Example F: Finite Element Beam with Nonlinear Support
This example considers a finite element model of a Beam (Euler-Bernoulli) that has a nonlinear attachment in one end and is clamped in the other. The same code can be used for simulating 3 different types of nonlinear attachments:
- A simple cubic spring of the form $\beta u^3$.
- A stiff string attachment with reaction force of the form $T \frac{w_T}{\sqrt{\ell_s^2+w_T^2}}$
- A non-smooth frictional joint.
Preamble: Load Packages
using GLMakie
using LinearAlgebra
using Random
using NonlinearSolve
using SparseArrays
using Revise
using juliajim.HARMONIC
using juliajim.MDOFUTILS
using juliajim.CONTINUATIONSetup System
Ey = 70e9;
rho = 2700.;
w, h = 3*25e-3, 12.5e-3;
ell = 1.0;
ρA = rho*w*h;
EI = Ey*w*h^3/12;Build Finite Element Modela
Ne = 10; # Number of elements
Nn = Ne+1; # Number of nodes
Xn = range(0, ell, Nn);
Me(Le) = ρA*Le/420*[156 22Le 54 -13Le;
22Le 4Le^2 13Le -3Le^2;
54 13Le 156 -22Le;
-13Le -3Le^2 -22Le 4Le^2];
Ke(Le) = EI/Le^3*[12 6Le -12 6Le;
6Le 4Le^2 -6Le 2Le^2;
-12 -6Le 12 -6Le;
6Le 2Le^2 -6Le 4Le^2];
M = zeros(2Nn, 2Nn);
K = zeros(2Nn, 2Nn);
for ei in 1:Ne
is = 2(ei-1)+1;
ie = 2(ei+1);
M[is:ie, is:ie] += Me(diff(Xn[ei:ei+1])[1]);
K[is:ie, is:ie] += Ke(diff(Xn[ei:ei+1])[1]);
end
F = [zeros(2Nn-2); 1.;0.];Apply Clamped Boundary Condition
Lb = I(2Nn)[:, 3:end];
Mb = Lb'M*Lb;
Kb = Lb'K*Lb;
Fb = Lb'F;
Wn = sqrt.(eigvals(Kb, Mb));
Zts = [0.2e-2, 0.1e-2];
ab = [1 ./2Wn[1:2] Wn[1:2]/2]\Zts;
Cb = ab[1]*Mb + ab[2]*Kb;Choose and Setup Nonlinearity
You can try out this example with any of the following nonlinearities. The setup is to have a tip nonlinearity either as a Cubic spring, (axially) stiffened string, or a frictional damper.
Cubic Spring
β = 500.0; # Nonlinearity
fnl = (t, u, ud) -> return β.*u.^3, 3β.*u.^2, zeros(size(u));
typ = :Inst;Stiffened String
Ts = 1e2;
ls = 0.10;
fnl = (t, u, ud) -> return Ts.*u./sqrt.(ls^2 .+u.^2),
Ts*ls^2 ./sqrt.(ls^2 .+u.^2).^3, zeros(size(u));
typ = :Inst;Frictional Support
kt = 500;
fs = 1e-2kt;
fnl = (t,u,up,fp)-> if all(abs.(fp+kt*(u-up)).<fs)
return fp+kt*(u-up), kt*ones(size(u)), -kt*ones(size(u)), ones(size(u)); else
return fs*sign.(fp+kt*(u-up)), zeros(size(u)), zeros(size(u)), zeros(size(u));
end
typ = :Hyst;
mdl = MDOFGEN(Mb, Cb, Kb);
mdl = ADDNL(mdl, typ, fnl, Float64.(Lb[end-1:end-1,:]));Setup HB
h = 0:5;
N = 256;
Nhc = NHC(h);
_, _, zinds, rinds, iinds = HINDS(mdl.Ndofs, h);
Fl = zeros(Nhc*mdl.Ndofs);
Fl[rinds[1:mdl.Ndofs]] = Fb;
Om0 = 10.0;
Om1 = 450.0;
dOm = 10.0;Linear Initial Guess
E, _ = HARMONICSTIFFNESS(mdl.M, mdl.C, mdl.K, Om0, h);
U0_(Fa) = E\ (Fa*Fl);Setup nonlinear function
fun(Fa) = NonlinearFunction((r,u,p)->HBRESFUN!([u;p], mdl, Fa*Fl, h, N; R=r),
jac=(J,u,p)->HBRESFUN!([u;p], mdl, Fa*Fl, h, N; dRdU=J),
paramjac=(Jp,u,p)->HBRESFUN!([u;p], mdl, Fa*Fl, h, N; dRdw=Jp));Forced Response Continuation
Continuation: The main parameters that can be changed are the step size (dOm here) and the angopt parameter.
angopt represents the desired angle between the secant and the tangent in a scaled space. Smaller angles represent that the secant is close to the tangent, i.e., the response curve is approximately a straight line.
We scale the vectors by the tangent so that the tangent at the previous point is represented by the orthotropic vector normalize(ones(N)). We exclude those unknowns that are close to zero.
cpars = (parm=:arclength, nmax=4000, Dsc=:auto, angopt=deg2rad(10), itopt=4);
Om0 = 40.0;
Om1 = 100.0;
dOm = 5.0;
solss = [];
Uhs = [];
Famps = [0.5, 1., 2., 4., 8., 16.];
for Famp in Famps
sols, _, _, _, _ = CONTINUATE(U0_(Famp), fun(Famp), [Om0, Om1], dOm; cpars...);
Uh = [Lb[end-1,:]'reshape(u, mdl.Ndofs,:) for u in sols.u];
push!(solss, sols)
push!(Uhs, Uh);
end
Algorithm: NewtonRaphson(
descent = NewtonDescent(),
autodiff = AutoForwardDiff(),
vjp_autodiff = AutoFiniteDiff(
fdtype = Val{:forward}(),
fdjtype = Val{:forward}(),
fdhtype = Val{:hcentral}(),
dir = true
),
jvp_autodiff = AutoForwardDiff(),
concrete_jac = Val{false}()
)
---- ------------- -----------
Iter f(u) inf-norm Step 2-norm
---- ------------- -----------
0 6.45228366e-02 0.00000000e+00
1 4.87498930e-13 1.50635845e-04
Final 4.87498930e-13
----------------------
2. 43.08 with step 3.2255 (0.1004) converged in 2 iterations. 0.559375 Δα.
3. 47.17 with step 4.0945 (0.0990) converged in 1 iterations. 0.843030 Δα.
4. 52.58 with step 5.4375 (0.1238) converged in 2 iterations. 1.085151 Δα.
5. 58.07 with step 5.6071 (0.1185) converged in 2 iterations. 1.554310 Δα.
6. 63.14 with step 5.6228 (0.1134) converged in 3 iterations. 2.310465 Δα.
7. 66.12 with step 4.0433 (0.0938) converged in 3 iterations. 2.773627 Δα.
8. 67.57 with step 2.0628 (0.0776) converged in 2 iterations. 2.591773 Δα.
9. 68.40 with step 1.1501 (0.0743) converged in 2 iterations. 2.248431 Δα.
10. 68.90 with step 0.6718 (0.0711) converged in 2 iterations. 1.950348 Δα.
11. 69.23 with step 0.4206 (0.0681) converged in 2 iterations. 1.710645 Δα.
12. 69.46 with step 0.2797 (0.0651) converged in 2 iterations. 1.516055 Δα.
13. 69.62 with step 0.1954 (0.0623) converged in 2 iterations. 1.353786 Δα.
14. 69.75 with step 0.1420 (0.0597) converged in 2 iterations. 1.214359 Δα.
15. 69.84 with step 0.1067 (0.0571) converged in 2 iterations. 1.091140 Δα.
16. 69.89 with step 0.0823 (0.0547) converged in 5 iterations. 10.613445 Δα.
17. 69.90 with step 0.0118 (0.0381) converged in 2 iterations. 0.771576 Δα.
18. 69.92 with step 0.0123 (0.0365) converged in 2 iterations. 1.095388 Δα.
19. 69.93 with step 0.0132 (0.0349) converged in 2 iterations. 1.076621 Δα.
20. 69.94 with step 0.0148 (0.0334) converged in 2 iterations. 0.320000 Δα.
21. 69.96 with step 0.0172 (0.0360) converged in 2 iterations. 0.579990 Δα.
22. 69.98 with step 0.0183 (0.0353) converged in 2 iterations. 0.628849 Δα.
23. 70.00 with step 0.0178 (0.0343) converged in 2 iterations. 0.700478 Δα.
24. 70.02 with step 0.0197 (0.0328) converged in 2 iterations. 0.465547 Δα.
25. 70.04 with step 0.0230 (0.0332) converged in 2 iterations. 0.112645 Δα.
26. 70.08 with step 0.0353 (0.0442) converged in 2 iterations. 0.210629 Δα.
27. 70.13 with step 0.0479 (0.0513) converged in 2 iterations. 0.443252 Δα.
28. 70.18 with step 0.0482 (0.0524) converged in 2 iterations. 0.773751 Δα.
29. 70.24 with step 0.0542 (0.0502) converged in 2 iterations. 0.380234 Δα.
30. 70.30 with step 0.0603 (0.0525) converged in 2 iterations. 0.589047 Δα.
31. 70.37 with step 0.0630 (0.0514) converged in 2 iterations. 0.641148 Δα.
32. 70.44 with step 0.0646 (0.0497) converged in 2 iterations. 0.555481 Δα.
33. 70.51 with step 0.0669 (0.0491) converged in 3 iterations. 0.310947 Δα.
34. 70.57 with step 0.0662 (0.0467) converged in 2 iterations. 0.156553 Δα.
35. 70.64 with step 0.0717 (0.0577) converged in 2 iterations. 0.875837 Δα.
36. 70.71 with step 0.0701 (0.0552) converged in 2 iterations. 0.140657 Δα.
37. 70.79 with step 0.0826 (0.0698) converged in 2 iterations. 0.880160 Δα.
38. 70.86 with step 0.0743 (0.0668) converged in 2 iterations. 0.887401 Δα.
39. 70.92 with step 0.0623 (0.0640) converged in 3 iterations. 0.899230 Δα.
40. 70.95 with step 0.0397 (0.0529) converged in 2 iterations. 1.050851 Δα.
41. 71.03 with step 0.0279 (0.0507) converged in 5 iterations. 32.066312 Δα.
42. 71.09 with step 0.0572 (0.0353) converged in 2 iterations. 1.131461 Δα.
43. 71.16 with step 0.0632 (0.0338) converged in 2 iterations. 1.213368 Δα.
44. 71.24 with step 0.0699 (0.0324) converged in 2 iterations. 1.296244 Δα.
45. 71.32 with step 0.0774 (0.0310) converged in 2 iterations. 1.380473 Δα.
46. 71.42 with step 0.0857 (0.0296) converged in 2 iterations. 1.466589 Δα.
47. 71.52 with step 0.0950 (0.0284) converged in 2 iterations. 1.555079 Δα.
48. 71.64 with step 0.1054 (0.0271) converged in 2 iterations. 1.646379 Δα.
49. 71.77 with step 0.1171 (0.0260) converged in 2 iterations. 1.740879 Δα.
50. 71.91 with step 0.1302 (0.0249) converged in 2 iterations. 1.838913 Δα.
51. 72.07 with step 0.1448 (0.0238) converged in 2 iterations. 1.940740 Δα.
52. 72.25 with step 0.1612 (0.0228) converged in 2 iterations. 2.046517 Δα.
53. 72.45 with step 0.1796 (0.0218) converged in 2 iterations. 2.156260 Δα.
54. 72.67 with step 0.2001 (0.0209) converged in 2 iterations. 2.269778 Δα.
55. 72.92 with step 0.2242 (0.0201) converged in 2 iterations. 2.386968 Δα.
56. 73.21 with step 0.2611 (0.0201) converged in 2 iterations. 2.510131 Δα.
57. 73.55 with step 0.3061 (0.0201) converged in 2 iterations. 2.640251 Δα.
58. 73.95 with step 0.3609 (0.0201) converged in 2 iterations. 2.775079 Δα.
59. 74.43 with step 0.4277 (0.0201) converged in 2 iterations. 2.910073 Δα.
60. 75.00 with step 0.5084 (0.0201) converged in 2 iterations. 3.037330 Δα.
61. 75.68 with step 0.6045 (0.0201) converged in 2 iterations. 3.144838 Δα.
62. 76.48 with step 0.7157 (0.0201) converged in 2 iterations. 3.217257 Δα.
63. 77.40 with step 0.8388 (0.0201) converged in 2 iterations. 3.239647 Δα.
64. 78.45 with step 0.9678 (0.0201) converged in 2 iterations. 3.204164 Δα.
65. 79.61 with step 1.0943 (0.0201) converged in 2 iterations. 3.112850 Δα.
66. 80.89 with step 1.2111 (0.0201) converged in 2 iterations. 2.979171 Δα.
67. 82.25 with step 1.3140 (0.0201) converged in 2 iterations. 2.821155 Δα.
68. 83.69 with step 1.4024 (0.0201) converged in 2 iterations. 2.655048 Δα.
69. 85.20 with step 1.4780 (0.0201) converged in 2 iterations. 2.492225 Δα.
70. 86.76 with step 1.5435 (0.0201) converged in 2 iterations. 2.339167 Δα.
71. 88.38 with step 1.6014 (0.0201) converged in 2 iterations. 2.198835 Δα.
72. 90.05 with step 1.6540 (0.0201) converged in 2 iterations. 2.071961 Δα.
73. 91.76 with step 1.7026 (0.0201) converged in 1 iterations. 1.959419 Δα.
74. 93.96 with step 2.1858 (0.0251) converged in 2 iterations. 1.832348 Δα.
75. 96.13 with step 2.1529 (0.0240) converged in 1 iterations. 1.724103 Δα.
76. 98.90 with step 2.7654 (0.0300) converged in 2 iterations. 1.607337 Δα.
77. 101.64 with step 2.7275 (0.0287) converged in 1 iterations. 1.509518 Δα.
Algorithm: NewtonRaphson(
descent = NewtonDescent(),
autodiff = AutoForwardDiff(),
vjp_autodiff = AutoFiniteDiff(
fdtype = Val{:forward}(),
fdjtype = Val{:forward}(),
fdhtype = Val{:hcentral}(),
dir = true
),
jvp_autodiff = AutoForwardDiff(),
concrete_jac = Val{false}()
)
---- ------------- -----------
Iter f(u) inf-norm Step 2-norm
---- ------------- -----------
0 1.29045673e-01 0.00000000e+00
1 9.74997860e-13 3.01271691e-04
Final 9.74997860e-13
----------------------
2. 42.27 with step 2.3807 (0.0863) converged in 2 iterations. 0.471795 Δα.
3. 45.82 with step 3.5605 (0.0872) converged in 2 iterations. 0.799666 Δα.
4. 49.39 with step 3.5803 (0.0835) converged in 2 iterations. 0.950012 Δα.
5. 52.97 with step 3.6205 (0.0799) converged in 2 iterations. 1.168824 Δα.
6. 56.52 with step 3.6314 (0.0765) converged in 2 iterations. 1.473245 Δα.
7. 59.86 with step 3.5407 (0.0732) converged in 3 iterations. 1.854114 Δα.
8. 62.36 with step 2.7809 (0.0606) converged in 3 iterations. 2.185221 Δα.
9. 64.07 with step 1.9829 (0.0501) converged in 2 iterations. 2.324973 Δα.
10. 65.36 with step 1.5290 (0.0480) converged in 2 iterations. 2.290320 Δα.
11. 66.30 with step 1.1260 (0.0459) converged in 2 iterations. 2.169102 Δα.
12. 66.99 with step 0.8231 (0.0439) converged in 2 iterations. 2.023596 Δα.
13. 67.51 with step 0.6096 (0.0420) converged in 2 iterations. 1.882750 Δα.
14. 67.91 with step 0.4607 (0.0402) converged in 2 iterations. 1.755905 Δα.
15. 68.22 with step 0.3556 (0.0385) converged in 2 iterations. 1.644426 Δα.
16. 68.47 with step 0.2799 (0.0369) converged in 2 iterations. 1.547021 Δα.
17. 68.67 with step 0.2244 (0.0353) converged in 2 iterations. 1.461756 Δα.
18. 68.84 with step 0.1827 (0.0338) converged in 2 iterations. 1.386741 Δα.
19. 68.98 with step 0.1508 (0.0323) converged in 2 iterations. 1.320335 Δα.
20. 69.09 with step 0.1261 (0.0309) converged in 2 iterations. 1.261168 Δα.
21. 69.19 with step 0.1065 (0.0296) converged in 2 iterations. 1.208118 Δα.
22. 69.28 with step 0.0909 (0.0283) converged in 1 iterations. 1.160039 Δα.
23. 69.32 with step 0.1022 (0.0354) converged in 5 iterations. 22.882360 Δα.
24. 69.33 with step 0.0088 (0.0247) converged in 2 iterations. 1.628805 Δα.
25. 69.34 with step 0.0074 (0.0236) converged in 2 iterations. 0.020294 Δα.
26. 69.35 with step 0.0127 (0.0403) converged in 2 iterations. 0.904729 Δα.
27. 69.36 with step 0.0136 (0.0386) converged in 2 iterations. 1.500823 Δα.
28. 69.38 with step 0.0161 (0.0370) converged in 2 iterations. 1.050117 Δα.
29. 69.40 with step 0.0168 (0.0354) converged in 2 iterations. 1.894919 Δα.
30. 69.42 with step 0.0192 (0.0339) converged in 2 iterations. 1.408861 Δα.
31. 69.44 with step 0.0201 (0.0324) converged in 2 iterations. 1.722518 Δα.
32. 69.47 with step 0.0230 (0.0310) converged in 2 iterations. 0.267034 Δα.
33. 69.50 with step 0.0269 (0.0345) converged in 2 iterations. 1.686369 Δα.
34. 69.53 with step 0.0298 (0.0330) converged in 2 iterations. 0.512055 Δα.
35. 69.56 with step 0.0344 (0.0330) converged in 2 iterations. 0.369572 Δα.
36. 69.61 with step 0.0417 (0.0346) converged in 2 iterations. 0.162984 Δα.
37. 69.67 with step 0.0592 (0.0424) converged in 2 iterations. 0.110374 Δα.
38. 69.76 with step 0.0861 (0.0566) converged in 2 iterations. 0.319671 Δα.
39. 69.88 with step 0.1095 (0.0610) converged in 2 iterations. 0.330000 Δα.
40. 70.02 with step 0.1321 (0.0653) converged in 2 iterations. 0.491227 Δα.
41. 70.18 with step 0.1494 (0.0656) converged in 2 iterations. 0.548353 Δα.
42. 70.35 with step 0.1542 (0.0649) converged in 2 iterations. 0.896755 Δα.
43. 70.52 with step 0.1645 (0.0621) converged in 2 iterations. 0.526889 Δα.
44. 70.69 with step 0.1662 (0.0618) converged in 2 iterations. 0.514681 Δα.
45. 70.85 with step 0.1658 (0.0617) converged in 2 iterations. 0.636745 Δα.
46. 70.99 with step 0.1499 (0.0598) converged in 2 iterations. 0.550294 Δα.
47. 71.14 with step 0.1480 (0.0591) converged in 3 iterations. 0.529826 Δα.
48. 71.26 with step 0.1298 (0.0511) converged in 2 iterations. 0.942800 Δα.
49. 71.36 with step 0.1075 (0.0489) converged in 2 iterations. 0.345434 Δα.
50. 71.46 with step 0.1024 (0.0520) converged in 2 iterations. 0.863034 Δα.
51. 71.57 with step 0.0747 (0.0497) converged in 5 iterations. 4.091343 Δα.
52. 71.68 with step 0.1015 (0.0347) converged in 2 iterations. 1.190121 Δα.
53. 71.80 with step 0.1114 (0.0332) converged in 2 iterations. 1.252479 Δα.
54. 71.94 with step 0.1223 (0.0318) converged in 2 iterations. 1.316930 Δα.
55. 72.08 with step 0.1344 (0.0304) converged in 2 iterations. 1.383640 Δα.
56. 72.25 with step 0.1476 (0.0291) converged in 2 iterations. 1.452750 Δα.
57. 72.42 with step 0.1622 (0.0278) converged in 2 iterations. 1.524361 Δα.
58. 72.62 with step 0.1782 (0.0267) converged in 2 iterations. 1.598526 Δα.
59. 72.83 with step 0.1958 (0.0255) converged in 2 iterations. 1.675231 Δα.
60. 73.07 with step 0.2151 (0.0244) converged in 2 iterations. 1.754374 Δα.
61. 73.33 with step 0.2362 (0.0234) converged in 2 iterations. 1.835732 Δα.
62. 73.61 with step 0.2592 (0.0224) converged in 2 iterations. 1.918926 Δα.
63. 73.92 with step 0.2842 (0.0214) converged in 2 iterations. 2.003375 Δα.
64. 74.26 with step 0.3112 (0.0205) converged in 2 iterations. 2.088252 Δα.
65. 74.63 with step 0.3401 (0.0196) converged in 2 iterations. 2.172435 Δα.
66. 75.04 with step 0.3708 (0.0188) converged in 2 iterations. 2.254478 Δα.
67. 75.47 with step 0.4030 (0.0180) converged in 2 iterations. 2.332616 Δα.
68. 75.95 with step 0.4376 (0.0173) converged in 2 iterations. 2.404942 Δα.
69. 76.48 with step 0.4925 (0.0173) converged in 2 iterations. 2.470593 Δα.
70. 77.08 with step 0.5539 (0.0173) converged in 2 iterations. 2.527455 Δα.
71. 77.75 with step 0.6215 (0.0173) converged in 2 iterations. 2.572018 Δα.
72. 78.49 with step 0.6945 (0.0173) converged in 2 iterations. 2.599877 Δα.
73. 79.31 with step 0.7713 (0.0173) converged in 2 iterations. 2.607811 Δα.
74. 80.21 with step 0.8501 (0.0173) converged in 2 iterations. 2.594118 Δα.
75. 81.19 with step 0.9285 (0.0173) converged in 2 iterations. 2.559138 Δα.
76. 82.23 with step 1.0045 (0.0173) converged in 2 iterations. 2.505252 Δα.
77. 83.35 with step 1.0763 (0.0173) converged in 2 iterations. 2.436337 Δα.
78. 84.53 with step 1.1430 (0.0173) converged in 2 iterations. 2.356924 Δα.
79. 85.76 with step 1.2042 (0.0173) converged in 2 iterations. 2.271408 Δα.
80. 87.04 with step 1.2601 (0.0173) converged in 2 iterations. 2.183523 Δα.
81. 88.38 with step 1.3114 (0.0173) converged in 2 iterations. 2.096325 Δα.
82. 89.75 with step 1.3586 (0.0173) converged in 2 iterations. 2.011674 Δα.
83. 91.17 with step 1.4023 (0.0173) converged in 2 iterations. 1.930663 Δα.
84. 92.62 with step 1.4433 (0.0173) converged in 2 iterations. 1.854024 Δα.
85. 94.12 with step 1.4820 (0.0173) converged in 1 iterations. 1.783293 Δα.
86. 96.03 with step 1.8988 (0.0216) converged in 2 iterations. 1.698724 Δα.
87. 97.91 with step 1.8670 (0.0206) converged in 1 iterations. 1.624222 Δα.
88. 100.31 with step 2.3938 (0.0258) converged in 2 iterations. 1.539279 Δα.
Algorithm: NewtonRaphson(
descent = NewtonDescent(),
autodiff = AutoForwardDiff(),
vjp_autodiff = AutoFiniteDiff(
fdtype = Val{:forward}(),
fdjtype = Val{:forward}(),
fdhtype = Val{:hcentral}(),
dir = true
),
jvp_autodiff = AutoForwardDiff(),
concrete_jac = Val{false}()
)
---- ------------- -----------
Iter f(u) inf-norm Step 2-norm
---- ------------- -----------
0 2.58091346e-01 7.75740819e-09
1 1.94999572e-12 6.02543382e-04
Final 1.94999572e-12
----------------------
2. 41.50 with step 1.5623 (0.0699) converged in 1 iterations. 0.379159 Δα.
3. 45.25 with step 3.7630 (0.0936) converged in 2 iterations. 0.771245 Δα.
4. 48.98 with step 3.7626 (0.0896) converged in 2 iterations. 0.910673 Δα.
5. 52.69 with step 3.7796 (0.0857) converged in 2 iterations. 1.110609 Δα.
6. 56.28 with step 3.7368 (0.0820) converged in 2 iterations. 1.367938 Δα.
7. 59.52 with step 3.5305 (0.0785) converged in 3 iterations. 1.634660 Δα.
8. 61.84 with step 2.6352 (0.0650) converged in 3 iterations. 1.801029 Δα.
9. 63.41 with step 1.8161 (0.0537) converged in 2 iterations. 1.829975 Δα.
10. 64.61 with step 1.4053 (0.0514) converged in 2 iterations. 1.772725 Δα.
11. 65.53 with step 1.0663 (0.0492) converged in 2 iterations. 1.680833 Δα.
12. 66.23 with step 0.8111 (0.0471) converged in 2 iterations. 1.581774 Δα.
13. 66.78 with step 0.6249 (0.0451) converged in 2 iterations. 1.487571 Δα.
14. 67.21 with step 0.4894 (0.0432) converged in 2 iterations. 1.402207 Δα.
15. 67.56 with step 0.3897 (0.0413) converged in 2 iterations. 1.326264 Δα.
16. 67.84 with step 0.3151 (0.0395) converged in 2 iterations. 1.259073 Δα.
17. 68.08 with step 0.2585 (0.0378) converged in 2 iterations. 1.199603 Δα.
18. 68.19 with step 0.2147 (0.0362) converged in 4 iterations. 23.520305 Δα.
19. 68.21 with step 0.0217 (0.0272) converged in 2 iterations. 3.873629 Δα.
20. 68.22 with step 0.0144 (0.0260) converged in 2 iterations. 3.862056 Δα.
21. 68.23 with step 0.0098 (0.0249) converged in 2 iterations. 1.752757 Δα.
22. 68.24 with step 0.0078 (0.0238) converged in 2 iterations. 0.767922 Δα.
23. 68.25 with step 0.0070 (0.0228) converged in 2 iterations. 0.066222 Δα.
24. 68.26 with step 0.0106 (0.0346) converged in 2 iterations. 0.764811 Δα.
25. 68.27 with step 0.0107 (0.0331) converged in 2 iterations. 1.342003 Δα.
26. 68.28 with step 0.0124 (0.0317) converged in 2 iterations. 1.080188 Δα.
27. 68.30 with step 0.0128 (0.0304) converged in 2 iterations. 2.406859 Δα.
28. 68.31 with step 0.0145 (0.0291) converged in 2 iterations. 1.764447 Δα.
29. 68.33 with step 0.0168 (0.0278) converged in 2 iterations. 0.667039 Δα.
30. 68.35 with step 0.0178 (0.0268) converged in 2 iterations. 1.011410 Δα.
31. 68.37 with step 0.0191 (0.0256) converged in 2 iterations. 1.134211 Δα.
32. 68.39 with step 0.0210 (0.0245) converged in 2 iterations. 1.046793 Δα.
33. 68.41 with step 0.0229 (0.0235) converged in 1 iterations. 0.735363 Δα.
34. 68.45 with step 0.0326 (0.0293) converged in 2 iterations. 0.770469 Δα.
35. 68.48 with step 0.0362 (0.0281) converged in 2 iterations. 0.407118 Δα.
36. 68.53 with step 0.0394 (0.0291) converged in 2 iterations. 0.665864 Δα.
37. 68.57 with step 0.0442 (0.0280) converged in 2 iterations. 0.104213 Δα.
38. 68.64 with step 0.0678 (0.0379) converged in 2 iterations. 0.273549 Δα.
39. 68.73 with step 0.0884 (0.0419) converged in 2 iterations. 0.082181 Δα.
40. 68.89 with step 0.1405 (0.0602) converged in 2 iterations. 0.531060 Δα.
41. 69.07 with step 0.1740 (0.0598) converged in 2 iterations. 0.352926 Δα.
42. 69.30 with step 0.2133 (0.0633) converged in 2 iterations. 0.540412 Δα.
43. 69.56 with step 0.2338 (0.0628) converged in 2 iterations. 0.949488 Δα.
44. 69.84 with step 0.2584 (0.0601) converged in 2 iterations. 0.757840 Δα.
45. 70.12 with step 0.2705 (0.0575) converged in 2 iterations. 0.739959 Δα.
46. 70.40 with step 0.2693 (0.0551) converged in 2 iterations. 1.030438 Δα.
47. 70.68 with step 0.2713 (0.0527) converged in 2 iterations. 0.267058 Δα.
48. 70.98 with step 0.3015 (0.0586) converged in 2 iterations. 0.199348 Δα.
49. 71.34 with step 0.3502 (0.0688) converged in 2 iterations. 0.067205 Δα.
50. 71.91 with step 0.5912 (0.1041) converged in 3 iterations. 0.302926 Δα.
51. 72.41 with step 0.5352 (0.0996) converged in 3 iterations. 0.391801 Δα.
52. 72.96 with step 0.3916 (0.0907) converged in 5 iterations. 7.176354 Δα.
53. 73.48 with step 0.4277 (0.0633) converged in 2 iterations. 1.311995 Δα.
54. 74.13 with step 0.5376 (0.0606) converged in 2 iterations. 1.438733 Δα.
55. 74.96 with step 0.6835 (0.0580) converged in 2 iterations. 1.578690 Δα.
56. 76.03 with step 0.8781 (0.0555) converged in 2 iterations. 1.728148 Δα.
57. 77.42 with step 1.1352 (0.0531) converged in 2 iterations. 1.875583 Δα.
58. 79.18 with step 1.4642 (0.0508) converged in 2 iterations. 1.997912 Δα.
59. 81.35 with step 1.8538 (0.0486) converged in 3 iterations. 2.062106 Δα.
60. 83.52 with step 1.9519 (0.0402) converged in 2 iterations. 2.058081 Δα.
61. 85.91 with step 2.2074 (0.0385) converged in 2 iterations. 1.995810 Δα.
62. 88.45 with step 2.4033 (0.0369) converged in 2 iterations. 1.901290 Δα.
63. 91.08 with step 2.5388 (0.0353) converged in 2 iterations. 1.796091 Δα.
64. 93.78 with step 2.6234 (0.0338) converged in 2 iterations. 1.692931 Δα.
65. 96.50 with step 2.6699 (0.0323) converged in 2 iterations. 1.598282 Δα.
66. 99.22 with step 2.6896 (0.0309) converged in 2 iterations. 1.514277 Δα.
67. 101.94 with step 2.6904 (0.0296) converged in 2 iterations. 1.440796 Δα.
Algorithm: NewtonRaphson(
descent = NewtonDescent(),
autodiff = AutoForwardDiff(),
vjp_autodiff = AutoFiniteDiff(
fdtype = Val{:forward}(),
fdjtype = Val{:forward}(),
fdhtype = Val{:hcentral}(),
dir = true
),
jvp_autodiff = AutoForwardDiff(),
concrete_jac = Val{false}()
)
---- ------------- -----------
Iter f(u) inf-norm Step 2-norm
---- ------------- -----------
0 5.16182693e-01 9.25386577e-03
1 3.89999144e-12 1.20508676e-03
Final 3.89999144e-12
----------------------
2. 40.90 with step 0.9258 (0.0538) converged in 1 iterations. 0.291677 Δα.
3. 43.83 with step 2.9454 (0.0746) converged in 2 iterations. 0.729969 Δα.
4. 46.70 with step 2.9020 (0.0714) converged in 2 iterations. 0.818566 Δα.
5. 49.53 with step 2.8715 (0.0683) converged in 2 iterations. 0.934514 Δα.
6. 52.28 with step 2.8162 (0.0654) converged in 2 iterations. 1.071082 Δα.
7. 54.87 with step 2.7048 (0.0626) converged in 2 iterations. 1.214792 Δα.
8. 57.22 with step 2.5110 (0.0599) converged in 2 iterations. 1.341660 Δα.
9. 59.26 with step 2.2285 (0.0573) converged in 2 iterations. 1.424836 Δα.
10. 60.95 with step 1.8869 (0.0549) converged in 2 iterations. 1.452244 Δα.
11. 62.31 with step 1.5406 (0.0525) converged in 2 iterations. 1.433147 Δα.
12. 63.40 with step 1.2338 (0.0503) converged in 2 iterations. 1.386316 Δα.
13. 64.27 with step 0.9835 (0.0481) converged in 2 iterations. 1.327632 Δα.
14. 64.98 with step 0.7876 (0.0461) converged in 2 iterations. 1.266643 Δα.
15. 65.55 with step 0.6366 (0.0441) converged in 2 iterations. 1.208047 Δα.
16. 65.92 with step 0.5201 (0.0422) converged in 3 iterations. 1.889294 Δα.
17. 66.01 with step 0.1163 (0.0349) converged in 2 iterations. 3.268071 Δα.
18. 66.07 with step 0.0716 (0.0334) converged in 2 iterations. 2.445691 Δα.
19. 66.11 with step 0.0496 (0.0320) converged in 2 iterations. 2.585266 Δα.
20. 66.14 with step 0.0337 (0.0306) converged in 2 iterations. 1.693099 Δα.
21. 66.17 with step 0.0255 (0.0293) converged in 2 iterations. 1.962784 Δα.
22. 66.18 with step 0.0192 (0.0280) converged in 2 iterations. 1.666704 Δα.
23. 66.20 with step 0.0145 (0.0268) converged in 2 iterations. 1.341214 Δα.
24. 66.21 with step 0.0118 (0.0257) converged in 1 iterations. 0.929808 Δα.
25. 66.22 with step 0.0134 (0.0321) converged in 2 iterations. 2.211372 Δα.
26. 66.23 with step 0.0105 (0.0307) converged in 2 iterations. 1.256457 Δα.
27. 66.24 with step 0.0090 (0.0294) converged in 2 iterations. 0.775712 Δα.
28. 66.25 with step 0.0080 (0.0281) converged in 1 iterations. 0.263960 Δα.
29. 66.26 with step 0.0108 (0.0396) converged in 2 iterations. 0.072945 Δα.
30. 66.28 with step 0.0163 (0.0586) converged in 2 iterations. 0.452208 Δα.
31. 66.30 with step 0.0191 (0.0596) converged in 2 iterations. 0.264442 Δα.
32. 66.32 with step 0.0257 (0.0664) converged in 2 iterations. 0.783245 Δα.
33. 66.36 with step 0.0315 (0.0636) converged in 2 iterations. 1.227351 Δα.
34. 66.41 with step 0.0428 (0.0608) converged in 2 iterations. 0.735934 Δα.
35. 66.46 with step 0.0498 (0.0582) converged in 2 iterations. 1.584767 Δα.
36. 66.53 with step 0.0592 (0.0557) converged in 2 iterations. 0.694511 Δα.
37. 66.62 with step 0.0764 (0.0534) converged in 2 iterations. 0.589607 Δα.
38. 66.73 with step 0.0956 (0.0523) converged in 3 iterations. 0.584315 Δα.
39. 66.83 with step 0.1057 (0.0444) converged in 2 iterations. 0.110805 Δα.
40. 67.01 with step 0.1595 (0.0593) converged in 2 iterations. 0.683353 Δα.
41. 67.21 with step 0.1771 (0.0569) converged in 3 iterations. 1.248877 Δα.
42. 67.41 with step 0.1758 (0.0471) converged in 2 iterations. 1.314261 Δα.
43. 67.61 with step 0.1970 (0.0451) converged in 2 iterations. 0.929315 Δα.
44. 67.83 with step 0.2031 (0.0432) converged in 2 iterations. 1.363796 Δα.
45. 68.06 with step 0.2161 (0.0413) converged in 2 iterations. 0.294544 Δα.
46. 68.33 with step 0.2487 (0.0451) converged in 3 iterations. 0.036988 Δα.
47. 68.79 with step 0.4398 (0.0711) converged in 2 iterations. 0.221681 Δα.
48. 69.38 with step 0.5448 (0.0819) converged in 2 iterations. 0.311738 Δα.
49. 70.09 with step 0.6380 (0.0885) converged in 3 iterations. 0.376325 Δα.
50. 70.77 with step 0.6486 (0.0812) converged in 2 iterations. 0.278139 Δα.
51. 71.64 with step 0.8075 (0.0896) converged in 2 iterations. 0.405297 Δα.
52. 72.66 with step 0.9585 (0.0928) converged in 3 iterations. 0.376598 Δα.
53. 73.65 with step 0.9885 (0.0852) converged in 3 iterations. 0.097776 Δα.
54. 74.76 with step 1.2254 (0.1060) converged in 3 iterations. 0.235233 Δα.
55. 76.27 with step 1.1322 (0.1069) converged in 2 iterations. 1.240377 Δα.
56. 78.55 with step 1.6777 (0.1023) converged in 3 iterations. 1.416776 Δα.
57. 81.40 with step 2.2021 (0.0847) converged in 3 iterations. 1.572263 Δα.
58. 84.61 with step 2.6719 (0.0700) converged in 3 iterations. 1.648992 Δα.
59. 87.91 with step 2.9312 (0.0579) converged in 3 iterations. 1.645375 Δα.
60. 91.08 with step 2.9381 (0.0479) converged in 2 iterations. 1.598169 Δα.
61. 94.44 with step 3.1996 (0.0459) converged in 2 iterations. 1.527791 Δα.
62. 97.95 with step 3.3861 (0.0439) converged in 2 iterations. 1.449426 Δα.
63. 101.55 with step 3.5111 (0.0420) converged in 2 iterations. 1.372949 Δα.
Algorithm: NewtonRaphson(
descent = NewtonDescent(),
autodiff = AutoForwardDiff(),
vjp_autodiff = AutoFiniteDiff(
fdtype = Val{:forward}(),
fdjtype = Val{:forward}(),
fdhtype = Val{:hcentral}(),
dir = true
),
jvp_autodiff = AutoForwardDiff(),
concrete_jac = Val{false}()
)
---- ------------- -----------
Iter f(u) inf-norm Step 2-norm
---- ------------- -----------
0 1.03236539e+00 2.09241099e+05
1 7.79998288e-12 2.41017353e-03
Final 7.79998288e-12
----------------------
2. 40.56 with step 0.5685 (0.0421) converged in 1 iterations. 0.229053 Δα.
3. 42.87 with step 2.3312 (0.0605) converged in 2 iterations. 0.694384 Δα.
4. 45.11 with step 2.2691 (0.0580) converged in 2 iterations. 0.750928 Δα.
5. 47.29 with step 2.2137 (0.0555) converged in 2 iterations. 0.820007 Δα.
6. 49.39 with step 2.1461 (0.0531) converged in 2 iterations. 0.895370 Δα.
7. 51.39 with step 2.0550 (0.0508) converged in 2 iterations. 0.970625 Δα.
8. 53.25 with step 1.9345 (0.0487) converged in 2 iterations. 1.038739 Δα.
9. 54.95 with step 1.7851 (0.0466) converged in 2 iterations. 1.093108 Δα.
10. 56.47 with step 1.6137 (0.0446) converged in 2 iterations. 1.129439 Δα.
11. 57.81 with step 1.4318 (0.0427) converged in 2 iterations. 1.146982 Δα.
12. 58.97 with step 1.2522 (0.0408) converged in 2 iterations. 1.148150 Δα.
13. 59.98 with step 1.0847 (0.0391) converged in 2 iterations. 1.136990 Δα.
14. 60.85 with step 0.9351 (0.0374) converged in 2 iterations. 1.146748 Δα.
15. 61.32 with step 0.5910 (0.0358) converged in 2 iterations. 3.262078 Δα.
16. 61.66 with step 0.3847 (0.0343) converged in 2 iterations. 1.715432 Δα.
17. 61.92 with step 0.2948 (0.0328) converged in 2 iterations. 1.432713 Δα.
18. 62.14 with step 0.2331 (0.0314) converged in 2 iterations. 1.242047 Δα.
19. 62.31 with step 0.1889 (0.0300) converged in 2 iterations. 1.105050 Δα.
20. 62.46 with step 0.1590 (0.0288) converged in 2 iterations. 1.330216 Δα.
21. 62.59 with step 0.1349 (0.0275) converged in 1 iterations. 1.355686 Δα.
22. 62.72 with step 0.1504 (0.0344) converged in 2 iterations. 1.271078 Δα.
23. 62.84 with step 0.1245 (0.0329) converged in 1 iterations. 0.961363 Δα.
24. 62.97 with step 0.1380 (0.0412) converged in 2 iterations. 0.945014 Δα.
25. 63.08 with step 0.1172 (0.0394) converged in 2 iterations. 1.032814 Δα.
26. 63.17 with step 0.0990 (0.0377) converged in 2 iterations. 1.040070 Δα.
27. 63.25 with step 0.0837 (0.0361) converged in 1 iterations. 0.695350 Δα.
28. 63.34 with step 0.0948 (0.0451) converged in 1 iterations. 0.696133 Δα.
29. 63.44 with step 0.1049 (0.0564) converged in 2 iterations. 0.682815 Δα.
30. 63.52 with step 0.0888 (0.0542) converged in 2 iterations. 0.701672 Δα.
31. 63.59 with step 0.0756 (0.0518) converged in 2 iterations. 0.705119 Δα.
32. 63.65 with step 0.0649 (0.0496) converged in 2 iterations. 0.684164 Δα.
33. 63.70 with step 0.0563 (0.0476) converged in 1 iterations. 0.631185 Δα.
34. 63.76 with step 0.0646 (0.0601) converged in 2 iterations. 0.556753 Δα.
35. 63.82 with step 0.0572 (0.0594) converged in 2 iterations. 0.522297 Δα.
36. 63.87 with step 0.0517 (0.0591) converged in 1 iterations. 0.464874 Δα.
37. 63.92 with step 0.0611 (0.0772) converged in 2 iterations. 0.431005 Δα.
38. 63.98 with step 0.0557 (0.0792) converged in 2 iterations. 0.409447 Δα.
39. 64.02 with step 0.0514 (0.0818) converged in 2 iterations. 0.385584 Δα.
40. 64.07 with step 0.0482 (0.0854) converged in 1 iterations. 0.347346 Δα.
41. 64.12 with step 0.0585 (0.1157) converged in 2 iterations. 0.341433 Δα.
42. 64.17 with step 0.0558 (0.1231) converged in 2 iterations. 0.342753 Δα.
43. 64.22 with step 0.0538 (0.1310) converged in 2 iterations. 0.340985 Δα.
44. 64.27 with step 0.0519 (0.1395) converged in 2 iterations. 0.359394 Δα.
45. 64.32 with step 0.0485 (0.1473) converged in 2 iterations. 0.273358 Δα.
46. 64.36 with step 0.0490 (0.1630) converged in 2 iterations. 0.286834 Δα.
47. 64.41 with step 0.0479 (0.1788) converged in 2 iterations. 0.221029 Δα.
48. 64.45 with step 0.0513 (0.2059) converged in 2 iterations. 0.260389 Δα.
49. 64.50 with step 0.0465 (0.2107) converged in 2 iterations. 0.142716 Δα.
50. 64.54 with step 0.0440 (0.2107) converged in 2 iterations. 0.104460 Δα.
51. 64.58 with step 0.0419 (0.2107) converged in 3 iterations. 0.023317 Δα.
52. 64.62 with step 0.0424 (0.2107) converged in 3 iterations. 0.195134 Δα.
53. 64.67 with step 0.0463 (0.2107) converged in 3 iterations. 0.429388 Δα.
54. 64.72 with step 0.0477 (0.1889) converged in 3 iterations. 0.093837 Δα.
55. 64.79 with step 0.0619 (0.2107) converged in 3 iterations. 0.165587 Δα.
56. 64.86 with step 0.0697 (0.2107) converged in 3 iterations. 0.448265 Δα.
57. 64.93 with step 0.0711 (0.1874) converged in 3 iterations. 0.260745 Δα.
58. 65.02 with step 0.0749 (0.1849) converged in 3 iterations. 0.138781 Δα.
59. 65.13 with step 0.0980 (0.2104) converged in 3 iterations. 0.221836 Δα.
60. 65.26 with step 0.1154 (0.2107) converged in 2 iterations. 0.317741 Δα.
61. 65.43 with step 0.1425 (0.2107) converged in 3 iterations. 0.340706 Δα.
62. 65.62 with step 0.1627 (0.1971) converged in 2 iterations. 0.413558 Δα.
63. 65.87 with step 0.2104 (0.2034) converged in 2 iterations. 0.434896 Δα.
64. 66.21 with step 0.2813 (0.2082) converged in 2 iterations. 0.436509 Δα.
65. 66.67 with step 0.3739 (0.2107) converged in 2 iterations. 0.465429 Δα.
66. 67.29 with step 0.5044 (0.2107) converged in 3 iterations. 0.489797 Δα.
67. 68.06 with step 0.6194 (0.1846) converged in 2 iterations. 0.544724 Δα.
68. 69.16 with step 0.8775 (0.1828) converged in 3 iterations. 0.615856 Δα.
69. 70.49 with step 1.0815 (0.1542) converged in 2 iterations. 0.669320 Δα.
70. 72.29 with step 1.4637 (0.1484) converged in 3 iterations. 0.705259 Δα.
71. 74.26 with step 1.6793 (0.1227) converged in 3 iterations. 0.699922 Δα.
72. 76.25 with step 1.7741 (0.1016) converged in 2 iterations. 0.427547 Δα.
73. 78.46 with step 2.1076 (0.1043) converged in 3 iterations. 0.123888 Δα.
74. 81.57 with step 2.3244 (0.1221) converged in 6 iterations. 0.190219 Δα.
75. 85.68 with step 3.1726 (0.1083) converged in 3 iterations. 1.284475 Δα.
76. 90.31 with step 3.8745 (0.0896) converged in 3 iterations. 1.331991 Δα.
77. 95.04 with step 4.2384 (0.0741) converged in 3 iterations. 1.316650 Δα.
78. 99.55 with step 4.2263 (0.0613) converged in 2 iterations. 1.273100 Δα.
79. 104.35 with step 4.5913 (0.0587) converged in 2 iterations. 1.215689 Δα.
Algorithm: NewtonRaphson(
descent = NewtonDescent(),
autodiff = AutoForwardDiff(),
vjp_autodiff = AutoFiniteDiff(
fdtype = Val{:forward}(),
fdjtype = Val{:forward}(),
fdhtype = Val{:hcentral}(),
dir = true
),
jvp_autodiff = AutoForwardDiff(),
concrete_jac = Val{false}()
)
---- ------------- -----------
Iter f(u) inf-norm Step 2-norm
---- ------------- -----------
0 2.06473077e+00 9.05589509e-145
1 1.55999658e-11 4.82034705e-03
Final 1.55999658e-11
----------------------
2. 40.43 with step 0.4373 (0.0370) converged in 1 iterations. 0.201135 Δα.
3. 42.41 with step 2.0011 (0.0542) converged in 2 iterations. 0.656395 Δα.
4. 44.32 with step 1.9389 (0.0523) converged in 2 iterations. 0.693220 Δα.
5. 46.15 with step 1.8666 (0.0501) converged in 2 iterations. 0.735642 Δα.
6. 47.89 with step 1.7842 (0.0479) converged in 2 iterations. 0.778470 Δα.
7. 49.51 with step 1.6884 (0.0459) converged in 2 iterations. 0.992330 Δα.
8. 50.85 with step 1.4566 (0.0439) converged in 2 iterations. 1.625689 Δα.
9. 52.00 with step 1.2355 (0.0420) converged in 2 iterations. 1.272432 Δα.
10. 52.98 with step 1.0549 (0.0402) converged in 2 iterations. 1.151735 Δα.
11. 53.82 with step 0.9030 (0.0385) converged in 2 iterations. 1.094929 Δα.
12. 54.54 with step 0.7753 (0.0369) converged in 2 iterations. 1.044922 Δα.
13. 55.17 with step 0.6668 (0.0353) converged in 2 iterations. 0.932241 Δα.
14. 55.72 with step 0.5802 (0.0338) converged in 2 iterations. 0.972805 Δα.
15. 56.19 with step 0.5050 (0.0323) converged in 2 iterations. 0.880225 Δα.
16. 56.61 with step 0.4445 (0.0309) converged in 2 iterations. 0.872643 Δα.
17. 56.99 with step 0.3920 (0.0296) converged in 2 iterations. 0.826806 Δα.
18. 57.32 with step 0.3484 (0.0283) converged in 1 iterations. 0.776485 Δα.
19. 57.70 with step 0.4069 (0.0354) converged in 2 iterations. 0.821637 Δα.
20. 58.04 with step 0.3570 (0.0339) converged in 2 iterations. 0.745602 Δα.
21. 58.34 with step 0.3161 (0.0324) converged in 2 iterations. 0.727660 Δα.
22. 58.61 with step 0.2807 (0.0310) converged in 1 iterations. 0.700898 Δα.
23. 58.92 with step 0.3280 (0.0388) converged in 2 iterations. 0.695673 Δα.
24. 59.20 with step 0.2887 (0.0372) converged in 2 iterations. 0.649438 Δα.
25. 59.44 with step 0.2585 (0.0359) converged in 1 iterations. 0.640137 Δα.
26. 59.73 with step 0.3035 (0.0453) converged in 2 iterations. 0.631905 Δα.
27. 59.98 with step 0.2693 (0.0439) converged in 2 iterations. 0.591401 Δα.
28. 60.21 with step 0.2427 (0.0430) converged in 2 iterations. 0.577349 Δα.
29. 60.42 with step 0.2205 (0.0422) converged in 1 iterations. 0.569687 Δα.
30. 60.67 with step 0.2605 (0.0539) converged in 2 iterations. 0.543192 Δα.
31. 60.89 with step 0.2353 (0.0534) converged in 2 iterations. 0.526453 Δα.
32. 61.09 with step 0.2150 (0.0531) converged in 2 iterations. 0.510153 Δα.
33. 61.28 with step 0.1969 (0.0531) converged in 2 iterations. 0.490357 Δα.
34. 61.45 with step 0.1824 (0.0534) converged in 2 iterations. 0.475790 Δα.
35. 61.61 with step 0.1701 (0.0539) converged in 2 iterations. 0.462451 Δα.
36. 61.76 with step 0.1596 (0.0547) converged in 2 iterations. 0.462242 Δα.
37. 61.91 with step 0.1488 (0.0555) converged in 1 iterations. 0.428932 Δα.
38. 62.08 with step 0.1820 (0.0732) converged in 2 iterations. 0.416344 Δα.
39. 62.24 with step 0.1703 (0.0754) converged in 2 iterations. 0.401912 Δα.
40. 62.39 with step 0.1598 (0.0782) converged in 2 iterations. 0.385716 Δα.
41. 62.53 with step 0.1514 (0.0816) converged in 2 iterations. 0.372359 Δα.
42. 62.66 with step 0.1442 (0.0856) converged in 2 iterations. 0.360175 Δα.
43. 62.79 with step 0.1380 (0.0903) converged in 2 iterations. 0.362008 Δα.
44. 62.91 with step 0.1313 (0.0952) converged in 2 iterations. 0.335748 Δα.
45. 63.03 with step 0.1261 (0.1017) converged in 2 iterations. 0.313301 Δα.
46. 63.15 with step 0.1228 (0.1098) converged in 2 iterations. 0.316005 Δα.
47. 63.25 with step 0.1185 (0.1185) converged in 2 iterations. 0.292513 Δα.
48. 63.36 with step 0.1154 (0.1295) converged in 2 iterations. 0.274084 Δα.
49. 63.47 with step 0.1131 (0.1433) converged in 2 iterations. 0.261161 Δα.
50. 63.57 with step 0.1116 (0.1599) converged in 2 iterations. 0.248711 Δα.
51. 63.67 with step 0.1092 (0.1800) converged in 2 iterations. 0.228031 Δα.
52. 63.76 with step 0.0976 (0.1848) converged in 2 iterations. 0.219495 Δα.
53. 63.83 with step 0.0843 (0.1848) converged in 2 iterations. 0.196899 Δα.
54. 63.90 with step 0.0740 (0.1848) converged in 2 iterations. 0.196099 Δα.
55. 63.96 with step 0.0650 (0.1848) converged in 2 iterations. 0.172421 Δα.
56. 64.01 with step 0.0572 (0.1848) converged in 2 iterations. 0.152714 Δα.
57. 64.06 with step 0.0512 (0.1848) converged in 2 iterations. 0.247466 Δα.
58. 64.10 with step 0.0455 (0.1848) converged in 2 iterations. 0.217642 Δα.
59. 64.14 with step 0.0407 (0.1848) converged in 2 iterations. 0.196270 Δα.
60. 64.18 with step 0.0371 (0.1848) converged in 2 iterations. 0.233668 Δα.
61. 64.21 with step 0.0333 (0.1848) converged in 2 iterations. 0.211137 Δα.
62. 64.24 with step 0.0301 (0.1848) converged in 2 iterations. 0.173353 Δα.
63. 64.26 with step 0.0274 (0.1848) converged in 2 iterations. 0.166339 Δα.
64. 64.29 with step 0.0248 (0.1848) converged in 2 iterations. 0.140462 Δα.
65. 64.31 with step 0.0234 (0.1848) converged in 2 iterations. 0.207421 Δα.
66. 64.33 with step 0.0213 (0.1848) converged in 2 iterations. 0.172832 Δα.
67. 64.35 with step 0.0197 (0.1848) converged in 2 iterations. 0.175081 Δα.
68. 64.36 with step 0.0180 (0.1848) converged in 2 iterations. 0.119335 Δα.
69. 64.38 with step 0.0171 (0.1848) converged in 2 iterations. 0.135922 Δα.
70. 64.40 with step 0.0163 (0.1848) converged in 2 iterations. 0.145457 Δα.
71. 64.41 with step 0.0153 (0.1848) converged in 2 iterations. 0.074904 Δα.
72. 64.42 with step 0.0146 (0.1848) converged in 2 iterations. 0.087754 Δα.
73. 64.44 with step 0.0141 (0.1848) converged in 2 iterations. 0.099747 Δα.
74. 64.45 with step 0.0136 (0.1848) converged in 2 iterations. 0.110824 Δα.
75. 64.46 with step 0.0133 (0.1848) converged in 2 iterations. 0.143857 Δα.
76. 64.48 with step 0.0124 (0.1848) converged in 2 iterations. 0.024111 Δα.
77. 64.49 with step 0.0122 (0.1848) converged in 2 iterations. 0.029814 Δα.
78. 64.50 with step 0.0121 (0.1848) converged in 2 iterations. 0.049458 Δα.
79. 64.51 with step 0.0118 (0.1848) converged in 2 iterations. 0.018981 Δα.
80. 64.52 with step 0.0118 (0.1848) converged in 2 iterations. 0.060565 Δα.
81. 64.53 with step 0.0116 (0.1848) converged in 2 iterations. 0.011207 Δα.
82. 64.54 with step 0.0106 (0.1848) converged in 2 iterations. 0.140793 Δα.
83. 64.55 with step 0.0113 (0.1848) converged in 3 iterations. 0.362597 Δα.
84. 64.56 with step 0.0095 (0.1708) converged in 2 iterations. 0.056245 Δα.
85. 64.57 with step 0.0101 (0.1848) converged in 2 iterations. 0.063854 Δα.
86. 64.58 with step 0.0099 (0.1848) converged in 2 iterations. 0.052736 Δα.
87. 64.59 with step 0.0098 (0.1848) converged in 2 iterations. 0.043079 Δα.
88. 64.60 with step 0.0107 (0.1848) converged in 2 iterations. 0.263625 Δα.
89. 64.61 with step 0.0098 (0.1848) converged in 2 iterations. 0.001287 Δα.
90. 64.62 with step 0.0098 (0.1848) converged in 2 iterations. 0.026474 Δα.
91. 64.63 with step 0.0099 (0.1848) converged in 2 iterations. 0.056167 Δα.
92. 64.64 with step 0.0101 (0.1848) converged in 2 iterations. 0.090241 Δα.
93. 64.65 with step 0.0103 (0.1848) converged in 2 iterations. 0.128503 Δα.
94. 64.67 with step 0.0106 (0.1848) converged in 2 iterations. 0.170709 Δα.
95. 64.68 with step 0.0109 (0.1848) converged in 2 iterations. 0.210819 Δα.
96. 64.69 with step 0.0122 (0.1848) converged in 2 iterations. 0.283823 Δα.
97. 64.70 with step 0.0127 (0.1848) converged in 2 iterations. 0.200714 Δα.
98. 64.71 with step 0.0131 (0.1848) converged in 2 iterations. 0.125389 Δα.
99. 64.73 with step 0.0136 (0.1848) converged in 2 iterations. 0.039922 Δα.
100. 64.74 with step 0.0131 (0.1848) converged in 2 iterations. 0.193316 Δα.
101. 64.76 with step 0.0144 (0.1848) converged in 2 iterations. 0.078750 Δα.
102. 64.77 with step 0.0148 (0.1848) converged in 2 iterations. 0.096153 Δα.
103. 64.79 with step 0.0161 (0.1848) converged in 2 iterations. 0.022483 Δα.
104. 64.81 with step 0.0157 (0.1848) converged in 2 iterations. 0.171399 Δα.
105. 64.82 with step 0.0169 (0.1848) converged in 2 iterations. 0.094560 Δα.
106. 64.84 with step 0.0174 (0.1848) converged in 2 iterations. 0.125335 Δα.
107. 64.86 with step 0.0179 (0.1848) converged in 2 iterations. 0.178810 Δα.
108. 64.88 with step 0.0195 (0.1848) converged in 2 iterations. 0.097008 Δα.
109. 64.90 with step 0.0204 (0.1848) converged in 2 iterations. 0.126964 Δα.
110. 64.93 with step 0.0215 (0.1848) converged in 2 iterations. 0.161308 Δα.
111. 64.95 with step 0.0237 (0.1848) converged in 2 iterations. 0.112966 Δα.
112. 64.98 with step 0.0257 (0.1848) converged in 2 iterations. 0.145583 Δα.
113. 65.01 with step 0.0279 (0.1848) converged in 2 iterations. 0.175567 Δα.
114. 65.04 with step 0.0304 (0.1848) converged in 2 iterations. 0.202995 Δα.
115. 65.08 with step 0.0333 (0.1848) converged in 2 iterations. 0.237516 Δα.
116. 65.12 with step 0.0376 (0.1848) converged in 2 iterations. 0.205013 Δα.
117. 65.16 with step 0.0422 (0.1848) converged in 2 iterations. 0.197663 Δα.
118. 65.21 with step 0.0468 (0.1848) converged in 2 iterations. 0.227423 Δα.
119. 65.27 with step 0.0534 (0.1848) converged in 2 iterations. 0.225635 Δα.
120. 65.33 with step 0.0598 (0.1848) converged in 2 iterations. 0.247124 Δα.
121. 65.41 with step 0.0681 (0.1848) converged in 2 iterations. 0.177423 Δα.
122. 65.49 with step 0.0786 (0.1848) converged in 2 iterations. 0.170167 Δα.
123. 65.59 with step 0.0896 (0.1848) converged in 2 iterations. 0.217232 Δα.
124. 65.71 with step 0.1055 (0.1848) converged in 2 iterations. 0.193899 Δα.
125. 65.84 with step 0.1221 (0.1848) converged in 2 iterations. 0.255777 Δα.
126. 66.01 with step 0.1447 (0.1848) converged in 2 iterations. 0.262336 Δα.
127. 66.20 with step 0.1739 (0.1848) converged in 2 iterations. 0.267175 Δα.
128. 66.44 with step 0.2098 (0.1848) converged in 2 iterations. 0.298968 Δα.
129. 66.74 with step 0.2561 (0.1848) converged in 2 iterations. 0.332326 Δα.
130. 67.11 with step 0.3177 (0.1848) converged in 2 iterations. 0.375679 Δα.
131. 67.58 with step 0.4028 (0.1848) converged in 2 iterations. 0.402740 Δα.
132. 68.21 with step 0.5192 (0.1848) converged in 2 iterations. 0.450428 Δα.
133. 69.05 with step 0.6896 (0.1848) converged in 2 iterations. 0.483777 Δα.
134. 70.22 with step 0.9411 (0.1848) converged in 2 iterations. 0.527441 Δα.
135. 71.90 with step 1.3184 (0.1839) converged in 2 iterations. 0.585131 Δα.
136. 74.34 with step 1.8828 (0.1802) converged in 3 iterations. 0.650725 Δα.
137. 77.29 with step 2.3527 (0.1507) converged in 3 iterations. 0.717139 Δα.
138. 80.57 with step 2.7548 (0.1247) converged in 4 iterations. 0.720586 Δα.
139. 83.57 with step 2.2295 (0.0935) converged in 106 iterations. 0.656416 Δα.
140. 85.23 with step 1.5922 (0.0475) converged in 2 iterations. 0.480261 Δα.
141. 86.95 with step 1.6759 (0.0479) converged in 3 iterations. 0.130899 Δα.
142. 89.20 with step 2.0277 (0.0553) converged in 2 iterations. 1.069541 Δα.
143. 91.70 with step 2.2760 (0.0529) converged in 2 iterations. 1.098074 Δα.
144. 94.43 with step 2.5225 (0.0507) converged in 2 iterations. 1.115555 Δα.
145. 97.39 with step 2.7553 (0.0485) converged in 2 iterations. 1.121641 Δα.
146. 100.53 with step 2.9641 (0.0464) converged in 2 iterations. 1.117441 Δα.Stability Certification
E0, _ = HARMONICSTIFFNESS(zeros(mdl.Ndofs,mdl.Ndofs), -2mdl.M,
zeros(mdl.Ndofs, mdl.Ndofs), [1.], h);
indsh1 = [rinds[1:mdl.Ndofs]; iinds[1:mdl.Ndofs]];
E0 = collect(E0);
stabs = [];
for (Famp,sols) in zip(Famps,solss)
stab = zeros(length(sols));
for iw in 1:length(sols)
J = zeros(mdl.Ndofs*Nhc, mdl.Ndofs*Nhc);
fun(Famp).jac(J, sols.u[iw], sols.p[iw])
eVs = eigvals(J[indsh1,indsh1], sols.p[iw]*E0[indsh1,indsh1]);
stab[iw] = sum(real(eVs).>=0);
end
push!(stabs, stab);
endPlot Forced Response
We now plot out the forced response of the system showing the characteristic frictional softening-dampening behavior.
set_theme!(theme_latexfonts())
fsz = 18;
fig = Figure(fontsize=fsz);
ax = Axis(fig[1, 1], xlabel="Excitation Frequency (rad/s)",
ylabel="Response (m)", yscale=log10);
for (Famp, Uh, sols, stab) in zip(Famps, Uhs, solss, stabs)
scatterlines!(ax, sols.p./(stab.==0), norm.(Uh)/Famp, label="F = $Famp")
scatterlines!(ax, sols.p./(stab.!=0), norm.(Uh)/Famp)
end
xlims!(ax, Om0, Om1)
ylims!(ax, 3e-4, 1e-1)
Legend(fig[0, 1], ax, nbanks=3, tellheight=true, tellwidth=false)
fig
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