Method of Weighted Residuals

1. Method of Weighted Residuals Lecture Notes Dr. Rakhmad Arief Siregar Universiti Malaysia Perlis

2. Introduction • In Numerical Analysis Subject (ENT 258), some basic concepts of FEM in term of line element has been introduced. • The displacement-force relations for the line elements are straightforward as there relations are readily described using only the concept of elementary strength of materials. • To extend the method of finite element analysis to more general application, additional mathematical technique are required. • Here, one of methods called as Method of weighted residuals will be discussed.

3. Method of Weighted Residuals • To solve many engineering problems that have complexities in geometry and loading, the approximate technique is required. • The Method of Weighted Residuals (MWR) is • An approximate technique for solving boundary value problem. • Utilizes trial function satisfying the prescribed boundary condition. • An integral formulation to minimize error over problem domain

4. Method of Weighted Residuals • We are given a body B with boundary S shown below • The boundary is divided into two regions, • Su with essential boundary condition • Sf with natural boundary condition

5. Method of Weighted Residuals • The essential boundary condition Su are specifications of solution on the boundary (i.e. known boundary displacement) • Natural boundary conditions St are specifications of derivatives of solution (e.i. surface tractions)

6. Method of Weighted Residuals • In this lecture, we will present the weighted residual methods in ways that require the approximating function to satisfy both the essential and natural boundary conditions. • In the next lecture, we will show how the weak form of the differential equation can be used to develop methods that loosen this requirement, so that only the essential boundary conditions must be satisfied by our approximating function. • The basic step in weighted residual methods is to assume a solution of the form: (Eq. 5.3 page 132)

7. How Weighted Residuals Work? • Let us first demonstrate how weighted residuals work using a bar subjected to body and end loads Uniform bar with body and tip loads

8. Uniform bar with body and tip loads • For static equilibrium, the summation of the forces is zero:

9. Uniform bar with body and tip loads • Rearranging and assuming constant area: • Taking the limit as Δx →0 :

10. Uniform bar with body and tip loads

11. Uniform bar with body and tip loads Assuming Young’s modulus E is constant, with f B (x)=b ,gives, with boundary conditions:

12. Solving with MRW • For the weight residual formulation, we first choose a weighting function w (x ) , multiply it by the weighting function: and then integrate over the entire body:

13. Solving with MRW • This is called the weighted residual formulation. • It is called this because if we assume an approximate solution un(that satisfies all boundary conditions) then

14. Solving with MRW • Instead, we have an error (residual) that is a function of x . Thus integration is really a weighting of the residual over the body: • We have taken the error (residual), multiplied by a weighting function and set the weighted integral to zero.

15. Using One Term Approximation • To see how this allows us to obtain values for the coefficients, let us do an example. • We will pick the approximate function to satisfy all boundary conditions. Recalled the boundary conditions: Satisfy or not?

16. ? ? Using One Term Approximation • Taking the derivatives:

17. Using One Term Approximation • Substitute the second derivative into residual formulation to obtain the residual, • The result will be:

18. Using One Term Approximation • For simplicity, pick A= E = P = L =b =1. Then it becomes:

19. Using One Term Approximation • Figure 2-3 plots the residual for different values of a1. Residual over body

20. Collocation Method • In this case, we force the residual to be zero at a specific location (“nail-down” method). • That is: • This is equivalent to selecting the Dirac delta function as the weighting function.

21. Collocation Method • For any function f (x), • That is, the weighted residual is

22. Collocation Method • If we pick xi = 0.5, that is, we want the residual to equal zero at the midpoint • Then inspection of Figure 2-3 shows that a value of a1 ≈ 0.60 satisfies that condition.

23. Collocation Method • Solving exactly,

24. Subdomain Method • Alternately, let us weight the residual uniformly over the interval (“glue” method). • That is: • That is, the weighted residual is

25. Subdomain Method • By inspection, we can again see that a value of a1 =0.6 will result in approximately equal areas above and below the x axis.

26. Subdomain Method • Let us solve exactly:

27. Least Squares Method • In least squares, we require that the squared residual be minimized with respect to the adjusting parameter, i.e.,

28. Least Squares Method • In least squares, we require that the squared residual be minimized with respect to the adjusting parameter, i.e., • That is, the weighted residual is

29. Least Squares Method • This is equivalent to selecting as the weighting function. • For our particular problem, there is only one adjusting parameter a1

30. Galerkin Method • Finally, if we use the same function for the weighting function • As we used for the approximating function (except that we do not include the term in the approximating function that satisfies the essential boundary conditions, which does not enter into this case)

31. Galerkin Method • Then we have:

32. Comparison of Solutions

33. One dimensional case • Given differential equation form: • Boundary condition • MWR seeks an approximate solution as:

34. One dimensional case • Where: • y* : approximate solution • ci : unknown constant parameters to be determine • Ni(x) trial function

35. One dimensional case • The trial functions are continues over the domain of interest and the specified boundary condition. • On substitution of assumed solution, a residual error results as: • However the residual is also a function of the unknown parameter ci • MWR required the ci be evaluated:

36. One dimensional case • Several variation of MWR exist in how the weight factors are determined or selected. • The most common is Galerkin’s method • In Galerkin’s MWR the weight function are chosen to be identical to the trial function that is • Therefore, the unknown parameters are determine via:

37. Example 1 • Use Galerkin’s method of weighted residual to obtain an approximate solution of the differential equation • With boundary conditions y(0)=y(1)=0

38. Solution 1

39. Solution 2 (minimizing the errors)

40. The Galerkin Finite Element Method

41. Fig. 5.4a

42. Fig. 5.4b

43. Fig. 5.5

44. Fig. 5.6

45. Fig. 5.7

46. Fig. 5.8a

47. Fig. 5.8b

48. Fig. 5.9

49. Fig. P5.11

50. Fig. P5.12