Finite Elements for Fluids Prof. Sheu

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1 Finite Elements for Fluids Prof. Sheu Yannick Deleuze Taipei, April 28, 26 Homework from April 7, 26 Use the code from the exercices. Write a small report with comments and discussion on the results and provide the code used. Convection-diffusion equation with u = ( u max, u max ). (uφ) (ν φ) = in [, ] [, ], φ = on Γ Γ 4 () φ = on Γ 2 Γ 3. Write the variational formulation of the convection-diffusion equation. 2. Plot the solution for different value of h = N K K h k of the convection-diffusion equation using the non-stabilized method, the artificial viscosity method and the SUPG methods in the case P e = max K (P e K ) < and P e >. 3. In the case, P e <, compare the solutions from each stabilized method with the non-stabilized method using the H norm. Stokes equation. Write the variational formulation of the Stokes equations Re 2 u + p = f in [, L] [, ], u + εp =, u = (y( y), ) on Γ, u = (, ) on Γ 2 4, (2) Re u n + p n = (, ) on Γ Analytical validation in the square [, ] [, ] with the V norm according to the analytical solution u = y( y), u 2 =, p = 2 ( x), f =. Re Plot the error for different values of h = max K h K for the P2-P, Pb-P, and P-P elements. 3. Solve the Stokes equation in the domain of your choice with the boundary condition of your choice. Give the variational formulation and a plot of the solution. Department of Engineering Science and Ocean Engineering, National Taiwan University ydeleuze@ntu.edu.tw

2 2 Correction 2. Convection-diffusion equation Let V ϕ = {w H () w = ϕ on Γ Γ 4, w = on Γ 2 Γ 3 }. One multiplies () by a test function v V and integrate over the entire domain. The variational formulation of the convection-diffusion problem becomes find φ V such that v V : a(φ, v) = (3) where the bilinear form 2.. Solutions for P e < and P e > Let ν = and U inf = 5. a(φ, v) = φ u v + ν φ v. Solutions for P e = The usual Galerkin method (non-stabilized) solution features a oscillations. From figure, one can observe that the artificial viscosity and SUPG method can give stabilized solutions without oscillation. From the cuts along the x and y directions in figure (e)-(f), one can see that the SUPG (2) method give the better approximation. Solutions for P e = 2483 The usual Galerkin method (non-stabilized) solution give the best approximation. From the cuts along the x and y directions in figure 2 (e)-(f), one can see that the SUPG (2) method give the approximation with less artificial viscosity Error for SUPG and artificial methods when P e < Please see figure 2 for the comparison of the solution obtained by the different methods. The error between the artificial and SUPG methods and the the Galerkin (non-stabilized) is given in figure 3. The difference methods are of order when using P finite elements. One can see that the SUPG (2) method give the approximation with less artificial viscosity FreeFem++ code / Generating Mesh / 2 r e a l L=; 3 border b ( t =,) {x=; y=t ; l a b e l =4;} 4 border b2 ( t =,L) {x=t ; y=; l a b e l =;} 5 border b3 ( t =,) {x=l ; y=t ; l a b e l =2;} 6 border b4 ( t=l, ) {x=t ; y=; l a b e l =3;} 7 r e a l Dx=., Dy=.; 8 mesh Th= buildmesh ( b ( c e i l (. /Dy) )+b2 ( c e i l (L/Dx) )+b3 ( c e i l (. /Dy) )+b4 ( c e i l (L/Dx) ) ) ; 9 p l o t (Th, wait =) ; / F i n i t e Element Spaces / 2 f e s p a c e Vh(Th, P) ; 3 f e s p a c e Xh(Th, P) ; 4 5 / Problem parameters / 6 r e a l nu =.; 7 Vh u=, v= ; 8 r e a l Uinf = max( u [ ]. l i n f t y, v [ ]. l i n f t y ) ; 9 r e a l U2 = s q r t ( uˆ2+v ˆ2) ; 2 p l o t ( [ u, v ], wait =, cmm= flow f i e l d [ u, v ], U i n f= +Uinf, c o e f =.) ; 2 r e a l [ i n t ] v i s o (. 3 :. :. ) ; //define fixed isolines / Macros / 24 macro a ( phih, vh ) int2d (Th) ( nu ( dx ( phih ) dx ( vh ) + dy ( phih ) dy ( vh ) ) + ( u dx ( phih ) + v dy ( phih ) ) vh ) //eom / Defining t h e v a r i a t i o n a l problem / 27 v a r f CD ( phi,w) = 28 a ( phi,w) 29 +on (, 4, phi =)+on ( 2, 3, phi =) ; 3 2

3 3 v a r f CDstab ( phi,w) = 32 a ( phi,w) 33 + int2d (Th) ( Uinf htriangle ( dx ( phi ) dx (w) + dy ( phi ) dy (w) ) ) //artifical viscosity 34 + on (, 4, phi =)+on ( 2, 3, phi =) ; r e a l d e l t a =. ; 37 Xh tau = htriangle /U2 ; 38 v a r f CDsupg( phi,w) = 39 a ( phi,w) 4 + int2d (Th) ( d e l t a ( nu ( dxx ( phi )+dyy ( phi ) )+u dx ( phi )+v dy ( phi ) ) //SUPG 4 tau ( u dx (w) + v dy (w) ) ) 42 +on (, 4, phi =)+on ( 2, 3, phi =) ; / P e c l e t Number / 45 Xh hk = htriangle ; 46 Xh PeK = Uinf hk / 2. / nu ; 47 r e a l Pe = PeK [ ]. max ; 48 cout << Pe = + Pe << endl ; 49 p l o t (PeK, wait =, f i l l =, value =,cmm= P e c l e t number : Pe= +Pe, ps= PeK +Pe+. eps ) ; 5 5 / S o l v i n g t h e problems / 52 Vh phi ; 53 matrix A = CD(Vh, Vh, s o l v e r = UMFPACK, f a c t o r i z e =) ; // assemble the FE matrix 54 r e a l [ i n t ] b = CD(,Vh) ; // Dirichlet boundary condition 55 phi [ ] = Aˆ b ; //solve the problem 56 p l o t ( phi, wait =, f i l l =, cmm= phi, v i s o=viso, ps= phi +Pe+. eps ) ; Vh p h i s t a b ; 59 A = CDstab (Vh, Vh, s o l v e r = UMFPACK, f a c t o r i z e =) ; // assemble the FE matrix 6 b = CDstab (, Vh) ; // Dirichlet boundary condition 6 p h i s tab [ ] = Aˆ b ; //solve the problem 62 p l o t ( phistab, wait =, f i l l =, cmm= phi + a r t i f i c i a l v i s c o s i t y, v i s o=viso, ps= a r t i f i c i a l v i s c o s i t y +Pe+. eps ) ; Vh phisupg ; 65 A = CDsupg (Vh, Vh, s o l v e r = UMFPACK, f a c t o r i z e =) ; // assemble the FE matrix 66 b = CDsupg (, Vh) ; // Dirichlet boundary condition 67 phisupg [ ] = Aˆ b ; //solve the problem 68 p l o t ( phisupg, wait =, f i l l =, cmm= phi + SUPG, v i s o=viso, ps= SUPG +Pe+. eps ) ; 69 7 tau = htriangle nu / 2. /U2 ; 7 Vh phisupg2 ; 72 A = CDsupg (Vh, Vh, s o l v e r = UMFPACK, f a c t o r i z e =) ; // assemble the FE matrix 73 b = CDsupg (, Vh) ; // Dirichlet boundary condition 74 phisupg2 [ ] = Aˆ b ; //solve the problem 75 p l o t ( phisupg2, wait =, f i l l =, cmm= phi + SUPG ( 2 ), v i s o=viso, ps= SUPG2 +Pe+. eps ) ; / Compare t h e d i f f e r e n t s o l u t i o n s / 79 //h errors 8 macro HNorm( u ) ( s q r t ( int2d (Th) ( uˆ2 + ( dx ( u ) ) ˆ2 + ( dy ( u ) ) ˆ2 ) ) ) //eom 8 { 82 Vh e r r s t a b = phi p h i s t ab ; 83 r e a l errstabh = HNorm( e r r s t a b ) ; 84 Vh errsupg = phi phisupg ; 85 r e a l errsupgh = HNorm( errsupg ) ; 86 Vh errsupg2 = phi phisupg2 ; 87 r e a l errsupg2h = HNorm( errsupg2 ) ; 88 ofstream f f ( e r r. s o l, append ) ; 89 f f << hk [ ]. max << << Pe << << er rs ta bh << << errsupgh << << errsupg2h << endl ; 9 } 9 92 // Plot the solution along two cuts parallel to the OX and OY axis 93 i n t Npts =; 94 r e a l [ i n t ] ox ( Npts ), ox2 ( Npts ) ; 95 r e a l [ i n t ] oy ( Npts ), oy2 ( Npts ) ; 3

4 96 f o r ( i n t j =; j<npts ; j++) { 97 // to obtain a OY-axis 98 oy [ j ] =. ; 99 oy2 [ j ]=.+.5 j ; } ; f o r ( i n t j =; j <; j++) { 2 // to obtain a OX-axis 3 ox [ j ]=.+ j. 5 ; 4 ox2 [ j ] =. ; 5 } ; 6 7 //solution along x=.5 8 { 9 r e a l xx =. 5 ; Npts = c e i l (. /Dy) ; ofstream f f ( xcut +Pe+. s o l ) ; 2 r e a l [ i n t ] phiv ( Npts ), phiy ( Npts ) ; 3 r e a l [ i n t ] phistabv ( Npts ), phistaby ( Npts ) ; 4 r e a l [ i n t ] phisupgv ( Npts ), phisupgy ( Npts ) ; 5 r e a l [ i n t ] phisupg2v ( Npts ), phisupg2y ( Npts ) ; 6 f o r ( i n t j =; j<npts ; j++) { 7 phiy [ j ]=Dy j ; 8 phiv [ j ]= phi ( xx, phiy [ j ] ) ; 9 phistabv [ j ]= p h i s t ab ( xx, phiy [ j ] ) ; 2 phisupgv [ j ]= phisupg ( xx, phiy [ j ] ) ; 2 phisupg2v [ j ]= phisupg2 ( xx, phiy [ j ] ) ; 22 f f << phiy [ j ] << << phiv [ j ] << << phistabv [ j ] << << phisupgv [ j ] << << phisupg2v [ j ] << endl ; 23 } ; 24 p l o t ( [ phisupg2y, phisupg2v ], [ phisupgy, phisupgv ], [ phistaby, phistabv ], [ phiy, phiv ], [ ox, ox2 ], [ oy, oy2 ], wait =, cmm = s o l u t i o n s along x=.5 ) ; 25 } //solution along y=.5 29 { 3 r e a l yy =. 5 ; 3 Npts = c e i l (. /Dx) ; 32 ofstream f f ( ycut +Pe+. s o l ) ; 33 r e a l [ i n t ] phiv ( Npts ), phix ( Npts ) ; 34 r e a l [ i n t ] phistabv ( Npts ), phistaby ( Npts ) ; 35 r e a l [ i n t ] phisupgv ( Npts ), phisupgy ( Npts ) ; 36 r e a l [ i n t ] phisupg2v ( Npts ), phisupg2y ( Npts ) ; 37 f o r ( i n t j =; j<npts ; j++) { 38 phix [ j ]=Dx j ; 39 phiv [ j ]= phi ( phix [ j ], yy ) ; 4 phistabv [ j ]= p h i s t ab ( phix [ j ], yy ) ; 4 phisupgv [ j ]= phisupg ( phix [ j ], yy ) ; 42 phisupg2v [ j ]= phisupg2 ( phix [ j ], yy ) ; 43 f f << phix [ j ] << << phiv [ j ] << << phistabv [ j ] << << phisupgv [ j ] << << phisupg2v [ j ] << endl ; 44 } ; 45 p l o t ( [ phisupg2y, phisupg2v ], [ phisupgy, phisupgv ], [ phistaby, phistabv ], [ phix, phiv ], [ ox, ox2 ], [ oy, oy2 ], wait =, cmm = s o l u t i o n s along y=.5 ) ; 46 } 4

5 phi phi + artificial viscosity (a) Non-stabilized (b) Artificial viscosity phi + SUPG phi + SUPG (2) (c) SUP G() (d) SUP G(2) non-stabilized Artificial viscosity non-stabilized Artificial viscosity SUPG- SUPG-2 SUPG- SUPG-2 Phi.4 Phi y x (e) Solutions along x =.5 (f) Solutions along y =.5 Figure : P e =

6 phi + artificial viscosity phi (a) Non-stabilized (b) Artificial viscosity phi + SUPG phi + SUPG (2) (c) SU P G() (d) SU P G(2) non-stabilized Artificial viscosity SUPG- SUPG-2 Phi Phi non-stabilized Artificial viscosity SUPG- SUPG y.7.9. (e) Solutions along x = x.7 (f) Solutions along y =.5 Figure 2: P e =

7 Artificial viscosity SUPG- SUPG-2 x u-uh H h Figure 3: Error u non stabilized u i with i =artificial viscosity, SUPG(), SUPG(2) 7

8 2.2 Stokes equations 2.2. Variational formulation Let the space V φ = { (w, r) [H ()] 2 L 2 () w Γ = φ, w Γ2 4 = }. Then the variational formulation of problem (2) is find (u, p) V g such that for all (v, q) V Re with g(x, y) = (y( y), ) Analytic validation ( u v + u 2 v 2 ) p( x v + y v 2 ) + ( x u + y u 2 ) q + εpq = (f v + f 2 v 2 ), From the figures 4 and 5, one can see that the solution improves with h and ε. In figure 4, the approximation of the solution given by the P 2 -P method is limited by the choice of ε. n figure 5, the approximations of the solution given by the P b-p and P -P methods are limited by the choice of h. (4) P2-P Pb-P P-P order.. (uh-u,ph-p) V e-6 e-8 e- e-2 e-4... h Figure 4: Error (u h u, p h p) V with respect to the mesh size h with ε = 3. P2-P Pb-P P-P order. e-6 e-8 (uh-u,ph-p) V e- e-2 e-4 e-6 e-8 e-2 e-2 e-8 e-6 e-4 e-2 e- e-8 e-6.. h Figure 5: Error (u h u, p h p) V with respect to the parameter ε with h =

9 2.2.3 FreeFem++ code / Generating Mesh / 2 r e a l L=; 3 border b ( t =,) {x=; y=t ; l a b e l =;} //Gamma 4 border b2 ( t =,L) {x=t ; y=; l a b e l =2;} //Gamma2 5 border b3 ( t =,) {x=l ; y=t ; l a b e l =3;} //Gamma3 6 border b4 ( t=l, ) {x=t ; y=; l a b e l =4;} //Gamma4 7 r e a l Dx=., Dy=.; 8 mesh Th= buildmesh ( b ( c e i l (. /Dy) )+b2 ( c e i l (L/Dx) )+b3 ( c e i l (. /Dy) )+b4 ( c e i l (L/Dx) ) ) ; 9 p l o t (Th, wait =) ; / F i n i t e Element Spaces / 2 f e s p a c e Vh(Th, [ P2, P2, P ] ) ; 3 f e s p a c e Vhb(Th, [ Pb, Pb, P ] ) ; 4 f e s p a c e VhStab (Th, [ P, P, P ] ) ; 5 f e s p a c e P2h (Th, P2) ; // for plots 6 f e s p a c e Ph(Th, P) ; // for plots 7 f e s p a c e Pbh(Th, Pb ) ; // for plots 8 f e s p a c e Xh(Th, P) ; 9 2 / macros / 2 macro grad ( u ) [ dx ( u ), dy ( u ) ] //eom - IMPORTANT : macro ends with a comments / Defining t h e v a r i a t i o n a l problem / 24 func g=y ( y ) ; 25 r e a l f =., f 2 =.; 26 r e a l Re = ; 27 r e a l nu =. / Re ; 28 r e a l e p s i l o n = e 3; 29 v a r f b i l i n e a r ( [ u, u2, p ], [ v, v2, q ] ) = //P2-P or Pb-P fomulation 3 int2d (Th) ( nu ( dx ( u ) dx ( v ) + dy ( u ) dy ( v ) 3 + dx ( u2 ) dx ( v2 ) + dy ( u2 ) dy ( v2 ) ) 32 p q e p s i l o n 33 p dx ( v ) p dy ( v2 ) 34 dx ( u ) q dy ( u2 ) q 35 ) 36 + on (, u=g, u2=) // boundary conditions 37 + on ( 2, 4, u=,u2=) 38 ; 39 4 r e a l d e l t a =.; 4 Xh hk2 = htriangle htriangle ; 42 v a r f b i l i n e a r S t a b ( [ u, u2, p ], [ v, v2, q ] ) = //Stabilized P-P formulation 43 int2d (Th) ( nu ( dx ( u ) dx ( v ) + dy ( u ) dy ( v ) 44 + dx ( u2 ) dx ( v2 ) + dy ( u2 ) dy ( v2 ) ) 45 + p q e p s i l o n 46 + d e l t a hk2 ( dx ( p ) dx ( q )+dy ( p ) dy ( q ) ) 47 p dx ( v ) p dy ( v2 ) 48 + dx ( u ) q+ dy ( u2 ) q 49 ) 5 + on (, u=g, u2=) // boundary conditions 5 + on ( 2, 4, u=,u2=) 52 ; v a r f l i n e a r ( [ u, u2, p ], [ v, v2, q ] ) = int2d (Th) ( f v+f 2 v2 ) ; / S o l v i n g t h e problem / 57 r e a l time=c l o c k ( ) ; 58 Vh [ u, v, p ] ; 59 matrix A = b i l i n e a r (Vh, Vh, s o l v e r=umfpack) ; // assemble the FE matrix 6 r e a l [ i n t ] bc = b i l i n e a r (,Vh) ; // Dirichlet boundary condition 6 r e a l [ i n t ] b = l i n e a r (,Vh) ; // assemble the right hand side vector 62 b += bc ; // modifying the entires for the Dirichlet BC. 63 u [ ] = Aˆ b ; //solve the problem 64 time = c l o c k ( ) time ; 65 cout << Mixed f o r m u l a t i o n P2 P : S o l v e r time = + time << endl ; 9

10 66 67 r e a l timeb=c l o c k ( ) ; 68 Vhb [ ub, vb, pb ] ; 69 matrix Ab = b i l i n e a r (Vhb, Vhb, s o l v e r=umfpack) ; // assemble the FE matrix 7 r e a l [ i n t ] bcb = b i l i n e a r (,Vhb) ; // Dirichlet boundary condition 7 r e a l [ i n t ] bb = l i n e a r (,Vhb) ; // assemble the right hand side vector 72 bb += bcb ; // modifying the entires for the Dirichlet BC. 73 ub [ ] = Abˆ bb ; //solve the problem 74 timeb = c l o c k ( ) timeb ; 75 cout << Mixed f o r m u l a t i o n Pb P : S o l v e r time = + timeb << endl ; r e a l times=c l o c k ( ) ; 78 VhStab [ uss, vss, pss ] ; 79 matrix As = b i l i n e a r S t a b ( VhStab, VhStab, s o l v e r = UMFPACK, f a c t o r i z e =) ; // assemble the FE matrix 8 r e a l [ i n t ] bcs = b i l i n e a r S t a b (, VhStab ) ; // Dirichlet boundary condition 8 r e a l [ i n t ] bs = l i n e a r (, VhStab ) ; // assemble the right hand side vector 82 bs += bcs ; // modifying the entires for the Dirichlet BC. 83 uss [ ] = Asˆ bs ; //solve the problem 84 times = c l o c k ( ) times ; 85 cout << S t a b i l i z e d P P f o r m u l a t i o n : S o l v e r time = + times << endl ; / V i s u a l i z e t h e s o l u t i o n / 88 p l o t ( c o e f =.3,cmm= Mixed f o r m u l a t i o n P2 P : Flow f i e l d [ u, v ], [ u, v ], value =, wait =) ; 89 p l o t (cmm= Mixed f o r m u l a t i o n P2 P : P r e s s u r e p, p, value =, f i l l =, wait =) ; 9 P2h V = s q r t ( u u+v v ) ; 9 p l o t (cmm= Mixed f o r m u l a t i o n P2 P : Fluid v e l o c i t y magnitude V,V, value =, f i l l =, wait =) ; p l o t ( c o e f =.3,cmm= Mixed f o r m u l a t i o n Pb P : Flow f i e l d [ u, v ], [ ub, ub ], value =, wait =) ; 94 p l o t (cmm= Mixed f o r m u l a t i o n Pb P : P r e s s u r e p, pb, value =, f i l l =, wait =) ; 95 Pbh Vb = s q r t ( ub ub+vb vb ) ; 96 p l o t (cmm= Mixed f o r m u l a t i o n Pb P : Fluid v e l o c i t y magnitude V,Vb, value =, f i l l =, wait =) ; p l o t ( c o e f =.3,cmm= S t a b i l i z e d P P f o r m u l a t i o n : Flow f i e l d [ u, v ], [ uss, vss ], value =, wait =) ; 99 p l o t (cmm= S t a b i l i z e d P P f o r m u l a t i o n : P r e s s u r e p, pss, value =, f i l l =, wait =) ; Ph Vs = s q r t ( uss uss+vss vss ) ; p l o t (cmm= S t a b i l i z e d P P f o r m u l a t i o n : Fluid v e l o c i t y magnitude V, Vs, value =, f i l l =, wait =) ; 2 3 / C o m p a t i b i l i t y c o n d i t i o n / 4 r e a l uin = intd (Th, ) ( g N. x ) ; 5 r e a l uout = intd (Th, 3 ) ( u N. x+v N. y ) ; 6 r e a l erru = s q r t ( ( uin+uout) ( uin+uout) ) ; 7 cout << endl << uin ( uout) L2 = + erru << endl << endl ; 8 9 / A n a l y t i c a l v a l i d a t i o n / func ue = y ( y ) ; 2 func dyue = 2 y ; 3 func ve =. ; 4 func pe = 2. / Re ( x ) ; 5 6 macro NuH( u ) ( s q r t ( int2d (Th) ( ( u ue ) (u ue ) + dx ( u ) dx ( u )+ ( dy ( u ) dyue ) ( dy ( u ) dyue ) ) ) ) //eom 7 macro NvH( v ) ( s q r t ( int2d (Th) ( v v + dx ( v ) dx ( v )+ dy ( v ) dy ( v ) ) ) ) //eom 8 macro NpL2( p ) ( s q r t ( int2d (Th) ( ( p pe ) (p pe ) ) ) ) //eom 9 macro NV(u, v, p ) ( s q r t ( u u + v v + p p ) ) //eom 2 2 //Mixed formulation 22 r e a l errunormh = NuH( u ) ; 23 r e a l errvnormh = NvH( v ) ; 24 r e a l errpnorml2 = NpL2( p ) ; 25 r e a l errnormv = NV( errunormh, errvnormh, errpnorml2 ) ; 26 cout << endl << Mixed f o r m u l a t i o n P2 P << endl ;

11 27 cout << uh u H = + errunormh << endl ; 28 cout << u2h u2 H = + errvnormh << endl ; 29 cout << ph p L2 = + errpnorml2 << endl ; 3 cout << ( uh u, ph p ) V = + errnormv << endl << endl ; 3 32 r e a l errubnormh = NuH( ub ) ; 33 r e a l errvbnormh = NvH( vb ) ; 34 r e a l errpbnorml2 = NpL2( pb ) ; 35 r e a l errnormvb = NV( errubnormh, errvbnormh, errpbnorml2 ) ; 36 cout << endl << Mixed f o r m u l a t i o n Pb P << endl ; 37 cout << uh u H = + errubnormh << endl ; 38 cout << u2h u2 H = + errvbnormh << endl ; 39 cout << ph p L2 = + errpbnorml2 << endl ; 4 cout << ( uh u, ph p ) V = + errnormvb << endl << endl ; 4 42 r e a l errusnormh = NuH( uss ) ; 43 r e a l errvsnormh = NvH( vss ) ; 44 r e a l errpsnorml2 = NpL2( pss ) ; 45 r e a l errnormvs = NV( errusnormh, errvsnormh, errpsnorml2 ) ; 46 cout << endl << S t a b i l i z e d P P f o r m u l a t i o n << endl ; 47 cout << uh u H = + errusnormh << endl ; 48 cout << u2h u2 H = + errvsnormh << endl ; 49 cout << ph p L2 = + errpsnorml2 << endl ; 5 cout << ( uh u, ph p ) V = + errnormvs << endl << endl ;