Philippe Bouillard

Dispersion and pollution of meshless solutions for the Helmholtz equation

It is well known today that the standard finite element method (FEM) is unreliable to compute approximate solutions of the Helmholtz equation for high wavenumbers due to the pollution effect, consisting mainly of the dispersion, i.e. the numerical wavelength is longer than the exact one. Unless highly refined meshes are used, FEM solutions lead to unacceptable solutions in terms of precision, while the use of very refined meshed increases the cost in terms of computational times. The paper presents an application of the element-free Galerkin method (EFGM) and focuses on the dispersion analysis in 2D. It shows that it is possible to choose the parameters of the method in order to minimize the dispersion and to get extremely good results in comparison with the stabilized FEM. Moreover, the present meshless formulation is not restricted to regular distribution of nodes and a simple but real-life problem is investigated in order to show the improvement in the accuracy of the numerical results w.r. FEM results.

A residual a posteriori error estimator for the finite element solution of the Helmholtz equation

This paper suggests an expression of a residual-based a posteriori error estimation for a Galerkin finite element discretisation of the Helmholtz operator applied in acoustics. In the first part, we summarise the finite element approach and the a priori estimates. The new residual estimator is then formulated, illustrated and tested on two one-dimensional model problems with closed form solution.

Multi-objective reliability-based optimization with stochastic metamodels

This paper addresses continuous optimization problems with multiple objectives and parameter uncertainty defined by probability distributions. First, a reliability-based formulation is proposed, defining the nondeterministic Pareto set as the minimal solutions such that user-defined probabilities of nondominance and constraint satisfaction are guaranteed. The formulation can be incorporated with minor modifications in a multiobjective evolutionary algorithm (here: the nondominated sorting genetic algorithm-II). Then, in the perspective of applying the method to large-scale structural engineering problems—for which the computational effort devoted to the optimization algorithm itself is negligible in comparison with the simulation—the second part of the study is concerned with the need to reduce the number of function evaluations while avoiding modification of the simulation code. Therefore, nonintrusive stochastic metamodels are developed in two steps. First, for a given sampling of the deterministic variables, a preliminary decomposition of the random responses (objectives and constraints) is performed through polynomial chaos expansion (PCE), allowing a representation of the responses by a limited set of coefficients. Then, a metamodel is carried out by kriging interpolation of the PCE coefficients with respect to the deterministic variables. The method has been tested successfully on seven analytical test cases and on the 10-bar truss benchmark, demonstrating the potential of the proposed approach to provide reliability-based Pareto solutions at a reasonable computational cost.

Superconvergent patch recovery technique for the finite element method in acoustics

A posteriori error estimation has become very popular, mainly in linear elasticity. A robust implementation of the superconvergent patch recovery technique of O. C. Zienkiewicz and J. Z. Zhu is presented for acoustic finite element analyses: the original concepts are extended to complex variables, and both local and global behaviours of the recovery procedure and the error estimation are studied. The numerical tests confirm the improvement of the rates of convergence for the recovered solution and also show the reliability of the error estimator except at frequencies corresponding either to the analytical or to the finite element eigenfrequencies.

Improved sensitivity analysis by a coupled FE–EFG method

Abstract In this paper, we modify the sensitivity analysis of a structural optimization process by using a coupled finite element–element-free Galerkin method. The aim is to improve the sensitivity analysis and to avoid a mesh degradation that can occur when the design variables are perturbed while using classical finite element method. The idea is to replace the finite element mesh by EFGM nodes in areas where the sensitivities of the structural responses have to be computed. The proposed methodology is illustrated by a realistic numerical example for which the finite element method cannot give correct results.

Admissible fields and error estimation for acoustic FEA with low wavenumbers

Abstract A posteriori error estimation in constitutive law , that has been mainly developed for stress analysis, can be applied to acoustic finite element analyses with low wavenumbers. The mathematical background is developed and the concept of admissible solutions is defined for the acoustic problem considering Dirichlet, Neumann and mixed boundary conditions. Particular attention is devoted to the calculation of the admissible acoustic velocity which must be of order p +1 to satisfy the Helmholtz equation. Numerical analyses with linear triangles show very encouraging results with low wavenumbers, namely the estimated error converges in O ( h p ), i.e. the same order as the exact error, and the distribution of the exact error is particularly well estimated, i.e. the areas containing concentrations of error are correctly identified but locally overestimated. Moreover, the estimated global absolute error always overestimates the exact error.

PAMUC II for multicriteria optimization of mechanical designs with expert rules

This study addresses the problem of optimizing mechanical components during the first stage of the design process. While a previous work focused on parametrized designs with fixed configurations - which led to the development of the PAMUC (Preferences Applied to Multiobjectivity and Constraints) method, for solving multicriteria constrained problems within evolutionary algorithms (EAs), the models analyzed in this work are enriched by the presence of topological variables enabling to consider simultaneously different configurations. Therefore, in order to create optimal but also realistic designs, i.e. fulfilling not only technical requirements but also technological constraints (e.g. related to the machining or the assembly), which are more naturally expressed in terms of rules, an original approach is proposed, named PAMUC II. It consists in integrating an inference engine within the EA, and repairing the individuals (with a given probability of replacement) violating the technological constraints (written as 0-order logical rules). PAMUC II is illustrated on a mechanical benchmark and an industrial case: the multicriteria optimization of a poppet valve design from the VINCI engine of launcher Ariane 5. Results show the efficiency of the proposed method to provide at once optimal and feasible designs.

A Meshless Approach for 2D Vibro-Acoustic Problems

The Element-Free Galerkin method seems to be suitable to obtain very accurate results for acoustic uncoupled problem. For vibro-acoustic problems using a coupled finite element - element-free Galerkin formulation, it has already been shown that the major part of the error on the discretisation is due to the finite element discretisation of the structure. Thus, in this paper, we propose to improve the vibroacoustic simulation by coupling an EFG method in the fluid to a partition of unity method in the solid. The paper shows that, for this latter, major difficulties have to be solved: the numerical quadrature and the continuity of the displacements for non planar shells.

Design and Implementation of a Flexible Guiding System in Translation

In the field of micro-mechanics, it is often advisable to replace conventional joints by flexible ones. The working principle of such joints is based on their elastic behaviour; they have multiple advantages like no backlash, no friction, and they can be miniaturized at low cost [1]

Behaviour of Flexure Hinges for Use as Articulations in High Precision Mechanisms

For micromechanical engineering purpose, the classical articulations have reached their limits in terms of precision: their movement is corrupted by backlash and imprecisions in assembly. It also becomes increasingly difficult to realise the assembly within a reasonable cost.