Introduction
Multi-material interpolation scheme
Tab.1 Expansion of the proposed interpolation scheme for three phases (M = 2) |
Density variables | Young’s modulus |
---|---|
, | |
, | |
, |
Deterministic topology optimization
Robust topology optimization
Uncertainty quantification and propagation
Multi-material interpolation scheme under uncertainty
Calculation of statistical moments
Sensitivity analysis
Numerical examples
Tab.2 Material properties of the 4-phase examples |
Material | Color | Mean Young’s modulus/MPa | Poisson’s ratio | Volume ratio constraint |
---|---|---|---|---|
Void | ‒ | 1.0e‒8 | 0.3 | ‒ |
Soft | Green | 500 | 0.3 | 10% |
Medium | Blue | 1000 | 0.3 | 10% |
Hard | Yellow | 2000 | 0.3 | 10% |
Materials uncertainty
Tab.3 Mean and standard deviation of the compliance (N∙mm) results under material uncertainty using UDR with 8 quadrature nodes and Monte Carlo with 10000 samples |
Solution type | Statistical moment | UDR/quadrature | Monte Carlo | Difference/% |
---|---|---|---|---|
Deterministic (b = 0) | Mean | 0.43036 | 0.43027 | 0.02 |
Std. Dev. | 0.02063 | 0.02087 | 1.13 | |
Robust (b = 10) | Mean | 0.46244 | 0.46262 | 0.04 |
Std. Dev. | 0.02018 | 0.02072 | 2.61 |
Std. Dev.: Standard deviation |
Load uncertainty
Tab.4 Compliance (N∙mm) results under load uncertainty using UDR with 8 quadrature nodes and Monte Carlo with 10000 samples |
Solution type | Statistical moment | UDR/quadrature | Monte Carlo | Difference/% |
---|---|---|---|---|
Deterministic (b = 0) | Mean | 1.02900 | 1.04167 | 1.22 |
Std. Dev. | 0.82306 | 0.92341 | 10.87 | |
Robust (b = 1) | Mean | 0.74000 | 0.74949 | 1.27 |
Std. Dev. | 0.42108 | 0.49094 | 14.23 | |
Robust (b = 10) | Mean | 0.76359 | 0.77067 | 0.92 |
Std. Dev. | 0.38892 | 0.44047 | 11.70 |
Std. Dev.: Standard deviation |
Materials and load uncertainties
Tab.5 Compliance (N∙mm) results under material and load uncertainties using UDR with 8 quadrature nodes and Monte Carlo with 10000 samples |
Solution type | Statistical moment | UDR/quadrature | Monte Carlo | Difference/% |
---|---|---|---|---|
Deterministic (b = 0) | Mean | 1.03223 | 1.03533 | 0.30 |
Std. Dev. | 0.82331 | 0.92008 | 10.52 | |
Robust (b = 1) | Mean | 0.74380 | 0.76006 | 2.14 |
Std. Dev. | 0.42198 | 0.49447 | 14.66 | |
Robust (b = 10) | Mean | 0.77624 | 0.78349 | 0.93 |
Std. Dev. | 0.39135 | 0.45541 | 14.07 |
Std. Dev.: Standard deviation |