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

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

Revisit: matrix assembly Global to Local

Stiffness and Mass Matrix K_{ij} = \int^{L}_{0} p(x) \frac{d{\color[rgb]{0.000000,0.000000,1.000000}\phi_i(x)}}{dx}\frac{d{\color[rgb]{1.000000,0.000000,0.000000}\phi_j}(x)}{dx} dx 1 2 3 4 5

1 2 3 4 5 1 2 3 4 5 M_{22} = \int^{L}_{0} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_2(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_2(x)} dx  M_{2,2}= \int_{\textcircled{2}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_2^{\textcircled{2}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_2^{\textcircled{2}}(x)} dx + \int_{\textcircled{3}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_1^{\textcircled{3}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_1^{\textcircled{3}}(x)} dx

1 2 3 4 5 1 2 3 4 5 M_{22} = \int^{L}_{0} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_2(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_2(x)} dx  M_{2,2}= \int_{\textcircled{2}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_2^{\textcircled{2}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_2^{\textcircled{2}}(x)} dx + \int_{\textcircled{3}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_1^{\textcircled{3}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_1^{\textcircled{3}}(x)} dx

1 2 3 4 5 M_{22} = \int^{L}_{0} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_2(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_2(x)} dx  M_{2,2}= \int_{\textcircled{2}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_2^{\textcircled{2}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_2^{\textcircled{2}}(x)} dx + \int_{\textcircled{3}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_1^{\textcircled{3}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_1^{\textcircled{3}}(x)} dx 1 2 3 4 5

1 2 3 4 5 M_{2,3} = \int^{L}_{0} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_2(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_3(x)} dx  M_{2,3}= \int_{\textcircled{3}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_1^{\textcircled{3}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_2^{\textcircled{3}}(x)} dx 1 2 3 4 5

1 2 3 4 5 1 2 3 4 5 M_{3,3}=\int_{\textcircled{3}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_3(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_3(x)} dx \int_{\textcircled{3}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_2^{\textcircled{3}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_2^{\textcircled{3}}(x)} dx  M_{2,2}= \int_{\textcircled{1}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_2(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_2(x)} dx + \int_{\textcircled{2}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_2(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_2(x)} dx

1 2 3 4 5 1 2 3 4 5 M_{22} = \int^{L}_{0} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_2(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_2(x)} dx  M_{2,2}= \int_{\textcircled{1}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_2(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_2(x)} dx + \int_{\textcircled{2}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_2(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_2(x)} dx

1 2 3 4 5 1 2 3 4 5 M_{22} = \int^{L}_{0} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_2(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_2(x)} dx  M_{3,3}= \int_{\textcircled{3}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_2^{\textcircled{3}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_2^{\textcircled{3}}(x)} dx + \int_{\textcircled{4}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_1^{\textcircled{4}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_1^{\textcircled{4}}(x)} dx

Recap: Global to Local

1 2 3 4 5 M_{22} = \int^{L}_{0} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_2(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_2(x)} dx  M_{2,2}= \int_{\textcircled{2}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_2^{\textcircled{2}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_2^{\textcircled{2}}(x)} dx + \int_{\textcircled{3}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_1^{\textcircled{3}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_1^{\textcircled{3}}(x)} dx 1 2 3 4 5

1 2 3 4 5

1 2 3 4 5 1 2 3 4 5 M_{22} = \int^{L}_{0} q(x) {\color[rgb]{0.000000,0.000000,1.000000}\phi_2(x) }{\color[rgb]{1.000000,0.000000,0.000000}\phi_2(x)} dx  M_{3,3}= \int_{\textcircled{3}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_2^{\textcircled{3}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_2^{\textcircled{3}}(x)} dx + \int_{\textcircled{4}} q(x) {\color[rgb]{0.000000,0.000000,1.000000}N_1^{\textcircled{4}}(x) }{\color[rgb]{1.000000,0.000000,0.000000}N_1^{\textcircled{4}}(x)} dx

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

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

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)

Bandwidth of Matrices K_{ij} = \int^{L}_{0} p(x) \frac{d{\color[rgb]{0.000000,0.000000,1.000000}\phi_i(x)}}{dx}\frac{d{\color[rgb]{1.000000,0.000000,0.000000}\phi_j}(x)}{dx} dx What is the bandwidth of the matrices for pth order FEA element?

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

Numerical Integration/Quadrature

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

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

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