1. 2/16/2017 Prof Xin (Cindy) Wang
MAE 4160/5160 Lecture 10
2/16/2017
Prof Xin (Cindy) Wang

2. Outline for today’s lecture
Continuous systems
Revisit: Assembly of stiffness matrix
Summarize: numerical quadrature
Finite Element in 2D
Strong form and Weak form in 2D
Vector Calculous: gradient, divergence, directional derivative, Laplacian
Green’s Theorem (Special case of divergence theorem)
2D linear elements

3. Revisit: matrix assembly
Global to Local

4. Stiffness and Mass Matrix
K_{ij} = \int^{L}_{0} p(x) \frac{d{\color[rgb]{ , , }\phi_i(x)}}{dx}\frac{d{\color[rgb]{ , , }\phi_j}(x)}{dx} dx 1 2 3 4 5

5. 1 2 3 4 5 1 2 3 4 5
M_{22} = \int^{L}_{0} q(x) {\color[rgb]{ , , }\phi_2(x) }{\color[rgb]{ , , }\phi_2(x)} dx
 M_{2,2}= \int_{\textcircled{2}} q(x) {\color[rgb]{ , , }N_2^{\textcircled{2}}(x) }{\color[rgb]{ , , }N_2^{\textcircled{2}}(x)} dx +
\int_{\textcircled{3}} q(x) {\color[rgb]{ , , }N_1^{\textcircled{3}}(x) }{\color[rgb]{ , , }N_1^{\textcircled{3}}(x)} dx

6. 1 2 3 4 5 1 2 3 4 5
M_{22} = \int^{L}_{0} q(x) {\color[rgb]{ , , }\phi_2(x) }{\color[rgb]{ , , }\phi_2(x)} dx
 M_{2,2}= \int_{\textcircled{2}} q(x) {\color[rgb]{ , , }N_2^{\textcircled{2}}(x) }{\color[rgb]{ , , }N_2^{\textcircled{2}}(x)} dx +
\int_{\textcircled{3}} q(x) {\color[rgb]{ , , }N_1^{\textcircled{3}}(x) }{\color[rgb]{ , , }N_1^{\textcircled{3}}(x)} dx

7. 1 2 3 4 5
M_{22} = \int^{L}_{0} q(x) {\color[rgb]{ , , }\phi_2(x) }{\color[rgb]{ , , }\phi_2(x)} dx
 M_{2,2}= \int_{\textcircled{2}} q(x) {\color[rgb]{ , , }N_2^{\textcircled{2}}(x) }{\color[rgb]{ , , }N_2^{\textcircled{2}}(x)} dx +
\int_{\textcircled{3}} q(x) {\color[rgb]{ , , }N_1^{\textcircled{3}}(x) }{\color[rgb]{ , , }N_1^{\textcircled{3}}(x)} dx 1 2 3 4 5

8. 1 2 3 4 5
M_{2,3} = \int^{L}_{0} q(x) {\color[rgb]{ , , }\phi_2(x) }{\color[rgb]{ , , }\phi_3(x)} dx
 M_{2,3}= \int_{\textcircled{3}} q(x) {\color[rgb]{ , , }N_1^{\textcircled{3}}(x) }{\color[rgb]{ , , }N_2^{\textcircled{3}}(x)} dx 1 2 3 4 5

9. 1 2 3 4 5 1 2 3 4 5
M_{3,3}=\int_{\textcircled{3}} q(x) {\color[rgb]{ , , }\phi_3(x) }{\color[rgb]{ , , }\phi_3(x)} dx
\int_{\textcircled{3}} q(x) {\color[rgb]{ , , }N_2^{\textcircled{3}}(x) }{\color[rgb]{ , , }N_2^{\textcircled{3}}(x)} dx
 M_{2,2}= \int_{\textcircled{1}} q(x) {\color[rgb]{ , , }\phi_2(x) }{\color[rgb]{ , , }\phi_2(x)} dx +
\int_{\textcircled{2}} q(x) {\color[rgb]{ , , }\phi_2(x) }{\color[rgb]{ , , }\phi_2(x)} dx

10. 1 2 3 4 5 1 2 3 4 5
M_{22} = \int^{L}_{0} q(x) {\color[rgb]{ , , }\phi_2(x) }{\color[rgb]{ , , }\phi_2(x)} dx
 M_{2,2}= \int_{\textcircled{1}} q(x) {\color[rgb]{ , , }\phi_2(x) }{\color[rgb]{ , , }\phi_2(x)} dx +
\int_{\textcircled{2}} q(x) {\color[rgb]{ , , }\phi_2(x) }{\color[rgb]{ , , }\phi_2(x)} dx

11. 1 2 3 4 5 1 2 3 4 5
M_{22} = \int^{L}_{0} q(x) {\color[rgb]{ , , }\phi_2(x) }{\color[rgb]{ , , }\phi_2(x)} dx
 M_{3,3}= \int_{\textcircled{3}} q(x) {\color[rgb]{ , , }N_2^{\textcircled{3}}(x) }{\color[rgb]{ , , }N_2^{\textcircled{3}}(x)} dx +
\int_{\textcircled{4}} q(x) {\color[rgb]{ , , }N_1^{\textcircled{4}}(x) }{\color[rgb]{ , , }N_1^{\textcircled{4}}(x)} dx

12. Recap: Global to Local

13. 1 2 3 4 5
M_{22} = \int^{L}_{0} q(x) {\color[rgb]{ , , }\phi_2(x) }{\color[rgb]{ , , }\phi_2(x)} dx
 M_{2,2}= \int_{\textcircled{2}} q(x) {\color[rgb]{ , , }N_2^{\textcircled{2}}(x) }{\color[rgb]{ , , }N_2^{\textcircled{2}}(x)} dx +
\int_{\textcircled{3}} q(x) {\color[rgb]{ , , }N_1^{\textcircled{3}}(x) }{\color[rgb]{ , , }N_1^{\textcircled{3}}(x)} dx 1 2 3 4 5

14. 1 2 3 4 5

15. 1 2 3 4 5 1 2 3 4 5
M_{22} = \int^{L}_{0} q(x) {\color[rgb]{ , , }\phi_2(x) }{\color[rgb]{ , , }\phi_2(x)} dx
 M_{3,3}= \int_{\textcircled{3}} q(x) {\color[rgb]{ , , }N_2^{\textcircled{3}}(x) }{\color[rgb]{ , , }N_2^{\textcircled{3}}(x)} dx +
\int_{\textcircled{4}} q(x) {\color[rgb]{ , , }N_1^{\textcircled{4}}(x) }{\color[rgb]{ , , }N_1^{\textcircled{4}}(x)} dx

16. Reverse the order: local to global
Implementation:
Reverse the order: local to global

17. 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

18. PseudoCode for Matrix Assembly
For every element
If Boundary element
Take care of boundary conditions
Else
For every local basis i in element
For every local basis j in element
Integrate {q(x) * Ni(x)Nj(x) } over elelment
Find out which global basis i and global basis j
Add to existing K(global basis i, global basis j)

19. Bandwidth of Matrices
K_{ij} = \int^{L}_{0} p(x) \frac{d{\color[rgb]{ , , }\phi_i(x)}}{dx}\frac{d{\color[rgb]{ , , }\phi_j}(x)}{dx} dx
What is the bandwidth of the matrices for pth order FEA element?

20. 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

21. Numerical Integration/Quadrature

22. Gaussian Quadrature
Integrates p orders of polynomials exactly for given m!

23. Solutions of matrix-vector equation
a = \begin{bmatrix}
a_1 \\
a_2 \\
\vdots\\
a_n
\end{bmatrix}
\tilde{u}(x) = \sum^n_{i=1} a_i\phi_i(x) 5 2 3 4 1

24. FEA – Road Map Strong form 1 2 3 4 5 Weak form
Weak form
FEA basis expansion (Meshing)
FEA matrix-vector assembly
Solve for K a + M a = f