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Finite element method.

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Presentation on theme: "Finite element method."— Presentation transcript:

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?

11

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

13 definition

14

15

16

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

31

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

36

37 is a Dirichlet matrix Known & unknown

38

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

41

42 Finer mesh

43


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