Displacement-based multiscale modeling of fiber-reinforced composites by means of proper orthogonal decomposition

Advanced Modeling and Simulation in Engineering Sciences

Table 2 Relative displacement and relative stress deviations of the multiscale problem with a linear elastic RVE

	\(u_x\)	\(u_y\)	\(u_z\)
Min (error)	0	0	0
Max (error)	0	0	\(1.86\cdot 10^{-8}\)

	\(\sigma _{xx}\)	\(\sigma _{yy}\)	\(\sigma _{zz}\)	\(\sigma _{xy}\)	\(\sigma _{yz}\)	\(\sigma _{zx}\)
Min (error)	0	0	\(1.98\cdot 10^{-14}\)	0	\(4.58\cdot 10^{-9}\)	\(2.05\cdot 10^{-8}\)
Max (error)	\(9.06 \cdot 10^{-10}\)	\(8.52\cdot 10^{-10}\)	\(1.16\cdot 10^{-9}\)	\(6.17\cdot 10^{-10}\)	\(6.30\cdot 10^{-5}\)	\(1.89\cdot 10^{-3}\)

The table gives the extreme values of the relative deviation \(|\Delta (*)|^*=|\overline{(*)} - (*)|/\mathrm{{max}}(|(*)|)\) of each displacement component \(\overline{u_i}\) and stress field \(\overline{\sigma }_{ij}\) computed in the FEPOD computation in comparison with the corresponding values \((*)\) of the reference FE\(^2\) computation for the multiscale example with linear elastic material behavior