“…The physical insight of porosities in the shells is revealed by the porosity volume fraction and porosity distribution profiles. Some studies that presented exact solutions for other focuses using the Hamiltonian system have been reported (Chen and Zhao, 2009;Kattimani and Ray, 2018;Lim and Xu, 2010;Ni et al, 2017;Sun et al, 2013Sun et al, , 2014aSun et al, , 2014bWang and Qin, 2007;Yao et al, 2009).…”