Philippe Destuynder

Explicit error bounds in a conforming finite element method

The goal of this paper is to define a procedure for bounding the error in a conforming finite element method. The new point is that this upper bound is fully explicit and can be computed locally. Numerical tests prove the efficiency of the method. It is presented here for the case of the Poisson equation and a first order finite element approximation.

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.

Explicit error bounds for a nonconforming finite element method

Let u be the solution of the following model:

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

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Few Remarks on the Controllability of an Aeroacoustic Model Using Piezo-Devices

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A dual algorithm for denoising and preserving edges in image processing

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A penalty duality method for the budiansky-sanders shell model

where f is a given function in L2(\Omega)

A mixed finite element for shell model with free edge boundary conditions Part 3. Numerical aspects

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