1. finite element
method

2. V = V1
V = V2
symmetry
Boundary conditions

3. Assume that

4. Locations & voltages are known!??
Ve satisfies Laplace’s equation

5. V1 V2 V3

6. V1 V2 V3

7. calculate V1 V3 V2 Internal voltage distribution
Known voltages and locations
calculate

8. definition
Area of triangle?

9. Show that the determinant of the matrix is equal to twice the area of a triangle.

10. x y 1 2 3
Area of triangle?

12. The determinant of this matrix is equal to twice the area of a triangle.

13. definition

17. Recall that this is twice the area Of the triangle
subscripts
= 1
Recall that this is twice the area Of the triangle

18. subscripts
= 0

19. Voltage distribution
Within the triangle
Is determined by the
Voltages at the
Corners and the
locations of the
corners are known!

20. V3= 1
V1= 0
V2= 1

21. MATLAB
V1= 0
V2= 1
V3= 1
V1= 0
V2= 1
V3= 1

22. V1= 0
V2= 1
V3= 1

23. V1= 0
V2= 1
V3= 1

24. constants
is a Dirichlet matrix

25. V1= 0
V2= 1
V3= 1
is a Dirichlet matrix

26. V1= 0
V2= 1
V3= 1

27. V6= 3
V4= 1
V5= 1

28. V6= 3
V4= 1
V5= 1

29. V1= 0
V2= 1
V3= 1
V6= 3
V4= 1
V5= 1

30. V1= 0
V2= 1
V3= 1
V6= 3
V4= 1
V5= 1

32. V1= 0
V2= 1
V3= 1
V6= ???
V4= 1
V5= 1

33. In order to minimize this energy, V must be a solution of Laplace’s equation.
Let there be another solution U that satisfies the boundary conditions. Linear media implies superposition!

34. Either 1) V is specified (U = 0) or 2) the normal derivative of V = 0
Either 1) V is specified (U = 0) or 2) the normal derivative of V = 0.<-- symmetry

35. Minimum with the
Actual solution

37. is a Dirichlet matrix
Known & unknown

39. symmetry
symmetry

40. V = 10 V = 0 Locate nodes # of nodes = 11 2) Draw triangles
3) Identify nodes of triangles
# of elements = 12
V = 0
4) Specify the known potentials at the nodes = 8
5) Specify the initial conditions at the 3 internal nodes
6) Calculate the potentials at the 3 internal nodes

42. Finer mesh