An element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems. In this method, moving least-squares interpolants are used to construct the trial and test functions for the variational principle (weak form); the dependent variable and its gradient are continuous in the entire domain. In contrast to an earlier formulation by Nayroles and coworkers, certain key differences are introduced in the implementation to increase its accuracy. The numerical examples in this paper show that with these modifications, the method does not exhibit any volumetric locking, the rate of convergence can exceed that of finite elements significantly and a high resolution of localized steep gradients can be achieved. The moving least-squares interpolants and the choices of the weight function are also discussed in this paper.

Element‐free Galerkin methods

https://hal.archives-ouvertes.fr/hal-01513346/document

Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement

Meshless methods: An overview and recent developments

A new continuous reproducing kernel interpolation function which explores the attractive features of the flexible time-frequency and space-wave number localization of a window function is developed. This method is motivated by the theory of wavelets and also has the desirable attributes of the recently proposed smooth particle hydrodynamics (SPH) methods, moving least squares methods (MLSM), diffuse element methods (DEM) and element-free Galerkin methods (EFGM). The proposed method maintains the advantages of the free Lagrange or SPH methods; however, because of the addition of a correction function, it gives much more accurate results. Therefore it is called the reproducing kernel particle method (RKPM). In computer implementation RKPM is shown to be more efficient than DEM and EFGM. Moreover, if the window function is C∞, the solution and its derivatives are also C∞ in the entire domain. Theoretical analysis and numerical experiments on the 1D diffusion equation reveal the stability conditions and the effect of the dilation parameter on the unusually high convergence rates of the proposed method. Two-dimensional examples of advection-diffusion equations and compressible Euler equations are also presented together with 2D multiple-scale decompositions.

Reproducing kernel particle methods

Built upon the two original books by Mike Crisfield and their own lecture notes, renowned scientist Rene de Borst and his team offer a thoroughly updated yet condensed edition that retains and builds upon the excellent reputation and appeal amongst students and engineers alike for which Crisfield's first edition is acclaimed. Together with numerous additions and updates, the new authors have retained the core content of the original publication, while bringing an improved focus on new developments and ideas. This edition offers the latest insights in non-linear finite element technology, including non-linear solution strategies, computational plasticity, damage mechanics, time-dependent effects, hyperelasticity and large-strain elasto-plasticity. The authors' integrated and consistent style and unrivalled engineering approach assures this book's unique position within the computational mechanics literature.

Non-Linear Finite Element Analysis of Solids and Structures: de Borst/Non-Linear Finite Element Analysis of Solids and Structures

This paper describes the new “diffuse approximation” method, which may be presented as a generalization of the widely used “finite element approximation” method. It removes some of the limitations of the finite element approximation related to the regularity of approximated functions, and to mesh generation requirements. The diffuse approximation method may be used for generating smooth approximations of functions known at given sets of points and for accurately estimating their derivatives. It is useful as well for solving partial differential equations, leading to the so called “diffuse element method” (DEM), which presents several advantages compared to the “finite element method” (FEM), specially for evaluating the derivatives of the unknown functions.

Generalizing the finite element method: Diffuse approximation and diffuse elements

Moving least squares response surface approximation: Formulation and metal forming applications

This work presents a general and efficient way of computing both diffuse and full derivatives of shape functions for meshless methods based on moving least-squares approximation (MLS) and interpolation. It is an extension of the recently introduced consistency approach based on Lagrange multipliers which provides a general framework for constrained MLS along with robust algorithms for the computation of shape functions and their diffuse derivatives. The particularity of the proposed algorithms is that they do not involve matrix inversion or linear system solving. The previous approach is limited to diffuse derivatives of the shape functions and not their full derivatives which are usually much more expensive to obtain. In the present paper we propose to efficiently compute the full derivatives by a new algorithm based on the formal differentiation of the previous one. In this way, we obtain a unified low-cost consistent methodology for evaluating the shape functions and both their diffuse and full derivatives. In the second part of the paper we introduce explicit forms of MLS shape functions in 1D, 2D and 3D for an arbitrary number of nodes. These forms are especially useful for comparing finite element and MLS approximations. Finally we present a general architecture of an MLS program. Copyright © 2000 John Wiley & Sons, Ltd.

Explicit form and efficient computation of MLS shape functions and their derivatives

This paper proposes a novel approach for multi-objective optimization when the criteria of interest rely on a functional output from an expensive-to-evaluate numerical simulator. More specifically, the proposed method is developed in the frame of an automotive application. The aim of this application is to design the shape of an intake port in order to maximize the mass flow (denoted by Q) and the tumble (denoted by T), which both depend on a 3D velocity field obtained by numerical flow simulation. Since the considered flow simulator is time-consuming, using regular multi-objective genetic algorithms (MOGA) directly on integral quantities depending on the simulator output is prohibitive. Three different Reduced Order Models (ROMs) are presented. The first one consists in directly Kriging the integral quantities Q and T on the basis of the outputs computed at an initial design of experiments, and basing the optimization search on the sequentially obtained couples of response surfaces. The other methods explored in the present work consist in building a parametrized representation of the whole velocity field by different variants of the Proper Orthogonal Decomposition (POD). Instead of directly Kriging Q and T at un-sampled locations, the proposed technique is hence to proceed in two steps: first approximate the functional outcome by Kriging the POD coefficients, and then compute the integral quantities Q and T associated with the approximate 3D field. However, such an approach induces new difficulties since the truncated POD does not preserve the global (integrated) quantities, and that surrogate-based MOGA with this kind of POD are therefore likely to fail locating the (Q, T)-Pareto front accurately. This is what motivates to propose an original constrained POD method (called CPOD) meant to overcome the bias created by the truncation made in regular POD. More precisely, this means modifying the way of calculating the POD coefficients by imposing the integral quantities Q and T based on the truncated POD to match with the actual Q and T values obtained by flow simulation at the design of experiments. A detailed comparison of the Pareto sets obtained from the three ROMs demonstrates the interest of the CPOD approach.

Pierre Villon

Papers

Generalizing the finite element method: Diffuse approximation and diffuse elements

Moving least squares response surface approximation: Formulation and metal forming applications

Explicit form and efficient computation of MLS shape functions and their derivatives

Model reduction by CPOD and Kriging

Generalizing the fem: diffuse approximation and diffuse elements