![Tutorial #1 (cont’d)
// Define the weak formulation
solve Poisson(u, w, solver=LU)
= int2d(Th)( // The bilinear part
dx(w)*dx(u) + dy(w)*dy(u)
)
- int2d(Th)( // The right hand side
w*f
)
// The Dirichlet boundary condition
+ on(1,2,3,4, u=uexact)
;
// Plot the result
plot(u);
load "iovtk"
int[int] Order = [1];
string filename = "Poisson2D-square-Dirichlet.vtu";
savevtk(filename, Th, u, dataname="u", order=Order);
Weak form LHS
Weak form RHS
Boundary conditions
Plot the solution
Export solution in VTK format.
Available only as first-order elements.
27
Ω
𝜕𝑤
𝜕𝑥
𝜕𝑢
𝜕𝑥
+
𝜕𝑤
𝜕𝑦
𝜕𝑢
𝜕𝑦
𝑑𝑉 =
Ω
𝑤 𝑓 𝑑𝑉 +
Γ𝑁
𝑤 𝑡 𝑑𝐴](https://image.slidesharecdn.com/fem4cfd-week1-240601134348-6749b794/85/Introduction-to-CFd-analysis-FEM4CFD-Week1-pptx-27-320.jpg)
![“square” function
// The triangulated domain Th
// unit square, starting at (0,0)
mesh Th = square(10, 10);
// rectangle of size 10x5, starting at (0,0)
mesh Th = square(10, 10, [10*x, 5*y]);
// rectangle of size 5x10, starting at (-4,-2)
mesh Th = square(10, 10, [-4+5*x, -2+10*y]);
34
//display mesh
plot(Th);](https://image.slidesharecdn.com/fem4cfd-week1-240601134348-6749b794/85/Introduction-to-CFd-analysis-FEM4CFD-Week1-pptx-34-320.jpg)
![Access to matrix and vector
// Define the weak (variational) formulation
varf PoissonLHS(u, w)
= int2d(Th)( // The bilinear part
dx(w)*dx(u)
+ dy(w)*dy(u)
)
// The Dirichlet boundary condition
+ on(1,2,3,4, u=uexact)
;
matrix A = PoissonLHS(Vh, Vh); //stiffness matrix
cout << A << endl;
// Define the weak (variational) formulation
varf PoissonRHS(unused, w)
= int2d(Th)(
f*w
)
// The Dirichlet boundary condition
+ on(1,2,3,4, u=uexact)
;
Vh F;
//F[] is the vector associated to the function F
F[] = PoissonRHS(0,Vh);
cout << F[] << endl;
//u[] is the vector associated to the function u
u[] = A^-1*F[];
cout << u[] << endl;
43](https://image.slidesharecdn.com/fem4cfd-week1-240601134348-6749b794/85/Introduction-to-CFd-analysis-FEM4CFD-Week1-pptx-43-320.jpg)
Introduction to CFd analysis.FEM4CFD-Week1.pptx
- 1. A short course on FEM for CFD Week 1: Basics of FEM and FreeFEM, and Poisson equation Dr Chennakesava Kadapa Email: [email protected] 1
- 2. Contents • Introduction to the course • Poisson’s equation • Navier-Stokes equation • Introduction to Finite Element Method • Introduction to the FreeFEM library • Examples 2
- 3. Delivery mode • Lecture • Tutorial • Exercise 3
- 5. Incompressible Navier-Stokes equation 𝜌 𝜕𝒖 𝜕𝑡 + 𝒖 ∙ ∇ 𝒖 − 𝜇∆𝒖 + 𝛻𝑝 = 𝒇 𝛻 ∙ 𝒖 = 0 𝒖 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑝 = 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝜌 = 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝜇 = 𝑑𝑦𝑛𝑎𝑚𝑖𝑐 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦 Acceleration term Viscous term Pressure gradient Local Acceleration Convective Acceleration Body force 5
- 6. Course structure • Poisson’s equation −𝜇∆𝑢 = 𝑓 • Steady Advection-Diffusion equation 𝒂 ∙ ∇ 𝑢 − 𝜇∆𝑢 = 𝑓 • Unsteady Advection-Diffusion equation 𝜌 𝜕𝑢 𝜕𝑡 + 𝒂 ∙ ∇ 𝑢 − 𝜇∆𝑢 = 𝑓 • Stokes equation 𝜌 𝜕𝒖 𝜕𝑡 − 𝜇∆𝒖 + 𝛻𝑝 = 𝒇 𝛻 ∙ 𝒖 = 0 • Navier-Stokes equation 𝜌 𝜕𝒖 𝜕𝑡 + 𝒖 ∙ ∇ 𝒖 − 𝜇∆𝒖 + 𝛻𝑝 = 𝒇 𝛻 ∙ 𝒖 = 0 6 Week 1 Week 2 Week 3 Week 4,5,6 Week 4,5,6
- 8. Poisson’s equation • Is an elliptic PDE • Many applications • Diffusion, heat transfer, electrostatics, magnetostatics • Poisson’s equation is −∆𝑢 = 𝑓 where 𝑢 is the (scalar) field, 𝑓 is the source term and ∆ is the Laplacian operator. • Laplacian operator (∆) ∆𝑢 = ∇2𝑢 = ∇ ∙ ∇𝑢 = 𝜕2 𝑢 𝜕𝑥2 + 𝜕2 𝑢 𝜕𝑦2 + 𝜕2 𝑢 𝜕𝑧2 8
- 9. Poisson’s equation (Cont’d) • We can also write as −∇ ∙ (∇𝑢) = 𝑓 where ∇ is the gradient operator and ∇ ∙ is the divergence operator. • Gradient operator (∇) ∇= 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 𝑇 • Thus ∇ ∙ ∇𝑢 = 𝜕 𝜕𝑥 𝜕 𝜕𝑦 𝜕 𝜕𝑧 ∙ 𝜕𝑢 𝜕𝑥 𝜕𝑢 𝜕𝑦 𝜕𝑢 𝜕𝑧 = 𝜕2𝑢 𝜕𝑥2 + 𝜕2𝑢 𝜕𝑦2 + 𝜕2𝑢 𝜕𝑧2 9 ∇𝑢 = 𝜕𝑢 𝜕𝑥 𝜕𝑢 𝜕𝑦 𝜕𝑢 𝜕𝑧
- 10. Boundary conditions • Boundary conditions Dirichlet boundary condition: 𝑢 = 𝑢 on Γ𝐷 Neumann boundary condition: ∇𝑢 ∙ 𝒏 = 𝑡 on Γ𝑁 where 𝑢 is the specified solution field on the Dirichlet boundary Γ𝐷 𝑡 is the specified flux on the Neumann boundary Γ𝑁. 10
- 12. • Finite Element Method (FEM) is a numerical method for solving problems modelled by partial differential equations (PDEs). • FEM is not just for solid mechanics. • FEM has become the de-facto numerical method for many problems in science and engineering. • Simulation software and/or libraries based on FEM are available for • Solid mechanics – stress analysis • Structural dynamics • Wave propagation • Heat transfer • Fluid dynamics • Chemical diffusion • Electromagnetics 12 What is FEM? Check my papers for the details.
- 13. FEM offers a flexible numerical framework with a rich set of elements of different shapes, orders and families. Applicability to complex geometries. Different element shapes – triangle, quadrilateral, tetrahedron, brick, pyramid, wedge. Different element families – Lagrange family, Bézier family, Spectral elements, IGA, H-Div and H-Curl elements. Ease of extensions to higher-order approximations Ease of extension to Multiphysics problems Better accuracies compared to FVM for unstructured meshes 13 Why FEM?
- 14. Dates back to 1940s Originally used for problems in structural mechanics 1960-1990 was the golden age of FEM Since 1990, an explosion of industrial applications of FEM 14 History of FEM https://link.springer.com/article/10.1007/s11831-022-09740-9
- 15. • The points are called nodes. • The cells are called elements. • The variables are usually stored at nodes. Variable per cell is also possible. • The solution variables at nodes are called as degrees of freedom (DOFs). • The value at nodes (DOFs) can be a • Scalar: temperature, pressure, density • Vector: displacement, velocity • Tensor: deformation gradient, strain, stress 15 FEM terminology
- 17. Weak form • Poisson’s equation −∇ ∙ (∇𝑢) = 𝑓 • Weighted residual statement Ω 𝑤 𝑅 𝑑𝑉 = 0 • Splitting the integral − Ω 𝑤 ∇ ∙ ∇𝑢 𝑑𝑉 = Ω 𝑤 𝑓 𝑑𝑉 • After using the integrating by parts and divergence theorem Ω ∇𝑤 ∙ ∇𝑢 𝑑𝑉 − Γ 𝑤 ∇𝑢 ∙ 𝒏 𝑑𝐴 = Ω 𝑤 𝑓 𝑑𝑉 • Using boundary conditions Ω ∇𝑤 ∙ ∇𝑢 𝑑𝑉 = Ω 𝑤 𝑓 𝑑𝑉 + Γ𝑁 𝑤 𝑡 𝑑𝐴 Residual: 𝑅 = −∇ ∙ ∇𝑢 − 𝑓 Weak form 17
- 18. Weak form • Weak form in compact form Ω ∇𝑤 ∙ ∇𝑢 𝑑𝑉 = Ω 𝑤 𝑓 𝑑𝑉 + Γ𝑁 𝑤 𝑡 𝑑𝐴 • Weak form in component form Ω 𝜕𝑤 𝜕𝑥 𝜕𝑤 𝜕𝑦 𝜕𝑤 𝜕𝑧 ∙ 𝜕𝑢 𝜕𝑥 𝜕𝑢 𝜕𝑦 𝜕𝑢 𝜕𝑧 𝑑𝑉 = Ω 𝑤 𝑓 𝑑𝑉 + Γ𝑁 𝑤 𝑡 𝑑𝐴 Ω 𝜕𝑤 𝜕𝑥 𝜕𝑢 𝜕𝑥 + 𝜕𝑤 𝜕𝑦 𝜕𝑢 𝜕𝑦 + 𝜕𝑤 𝜕𝑧 𝜕𝑢 𝜕𝑧 𝑑𝑉 = Ω 𝑤 𝑓 𝑑𝑉 + Γ𝑁 𝑤 𝑡 𝑑𝐴 1D 2D 3D 18
- 19. Finite Element Method • In the FEM, we split the domain into smaller parts, called elements. • As the size of the elements is reduced, the approximation error is reduced. • Elements are described by their connections to nodes. • Discretisation 𝑢 = 𝑖=1 𝑛𝑒𝑛 𝑁𝑖𝑢𝑖 = 𝑁1 𝑁2 ⋯ 𝑁𝑛𝑒𝑛 𝑢1 𝑢2 ⋮ 𝑢𝑛𝑒𝑛 = 𝐍𝐮 𝑤 = 𝑖=1 𝑛𝑒𝑛 𝑁𝑖𝑤𝑖 = 𝐍𝐰 ● is a node. 19 ∇𝑢 = 𝜕𝑢 𝜕𝑥 𝜕𝑢 𝜕𝑦 𝜕𝑢 𝜕𝑧 = 𝜕𝑁1 𝜕𝑥 𝜕𝑁2 𝜕𝑥 ⋯ 𝜕𝑁𝑛𝑒𝑛 𝜕𝑥 𝜕𝑁1 𝜕𝑦 𝜕𝑁1 𝜕𝑧 𝜕𝑁2 𝜕𝑦 𝜕𝑁2 𝜕𝑧 ⋯ ⋯ 𝜕𝑁𝑛𝑒𝑛 𝜕𝑦 𝜕𝑁𝑛𝑒𝑛 𝜕𝑧 𝑢1 𝑢1 ⋮ 𝑢𝑛𝑒𝑛 = 𝜕𝑵 𝜕𝒙 𝐮 ∇𝑤 = 𝜕𝑵 𝜕𝒙 𝐰
- 20. Discretised problem • Weak form: Ω ∇𝑤 ∙ ∇𝑢 𝑑𝑉 = Ω 𝑤 𝑓 𝑑𝑉 + Γ𝑁 𝑤 𝑡 𝑑𝐴 • Using the discretization, the system of equations can be written as 𝐊 𝐮 = 𝐅 where, 𝐊 = 𝐀 e=1 𝐊𝒆; 𝐊𝒆 = Ω𝑒 𝜕𝑵 𝜕𝑥 𝑇 𝜕𝑵 𝜕𝑥 𝑑𝑉 𝐅 = 𝐀 e=1 𝐅𝒆; 𝐅𝒆 = Ω𝑒 𝑵𝑇𝑓𝑑𝑉 + Γ𝑁 𝑒 𝑵𝑇𝑡 𝑑𝐴 20
- 21. FEM ingredients We need to evaluate 𝐊 = 𝐀 e=1 𝐊𝒆; 𝐊𝒆 = Ω𝑒 𝜕𝑵 𝜕𝑥 𝑇 𝜕𝑵 𝜕𝑥 𝑑𝑉 𝐅 = 𝐀 e=1 𝐅𝒆 ; 𝐅𝒆 = Ω𝑒 𝑵𝑇 𝑓𝑑𝑉 + Γ𝑁 𝑒 𝑵𝑇 𝑡 𝑑𝐴 Information needed: • Mesh – nodal coordinates and connectivity • Element – basis functions • Stiffness matrix and force vector - numerical quadrature • Linear system – matrix solver (LU, CG, etc..) 21
- 23. Introduction to FreeFEM 23 • An opensource library for FEM. • Not an application/solver library like OpenFOAM. • Library that provides all the basic FEM ingredients. • To develop custom solvers. Well supported. Easy to install and use. Up-to-date tutorials. x Supports triangular and tetrahedral elements only. • Input through scripts • File extension “.edp” • Similar to C++ syntax • No need to compile • FreeFem++.exe <inp-script-file>
- 24. Tutorial #1 24
- 26. Tutorial #1 // The triangulated domain Th mesh Th = square(10,10); // The finite element space defined over Th is called here Vh fespace Vh(Th, P1); // Define u and w in the function space Vh u, w; // Define functions func uexact = (cosh(pi*y)-sinh(pi*y)/tanh(pi))*sin(pi*x); func f= 0; Mesh over a square or mapped square Finite element space. Element type, order etc. P1 is 3-noded triangle. Solution and test function Functions for use 26
- 27. Tutorial #1 (cont’d) // Define the weak formulation solve Poisson(u, w, solver=LU) = int2d(Th)( // The bilinear part dx(w)*dx(u) + dy(w)*dy(u) ) - int2d(Th)( // The right hand side w*f ) // The Dirichlet boundary condition + on(1,2,3,4, u=uexact) ; // Plot the result plot(u); load "iovtk" int[int] Order = [1]; string filename = "Poisson2D-square-Dirichlet.vtu"; savevtk(filename, Th, u, dataname="u", order=Order); Weak form LHS Weak form RHS Boundary conditions Plot the solution Export solution in VTK format. Available only as first-order elements. 27 Ω 𝜕𝑤 𝜕𝑥 𝜕𝑢 𝜕𝑥 + 𝜕𝑤 𝜕𝑦 𝜕𝑢 𝜕𝑦 𝑑𝑉 = Ω 𝑤 𝑓 𝑑𝑉 + Γ𝑁 𝑤 𝑡 𝑑𝐴
- 28. Tutorial #1 (cont’d) // Error real error = int2d(Th)( (uexact-u)^2 ); error = sqrt(error); 28 • Error 𝑒 = 𝑢𝑒𝑥𝑎𝑐𝑡 − 𝑢 • L2 norm of error 𝑒𝐿2 = Ω 𝑒2 𝑑𝑉 = Ω (𝑢𝑒𝑥𝑎𝑐𝑡 − 𝑢)2 𝑑𝑉
- 29. Exercise #1 29 • Different number of elements // The triangulated domain Th mesh Th = square(10,10); mesh Th = square(20,20); mesh Th = square(80,80); • Different orders of element // The finite element space defined over Th is called here Vh fespace Vh(Th, P1); fespace Vh(Th, P2); load "Element_P3"; fespace Vh(Th, P3); Solve the problem in Tutorial 1 using different number of elements and different orders of elements and observe the error.
- 30. Neumann BCs 30
- 31. Tutorial #2 31 Ω ∇𝑤 ∙ ∇𝑢𝑑𝑉 = Ω 𝑤 𝑓 𝑑𝑉 + Γ𝑁 𝑤 𝑡 𝑑𝐴
- 32. Tutorial #2 (cont’d) // Define the weak form solve Poisson(u, w, solver=LU) = int2d(Th)( // The bilinear part dx(w)*dx(u) + dy(w)*dy(u) ) - int2d(Th)( // Source term w*f ) - int1d(Th,3)( // Neumann BC term w*tbar ) + on(1,2,4, u=20) // Dirichlet BC ; 32 Ω 𝜕𝑤 𝜕𝑥 𝜕𝑢 𝜕𝑥 + 𝜕𝑤 𝜕𝑦 𝜕𝑢 𝜕𝑦 𝑑𝑉 = Ω 𝑤 𝑓 𝑑𝑉 + Γ𝑁 𝑤 𝑡 𝑑𝐴
- 34. “square” function // The triangulated domain Th // unit square, starting at (0,0) mesh Th = square(10, 10); // rectangle of size 10x5, starting at (0,0) mesh Th = square(10, 10, [10*x, 5*y]); // rectangle of size 5x10, starting at (-4,-2) mesh Th = square(10, 10, [-4+5*x, -2+10*y]); 34 //display mesh plot(Th);
- 35. Complex geometries – inbuilt functions int boundary = 1; border C01(t=0, 1){x=t; y=0.0; label=boundary;} border C02(t=0, 1){x=1; y=0.5*t; label=boundary;} border C03(t=0, 1){x=1.0-0.5*t; y=0.5; label=boundary;} border C04(t=0, 1){x=0.5; y=0.5+0.5*t; label=boundary;} border C05(t=0, 1){x=0.5-0.5*t; y=1.0; label=boundary;} border C06(t=0, 1){x=0.0; y=1-t; label=boundary;} int n = 20; mesh Th = buildmesh(C01(n) + C02(n/2) + C03(n/2) + C04(n/2) + C05(n/2) + C06(n)); plot(Th); C01 C02 C03 C04 C05 C06 35 https://doc.freefem.org/examples/mesh-generation.html
- 36. Complex geometries – using GMSH load "gmsh" // The triangulated domain Th mesh Th = gmshload("Lshaped-gmsh-mesh.msh"); int boundary = 1; mesh Th = gmshload("cylinder-gmsh-mesh.msh"); int inlet = 1; int outlet = 2; int topwall = 3; int bottomwall = 4; int cylinder = 5; 36 //display mesh plot(Th);
- 37. Exercises 37
- 38. Exercise #2 Solve the Poisson equation over the L-shaped domain. Use the mesh from “Lshaped-gmsh-mesh.msh”. 𝑢 = 0 on all boundaries. 𝑓 = 20 38
- 39. Solution for exercise #2 39
- 40. Exercise #3 Solve the Poisson equation for the following problem. Use the mesh from “cylinder-gmsh-mesh.msh”. 𝑢 = 10 𝜕𝑢 𝜕𝑦 = 0 𝜕𝑢 𝜕𝑦 = −2 𝑢 = 30 𝑓 = 0 𝑢 = 50 40
- 41. Solution for exercise #3 41
- 42. 42
- 43. Access to matrix and vector // Define the weak (variational) formulation varf PoissonLHS(u, w) = int2d(Th)( // The bilinear part dx(w)*dx(u) + dy(w)*dy(u) ) // The Dirichlet boundary condition + on(1,2,3,4, u=uexact) ; matrix A = PoissonLHS(Vh, Vh); //stiffness matrix cout << A << endl; // Define the weak (variational) formulation varf PoissonRHS(unused, w) = int2d(Th)( f*w ) // The Dirichlet boundary condition + on(1,2,3,4, u=uexact) ; Vh F; //F[] is the vector associated to the function F F[] = PoissonRHS(0,Vh); cout << F[] << endl; //u[] is the vector associated to the function u u[] = A^-1*F[]; cout << u[] << endl; 43
- 44. More practice with FreeFEM • FreeFEM - Getting started https://doc.freefem.org/tutorials/poisson.html • FreeFEM – Additional examples https://doc.freefem.org/tutorials/index.html 44
- 45. 45
- 46. 46