Solving Laplace’s Equation Using Finite Element Method

1. Solving Laplace’s Equation Using
Finite Element Method
Pedram Parchebafieh
Author:

2. Basic Steps of Finite Element
1. Discretization or subdivision of the domain.
2. Selection of the interpolation functions.
3. Formulation of the system equations.
4. Solution of the system of equations.

3. • The variational Method (Ritz Method)
Also known as the Rayleigh-Ritz method
Definition:
Self-adjoint:
The solution to (1) can be obtained by minimizing the functional:
(1)
can be approximated by:
Substituting (4) into (3).We obtain:
(3)
(4)

4. To minimize F we force its partial derivatives with respect to c_i to vanish.
Which can be written as the matrix equation
With the elements in [S] given by:
And elements in {b} given by:

5. It is evident that [S] is symmetric matrix. By invoking the self-adjoint property
of the operator L, elements in [S] can be written as:

6. • Galerkin’s Method
This method belongs to the family of weighted residual methods.
Substitution of For φ in (1) would the result in a nonzero residual.
Where R denote weighted residual integrals and w_i are chosen
weighted functions.
The weighting functions are then selected as:
So that (5) becomes:
(5)

7. This again leads to the matrix system given above, although now the matrix [S] is
not necessarily symmetric unless operator L is self-adjoint. if L is self-adjoint,
Galerkin’s method Results in the same system of equations as those given by the Ritz.
Different formulation are listed below:
•Point collocation method(point matching method)
•Least squares method
•Sub domain collocation method

8. • One-dimensional FEM
Discretization and Interpolation
Within the eth element φ(x) may be approximated by
Where Denotes the value of φ(x) atand and

9. Where Denotes the interpolation or basis functions given by:and
With Obviously,
And now we are ready to formulate
(6)

10. Substituting (6) into above and taking the derivative of F respect
To φ ,we obtain:
If α,β are constant or can be approximated by constants within each element, its matrix
elements can evaluated analytically and the result is:

11. where Denotes the corresponding values of α,β and f
One plate of a capacitor is biased to a constant potential and another plate is grounded
•One dimensional problem

12. • Two-Dimensional FEM
Once we have discretized the domain, we need to approximate the unknown
function φ within each element.
And it is obvious that

13. Where is
Area of the eth element

14. •Assembly to form the system of equations
Connectivity matrix
The general rule for this processes is to add to

15. Incorporation of boundary condition of Third kind
The general case with non-vanishing γ and q is
But in our special case we face with vanishing γ and q. in this situation the global K
will remain unchanged, but there is a restriction on imposing the Dirichlet BC.

16. •Domain Discretization
Number of triangles:11456Number of triangles:162

17. • Two-Dimensional Problems
• Two dimensional box with the Dirichlet BC
Number of triangles:11456Number of triangles:162

18. • A long hollow conducting cylinder that is divided into equal quarters, alternate
segments being held at potential + V and - V.

19. • A box with a perturbation.
V=1

20. • A box that the bottom and top edges satisfy the Dirichlet boundary condition and
the left and the right edges satisfy homogeneous Neumann condition.
Homogeneous Neumann BC
Homogeneous Neumann BC Dirichlet BC
Dirichlet BC

21. •The sphere is in the electric field.

22. 1. Jin, Jianming - The Finite Element Method in Electromagnetics (Wiley, 2ed., 2002)
2. Gennadiy Nikishkov - Programming Finite Elements in Java
3. Field Solutions on Computers - Finite-element Methods for Electromagnetics -
Stanley Humphries Jr. - 1997 2010
4. MATLAB documentation and related papers on the Web
REFERNCES