Michel Salaun

Mathematical Analysis of Thin Plate Models

I - Plate models for thin structures.- I.0 - A short description of the chapter.- I.1 - The three dimensionnal elastic-model.- I.1.1 - About the kinematics.- I.1.2 - About the Principle of virtual work.- I.1.3 - About the constitutive relationship.- I.1.4 - Existence uniqueness of the solution to the elastic model.- I.2 - The Kirchhoff-Love assumption.- I.3 - The Kirchhof f-Love plate model.- I.3.1 - Existence and uniqueness of a solution to the Kirchhof f-Love model.- I.3.2 - The local equations satisfied by the Kirchhoff-Love plate model.- I.3.3 - The transverse shear stress in Kirchhoff-Love theory.- I.4 - The Naghdi model revisited using mixed variational formulation.- I.4.1 - Existence and uniqueness of a solution to the revisited Naghdi model.- I.4.2 - Local equations of the Naghdi model.- I.5 - About the rest of the book.- References of Chapter I.- II - Variational formulations for bending plates.- II.0 - A brief summary of the chapter.- II. 1 - Why a mixed formulation for plates.- II.2 - The primal variational formulation for Kirchhoff-Love model.- II.2.1 - Double Stokes formula for plates.- II.2.2 - The variational formulation.- II.2.3 - Another variational formulation.- II.2.4 - Interest of formulation (II. 12).- II.3 - The Reissner-Mindlin-Naghdi model for plates.- II.3.1 - The penalty method applied to the Kirchhoff-Love model.- II.3.2 - A correction to the penalty method.- II.4 - Natural duality techniques for the bending plate model.- II.4.1 - A mixed variational formulation for Kirchhoff-Love model.- II.4.2 - Existence and uniqueness of solution to the mixed formulation.- II.4.3 - Computation of the deflection u3.- II.4.4 - How to be sure we solved the right model (interpretation of the model).- II.4.5 - What is the meaning of ? and when is it zero?.- II.4.6 - Non-homogeneous boundary conditions.- II.4.7 - The revisited modified Reissner-Mindlin-Naghdi model.- II.4.8 - Extension to a multi-connected boundary.- II.5 - A comparison between the mixed method and the one of section II.2.4.- References of Chapter II.- III - Finite element approximations for several plate models.- III.0 - A summary of the chapter.- III. 1 - Basic results in finite element approximation.- III. 1.1 - Several useful definitions.- III. 1.2 - A brief recall concerning error estimates.- III.2 - C1 elements.- III.3 - Primal finite element methods for bending plates.- III.4 - The penalty-duality finite element method for the bending plate model.- III.4.1 - Stability with respect to the penalty parameter of the R.M.N. solution.- III.4.2 - A finite element scheme and error estimates for the R.M.N. model.- III.4.3 - Practical aspects in solving the R.M.N. finite element model.- III.4.4 - About the famous QUAD4 element.- III.5 - Numerical approximation of the mixed formulation for a bending plate.- III.5.1 - General error estimates between (0,A) and (?, ?h).- III.5.2 - Theoretical estimates on u3 - u3h.- III.5.3 - A first choice of finite elements.- HI.5.4 - A second choice of finite elements.- References of Chapter III.- IV - Numerical tests for the mixed finite element schemes.- IV.0 - A brief description of the chapter.- IV. 1 - Precision tests for the mixed formulation.- IV. 1.1 - A recall of the equations to be solved.- IV. 1.2 - Numerical tests.- IV. 1.3 - A few remarks relative to the above numerical results.- IV.2 Vectorial and parallel algorithms for mixed elements.- IV.2.1 - Three strategies for solving the system (IV. 18).- IV.2.2 - Optimization of Crout factorization.- IV.2.3 - Optimization of node renumbering.- IV.2.4 - Numerical tests.- IV.3 - Concluding remarks.- References of Chapter IV.- V - A Numerical model for delamination of composite plates.- V.O - A brief description of the chapter.- V. 1 - What is delamination of thin multilayered plates.- V.2 - The three-dimensional multilayered composite plate model with delamination.- V.3 - A plate model for large delamination.- V.4 - The three-dimensional energy release rate.- V.4.1 - The energy release rate..- V.4.2 - The energy release rate for delaminated plates.- V.5 - The mechanical example and the numerical method.- V.5.1 - The specimen studied.- V.6 - Concluding remarks.- References of Chapter V.

A mixed finite element for shell model with free edge boundary conditions Part 1. The mixed variational formulation

Abstract After a very brief recall on general shell theory, we construct a mixed variational formulation based on the introduction of a new unknown: the rotation of the normal to the medium surface. In Koiter shell theory (for instance), this rotation can be expressed with respect to the three components of the displacement field of the medium surface and their derivatives. The Lagrange multiplier corresponding to this relation (known as the Kirchhoff-Love kinematical assumption), is also introduced as an independent unknown. There are two main difficulties: one is due to the differential geometry of surfaces and is rather technical; the other is to define correctly the dual space for the Kirchhoff-Love relation. The difficulty is similar to the one met in the characterization of the dual space of the Sobolev space: H 1 ( ω ) (ω being the medium surface of the shell), for which a boundary component appears except for clamped shells which is a very restrictive situation.

A mixed finite element for shell model with free edge boundary conditions Part 2. The numerical scheme

Abstract From the mixed variational formulation which was suggested in Part 1, we construct a finite element scheme. Then, a mathematical justification concerning the error estimate is developed. The most important point is to justify two compatibility conditions between the approximation of the kinematics and the stresses. Because of the structure of the shell operator they are not obvious at all and several restrictions on the mesh are necessary. Therefore, the classical result known in fluid mechanics for Stokes equations cannot be applied directly.

Integral method for the Stokes problem

Abstract We consider the bidimensional Stokes problem for incompressible fluids in stream function-vorticity formulation. For this problem, the classical finite elements method of degree one converges only in O ( h ) for the quadratic norm of the vorticity, if the domain is convex and the solution regular. We propose to use harmonic functions obtained by a simple layer potential to approach vorticity along the boundary. Numerical results are very satisfying and we prove that this new numerical scheme leads to an error of order O (h) for the natural norm of the vorticity and under more regularity assumptions from O (h 3/2 ) to O (h 2 ) for the quadratic norm of the vorticity. To cite this article: T. Abboud et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 71–76

Numerical Tests for the Mixed Finite Element Schemes for Bending Plates

The numerical performances of the finite element schemes discussed in the previous chapters are presented. The results obtained are compared to the ones of the QUAD 4 element of MacNeal. Additionnally the solution methods are evaluated, and the vector and parallel optimization has been used in order to deliver the best computational time.

Plate Models for Thin Structures

In order to simplify three-dimensional equations, mechanical assumptions can be used in order to derive a simpler model for thin structures. The Kirchhoff-Love model is obtained by this method. But the Naghdi formulation is also an interesting plate model. In this chapter, both are derived from the three-dimensional theory, using the Hellinger-Reissner mixed formulation.

Variational Formulations for Bending Plates

From the continuous point of view, the bending plate model involves a fourth-order operator. The goal of this chapter is to give several variational formulations of the plate model which only involve second order operators. First of all, we shall derive simple penalty methods based on the introduction of the rotation of the unit normal to the medium surface of the plate. But a major difficulty appears due to the boundary conditions on the edges of the plate. Then a more general mixed formulation is derived, for which several numerical applications are given in the other chapters. Let us go further in the details of the formulations which are explored. After a brief recall of the classical primal formulation, we focus historically on the first attempt to derive a mixed formulation. The idea is due to R. Glowinski. We extend it to more general boundary conditions.

Finite Element Approximations for Several Plate Models

After a brief reminder of finite elements methods we discuss the possible approximations of the penalty or modified penalty model which has been studied in Chapter II. The connections with the famous QUAD 4 element (and its numerous variants) are examined. This permits one to give a partial justification of this well-known element and an extension to general cases including triangles. Then the natural duality technique is used to construct a new kind of structural finite elements which are analyzed from the error point of view.

A Numerical Model for Delamination of Composite Multilayered Plates

After a brief recall on the laminated plate theory, a computational model for studying the delamination is presented. In order to avoid a too complicated mathematical justification (which is included in [1]), we base our developments on physical feelings. Then a mechanical example including comparison with experiments is discussed. All the chapter is a survey of a study performed five years ago with our friend Thierry Nevers. The extended versions can be found in the references [2], [3], [4], [5], [6].