Finite Element Discretization for Structural Domain Equations
Explore figures depicting numerical solutions, exact solutions, and finite element discretizations for structural domain equations. Understand concepts such as approximate solutions, interpolated solutions, and gradients, with comparisons between exact and approximate values.
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Presentation Transcript
Figure 2.1 1.4 1.2 1 u(x) 0.8 0.6 0.4 Exact solution Approx. solution with W = 1 Approx. solution with W = x 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x 1 Figure 2.2 0.4 0.3 0.2 0.2 R(x)W(x) R(x)W(x) 0 0.1 -0.2 0 -0.4 -0.1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x x (a) W = 1 (b) W = x
Figure 2.3 1.6 1.4 1.2 u(x), du/dx 1 0.8 uexact uapprox duexact/dx duapprox/dx 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 x Figure 2.4 0.1 0.05 0 u(x), du/dx -0.05 uexact uapprox duexact/dx duapprox/dx -0.1 -0.15 -0.2 -0.25 0 0.2 0.4 0.6 0.8 1 x
4 Figure 2.5 3 2 wexact wapprox wexact wapprox w, w 1 0 -1 -2 0 0.2 0.4 0.6 0.8 1 x Figure 2.6 F t s xy y xx x + + = 0 b x Structure t s xy x yy y + + = 0 b y (a) Structural domain with governing differential equations F Element (b) Finite element discretization
Figure 2.7 u(x) Nodes Approximate solution x Finite elements Exact solution Figure 2.8 u Exact solution Two elements Four elements Eight elements x n 1 1 2 n Figure 2.9 n 1 n+1 n 1 2 3 ui+1 ui i+1 i xi L(i) xi+1
Figure 2.10 u ui+2 d d u x ui ui+1 xi xi+1 xi+2 xi xi+1 xi+2 (a) Interpolated solution (b) Gradient of solution Figure 2.11 1 ( ) f ix - 1 i ( ) 1/ L xi+1 xi 2 xi xi 1 ( ) i - 1/ L ( ) x d f i dx Figure 2.12 1.0 1(x) 2(x) 3(x) (x) 0.5 0 0.5 x 1.0 0
Figure 2.13 1.6 1.2 u(x) 0.8 u-exact u-approx. 0.4 0 0 0.2 0.4 0.6 0.8 1 x 2 Figure 2.14 1.5 1 du/dx du/dx (exact) du/dx (approx.) 0.5 0 0 0.2 0.4 0.6 0.8 1 x 1( ) 2( ) x N x N Figure 2.15 Element e i x j x x ( ) e L
Figure 2.16 0.08 u-approx. u-exact 0.06 u(x) 0.04 0.02 0 0 0.2 0.4 0.6 0.8 1 x Figure 2.17 F3 F2 3 2 r r 1 4 F1 F4 Equilibrium Virtual displacement Figure 2.18 xu x u mg mg
Figure 2.19 x E, A(x) F Bx L u2 Figure 2.20 F3 1 3 2 1 F3 3 2 u3 u1 bx Figure 2.21 F u1 u2 u3 Figure P2.17 1 2 3 1 m 1 m x Figure P2.18 x f L
Figure P2.19 x E A L q Figure P2.20 L = 1 m Figure P2.21 0.3 106 N 0.3 m Figure P2.22 RL RR F 0.3 m 0.4 m 0.3 m q Figure P2.23 LT q = cx E, A Figure P2.26 L x, u
Figure P2.27 y 1 2 x x = 1 x = -1 u1 u2 u3 Figure P2.28 1 2 3 1/4 3/4 x u1 u2 u3 1 2 3 x = -1 x = 0 x = 1 q x
