Piotr Breitkopf

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

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.

Multidisciplinary Design Optimization in Computational Mechanics

This book provides a comprehensive introduction to the mathematical and algorithmic methods for the Multidisciplinary Design Optimization (MDO) of complex mechanical systems such as aircraft or car engines. We have focused on the presentation of strategies efficiently and economically managing the different levels of complexity in coupled disciplines (e.g. structure, fluid, thermal, acoustics, etc.), ranging from Reduced Order Models (ROM) to full-scale Finite Element (FE) or Finite Volume (FV) simulations. Particular focus is given to the uncertainty quantification and its impact on the robustness of the optimal designs. A large collection of examples from academia, software editing and industry should also help the reader to develop a practical insight on MDO methods.

Numerical material representation using proper orthogonal decomposition and diffuse approximation

From numerical point of view, analysis and optimization in computational material engineering require efficient approaches for microstructure representation. This paper develops an approach to establish an image-based interpolation model in order to efficiently parameterize microstructures of a representative volume element (RVE), based on proper orthogonal decomposition (POD) reduction of density maps (snapshots). When the parameters of the RVE snapshot are known a priori, the geometry and topology of individual phases of a parameterized snapshot is given by a series of response surfaces of the projection coefficients in terms of these parameters. Otherwise, a set of pseudo parameters corresponding to the detected dimensionality of the data set are taken from learning the manifolds of the projection coefficients. We showcase the approach and its potential applications by considering a set of two-phase composite snapshots. The choice of the number of retained modes is made after considering both the image reconstruction errors as well as the convergence of the effective material constitutive behavior obtained by numerical homogenization.

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.

Architecture des logiciels et langages de modélisation

ABSTRACT The article points out major difficulties emerging in the software developement for computational engineering purposes. Solutions are suggesteed, some of them are not definitive. The common characteritics of existing programs are described; the problems encountered are detailed. Finally the solutions adopted for the SIC program general architecture are presented. SIC is developed at UTC, since 1985 in cooperation with industry partners and scientific laboratories from Marseille, Grenoble, Montpellier, Poitiers.

Asynchronous evolutionary shape optimization based on high-quality surrogates: application to an air-conditioning duct

Multi-processor HPC tools have become commonplace in industry and research today. Evolutionary algorithms may be elegantly parallelized by broadcasting a whole population of designs to an array of processors in a computing cluster or grid. However, issues arise due to synchronization barriers: subsequent iterations have to wait for the successful execution of all jobs of the previous generation. When other users load a cluster or a grid, individual tasks may be delayed and some of them may never complete, slowing down and eventually blocking the optimization process. In this paper, we extend the recent “Futures” concept permitting the algorithm to circumvent such situations. The idea is to set the default values to the cost function values calculated using a high-quality surrogate model, progressively improving when “exact” numerical results are received. While waiting for the exact result, the algorithm continues using the approximation and when the data finally arrives, the surrogate model is updated. At convergence, the final result is not only an optimized set of designs, but also a surrogate model that is precise within the neighborhood of the optimal solution. We illustrate this approach with the cluster optimization of an A/C duct of a passenger car, using a refined CFD legacy software model along with an adaptive meta-model based on Proper Orthogonal Decomposition (POD) and diffuse approximation.

Object-Oriented Approach and Distributed Finite Element Simulations

ABSTRACT We present an application of object-oriented approach in the context of distributed computing in the field of structural engineering problems. In this work, conducted within the framework of a general purpose finite element code, we consider two types of distributed algorithms: the cooperation of heterogeneous computing systems and an algorithm for distributing the resolution of the finite element problem. In the first case, the major issue is the transparent distribution of the data base involving data structures and algorithms. In the first part of the present work, we present DDSM (Distributed Data Structures Manager) dealing with this first issue. The second case addressed is that of solution of linear systems by a domain decomposition direct method. Performance results given are those of a Cray T3D system. Communications and process control are implemented using the PVM library.

An algorithm for construction of iso-valued surfaces for finite elements

Finite element analysis of 3D phenomena gives as a result a set of function values specified on the nodes of the mesh. Various modes of graphical representation of such results are possible, including contour plots on cross-sections and surfaces of constant field values. In this paper, we propose an algorithm for the generation of such models. The continuous surfaces representing constant field values are obtained element-by-element by linear interpolation of nodal values. The normalized gradient of computed values is used for surface smoothing and shading. The method uses shaded polygon and shaded triangle strip primitives to meet with industry standards for graphical equipment.

Proper orthogonal decomposition with high number of linear constraints for aerodynamical shape optimization

Shape optimization involving finite element analysis in engineering design is frequently hindered by the prohibitive cost of function evaluations. Reduced-order models based on proper orthogonal decomposition (POD) constitute an economical alternative. However the truncation of the POD basis implies an error in the calculation of the global values used as objectives and constraints which in turn affects the optimization results. In our former contribution (Xiao and Breitkopf, 2013), we have introduced a constrained POD projector allowing for exact linear constraint verification for a reduced order model. Nevertheless, this approach was limited a to relatively low numbers constraints. Therefore, in the present paper, we propose an approach for a high number of constraints. The main idea is to extend the snapshot POD by introducing a new constrained projector in order to reduce both the physical field and the constraint space. This allows us to search for the Pareto set of best compromises between the projection and the constraint verification errors thereby enabling fine-tuning of the reduced model for a particular purpose. We illustrate the proposed approach with the reduced order model of the flow around an airfoil parameterized with shape variables.