Aziz Hamdouni

A classification of thin plate models by asymptotic expansion of non-linear three-dimensional equilibrium equations

Abstract We present in this paper a new constructive method of classification of two-dimensional plate models from the applied forces level and the geometrical data. This approach which uses asymptotic methods is based on a dimensional analysis of the non-linear equilibrium equations. This dimensional analysis leads to dimensionless numbers which reflect the geometry of the structure and the applied forces. For a given forces level, the order of magnitude of the displacements and the corresponding two-dimensional model are deduced by asymptotic expansion of the three-dimensional equations. For decreasing forces level, we obtain successively the non-linear membrane model, another membrane model, the usual non-linear plate model and the linear Kirchhoff–Love model.

A priori reduction method for solving the two-dimensional Burgers’ equations

Abstract The two-dimensional Burgers’ equations are solved here using the A Priori Reduction method. This method is based on an iterative procedure which consists in building a basis for the solution where at each iteration the basis is improved. The method is called a priori because it does not need any prior knowledge of the solution, which is not the case if the standard Karhunen–Loeve decomposition is used. The accuracy of the APR method is compared with the standard Newton–Raphson scheme and with results from the literature. The APR basis is also compared with the Karhunen–Loeve basis.

Comparison of some Lie-symmetry-based integrators

Lie-symmetry based integrators are constructed in order to preserve the local invariance properties of the equations. The geometrical methods leading to discretized equations for numerical computations involve many different concepts. Therefore they give rise to numerical schemes that vary in the accuracy, in the computational cost and in the implementation. In this paper a comparison is made between some alternative Lie-symmetry based methods illustrated on the example of the Burgers equation. The importance of the symmetry preservation is numerically highlighted.

A mathematical and numerical study of the sensitivity of a reduced order model by POD (ROM–POD), for a 2D incompressible fluid flow

Abstract In this work, we present contributions concerning a mathematical study of the sensitivity of a reduced order model (ROM) by the proper orthogonal decomposition (POD) technique applied to a quasi-linear parabolic equation. In particular, we apply our theoretical study to the Navier–Stokes equations for a 2D incompressible fluid flow. We present a numerical test of our theoretical result, for an unsteady fluid flow in a channel around a circular cylinder.

Low order dynamical system for fluid-rigid body interaction problem using POD method

This paper describes the Reduced Order Modeling (ROM) for fluid rigid body interaction problem and discusses Proper Orthogonal Decomposition (POD) utilisation. The principal difficulty for using POD being the moving domains, a referenced fixed domain is introduced. The POD is applied for the velocity field obtained on the fixed domain. Then a method to reduce dynamical system in rigid body fluid interaction is developed. This method uses the fictitious domain method approach, which consists in treating the entire fluid-solid domain as a fluid. The rigid body is considered as a fluid, by using a high viscosity which can play the role of a penalisation factor of the rigidity constraint, and by adding a distributed Lagrange multiplied associated to this constraint in the weak formulation. The fluid flow problem is then formulated on the reference domain and POD modes are used in the weak formulation. The results are compared with computational solution and discussed.

Consequences of Symmetries on the Analysis and Construction of Turbulence Models

Since they represent fundamental physical properties in turbulence (conservation laws, wall laws, Kolmogorov energy spectrum, ...), symmetries are used to analyse common turbulence models. A class of symmetry preserving turbulence models is proposed. This class is refined such that the models respect the second law of thermodynamics. Finally, an example of model belonging to the class is numerically tested.

An asymptotic elastic–plastic plate model for moderate displacements and strong strain hardening

Abstract We propose in this paper to generalize to elastic–plastic plates the constructive asymptotic approach developed by the authors for elastic plates and shells. A dimensional analysis of three-dimensional equations makes appear dimensionless numbers characterizing the problem. Then using the geometric and mechanical data, these dimensionless numbers are linked to the perturbation parameter (the relative thickness of the plate) to obtain a one scale problem. For standard generalized materials with strong strain hardening, subjected to usual force levels, the asymptotic expansion of equations leads to an elastic–plastic plate model valid for moderate displacements. The constructive approach developed enables to specify the domain of validity of the asymptotic elastic–plastic plate model obtained thanks to the dimensional numbers introduced.

Time integration algorithm based on divergent series resummation, for ordinary and partial differential equations

Borels technique of divergent series resummation is transformed into a numerical code and used as a time integration scheme. It is applied to the resolution of regular and singular problems arising in fluid mechanics. Its efficiency is compared to those of classical discretization schemes.

Reduced-order modelling by POD-multiphase approach for fluid-structure interaction

This paper describes the Reduced Order Modeling (ROM) for fluid rigid body interaction problem and discusses Proper Orthogonal Decomposition (POD) utilisation. The principal difficulty for using POD being the moving domains, a referenced fixed domain has been introduced. The POD has been applied for the velocity field obtained on the fixed domain. Then a method to reduce dynamical system for rigid body fluid interaction has been developed. This method consists in treating the entire fluid-solid domain as a fluid. The rigid body has then been considered as a fluid, by using a high viscosity which can play the role of a penalisation factor of the rigidity constraint. The fluid flow problem is then formulated on the reference domain and POD modes have been used in the weak formulation.

Proper orthogonal decomposition investigation in fluid structure interaction

This paper describes Reduced Order Modeling (ROM) in Fluid Structure Interaction (FSI) and discusses Proper Orthogonal Decomposition (POD) utilization. The ROM method was selected because its performance in fluid mechanics. The principal problems of its application in FSI are due the space character of the modes resulting from the POD whereas domains are mobile. To use POD in moving domain, a charateristic function of fluid is introduced in order to work on a fixed rigid domain, and the global velocity (fluid and structure) is studied. The POD modes efficiency is tested to reconstruct velocity field in one and two-dimensional FSI case. Then reducing dynamic system using POD is introduced in moving boundaries problem. In addition, the one dimensional case of Burgers equation coupled with spring equation is tested.