Lecture of FreeFEM Update:2015/7/29

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Brittany Sharyl Powell
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1 Lecture of FreeFEM Update:2015/7/29

2 About this lecture The goal of this lecture is: At the first, 3~4 participants are grouped, They understand how to use throughout total 10 problems, They should teach each other in the same group They can search anything from internet.

3 Schedule 9:00-10:20 Running Sample files and Mesh Generating 10:30-11:50 Poisson equations 12:00-13:30 13:30-14:50 Lunch Time Convection-Diffusion equations 15:00-16:20 Navier-Stokes equations 16:30-17:30 Free Time One lecture is 80 minutes in which, 30 minutes is used for explanations from me, 50 minutes is used for exercise.

4 Grouping and notations Grouping 3~4 participants are grouped. Notations About evaluations from A. Prof. J. Masamune

5 Schedule 9:00-10:20 Running Sample files and Mesh Generating 10:30-11:50 Poisson equations 12:00-13:30 Lunch Time 13:30-14:50 Convection-Diffusion equations 15:00-16:20 Navier-Stokes equations 16:30-17:30 Free Time

6 Running Sample files Laplace.edp diffusion.edp convection.edp tunnel.edp LapComplexEigenValue.edp Mesh_square.edp Mesh_circle.edp Mesh_circle_in_square.edp I show you how to generate finite element meshes

7 Mesh generation(mesh_square.edp) border a0(t=1,0){ x=0; y=t; label=1;} border a1(t=0,1){ x=t; y=0; label=2;} border a2(t=0,1){ x=1; y=t; label=3;} border a3(t=1,0){ x=t; y=1; label=4;} int n=5; Mesh Th =buildmesh( a0(10*n)+a1(10*n)+a2(10*n)+a3(10*n));

8 Mesh generation(mesh_square.edp) border a0(t=1,0){ x=0; y=t; label=1;} border a1(t=0,1){ x=t; y=0; label=2;} border a2(t=0,1){ x=1; y=t; label=3;} border a3(t=1,0){ x=t; y=1; label=4;} int n=5; Mesh Th =buildmesh( a0(10*n)+a1(10*n)+a2(10*n)+a3(10*n));

9 Command for definitions of borders border a0(t=1,0){ x=0; y=t; label=1;}

10 border a0(t=1,0){ x=0; y=t; label=1;} The name of border. You can use any words which you want.

11 border a0(t=1,0){ x=0; y=t; label=1;} A parameter range from 1 to 0. You can use any number which you want.

12 positions of borders (Like position vector?) border a0(t=1,0){ x=0; y=t; label=1;} y (0,1) o (0,0) x

13 positions of borders (Like position vector?) border a0(t=0,1){ x=0; y=t; label=1;} y (0,1) o (0,0) x

14 y (0,1) border a3(t=1,0){ x=t; y=1; label=4;} (1,1) border a0(t=1,0){ x=0; y=t; label=1;} border a2(t=0,1){ x=1; y=t; label=3;} border a1(t=0,1){ x=t; y=0; label=2;} (0,0) (1,0) x

15 y Notatins: definitions of borders in the Anti-Crockwise direction should be done! (0,1) border a3(t=1,0){ x=t; y=1; label=4;} (1,1) border a0(t=1,0){ x=0; y=t; label=1;} border a2(t=0,1){ x=1; y=t; label=3;} border a1(t=0,1){ x=t; y=0; label=2;} (0,0) (1,0) x

16 Mesh generation(mesh_square.edp) border a0(t=1,0){ x=0; y=t; label=1;} border a1(t=0,1){ x=t; y=0; label=2;} border a2(t=0,1){ x=1; y=t; label=3;} border a3(t=1,0){ x=t; y=1; label=4;} int n=5; Mesh Th =buildmesh( a0(10*n)+a1(10*n)+a2(10*n)+a3(10*n));

17 Commands for definitions of finite element meshes Mesh Th =buildmesh( a0(10*n)+a1(10*n)+a2(10*n)+a3(10*n)); Commands for definitions of finite element meshes

18 The name of Mesh You can use any words which you want. Mesh Th =buildmesh( a0(10*n)+a1(10*n)+a2(10*n)+a3(10*n));

19 Fineness of the mesh: should be defined by integer Mesh Th =buildmesh( a0(10*n)+a1(10*n)+a2(10*n)+a3(10*n));

20 Mesh generation(mesh_square.edp) border a0(t=1,0){ x=0; y=t; label=1;} border a1(t=0,1){ x=t; y=0; label=2;} border a2(t=0,1){ x=1; y=t; label=3;} border a3(t=1,0){ x=t; y=1; label=4;} int n=5; Mesh Th Making boundaries =buildmesh( a0(10*n)+a1(10*n)+a2(10*n)+a3(10*n)); Generating finite elements(triangles)

21 To define inside boundary Mesh_circle_in_square.edp border a0(t=1,0){ x=0; y=t; label=1;} border a1(t=0,1){ x=t; y=0; label=2;} border a2(t=0,1){ x=1; y=t; label=3;} border a3(t=1,0){ x=t; y=1; label=4;} border a4(t=0,-2*pi){ x= *cos(t); y= *sin(t); label=1;} int n=5; mesh Th= buildmesh( a0(10*n)+a1(10*n)+a2(10*n)+a3(10*n)+a4(10*n));

22 y Notatins: definitions of outside borders in the Anti-Crockwise direction should be done! (0,1) border a3(t=1,0){ x=t; y=1; label=4;} (1,1) border a0(t=1,0){ x=0; y=t; label=1;} border a2(t=0,1){ x=1; y=t; label=3;} border a1(t=0,1){ x=t; y=0; label=2;} (0,0) (1,0) x

23 y Notatins: definitions of inside borders in the Crockwise direction should be done! (0,1) (1,1) border a4(t=0,-2*pi){ x= *cos(t); y= *sin(t); label=1;} (0,0) (1,0) x

24 Problem1 y o x 5 10

25 Problem2 y o 1 2 x -1

26 Schedule 9:00-10:20 Running Sample files and Mesh Generating 10:30-11:50 Poisson equations 12:00-13:30 13:30-14:50 Lunch Time Convection-Diffusion equations 15:00-16:20 Navier-Stokes equations 16:30-17:30 Free Time

27 Strong form of Poisson eq. Γ R 2 Δu f = 0 in u = 0 on Γ

28 Weak form of Poisson eq. = lhs = Δu f v d = 0 u vd + x x + y y fvd v d + fvd = x x v + y y v d + x = v + x y v d + v x x + v y y = v + Γ x y v ndγ + v x x + v y y = Γ n vdγ + v x x + v y y v x + v y y d + fvd d + fvd d + fvd d + fvd

29 Strong form and Weak form of Poisson eq. Γ R 2 Δu f = 0 in u = 0 on Γ x v x + y v y d Γ n vdγ + fvd = 0

30 Boundary Conditions x v x + y v y d Γ n vdγ + fvd = 0 In the case of Dirichlet condition Let the trial function v be 0 on the boundary, we have v x x + v y y d + fvd = 0 But commands for the Dirichlet condition should be written in the source code in FreeFEM++. In the case of Neumann condition Substitute n v x x + y = g into boundary integration term, we have v y d g vdγ + Γ fvd = 0

31 1 Γ 1 Poisson equation: Laplace.edp o y Γ 4 R 2 Γ 3 x Δu = 0 in u = 1 on Γ 1 u = 0 on Γ 2 n = 0 on Γ 3 n = 0 on Γ 4 Γ 2 1

32 Poisson equation: Laplace.edp fespace Vh(Th,P2); Vh uh,vh; problem laplace(uh,vh,solver=lu) = int2d(th)( dx(uh)*dx(vh) + dy(uh)*dy(vh) ) + on(1,uh=1) + on(2,uh=0) ; laplace; plot(uh); Definition of variable Running Numerical calculations Visualizations Definition of finite element space Weak forms and numerical scheme

33 Poisson equation: Laplace.edp Mesh Th=buildmesh( a0(10)+a1(10)+a2(10)+a3(10)); Commands for definitions of finite element space fespace Vh(Th,P2); Vh uh,vh;

34 Poisson equation: Laplace.edp Mesh Th=buildmesh( a0(10)+a1(10)+a2(10)+a3(10)); The name of finite element space You can use any words which you want. fespace Vh(Th,P2); Vh uh,vh;

35 Poisson equation: Laplace.edp Mesh Th=buildmesh( a0(10)+a1(10)+a2(10)+a3(10)); fespace Vh(Th,P2); Vh uh,vh;

36 Poisson equation: Laplace.edp Mesh Th=buildmesh( a0(10)+a1(10)+a2(10)+a3(10)); fespace Vh(Th,P1); Variables are calculated on 3 points on one triangle Vh uh,vh; P1 element

37 Poisson equation: Laplace.edp Mesh Th=buildmesh( a0(10)+a1(10)+a2(10)+a3(10)); fespace Vh(Th,P2); Variables are calculated on 6 points on one triangle Vh uh,vh; P2 element

38 Poisson equation: Laplace.edp Mesh Th=buildmesh( a0(10)+a1(10)+a2(10)+a3(10)); fespace Vh(Th,P1 or P2); P1: low cost and low accuracy P2: high cost and high accuracy P1 element Vh uh,vh; P2 element

39 Poisson equation: Laplace.edp Commands for definition of numerical calculations problem laplace(uh,vh,solver=lu) = int2d(th)( dx(uh)*dx(vh) + dy(uh)*dy(vh) ) + on(1,uh=1) + on(2,uh=0) ; laplace;

40 Poisson equation: Laplace.edp The name of the problem You can use any words which you want. problem laplace(uh,vh,solver=lu) = int2d(th)( dx(uh)*dx(vh) + dy(uh)*dy(vh) ) + on(1,uh=1) + on(2,uh=0) ; laplace;

41 Poisson equation: Laplace.edp Variables that you need to calculate problem laplace(uh,vh,solver=lu) = int2d(th)( dx(uh)*dx(vh) + dy(uh)*dy(vh) ) + on(1,uh=1) + on(2,uh=0) ; laplace;

42 Poisson equation: Laplace.edp Numerical Scheme LU decompositions problem laplace(uh,vh,solver=lu) = int2d(th)( dx(uh)*dx(vh) + dy(uh)*dy(vh) ) + on(1,uh=1) + on(2,uh=0) ; laplace;

43 Poisson equation: Laplace.edp Integral on finite elements Th If you want to Integrate in 2 dimension, you should int2d. int2d(th)( dx(uh)*dx(vh) + dy(uh)*dy(vh) ) Weak form of Poisson equation dx() and dy() mean derivatives with respect to x and y. x v x + y v y d

44 Poisson equation: Laplace.edp problem laplace(uh,vh,solver=lu) = int2d(th)( dx(uh)*dx(vh) + dy(uh)*dy(vh) ) + on(1,uh=1) + on(2,uh=0) ; 1 Γ 4 (Dirichlet conditions) Dirichlet condition u = 1 on Γ 1 u = 0 on Γ 2 border a0 Γ 1 Γ 3 border a0(t=1,0){ x=0; y=t; label=1;} border a1(t=0,1){ x=t; y=0; label=2;} Γ 2 border a1 1

45 Poisson equation: Laplace.edp problem laplace(uh,vh,solver=lu) = int2d(th)( dx(uh)*dx(vh) + dy(uh)*dy(vh) ) + on(1,uh=1) + on(2,uh=0) ; Neumann conditions x v x + y v y d Γ n vdγ + fvd = 0 n = 0 on Γ 3 n = 0 on Γ 4 Substitute n = 0 into the second term, You should not type anything.

46 If you want to use Neumann conditions x v x + y v y d g vdγ + Γ fvd = 0 Please type boundary integration as follows; ex) +int1d(th, (label number))(- g *vh) on the boundary border a1, for g =2, you should type +int1d(th, 2)(-2*vh) border a1(t=0,1){ x=t; y=0; label=2;}

47 Poisson equation: Laplace.edp Integral on finite elements Th If you want to Integrate in 2 dimension, you should int2d. int2d(th)( dx(uh)*dx(vh) + dy(uh)*dy(vh) ) Weak form of Poisson equation dx() and dy() mean derivatives with respect to x and y. x v x + y v y d

48 If you want to use Neumann conditions x v x + y v y d g vdγ + Γ fvd = 0 Please type boundary integration as follows; +int1d(th, label number)(- g *vh) ex) on the boundary border a1, for g =2, you should type +int1d(th, 2)(-2*vh) border a1(t=0,1){ x=t; y=0; label=2;}

49 Poisson equation: Laplace.edp problem laplace(uh,vh,solver=lu) = int2d(th)( dx(uh)*dx(vh) + dy(uh)*dy(vh) ) + on(1,uh=1) + on(2,uh=0) ; Dirichlet condition laplace; Numerical Calculation

50 Output with Paraview Laplace_output.edp Please add the following commands in the line where you want output data. load "iovtk"; string vtkoutputfile="./output.vtk";savevtk(vtkoutputf ile, Th, uh, dataname="poisson");

51 Problem3 1 Γ 1 y Γ 4 R 2 Γ 3 Δu = u in u = x x x on Γ 1 u = 0 on Γ 2 n = 0 on Γ 3 n = 1 on Γ 4 o x Γ 2 1

52 Answer of Problem3

53 y Problem4 (The same domain as Problem1) Γ top Γ in Γ circle Γ out Γ bottom x

54 Problem4 (The same domain as Problem1) Δu = sin x cos(y) in n = 0 on Γ top Γ bottom u = 1 on Γ in Γ out n = 1 on Γ circle

55 Answer of Problem4

56 Γ 1 z Problem 5 In the axi-symmetric Cylindrical coordinate 1 system: Γ 4 R 2 o 1 Γ 2 Γ 3 r Δu = 1 r Space integration In this system: 2 u r r r + 2 u z 2 = 0 in u = 1 on Γ 2 u = 0 on Γ 3 n = 1 on Γ 4 r r r = 2 u r r r + 2 u z 2 Δu vrd = 0

57 Answer of Problem 5

58 Schedule 9:00-10:20 Running Sample files and Mesh Generating 10:30-11:50 Poisson equations 12:00-13:30 13:30-14:50 Lunch Time Convection-Diffusion equations 15:00-16:20 Navier-Stokes equations 16:30-17:30 Free Time

59 Diffusion equation: diffusion.edp Γ 1 Re t 1 Re Δu = 0 un+1 u n 1 Δt Re Δun+1 = 0 u n+1 u n 1 Δt Re Δu wd = 0 u n+1 Δt n+1 un w Δt w + 1 Re un+1 w n w = 0 d

60 Convection equation:convection.edp + u u = 0 t un+1 u n + u = 0 Δt u n+1 u n + u u n wd = 0 Δt u n+1 Δt u n+1 w un Δt w + u un w d = 0 Δt w w Δt convect u, Δt, un d = 0 u n+1 = convect u, Δt, u n = 0

61 u n+1 Δt u n+1 Convection diffusion equation: convection_diffusion.edp t + u u 1 Re Δu = 0 un+1 u n + u u n 1 Δt Re Δun+1 = 0 u n+1 u n + u u n 1 Δt Re Δun+1 wd = 0 w un Δt w + un u n w + 1 Re un+1 w d Γ Δt w w Δt convect u, Δt, un + 1 Re un+1 w d Γ Re 1 Re 1 n+1 n w = 0 n+1 n w = 0

62 Problem6 (The same domain as Problem1) Δu = 0 in t Initial condition: u = exp 10 x y Boundary conditions: u = 0 on Γ top [0, t] u = 0 on Γ bottom [0, t] n = 0 onγ in [0, t] n = 0 on Γ out [0, t] u = 0 on Γ circle [0, t]

63 Problem7 (The same domain as Problem1) + f u Δu = 0 in t where f = ( y y 5, 0) Initial condition: u = exp 10 x y Boundary conditions: u = 0 on Γ top [0, t] u = 0 on Γ bottom [0, t] n = 0 onγ in [0, t] n = 0 on Γ out [0, t] u = 0 on Γ circle [0, t]

64 Schedule 9:00-10:20 Running Sample files and Mesh Generating 10:30-11:50 Poisson equations 12:00-13:30 13:30-14:50 Lunch Time Convection-Diffusion equations 15:00-16:20 Navier-Stokes equations 16:30-17:30 Free Time

65 Navier-Stokes equation u u t u 0 Smac method 1st step 2nd step 3rd step n u* u t u* q t p n 1 p n u p 1 Re u 1 Re n n n u u p u* q u ( u, v) 4th step u n 1 u* q t

66 1 st step on Smac method: Only x-direction component u t u u n Δt u u n w un Δt w + Δt Δt u Δt w w Δt convect un, Δt, u n + u u + p x 1 Re Δu = 0 + u n u n + pn x 1 Re Δu = 0 + u n u n + pn x 1 Re Δu wd = 0 un u n w + pn x w + 1 Re u w d Γ + pn x w + 1 Re u w d Γ 1 Re n w = 0 1 Re n w = 0

67 Boundary condition Γ wall Γ in Γ out Γ wall q u y( 1 y), v 0, n u 0, v 0, q 0 x q u 0, v 0, 0 n 0 on Γ in on Γ out on Γ wall

68 Problem8 (The same domain as Problem1) t + u u = p + 1 Δu in Re u = 0 in Parameters: Re = 10, Δt = 0.05 Initial condition: u = y y 5, v = 0 Boundary conditions: u = 0, v = 0, q n = 0 on Γ top [0, t] u = 0, v = 0, q n = 0on Γ bottom [0, t] u = y y 5, v = 0, q n = 0onΓ in [0, t] n = 0, v = 0, p = 0 on Γ out [0, t] u = 0, v = 0, q n = 0on Γ circle [0, t]

69 Problem9 (The same domain as Problem1) t + u u = p + 1 Δu in Re u = 0 in f + u f = Δf in t Parameters: Re = 10, Δt = 0.05 Initial condition: u = y y 5, v = 0, f = exp 10 x y Boundary conditions: u = 0, v = 0, q n = 0, f = 0 on Γ top [0, t] u = 0, v = 0, q n = 0, f = 0 on Γ bottom [0, t] u = y y 5, v = 0, q f = 0, n n = 0 onγ in [0, t] f = 0, v = 0, p = 0, n n = 0 on Γ out [0, t] u = 0, v = 0, q n = 0, f = 0 on Γ circle [0, t]

70 t + u u = p + 1 Δu in Re u = 0 in Parameters: Re = 10 5, Δt = Initial condition: u = 1, v = 0.4 Boundary conditions: n Problem10 (The same domain as Problem1) = 0, v n = 0, q = 0 on Γ top [0, t] u = 1, v = 0.4, q n = 0 on Γ bottom [0, t] u = 1, v = 0.4, q n = 0 onγ in [0, t] v = 0, n n = 0, q = 0 on Γ out [0, t] u = 0, v = 0, q n = 0 on Γ wing [0, t]

71 Borders for Problem 10 border a41(t=0.5*pi,1.5*pi){ x=1+2*cos(t); y=2*sin(t); label=1;} border a42(t=-0.5*pi,0.5*pi){ x=1+2*cos(t); y=2*sin(t); label=2;} border Splus(t=0,1){ x = t; y = *sqrt(t) *t *(t^2) *(t^3) *(t^4); label=5;} border Sminus(t=1,0){ x =t; y= -( *sqrt(t) *t *(t^2) *(t^3) *(t^4)); label=5;} int n=5; mesh Th= buildmesh( a41(30*n)+a42(30*n)+splus(60*n)+sminus(60*n));plot(th);

72 Output with Paraview Please add the following commands at the top of code load "iovtk"; Please add the following commands in the line where you want output data. String vtkoutputfile="./problem10_out"+n+".vtk"; savevtk(vtkoutputfile, Th, [u,v,0],p,dataname="velocity pressure");

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