Structural Optimization of a Guiding Roller Support using FEM and a Meshless Method and 3D Printing with PLA

F

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Structural Optimization of a Guiding

Roller Support using FEM and a

Meshless Method and 3D Printing with

PLA

Emanuel Bastos

Mestrado Integrado em Engenharia Mecânica

Orientador: Prof. Jorge Lino

Co-orientador: Prof. Jorge Belinha

Structural Optimization of a Guiding Roller Support

using FEM and a Meshless Method and 3D Printing with

PLA

Emanuel Bastos

Mestrado Integrado em Engenharia Mecânica

Resumo

Este projeto de dissertação aborda o tema otimização estrutural, focando mais especificamento no subtipo otimização topológica. A otimização topológica permite a produção de componentes, nos quais ocorre a alteração da sua topologia, segundo determinados objetivos e respeitando os requi-sitos impostos. Como exemplo de objetivo pode considerar-se a maximização da rigidez e como requisito uma redução de volume mínimo. Isto revela-se supra importante ao nível competitivo na indústria, bem como uma prática que contribui para reduzir o impacto ambiental negativo da extração e transformação de materiais.

A otimização topológica é aplicada num componente industrial, utilizando dois software difer-entes, um comercial, Abaqus e outro académico, FEMAS. Estes programas estão associados a diferentes formulações dos métodos e aplicam métodos numéricos distintos. Os métodos numéri-cos aplicados são o tradicional método de elementos finitos e um método pertencente ao grupo dos mais recentes métodos sem malha, o NNRPIM.

Durante o pré-processamento da otimização topológica é tida particular atenção na correta definição dos casos de carga e do domínio inicial, vulgarmente referido como volume da peça, e ainda na coloção das zonas de fixação do componente. Concluída a otimização, os resultados das otimizações são sujeitos a análises numéricas para perceber o seu comportamento em relação à peça original.

A formulação para a otimização topológica utilizada no caso do método sem malha é a Bidirec-tional Evolutionary Structural Optimization Method(BESO), enquanto que no caso do software comercial a formulação utilizada é Solid Isotropic Material with Penalization (SIMP).

O resultado relativo ao software comercial converge e consegue uma redução de 50% na massa, mantendo uma rigidez ligeiramente superior à peça original. O resultado da otimização topológica que utiliza o método numérico sem malha também converge, o que valida a aplicabilidade de métodos sem malha em otimização topológica. Fruto da discretização e da formulação do método, a convergência não é para um valor ótimo absoluto e a redução da massa é próxima dos 30%.

Os resultados provenientes dos diferentes métodos numéricos e programas utilizados, são pós-processados e utilizados na produção de protótipos através de fabrico aditivo. A tecnologia de fabrico aditivo e material utilizado são Fused Filament Fabrication (FFF) e Polylactic Acid (PLA), respetivamente.

Os protótipos são sujeitos a testes mecânicos, de modo a extrapolar o seu resultado para os re-sultados que obteria com um modelo em aço. No entanto, os rere-sultados destes testes não permitem extrapolar o comportamento destas peças para uma semelhante em aço, principalmente devido à anisotropia criada por este tipo de fabrico.

Palavras-Chave: Otimização topológica, SIMP, BESO, fabrico aditivo, FFF, Método de ele-mentos finitos, NNRPIM.

Abstract

This dissertation tackles structural optimization, more specifically the subchapter of topological optimization. The topological optimization allows for the production of components, where a change of topology occurs, following pdetermined objectives and respecting the necessary re-quirements. One of these objectives could be, for example, maximization of stiffness, and the reduction of the minimum volume could be an example of a requirement. This proves to be ex-tremely important at a competitive level in the industry, as well as a practice that contributes to reducing the negative environmental impact of the extraction and transformation of materials.

The topological optimization is applied to an industrial component, using two different soft-ware, one being a commercially available softsoft-ware, Abaqus, while the other is academic, FEMAS. Both these programs are associated with different methodology formulations and they both use different numerical methods. The applied methods are the traditional finite element method and a newer meshless method called NNRPIM.

During pre-processing of the topological optimization, particular attention and precision is paid to correctly define the cases concerning loads and the initial domain, more commonly refereed to as the part’s volume, and also the component attachment zones. With the optimization having been concluded, the optimization results are subjected to multiple numerical analysis, to more clearly understand their behaviour when compared to the original part.

The formulation for topological optimization used in the mesh-less method is Bidirectional Evolutionary Structural Optimization Method (BESO), while when concerning the commercial software, the formulation used is Solid Isotropic Material with Penalization (SIMP).

The results concerning the commercial software converge and achieve a 50% reduction in mass, while attaining a slightly superior stiffness when compared to the original part. The results of the topological optimization that use the mesh-less numerical method also converge, which supports the applicability of mesh-less methods in topological optimization. Because of the dis-cretization and the method’s formulation, this convergence is not towards an optimal and absolute value and the mass reduction is around 30%.

The results provided by the different numerical methods and programs used are post-processed and used in the production of prototypes, through additive manufacturing. The additive manufac-turing process and material used are Fused Filament Fabrication (FFF) and Polylactic Acid (PLA), respectively.

These prototypes are subjected to mechanical tests, in order to extrapolate those results to the results which would be obtained with a steel model. However, these results do not allow for the extrapolation of the behaviour of these parts to a similar steel model, mostly because of the anisotropy created by this type of manufacturing.

Keywords: Topology optimization, SIMP, BESO, Additive manufacturing, PLA, Finite ele-ment method, NNRPIM.

Acknowledgments

This work was developed during the present semester, however their meaning its not limited to this few months. For me, the realization of this work means the fulfilment of a dream. This would not be possible without the contributions of some who directly or indirectly played their role to play in this endeavour. In that sense, I express my heartiest gratitude:

• To my tutor, Prof. Jorge Lino, who gave me the opportunity to work in a forthcoming area as additive manufacturing and always gave me all the necessary support and material to do the best work possible;

• To Prof. Jorge Belinha for his expertise and availability to clarify all my doubts;

• To the entire team from LPDS from DEMec for all their support with my technical and theoretical questions;

• To my friends, for their friendship, companionship, and affability. Without them my aca-demic life would not have so many memories full of joy and longing;

• To my grandfather, for being my second father; • To my brother, my friend for life, always;

• To my father and mother, for being my example and my heroes.

The author truly acknowledges the work conditions provided by the Applied Mechanics Di-vision (SMAp) of the department of mechanical engineering (DEMec) of Faculty of Engineering of the University of Porto (FEUP), and by the MIT-Portugal project “MIT-EXPL/ISF/0084/2017”, funded by Massachusetts Institute of Technology (USA) and “Ministério da Ciência, Tecnologia e Ensino Superior - Fundação para a Ciência e a Tecnologia” (Portugal).

Additionally, the authors gratefully acknowledge the funding of Project NORTE-01-0145-FEDER-000022 - SciTech - Science and Technology for Competitive and Sustainable Industries, cofinanced by Programa Operacional Regional do Norte (NORTE2020), through Fundo Europeu de Desenvolvimento Regional (FEDER).

Finally, the author acknowledges the synergetic collaboration with the collaborators of “Com-putational Mechanics Research Laboratory CMech-Lab” (ISEP/FEUP/INEGI), and its director, Prof.Dr. Jorge Belinha, and its senior advisors, Prof.Dr. Renato Natal Jorge and Prof.Dr. Lúcia Dinis.

“Humildade, respeito e dedicação trouxeram-me até aqui Irei até onde me levarem!”

Contents

1 Introduction 1

1.1 General Aspects . . . 1

1.2 Objectives of the dissertation . . . 2

1.3 Work Structure . . . 2

2 Structural optimization 3 2.1 Sizing, shape and topology optimization . . . 3

2.2 Mathematical structural optimization problem . . . 4

3 Topology optimization 7 3.1 Material interpolation . . . 7

3.2 Numerical problems - Checkerboards . . . 8

3.2.1 Sensitivity Filter . . . 8

3.3 General problem formulation . . . 8

3.3.1 Minimum compliance . . . 9

3.3.2 Minimum volume . . . 9

3.3.3 General topology optimization flowchart . . . 10

3.4 Topology approaches . . . 10

3.4.1 Gradient based approaches . . . 11

3.4.2 Non-Gradient based approaches . . . 12

3.5 Solution methods . . . 13

3.5.1 Optimality Criteria method . . . 13

3.5.2 Mathematical programming methods . . . 14

4 Numerical methods 15 4.1 Finite element method . . . 15

4.1.1 Computational use of FEM . . . 16

4.1.2 Strong and weak forms . . . 16

4.2 Meshless methods . . . 17

4.2.1 Generic Aplication of a Meshless Method . . . 17

5 3D printing 25 5.1 Classification and technologies . . . 25

5.2 Material extrusion . . . 25

5.3 Fused Filament Fabrication (FFF) . . . 26

5.3.1 Materials . . . 27

5.4 FFF machines . . . 28

5.4.1 Machine formats . . . 29

5.5 3D printing in topology optimization . . . 30

5.5.1 Support structures design . . . 31

5.5.2 Infill design . . . 32

5.5.3 3D printing imposed material anisotropy . . . 32

6 Part description 33 6.1 Scope . . . 33

6.2 Analysis of the part’s functionality . . . 33

6.3 Part characteristics . . . 34

6.4 Loads and Safety factors . . . 35

7 Numerical results 37 7.1 Data needed before Topology optimization . . . 38

7.1.1 Direction and application point of the loads . . . 38

7.1.2 Material . . . 39 7.1.3 Stiffness . . . 39 7.2 Autodesk Fusion 360 . . . 39 7.2.1 Work model . . . 39 7.3 Abaqus . . . 41 7.3.1 Static analysis . . . 42

7.3.2 ABAQUS model - SOLIDWORKS . . . 43

7.3.3 Topology optimization . . . 43

7.4 FEMAS . . . 46

7.4.1 Definition of the load cases . . . 47

7.4.2 Optimization Model - Solidworks . . . 47

7.4.3 First optimization - FEMAS . . . 48

7.4.4 Second Optimization - FEMAS . . . 51

7.4.5 Third Optimization - FEMAS . . . 55

7.5 Numerical analysis of optimised models . . . 59

7.5.1 Model and volume mesh creation . . . 59

7.5.2 Numerical results . . . 59 8 Experimental work 63 8.1 Material-PLA . . . 64 8.2 Equipment used . . . 65 8.2.1 3D printer . . . 65 8.2.2 Testing Machine . . . 66 8.2.3 Instruments . . . 66 8.3 3D Printing pre-process . . . 68 8.3.1 Abaqus . . . 68 8.3.2 FEMAS . . . 68 8.4 3D printing . . . 69 8.4.1 Printer calibration . . . 69 8.4.2 Printing parameters . . . 69 8.5 Mechanical tests . . . 71 8.5.1 Procedure . . . 71

CONTENTS xi

9 Mechanical Results 73

9.1 Specimen analysis . . . 73

9.2 Results of the mechanical tests . . . 75

9.2.1 Mass of the specimens . . . 75

9.2.2 Stiffness extrapolation to EN S235 JR . . . 76

10 Conclusions 79 10.1 Future work . . . 81

A Appendix A 89 A.1 Technical drawings of the auxiliary parts . . . 89

A.2 Matlab script of the scatter with variable range of density . . . 93

A.3 Matlab script to generate STL. files . . . 94

List of Figures

2.1 Different types of structural optimization size a), shape b) and topology c) . . . . 4

3.1 Consequence of applying a sensitivity filter in a engine support . . . 8

3.2 Flowchart for general topology optimization . . . 10

4.1 Shell elements discretizing a hemisphere small section . . . 16

4.2 Solid domain a), Regular nodal discretization b), Irregular nodal discretization c) 18 4.3 Variable influence domain . . . 19

4.4 Voronoï diagram construction . . . 20

4.5 Delaunay triangulation . . . 20

4.6 first degree influence cell a), second degree influence cell b) . . . 21

4.7 Initial quadrilateral grid-cell a), Isoparametric square and 2×2 Gauss-Legendre quadrature application b), return to the initial quadrilateral shape c) . . . 22

4.8 Voronoï cell a), Middle points, MIi, and respective quadrilaterals b), Individual quadrilateral c) . . . 23

4.9 Voronoï cell a), Middle points, MIi, and respective triangle b), individual triangle c) 23 4.10 Integration points of triangular and quadrilateral cells . . . 23

4.11 Gauss-Legendre quadrature integration in sub-sub-cells . . . 24

5.1 Material extrusion process . . . 26

5.2 Fused Filament Fabrication . . . 27

5.3 BigRep industrial 3D printer . . . 29

5.4 Low cost 3D printer . . . 29

5.5 Cartesian and delta machines types . . . 30

5.6 Slope and straight wall supports . . . 31

5.7 Bridge-like support . . . 31

5.8 Overhang-free topology optimization with different minimum self-supporting angles 32 6.2 Guiding roller’s main parts . . . 34

6.1 Guiding roller’s assembly . . . 34

6.3 Guiding roller support structure top view a), bottom view b) . . . 35

7.1 Scheme of the work performed in this section . . . 37

7.2 Direction of the applied loads . . . 38

7.3 Application point of the loads. . . 38

7.4 7Bulk model used in Autodesk Fusion 360 . . . 40

7.5 Optimised model in Autodesk Fusion 360 according the different directions of the load, corresponding to F1a) , -F2b), F3c) and F4) d) . . . 40

7.6 Optimised merged model back view a), front view b). . . 41

7.7 Initial part statical analysis, Stresses a), displacements b) . . . 43

7.8 Model used in abaqus topology optimization, front view a, rear view b) . . . 43

7.9 Optimization 1,2 and 3: Front view of Abaqus stress distribution - a),c),e); Rear view Abaqus displacement distribution - b),d),f). . . 44

7.10 Optimization 4,5 and 6: Front view of Abaqus stress distribution - a),c),e); Con-tinue of rear view Abaqus displacement distribution - a),c),e) . . . 45

7.11 Stiffness evolution during topology optimization - a), specific stiffness evolution during topology optimization - b) . . . 46

7.12 a)-Loads at point A. b)-Loads at point B, c)-Applied loads in the support. . . 47

7.13 Solid works models created to FEMAS topology optimization. First model a), second model b) and third model c) . . . 48

7.14 Scatter plot of FEMAS - FEM optimization at iteration 8 . . . 49

7.15 Scatter plot of FEMAS - FEM first optimization at iteration 8, without low values of density . . . 50

7.16 Scatter plot of FEMAS - NNRPIM first optimization at iteration 7. . . 51

7.17 Scatter plot of FEMAS - NNRPIM first optimization at iteration 7, without low values of density. . . 51

7.18 Scatter plot of FEMAS - FEM second optimization at iteration 8 . . . 53

7.19 Scatter plot of FEMAS - FEM second optimization at iteration 8, without low values of density . . . 53

7.20 Scatter plot of FEMAS - NNRPIM second optimization at iteration 8 . . . 54

7.21 Scatter plot of FEMAS - NNRPIM second optimization at iteration 8, without low values of density . . . 55

7.22 Scatter plot of FEMAS - FEM third optimization at iteration 9 . . . 56

7.23 Scatter plot of FEMAS - FEM third optimization at iteration 9, without low values of density . . . 57

7.24 Scatter plot of FEMAS - FEM third optimization at iteration 8 . . . 58

7.25 Scatter plot of FEMAS - NNRPIM third optimization at iteration 8, without low values of density . . . 58

8.1 Scheme of the practical work performed . . . 64

8.2 3D printer used to print the models . . . 66

8.3 Machine used to realize the mechanical tests . . . 66

8.4 Mitutoyocaliper . . . 67

8.5 Scale used to measure the specimens weights . . . 67

8.6 Optimized model from Abaqus a), and smoothed model from Autodesk mesh-mixer b) . . . 68

8.7 Post-process of models obtained in FEMAS . . . 69

8.8 Load case F2 mechanical test placement . . . 71

8.9 Auxiliary test part to place the models parallel and perpendicular way to the ma-chine movement a), part to place the models inclined 45◦ to the machine move-ment b), Pin created to enable the transmission of the loads from the machine to the model c) . . . 72

9.1 All the optimized printed models . . . 73

9.2 Model from FEMAS - FEM a), Model from FEMAS - NNRPIM b) . . . 74

List of Tables

5.1 Table with aditive manufacturing processes according to ISO/ASTM 52900 . . . 25

6.1 Table with loads and stresses magnitudes for the regular and the static load case . 35 7.1 EN S235 JR steel properties . . . 39

7.2 Initial model analysis results . . . 42

7.3 Optimization results . . . 45

7.4 Density evolution during the first topology optimization using FEM in FEMAS . 49 7.5 Density evolution during the first topology optimization using NNRPIM in FEMAS 50 7.6 Density evolution during the second topology optimization using FEM in FEMAS 52 7.7 Density evolution during the second topology optimization using NNRPIM in FE-MAS . . . 54

7.8 Density evolution during the third topology optimization using FEM in FEMAS . 56 7.9 Density evolution during the third topology optimization using NNRPIM in FEMAS 57 7.10 Stiffness of the optimised and initial models calculated numerically . . . 60

7.11 Optimised models mass and reduction % comparing with the original model . . . 60

7.12 Specific stiffness of the optimised and initial models calculated numerically . . . 60

8.1 Formfutura Properties of premium PLA . . . 65

8.2 Formfutura premium PLA dimensions . . . 65

8.3 Testing machine specs . . . 67

8.4 Caliper Specifications . . . 67

8.5 HLD 300 scale specifications . . . 68

8.6 Specimens printing parameters . . . 70

9.1 Stiffness of 3D printed models . . . 75

9.2 Mass of the printed models . . . 75

9.3 Steel EN S235 JR extrapolation from PLA mechanical tests results . . . 76

9.4 Extrapolation error for all the load cases . . . 76

A.1 Results from the mechanical tests to all the load cases . . . 98

Acronyms and Symbols

Acronyms

ABS Acrylonitrile Butadiene Styrene AM Additive manufacturing

AD Average Deviation

BESO Bi-directional Evolutionary Structural Optimization Method CONLIN Convex Linearisation

ESO Evolutionary Structural optimization FEM Finite Element Method

FEMAS Finite Element and Meshless Analysis Software FFF Fused Filament Fabrication

MMA Method of Moving Asymptotes NEM Natural Element Method

NNRPIM Neighbour Radial Point Interpolation Method OC Optimality Criteria

PETG Polyethylene Terephthalate Glycol PLA Polylactic Acid

RPIM Radial Point Interpolation Method SIMP Solid Isotropic Material with Penalization SLP Sequential Linear Programming

SQP Sequential Quadratic Programming

Symbols

Chapter 1

Introduction

This chapter aims to contextualize the reader on the main theme of this dissertation. Firstly, an overview is given about structural optimization, its relevance and connection with additive manu-facturing. It follows the presentation of the general aspects, main objectives and work structure.

1.1

General Aspects

Structural optimization creates optimal design of load-carrying mechanical structures.

Within the theme of structural optimization there is topological optimization, as example of application, considering an initial design domain with supports and some loads, topology opti-mization seeks for the optimal structure to support the loads. This is done by defining zones, in the design domain, that should be material or voids, according to predetermined constraints. Nor-mally, weight reduction is the objective, subjected to displacements and stresses constrains. To solve this problem numerically, a domain discretization is performed and the problem is solved using a efficient numerical optimization method [1].

The huge importance that structural optimization is related to several factors, however the main one is the environmental factor. This is easily explained by the restrictions in the Carbon dioxide (CO2) emissions in all the industries. If less material is used to produce a new part, with the

same functionality as the old one, a reduction chain of emissions is created, from manufacturing processes until effective use.

Besides environmental factors, structural optimization leads to a reduction in time spent on development process, by reducing the number of design iterations in comparison with traditional approaches and, more important, provides a commercial advantage to companies that produce lighter parts, with better mechanical performance, in relation to the competition.

Nowadays, in product process development, together with structural optimization, additive manufacturing is also present. It can be used in the manufacturing process of the final product or, as usual, used to produce prototypes quickly. Many times the prototype material behaves similar to the final manufacturing material, making possible to mechanically test the prototypes and extrapolate results, predicting the final model performance.

Therefore, additive manufacturing together with structural optimization enhances the product development cost, velocity and quality. This thesis puts in practice the application of structural optimization with additive manufacturing prototyping.

1.2

Objectives of the dissertation

The main goal of this work is to apply topology optimization to a mechanical component in order to reduce the part weight and improve its mechanical behaviour.

It is intended to use different software and numerical methods in order to create different optimized solutions.

Another objective is to perform a comparison between the initial models and the optimized one, in order to compare their performances.

A practical approach of 3D printing is followed with the aim of testing and then extrapolate the results in order to validate the simulation results.

Lastly, it is desired to compare the different numerical approaches and software.

1.3

Work Structure

In chapter2, a briefly introduction about structural optimization is going to be performed, where the subroutines of structural optimization are going to be explained.

The next chapter,3, deals with topology optimization. Here the typical topology optimization problem will be introduced and also a review of the approaches, formulations and methods of solution will be made.

In chapter,4, will be done an introduction to numerical methods. Normally structural opti-mization uses finite element methods (FEM), however in this thesis, meshless methods are also going to be used.

The next chapter,5, introduces the 3D printing technologies and the used materials, paying particular attention to fused filament fabrication, which is used in this work.

In chapter6, the model used to apply topology optimization is introduced, the load and geo-metric restrictions as well.

In chapter,7, finished the theoretical concepts from the topics in study, the model optimization is performed with the commercial software and the results are registered.

In chapter8, the models are created, which are produced using FFF. In this chapter are also defined the material properties, the used machines and also all the production tools. Lastly, the procedure for the mechanical tests is also presented.

In the next chapter,9, the results from mechanical tests are analysed.

In last chapter,10, the main conclusions are presented as the comparisons between the different software and numerical methods.

Chapter 2

Structural optimization

Structural optimization can be defined as the optimal design creation of a mechanical part, auto-matically, based in structural considerations [1]. The goal is to achieve the best component subject to a given set of structural functional requirements and performance constrains. The conditions and requirements that define the operation of the component are the functional requirements, like the weight of the component, directions and magnitudes of the applied loads, location of the sup-ports and material properties. The performance constrains are the acceptable structural behaviour, like the maximum allowable stress, deflection, minimum heat dissipation rate, etc [2]. The re-quirements mentioned above are also nominated as design criteria. It is not possible to solve the structural optimization design criteria simultaneously, only a particular design criteria can be addressed to the problem [3].

2.1

Sizing, shape and topology optimization

Structural optimization can be divided into three different categories, depending on the geometric feature [2,4]:

• Sizing optimization: In this case the design variable (x) is a dimension of a structure, e.g., the cross section width of a beam.

• Shape optimization: The idea is to vary the shape, in a domain, to produce a contour or form of the boundary of the structure domain, which is the design variable. The integration domain for the differential equations is chosen in an optimal way. New boundaries are not formed because the connectivity of the structure is not changed.

• Topology optimization: The most widely applied form of structural optimization. The final design can be independent of the initial structure, typically associated to a large number of variables. In a discrete case, like a truss structure, it is achieved by taking off parts from the truss structure. In a continuum-type structure, the optimization results in areas ideally without thickness and others with a fixed maximum thickness. Normally computational

software are applied to solve structural optimization problems, associated to numerous dif-ferent methods [4]. This methods will be object of analysis in section3.

In figure2.1, it is possible to see the three different types of structural optimization mentioned above.

(a)

(b)

(c)

Figure 2.1: Different types of structural optimization size a), shape b) and topology c) [2].

2.2

Mathematical structural optimization problem

In the formulation of a structural optimization problem the following function and variables are present [4]:

Objective function (f): The objective function in structural optimization is used to classify the different designs. It is the function that we want to improve, it can be minimized, maximized or even set into a fixed value. The objective function can measure the weight, displacement or the effective stress, for example.

Design variable(x): It represents a vector or function that describes the conditions that change during optimization. It may be, for example, the thickness of a sheet or the area of a profile.

State variable(y): The state variable describes the state of a dynamic system. It represents the response of the structure, like displacement, strain or force, in the case of a mechanical system.

Moreover, the state variable depends on the design variable y(x). Considering this, the general structural optimization problem takes the form [5]:

minimize f(x, y(x)) (2.1) Subjected to         

Behavioural constrains on y(x) Design constrains on x

Equilibrium constraint

The behavioural constrains are related with the state variable y. Normally they use the notation g(y) ≤ 0, where g is a function that represents, e.g., the displacement. Design constrains involve

2.2 Mathematical structural optimization problem 5

the design variable x. Depending on the situation, the design variable x and the state variable y can be finite dimensional or have an infinite number of degrees-of-freedom. In the case where variables are finite dimensional, one talks of discrete parameter systems, e.g., a truss structure. If the variables have a infinite number of degrees-of-freedom, i.e., if they are functions or "fields", one talks of distributed parameter systems, also designated continuum problems [4].

When solving continuum problems by computational means, typically, a discretization is per-formed, resulting in a discrete parameter system. To distinguish these systems from truss structure systems, the after mentioned are called naturally discrete parameter systems. The discretization is performed because computer implementations of mechanical problems are based on algebra, which is finite dimensional. Finally, in these discretized systems, the equilibrium constraint is:

K(x)u(x) = f(x) (2.2)

where K(x) is the stiffness matrix, u is the displacement vector and f(x) is the force vector. Normally, in a continuum problem, the equilibrium constraint is a partial differential equation.

In the structural optimization formulation presented above, y and x are treated as independent variables (simultaneous formulation), since equilibrium, or more generally, the state problem is solved at the same time as the optimization problem. In many cases, like when K(x) is invertible for all x, the state problem defines y in case of a given x, e.g., u = u(x) = K(x)−1f(x). Considering u(x) a given function, which is substituted for the state variable, the equilibrium constraint can be left out of the structural optimization [4]. The resultant structural optimization formulation is the nested formulation[5]:    min x f(x) s.t. g(x, u(x)) ≤ 0 (2.3)

It is assumed that all the design constraints can be written as g(x, u(x)) ≤0. The structural op-timization nested formulation is solved by evaluating the derivatives of f and g with respect to x[4].

The objective function can be composed by multiple objectives, the multi objective function problem is [5]:

min

x f( f1(x, y), f2(x, y), ..., fn(x, y)) (2.4)

where n is the number of functions. The functions can have different levels of importance, and in that case they receive weights (w) according to their importance. A scalar formulation is:

f =

n

∑

fiwi (2.5)

and the total sum of the weights is:

n

∑

wi= 1 (2.6)

Varying the set of weights, a unique solution where no objective can be improved without worsen another can be obtained. This set is called the Pareto set [5].

Chapter 3

Topology optimization

Topology optimization was firstly introduced in 1988 by Bendsøe and Kikuchi [6]. Since then it has been through a tremendous development in the number of approaches to solve this problem [7]. It aims to achieve the best optimal layout of a structure within a specific region. There are some important variables that must be defined a priori. These are the applied loads, the design restrictions, like holes or solid areas, the volume of the structure and support conditions. The physical shape, size and connectivity of the structure are unknown for this problem [2].

3.1

Material interpolation

Being Ω the design domain, topology optimization seeks for a Ωmat⊂ Ω. The design variable

de-fined as x in chapter2is now represented by the density vector ρ containing the elemental densities ρe. The local stiffness tensor E can be formulated incorporating ρ as an integer formulation [2,5]:

E(ρ) = ρE0 (3.1) ρe=    1 i f e ∈ Ωmat 0 i f e ∈ Ω Ωmat

Together with the integer formulation of ρ a volume constraint is also needed:

Z

Ω

ρ dΩ = VolΩ(mat) ≤ V (3.2)

where V is the volume of the initial design domain. If ρe= 1 that element is considered to be

filled, when the opposite occurs, ρe= 0, the element is considered to be a void. In order to solve

the optimization problem with a gradient based solution strategy the integer problem is replaced by a continuous function, then some form of penalty is introduced to achieve the discrete values of 0 or 1. A well known and used method, Solid Isotropic Material with Penalization (SIMP), can be used to relax the integer, resulting in the following density function [2]:

E = ρpE0, ρ ∈ [ρmin, 1], p> 1 (3.3)

where p is the penalization factor. It will penalize intermediate densities resulting in a final design that has density 0 or 1 in all points, ρminis the lower density value limit in order to avoid

singularities.

3.2

Numerical problems - Checkerboards

One of the numerical problems that topology faces are the checkerboards , which are regions of alternating solid and void elements ordered in a chequerboard way [8]. It was thought that these regions could be the optimal design, however according Numerical Instabilities in topology optimization [8] they are the result of bad numerical discretization. There are some methods used to overcome this problem which are almost all based on heuristics.

3.2.1 Sensitivity Filter

When searching for a mesh dependent solution, a sensitivity filter should be used which will not add too much computational time or constrains. It is based on the weight average of the neighbourhood elements under a certain radius that the design sensitivity is modified [5]. This can be explained in detail in Checkerboard and minimum member size control in topology optimization [9]. From figure3.1ais possible to see this problem. After the application of a sensitivity filtering method it is possible to see the improved result in figure3.1b[9].

(a) (b)

Figure 3.1: Consequence of applying a sensitivity filter in a engine support [9].

3.3

General problem formulation

The nested structural optimization formulated in2.3can now be written as [5]: min

ρ

3.3 General problem formulation 9 Subjected to          0 ≤ ρ ≤ 1

State function constraint Manufacturing constraints

If SIMP interpolation method is used, ρ is a vector containing the element densities.

3.3.1 Minimum compliance

Compliance can be defined as the elastic strain energy of the structure induced by the load case. For example, the deformation in a structure subject to a load can be reduced by reinforcing the critical areas with material. The reinforcement will lead the structure to better global stiffness [10]. Normally, the formulation of topology optimization searches for the minimum compliance (maximum global stiffness). The compliance can be defined by:

C(ρ) = fTu (3.5)

where u solves the equilibrium equation:

K(ρ)u(x) = f(x) (3.6) K(ρ) is K(ρ) = n

∑

where K0i is the elemental stiffness matrix with the initial stiffness tensor E0. A volume constrain

needs to be place, otherwise the full design volume would be the result. To solve the optimization by a gradient based approach, derivatives of C(ρ) are needed.

3.3.2 Minimum volume

Instead of the compliance, the volume can be minimized: V(ρ) =

n

∑

ρ_ipVi0 (3.8)

Where V0is the initial volume. Here, a constrain for the maximum displacement or effective stress is imposed, this way the optimization will not remove all the volume.

The optimization is solved according the objective function and constrains, however, in cases that the objective function is formulated according the volume or weight, the derivatives are eval-uated for the constrains [5]. As an example, for a gradient based solution, if a displacement vector is imposed, the named state derivatives from u(ρ) are determined.

3.3.3 General topology optimization flowchart

In figure3.2in shown the general procedure to solve the optimization problem represented by a flowchart.

Figure 3.2: Flowchart for general topology optimization, adapted from [2]

3.4

Topology approaches

Nowadays topology optimization is related to different approaches to solve the optimization prob-lem. There are gradient-based topology optimization approaches and non-gradient-based ones.

3.4 Topology approaches 11

According to Sigmund [11], the gradient optimization approaches totally outperform the non-gradient based approaches in all aspects, due to their huge computational cost and, although they use a global search, it is unlikely they find a global optimum solution, being their only advantage the fact that they are easy to implement since do not make the use of gradients.

3.4.1 Gradient based approaches

The gradient based techniques for continuum problems are the homogenization method [6], the SIMP method [2,12], introduced above while explaining the general application of topology opti-mization, the Level-set Approach [13,14], the Bidirectional Evolutionary Structural Optimization Method (BESO) [15–17], etc. SIMP and BESO, winch are to approaches used to solve the opti-mization problem, result from the homogenization method. Only these three methods are briefly introduced.

3.4.1.1 Homogenization method

This method approach transforms topology optimization problems in size optimization problems. This is done by introducing a material density function in each element that is composed of an infi-nite number of periodically distributed holes. The homogenization method is applied to determine the macroscopic constitutive equation for the material with microscopic material constituents. The microstructures can be divided into methods based on rank laminate composites, where the ho-mogenization equation can be analytically solved, and methods based on microcells with internal voids, normally for this methods numerical methods are used [18,19].

3.4.1.2 SIMP method

The SIMP method has proven very popular in industrial software due to its simplicity and compu-tational efficiency. This method takes the density ρ of an element as the design variable to control where there should be material and voids. The equation3.9is used to define the element density:

ρe= xeρ0 (3.9)

where xeis the relative density and ρ0is the density of the reference material. The relative density

can vary between xe_minand 1 (0 < xe_min< xe≤ 1). xe

minis a very small number to avoid singularities

and in that case the element is empty, the value 1 means that the element consists in solid material. To avoid intermediate densities, which are undesirable in the optimal design, a penalty is applied when calculating the element stiffness:

Ke= (xe)pK0 (3.10)

where K0 is the stiffness matrix of an element consisting of solid material and p > 1 is the pe-nalization factor. Experience shows that in problems where volume constrain is active, in order to obtain 0 - 1 designs, p ≥ 3 is required [19]. In some simple optimization problems, both the

objective function and all of the constrains, are written as explicit functions of the design vari-ables. However, the same is impossible, or very difficult, for larger problems. The solution is to create a sequence of explicit sub-problems that are approximations of the original problem and solve these sub-problems instead. In other words, the majority of structural optimization problems are nonconvex problems and to solve that convex approximations are chosen [4]. To solve this, several mathematical methods were developed, including Sequential Linear Programming (SLP), The Sequential Quadratic Programming (SQP), Convex Linearisation (CONLIN), Optimality Cri-teria (OC) and Method of Moving Asymptotes (MMA). Methods involving OC are efficient for problems with a large number of variables and few of constrains. In the case of several constrains the SLP is used. These methods are explained in detail in3.5.

3.4.1.3 ESO(BESO) method

This method removes inefficient material from the structure in each interaction, in order to obtain the optimal solution. In a structure, any material under low stress is assumed to be inefficient, being removed [15–17]. To complete the optimization process is important to define the conditions under which an element is removed, being the Von Mises Stress the most commonly criteria that is used. The von Mises stress of an element σe

v is compared with the von Misses stress of the whole

structure σvM. After each iteration the points that satisfy3.11are deleted [19].

σ_ve/σ_vM≤ RRi, (3.11)

RRi is the rejection ratio for iteration i. In every numerical analysis points are removed using the

same rejection ratio until occurs convergence. An evolutionary rate ER is added to create a new rejection ratio RRi+1,

RRi+1= ER + RRi. (3.12)

The numerical analysis start again and, considering the new rejection ratio, points are removed until convergence is achieved again. The ESO method can raise some questions, about how to ensure the solution isn’t a local minimum and whether deleted elements can return. In order to answer these questions a so-called bidirectional ESO (BESO) has developed. This method evolution allows elements to be added and removed and, with that, achieve the global optimum solution.

σve/σvM≥ IRi, (3.13)

To identify the conditions under which material can be added to the structure, the inequality3.13

need to be satisfied, where the IRiis the inclusion ratio [19]. 3.4.2 Non-Gradient based approaches

The non-gradient-based optimization approaches are based in random processes which are genetic algorithms [3], artificial immune algorithms, simulated annealing, etc [11]. Here only the Genetic

3.5 Solution methods 13

algorithm is going to be briefly introduced as an example.

3.4.2.1 Genetic Algorithm method

This optimization procedure is based on the theory of natural selection, where the optimization occurs through an evolution of a chromosomes population. The chromosome is dived in genes that correspond to the design variables and the gene’s allele values control the values of the design variables. Each one of the different chromosomes corresponds to a possible optimum set of design variables. The way the algorithm works is by starting with a group of chromosomes and eval-uating, ussing a metric function, their "fitness". The ones fitting better the components designs are then used as parents. The parents designs are paired together creating child designs, these new population is subject to infrequent, random mutation as replace the parents population. Af-ter many iAf-terations the component designs with betAf-ter design characAf-teristics propagate into child generations resulting in an increase of the quality of the design component. A more detailed expla-nation about the application of the genetic algorithm method can be found in Structural Topology Optimization Via The Genetic Algorithm[3].

3.5

Solution methods

In this section some numerical solution methods used to solve the topology optimization ap-proaches described above, also refereed as optimization algorithm, are going to be introduced. These solutions techniques guide the search for the set of design variable values that satisfy the objective function and constrains [3].

3.5.1 Optimality Criteria method

The Optimality Criteria consists in an iterative redesign algorithm to generate the design of the component fulfilling a set of optimality criteria and defining the optimal structural behaviour. The optimality criteria method can be divided in two components. The first is the set of optimality cri-teria which defines the conditions for the optimality of a structure. Typically, differential equations are satisfied by the optimal design. The second component of the OC method is the iterative re-design task in which, at any iteration K, the rere-design relates the current of re-design variables values x(k)to a new set of design variables values x(k+1)achieving a design of better quality:

x(K+1)= f(x(k)) (3.14)

The iterative redesign procedure can be based in rigorous mathematical considerations such as the Kuhn-tucker conditions or based on either intuitive heuristics, like decide to remove material which is not fully stressed [3]. For more complicated problems, as cases where constrains of a non-structural nature should be considered, the use of mathematical programming methods could give a more direct way to convergence, because this method can converge to a non-optimal solution [4].

3.5.2 Mathematical programming methods

The mathematical techniques, sometimes referred as direct methods, search for the nearest local optimum result, corresponding to a set of optimum design variable values and a locally-optimum component design through an iterative process. In more detail, in an iteration k the set of design variables values, x(k) is used to calculate the values and gradients of the objective and constrain function. With this first step’s information a search direction d(k)and the "place" where the optimization algorithm should move, designed as step size α(k)are defined. Lastly, in3.15is

shown how to calculate the new design variables x(k+1):

x(k+1)= x(k)+ α(k)d(k) (3.15)

The mathematical algorithms using the gradient information can achieve the optimal design with-out using too much computational time, however because these algorithms search for the near-est locally-optimum peak in the search space, the main global peak could not be found during optimization. In order to overcome this problem the mathematical-based structural optimiza-tion should be executed with a variety of initial condioptimiza-tions increasing the chance of finding the globally-optimum design. There are some mathematical programming algorithms used in struc-tural optimization, because none of them is suitable for every problem [3].

In the Sequential Linear Programming (SLP) approximation at xk, the objective function and all constrained equation are linearised at the design xk. In SLP all expressions are explicit functions of x, which means that SLP is an explicit approximation to the nested formulation of the structural optimization problem.

The Sequential Quadratic Programming (SQP) works by adding a second order term in the Taylor expansion of the objective function in SLP, this way SQP is a better approximation of the original problem at the current design.

Convex Linearisation (COLIN), is more conservative than SLP, is an explicit convex and sep-arable approximation used in a wide range of structural optimization problems.

The Method of Moving Asymptotes (MMA) is a method where the "Conservatism" degree can be controlled and that is done by modifying the asymptotes during the iterations. The MMA approximation is first order, explicit, convex and also separable. In fact SLP and COLIN are special cases of MMA [4].

Chapter 4

Numerical methods

Normally topology optimizations approaches are formulated based on element design variables. The finite elements are used to represent the topology of structures, fields of displacement and strains, assuming that the element densities and relevant material properties are constant within each element [20]. However there are also other numerical methods that can be implemented in topology approaches, such as meshless methods, which are relatively simple to implement. These methods can provide sufficient solution accuracy for a certain type of problems, without require the need for mesh connectivity. There are several different meshless methods available, being each one adequate to solve a specific type of problem [21,22].

4.1

Finite element method

Finite Element Method (FEM) is well known tool used by engineers around the world. Before explaining the method it is important to refer to discrete and continuous problems. The first is obtained when using a finite number of well-defined components whilst in the continuous prob-lems the subdivision is continued indefinitely, this leads to differential equations which imply an infinite number of elements. Continuous problems can only can be solved using mathematical techniques which are usually limited for exact solutions. It was to overcome this problem that FEM was created [23].

Nowadays it is an indispensable technology in modelling and simulation of advanced engi-neering systems and can be described as a numerical method that finds an approximated solution of the distribution of field variables in the problem domain, which is very difficult to obtain an-alytically. When using FEM to solve a problem governed by differential equations, the first step is to discretize the problem domain into small elements, each one having a simple geometry, as shown in figure4.1.

The elements, also referred as sub-domains, are formed by nodes. In each element, a con-tinuous function of an unknown field variable is piecewise approximated using linear functions. Then, all the equations are assembled together with adjoining elements to form the global finite

Figure 4.1: Shell elements discretizing a hemisphere small section [24].

element equation for the whole problem domain. These equations can be solved easily to obtain the required field variable [24].

4.1.1 Computational use of FEM

Most of the times the engineering systems have a very complex domain with complicated bound-ary conditions. This, as referred earlier, leads to a difficulty in solving the governing differential equations via analytical means and, in practice, the problems are solved using a computer through the application of numerical methods. In FEM, the methods of domain discretization are the most popular. The procedure to apply FEM through the use of a computer follow the next steps:

• Modelling of the geometry;

• Creation of the mesh (discretization); • Specification of the material property;

• Specification of initial, load and boundary conditions.

4.1.2 Strong and weak forms

The problem can be solved using the strong form or the weak form. The first requires continuity on the field variables, because the functions defining these variables need to be differentiable up to the order of the partial differential equations. Usually, obtaining the exact solution for a strong form of the system equation is very difficult for practical engineering problems. The weak form is often an integral form which requires a weaker continuity of the field variables, this leads to a set of discretization system equations that give much more accurate results, specially in problems that offer an complex geometry [24].

4.2 Meshless methods 17

4.2

Meshless methods

Meshless methods are an alternative to the Finite Element Method (FEM), which is based in mesh interpolation. Usually, to find good results it is necessary to remesh the mesh and this is time consuming [25]. Meshless methods are used for solve partial differential equations, where the nodes can be arbitrarily distributed and, instead of using elements, the field functions are approx-imated with an influence domain around the node. The domains should overlap with each other as opposite to the FEM , where the domains, the "elements", can’t overlap [26,27]. A signifi-cant number of methods was developed in the last years, however the most relevant one for this topology optimization study is the Natural Neighbour Radial Point Interpolation Method (NNR-PIM) [28]. This results from the combination between two other meshless methods, the Natural Element Method (NEM) [29–31] and the Radial Point Interpolation Method (RPIM) [32,33]. The RPIM arised from Point Interpolation Method (PIM) [34], which consists in constructing poly-nomial interpolations, possessing the Kronecker delta property, considering a group of arbitrarily distributed points. The Kronecker delta property enables the imposition of boundary conditions. This method evolved to the RPIM in order to solve some numerical problems. The next evolution is the NNRPIM, where the influence domain used in the RPIM is changed to the influence-cell concept. The influence-cells are based in geometry and mathematical constructions such as the Voronoï diagrams and the Delaunay tessellation [26].

4.2.1 Generic Aplication of a Meshless Method

The majority of meshless methods follow a standard procedure, described in the next steps [26,35]: 1. Definition of the solid domain and contour of the problem geometry;

2. Identification of the essential and natural boundary conditions;

3. Numerically discretization by a nodal set, following a regular or irregular distribution, of the problem domain and boundary, figure4.2. The nodal density of the discretization and the spatial distribution are responsible for the method performance. Fine nodal distribution leads to better results however also leads to high computational costs. The correct distribu-tion of the nodes is also important to achieve good results. Locadistribu-tion with predictable stress concentrations need a higher nodal density distribution, figure4.2c;

4. Construction of a background integration mesh: It can be nodal dependent (NNRPIM) or independent (RPIM). Normally the integration mesh has the size of the problem domain, however it can be bigger than that without affecting much the results. In meshless methods Gaussian integration can be used, as in FEM, however other approaches to integrate the weak form are used, like the nodal integration. It needs the integration weight on each node given by the Voronoï diagrams;

5. Imposition of the nodal connectivity: In FEM the "elements" predefine the nodal connec-tivity, however in meshless methods, concentric areas or volumes are defined around the

(a) (b) (c)

Figure 4.2: Solid domain a), Regular nodal discretization b), Irregular nodal discretization c), adapted from [36].

point of interest, and the influence domain is constructed by the nodes inside that region. In order to obtain good results it is important to have the same number of nodes inside all the influence-domains. To accomplish that, the size of the influence domain should depend on the nodal density around the interest point;

6. Obtaining the field variables using the interpolation function. Consider, for example, the displacement field u, whose components uI = (u, v, w) at any interest point xI within the

problem domain. The displacement components are interpolated using the displacement of the nodes inside the influence domain of such interest point xI,

u(xI) = n

∑

φ (xI)u(xi) (4.1)

where u(pI) represents the displacement components of each node within the

influence-domain, n value is equal to the number of nodes inside the influence-domain of such interest point pI and φ (pI) is the interpolation function value of the ith node obtained using the n

nodes inside the influence-domain;

7. Establishing the equation system: In meshless methods, the interpolation function applied to the strong or the weak form formulation, formulate the discrete equations. If the Galerkin weak form is used, the discrete equations can be obtained, by applying to the differential equation governing the physical phenomenon, the weighted residual method of Galerkin. The system of equations is arranged in a local nodal matrix and assembled into a global equation system matrix. Then, the appropriate solver is used according to the type of anal-ysis. As an example, for a static problem, the displacement field can be obtained using a linear solution algorithm, such as the Gauss Elimination Method.

4.2.1.1 Influence-Domains

The nodal connectivity in meshless methods, usually, is obtained by the overlap of the influence-domains, which is the case of the Radial Point Interpolation Method (RPIM). The influence do-mains are created by searching enough nodes inside an area or volume, respectively for the 2D and the 3D problem. Their size and shape can be fixed or variable, as shown in figure4.3. It is

4.2 Meshless methods 19

recommended that all the influence domains should have the same number of nodes for better per-formance of the method, that is the reason why flexible size and shape of the influence domain is preferred. The literature recommends between 9 and 16 nodes for 2D problems. For a 3D problem the number should be between 27 and 70 nodes [26,35].

Figure 4.3: Variable influence domain [26].

4.2.1.2 Influence-cells

To determine directly the influence domains, it is possible to use the spatial collocation of the nodes discretizing the problem domain. In this case the influence domains is called influence cell, because it is determined based on the geometric and spatial relations between the Voronoï cells obtained from the Voronoï diagram of the nodal distribution. The Voronoï diagram procedure is shown in figure4.4, and follows the natural neighbour mathematical concept [37,38]. The procedure described is a two-dimensional diagram because it is easy to visualize, however this can be extrapolated to the three-dimensional space. Firstly the potential neighbours of the node n0are determined, then consider, for example, the node n4as potential neighbour. It is possible to

construct a normal vector from n4to n0, and then define a plane normal to that vector, as shown in

figure4.4b. The nodes that are outside the plane, and in the opposite direction of the normal vector, are eliminated as natural neighbours of the node n0. The process is repeated until all the neighbour

nodes are found, as shown in figure 4.4d. Afterwards, parallel planes are created between the neighbour nodes and n0. These new limits of the merged plans form the Voronoï cell, as shown

in figure4.4e . A similar procedure is used to determine the rest of the cells, as shown in figure

Figure 4.4: Voronoï diagram construction, [26].

The Delaunay triangulation is constructed by creating the nodes of the Voronoï cells with common boundaries, as shown in figure4.5.

4.2 Meshless methods 21

In the NNRPIM, the duality of the Delaunay triangles and the Voronoï cells is used to construct a nodal dependent background integration mesh, whilst the Voronoï diagram creates the "influence cells", enforcing the connectivity of the nodes and discretizing the problem domain. The use of influence cells concept respond to the difficulties of the influence domains in high discontinuous boundaries, which normally are not capable of assuring the same number of nodes inside the influence domains, because, instead of a radial distance, the Voronoï Diagram is used to determine the set of n nodes responsible for the interpolation of the interest point XI. The influence cells can

have different levels, e.g. in the figure4.6ait is shown a "first degree influence cell" and in figure

4.6bit is shown a "second degree influence cell".

(a) (b)

Figure 4.6: first degree influence cell a), second degree influence cell b) [26].

4.2.1.3 Numerical Integration

• Gaussian Quadrature Integration

The RPIM uses the Galerkin weak formulation, and the numerical integration process can be the Gauss-Legendre quadrature. This integration process divides the solid domain in a regular grid, where the created grid-cells are filled with integration points according to the Gauss-Legendre quadrature rule. As an application example consider the figure4.7. The ini-tial quadrilateral, shown in figure4.7a, is transformed in an isoparametric square and a 2×2 Gauss-Legendre quadrature is created, as presented in fig4.7b. Then, using isoparametric interpolation functions the Cartesian coordinates of the quadrature points are obtained, as shown is figure4.7c. To achieve the integration weight of the quadrature point, the isopara-metric weight of the quadrature point is multiplied with the inverse of the Jacobian matrix determinant, of the respective grid-cell [26].

Figure 4.7: Initial quadrilateral grid-cell a), Isoparametric square and 2×2 Gauss-Legendre quadrature application b), return to the initial quadrilateral shape c) [26].

The global numerical integration, considering the function F(x) defined in the domain Ω, can be expressed as the following sum:

Z Ω F(x)dΩ = ng

∑

wherewbiis the weigth of the integration point xi.

• Nodal base integration

The nodal base integration is a technique used to numerically integrate the integro-differential equations ruling the studied physical phenomenon. It can be applied to any meshless method, in one-, two- and three-dimensional domains [34,39,40]. The numerical inte-gration technique uses the nodal distribution spatial information, and achieves better results in comparison with the Gauss integration schemes, however, the needed extra computa-tional time does not compensate the increased accuracy of the final result. After the domain discretization, the Voronoï cells are determined, and they can be divided into small areas, which can be quadrilaterals or triangles, according an irregular ou regular nodal distribu-tion, respectively. In figure4.8a it is shown the Voronoï cell VIof the node nI, in an irregular

distribution. The middle points, MIi, between the neighbour nodes are determined, figure

4.8b, and quadrilateral sub-cells, SIi, are created, being the number of sub-cells the number

4.2 Meshless methods 23

Figure 4.8: Voronoï cell a), Middle points, MIi, and respective quadrilaterals b), Individual

quadri-lateral c) [26].

If the nodal distribution is regular, the same applies, however the sub-cells are triangles as shown in figure4.9.

Figure 4.9: Voronoï cell a), Middle points, MIi, and respective triangle b), individual triangle

c) [26].

To establish a simple integration scheme, a single integration point is inserted in the barycen-tre of the sub-cells, figure4.10, being the weight of each integration point the area of the respective sub-cell.

Another integration scheme, more general than the previous one is the Gauss-Legendre Integration Scheme. In this scheme sub cells can also be subdivided again, then using the Gauss-Legendre quadrature, the sub-quadrilaterals can be filled with k ×k integration points, figure4.11. The integration scheme can have different quadratures, in figure4.11is shown a 1 × 1 and 3 × 3 quadrature. The integration weight of each point xI is determined by the

following expression, b wi= wηwξ  A 4  (4.3) where Ais the area of the sub-quadriteral, and wη and wξ are the Gauss-Legendre

quadra-ture weigths for an isoparametric quadriteral cell, as previouly shown in figure4.7.

Figure 4.11: Gauss-Legendre quadrature integration in sub-sub-cells [26].

Other integration scheme is the nodal integration scheme, where just the integration point xI coincident with the barycentre of the Voronoï cell, VI and the weight of each integration

point,wbI, are needed. Unfortunately, the meshless shape functions are not integrated accu-rately with the nodal integration, therefore stabilization techniques are needed [41,42]. The stabilization techniques are going to increase the method computational time, which turns the method more inefficient [26].

Chapter 5

3D printing

5.1

Classification and technologies

Nowadays, 3D printing technology, also known as Additive Manufacturing, is present everywhere. From our home to the factories it is revolutionizing the way things are done. Additive manufac-turing is defined as a general term for all the technologies that are used to create physical objects, by successive addition of material. In 2015, the ISO/ASTM 52900 was created to define terms used in additive manufacturing (AM) technology. According to the standard, the different process categories are shown in the table5.1[43].

Table 5.1: Table with aditive manufacturing processes according to ISO/ASTM 52900 [43]. Process categories Description

Binder jetting In this process a liquid bonding agent is selectively deposited to join powder materials.

Directed energy deposition Process in which an energy source, like a laser, electron beam, or plasma arc, is used to melt the material being deposited.

Material extrusion Process in which the material is selectively dispensed through a nozzle or orifice.

Material jetting In this process droplets of feedstock material, like photopolymer resin and wax, are selectively deposited

Power bed fusion Process in which thermal energy selectively fuses regions of a power bed.

Sheet lamination In this process sheets of material are bonded to form a part. Vat photopolymerization Process in wich liquid photopolymer in a vat is selectively cured

by light-activated polymerization.

5.2

Material extrusion

Material extrusion is a process in which the material is selectively dispensed through a nozzle or orifice [43]. There are a variety of materials that can be used in this process, like ceramics,

composites, metal-filled clays, concrete, food, living cells suspended in hydrogel, melted thermo-plastics, etc. The material that goes through the nozzle, in a semi-liquid state, is deposited in the build plate according to the predefined pattern. Finished the layer, the extrusion head moves up, or the build plate moves down, according the machine type. Afterwards, another layer is extruded, bonding to the previous one (figure8.2). This process, most of the times, is done layer by layer, resulting in machines working as 2.5 axis machines. In fact, any material that fulfils the following requirements can be used in material extrusion process:

• The material can be transformed in a past state, in order to allow their extrusion,

• The material can bond or solidifies to the other material already dispensed, or build plate, The solidification can be physical, cooling, or chemical, e.j. photopolimerization [44]. This pro-cess, compared to other additive manufacturing processes, has low initial and running costs, it is an easily understandable printing technique and normally the equipment size is small. There are, however, some disadvantages like visible layers lines, some parts require supports and it is a slow process in the majority of times.

(a) (b) (c)

Figure 5.1: Material extrusion process.

5.3

Fused Filament Fabrication (FFF)

The FFF is the most common material extrusion technology process. The term FFF appeared as an alternative to the Fused Deposition Modelling (FDM), a term trademarked by Stratasys but with the same principle [45]. This process starts with the creation of a CAD model, which is converted to STL or IGES format files. STL is an abbreviation of stereolithography and IGES of Initial Graphics Exchange Specification. These files are imported into a software who slices the part in the different layers, produce the printing paths and create support and fixations to the build plate if necessary. The paths are sequences of g-code that the machine will read. There is a huge amount of printing parameters that can be defined in order to print the parts, however, according to the material type, there are default settings for the desired quality. Only after some months of experience the user can understand well all the consequences of all the parameters and

5.3 Fused Filament Fabrication (FFF) 27

change them to achieve defined requirements. In figure5.2is shown a scheme explaining the FFF extruding process. It is important to understand that this process will not produce parts with the

Figure 5.2: Fused Filament Fabrication [46].

same performance of an injected part in the same material. Even with 100 % infill there are gaps all over the part, and because the layered process resistance is different in different directions, the process imposes anisotropy.

5.3.1 Materials

The material, in filament form, usually has a diameter of 1.75mm or 2.85mm. The materials used in FFF are mostly thermoplastics because they are easy to use and produce good results. The most common thermoplastics utilized in FFF are: Polylactic Acid (PLA), Nylon, Acrylonitrile Butadiene Styrene (ABS) and Polyethylene Terephthalate Glycol (PETG) [47,48]:

5.3.1.1 PLA

PLA is a material derived from renewable resources as sugar-cane or cornstarch, also called "green plastic" [48]. Normally, Its melting temperature is within the range 180-230◦C and has an average

price of 26.5e [44]. Other great feature, is the fact it can be used with food and in the medical industry [48]. It is the mostly widely material used in FFF desktop printing.

5.3.1.2 Nylon

Nylon is a synthetic thermoplastic linear polyamide and well-known 3D printing filament because of its flexibility, durability, low friction and corrosion resistance [47,48]. It should be extruded at a temperature around 245◦Cand its price is the double of PLA.

5.3.1.3 ABS

It is one of the most accessible and cheap materials for FFF, has a long lifespan and is mechan-ically strong, however require a heated build plate (100◦C) when printing. Due to their higher melting point (210-260◦C), when compared with other materials, such as PLA, it has a tendency to experience warping. Another problem is the releasing of toxic fumes while printing [48]. In comparison with PLA, produced parts are more robust, less brittle, and more resistant to higher temperatures [44].

5.3.1.4 PETG

This material is very durable, impact resistant and recyclable, can be sterilised and has excellent layer adhesion. It has a combined functionality of ABS (Temperature resistant, durable) and PLA which is easy to print. It can also be used with food products and medical instruments [47,48]. Their melting temperature is around 240◦C.

5.4

FFF machines

FFF machines, or 3D printers, are divided in two very different types, each one developed accord-ing their application [49].

• Professional: These printers, normally have big dimensions, have a solid structure protect-ing their vital components, have a close source software and patented operative systems. Are also more reliable and of easy use (figure5.3)

• Non professional: Low cost printer, an example is shown in figure 5.4, have open source software and hardware, normally do not have a protection case and are small machines. They are not so reliable and not so accurate, however they are cheaper. It is possible to buy a reasonable printer for less than 200e. These machines are ideal to print thermoplastics, since they solidify at ambient temperature. Most of them are capable of printing PLA and PETG at their open configuration, if placed under an enclosure, they can also print ABS. The filament size is standard of 1.75 mm [44].