Alain Rassineux

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

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

Surface remeshing by local hermite diffuse interpolation

We propose a method to build a three-dimensional adapted surface mesh with respect to a mesh size map driven by surface curvature. The data needed to optimize the mesh have been reduced to an initial mesh. The building of a local geometrical model but continuous over the whole domain is based on a local Hermite diffuse interpolation calculated from the nodes of the initial mesh and from the normal vectors to the surface. The optimization procedures involve extracting from the surface mesh sets of triangles sharing the same node or the same edge and then remeshing the outer contour to a higher criterion (size or shape). These procedures may be used in order to refine or coarsen the mesh but also in a final step to enhance the shape quality of the elements. Examples demonstrate the ability of the method to create adapted meshes of complex surfaces while meeting high-quality standards and a good respect of the geometrical surface.

An Introduction to Moving Least Squares Meshfree Methods

We deal here with some fundamental aspects of a category of meshfree methods based on Moving Least Squares (MLS) approximation and interpolation. These include EFG, RKPM and Diffuse Elements. In this introductory text, we discuss different formulations of the MLS from the point of view of numerical precision and stability. We talk about the issues of both “diffuse” and “full” derivation and we give proof of convergence of both approaches. We propose different algorithms for the computation of MLS based shape functions and we give their explicit forms in 1D, 2D and 3D. The topics of weight functions, the interpolation property with or without singular weights, the domain decomposition and the numerical integration are also discussed. We formulate the integration constraint, necessary for a method to satisfy the linear patch test. Finally, we develop a custom integration scheme, which satisfies this integration constraint.

3D mesh adaptation. Optimization of tetrahedral meshes by advancing front technique

The objective of the paper is to present an automatic mesh generator which builds a mesh which respects the element densities prescribed as a result of an error sensitivity analysis. The desired new mesh is fully characterized by the distribution of the refinement levels on the initial mesh. Therefore, the most important criterion in the choice of a mesh generator comes from the ability to produce a mesh with imposed density while meeting ‘acceptable’ shape quality requirements. In this spirit, a new method for tetrahedron mesh generation and optimization with respect to both a shape and a size criterion is presented. The advancing front technique is used to mesh the whole volume. Then, optimization procedures make astute local use of the algorithm used to mesh the complete model. The process is iterative. The corresponding method involves extracting sub-volumes or shells from the tetrahedron mesh volume, which are then remeshed by an advancing front technique to improve their quality. The sub-volumes are constructed by determining the set of tetrahedra which touch the same node, edge or face. Local transformations are extended to non-convex sub-regions which gives a higher quality mesh. Industrial examples of relatively complex volumes are given, demonstrating that a high quality and optimized mesh can be obtained by the proposed method.

Improving surface meshing from discrete data by feature recognition

In this paper, we propose a method to identify, on a mesh, geometric primitives commonly used in mechanical parts (plane, sphere, cylinder, torus, cone) in order to improve the quality of the surface remeshing. We have already presented techniques to adapt an existing surface mesh based on a mesh-free technique denoted as diffuse interpolation. In this approach, a secondary local geometrical model is built from the mesh. From this model, principal curvatures are calculated and the type of surface can be determined from the computation of the curvatures. Some of the concepts presented here are original while others have been adapted from techniques used in reverse engineering. Our approach is not limited to feature recognition on meshes but has been extended to a set of points.

Prediction of serrated chip formation in orthogonal metal cutting by advanced adaptive 2D numerical methodology

In this work a complete numerical methodology combining ‘advanced’ thermo-elasto-viscoplastic constitutive equations accounting for mixed (isotropic and kinematic) non-linear hardening, thermal effects, isotropic ductile damage and contact with friction is proposed to simulate the 2D orthogonal cutting process of AISI4340 steel. First, the fully coupled constitutive equations are presented and their specificities highlighted. The relevant numerical aspects concerning both the local integration scheme as well as the global resolution strategy together with the 2D adaptive remeshing facility are discussed. This model is implemented into ABAQUS/EXPLICIT using the Vumat user subroutine and connected with an adaptive 2D meshing program. Application is made to the 2D orthogonal metal cutting by chip formation. A special care is put in the prediction of the primary shear band where the temperature, the strain and the damage are highly localised giving the serrated shape and possible segmentation of the ship.

A robust protocol for the creation of patient-specific finite element models of the musculoskeletal system from medical imaging data

A robust, accurate and flexible computational protocol to create personalised finite element models of musculoskeletal systems from medical images is presented. A contour-based geometrical reconstruction well suited to the complex and intrinsic character of living biological tissues and structures is used. The 3D surface model is decomposed into a number of stripes considered as quasi-developable domains which can be, therefore, meshed in a 2D parametric space. Once all boundary domains have been meshed, nodes are projected on a Hermite bicubic patch model derived from the segmentation points. A specific procedure to handle branching structures is proposed. In a final step, a tetrahedron mesh is generated and the mechanical properties could be assigned. Case studies using magnetic resonance imaging and computed tomography data are carried out and demonstrated. Results of the protocol are presented and discussed.

Springback assessment based on level set interpolation and shape manifolds in deep drawing

In this paper, we introduce an original shape representation approach for automatic springback characterization. It is based on the generation of parameterized Level Set functions. The central idea is the concept of the shape manifold representing the design domain in the reduced-order shape-space. Performing Proper Orthogonal Decomposition on the shapes followed by using the Diffuse Approximation allows us to efficiently reduce the problem dimensionality and to interpolate uniquely between admissible input shapes, while also determining the smallest number of parameters needed to characterize the final formed shape. We apply this methodology to the problem of springback assessment for the deep drawing operation of metal sheets.

A Modular Design for a Parallel Multifrontal Mesh Generator

The proposed approach consists in extending an existing sequential mesh generator in order to design a parallel mesh generator. A sequential three-dimensional mesh generator builds the internal mesh by applying advancing front techniques from a surface mesh defining the volume of the initial domain. We geometrically decompose the domain by recursively splitting it into subdomains using cutting planes. We run a sequential mesh generator code on different processors, each of them working on a single subdomain. The interface mesh compatibility on the two sides of two subdomains makes it possible to merge the results. We present in this paper our modular approach, an example of a interface mesh generation, and we specify the cutting plane decomposition method.

Feature Recognition and Remeshing by Local Hermite Diffuse Interpolation

We propose a method to adapt a three-dimensional surface mesh. The data needed to optimize the mesh have been reduced to an initial mesh. The building of the geometrical model is based on a local Hermite interpolation calculated from the nodes of the initial mesh and from the normal vectors to the surface calculated from the mesh. The determination of an interpolating local surface equation, however continuous over the domain, enables us to locate nodes on the surface with respect to the curvature during a refinement process. It also allows us to control the coarsening of the mesh. The local surface equation is used to compute the principal curvatures on the surface for two main purposes, error estimation and feature recognition..The remeshing procedures are used to refine or coarsen the mesh but also in a final step to enhance the shape quality of the elements.