Meshfree methods for conservation laws using kinetic approach and - PowerPoint PPT Presentation

Meshfree methods for conservation laws using kinetic approach and alternate least squares procedures Praveen. C TIFR Center for Applicable Mathematics, Bangalore praveen@math.tifrbng.res.in Meshfree-2011 Conference Dept. of Aerospace Engg. Indian Institute of Science, Bangalore 10-11 January, 2011 Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 1 / 62

Outline 1 Kinetic meshless method for conservation laws 2 Comparison with other schemes 3 A positive meshless method 4 Alternate least squares 5 LS formula leading to positive method 6 Third order scheme for divergence operator Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 2 / 62

Kinetic schemes • Exploit connection between Boltzmann and Euler/Navier-Stokes equations ∂F v ⋅ ∇ F = 0 ∂t + ⃗ ⇓ ∂U ∂t + div G = 0 • Kinetic scheme Upwind Discretized Boltzmann Upwind Moments Scheme for Boltzmann Equation Conservation Scheme Equation Law Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 3 / 62

Kinetic Meshless Method • 2-D Boltzmann equation ∂F ∂t + v x F x + v y F y = 0 • Point collocation approach • Least squares approximation at node ”0” d F 0 a j ( F j − F 0 ) + v y ∑ b j ( F j − F 0 ) = 0 d t + v x ∑ j j Leads to unstable scheme; no wave propagation effects Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 4 / 62

✌ ✞ ☞ ✍ ☛ ✡ ✠ ✟ ✝ ✆ ☎ ✄ ✁ � Kinetic Meshless Method • Upwinding through introduction of a mid-point state (Morinishi, Balakrishnan) n j s j P j ✆✂✝ ✄✂☎ ✌✂✍ F j/2 �✂✁ ✞✂✟ P o ☛✂☞ ✠✂✡ • Kinetic upwind approximation if ⃗ n j ≥ 0 F j / 2 = { F 0 v ⋅ ˆ if ⃗ n j < 0 v ⋅ ˆ F j Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 5 / 62

Kinetic Meshless Method • LS using mid-point states d t + v x ∑ a j ( F j / 2 − F 0 ) + v y ∑ b j ( F j / 2 − F 0 ) = 0 d F 0 j j • Semi-discrete scheme d t + ∑ [ a j ( GX j / 2 − GX 0 ) + b j ( GY j / 2 − GY 0 )] = 0 d U 0 • No stencil splitting, smaller stencil • Rotationally invariant • Nearly positive scheme - good stability properties • On Cartesian points, KMM reduces to finite volume method • Edge-based updating possible - speed up of 2 Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 6 / 62

Kinetic Meshless Method • LS using mid-point states d t + v x ∑ a j ( F j / 2 − F 0 ) + v y ∑ b j ( F j / 2 − F 0 ) = 0 d F 0 j j • Semi-discrete scheme d t + ∑[ a j ( GX j / 2 − GX 0 ) + b j ( GY j / 2 − GY 0 )] = 0 d U 0 • No stencil splitting, smaller stencil • Rotationally invariant • Nearly positive scheme - good stability properties • On Cartesian points, KMM reduces to finite volume method • Edge-based updating possible - speed up of 2 Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 6 / 62

✄ � ☎ ✁ Higher order scheme • Define left and right states at each mid-point using linear reconstruction along the ray j / 2 = V 0 + 1 2∆ ⃗ j / 2 = V j − 1 2∆ ⃗ V + V − r j ⋅ ∇ V 0 and r j ⋅ ∇ V j ✄✂☎ P j V − + V mid−point �✂✁ P o Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 7 / 62

Numerical order of accuracy ∂t + y∂u ∂x − x∂u ∂y = 0 ∂u y u=0 (0,1) u=u o x (0,0) (1,0) outflow boundary Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 8 / 62

Numerical order of accuracy Uniform and random point distributions Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 9 / 62

Numerical order of accuracy -1 -1 L1 L1 L2 L2 Linf Linf -1.2 -1.2 Curve fit Curve fit -1.4 -1.4 -1.6 -1.6 log(Error) -1.8 log(Error) -1.8 -2 -2 -2.2 -2.2 -2.4 -2.4 -2.6 -2.8 -2.6 -2.1 -2 -1.9 -1.8 -1.7 -1.6 -1.5 -2 -1.9 -1.8 -1.7 -1.6 -1.5 log(h) log(h) Point distribution L 1 L 2 L ∞ Uniform 2.27 2.21 1.97 Random 2.19 2.12 1.90 Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 10 / 62

Flow over Williams airfoil Free-stream Mach number = 0.15 Angle of attack = 0 Number of points = 6415 Points on main airfoil = 234 Points on flap = 117 n KMM q-LSKUM 3 0.19 - 4 4.86 0.02 5 6.84 0.12 6 80.45 71.36 7 6.50 27.79 8 0.16 0.69 9 - 0.02 Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 11 / 62

William’s airfoil Pressure coefficient MAIN AIRFOIL FLAP 10 6 q-LSKUM q-LSKUM KMM KMM EXACT EXACT 5 8 4 6 3 -Cp -Cp 4 2 2 1 0 0 -2 -1 0 0.2 0.4 0.6 0.8 1 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 x x Scheme C l C d S min S max − 1 . 535 × 10 − 3 1 . 031 × 10 − 2 q-LSKUM 3.0927 0.0197 − 4 . 99 × 10 − 4 7 . 246 × 10 − 3 KMM 3.7608 0.0069 Potential 3.736 0 0 0 Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 12 / 62

Flow over cylinder M ∞ = 0 . 38 and α = 0 No of points = 4111 Points on cylinder = 250 n 4 5 6 7 8 9 KMM 6.67 5.50 83.12 4.72 - - LSKUM - 0.19 82.56 16.78 0.44 0.02 q-LSKUM KMM q-LSKUM KMM Pressure Mach Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 13 / 62

Subsonic flow over cylinder q-LSKUM KMM entropy, min = -0.00634187, max = 0.0222672 entropy, min = -0.00138786, max = 0.000165191 q-LSKUM KMM Scheme C l C d S min S max q-LSKUM 0.0237 0.0324 -0.00634 0.022267 KMM 0.0006 0.0012 -0.00138 0.000165 Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 14 / 62

Suddhoo-Hall airfoil • Number of points = 14091 • On airfoils = 229, 196, 217, 157 • Mach = 0.2 and α = 0 1 2 3 4 KMM 0.5387 4.8095 2.0925 0.7065 Exact 0.5215 4.7157 2.0794 0.7216 % Error 3.3 1.9 0.6 -2.0 Circulation around different elements Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 15 / 62

Suddhoo-Hall airfoil 5 KMM Exact 4 3 2 -Cp 1 0 -1 -2 -3 -2 -1 0 1 2 3 4 x Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 16 / 62

NACA-0012 airfoil M ∞ = 0 . 85 ,α = 1 o Adapted points, 3777 Mach contours Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 17 / 62

Scramjet Intake - initial solution Inlet Mach number = 5 Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 18 / 62

Scramjet Intake - adapted solution Inlet Mach number = 5 Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 19 / 62

Cartesian Points NACA-0012 - coarse M ∞ = 1 . 2, α = 0 o 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Points Mach number (Mohan Varma) Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 20 / 62

Cartesian Points NACA-0012 - adapted M ∞ = 1 . 2, α = 0 o 2 1.5 1 0.5 0 -0.5 -1 -1.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Points Mach number (Mohan Varma) Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 21 / 62

Finite Point Method Morinishi, Lohner, etc. Conservation law ∂t + ∂f ∂x + ∂g ∂y = 0 ∂u Meshless finite difference using mid-point fluxes d t + ∑ α ij ( f ij − f i ) + ∑ β ij ( g ij − g i ) = 0 d u i j j Define vector ℓ ij = ( α ij ,β ij ) d t + ∑ [( α ij f ij + β ij g ij ) − ( α ij f i + β ij g i )] = 0 d u i �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� j flux along ℓ ij Use your favourite numerical flux function (Roe, KFVS, etc.) H α ij f ij + β ij g ij = H ( u i ,u j ; ℓ ij ) What is the direction of ℓ ij ? Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 22 / 62

A positive meshless method FM Report 2004-FM-16 Semi-discrete scheme d t = ∑ c ij ( u j − u i ) d u i j if c ij ≥ 0 then maxima do not increase and minima do not decrease. Under a CFL-condition this leads to a positive update scheme = ∑ k ij ≥ 0 u n + 1 k ij u n j , (1) i j is stable in maximum norm j ≤ u n + 1 ≤ max u n u n min i j j j Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 23 / 62

Conservation law ∂t + div ⃗ ⃗ Q ( u ) = 0 , Q ( u ) = ⃗ ⃗ a = ( a x ,a y ) ∂u au, Least squares approximation of derivatives ∂x ∣ = ∑ α ij ( u j − u i ) , ∂y ∣ = ∑ β ij ( u j − u i ) ∂u ∂u i j ∈ C i i j ∈ C i Central difference scheme div ⃗ Q ( u ) i = a x ∑ α ij ( u j − u i ) + a y ∑ β ij ( u j − u i ) j j is unstable since it does not account for wave propagation effects. Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 24 / 62

Upwind scheme I � � � � � � � � � � � � u j � � � � uij � � � � � � � � u i � � � � � � � � div ⃗ Q ( u ) i = 2 a x ∑ α ij ( u ij − u i ) + 2 a y ∑ β ij ( u ij − u i ) j j Now let θ ij be the angle between � � � → N i N j and the positive x -axis, ˆ n ij = ( cos θ ij , sin θ ij ) be the unit vector along � � � → N i N j and Praveen. C (TIFR-CAM) Meshfree methods Meshfree-2011 25 / 62