Professor Philippe G. Ciarlet

MAIN THEMES OF WORKS
 

 Numerical analysis of finite difference methods and general variational approximation methods: Discrete Green’s functions; discrete maximum principle and convergence; nonlinear problems of monotone type.

 Numerical analysis of the finite element method in general: Lagrange and Hermite interpolation theory in Rⁿ; discrete maximum principle and uniform convergence; curved finite elements ; numerical integration.

 Numerical analysis of the finite element method applied to problems in elasticity and fluid mechanics: Non-conforming finite elements and macro-elements for plate problems; mixed method for the biharmonic equation in fluid mechanics; finite element methods for shell problems.

 Modeling and mathematical analysis in three-dimensional linearized and nonlinear elasticity: Existence of solutions; incremental methods; modeling of constitutive equations; modeling of contact and noninterpenetration; weak form of Saint-Venant compatibility equations.

 Mathematical analysis of the von Kármán and generalized von Kármán equations: Existence, multiplicity, and bifurcation of solutions, equivalence with a "displacement" problem.

 Modeling of plates by the techniques of asymptotic analysis and singular perturbations: Convergence to a two-dimensional model in the linear case; justification of two-dimensional model, including the von Kármán equations, in the nonlinear case by formal asymptotic expansions; extension to “shallow shells”; “generalized” von Kármán and Marguerre-von Kármán equations.

 Modeling, mathematical analysis, and numerical simulation of “elastic multi-structures” that comprise junctions: Convergence to a “pluri-dimensional” model in the linear case; justification of the boundary conditions of clamping for a plate; vibrations of a multi-structure.

 Modeling and mathematical analysis of general shells: Existence theorems for two-dimensional linear shell models (W.T. Koiter, P.M. Naghdi, “flexural”, “membrane”); justification of two-dimensional linear shell models (“membrane” and “flexural” equations; W.T. Koiter’s equations) by the techniques of asymptotic analysis; existence theory for nonlinear shell equations; definition of a new nonlinear shell model “of Koiter’s type”; intrinsic shell equations.

 Differential geometry: Inequalities of Korn’s type on a surface; rigidity theorems and manifold theory; recovery and continuity of a manifold, with or without boundary, as a function of its metric tensor; nonlinear Korn inequality in an open set of Rⁿ ; recovery of a surface, with or without boundary, with prescribed first and second fundamental forms; continuity of a surface, with or without boundary, considered as a function of its two fundamental forms; nonlinear Korn inequality on a surface.

Intrinsic methods in elasticity: Weak Poincaré lemma and weak Saint Venant compatibility conditions; reformulation of linearized elasticity with the linearized strains as new unknowns; reformulation of linear shell theory with the linearized change of metric and change of curvature tensors as new unknowns; implementation as a finite element method with  "edge" finite elements; reformulation of nonlinear three-dimensional elasticity in terms of the Cauchy-Green strain tensor as the only unknown.