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Finite-Volumes I Sauro Succi Finite Volumes Real-life geometries: coordinate-free Courtesy of Prof. M. Porfiri, NYU Gauss theorem Given a Volume V, enclosed in a piecewise smooth boundary (surface) S, characterized by the normal n in each point; The flux of a is defined as IF S is regular and a is continuously differentiable, we have the following: Gauss theorem: conservation laws Gauss theorem: conservation laws Volume vs Surface average Gauss theorem: ADE Gauss theorem: Control Volume Conservativeness Flux(P,Q)+ Flux(Q, P) = 0 Colocated; Control Volume N n W w ne e E s S 1 1 Simple, but not good for surfint > geos VP uncoupled, hourglass Centers to Edges: Interpolate Fe = aPE FP + (1- aPE )FE aPE =1 aPE N n w s xe - x P = x E - xP ne e E Staggered 1 1 1 Laborious, no interpolation > simple geos No hourglass, VP coupled Discretized Gauss: Continuity Dt j P (t + dt) = j P (t) + Fk (t) å V k={e,n,w,s} Discretized Fluxes: Interpolation D F = ò DÑj × nˆk dS = Ak å dkQjQ Dk Sk Q D k Discretized Fluxes: General AUDt j P (t + dt) = j P (t) + LPQjQ (t) å V Q j P (t + dt) = j P (t) + CFL * å LPQjQ (t) Q Discretized Fluxes: Interpolation Ak D LPQ = å (akQ + dkQ ) UD k k={e,n,w,s} A Finite Volumes Sh <1 V J =U(j )j (U x Sx +U y Sy +Uz Sz )h V J || S <1 Topological issues C = hS /V >>1 l x ~ l y ~ lz ~ V S /V >>1 J << p / 2 J >> p / 2 1/3 FD<FV<FE Cartesian Non-Cartesian, Structured Unstructured 1d and 2d example: advection-diffusion FV: AD d=2 Dj P +(Fe +Fn + Fw + Fs )Dt = 0 Discretized Fluxes FkD = ò DÑj × nˆ k Sk dS One Dimension ¶tj = -U¶xj + D¶xxj Dx º DxP = xe - xw W w P e DtDy j P (t + Dt) - j P (t) = [-Uj + D¶xj ]ew DxDy E One Dimension W P w e E [Uj ]e = ae [Uj ]E + (1- ae )[Uj ]P [Uj ]w = aw [Uj ]W + (1- aw )[Uj ]P Linear Interp: Upwind: x E - xe ae = xE - xP x P - xw aw = xP - xW ae = 0 if Ue > 0 ae =1 if Ue < 0 aw =1 if Uw > 0 aw = 0 if Uw < 0 One Dimension W w P e E [D¶xj ]e = D[j E - j P ] /[xE - xP ] [D¶xj ]w = D[j P - jW ] /[xP - xW ] One Dimension W w P e E UDt j P (t + Dt) = j P (t) + LPQjQ (t) å Dx Q={W ,P,E} FluxPQ + FluxQP = 0 1d: example Give explicit expression of L_PQ and show that it reduces to standard FD for square finite volumes. Again, we don’t care about non-uniformity because the unknowns are cell averages (more physical) Two-dimensions Going to 2D: same principles more labour! Cartesian (Orthogonal) Trapezoidal (non-orthogonal) Unstructured (similar to FEM) Cartesian d=2 N {xi , y j } W E P xe - xw = DxP yn - ys = DyP V j d S i d U Structured cartesian: Advection fluxes Fe =UejeDyP Fn = Vnj n DxP D E x = x E - xP = DxP + DxE 2 D N y = yN - yP Fw =Uwj wDyP Fs = Vsj s DxP DW x = xW - xP D S y = yS - yP je = (1- aE )j P + aEj E aE = DxP DxP + DxE Structured cartesian: diffusive flux Adv _ Flux = -ae aEj E + -a w aW jW - an aNj N + as aSj S + [-a e (1- aE ) + a w (1- aW ) - an (1- aN ) + a s (1- aS )]j P ae =UeDyP an = VnDxP Adv _ Flux = å APQj Q Q={P.E,N,W ,S} dj P VP = APQjQ å dt Q={P.E,N,W ,S} Structured matrix: 5 nonzero entries Structured cartesian: Advection fluxes Fe =UejeDyP Fn = Vnj n DxP D E x = x E - xP = DxP + DxE 2 D N y = yN - yP Fw =Uwj wDyP Fs = Vsj s DxP DW x = xW - xP D S y = yS - yP je = (1- aE )j P + aEj E aE = DxP DxP + DxE 2d cartesian: diffusive fluxes je,x = (j P - j E ) / (xP - xE ) j e,y = 0 j n,y = (j N - j P ) / (yP - yE ) j n,x = 0 Structured cartesian: cieff’s Dif _ Flux = -de dEj E + -dw dW jW - dn d Nj N + ds dSj S + [-de (1- dE ) + dw (1- dW ) - dn (1- d N ) + ds (1- dS )]j P de = 2DeDyP / (DxP + DxE ) Dif _ Flux = å DPQj Q Q={P.E,N,W ,S} dj P VP = DPQj Q å dt Q={P.E,N,W ,S} Structured matrix: 5 nonzero entries Structured non-cartesian Structured Non-Cartesian Geometrical data Non-cartesian: structured NW NΕ N n W w C ne Non-orthogonal e Ε Fk = Jn An = J x Ax + J y Ay Dxe ¹ 0 s SW S se SΕ Dye ¹ Dyw ¹ DyP Still structured CEV = Centers/Edges/Vertices Co/Contravariant/Cartesian Co/Contravariant/Cartesian Co/Contravariant/Cartesian P = P(q , q , q ) 1 2 3 ¶P 1 ¶P 2 ¶P 3 dP = 1 dq + 2 dq + 3 dq ¶q ¶q ¶q Staggered NW NΕ N n W w C s SW S ne Non-orthogonal e Ε Fk = Jn An = J x Ax + J y Ay se SΕ CEV = Centers/Edges/Vertices Navier-Stokes (Compressible) Staggered FV Vertex-centered staggered N NW Ww SW n P s S NΕ e SΕ E Discretized Gauss: Continuity DrC + (Fe + Fn + Fw + Fs )Dt = 0 Discretized Convective Fluxes F Adv e = re [ueDye + veDxe ] Dxe = xne - xse Dye = yne - yse Same for north,west, south … Non-orthogonality issues (!) NW W w n ne C e s SW NΕ N S Ε se SΕ Discretized Gauss: Continuity DrC + (Fe + Fn + Fw + Fs )Dt = 0 Discretized midpoint (2nd order 8 neigh) F Adv e Dye = re [(une + use ) +...{vDxe }..] 2 Discretized Simpson (4th order, 8 neigh) F Adv e Dye = re [(une + 4ue + use ) +...{vDxe }..] 6 Discretized Convective Fluxes Fadv (e) = r(e)[u(e)Dy(e)+ v(e)Dx(e)] r(e) = ar(C)+ (1- a)r (E) x(E) - x(e) a= x(E) - x(C) u(e), v(e) = Interp[C, E] Discretized Gauss: Momentum_x DJ x,C + (Fx,e + Fx,n +Fx,w +Fx,s )Dt = 0 1 uC = (une + unw + use + usw ) 4 rC native Convective and Dissipative Fluxes F = [ ru + p]e Dye +[ ruv]e Dxe Adv x,e F Dif x,e 2 = me {ux,eDye +[¶x u +¶y v]e Dxe } uE - uC ¶ x ue = xE - xC vE - vC ¶ y ve = yE - yC Non-Linear (outer) iteration rP (t + Dt) = r P (t) + Dt å[APQuQ (t) + BPQ vQ (t)]rQ (t) Q å rP uP (t + Dt) = rP uP (t) + Dt [EPQuQ (t) + FPQ vQ (t)]rQ (t + Dt) + CPQ pQ (t + dt) Q={E,N,...} rP vP (t + Dt) = r P vP (t) + Dt å [GPQuQ (t + Dt) + H PQ vQ (t)]rQ (t + Dt) + DPQ pQ (t + dt) Q={E,N,...} Nonlinear (outer) iteration, k=0,1… Ax (t + Dt) = B(x )x (t) º b k+1 k k k Real-life geometries Courtesy of Prof. M. Porfiri, NYU Poly Example: Global: Cylindrical, Spherical, Local: Oblique Unstructured FV~FEM Cell vs Vertex Centered Vertex control elements Tk+1 Sk = {EkCk Ek+1} Ek+2 Ck+1 Tk Ek+1 Ck Ek Finite Volumes: summary Intuitive and physically sound Round-off Conservative (fluxin=-fluxout) Geo-topological ahead, laborious Interpolation to be decided (unlike FEM) Structured: Finite-Difference with non-smooth coordinates No-singularity (1/r for sherical coordinates) Commercially dominant (STAR-CD, FLUENT…)