timisation
Certains outils d’optimisation ont ´et´e discut´es. Le domaine de l’optimisation ´etant in- croyablement vaste, l’objectif de ce chapitre est simplement de pr´esenter les principales caract´eristiques de certains algorithmes d’optimisation et d’expliquer pourquoi elles sont importantes. Des crit`eres de classification pour les probl`emes d’optimisation ont ´egalement ´
et´e fournis et l’attention s’est concentr´ee sur les algorithmes d´eterministes et m´eta-heuristiques pour la solution d’un CNLPP. L’aspect le plus remarquable de cette discussion est que le ”meilleur algorithme” n’existe pas et que le choix de l’algorithme d’optimisation doit ˆ
etre soigneusement pris en compte en fonction du probl`eme rencontr´e. En particulier, il sera montr´e dans les chapitres suivants que de nombreux probl`emes d’OT sont encore r´esolus par des algorithmes d´eterministes, bien qu’ils soient non convexes. `A la lumi`ere des sujets li´es aux m´eta-heuristiques, on peut s’interroger sur la pertinence de ce choix : un algorithme m´eta-heuristique pourrait mieux explorer le domaine de la conception et r´eduire la possibilit´e de tomber sur une solution pseudo-optimale. L’objection ´evidente est que les probl`emes d’OT sont caract´eris´es par une quantit´e ´enorme de variables et qu’une solution compl`ete au moyen d’une approche purement m´eta-heuristique pourrait ˆ
etre impossible, mˆeme pour un probl`eme 2D simple. Ainsi, lorsque le probl`eme d’opti- misation est non convexe, l’utilisation de m´eta-heuristiques n’est pas syst´ematiquement la bonne r´eponse pour le r´esoudre. Par exemple, fournir un point de d´epart appropri´e et utiliser une m´ethode d´eterministe pourraient ˆetre des bon choix : dans ce cas, le crit`ere d´eterminant pour s´electionner une estimation initiale significative des variables de concep- tion est fourni par la connaissance des ph´enom`enes li´es au probl`eme lui-mˆeme. Un autre choix est l’utilisation d’outils m´eta-heuristiques - d´eterministes hybrides : dans un pre- mier temps, le domaine est ´etudi´e par une m´ethode m´eta-heuristique et, lorsqu’une zone int´eressante est identifi´ee, la solution optimale de l’algorithme m´eta-heuristique est uti-
A NURBS-based Topology
Optimisation Algorithm
4.1
Introduction
Non-Uniform Rational Basis Splines (NURBS) are extremely versatile mathematical en- tities, which are used to solve several engineering problems. As stated in Chapter 2, they constitute the foundations of Computer Aided Design (CAD) for surfaces and curves modelling [66–68]. The utilisation of NURBS has shown a considerably growth over the years: nowadays, they are employed in problems of different nature, such as optimisa- tion strategies for curve/surface fitting [95–97] or meta-modelling [98, 99]. Moreover, an increasing number of work is carried out, wherein NURBS entities are associated to the structural analysis of mechanical components. A smart application of NURBS surfaces has been shown in [93, 94, 100] for optimising mechanical properties of variable angle tow (VAT) composites. NURBS also constitute the basis of the relatively new concept of iso- geometric analysis [101], which represents a challenging integration of CAD entities into the Finite Element (FE) method. When looking at the consistent amount of contributions dealing with NURBS curves and surfaces (of which the aforementioned works constitute just an example), one can imagine to exploit their potential as well as their interesting properties (refer to Chapter 2) in the domain of TO.
Recent research efforts, finalised to fill in some gaps of current TO methods, have direc- ted their attention towards NURBS entities. As known, the main source of the drawbacks related to density-based TO methods is the lack of a geometric entity dedicated to the description of the topology: the mesh of the part to be optimised and the pseudo-density function provide information about both the performance of the structure (according to the problem at hand) and the topology (refer to Chapter 1). The LSM represents a first attempt to overcome this difficulty by introducing geometric entities to properly describe the topology. In this context, some authors tried to integrate the advantages of NURBS.
B-Spline curves constitute the shape function of the special class of FE employed in [102]. In the procedure discussed in [103], the LSM is coupled to the isogeometric analysis by means of the NURBS formalism for both LSF parametrisation and objective function computation. However, the intrinsic difficulties related to the LSM and the sensitivity of the solution to the starting guess of the LSF (see Chapter 1) still constitute a major issue that circumvents the wide spread of LSM-based TO strategies. Therefore, it is not surprising that pioneering TO density-based strategies [5, 34, 41] are still widely studied research topics. An interesting and promising enhancement of the SIMP method, based on the application of B-Spline entities, has been provided in [104] and [105] for 2D and 3D applications, respectively. In these studies, the fictitious density field has been related to a B-Spline surface/hyper-surface. The B-Spline formalism permits to take advantage of an implicitly defined filter zone, whose size depends on the B-Spline parameters. Therefore, the well-known checker-board and mesh dependence effects can be overcome without any dedicated strategy, e.g. distance based filters [5] or projection methods [10], [42].
This Chapter introduces an innovative TO method and the related algorithm, based on a further development and a generalisation of the method proposed in [104, 105]. Here, the pseudo-density field characterising the SIMP method is related to a NURBS surface/hyper-surface for 2D and 3D applications, respectively. For 2D problems, each point of the NURBS control net is then characterised by three coordinates of which two are Cartesian coordinates and the third one is the pseudo-density. For 3D problems, a 4D hyper-surface is used to represent the fictitious density field: each point constituting the control hyper-net has three Cartesian coordinates and the fourth coordinate is the pseudo-density. The proposed approach is called NURBS-based SIMP method.
The impact of other parameters, such as the NURBS weights, is investigated and suitable comparisons are carried out between solutions of TO problems obtained through B-Spline and NURBS.
Furthermore, the discussion is not restrained only to the beneficial implicit filter zone provided by the NURBS formalism. It is well-known that one of the main shortcom- ing of density-based methods is the time consuming postprocessing phase, necessary to rebuild the boundary of the optimum topology of the structure starting from a FE “pixel- ised”/“voxelised” domain (providing the required smoothness). A careful description of the geometry is crucial in TO not only to save time in postprocessing but, mostly, to en- sure that the optimum shape of the component (rebuilt at the end of TO) could meet the design constraints. It will be shown that the NURBS-based approach can easily provide fully CAD-compatible optimised geometries and that optimisation constraints are met on the actual reassembled geometry in 2D.
As far as 3D applications are concerned, although the NURBS hyper-surfaces are geometric, potentially CAD-compatible entities, the geometry reconstruction/assembly phase of the structure boundary after optimisation in terms of CAD surfaces still remains a challenge for 3D problems. Currently, the 3D topology is obtained after an intersection
operation between the 4D NURBS hyper-surface and a suitable hyper-plane. Hence, the resulting geometry is described by a “well-defined” Standard Tessellation Language (STL) CAD native format (i.e. an STL file without missing or degenerated triangles) representing the boundary of the optimum topology. The difficulties related to the full CAD-compatibility of 3D structures will be discussed as well.
The Chapter follows this outline: firstly, the mathematical statement of the NURBS- based TO method is provided for both 2D and 3D applications. The classic problem of compliance minimisation with an equality constraint on the volume is considered in this discussion. Secondly, the algorithm related to the NURBS-based approach is presented and its capabilities are discussed, together with its limitations. Then, meaningful results are shown on standard benchmarks. The contribution of the weights appearing in the NURBS formulation is investigated and results are compared to those obtained using B- Spline entities. Moreover, a sensitivity analysis to the NURBS discrete parameters (i.e. the degrees and the number of control points) is carried out. Results provided by the proposed algorithm are validated by means of the well-established TO software Altair OptiStruct R. Finally, conclusions and perspectives on the contributions of the NURBS-
based SIMP method are given.
The contents of this Chapter refer to the articles [106–109].