 × 1000 mesh KOKH (CEA) Int MultiMat Fl. Guidel May 2011 64 / 110
Kelvin-Helmholtz Instability (2) upwind anti-diffusive KOKH (CEA) Int MultiMat Fl. Guidel May 2011 65 / 110
Kelvin-Helmholtz Instability (3) upwind anti-diffusive KOKH (CEA) Int MultiMat Fl. Guidel May 2011 66 / 110
Kelvin-Helmholtz Instability (4) upwind anti-diffusive KOKH (CEA) Int MultiMat Fl. Guidel May 2011 67 / 110
Kelvin-Helmholtz Instability (5) upwind anti-diffusive KOKH (CEA) Int MultiMat Fl. Guidel May 2011 68 / 110
Kelvin-Helmholtz Instability (6) upwind anti-diffusive KOKH (CEA) Int MultiMat Fl. Guidel May 2011 69 / 110
Kelvin-Helmholtz Instability (7) upwind anti-diffusive KOKH (CEA) Int MultiMat Fl. Guidel May 2011 70 / 110
Kelvin-Helmholtz Instability (8)
Kelvin-Helmholtz Instability (9): Numerical Diffusion 70 upwind anti-diffusive 60 50 Diffusion Cells (%) 40 30 20 10 0 0 2000 4000 6000 8000 10000 time steps KOKH (CEA) Int MultiMat Fl. Guidel May 2011 72 / 110
Kelvin-Helmholtz Instability (10): Evolution of the Kinetic �� 1 2 ρ u 2 Energy in the x 2 -Direction t �→ 2 d x 1 d x 2 0.007 upwind anti-diffusive 0.006 kinetic energy in the x 2 direction (J) 0.005 0.004 0.003 0.002 0.001 0 0 1 2 3 4 5 6 7 8 t (s) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 73 / 110
Outline Parallel Implementation 5 Goals & Methods TRITON: Parallel Implementation Speed-Up Results KOKH (CEA) Int MultiMat Fl. Guidel May 2011 74 / 110
Parallel Implementation: about Goals & Methods Numerical Tools for Physical Modelization in an Ideal World Taking advantage of the increasing CPU power and mature distributed computing techniques Series of “fine Data ➡ ➡ “average scale” Processing scale” Models simulations Methodology / Guidelines Large meshes / Scalability “Reasonably Simple” Choices The above features condition our choices regarding numerical methods but also regarding physical models. We hope that simplicity will provide us computational power through scalability.
Parallel Algorithm TRITON Code TRITON: 3D parallel code developped at CEA Saclay (DM2S/SFME/LETR) dedicated to compressible with interface Code History A first 3D “Toy Code” developped in collaboration with R. Tuy Parallel version 0 by Ph. Fillion (CEA) and R. Tuy The present algorithm based on the work of Ph. Huynh and M. Flores (CS software support team of the CCRT) “Natural Choice” Domain Decomposition Distributed memory KOKH (CEA) Int MultiMat Fl. Guidel May 2011 76 / 110
Managing the Communications Communication Process Fictitious Cell Physical Boundary Fictitious Cell communication 0.9 0.9 communication 8.2 8.2 Subdomain 1 Subdomain 2 3: Communication 1: Communication 2: Computation 4: Computation Wait until step 1 is Inter-subdomain Compute the inner Compute the fluxes at non-blocking fluxes in the over. the subdomains communications subdomains boundaries
Speed Up Results Preliminary Tests Only!! These results must be confirmed and refined. Tests performed with the cluster PLATINE at CCRT. 1 2 50 100 200 Number of CPUs 405.45 201.7 9.78 4.9 2.06 Averaged Elapsed Time (s) 1 2.01 41.46 82.74 196.82 Speed-Up 100 100.51 82.91 82.74 98.41 Speed-Up (%) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 78 / 110
Outline 3D Simulation 6 KOKH (CEA) Int MultiMat Fl. Guidel May 2011 79 / 110
Example of 3D Simulation Performed with TRITON Context of The Test Safety Study: Acid Test (Engineer Study) Blind Test: no parameter tuning was allowed. 3D Gas Bulk Rising Towards a Free Surface Domain restricted to a quarter of space (due to time constraints) 54 × 54 × 400 = 1 166 400 cells mesh Test performed on a 1024 nodes cluster (PLATINE) at CCRT “Wall clock” time: about 60 h (Joint work with CEA coworkers: O. Gr´ egoire & P. Salvatore) KOKH (CEA) Int MultiMat Fl. Guidel May 2011 80 / 110
3D Gas Bulk Rising (1)
3D Gas Bulk Rising (2)
3D Gas Bulk Rising (3)
3D Gas Bulk Rising (4)
3D Gas Bulk Rising (5)
3D Gas Bulk Rising (6)
3D Gas Bulk Rising (7)
3D Gas Bulk Rising (8) Averaged Volume Fraction Estimate �� < z > ( x 3 , t )= z ( x 1 , x 2 , x 3 , t ) d x 1 d x 2 Comparison with Theoretical Estimates The gas bulk reaches its asymptotic velocity quasi-instantaneously The rising velocity measured on the 3D results provides a match with the theoretical results with a 2 . 6 × 10 − 2 relative error.
3D Gas Bulk Rising (9) Particle Dynamics within the Flow Use of the DSMC code (Lagrangian Particle Dynamics) for simulating the motion of particles within the two-phase flow. Chaining both codes allowed to precisely corroborate engineers estimates.
3D Gas Bulk Rising (10) Estimate of the global residual mass of particle within the gas bulk
3D Gas Bulk Rising (11) Estimate of the residual mass of particle within the gas bulk group by group
Outline High-Order Strategy 7 KOKH (CEA) Int MultiMat Fl. Guidel May 2011 92 / 110
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