Tengiz Buchukuri, Roland Duduchava

Solvability and Numerical Approximation of the Shell Equation Derived by the Γ-Convergence

abstract:
A mixed boundary value problem for the Lam\'e equation in a thin layer $\Omega^h=\mathcal{C}\times[-h,h]$ around a surface $\mathcal{C}$ with the Lipshitz boundary is investigated. The main goal is to find out what happens when the thickness of the layer tends to zero, $h\to0$. To this end, we reformulate BVP into an equivalent variational problem and prove that the energy functional has the $\Gamma$-limit of the energy functional on the mid-surface $\mathcal{C}$. The corresponding BVP on $\mathcal{C}$, considered as the $\Gamma$-limit of the initial BVP, is written in terms of G\"unter's tangential derivatives on $\mathcal{C}$ and represents a new form of the shell equation. It is shown that the Neumann boundary condition from the initial BVP on the upper and lower surfaces transforms into the right-hand side of the basic equation of the limit BVP. The finite element method is established for the obtained BVP.

Mathematics Subject Classification: 35J05, 35J20, 53A05, 80A20

Key words and phrases: Hypersurface, Günter's derivatives, Lam\'e equation, $\Gamma$-convergence, shell equation