Mathematical Models and Methods in Applied Sciences

We first establish that the linearized change of metric and change of curvature tensors, with components in L² and H^-1 respectively, associated with a displacement field, with components in H¹, of a surface S immersed in ℝ³ must satisfy in the distributional sense compatibility conditions that may be viewed as the linear version of the Gauss and Codazzi-Mainardi equations. These compatibility conditions, which are analogous to the familiar Saint Venant equations in three-dimensional elasticity, constitute the Saint Venant equations on the surface S.

We next show that these compatibility conditions are also sufficient, i.e. that they in fact characterize the linearized change of metric and the linearized change of curvature tensors in the following sense: If two symmetric matrix fields of order two defined over a simply-connected surface S ⊂ ℝ³ satisfy the above compatibility conditions, then they are the linearized change of metric and linearized change of curvature tensors associated with a displacement field of the surface S, a field whose existence is thus established.

The proof provides an explicit algorithm for recovering such a displacement field from the linearized change of metric and linearized change of curvature tensors. This algorithm may be viewed as the linear counterpart of the reconstruction of a surface from its first and second fundamental forms.

Finally, we show how these results can be applied to the "intrinsic theory" of linearly elastic shells, where the linearized change of metric and change of curvature tensors are the new unknowns. These new unknowns solve a quadratic minimization problem over a space of tensor fields whose components, which are only in L², satisfy the Saint Venant compatibility conditions on a surface in the sense of distributions.

We consider optimal design problems for semiconductor devices which are simulated using the energy transport model. We develop a descent algorithm based on the adjoint calculus and present numerical results for a ballistic diode. Furthermore, we compare the optimal doping profile with results computed based on the drift diffusion model. Finally, we exploit the model hierarchy and test the space mapping approach, especially the aggressive space mapping algorithm, for the design problem. This yields a significant reduction of numerical costs and programming effort.

In this paper, we give a global existence theorem of weak solutions to model equations governing interaction fluid structure in a two-dimensional layer, cf. Refs. 8 and 14. To our knowledge this is the first existence theorem of global in time solutions for such model. The interest of our result is double because, first, we change the original initial value problem by deleting one initial condition, second, we construct a solution through the classical Galerkin method for which several computing codes have been constructed.

We consider a class of abstract second-order evolutionary inclusions involving a Volterra-type integral term, for which we prove an existence and uniqueness result. The proof is based on arguments of evolutionary inclusions with monotone operators and the Banach fixed point theorem. Next, we apply this result to prove the solvability of a class of second-order integrodifferential hemivariational inequalities and, under an additional assumption, its unique solvability. Then we consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with a viscoelastic constitutive law involving a long memory term and the contact is modelled with subdifferential boundary conditions. We derive the variational formulation of the problem which is of the form of an integrodifferential hemivariational inequality for the displacement field. Then we use our abstract results to prove the existence of a unique weak solution to the frictional contact model.

In this paper, the Oberbeck–Boussinesq approximation is identified as a singular limit of the full Navier–Stokes–Fourier system provided the Mach and Froude numbers tend to zero. The result holds for any ill-prepared initial data and without any restrictions imposed on the length of the time interval. In particular, it is shown that the velocity converges almost everywhere, the oscillations of the sound waves being effectively damped by the presence of a "wavy bottom" of the physical domain.