This paper is devoted to the theoretical and numerical study of singular perturbation problems for elliptic inhibited shells. We present a reduction of the classical membrane equations to a partial differential equation with respect to the bending displacement, which is well adapted to the study of singularities of the limit problem. For a discontinuous loading or when the boundary of the loading domain presents corners, we put in a prominent position the existence of two kinds of singularities. One of them is not classical; it reduces to a logarithmic point singularity at the corner of the loading domain. To finish numerical simulations are performed with a finite element software coupled with an anisotropic adaptive mesh generator. They enable to visualize precisely the singularities predicted by the theory with only a very small number of elements.

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The entire glass sheet forming process consists of heating and forming a glass sheet and cooling and tempering it afterwards. For the first step, the glass sheet is heated using a local radiative source and deforms by sagging. In the thermo-mechanical calculations, temperature dependent glass viscosity, heat exchange with the ambient air and radiative source effects should be considered. A two-dimensional finite element model with plane deformation assumptions is developed. Using the PI-Approximation, the formulation and numerical resolution of the Radiative Transfer Equation (RTE) are performed on the glass domain as it changes over time to estimate the flux of the radiative body at each position in the glass. In the next step, the sheet is cooled. Narayanaswamy's model is used to describe the temperature dependent stress relaxation and the structural relaxation. The RTE is again solved using the P1-Approximation to consider the internal radiative effects during the cooling. There is a discussion using the P1-Approximation and comparing the results to other existing methods for the temperature changes of the glass throughout the forming process, for the deformed shape at the end of the forming step and for the residual stresses after tempering

This paper deals with elliptic shell problems using the Koiter shell model. When the shell is well-inhibited, the limit membrane problem satisfies the Shapiro-Lopatinskii condition and we have a classical singular perturbation problem. In a previous paper, the existence of two kinds of singularities was put in a prominent position for this kind of problem. Conversely, for ill-inhibited shells (when a part of the boundary is free), the limit problem does not satisfy the Shapiro-Lopatinskii condition. Complexification phenomenon appears when the thickness approaches zero, leading to large oscillations corresponding to a new kind of instability on the free boundary. To complete the theoretical analysis, numerical simulations are performed with a finite element software coupled with an anisotropic adaptive mesh generator. This enables us to visualize precisely the singularities and the instabilities predicted by the theory with only a small number of elements.