The MKS is a localization technique which allows for the computationally efficient prediction of some local field given some homogenized response at the macroscale. This study investigates the effect of the careful selection of generalized spherical harmonic (GSH) basis functions on the accuracy of the prediction of total strain fields in polycrystalline titanium under elastic-plastic loading.

1. Selection of GSH basis
functions for the MKS
prediction of total strain
Noah Paulson
May 26th, 2015
Advisors: Surya R. Kalidindi, David L.
McDowell, Donald S. Shih (Boeing)
1

2. Basis Selection Procedure
2
1. MKS calibration/validation performed with GSH for 𝑙 in [0,2,4] with the
addition of single extra basis function
i. Cycle through all basis functions for 𝑙 in [6,7,8]
2. Extra basis functions ranked by mean/maximum error characteristics
3. Basis functions successively added starting with those with best error
characteristics
4. Error improvements validated with different calibration/validation MVE
sets (cross-validation)
Note:
• 𝐶 =
𝑟𝑒𝑎𝑙 𝑇𝑙
𝑚𝑛
𝑙, 𝑚, 𝑛, 𝜑1, Φ, 𝜑2 for − 𝑙 ≤ 𝑚 ≤ 0
𝑖𝑚𝑎𝑔 𝑇𝑙
𝑚𝑛
𝑙, 𝑚, 𝑛, 𝜑1, Φ, 𝜑2 for 0 < 𝑚 ≤ 𝑙
• This maintains basis orthogonality

3. Basis Selection Mean Error
3
𝑒𝑟𝑟 ≡
𝜀 𝐹𝐸𝑀
− 𝜀 𝑀𝐾𝑆
𝜀11
𝐹𝐸𝑀
Mean Error: average of
error in all voxels in all
MVEs
0.490%
0.495%
0.500%
0.505%
0.510%
0.515%
Error
GSH Basis Functions (Cumulative)

4. Basis Selection Maximum Error
4
𝑒𝑟𝑟 ≡
𝜀 𝐹𝐸𝑀
− 𝜀 𝑀𝐾𝑆
𝜀11
𝐹𝐸𝑀
Maximum Error:
average of error in
voxel with maximum
error in all MVEs5.050%
5.100%
5.150%
5.200%
5.250%
5.300%
5.350%
5.400%
5.450%
Error
GSH Basis Functions (Cumulative

5. 5
Conclusions
• Careful addition of basis functions can improve MKS error characteristics for
prediction of total strain fields
• As number of terms added increases, rate of error reduction slows
• Perhaps enlarging the SVE set this effect could be abated
• Pattern in terms with best error characteristics:
• 𝑙 is even
• 𝑚 is even, 𝑚 < 0
• 𝑛 = 1