1. Introduction to the Finite Element Method Spring 2010

6. Numerical Solution of Boundary Value Problems Weighted Residual Methods

15. Collocation Method

16. Example Problem

17. The bar tensile problem

18. Bar application Applying the collocation method

19. In Matrix Form Solve the above system for the “generalized coordinates” a i to get the solution for u(x)

25. The Subdomain Method

26. Bar application Applying the subdomain method

27. In Matrix Form Solve the above system for the “generalized coordinates” a i to get the solution for u(x)

29. The Galerkin Method

30. Bar application Applying Galerkin method

31. In Matrix Form Solve the above system for the “generalized coordinates” a i to get the solution for u(x)

33. Substituting with the approximate solution:

34. Substituting with the approximate solution: (Int. by Parts) Zero!

40. Exact Solution

41. The Finite Element Method 2 nd order DE’s in 1-D

48. Polynomial Approximation

53. What happens for adjacent elements?

58. Two–Element example

60. Performing Integration: Note that if the integration is evaluated from 0 to h e , where h e is the element length, the same results will be obtained .

61. Two–Element bar example

62. Applying Boundary Conditions

69. Bars and Trusses

70. Objectives

79. Equation of Motion

80. Transformation Matrix

82. Recall Where:

83. Element Stiffness Matrix in Global Coordinates

84. Element Stiffness Matrix in Global Coordinates

86. Element Equations

87. Assembly Procedure

88. Global Force Vector Remember! NO distributed load is applied to a truss

89. Boundary Conditions Remove the corresponding rows and columns Continue! (as before)

90. Results

91. Postcomputation

92. Postcomputation

109. Beams and Frames

114. Beam Interpolation Function

115. Beam Interpolation Function

116. Beam Interpolation Function

117. Beam Interpolation Function

118. Beam Interpolation Function

119. Interpolation Functions

123. Use of Symbolic Manipulator Beam Example

125. Two Dimensional Elements

129. Let’s follow the same procedure!

130. 2-D Interpolation Function

131. 2-D Interpolation Function

132. 2-D Interpolation Function

133. How does this look like?

134. 2-D Interpolation Functions

135. 2-D Interpolation Functions

136. Example: Laplace Equation

137. Example: Laplace Equation Applying the Galerkin method and integrating by parts, the element equation becomes

138. The Element Equaiton

139. The Logistic Problem!

142. 2-D Example

143. 2-D Example

144. For Element #5 Global Node Number Local Node Number 5 1 6 2 9 3 8 4

145. Contribution of element #5 to global matrix 12 11 10 9 8 7 6 5 4 3 2 1 1 2 3 4 1,3 1,4 1,2 1,1 5 2,3 2,4 2,2 2,1 6 7 4,3 4,4 4,2 4,1 8 3,3 3,4 3,2 3,1 9 10 11 12

147. Elements Register: Global Numbering Node Number Element Number 4 3 2 1 4 5 2 1 1 7 8 5 4 2 10 11 8 7 3 5 6 3 2 4 8 9 6 5 5 11 12 9 8 6