A NURBS-based Topology Optimisation Algorithm 

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The Additive Layer Manufacturing Technology

ALM has experienced an important spread in the last decades and several technologies have been developed. Each technology is suited for a speci c application and for well-de ned materials. Beyond the peculiar di erences characterising the ALM technologies, the procedure that allows for obtaining a component is composed of several steps, as described here below [14].
1. CAD Model. The CAD model of the component to be produced is considered. It could be the result of an optimisation process or simply the best trade-o choice made by the designer in order to meet the technical speci cations.
2. Conversion to standard le format. The CAD model boundary should be converted into a standard le format containing its discretised form. Generally, an STL le is chosen but other, more complete le formats are available (e.g. the AMF le, which relies on the de nition of curved triangles and it has been conceived for ALM applications).
3. Transfer to ALM Machine and File Manipulation. The standard le is trans-ferred to the ALM machine computer. Here, the le is modi ed according to precise requirements such as orientation, facets \cleaning » (i.e. suppression of degenerated triangles), etc.. The eventual volume of support structures and its topology are established during this phase as well.
4. Machine Set-up. This task consists of assigning all the parameters characterising the ALM technology at hand. Some among them are easy to set but others have contrasting e ects on the nal component, as remarked later on.
5. Build. The actual building phase is an automatic task for the most of ALM ma-chines nowadays.
6. Removal and Post-processing. Once removed from the machine, the piece should be carefully cleaned up (typically, surface treatments are necessary) and, eventually, useless support structures must be removed.
As it can be deduced from the previous list, the production of a component through ALM is not only matter of manufacturing. It requires a deep and skilled knowledge of the machine parameters and of their impact on the physical phenomena involved in the process. Nevertheless, handling CAD les and managing the correct information ow to an STL/AMF le is a fundamental task as well. Moreover, the previous scheme should be tailored for each speci c ALM technology, as it can be easily deduced from the high amount of variants for this manufacturing process, discussed in the following.

A classi cation of ALM processes

In this brief bibliography on ALM technologies, the classi cation discussed in [14] is con-sidered. The adjectives \direct » and \indirect » are used according to the de nition given in [15]. In indirect technologies the bonding function is performed by a binder material, di erent from the bulk material. Conversely, in direct methods, the bulk material is also responsible of the binding function.
Vat photo-polymerization processes exploit the reactivity of photo-polymers to ra-diation in the ultra-violet (UV) range of wavelengths: under these conditions, photo-polymers undergo a complex suite of chemical reactions leading to solidi cation.
Powder bed fusion processes are perhaps the most suited for structural applications. Many processes belong to this category, as Selective Laser Sintering (SLS), Selective Laser Melting (SLM) and Electron Beam Melting (EBM). Regardless to the speci c technology, usually an energy source is used to induce the partial or total melting of powder particles and prescribed mechanisms deal with the uniform spreading of the powder and with the provision of new powder layers (generally a screaper). Various materials can be employed in powder bed systems, such as metals, thermoplastics and ceramics.
Extrusion-based systems are commonly used mainly because of the relatively low cost of the machine and because of the simplicity of the set-up. The principle consists of melting the bulk material in a suitable chamber and pushing it out through a nozzle. The bonding mechanism can be realised by means of the simple solidi cation of the material itself (direct method) or by means of secondary build materials (indirect method). The most common extrusion-based ALM technology is the Fused Deposition Modelling (FDM) to produce ABS (Acrylonitrile Butadiene Styrene) components.
Sheet Lamination processes, as LOM (Laminated Object Manufacturing), involve a layer-by-layer lamination of material sheets and some bonding mechanism is exploited (adhesives, thermal bonding, clamping, ultrasonic welding). Obviously, this method is really sensitive to the variation of strength in the direction normal to the layers.
Promising processes are those belonging to the class of Material Jetting and Binder Jetting. In the former case, the bulk material is directly dispensed from a print-head, whilst in the latter case the bulk material is a powder substrate and a suitable binder is printed on it.
Directed Energy Deposition (DED) machines allow for producing parts by melting materials as they are being deposited: to this aim, the print-head is constituted of a heat source (e.g. a laser beam or an electron beam) and a material powder feeder. This technology is mostly employed for metal parts, but polymers and ceramics can be used as well. When compared to powder bed fusion processes, DED machines can achieve directional solidi ed and single crystal structures, multi-material components and fully dense parts. However, the accuracy is not very high and surface roughness is not satisfying for engineering applications. Moreover, DED process are not able to produce small-scale features.
Direct Write (DW) technologies are well suited for very small scale components. Typ-ical applications of this emerging additive manufacturing process involve the thermal and strain sensors production and antenna fabrication.
In this manuscript, the attention is focused on the metal powder bed technology, since it is the most suited for engineering automotive and aerospace applications. All metal materials that can be welded are prone to ALM [15]. Among the powder bed processes, the most widespread are undoubtedly SLS, SLM and EBM. In SLS, the metal powder is rstly spread in a layer and then selectively scanned by a laser to build each layer. SLS can work either as direct method by partially melting the metal particles or as indirect method by melting a low-melting-point binder (phenolic polymers or low-melting-point metals). Anyway, a post-processing phase concerning the removal of polymer binder, such as thermal sintering, liquid-metal in ltration or hot isostatic pressing (HIP), is necessary. The most important issue related to SLS is the impossibility to reach fully dense parts: consequently, porosity heavily a ects mechanical properties of SLS structures. SLM and EBM can partially overcome this issue because of the improved energy own through the heat source, which is a laser beam for SLM and an electron beam for EBM. In any case, the bonding mechanism is the melting metal itself. Nearly fully dense metal parts exhibiting as good mechanical properties as the bulk material can be produced.

The Selective Laser Melting Technology

Here below, a more detailed description of the SLM technology is provided because it has been chosen as a reference ALM technology in the context of the FUTURPROD project. In the framework of SLM, researchers have investigated the mechanical properties of components obtained by using di erent materials: stainless steels [16{18], several alu-minium alloys (often silicon-based alloys) [19,20], titanium alloys [21], hastelloys [22] and Inconel [23]. It is evident that the physical-chemical phenomena involved in the SLM process have a strong e ect on the properties of the manufactured part. The parameters controlling the process have been identi ed by Spears and Gold [24]: 50 parameters can be identi ed and only 12 can be controlled by the user. The parameters are classi ed as laser and scanning parameters (e.g. the laser power supply, scan speed and path), powder material properties (bulk density, material absorptivity), powder bed properties (layer thickness, powder bed temperature, thermal conductivity) and build environment para-meters (inert atmosphere gasses). Therefore, the produced component is always the result of the interaction of all these parameters. Being impossible to provide a complete picture of the whole set of process parameters and the related in uence on the resulting part, it has been decided to take a pragmatic approach and to summarise the most important defects appearing in SLM structures, with a particular attention on their causes.
Porosities. The presence of porosities cannot be totally eliminated in SLM and they constitute internal defects, prone for crack onset. The porosity is primary a ected by the laser parameters (scanning speed, laser power, lasing strategy) and powder parameters (preheating temperature, the layer thickness). Too low laser energies do not allow for a complete melting of the powder particles, which can be trapped in the surrounding melting pot, whilst a too high laser energy could lead to trap some gas portions, resulting in spherical cavities. The quality of the powder itself is important (impurities should be eliminated, especially for those technology that make use of recycle), together with the absorption rate of the laser energy.
Hot Cracking Defects. This phenomenon is typical of welded aluminium al-loys and it depends on the severe thermal gradient undergone by the heat a ected zone. Limiting the temperature gradient would be useful to solve this problem. Improvements can be achieved thanks to the composition of the alloy at hand: typically, low melting temperature elements lead to a decrease of the hot cracking phenomenon. It should be pointed out that hot cracking is an evident consequence of the thermal history of the component but other possible issues are related to the strong thermal gradient. For instance, the built part can exhibit deformations or residual stresses [25]. Thermal e ects can involve very local phenomena (such as the material delamination when a zone is scanned by the laser) or global phe-nomena: the latter is the case of variation of processing temperature. Generally, cross-sections of the part are scanned with paths parallel to each other. In order to achieve a compromise between material wetting conditions and uniform temper-atures, Kruth et al. [16] have demonstrated that better results can be achieved in terms of distortion by scanning a steel plate dividing it in small rectangular sectors rather than following parallel paths.
Anisotropy. It is not di cult to guess that ALM structures, being produced according to a preferential build direction, exhibit an overall anisotropic behaviour. Indeed, this fact is not crucial in the case of SLM structure, whilst it is much more critical and evident in extrusion-based technologies. In any case, reducing thermal gradients and suitably orienting the component in the machine chamber are good practices allowing for an almost isotropic behaviour.
Surface Roughness. The surface quality is far to be appropriate for a struc-ture produced by SLM and supposed to satisfy durability requirements. Generally speaking, nishing and polishing post-processing phases are necessary. The porosity is one of the main causes of the poor quality of surfaces. Poor surface quality is also due to laser parameters, as the power and the scanning rate. Moreover, the orientation of the piece in the machine has a strong impact on the surface quality as well: the scale e ect occurs for all those surfaces which are not perpendicular or parallel to the basement plane [26,27]. Finally, surface roughness is also a ected by the balling phenomenon, occurring when the molten material does not wet the un-derlying substrate because of the surface tension, which tends to deform the liquid into a sphere. This results in a rough and bead-shaped surface, hindering a smooth layer deposition and decreasing the density of the produced part.
Beyond the parameters directly related to the machine, there are other parameters that depend on the designer’s sensibility and experience, such as the support material creation and the piece orientation with respect to both the scraper and the machine basement. The e ect of the orientation of the component in the machine chamber has already been mentioned for the surface roughness and for the occurrence of anisotropy. However, the orientation has a strong impact also on other features characterising the process and the quality of the nal component. If the part has a too broad surface perpendicular to the scraper sweep direction, this one could hit the component during production and could make it move (shock), being the powder bed absolutely not sti enough to prevent this movement. The orientation of the part is still basic to properly cover it with powder. Its compactness is about 60%, so a material withdrawal occurs during melting. Then, an amount of excess powder is required but, if the highest dimension is disposed along the scraper sweep direction, the powder exhaustion will be really probable. Accordingly, a good practice would be tilting the mechanical component with respect to the sweeping direction of the scraper. Manufacturing time is strongly a ected by the part positioning in the machine as well: it depends on the laser power and on the number of layers to be deposed. It is worthy to dispose the part in such a way that its highest dimension lies on the basement plane.
Further technological requirements to be investigated are those related to the pres-ence of support structures. When one zone of the mechanical part overcomes a critical angle of inclination (refer to Fig. 1.1), a support structure is necessary in order to avoid material breakdown or instability. Cloots et al. [28] distinguish supported overhangs and unsupported overhangs.
Figure 1.1 { Critical Angle for supported overhangs.
Besides the e ective support role, these structures are really useful in order to improve the quality of the nal part:
supports can be used as interfaces between the base-plate and the part; in this way, e orts are better distributed inside the structure, the part can be removed by the machine without being marked and, nally, it can be lifted up on the basement (so the irregular rst layers are not included in the piece manufacturing);
the centre of gravity of the mechanical part changes during ALM production, so a support structure could be basic to stabilise the component;
concerning residual stresses induced by the gradient of temperature, a suitable pla-cing of supports during manufacturing can sti some critical regions in order to meet geometrical constraints after the cooling phase; therefore, a thermal treatment for stress relaxation is applied, holding the support in the initial position and, un-der these conditions, the support structure can be removed since no high residual stresses are in the mechanical part.
Often a lattice structure is chosen to create supports [29]: selective connection points permit the e orts transmission, heat dissipation and an easier separation in the post-processing phase. Moreover, the grid, no-walled, structure assures a suitable sti ness without wasting powder which is not embedded. Because of necessary post-processing, functional surfaces have not to be put in contact with the support but they must be perpendicular or parallel to the scraper sweeping direction. However the optimisation of the support structure (in terms of minimum support volume, best heat conductivity) is an active research topic [28,30,31] and it should be included in the design ow of an ALM structure.

Topology Optimisation Methods

In the last three decades, Topology Optimisation (TO) has gained an increasing degree of interest in both academic and industrial elds. The aim of TO for structural applic-ations is to distribute one or more material phases in a prescribed domain in order to satisfy the requirements for the problem at hand. Usually, the design problem is formu-lated as a Constrained Non-Linear Programming Problem (CNLPP), wherein a given cost (or objective) function must be minimised and, meanwhile, the full set of optimisation constraints has to be met.
In this literature review, only density-based methods and the Level Set Method (LSM) are discussed. Other TO methodologies, such as the Evolutionary Structural Optimisation (ESO) [32] and the Phase-Field method [33] are not considered here.

Density-Based Methods

Classically, rst TO methods were based on a Finite Elements (FE) description of the design domain [34]. The basic idea consists of de ning a continuous ctitious density function (or pseudo-density function) varying between zero and one on the computation domain. The pseudo-density function is evaluated at the centroid of each element of a prede ned mesh and provides information about the topology: \void » and \solid » phases are associated to the lower and upper bounds of the density function, i.e. zero and one, respectively. Meaningless \gray » elements (related to intermediate values of the density function) are allowed but penalised during optimisation in order to achieve a \clear » solid-void nal design. Thus, mechanical properties of each element are computed (and penalised) according to the local density value. Several penalisation schemes have been developed for evaluating mechanical properties, e.g. Solid Isotropic Material with Penalisation (SIMP) or Rational Approximation of Material Properties (RAMP) [5].
The mathematical statement of the classic SIMP method is brie y described in the following. Without loss of generality, the discussion focuses on 3D TO problems. More precisely, the mathematical formulation is here limited, for the sake of clarity, to the problem of minimising the compliance of a structure, subject to an equality constraint on the volume.
Let D R3 be a compact subset in the 3D Euclidean space, in which a Cartesian orthogonal frame O(x1; x2; x3) is de ned:
D = f(x1; x2; x3) 2 R3jx1 2 [0; a1]; x2 2 [0; a2]; x3 2 [0; a3]g; (1.1)
where a1, a2 and a3 are three reference lengths of the domain (related to the problem at hand), de ned along x1, x2 and x3 axes, respectively. The distribution of a given isotropic \heterogeneous material » (i.e. the de nition of void and material zones) in the design domain D is sought in order to minimise the virtual work of external loads applied to the structure and, meanwhile, to meet a suitable volume equality constraint. Let D be the material domain. In the SIMP approach, is determined by means of a ctitious density function (x1; x2; x3) 2 [0; 1] de ned over the whole design domain D. Such a density eld is related to the material distribution: (x1; x2; x3) = 0 means absence of material, whilst (x1; x2; x3) = 1 implies completely dense bulk material (refer to Fig. 1.2).
The density eld a ects the sti ness tensor Eijkl(x1; x2; x3), which is variable over the domain D, according to the following formula: Eijkl( (x1; x2; x3)) = (x1; x2; x3) Eijkl0; i; j; k; l = 1; 2; 3; (1.2) where Eijkl0 is the sti ness tensor of the bulk isotropic material and > 1 is a suitable parameter that aims at penalising all the meaningless densities between 0 and 1. The power law of Eq. (1.2) is the most widely used in the SIMP framework when the problem of minimum compliance with an equality volume constraint is faced. The choice of other penalisation schemes should be carefully assessed according to the particular problem at hand [5].
Considering the FE formulation of the equilibrium problem for a linear elastic static analysis in the global reference frame, let fdg be the vector of the overall displacements and rotations (referred to as degrees of freedom – DOFs) and ffg the vector of applied generalised nodal forces. The relationship between fdg and ffg is [K] fdg = ffg ; (1.3) where [K] is the global sti ness matrix of the structure. Accordingly, the compliance of the structure is computed as c = fdgT [K] fdg : (1.4)
Taking into account Eq. (1.2), [K] can be expressed as [K] = e [Ke0]; (1.5) =1 where e is the ctitious density computed at the centroid of the generic mesh element e, Ne the total number of elements, whilst [K0e] is the non-penalised element sti ness matrix expanded over the full set of DOFs of the structure.
The problem of minimising the compliance of a 3D structure subject to a constraint on the overall volume can be stated as follows:
min c( e);
subject to:
8 V ( e) e=1 eVe
> [K]fdg = ffgN;e (1.6)
Vref = Vref = ;
In Eq. (1.6), Vref is a reference volume, V ( e) is the volume of the material domain , while is the xed volume fraction; Ve is the volume of element e and min represents the lower bound, imposed to the density eld in order to prevent any singularity for the solution of the equilibrium problem. The design variables of the TO problem in the classic SIMP framework are the ctitious densities de ned at the centroid of each element: therefore the overall number of design variables is equal to Ne.
It is well-established that the classic SIMP problem (1.6) is ill-posed [5]. As matter of fact, the topologies proposed by the SIMP method change when a di erent mesh size is used; this is due to the fact that the greater the number of holes in the structure (by keeping constant its volume) the better the structure performance is. The limit of this process is a structural variation at the microscopic scale, that cannot be caught by an isotropic material description. It results in a numerical instability where a higher number of holes appears if a ner mesh is used. Several techniques can be adopted to overcome this problem [5]. The rst one is trivially a perimeter control: a constraint on the maximum value of the perimeter results, de facto, in a limitation in the number of holes. Alternatively, a constraint on the spatial gradient of the density function plays a similar role. However, the most popular choice made by TO algorithms developers is to eliminate the mesh dependency by means of a ltering operation. Indeed, when a density value is associated to a mesh element, a priori there is no inter-dependence among contiguous elements. This fact leads to the well-known checker-board layout of material distribution, which is an arti cial high sti ness solution (but totally meaningless from a physical viewpoint) [5]. Therefore, lters can be used in order to establish a ctitious dependence among adjacent elements. For instance, distance-based lters are very often employed. The use of lters in TO allows for overcoming both the mesh dependence issue and the occurrence of checker-board patterns. It can be shown that all the previous techniques result, at the end, in establishing a minimum length scale in the design, as discussed in [35] and in Chapter 5 of this manuscript.
It should be highlighted that TO problems are non-convex in general. Therefore, if a gradient-based algorithm is chosen in order to update the design variables at each iteration (i.e. to provide a new density distribution), probably the retrieved result will be a local optimum and not a global optimum. Nevertheless, global strategies allowing for a better exploration of the design domain would fail and they are strongly not advisable because of the high number of design variables, usually characterising TO problems [36]. A trade-o solution to this problem could be employing so-called continuation methods, wherein an arti cial convex or quasi-convex form of the problem is solved at the beginning and it is progressively changed into the original non-convex problem. In this case, gradient-based algorithms are well suited to perform the solution search.
The advantage of using mathematical programming in TO instead of zero-order al-gorithms (e.g. meta-heuristics) is the possibility to exploit the information provided by the derivatives of objective/constraints functions with respect to the whole set of design variables for the solution search. In the case of problem (1.6), the derivatives of the com-pliance and of the volume are reported here below for the sake of completeness (see [34] for more details).
An interesting alternative to the more rigorous (and time consuming) mathematical programming strategy is the Optimality Condition [5]. The idea is to exploit the neces-sary conditions of optimality (as discussed in Chapter 3) in order to develop an e cient (although heuristic) updating scheme for element densities. These approaches have been widely tested in literature [37, 38]. Of course, the evident shortcoming of the optimality condition approach is that it is not general and an ad hoc rule should be provided for whatever constraint or objective function. In the case of compliance minimisation with a volume equality constraint, the criterion is easy to develop because the optimum design should be fully stressed or nearly fully stressed. However, developing heuristic criteria is not straightforward for other mechanical quantities or for manufacturing-oriented con-straints. It is noteworthy that one of the most common algorithm used in literature is the well-known Method of Moving Asymptotes (MMA) [39, 40], but it represents just one among the possible optimisation algorithms suited for TO problems resolution. A more thorough discussion on mathematical programming algorithms can be found in Chapter 3 of this manuscript.
A summary of advantages and drawbacks of the SIMP method (that can be assumed as the reference density-based method) is provided here below.

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Advantages

The SIMP method is relatively easy to understand and it can be implemented in very compact scripts [41].
The robustness of density based methods has been widely tested in literature. The considerable amount of bibliographic references provided in the following of this manuscript proves that the SIMP method is extremely e cient and versatile for several objective/constraint function implementation.
The reliability of the SIMP method has made possible to embed this algorithm in well-established software packages (Altair OptiStruct R [11], TOSCA [12]), currently constituting the reference for TO in the industrial eld.

Drawbacks

Since the pioneering works on TO, di erent strategies were proposed during the years in order to overcome classic TO drawbacks, such as checker-board e ect and mesh dependence. Projection methods have been used in [10] and their robustness has been investigated in [42]. In these methods, the design variables are the values of the ctitious density function at the mesh nodes. The element densities are obtained by means of a suitable Heaviside step function-based projection. Such a projection can be chosen in such a way to impose a minimum length scale or other kind of constraints (e.g. maximum length scale [43]). However, the well-posedeness of the problem is subject to an arti cial choice (the lter size, the lter type, the projection method, etc.).
In spite of their relative simplicity, the SIMP method provides a FE-based descrip-tion of the nal geometry and suitable postprocessing must be forecast in order to obtain a smooth CAD-compatible design. This shortcoming involves both the mathematical nature and the lack of e ective tools to interpret the nal design in terms of CAD entities.
There is no possibility to keep control of the boundary of the current topology during optimisation. This issue is related to the previous point and the common cause is the lack of a purely geometric entity describing the topology.

The Level-Set Method

More recently, an alternative TO method, known as Level-Set Method (LSM) has been developed and successfully applied to solve TO problems [9, 44, 45]. Indeed, the LSM ori-ginates by the exigency of solving mathematical and physical problems involving boundary evolutions (evolution of interfaces in multi-phase ows, image segmentation, etc.) [1]. Sev-eral LSM-based algorithms have been investigated in the TO framework [46]. The basic idea is to describe the topology of the component through a suitable Level-Set Function (LSF). The sign of the LSF can be conventionally associated to material or void zones, while the zero value represents the boundary of the optimised structure:
8 (x) = 0 , x 2 @ r; @ ; (1.11)
> (x) > 0 x
, x 2 D ( @ ):
< (x) < 0 , 2 r [
In Eq. (1.11), is the LSF de ned on the computational domain D, is the material  domain, whilst @ is the boundary. An example of LSF for 2D structures, which is represented by means of a suitable 3D surface, is shown in Fig. 1.4. Analogously, the LSF is a 4D hyper-surface for the optimisation of 3D components and its 0-level set (@ according to this notation) is constituted of 3D surfaces. Therefore, the LSM o ers an implicit de nition of the topology boundary, since the LSF is parametrised on the computational domain. Whatever transformation of a 2D or 3D material domain (eventually implying modi cations of the connectivity of the domain) can be caught thanks to a LSF that can be represented in a domain whose dimension is increased by one with respect to that of the computational domain D.
Although several versions of the LSM have been proposed in literature [9,46,47], they share the same general logical procedure, whose owchart is shown in Fig. 1.5. The most important feature is that the description of the topology is now related to the LSF and no more to the underlying mesh (decoupling e ect): the FE model and the related mesh are necessary in order to perform the physical responses evaluation. Each step of the LSM is brie y described in the following.

Table of contents :

Introduction 
The Thesis Context and the FUTURPROD Project
Optimisation, Design and CAD in the ALM Framework
Thesis Objectives
Thesis Outline
Introduction 
Contexte de la thèse et le projet FUTURPROD
Intégration de l’optimisation, de la conception et de la CAO
Objectifs de la thèse
Structure de la thèse
1 Literature Review 
1.1 Introduction to the Literature Review
1.2 The Additive Layer Manufacturing Technology
1.2.1 A classification of ALM processes
1.2.2 The Selective Laser Melting Technology
1.3 Topology Optimisation Methods
1.3.1 Density-Based Methods
1.3.2 The Level-Set Method
1.4 Implementation of Manufacturing Constraints in Topology Optimisation
1.5 Conclusions on the Literature Review
1 Revue de la littérature
1.1 Introduction de la revue de la littérature
1.2 La technologie de fabrication additive par couche (Additive Layer Manufacturing)
1.2.1 Classification des processus ALM
1.2.2 La technologie de fusion sélective par laser (SLM)
1.3 Méthodes d’Optimisation Topologique
1.3.1 Méthodes basées sur la densité
1.3.2 La Méthode Level-Set
1.4 Implémentation des contraintes de l’ALM dans l’OT
1.5 Conclusions de la revue de la littérature
2 Fundamentals of Geometrical Modelling 
2.1 Introduction to the Fundamentals of Geometrical Modelling
2.2 The NURBS curves theory
2.3 The NURBS surfaces theory
2.4 The NURBS hyper-surfaces theory
2.5 Conclusions on the NURBS entities theory
2 Principes fondamentaux de la modélisation géométrique 
2.1 Introduction aux Principes fondamentaux de la modélisation géométrique
2.2 La théorie des courbes NURBS
2.3 La théorie des surfaces NURBS
2.4 La théorie des hyper-surfaces NURBS
2.5 Conclusions sur la theorie des entites NURBS
3 Optimisation Methods and Algorithms 
3.1 Introduction to Optimisation Methods
3.2 Deterministic Methods for CNLPP
3.2.1 Generalities on Deterministic Methods
3.2.2 Optimality Conditions for CNLPP
3.2.3 Deterministic Algorithms for CNLPP
3.3 Meta-heuristic Methods for CNLPP
3.3.1 Generalities on Meta-heuristics
3.3.2 The Genetic Algorithm BIANCA
3.3.3 The MATLAB version of BIANCA
3.4 Conclusions on Optimisation Methods and Algorithms
4 A NURBS-based Topology Optimisation Algorithm 
4.1 Introduction
4.2 Mathematical Formulation of the NURBS-based Topology Optimisation Method
4.3 The algorithm SANTO (SIMP And NURBS for Topology Optimisation)
4.4 Results Discussion
4.4.1 The 2D benchmark
4.4.2 Results for 2D problems: sensitivity to the NURBS surface degrees, number of control points and weights
4.4.3 Results for 2D problems: comparison between classical and NURBSbased SIMP approaches
4.4.4 Results for 2D problems: in uence of Non-Design Regions
4.4.5 Results for 2D problems: in uence of a symmetry constraint
4.4.6 The 3D benchmarks
4.4.7 Results for 3D problems: sensitivity to the NURBS hyper-surface degrees, number of control points and weights
4.4.8 Results for 3D problems: comparison between classical and NURBSbased SIMP approaches
4.5 Conclusions and Perspectives on the NURBS-based TO Algorithm
5 Geometrical Constraints Implementation in the Framework of the NURBS- based Topology Optimisation Algorithm 
5.1 Introduction to Geometrical Constraints in Topology Optimisation
5.2 Poulsen’s Formulation of the Minimum Length Scale Constraint
5.2.1 Mathematical Statement of Poulsen’s Minimum Length Scale constraint
5.2.2 Poulsen’s Minimum Length Scale constraint: numerical results
5.3 Minimum Length Scale control in the NURBS-based TO Algorithm
5.3.1 Minimum length scale resulting from B-Spline entities
5.3.2 Some remarks about the proposed approach
5.3.3 The effects of the NURBS weights on the minimum length scale
5.3.4 Results: Minimum length scale in 2D
5.3.5 Results: Minimum length scale in 3D
5.3.6 The effects of a non-uniform knot vector on the minimum length scale
5.4 The Maximum Length Scale
5.4.1 Mathematical Statement of the Maximum Length Scale constraint
5.4.2 Results: Maximum Length Scale in 2D
5.4.3 Results: Maximum Length Scale in 3D
5.5 The Minimum Curvature Radius
5.5.1 Mathematical Statement of the Minimum Curvature Radius constraint
5.5.2 Results on the application of the Minimum Curvature Radius constraint
5.6 Conclusions and Perspectives on Geometrical Constraints in the NURBSbased TO algorithm
6 Eigenvalue Problems in the Framework of the NURBS-Based Topology Optimisation Algorithm 
6.1 Introduction
6.2 Eigenvalue Buckling Problems
6.2.1 First buckling load maximisation in the classic SIMP framework
6.2.2 Mathematical formulation of buckling problems in the NURBSbased TO algorithm
6.2.3 Discussion on numerical aspects
6.2.4 Results
6.3 Eigen-frequencies Problems
6.3.1 First Eigen-frequency maximisation in the classic SIMP framework 235
6.3.2 Mathematical formulation of eigen-frequencies problems in the NURBSbased TO algorithm
6.3.3 Discussion on numerical aspects
6.3.4 Results
6.4 Conclusions and Perspectives
7 General Fitting Techniques for Curves and Surfaces Reconstruction 
7.1 Introduction to Curve and Surface Reconstruction
7.2 A General Hybrid Optimisation Strategy for Curve Fitting in the NURBS Framework
7.2.1 Mathematical Formulation of the Curve Fitting Problem
7.2.2 Numerical Strategy
7.2.3 Studied Cases and Results for curve tting
7.3 Surface Reconstruction in the NURBS Framework
7.3.1 Surface Parametrisation
7.3.2 Surface Fitting
7.3.3 Results on Surface Fitting
7.4 Conclusions and Perspectives on Approximation Problems
General Conclusions
Perspectives of this Thesis
Conclusions et Perspectives

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