Figure B.16: The error of the solution as a function of iteration number for the nonlinear solver
B.3 Tensile ratio of 3
Standard plate
1 Stationary Solver 1 in Solver 1 started at 19-jun -2014 11:06:38.
2 Nonlinear solver
3 Number of degrees of freedom solved for: 34708.
4 Nonsymmetric matrix found .
5 Scales for dependent variables :
6 comp1.u: 6.1e -05
7 comp1.XY: 10
8 Iter ErrEst Damping Stepsize #Res #Jac #Sol LinErr LinRes
9 1 6.9e -13 1.0000000 0.67 2 1 2 7.6e -11 8.8e -14
10 2 5.7e -16 1.0000000 6.9e -13 3 2 4 3.8e -11 5.4e -16
11 Stationary Solver 1 in Solver 1: Solution time: 2 s
12 Physical memory : 1.03 GB
13 Virtual memory : 5.45 GB
Figure B.17: v. Mises stress
Figure B.18: The error of the solution as a function of iteration number for the nonlinear solver
B.3 Tensile ratio of 3 99
Optimized plate
1 ============================================================
2 Number of vertex elements : 5
3 Number of boundary elements : 192
4 Number of elements : 2121
5 Minimum element quality : 0.4294
6 Number of vertex elements : 5
7 Optimization Solver 1 in Solver 2 started at 19-jun -2014 03:18:29.
8 Optimization solver (SNOPT )
9 Analytic gradient with the adjoint method .
10 Warning : New constraint force nodes detected : These are not stored .
11 Itns Major Minor Step nPDE Error Objective
12 2 0 2 - 1 1.1 236.6
48 Number of optimization variables : 3.
49 Number of objective function evaluations : 120.
50 Number of Jacobian evaluations : 118.
51 Final objective function value : 209.9367235.
52 Warning : Requested accuracy could not be achieved .
53 Optimization Solver 1 in Solver 2: Solution time: 377 s (6 minutes , 17 seconds )
54 Physical memory : 1.09 GB
55 Virtual memory : 5.45 GB
Iteration # σvM,max [MPa] U [J/m3] Utot [J] Area [m2] q1 [-] q2 [-] q3 [-]
1 8.82 27.03 236.65 87.54 0.62 0.30 -0.73
2 7.46 25.71 225.14 87.59 1.37 0.61 -1.40
3 4.88 23.57 207.62 88.10 3.67 1.10 -2.90
4 4.92 23.95 209.63 87.53 5.20 1.35 -3.24
5 4.77 24.02 210.02 87.44 5.11 1.23 -3.15
6 4.88 24.00 210.01 87.49 4.51 0.68 -2.82
7 4.53 24.01 209.96 87.43 5.01 1.00 -3.05
8 4.50 24.01 209.93 87.43 5.13 0.72 -3.01
9 4.48 24.01 209.94 87.43 5.12 0.74 -3.02
10 5.78 24.04 210.34 87.48 5.03 2.00 -3.34
11 4.48 24.01 209.94 87.43 5.12 0.74 -3.02
12 4.48 24.01 209.94 87.43 5.12 0.75 -3.02
13 4.72 24.01 209.97 87.44 4.94 0.52 -2.90
14 4.49 24.01 209.94 87.43 5.12 0.74 -3.02
15 4.49 24.01 209.94 87.43 5.12 0.74 -3.02
16 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
17 4.48 24.01 209.94 87.43 5.12 0.75 -3.02
18 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
19 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
20 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
21 4.49 24.01 209.94 87.43 5.12 0.74 -3.02
22 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
23 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
24 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
25 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
26 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
27 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
28 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
29 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
30 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
31 4.48 24.01 209.94 87.43 5.12 0.75 -3.02
32 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
33 4.48 24.01 209.94 87.43 5.12 0.75 -3.02
34 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
35 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
36 4.49 24.01 209.94 87.43 5.12 0.75 -3.02
Table B.19: Values for each iteration during optimization
B.3 Tensile ratio of 3 101
Figure B.20: Surface v. Mises stress
Figure B.21: Surface v. Mises stress showing the whole plate
Figure B.22: Graph comparing the v. Mises stress for the original shape with the optimized design
Figure B.23: The error of the solution as a function of iteration number for the optimization solver
B.3 Tensile ratio of 3 103
Figure B.24: The error of the solution as a function of iteration number for the nonlinear solver
APPENDIX C
Fillet
C.1 Standard
1 ============================================================
2 Number of vertex elements : 7
3 Number of boundary elements : 96
4 Number of elements : 1000
5 Minimum element quality : 0.893
6 Stationary Solver 1 in Solver 1 started at 24-jun -2014 00:18:20.
7 Nonlinear solver
8 Number of degrees of freedom solved for: 8388.
9 Nonsymmetric matrix found .
10 Scales for dependent variables :
11 comp1.RZ: 32
12 comp1.u: 1.2e -05
13 Iter ErrEst Damping Stepsize #Res #Jac #Sol LinErr LinRes
14 1 1.5e -14 1.0000000 0.67 2 1 2 2e -12 1.4e -14
15 2 3.2e -16 1.0000000 1.5e -14 3 2 4 3.7e -11 5.1e -16
16 Stationary Solver 1 in Solver 1: Solution time: 0 s
17 Physical memory : 821 MB
18 Virtual memory : 5224 MB
Figure C.1: v. Mises stress in 2D
Figure C.2: v. Mises stress in 3D
C.2 One summation 107
Figure C.3: The error of the solution as a function of iteration number for the nonlinear solver
C.2 One summation
1 ============================================================
2 Optimization Solver 1 in Solver 2 started at 24-jun -2014 01:21:15.
3 Optimization solver (SNOPT )
4 Analytic gradient with the adjoint method .
5 Itns Major Minor Step nPDE Error Objective
6 2 0 2 - 1 0.569 1.92e+05
17 Number of optimization variables : 3.
18 Number of objective function evaluations : 13.
19 Number of Jacobian evaluations : 11.
20 Final objective function value : 186574.0434.
21 Optimality conditions satisfied .
22 Optimization Solver 1 in Solver 2: Solution time: 12 s
23 Physical memory : 881 MB
24 Virtual memory : 5296 MB
Figure C.4: v. Mises stress in 1D
Figure C.5: v. Mises stress in 2D
C.2 One summation 109
Figure C.6: v. Mises stress in 3D
Figure C.7: The error of the solution as a function of iteration number for the nonlinear solver
Figure C.8: The error of the solution as a function of iteration number for the optimization solver
C.2 One summation 111
Iteration#σvM,max[MPa]Utot[kJ]U[J/m3 ]Area[m2 ]p1[-]p2[-]q1[-] 16.57191.9810.30524.990.92-0.46-0.36 26.40191.3910.25524.991.84-0.93-0.70 35.37189.0210.06524.787.44-3.69-2.73 44.75187.959.97524.4614.90-7.32-4.15 54.62187.009.88524.9019.99-8.38-5.23 64.49186.699.85525.0023.45-8.55-7.37 74.41186.609.85524.9926.00-8.12-10.28 84.43186.599.85524.9926.00-7.79-10.90 94.43186.589.85525.0026.00-7.66-11.14 104.44186.579.85525.0026.00-7.60-11.24 114.44186.579.85525.0026.00-7.60-11.24 TableC.9:Valuesforeachiterationduringoptimization
C.3 Two summations
1 ============================================================
2 Optimization Solver 1 in Solver 2 started at 24-jun -2014 01:25:27.
3 Optimization solver (SNOPT)
4 Analytic gradient with the adjoint method .
5 Itns Major Minor Step nPDE Error Objective
6 3 0 3 - 1 0.569 1.919e+05
16 13 10 1 1.00 11 0.00133 1.865e+05
17 14 11 1 1.00 13 0.00252 1.865e+05
18 15 12 1 0.01 17 0.000688 1.865e+05
19 16 13 1 0.10 21 8.22e -05 1.865e+05
20 Number of optimization variables : 4.
21 Number of objective function evaluations : 23.
22 Number of Jacobian evaluations : 21.
23 Final objective function value : 186485.925.
24 Optimality conditions satisfied .
25 Optimization Solver 1 in Solver 2: Solution time: 18 s
26 Physical memory : 880 MB
27 Virtual memory : 5301 MB
C.3 Two summations 113
Figure C.10: v. Mises stress in 1D
Figure C.11: v. Mises stress in 2D
Figure C.12: v. Mises stress in 3D
Figure C.13: The error of the solution as a function of iteration number for the nonlinear solver
C.3 Two summations 115
Figure C.14: The error of the solution as a function of iteration number for the optimization solver
Iteration#σvM,max[MPa]Utot[kJ]U[J/m3 ]Area[m2 ]p1[-]p2[-]q1[-]q2[-] 16.75191.9010.29525.000.78-0.39-0.310.53 26.73191.2610.24524.991.60-0.79-0.620.99 35.49188.6210.02524.838.03-3.21-3.621.14 45.07187.519.93524.8213.91-5.30-5.741.79 54.76187.069.89524.8319.44-7.14-7.270.92 64.41186.819.87524.8824.64-8.49-8.68-0.25 74.32186.589.85525.0026.49-8.55-9.75-0.69 84.15186.519.84524.9930.33-8.25-13.39-1.96 94.13186.499.84525.0031.57-8.11-14.64-2.12 104.14186.499.84525.0031.76-8.06-14.89-2.01 114.14186.499.84525.0031.72-8.06-14.85-1.97 124.15186.499.84525.0031.67-8.08-14.77-1.91 134.15186.499.84525.0031.70-8.06-14.84-1.97 144.14186.499.84525.0031.72-8.06-14.85-1.97 TableC.15:Valuesforeachiterationduringoptimization
C.4 Three summations 117
C.4 Three summations
Figure C.16: v. Mises stress in 1D
Figure C.17: v. Mises stress in 2D
Figure C.18: v. Mises stress in 3D
C.4 Three summations 119
Figure C.19: The error of the solution as a function of iteration number for the nonlinear solver
Figure C.20: The error of the solution as a function of iteration number for the optimization solver
C.4 Three summations 121
Iteration#σvM,max[MPa]Utot[kJ]U[J/m3 ]Area[m2 ]p1[-]p2[-]q1[-]q2[-]q3[-] 16.84191.8910.29525.000.71-0.40-0.330.510.28 26.86191.2310.24524.991.55-0.84-0.680.960.44 35.43188.8810.04524.837.78-3.26-2.981.81-1.18 45.09187.579.93524.7614.53-5.69-6.221.001.26 54.90187.009.88524.8719.43-7.20-7.291.470.63 64.64186.819.86524.9123.44-8.38-8.160.720.74 74.49186.649.85524.9925.50-8.66-8.950.200.63 84.23186.569.84525.0028.60-8.76-10.98-1.050.14 94.05186.509.84525.0031.92-8.52-13.89-2.46-0.55 103.98186.489.84524.9934.73-8.09-16.79-3.57-1.18 113.99186.459.84525.0035.35-7.88-17.64-3.67-1.25 124.01186.459.84525.0035.33-7.71-17.99-3.52-1.16 134.02186.449.84525.0035.24-7.65-18.02-3.48-1.13 144.02186.449.84525.0035.14-7.64-17.97-3.47-1.12 154.03186.459.84525.0034.86-7.50-18.02-3.43-1.10 164.02186.449.84525.0035.09-7.63-17.95-3.46-1.12 174.02186.449.84525.0035.05-7.63-17.91-3.45-1.13 184.02186.449.84525.0035.01-7.61-17.93-3.45-1.13 194.02186.449.84525.0035.07-7.63-17.93-3.46-1.12 TableC.21:Valuesforeachiterationduringoptimization
C.5 Four summations
1 ============================================================
2 Optimization Solver 1 in Solver 2 started at 24-jun -2014 01:03:00.
3 Optimization solver (SNOPT)
4 Analytic gradient with the adjoint method .
5 Itns Major Minor Step nPDE Error Objective
6 5 0 5 - 1 0.521 1.919e+05
17 16 11 1 0.38 13 0.0182 1.865e+05
18 17 12 1 1.00 14 0.0153 1.864e+05
19 18 13 1 1.00 15 0.018 1.864e+05
20 19 14 1 1.00 16 0.0107 1.864e+05
21 20 15 1 1.00 17 0.000933 1.864e+05
22 21 16 1 1.00 19 0.00347 1.864e+05
23 22 17 1 1.00 20 0.0013 1.864e+05
24 23 18 1 1.00 22 0.0031 1.864e+05
25 24 19 1 0.32 24 0.000521 1.864e+05
26 25 20 1 1.00 26 0.00098 1.864e+05
27 26 21 1 0.01 30 0.0004 1.864e+05
28 27 22 1 0.00 34 7.01e -05 1.864e+05
29 Number of optimization variables : 6.
30 Number of objective function evaluations : 36.
31 Number of Jacobian evaluations : 34.
32 Final objective function value : 186426.2261.
33 Optimality conditions satisfied .
34 Optimization Solver 1 in Solver 2: Solution time: 28 s
35 Physical memory : 879 MB
36 Virtual memory : 5299 MB
C.5 Four summations 123
Figure C.22: v. Mises stress in 1D
Figure C.23: v. Mises stress in 2D
Figure C.24: v. Mises stress in 3D
Figure C.25: The error of the solution as a function of iteration number for the nonlinear solver
C.5 Four summations 125
Figure C.26: The error of the solution as a function of iteration number for the optimization solver
Iteration#σvM,max[MPa]Utot[kJ]U[J/m3 ]Area[m2 ]p1[-]p2[-]q1[-]q2[-]q3[-]q4[-] 17.01191.8610.29525.000.63-0.36-0.300.450.250.46 27.10191.1710.23524.991.47-0.78-0.640.910.350.75 35.66188.8410.04524.837.60-3.18-3.062.65-0.77-0.16 45.31187.469.92524.8613.48-5.01-5.811.700.301.04 55.18187.139.89524.9018.07-6.43-7.811.501.770.21 64.96186.839.87524.9122.70-7.69-9.071.251.170.75 74.78186.679.85524.9825.01-8.23-9.710.591.240.76 84.57186.619.85524.9926.90-8.54-10.43-0.071.010.65 94.18186.549.84525.0030.08-8.75-12.16-1.690.070.22 104.04186.509.84525.0031.46-8.60-13.35-2.37-0.54-0.11 113.95186.669.86524.9337.63-7.60-19.39-5.69-3.37-1.58 123.88186.459.84525.0034.59-8.11-16.44-3.64-1.69-0.75 133.90186.449.84525.0035.80-7.94-17.73-3.71-1.69-0.71 143.91186.439.84525.0036.79-7.80-18.74-4.04-1.76-0.64 153.92186.439.84525.0037.58-7.73-19.50-4.40-1.80-0.56 163.92186.439.84525.0037.75-7.75-19.58-4.47-1.83-0.57 173.91186.439.84525.0037.87-7.87-19.44-4.49-1.85-0.60 183.91186.439.84525.0038.03-7.85-19.59-4.53-1.86-0.60 193.92186.439.84525.0038.25-7.77-19.91-4.62-1.90-0.60 203.91186.439.84525.0038.07-7.83-19.67-4.54-1.87-0.60 213.91186.439.84525.0037.99-7.84-19.58-4.50-1.86-0.60 223.91186.439.84525.0038.04-7.82-19.66-4.54-1.87-0.60 233.91186.439.84525.0038.07-7.83-19.67-4.54-1.87-0.60 TableC.27:Valuesforeachiterationduringoptimization
C.6 Five summations 127
C.6 Five summations
1 ============================================================
2 Optimization Solver 1 in Solver 2 started at 24-jun -2014 00:57:39.
3 Optimization solver (SNOPT )
4 Analytic gradient with the adjoint method .
5 Itns Major Minor Step nPDE Error Objective
6 6 0 6 - 1 0.487 1.919e+05
32 Number of optimization variables : 7.
33 Number of objective function evaluations : 41.
34 Number of Jacobian evaluations : 39.
35 Final objective function value : 186418.0779.
36 Optimality conditions satisfied .
37 Optimization Solver 1 in Solver 2: Solution time: 33 s
38 Physical memory : 860 MB
39 Virtual memory : 5283 MB
Figure C.28: v. Mises stress in 1D
Figure C.29: v. Mises stress in 2D
C.6 Five summations 129
Figure C.30: v. Mises stress in 3D
Figure C.31: The error of the solution as a function of iteration number for the nonlinear solver
Figure C.32: The error of the solution as a function of iteration number for the optimization solver
C.6 Five summations 131
Iteration#σvM,max[MPa]Utot[kJ]U[J/m3 ]Area[m2 ]p1[-]p2[-]q1[-]q2[-]q3[-]q4[-]q5[-] 17.11191.8510.29525.000.58-0.36-0.310.430.220.430.32 27.22191.1510.23524.991.43-0.82-0.670.910.320.680.45 35.53188.8210.04524.817.90-3.38-3.062.72-0.56-0.03-0.61 45.38187.479.92524.8513.96-5.29-6.051.920.420.460.77 56.36187.569.92524.8319.92-7.10-8.090.601.662.39-0.77 65.18186.879.87524.9120.51-7.34-8.190.871.610.890.32 74.88186.849.87524.8625.93-8.80-9.590.430.991.070.91 84.86186.609.85525.0026.07-8.64-9.750.161.101.010.80 94.73186.589.85525.0026.30-8.43-10.30-0.261.000.870.62 104.36186.569.85524.9827.89-7.90-12.36-1.540.360.370.18 114.19186.499.84525.0029.70-7.82-13.69-2.17-0.180.130.09 123.80186.669.86524.9040.00-6.55-22.74-6.44-4.01-1.84-0.87 133.85186.439.84525.0036.59-7.45-19.04-4.45-2.13-0.97-0.41 143.84186.429.84525.0038.36-7.60-20.09-4.81-2.31-1.06-0.43 153.84186.429.84525.0039.80-7.69-20.99-5.19-2.48-1.11-0.42 163.85186.429.84525.0040.00-7.69-21.16-5.28-2.49-1.09-0.41 173.86186.429.84525.0040.00-7.68-21.20-5.27-2.44-1.03-0.38 183.86186.429.84525.0040.00-7.65-21.24-5.27-2.44-1.03-0.38 193.86186.429.84525.0039.71-7.49-21.31-5.25-2.46-1.01-0.40 203.86186.429.84525.0040.00-7.60-21.33-5.28-2.46-1.04-0.39 213.86186.429.84525.0040.00-7.67-21.22-5.23-2.43-1.04-0.38 223.86186.429.84525.0040.00-7.61-21.31-5.27-2.46-1.03-0.39 233.86186.429.84525.0040.00-7.56-21.41-5.28-2.48-1.04-0.40 243.86186.429.84525.0040.00-7.60-21.34-5.27-2.46-1.04-0.39 253.86186.429.84525.0040.00-7.67-21.22-5.25-2.42-1.02-0.37 263.86186.429.84525.0040.00-7.60-21.33-5.27-2.46-1.03-0.39 TableC.33:Valuesforeachiterationduringoptimization
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