1. MATH 212 NE 217 Douglas Wilhelm Harder Department of Electrical and Computer Engineering University of Waterloo Waterloo, Ontario, Canada Copyright © 2011 by Douglas Wilhelm Harder. All rights reserved. Advanced Calculus 2 for Electrical Engineering Advanced Calculus 2 for Nanotechnology Engineering Finite-Element Methods in Two Dimensions

2. Finite-element Methods in Two Dimensions 2 Outline This topic discusses an introduction to finite-element methods in two dimensions –What we did in one dimension –Dividing up a 2-D region –Defining planar test functions on those regions –Converting the test functions to systems of linear equations –Solving the system –Examples

3. Finite-element Methods in Two Dimensions 3 Outcomes Based Learning Objectives By the end of this laboratory, you will: –You will understand dividing regions into triangular sub-regions (tessellations) –Understand the mathematical theory for performing finite-element methods in two dimensions

4. Finite-element Methods in Two Dimensions 4 Integration-by-Parts in Higher Dimensions Before we start, we need a higher-dimensional version of integration by parts: –Using the properties of the gradient: –Now use a corollary of Stokes’ theorem: Ref: http://en.wikipedia.org/wiki/Vector_calculus_identities

5. Finite-element Methods in Two Dimensions 5 Integration-by-Parts in Higher Dimensions This variation has the product of two scalar-valued functions replaced with a product of scalar- and vector-valued functions –Using the properties of the gradient: –Now use a corollary of Stokes’ theorem:

6. Finite-element Methods in Two Dimensions 6 Poisson’s Equation in Two Dimensions At this point, we have generalized the finite-difference method to allow arbitrarily spaced points in one dimension to solve systems such as Poisson’s equation: The ultimate goal is to do this in 2 and 3 dimensions –In order to achieve this, we need one further idea

7. Finite-element Methods in Two Dimensions 7 Extending Test Functions In the previous –Use triangles to divide up the region instead of intervals Tessellations –Approximate the solution using interpolating polynomials Intervals Tessellation

8. Finite-element Methods in Two Dimensions 8 Partitioning a Region Given a region R, a partial differential equation defines conditions on the boundary ∂R and we attempt to find an approximation of the solution in the region

9. Finite-element Methods in Two Dimensions 9 Partitioning a Region We have seen how using finite differences, we: –Create a uniform grid –Assign points to either the boundary or the interior of the region

10. Finite-element Methods in Two Dimensions 10 Partitioning a Region Unfortunately, uniform grids make a poor approximation of most real-world situations

11. Finite-element Methods in Two Dimensions 11 Partitioning a Region It would be much more desirable to allow the user to select the placement of points where appropriate: –Non-linear boundaries can be fitted –Points can be concentrated at regions of interest Let N be the number interior points – in this case, 22

12. Finite-element Methods in Two Dimensions 12 Partitioning a Region In one dimension, the regions between points were intervals –Similarly, we must divide the region into a number of sub-regions

13. Finite-element Methods in Two Dimensions 13 Partitioning a Region Our goal will be to find approximations of the solution on the interior of the region R

14. Finite-element Methods in Two Dimensions 14 Partitioning a Region We will divide the region into triangular elements: –This forms a connected graph –The collection of all triangles is a tessellation –Tessella is the Latin word for a small piece of a mosaic

15. Finite-element Methods in Two Dimensions 15 Partitioning a Region From wikipedia: a tessellation of a 2D magnetostatic configuration User: Zureks

16. Finite-element Methods in Two Dimensions 16 Defining Test Functions In one dimension, for each interior point, x k, we used the two adjacent intervals upon which we defined the test functions –The support of the test function  6 (x) is the interval [x 5, x 7 ] 6(x)6(x) 7(x)7(x)

17. Finite-element Methods in Two Dimensions 17 Defining Test Functions In two dimensions, given the point x k, we will define a test function on the region comprised of triangles adjacent to x k –Call that region R k

18. Finite-element Methods in Two Dimensions 18 Defining Test Functions For ease of reference, given a point x k, we will number the adjacent points, in this case: x k,1 x k,2 x k,3 x k,4 x k,5 x k,6

19. Finite-element Methods in Two Dimensions 19 Defining Test Functions Similarly, we will label each triangles with R k, in this case: R k,1 R k,2 R k,3 R k,4 R k,5 R k,6

20. Finite-element Methods in Two Dimensions 20 Defining Test Functions On each region R k, we will define a piecewise planar test function –The test function is  k (x, y) –It is defined to be zero outside R k –It is also useful to have it zero on the boundary ∂R k

21. Finite-element Methods in Two Dimensions 21 Defining Test Functions The formula for each plane  k, ℓ (x, y) defined on R k, ℓ is a plane of the form  x +  y +  –To find it, we use linear algebra

22. Finite-element Methods in Two Dimensions 22 Finding Interpolating Planes Given three points in the plane (x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ) if we want to find the interpolating plane  x +  y +  that passes through these points z 1, z 2, and z 3, respectively, we must solve the system of linear equations

23. Finite-element Methods in Two Dimensions 23 Finding Interpolating Planes Rewritten using matrices and vectors, we must solve:

24. Finite-element Methods in Two Dimensions 24 Finding Interpolating Planes Example, given the points in the plane (1.3, 5.4), (2.9, 7.0) and (6.5, 4.9)

25. Finite-element Methods in Two Dimensions 25 Finding Interpolating Planes Example, given the points in the plane (1.3, 5.4), (2.9, 7.0) and (6.5, 4.9) we want to find the polynomial  x +  y +  that passes through the points 2.8, 1.6 and 3.9, respectively

26. Finite-element Methods in Two Dimensions 26 Finding Interpolating Planes Example, given the points in the plane (1.3, 5.4), (2.9, 7.0) and (6.5, 4.9) we want to find the polynomial  x +  y +  that passes through the points 2.8, 1.6 and 3.9, respectively We must therefore solve >> [1.3 5.4 1; 2.9 7.0 1; 6.5 4.9 1] \ [2.8 1.6 3.9]' ans = 0.1272 -0.8772 7.3715

27. Finite-element Methods in Two Dimensions 27 Finding Interpolating Planes From >> [1.3 5.4 1; 2.9 7.0 1; 6.5 4.9 1] \ [2.8 1.6 3.9]' ans = 0.1272 -0.8772 7.3715 we have that the interpolating plane is 0.1272x – 0.8772y + 7.3715

28. Finite-element Methods in Two Dimensions 28 Using the Test Function Now, going back to the problem: –We have the test function  k (x, y) and we have a function V(x, y) that is known to satisfy the formula V(x, y) = 0 As we did in one dimension, it must therefore be true that

29. Finite-element Methods in Two Dimensions 29 Using the Test Function Again, let’s consider Poisson’s equation: We defined and thus our integral is

30. Finite-element Methods in Two Dimensions 30 Using the Test Function Expanding the integral, we get where the right-hand side is reasonably easy to calculate The left-hand side, however, requires further calculus

31. Finite-element Methods in Two Dimensions 31 Using the Test Function Using the 2-dimensional equivalent of integration-by-parts, we get:

32. Finite-element Methods in Two Dimensions 32 Using the Test Function Using the 2-dimensional equivalent of the fundamental theorem of calculus: However, recall that we specifically chose a test function that is zero on the boundary ∂R k –Therefore, the first term disappears! 0

33. Finite-element Methods in Two Dimensions 33 Using the Test Function Therefore, we have the easier integral: and therefore we have

34. Finite-element Methods in Two Dimensions 34 Using the Test Function We can calculate the right-hand side: both are known The left-hand side has the unknown function u(x, y) …

35. Finite-element Methods in Two Dimensions 35 Using the Test Function We can calculate the right-hand side: both are known First, rewrite the left-hand side as integrals over each sub-region Let n k be the number of triangles touching x k – in this case, 6

36. Finite-element Methods in Two Dimensions 36 Finding the Test Function Gradient The test function defined on each sub-region is planar, that is thus we can easily calculate:

37. Finite-element Methods in Two Dimensions 37 Finding the Test Function Gradient To find the gradient of this plane, we must find the vector

38. Finite-element Methods in Two Dimensions 38 Finding the Test Function Gradient Solving for  is as follows:

39. Finite-element Methods in Two Dimensions 39 Finding the Test Function Gradient Solving for  is as follows:

40. Finite-element Methods in Two Dimensions 40 Approximating the Solution Gradient The next step is to approximate the unknown

41. Finite-element Methods in Two Dimensions 41 Approximating the Solution Gradient The actual unknown function u(x, y) an unknown and likely non- linear function –You don’t get paid $100 000/a to find a linear function…

42. Finite-element Methods in Two Dimensions 42 Approximating the Solution Gradient Recall from 1-D finite element methods: –We approximated the solution with an unknown piecewise linear function

43. Finite-element Methods in Two Dimensions 43 Approximating the Solution Gradient We are attempting to approximate u(x, y) at a number of interior points where the solution is defined on the border

44. Finite-element Methods in Two Dimensions 44 Approximating the Solution Gradient We can do the same as we did in one dimension: –Approximate the solution by a piecewise planer approximation

45. Finite-element Methods in Two Dimensions 45 Approximating the Solution Gradient As before, we can find the interpolating polynomial:

46. Finite-element Methods in Two Dimensions 46 Approximating the Solution Gradient To find the gradient of this triangle, we must find the vector

47. Finite-element Methods in Two Dimensions 47 Approximating the Solution Gradient Solving for a is as follows:

48. Finite-element Methods in Two Dimensions 48 Approximating the Solution Gradient Solving for b is as follows:

49. Finite-element Methods in Two Dimensions 49 Approximating the Double Integral Thus, we can find both gradients: If you look back, all values are either known –The positions of the points x k, x k,1 and x k,2 : x 1, x 2, x 3, y 1, y 2, y 3 or unknown constants u k, u k,1 and u k,2 : –Take them out of the integral

50. Finite-element Methods in Two Dimensions 50 Approximating the Double Integral Finally, we note that the right-hand integral is just the area of the region R k,1, call it A k,1 :

51. Finite-element Methods in Two Dimensions 51 Approximating the Double Integral Thus, for each interior point x k, we must calculate: For each x k, there are n k adjacent triangles R k,ℓ, and each of these is associated given three points, call them x 1 = (x 1, y 1 ), x 2 = (x 2, y 2 ), x 3 = (x 3, y 3 ) and the unknowns associated with, call them u 1, u 2 and u 3 : –Each interior point ends up with n k linear equations that are summed together forming a single linear equation

52. Finite-element Methods in Two Dimensions 52 Approximating the Double Integral If the ratio is a constant, the right-hand integral can be simplified further: where A k is the area of the region R k

53. Finite-element Methods in Two Dimensions 53 Finding the points This will produce N linear equations and unknowns –We can solve this system of linear equations for the unknown values u 1, u 2, u 3, …, u N

54. Finite-element Methods in Two Dimensions 54 Code For 2-D Finite Elements A set of functions are available on the web site that set a finite element system and solve for the interior values –It assumes the right-hand side of Poisson’s equation is constant fe_system( rho, pts ) Create a data structure for the above equation by passing in a set of points (interior and boundary) with values for the boundary points and -Inf for the interior points fe_update( obj, pt, alist ) Indicate the tessellation around an interior point by passing a list of neighbouring points fe_solve( obj ) Having called fe_update for each interior point, now solve the system of equations returning the initial argument pts with all values -Inf replaced with values fe_coeffs( x1, x2, x3 ) A helper function

55. Finite-element Methods in Two Dimensions 55 Example 1 Consider the following region and we are trying to approximate Laplace’s equation (  = 0 ) with the following region and boundary values:

56. Finite-element Methods in Two Dimensions 56 Example 1 We can define the system as follows: pts1 = [ 0 -1 -1 % 1 -1 0 1 % 2 0 0 -Inf % 3 1 0 1 % 4 -1 1 0 % 5 0 1 -Inf % 6 0 2 -4]'; % 7 obj1 = fe_system( 0, pts1 );

57. Finite-element Methods in Two Dimensions 57 Example 1 We define the tessellation around the point 3 by listing the adjacent points in order (counter-clockwise): obj1 = fe_update( obj1, 3, [4 6 2 1] ); Note, Matlab does not have proper objects, thus we simulate object-oriented programming by replacing obj1.fe_update( 3, [4, 6, 2, 1] ) with obj1 = fe_update( obj1, 3, [4, 6, 2, 1] )

58. Finite-element Methods in Two Dimensions 58 Example 1 Similarly, we define the tessellation around the point 6 by listing the adjacent points in order (counter-clockwise): obj1 = fe_update( obj1, 6, [4 7 5 2 3] );

59. Finite-element Methods in Two Dimensions 59 Example 1 Having defined the system, we solve the system and plot the result: u1 = fe_solve( obj1 ) u1 = 0 -1.0000 0 1.0000 -1.0000 0 0 -1.0000 0 0 0 1.0000 1.0000 2.0000 -1.0000 1.0000 -0.1053 1.0000 0 -1.4211 -4.0000 plot3( u1(1,: ), u1(2,:), u1(3,:), 'ko' ) The plot was augmented in CorelDRAW!

60. Finite-element Methods in Two Dimensions 60 Example 2 Consider the following region and we are trying to approximate Poisson’s equation with  = 4, the following region and boundary values:

61. Finite-element Methods in Two Dimensions 61 Example 2 We can define the system as follows: pts2 = [-2 -2 8 % 1 1 -2 5 % 2 -3 -1 10 % 3 0 -1 -Inf % 4 3 -1 10 % 5 -1 0 -Inf % 6 2 0 -Inf % 7 -3 1 10 % 8 1 1 –Inf % 9 3 1 10 % 10 -2 2 8 % 11 0 2 -Inf % 12 1 3 10]'; % 13 obj2 = fe_system( 4, pts2 );

62. Finite-element Methods in Two Dimensions 62 Example 2 We then define the various tessellations: obj2 = fe_update( obj2, 4, [1 2 7 9 6] ); obj2 = fe_update( obj2, 6, [1 4 9 12 11 8 3] ); obj2 = fe_update( obj2, 7, [2 5 10 9 4] ); obj2 = fe_update( obj2, 9, [4 7 10 13 12 6] ); obj2 = fe_update( obj2, 12, [6 9 13 11] );

63. Finite-element Methods in Two Dimensions 63 Example 2 Finally, we solve the system: u2 = fe_solve( obj2 ) u = Columns 1 through 8 -2.0000 1.0000 -3.0000 0 3.0000 -1.0000 2.0000 -3.0000 -2.0000 -2.0000 -1.0000 -1.0000 -1.0000 0 0 1.0000 8.0000 5.0000 10.0000 1.8256 10.0000 1.2883 4.6679 10.0000 Columns 9 through 13 1.0000 3.0000 -2.0000 0 1.0000 1.0000 1.0000 2.0000 2.0000 3.0000 2.9111 10.0000 8.0000 5.5663 10.0000 plot3( u2(1,:), u2(2,:), u2(3,:), 'o' )

64. Finite-element Methods in Two Dimensions 64 Example 2 Here we see the boundary points and the actual solution (light blue) and the approximations (in red)

65. Finite-element Methods in Two Dimensions 65 Summary This topic discusses an introduction to finite-element methods in two dimensions –We generalized the test functions in one dimension –Created a tessellation of the region R –Defined planar test functions Found the gradient of the planes making the test functions –Approximated the solution with piecewise planar sections –Created a linear equation This is done for each interior point –The resulting system of linear equations may be solved –Two examples…

66. Finite-element Methods in Two Dimensions 66 What’s Next? The next step is to go to three dimensions –Divide the region into tetrahedral regions –3-dimensional tessellations Tessellation User: Tom Ruen Robert Webb's Great Stella software http://www.software3d.com/Stella.html

67. Finite-element Methods in Two Dimensions 67 References [1]Glyn James, Advanced Modern Engineering Mathematics, 4 th Ed., Prentice Hall, 2011, §§9.2-3.

68. Finite-element Methods in Two Dimensions 68 Usage Notes These slides are made publicly available on the web for anyone to use If you choose to use them, or a part thereof, for a course at another institution, I ask only three things: –that you inform me that you are using the slides, –that you acknowledge my work, and –that you alert me of any mistakes which I made or changes which you make, and allow me the option of incorporating such changes (with an acknowledgment) in my set of slides Sincerely, Douglas Wilhelm Harder, MMath dwharder@alumni.uwaterloo.ca