One-Parameter Residual Equations

Transcription

1 4 One-Parameter Residual Equations NFEM Ch 4 Slide 1

2 Total Force Residual Equation with One Control Parameter r(u, ) = 0 total force residual state vector staging parameter = single control parameter Derivatives with respect to will be often abbreviated with primes. For example u = u r = 2 r 2 = 1 = 0 NFEM Ch 4 Slide 2

3 Residual Derivatives Parametric representation of state and control as functions of pseudotime t : u = u(t) = (t) First two derivatives wrt t in matrix form:... r = K u q r = K u + K u q q in which K = r u, q = r = r NFEM Ch 4 Slide 3

4 Rate (Incremental) Equations These are obtained by setting residual derivatives to zero. First order incremental equation:... r = 0 or K u = q Second order incremental equation: r = 0 or K u + K u = q + q Both are systems of ordinary differential equations (ODE) in pseudotime t NFEM Ch 4 Slide 4

5 Incremental Velocity The first order rate equation introduced in last slide is... r = 0 or K u = q If K is nonsingular (a regular equilibrium point)... def. 1 u = K q = u' = v in which 1 v = u' = K q is called the incremental velocity NFEM Ch 4 Slide 5

6 Separable Residuals and Proportional Loading The balanced force residual form of r(u, ) = 0 is p(u) = f(u, ) If the external forces do not depend on the state p(u) = f( ) the residual is called separable. Furthermore if f is linear in, the loading is said to be proportional. NFEM Ch 4 Slide 6

7 Response Plots: Positive and NegativeTraversal Sense Response curve r = 0 (primary equilibrium path) u + Pseudo-time t increasing Pseudo-time t decreasing u t P t + Positive tangent vector Negative tangent vector u NFEM Ch 4 Slide 7

8 Response Visualization P(u, ) t + normal "hyperplane" at P P(u, ) t + positive tangent vector at P positive tangent vector at P equilibrium path r = 0 equilibrium path r = 0 u u Incremental flow Flow orthogonal envelope NFEM Ch 4 Slide 8

9 Example of Previous Lecture A E,A constant B A C' P = EA B θ θ u C k = β EA L k L 2L L NFEM Ch 4 Slide 9

10 Example Problem Response With a Compressible Initial Strain Load factor = P/EA Response using Green-Lagrange strain measure and e = β = 1 β = 1/10 β = 1/100, 1/1000 (indistinguishable at plot scale) Dimensionless displacement µ = u/l NFEM Ch 4 Slide 10

11 Load factor = P/EA Incremental Flow for Previous Response Dimensionless displacement µ = u/l Done by setting r(µ, ) = r c(a constant) value), solving for = (µ, r c), and plotting response curves for sample values of r shown on Figure c NFEM Ch 4 Slide 11

12 Incremental Flow Using Contour Plots r=µ*(β+(2*e0+µ^2)/sqrt[1+µ^2])-; r=simplify[r/.{β->1/10,e0->-0.2}]; ContourPlot[-r,{µ,-1,1},{,-0.25,.25},PlotPoints->101]; ContourPlot[-Sqrt[Abs[r]],{µ,-1,1},{,-0.25,.25},PlotPoints->101]; ContourPlot[-Sqrt[Abs[r]],{µ,-1,1},{,-0.25,.25},PlotPoints->301]; Does not require solving for (which may be inconvenient or impossible), but care must be taken to get reasonably good grading near r = 0, as well as sufficient plot resolution there NFEM Ch 4 Slide 12

13 Incremental Flow for Multiple DOF Problem r = 0 r = 0 u2 u2 u 1 u 1 NFEM Ch 4 Slide 13

14 Tangent Vector and Normal Hyperplane Normal.. hyperplane T v u + = 0 at P P t + + sense of increasing t P t + Equilibrium path r = 0 u 2 u 1 NFEM Ch 4 Slide 14

15 ArcLength Distance Normal hyperplane at P P s Q Positive tangent direction Equilibrium path r = 0 u 2 u 1 NFEM Ch 4 Slide 15

16 Unnormalized t = Tangent Vector [ ] u = [ ] v 1 P P Normal.. hyperplane T v u + = 0 at P t + + sense of increasing t r = 0 Normalized unit tangent vector with positive sense [ ] + v/f t u = 1/f in which u 1 t + Equilibrium path u 2 f = t =+ t 2 =+ 1 + v T v is a normalization factor NFEM Ch 4 Slide 16

17 ArcLength Distance and Orthogonal Flow Normal hyperplane at P The orthogonal hyperplane equation is (see Figure) P s Q Positive tangent direction where u = u u v T u + = 0 P = are increments from P. Dividing by s and t and passing to the limit we get v T.. u + = 0 as the differential equation for the orthogonal flow envelope P u 1 r = 0 Equilibrium path u 2 NFEM Ch 4 Slide 17