Introduction
Density-based topology optimization methods
Super-resolution convolutional neural network
Network architecture
Patch extraction and representation
Nonlinear mapping
Image reconstruction
Network calculation steps
Training
Details of the optimization
High-resolution filter
Tab.1 Model data of different high-resolution transformations |
Method | Enhancement mode | Model size | FEA mesh | Filter radius | Output resolution | |
---|---|---|---|---|---|---|
Low-resolution | Basic model | 200×100 | 200×100 | 3 | 200×100 | |
Intuitive choice | Large-scale | 800×400 | 800×400 | 3 | 800×400 | |
High-precision | 200×100 | 800×400 | 12 | 800×400 | ||
Training set | Large-scale | 800×400 | 800×400 | 3 | 800×400 | |
High-precision | 200×100 | 800×400 | 15 | 800×400 |
Pooling strategy
Numerical implementation
Combination treatment of 3D models
Numerical examples
2D numerical examples
Fig.8 High-resolution images of the barrier structure under various strategies. Results obtained by traditional methods: (a) Low-resolution, 210×210, rmin= 4, cobj = 91.495; (b) high-precision, 840×840, rmin = 19, cobj = 97.613; (c) large-scale, 840×840, rmin = 4, cobj = 84.179. (d) Result post-processed by SRCNN, 840×840, rmin = 4, cobj = 1.2×108. Results obtained by HRTO: (e) High-precision, 840×840, rmin = 4, cobj = 81.168; (f) large-scale, 840×840, rmin = 0.25, cobj= 83.348. rmin and cobj represent the filter radius and the objective function, respectively. |
Fig.9 High-resolution images of the sandwich structure under various strategies. Results obtained by traditional methods: (a) Low-resolution, 200×140, rmin = 4, cobj= 2.0214; (b) high-precision, 800×560, rmin = 19, cobj = 2.0715; (c) large-scale, 800×560, rmin = 4, cobj = 1.9335. (d) Result post-processed by SRCNN, 800×560, rmin = 4, cobj= 2.1632. Results obtained by HRTO: (e) High-precision, 800×560, rmin = 4, cobj = 1.8352; (f) large-scale, 800×560, rmin = 0.25, cobj = 1.9201. rmin and cobj indicate the filtering radius and objective function, respectively. |
Tab.2 Alternative optimization parameters |
Basic resolution | Target volume | Filter radius | Upscaling factor |
---|---|---|---|
100×50 | 0.3 | 1 | 2 |
120×60 | 0.4 | 2 | 3 |
140×70 | 0.5 | 3 | 4 |
160×80 | 0.6 | 4 | – |
180×90 | 0.7 | 5 | – |
Fig.11 The influence of each optimization parameter of 2D designs on the objective. The influence of (a) number of elements, (b) volume fraction, (c) filter radius, and (d) upscaling factor on the objective under the high-precision situation. The influence of (e) number of elements, (f) volume fraction, (g) filter radius, (h) upscaling factor on the objective under the large-scale situation. |
Tab.3 MBB beam efficiency of the conventional method and HRTO method |
Method | Enhancement mode | Output resolution | I.T./s | It. | S.T./s | Max. ram/GB |
---|---|---|---|---|---|---|
Low-resolution | Basic model | 200×100 | 0.0994 | 606 | 0.3328 | 0.0100 |
Conventional | High-precision | 800×400 | 209.4000 | 8174 | 20.0261 | 2.5303 |
Large-scale | 800×400 | 1.6530 | 4231 | 7.5468 | 0.1529 | |
HRTO | High-precision | 800×400 | 0.1138 | 1092 | 2.7686 | 0.1621 |
Large-scale | 800×400 | 0.0256 | 474 | 2.7592 | 0.1621 |
Tab.4 Efficiency data reduction ratio of the conventional method and HRTO method |
Enhancement mode | Reduction ratio/% | |||
---|---|---|---|---|
I.T. | It. | S.T. | Max. ram | |
High-precision | 99.95 | 86.64 | 86.18 | 93.59 |
Large-scale | 98.45 | 88.80 | 63.44 | –6.08 |
Tab.5 Efficiencies of the HRTO method at different resolutions |
Basic resolution | Output resolution | Efficiency of conventional method | Efficiency of HRTO | ||||
---|---|---|---|---|---|---|---|
I.T./s | S.T./s | I.T./s | Reduction ratio/% | S.T./s | Reduction ratio/% | ||
High-precision | |||||||
100×50 | 400×200 | 15.19 | 4.591 | 0.023 | 99.85 | 0.660 | 85.63 |
120×60 | 480×240 | 24.89 | 7.166 | 0.039 | 99.84 | 0.905 | 87.37 |
140×70 | 560×280 | 37.55 | 7.068 | 0.054 | 99.86 | 1.254 | 82.25 |
160×80 | 640×320 | 58.13 | 9.790 | 0.073 | 99.88 | 1.719 | 82.44 |
180×90 | 720×360 | 121.0 | 14.14 | 0.088 | 99.93 | 2.115 | 85.04 |
200×100 | 800×400 | 209.4 | 20.03 | 0.114 | 99.95 | 2.769 | 86.18 |
Large-scale | |||||||
100×50 | 400×200 | 0.539 | 1.710 | 0.006 | 98.85 | 0.695 | 59.37 |
120×60 | 480×240 | 0.731 | 2.607 | 0.007 | 99.00 | 0.899 | 65.51 |
140×70 | 560×280 | 0.746 | 2.794 | 0.010 | 98.66 | 1.230 | 56.00 |
160×80 | 640×320 | 1.015 | 3.935 | 0.014 | 98.66 | 1.631 | 58.54 |
180×90 | 720×360 | 1.342 | 5.089 | 0.021 | 98.41 | 2.015 | 60.40 |
200×100 | 800×400 | 1.653 | 7.547 | 0.026 | 98.45 | 2.759 | 63.44 |
3D numerical examples
Tab.6 Comparison of acceleration ratios of three algorithms |
Method | Acceleration rate | |
---|---|---|
2D model | 3D model | |
FCM-based | 2.9 | 32 |
MsFEM-based | 17 | 50 |
HRTO | 24–54 | 79 |