approaches using meshless methods
Thesis submitted to Faculdade de Engenharia da Universidade do Porto as a requirement to obtain the degree of MSc in Mechanical
Engineering
Author:
Vítor Manuel Mota da Cunha
Under the supervision of:
Professor Jorge Américo Oliveira Pinto Belinha
Professor Lúcia Maria de Jesus Simas Dinis
Professor Renato Manuel Natal Jorge
Ao Professor Jorge Belinha pela sua orientação e ajuda durante o período de elabo-ração da tese e por toda a disponibilidade e paciência que sempre revelou e me permitiu ultrapassar as dificuldades encontradas ao longo deste trabalho.
Ao Luís Ramalho que se mostrou sempre disponível para escutar as minhas dúvidas e foi fundamental para resolver algumas dúvidas existentes neste trabalho.
Aos meus pais e ao meu irmão por todo o apoio e confiança que me deram nos momentos mais difíceis ao longo destes cinco anos.
Um agradecimento especial ao João Sousa pela ajuda na melhoria da escrita deste trabalho, nomeadamente no domínio da língua inglesa.
A todos os meus amigos, tanto os de Lousada como os que conheci ao longo destes anos na FEUP, pelos momentos que passamos ao longo destes cinco anos e por toda a sua paciência em ouvir as minhas queixas e dúvidas relativamente à minha capacidade e meu futuro.
The author truly acknowledges the work conditions provided by the Applied Me-chanics Division (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 Technol-ogy (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 Sus-tainable 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 collabora-tors of "Computational 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.
A previsão da propagação de fenda num modelo é um importante problema de engenharia. Neste trabalho são utilizados três métodos numéricos: o finite element method (FEM), o radial point interpolation method (RPIM) e o natural neighbour radial point interpolation method (NNRPIM).
Alguns defeitos podem ser associados ao método dos elementos finitos (MEF), tais como a obtenção de resultados com bastante erro quando são utilizadas geometrias complexas. Portanto, os métodos sem malha são apresentados como uma solução para esse problema. Um dos objetivos deste trabalho é concordar com a hipótese dos métodos sem malha serem uma solução viável para o método dos elementos finitos.
Este trabalho tem como base um código anteriormente desenvolvido no Compu-tational Mechanics Research Laboratory (CMECH). Esse algoritmo original extende iterativamente uma fenda existente até que um comprimento máximo de fenda (previ-amente estabelecido) seja alcançado e o caminho da fenda é calculado usando o critério de máxima tensão tangencial. Em relação a este algoritmo, neste trabalho várias al-terações são feitas com o objetivo de melhorar os resultados obtidos com o algoritmo original. Tais mudanças incluem a modificação do comprimento e forma do campo de tensão na ponta da fenda, modificação da função de peso responsável por suavizar o campo de tensão na vizinhança da ponta da fenda e uma introdução de dano anterior à propagação da ruptura da fenda existente.
De modo a averiguar a validade do algoritmo original, as abordagens implemen-tadas e o próprio software, foram realizados diversos testes de exemplos de referência da mecânica da fratura com diferentes modos de fratura e um exemplo de convergência de uma placa quadrada com um furo central foi realizado com o objetivo de garantir a qualidade do software usado, o FEMAS, desenvolvido no CMECH (cmech.webs.com).
Palavras-chave: Mecânica da Fratura; Propagação de fenda; Métodos Numéricos;
The prediction of the crack propagation in a model is an important engineering problem. In this work three numerical methods are used: the finite element method (FEM), the radial point interpolation method (RPIM) and the natural neighbour radial point interpolation method (NNRPIM).
Some flaws can be associated with the finite element method (FEM), such as ob-taining high errors when complex geometries are used. Therefore, meshless methods are presented as a solution for this problem. One of the goals in this work is to comply with the assumption of meshless methods as a viable solution for the finite element method.
This work uses a previously developed code in Computational Mechanics Research Laboratory (CMECH). That original algorithm extends an existing crack iteratively until a maximum crack length (previously established) is achieved and the crack path is calculated using the maximum tangential stress criterion. Regarding this algorithm, in this work several changes are made with the goal to improve the results obtained with the original one. Such changes include the modification of the crack tip stress field’s length and shape, modification of the weight fucntion responsible to smoothen the stress field in the vicinity of the crack tip and an introduction of damage previous to the rupture propagation of the existing crack.
In order to ascertain the validity of the original algorithm, the tested approaches, and the software itself, several fracture mechanics benchmark tests were performed regarding different fracture modes and a convergence example regarding a square plate with a central hole was performed with the goal to assure the quality of the used software, FEMAS developed in CMECH (cmech.webs.com).
Keywords: Fracture Mechanics; Crack propagation; Numerical Methods; FEM;
1 Introduction 1
1.1 Meshless Method . . . 1
1.2 Fracture Mechanics and Numerical Methods . . . 2
1.3 Purpose of this work . . . 3
1.4 Thesis Outline . . . 3
2 Solid Mechanics 5 2.1 Fundamentals . . . 5
2.1.1 Stresses and Strains . . . 6
2.1.2 Principal Stresses and Strains . . . 6
2.1.3 Constitutive Equations . . . 8
2.2 Galerkin Weak Formulation . . . 9
2.2.1 Discrete System of Equations . . . 10
2.2.2 Stiffness Matrix . . . 10
2.2.3 Force Vector . . . 11
2.3 Fracture Mechanics . . . 12
2.3.1 Crack Propagation Criteria . . . 16
3 Numerical Methods 19 3.1 Finite Element Method . . . 19
3.1.1 Mesh Generation . . . 19
3.1.2 Integration Points . . . 19
3.1.3 Shape Function Construction . . . 20
3.1.4 Stiffness Matrix . . . 22
3.1.6 Displacements, Strains and Stresses . . . 23
3.2 Meshless Methods . . . 23
3.2.1 Nodal Generation and Connectivity . . . 23
3.2.2 Influence-Domains . . . 24
3.2.3 Influence-Cells and Natural Neighbours . . . 26
3.2.4 Shape Functions . . . 29
4 Computational Fracture Mechanics 35 4.1 Finite Element Method . . . 35
4.2 Meshless Methods . . . 36
4.2.1 Visibility Method . . . 37
4.2.2 Diffraction Method . . . 37
4.2.3 Transparency method . . . 39
4.2.4 Shape Function Enrichment . . . 40
5 Crack Opening Path Approaches 41 5.1 Parametric study with cubic weight function . . . 41
5.2 Exclusion of the nodes near the crack tip . . . 43
5.3 Damage Zone Consideration . . . 44
6 Numerical Examples 47 6.1 Convergence Example . . . 47
6.1.1 Convergence test . . . 49
6.2 Fracture Mechanics Examples . . . 60
6.2.1 Mode II Shear Loading . . . 61
6.2.2 Three Point Bending of a Beam . . . 69
7 Conclusions and Further Improvements 79 7.1 Conclusions . . . 79
2.1 Continuous solid subjected to volume forces and external forces. . . 5
2.2 Fundamental modes of deformation in fracture mechanics. . . 12
2.3 Scheme of the infinite cracked plate problem. . . 13
2.4 Radial Coordinate System in the crack tip.s . . . 16
3.1 Quadrilateral element with natural coordinate system. . . 20
3.2 Example of a Voronoï diagram. . . 24
3.3 (a) Fixed rectangular shaped influence-domain; (b) Fixed circular shaped influence-domain; (c) Flexible circular shaped influence-domain. . . 25
3.4 Voronoï cell construction: (a) initial set of potential neighbour nodes of n0; (b) First trial plane; (c) Final trial cell containing only the natural neighbours of node n0; (d) Voronoï cell V0 . . . 27
3.5 Construction of the Delaunay triangles: (a) initial Voronoï diagram; (b) respective Delaunay triangulation. . . 28
3.6 (a) Voronoï cell and its respective PIi intersection points. (b) Middle points MIiand the respective generated triangles. (c) Generated triangle. 28 3.7 (a) Voronoï cell and its respective PIi intersection points. (b) Middle points MIi and the respective generated quadrilaterals. (c) Generated quadrilateral. . . 28
3.8 Integration Points’ generation: (a) Triangular sub-cell; (b) Quadrilat-eral sub-cell. . . 29
4.1 Influence domains near a discontinuity using the visibility method. The shaded areas are excluded from the influence domain. . . 37
4.2 Influence domain of node I, when considering the diffraction method. . 38
4.3 Scheme of the calculation of s(x) using the diffraction method. . . . 39
4.4 Scheme of the calculation of s(x) using the transparency method. . . . 40 5.1 Scheme of node selection in the crack tip vicinity on the original algorithm. 42
5.2 Scheme of node selection after applying the Parametric Study with cubic
weight function approach. . . 42
5.3 Representation of the applied weight function. . . 43
5.4 Scheme of node selection after applying the Exclusion of the nodes near the crack tip approach. . . 44
5.5 Scheme of the damage zone representation in Linear Elastic Fracture Mechanics. . . 45
5.6 Scheme of node selection after applying the Plastic Zone Consideration approach. . . 45
6.1 Scheme of the used convergence example. . . 47
6.2 Used meshes in the 2D convergence example: (a) Sparsest Mesh (b) Second most sparse mesh (c) Most refined mesh. . . 48
6.3 Used meshes in the 3D convergence example: (a) Sparsest Mesh (b) Second most sparse mesh (c) Most refined mesh. . . 48
6.4 Results obtained for the displacements in the 2D model. . . 50
6.5 Results obtained for stresses in the 2D example: (a) Normal component of stress in point E,σxxE (b) Normal component of stress in point D,σyyD. 51 6.6 (a) Normal stress σxx obtained along the axis x = 0 (b) Normal stress σyy obtained along the axis y = 0. . . . 51
6.7 Obtained colourmaps using the sparsest mesh regarding Von Mises’ Equivalent Stress [P a]: (a) FEM; (b) RPIM; (c) NNRPIM. . . . 57
6.8 Obtained colourmaps using the most refined mesh regarding Von Mises’ Equivalent Stress [P a]: (a) FEM; (b) RPIM; (c) NNRPIM. . . . 57
6.9 Results obtained for the displacements in the 3D model. . . 58
6.10 Scheme of Mode II Shear Loading example [m]. . . . 61
6.11 Mode II Shear Loading example’s used mesh. . . 62
6.12 Crack path prediction for Mode II shear loading example. . . 63
6.13 Mode II shear loading final mesh a) FEM, b)RPIM, c) NNRPIM. . . . 63
6.14 (a) Crack path prediction applying the parametric study with a cubic weight function - FEM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 64
6.15 (a) Crack path prediction applying the parametric study with a cubic weight function - RPIM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 65
6.16 (a) Crack path prediction applying the parametric study with a cubic weight function - NNRPIM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 65
6.17 (a) Crack path prediction applying the exclusion of nodes near crack tip - FEM; (b) Distance in each iteration between FEM original solution
and the obtained solution. . . 66
6.18 (a) Crack path prediction applying the exclusion of nodes near crack tip - RPIM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 66
6.19 (a) Crack path prediction applying the exclusion of nodes near crack tip - RPIM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 67
6.20 (a) Crack path prediction applying the damaged zone consideration -FEM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 68
6.21 (a) Crack path prediction applying the damaged zone consideration -RPIM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 68
6.22 (a) Crack path prediction applying the damaged zone consideration -NNRPIM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 69
6.23 Scheme of three point bending of a beam example [mm]. . . . 70
6.24 Three Point Bending of a Beam’s used mesh. . . 70
6.25 Crack path prediction for three point bending example. . . 71
6.26 Three Point Bending final mesh a) FEM, b)RPIM, c) NNRPIM. . . 72
6.27 (a) Crack path prediction applying the parametric study with a cubic weight function - FEM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 73
6.28 (a) Crack path prediction applying the parametric study with a cubic weight function - RPIM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 73
6.29 (a) Crack path prediction applying the parametric study with a cubic weight function - NNRPIM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 74
6.30 (a) Crack path prediction applying the exclusion of nodes near crack tip - FEM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 75
6.31 (a) Crack path prediction applying the exclusion of nodes near crack tip - RPIM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 75
6.32 (a) Crack path prediction applying the exclusion of nodes near crack tip - NNRPIM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 76 6.33 (a) Crack path prediction applying the damaged zone consideration
-FEM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 77 6.34 (a) Crack path prediction applying the damaged zone consideration
-RPIM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 77 6.35 (a) Crack path prediction applying the damaged zone consideration
-NNRPIM; (b) Distance in each iteration between FEM original solution and the obtained solution. . . 78
3.1 Gaussian quadrature coordinates and weights. . . 20 6.1 Parameters used in meshless methods. . . 49 6.2 Results for the vertical displacement in point A, vA[m], for the 2D model. 52
6.3 Results for the vertical displacement in point B, vB [m], for the 2D model. 52
6.4 Results for the horizontal displacement in point B, uB [m], for the 2D
model. . . 53 6.5 Results for the horizontal displacement in point C, uC [m], for the 2D
model. . . 53 6.6 Results obtained for the normal component of stress in point E, σxxE [M P a],
for the 2D model. . . 53 6.7 Results obtained for the normal component of stress in point D, σyyD [M P a],
for the 2D model. . . 54 6.8 Results obtained from the difference between FEMAS’ results and
AN-SYS’ results for the vertical displacement in point A, vA. . . 54
6.9 Results obtained from the difference between FEMAS’ results and AN-SYS’ results for the vertical displacement in point B, vB. . . 55
6.10 Results obtained from the difference between FEMAS’ results and AN-SYS’ results for the horizontal displacement in point B, uB. . . 55
6.11 Results obtained from the difference between FEMAS’ results and AN-SYS’ results for the horizontal displacement in point C, uC. . . 55
6.12 Results obtained from the difference between FEMAS’ results and AN-SYS’ results for the horizontal component of stress in point E, σxxE. . . 56
6.13 Results obtained from the difference between FEMAS’ results and AN-SYS’ results for the vertical component of stress in point D, σyyD. . . . 56
6.14 Thickness of the example compared with the number of nodes.s . . . . 58 6.15 Results for the vertical displacement in point A, vA[m], for the 3D model. 59
6.17 Results for the horizontal displacement in point B, uB [m], for the 3D
model. . . 59 6.18 Results for the horizontal displacement in point C, uC [m], for the 3D
Introduction
In this work, Meshless Methods and the Finite Element Method (FEM) are used in order to solve fracture mechanics examples and to predict the behaviour of crack growth. Regarding the meshless methods, this work considered two distinct radial point interpolation techniques: the Radial Point Interpolation Method (RPIM) and the Natural Neighbour Radial Point Interpolation Method (NNRPIM).
1.1 Meshless Method
In recent years, meshless methods came to attention, particularly in the engineering community as a viable alternative to the Finite Element Method (FEM) [1]. Finite Element Method has been used to a great extent in many fields with both academic and industrial applications. However it possesses some limitations. As a result of its mesh-based interpolation, when distorted meshes or complex geometries are used, the results obtained can present high errors [2].
Meshless methods were created in order to overcome most difficulties associated with the use of a mesh to construct the approximation required to solve the problem [3]. One of the oldest meshless methods is Smoothed Particle Hydrodynamics (SPH). This method was originally created to solve astrophysics problems with particular emphasys in fluid mechanics and it was based on a strong form instead of a regular weak form [4]. The Element-free Galerkin Method (EFGM) [5] is one of the most popular meshless methods and it is based in a weak form. This method consists of an evolution made by Belytschko of the Diffuse Element Method (DEM), which was the first meshless method to use the Moving Least Squares (MLS) in the construction of the approximation functions [6].
Several other meshless methods were developed, such as the Meshless Local Petrov-Galerkin method (MLPG) [7], which is based on a local weak form instead of a global weak form, the Finite Point Method (FPM) [8] and the Method of Finite Spheres (FSM) [9].
Despite being applied in computational mechanics with relative success, there were some problems that were not solved, such as the difficulty in imposition of essential
and natural boundary conditions as a result of the lack of the delta Kronecker property through the use of approximation functions [2]. In order to address this problem, a new method was developed, the Point Interpolation Method (PIM). This method constructs the shape functions using polynomial functions. The obtained shape functions possess the delta Kronecker property [10]. PIM only used polynomials as its interpolation functions, which can be obtained very efficiently and the essential boundary conditions can be easily imposed, as done in FEM [11].
Proposed by Liu [12], an evolution of PIM was created. The new method evolved the Point Interpolation Method by adding a radial basis functions.It was used to solve partial differential equations. Thus, the Radial Point Interpolation Method (RPIM) was created. This method combines the Galerkin weak form with radial basis functions, creating a Radial PIM. Its main differences from PIM are the fact that the interpolation is carried within an influence domain instead of a global domain, and its system matrix is sparse and banded, making it suitable for large scale problems [12].
Using the basic principles of RPIM, a new meshless method was created, the Nat-ural Neighbour Radial Point Interpolation Method (NNRPIM). This method has the advantage of not relying on a background mesh for interpolation purposes. If a mesh-less method depends on a secondary background mesh, it is not considered a truly mesh free method [2]. To construct the interpolation functions, this method relies on the mathematical concepts of Voronoï diagrams and Delauney tessellation. Voronoï cells are a set of influence cells constructed from a set of nodes and the Delaunay triangles consist in the dual of Voronoï cells and are used to create a background inte-gration mesh to use in the numerical inteinte-gration of NNRPIM. In NNRPIM, a random distribution of nodes is allowed without hindering the convergence of the method [1].
1.2 Fracture Mechanics and Numerical Methods
Numerical methods have been applied in fracture mechanics problems for a long time now, the most common problem being the prediction of the crack path of an example.
FEM has been used since the 1970s [13] to predict the crack path, as it was the most trustworthy method available. As it was mentioned before, FEM still has some flaws, such as a particular difficulty with problems containing complex geometries, where it is difficult to align element edges with the edges of the domain without de-forming the elements. An example of those geometries are examples containing cracks. One strategy to deal with these flaws with mesh-based methods would be remeshing. However, remeshing is costly and still is difficult in three dimensions. An alternative to remeshing in mesh-based method is the Extended Finite Element Method (X-FEM), which enriches the approximation space in order to capture discontinuities [3]. XFEM enriches the normal FEM shape functions with a continuous encrichment function or a discontinuous function with the goal to approximate the solution near an interface. Notice that this functions only exist in the element’s nodes that intersect the interface. This allows XFEM to accommodate elements that do not conform in the interface [14].
Several authors have already appplied XFEM in order to predict the crack path of a problem with accurate results, namely Moës et al. [15], Fries [16], and Geniaut et al. [17], being that the last one will be used in this thesis as a reference solution for the Three Point Bending example.
Other methods have been successfully applied in fracture mechanics, as for ex-ample, the Boundary Element Method (BEM) and Boundary Element Free Method (BEFM), applied by [18] and [19].
In this work, all of fracture mechanics analysis will be related with the crack path prediction of some examples. The prediction of a crack path in a fracture mechanics problem is an iterative process in which the mesh has to be updated for each iteration. In order to be able to define the crack path propagation direction for each iteration, the stress and displacement field has to be obtained. In fracture mechanics, there are many criterions that are able to define the crack propagation direction.
In this work, the maximum circumferential stress criterion is the fracture mechanics criterion used. Nevertheless, numerical enhancements are tested.
1.3 Purpose of this work
The main purpose of this work is to apply different algorithms and approaches to fracture mechanics analysis of benchmark examples. Initially, the original algorithm, already developed, will be tested to ensure that algorithm is robust and can produce accurate results while using different numerical methods and different examples.
After that, different approaches will be implemented, mainly regarding different ways of stress calculations with the goal of obtaining improved crack path results. The obtained solutions are always compared with theoretical solutions and the solutions obtained with the original algorithm, in order to evaluate its improvement.
All approaches developed in this work were applied in numerical simulations, while the algorithm changes were programed in the commercial software Matlab, using the academic software FEMAS (cmech.webs.com).
1.4 Thesis Outline
This thesis is divided into seven chapters.
In the first chapter, the thesis is introduced and a brief state of the art is presented, with a brief overview of the goals of this thesis. An introduction of numerical methods and fracture mechanics is presented.
The second chapter presents solid mechanics fundamentals, the discretization of solid mechanics with a presentation of fundamental concepts of fracture mechanics, and a brief presentation of propagation criterions found in literature.
Chapter three consists in the presentation of all numerical methods applied in this work- FEM, RPIM, and NNRPIM, and its formulations.
Chapter four presents an overview of the already existing algorithms and solutions used to numerically predict the crack propagation.
In the fifth chapter, the studied and applied approaches are presented.
Chapter six presents the studied approaches applied in several benchmark tests and its results.
In the seventh and final chapter, the conclusions of this work are presented and discussed, along with the presentation of future improvements to this work.
Solid Mechanics
In this chapter, the mechanical fundamentals behind the numerical applications made in this work are presented. Firstly, the fundamentals of Solid Mechanics are presented. Following it, the Galerkin Weak Formulation and its equation system are presented. Next, a presentation of Fractures Mechanics is made, where its fundamen-tals and crack propagation theories are presented.
2.1 Fundamentals
Solid Mechanics and Structural Mechanics deal with, for a given solid and its boundary conditions, the relationship between the obtained stress and strain and, the relationship between strain and displacements [1].
In this work, it will be assumed that the analysed solids on which exterior forces will be applied, will operate within the linear-elastic field. Elastic behaviour means that the material returns to its initial shape after the applied loads are removed, which means that plastic deformation is not assumed. Linear refers to the relationship between stresses and strains that are geometrically linear [20].
0 z x y G Γu Ω ¯t Γt b
In figure 2.1, it is possible to observe a solid subjected to volume forces and external forces. Where Ω represents its domain, subject to a body force b and a boundary Γ, with a force ¯t applied in Γt, and its movement constrained in Γu. The movement
con-straint is considered a natural boundary condition and the applied force is considered an essential boundary condition [1].
2.1.1 Stresses and Strains
When a solid is subjected to exterior forces, it deforms and internal forces are created, which caracterize, in each point of the solid, the stress state of that point. The intensity of these forces is usually measured by a number that reflects the value of that force through each unit of area [20].
In order to define the stress on a given point, the stress tensor is used:
h σi= σxx σxy σxz σyx σyy σyz σzx σzy σzz (2.1.1)
In 2.1.1, the Cauchy stress tensor is defined. It is a symmetric tensor because σij = σji and it represents the stress state of a given point [1].
The stress state can also be defined using the Voigt notation. The Voigt notation expresses the stress state of a point in a column vector, which can be easier to apply in computational mechanics [1]:
n
σo=nσxx σyy σzz σxy σyz σzx
oT
(2.1.2)
The strain, in the matrix form, is defined as:
h εi= εxx εxy εxz εyx εyy εyz εzx εzy εzz (2.1.3)
Since it possesses the same properties as the stress tensor, it can also be written using the Voigt notation:
n
εo=nεxx εyy εzz εxy εyz εzx
oT
(2.1.4)
2.1.2 Principal Stresses and Strains
For any given stress matrix, it is possible to find a direction, in which the resulting stress vector t(n) is parallel with the normal vector n. When this happens, the shear
stress component of the matrix is zero, τ = 0 [20]. t(n) = n · σij = h n1 n2 n3 i σxx σxy σxz σyx σyy σyz σzx σzy σzz (2.1.5)
Whatever the stress state is at a given point, there are always three possible planes mutually orthogonal between each other, and those planes have associated between them three distinct principal stresses [20].
Stress tensor is a physical quantity, independent of the chosen coordinate system, therefore, there are invariants associated with it that are also independent of the chosen coordinate system. Since it is a second order tensor, it contains eigenvalues, in which it is possible to determine the principal stresses. If equation 2.1.5 is applied, knowing that t(n)i = σijnj and ni = δijnj, where δij refers to the Kronecker delta, an
homegeneous system with three linear equations for three unknown nj is obtained. To
obtain the unknown nj, the obtained matrix determinant needs to be equal to zero,
therefore [1]: t(n)i = λni → σijnj = λδijnj → (σij− λδij)nj = 0 (2.1.6) In matrix formulation: σij − λδij = σxx− λ σxy σxz σyx σyy − λ σyz σzx σzy σzz− λ = 0 (2.1.7)
Being λ a constant of proportionality, representing the magnitude of the principal stresses, the solution of equation 2.1.7 leads to a cubic equation, represented below:
σij − λδij = −λ 3+ I 1λ2− I2λ + I3 = 0 (2.1.8) With equation 2.1.8, the invariants I1,I2, and I3 of stress matrix are defined by:
I1 = σkk (2.1.9)
I2 = 1
2(σiiσjj− σijσji) (2.1.10)
I3 = det(σij) (2.1.11)
The three roots of equation 2.1.8,λ1 = σ1, λ2 = σ2, and λ3 = σ3, are the principal stresses and are unique. The stress invariants have always the same value regardless the orientation of the coordinate system. In order to obtain the principal directions, equation 2.1.6 must be solved, replacing λ by each principal stress, and, in so doing, obtaining the direction corresponding to that principal stress [1].
In order to obtain the principal strains, the same process that was shown before should be applied, but the stress tensor must be substituted by the strain tensor.
2.1.3 Constitutive Equations
In order to obtain stresses through calculation of strains, or otherwise, a relation-ship between both must be assumed. To obtain the relationrelation-ship, the constitutive matrix, c, is used. The relationship between stresses and strains is given by Hooke’s Law, represented in equations below:
σ = cε (2.1.12)
ε = c−1σ (2.1.13)
Being that s = c−1, and that it is known as compliance matrix, for a three-dimensional case with an anisotropic material the matrix is defined by [1]:
s = 1 Exx − νyx Eyy νzx Ezz 0 0 0 −νxy Exx 1 Eyy − νzy Ezz 0 0 0 −νxz Exx − νyz Eyy 1 Ezz 0 0 0 0 0 0 G1 xy 0 0 0 0 0 0 G1 yz 0 0 0 0 0 0 G1 zx (2.1.14)
Where Eii represents the Young’s Modulus of the material in the i direction, νij
is the Poisson’s ratio of the material, which represents the ratio of deformation in direction j when a force is applied in direction i, and Gij represents the shear modulus
and it is defined by:
Gij =
Eii
2(1 + vij)
(2.1.15)
For a 2D plane stress case, which is applicable in examples with small thicknesses, and it will applied in this work. So, it assumes that all stresses in z direction are null, so σzz = σzx= σzy = 0, which leads to the following compliance matrix:
s = 1 Exx − νyx Eyy 0 −νxy Exx 1 Eyy 0 0 0 G1 xy (2.1.16)
Another possible simplification is the assumption of a 2D plane strain, in which the strains in z direction are considered null, so εzz = εzx = εzy = 0 , which results in
the matrix represented below:
s = 1 Exx − νzxνxz Exx − νyx Eyy − νzxνyz Eyy 0 −νxy Exx − νzyνxz Exx 1 Eyy − νzyνyz Eyy 0 0 0 G1 xy (2.1.17)
2.2 Galerkin Weak Formulation
Strong formulation consists in the partial differential system equations governing a studied problem. Solving these equations provides the exact solution to that problem. On the other hand, the weak form requires a weaker consistency on the used interpo-lation functions. Ideally, the exact solution would be obtained from the strong form, however, in complex engineering problems, it is complicated to obtain the solution from the strong form [1].
Weak formulations produce an algebraic system of equations and give a discretized system of equations, leading to more accurate results. Therefore, the weak form is the preferred way to obtain a solution.
Considering a solid as represented in figure 2.1, subjected to the same boundary conditions as represented before, the Galerkin weak formulation can be applied. The Galerkin weak formulation is a variational principle that is based on an energy prin-ciple. There are many forms to obtain the system of equations such as the Virtual Work Principle and Hamilton’s Principle [21]. In this work, the system of equations will be obtained through the minimization of the Lagrangian functional L [1]:
L = T − U + Wf (2.2.1)
Where T represents the kinetic energy, U represents the strain energy and Wf
represents the work produced by the external forces. Kinetic energy can be defined by [1]: T = 1 2 Z Ω ρ ˙uTu dΩ˙ (2.2.2)
Where ˙u is the displacement first derivative with respect to time, representing the
velocity and ρ is the mass density of the solid. For elastic materials, as this case, the strain energy is given by:
U = 1 2
Z
Ω
εTσ dΩ (2.2.3)
Where ε is the strain vector and σ is the stress vector. Finally, the work produced by external forces is given by:
Wf = Z Ω uTb dΩ + Z Γt uT¯t dΓ (2.2.4)
Where, as referred before, u represents the displacement, b the body forces and, Γt represents the boundary where the external forces ¯t are applied [1]. Since only
static problems are studied in this work, after several mathematical manipulations represented in [1], the weak form of Galerkin for a static problem is:
− Z Ω δεTσ dΩ + Z Ω δuTb dΩ + Z Γt δuT¯t dΓ = 0 (2.2.5)
2.2.1 Discrete System of Equations
For FEM and the meshless methods analysed in this work (RPIM and NNR-PIM), the discrete equations are obtained from the virtual work principle by using the method’s shape functions as test functions. In these methods, the solid domain has to be discretized into nodes. In FEM, the connection between nodes is imposed by the nodes belonging to the same element and the interaction between common nodes belonging to adjacent elements [21]. In meshless methods, the creation of an "influence domain" imposes the nodes connectivity between neighbouring nodes and the overlap-ping of those domains [1]. The test function for FEM and meshless methods is given by [1]: u(xI) = n X i=1 φi(xI)ui (2.2.6)
where φi(xI) represents the shape function for each method and ui represents the
nodal displacements of the n nodes belonging to the element or influence domain of the interest node xI. The shape functions for the methods that will be analysed in
this work have to satisfy the condition presented by [1]:
φi(xj) = δij (2.2.7)
being δij the Kronecker delta, where δij = 1 if i = j and δij = 0 if i 6= j. Notice, as
mentioned before, that not all meshless methods’ shape functions possess this property.
2.2.2 Stiffness Matrix
FEM and Meshless formulations are established in a weak form of the differential equation obtained for the static case. This means that equation 2.2.5 has to be used and the minimization of the Lagrangian functional has to be made [1]:
L = Z Ω dεTσ dΩ − Z Ω duTb dΩ − Z Γt duT¯t dΓ = 0 (2.2.8)
The virtual deformation, dε, is defined by:
dε = Bdu (2.2.9)
Where B represents the deformation matrix. Thus, the first term of equation 2.2.8 can be expressed as
L1 =
Z
Ω
duTBTσ dΩ (2.2.10)
For a two dimensional problem in a linear case, the deformation matrix B can be written as, B = ∂φ1 ∂x 0 · · · ∂φn ∂x 0 0 ∂φ1 ∂y · · · 0 ∂φn ∂y ∂φ1 ∂y ∂φ1 ∂x · · · ∂φn ∂y ∂φn ∂x (2.2.11)
Where n represents the number of nodes of an element in the case of FEM or an influence domain in the case of a meshless method [1].
The stiffness matrix K can be obtained by considering the variation of virtual work of equation 2.2.10 in order to the displacement du [1]:
dL1 = d " Z Ω BTσ dΩ # (2.2.12)
Which can be developed as, dL1 = Z Ω dBTσ dΩ + Z Ω BTdσ dΩ (2.2.13)
Considering only small deformations, which make dB = 0, stiffness matrix is ob-tained: dL1 = Z Ω BTdσ dΩ = K (2.2.14)
2.2.3 Force Vector
There are two other integrals in equation 2.2.8 which correspond to the body forces and external forces, and their virtual can be expressed by:
dL2 = d " Z Ω b · duT dΩ # = fb (2.2.15) and dL3 = d " Z Γ ¯t · duT dΓ # = ft¯ (2.2.16)
Considering that the total force vector, f , is defined as:
f = fb+ f¯t (2.2.17)
The total force vector f can be written in matrix form as [1]:
f = Z Ω HTb dΩ + Z Γ HT¯t dΓ (2.2.18)
Where, H is the shape function matrix and for a 2D problem is defined by:
H = " φ1 0 · · · φn 0 0 φ1 · · · 0 φn # (2.2.19)
2.3 Fracture Mechanics
The field that deals with fracture and failure processes in engineering is called Fracture Mechanics. In this field, it is assumed that every component always possesses flaws due to possible errors in manufacturing. This defects can appear in the course of service loading of a given component. Fracture Mechanics can be divided into many subfields, depending on the deformation behaviour of the material, the type of loading, and the crack behaviour [22].
In this work, only Linear Elastic Fracture Mechanics will be analysed, using brit-tle materials. One of the first contributions for the study of Fracture Mechanics was Griffith, that in the 1920’s was studying the reason why the mechanical resistance of any material was lower than the theoretical one. Griffith realized that the lower resis-tance of materials was due to the presence of small flaws in materials and, therefore, deducted the formulas for the energy release rate when an element of the crack tip fractures [23].
Fracture Mechanics can be divided into crack opening modes. There are three modes, mode I, mode II, and mode III, dependable on how the loading is applied to the body with a crack. All three modes are represented in figure 2.2 and can be defined as [22]:
• Mode I: Opening Mode : Crack opening is perpendicular to crack plane, usually caused by tensile loading.
• Mode II: In-plane sliding mode: Crack faces are displaced on their plane, normal to the crack front. This mode correlates to a transversal shear loading.
• Mode III: Out-of-plane tearing mode: Crack faces are displaced on their plane, parallel to the crack front, related to anti-plane longitudinal shear loading.
Figure 2.2: Fundamental modes of deformation in fracture mechanics.
Every type of crack deformation in most-real world problems can be analysed as a superposition of these three basic modes, obtaining a mixed mode loading, that can
be a combination of two of the modes or even all of them [22]. In this work, only Mode I, Mode II or mixed modes between these two will be analysed, since only 2D models will be studied.
As mentioned before, one of the first contributions for the analysis of Linear Elastic Fracture Mechanics was made by Griffith with the concept of energy release rate [24], related with the Griffith’s failure criterion, which is a crack path criterion, determining when and where the crack grows.
The example studied by Griffith was the infinite plate subjected to tensile loading, as represented in figure below.
w y x 2a σ σ
Figure 2.3: Scheme of the infinite cracked plate problem.
In 1921, Griffith deducted that the strain energy had the following relation with crack length, a, as [24]: U = σ 2V 2E − σ2Bπa2 2E (2.3.1)
Where V represents the volume of the plate, B represents its thickness, and E the Young’s Modulus of the material.
If the energy required to break the atomic bond is considered, it must be added to the strain energy in order to obtain the total energy in the system. The energy required to break the atomic bonds is given by:
W = 2γaB (2.3.2)
surface area. The total energy in the system is given by: E = W + U = 2γaB +σ 2V 2E − σ2Bπa2 2E (2.3.3)
When the crack length is short, the total energy of the system is increasing with the increasing length, which means that energy has to be added to the system. When the crack is larger, no additional energy is required in order for it to grow. The crack propagation condition is:
∂E ∂a = 0 (2.3.4) Therefore, 2γB − σ 2Bπa 2E = 0 (2.3.5)
It is then possible to deduct a critical value of stress, in which the crack will start to grow:
σc=
s
2γE
πa (2.3.6)
Assuming that 2γ is the critical value of energy release rate, applying the Griffith’s criterion, then the expression for critical stress, σc can be written as:
σc=
s
GcE
πa (2.3.7)
Another very important work in Fracture Mechanics is the Wastergaard Formu-lation [25] used to calculate the stress and strain fields in the infinite cracked plate program as represented in figure 2.3. Initially, Airy’s stress equation is found:
∇4φ = 0 (2.3.8)
The solution is found using a complex variable, z = x + iy, therefore:
¯ Z = ∂ ¯ ¯ Z ∂z Z = ∂ ¯Z ∂z (2.3.9)
Considering the real and complex parts of Z function:
Z = RZ + iIZ (2.3.10)
Where RZ and IZ represent the real and imaginary parts of Z, respectively. There-fore, Westergaard proposes the following stress function [25]:
As known from Elasticity Theory, σxx = ∂2φ ∂y2 (2.3.12) σyy = ∂2φ ∂x2 (2.3.13) τxy = − ∂2φ ∂x∂y (2.3.14)
From the equations stated before, it results:
σxx = RZ + yIZ0 (2.3.15)
σxx = RZ − yIZ0 (2.3.16)
τxy = −yRZ0 (2.3.17)
Notice that when x >> a then σyy ' σ and σyy = 0 when |x| < a, and after some
mathematical manipulation seen in [25], it is stated that: Z = q σ
1 − (a z)2
(2.3.18)
and Z0 its derivative in order to z, the following equation for σyy is obtained:
σyy = R " σ q 1 − (a x)2 # (2.3.19)
Another important work in the development of Fracture Mechanics was Irwin’s Stress Intensity factor and his functions to determine the stresses near crack tip [26]. Following equations of Westergaard for the stress fiels, Irwin deducted that by changing the coordinate system to radial coordinates in the origin of the crack tip, as seen in figure 2.4, and assuming that crack length was higher than its radius,a >> r, then the following stress equations for the infinite plate with a crack example would be obtained: σxx = KI √ 2πr cos θ 2 1 − sin π 2 sin 3π 2 (2.3.20) σyy = KI √ 2πr cos θ 2 1 + sin π 2 sin 3π 2 ! (2.3.21) τxy = K √ 2πrcos θ 2sin θ 2cos 3θ 2 (2.3.22)
r
θ
crack tip
Figure 2.4: Radial Coordinate System in the crack tip.s
And the stress intensity factor for Mode I loading, KI, is given by:
KI = σ
√
πa (2.3.23)
2.3.1 Crack Propagation Criteria
Predicting how a crack propagates is one of the most important fields in Fracture Mechanics. To predict the crack propagation of a problem, two features have to be determined. Firstly, if there is enough loading to start crack propagation, then (if it is propagating) in which direction the propagation occurs. Crack propagation criteria can be divided into three major groups: energy based, stress based, and strain based [27].
The energy based criteria use energy concepts in order to determine the energy around the crack tip and determine if the crack is propagating and its direction of propagation. Energy based criteria lead to more realistic results regarding crack prop-agation than stress based criteria, because they take into account energy dissipation during fracture. However, stress based criteria are easier to implement due to their simplicity [28]. As previously mentioned, the first energy based criterion was proposed by Griffith in 1921 [24]. However, it is only used in pure Mode I situations and it is not able to determine the crack propagation direction.
This criterion was later developed by [29], in which it assumes that the crack propagates in the direction where the Strain Energy Release Rate has its maximum value and it is able to be applied in a mixed mode fracture. Another very common criterion was developed by Sih [30] and it is the Strain Energy Density criterion, in which it states that the crack propagates in the direction with the minimum strain energy density.
Stress based criterions, as mentioned before, are easier to study and apply than energy based criterion. The introduction of Stress Intensity Factor by Irwin was one of the first stress based criteria introduced, as it was possible to determine if a crack is growing if the SIF is higher than the fracture toughness of the used material. But, like Griffith’s criterion, it does not predict the direction of crack propagation. One of the most commonly used and undemanding to understand and implement is the Maximum Tangential Stress criterion, proposed by Erdogan and Sih [31]. This criterion assumes that the crack propagates in the perpendicular direction to the maximum tangential stress in the crack tip. This criterion can be easily implemented using various alter-natives, being the most common one, and applied in this work, the one that considers that the direction of the maximum tangential stress corresponds to the first princi-pal stress direction at the crack tip, therefore, since the second principrinci-pal direction of
stress is perpendicular to the first principal direction, the crack will propagate along the second principal direction of stress.
In some materials, the strain based criteria can produce better results than the energy based and the stress based criterion [28]. One of the most commonly used strain based criteria is the Maximum Tangential Strain criterion proposed by Chang [32]. It considers that the crack propagates in the direction of the maximum tangential strain, which produces good results in both sharp and rounded crack tips.
Numerical Methods
3.1 Finite Element Method
The Finite Element Method (FEM) is currently the most common numerical method in solid mechanics, mainly due to the fact that it is one of the most effi-cient methods for numerical calculations [22]. Even though, many numerical methods were known since the XVIII century, their development has only occurred with the appearence of modern computer science [21]
3.1.1 Mesh Generation
FEM can be applied to both 2D and 3D problems, but since this work will only approach 2D fracture examples, only 2D formulations will be analysed. In order to build a mesh in FEM, the domain has to be divided along its x and y directions to firstly create the nodes for the division of the domain. After the creation of the nodes, the elements containing the created nodes will be established. 2D elements can be triangular or quadrilateral with distinct numbers of nodes. In this work, only triangular elements will be used because the used models have irregular shapes, therefore, it is difficult to create undeformed quadrilateral elements in irregular shapes.
3.1.2 Integration Points
After obtaining and organizing the elements that will be part of the used mesh, the next step, in order to complete the discretization of the problem, is to create integration points for each element. In order to interpolate complex geometries, a different coordinate is used. This coordinate system uses natural or local coordinates, where an example of a quadrilateral element can be seen in figure 3.1.
Figure 3.1: Quadrilateral element with natural coordinate system.
The number of integration points can be chosen according to the number of nodes of each element, the coordinates of the integration points, and its weights can be seen in table 3.1.
Table 3.1: Gaussian quadrature coordinates and weights.
Number of points Coordinates Weights
1 0 2 2 ±q1 3 1 3 0 8 9 ±q3 5 5 9
3.1.3 Shape Function Construction
The construction of the shape functions of an element is dependent on the number of nodes on an element. Considering the local coordinates of an element, ξ and η, firstly, the created shape functions have to respect two conditions:
• Condition of nodal compatibility:
Ni(ξj, ηj) = δij = 1, i = j 0, i 6= j (3.1.1)
• Rigid body condition
n
X
i=1
A simple way to define the shape functions in local coordinates is to create a matrix
P, with n × n size, where n is the number of nodes in the element. For a quadrilateral
element with 4 nodes, the matrix P is defined by [33] :
P = 1 ξ1 η1 ξ1η1 1 ξ2 η2 ξ2η2 1 ξ3 η3 ξ3η3 1 ξ4 η4 ξ4η4 (3.1.3)
Notice that the polynomial order of the polynomial function depends on the number of nodes of the element. The polynomial order of the polynomial function must respect the tirangle of Pascal. Therefore, the matrix is dependant on the Pascal triangle to choose its terms, related with its number of nodes [21]. Considering that the elements used in the defined mesh of a given problem are the same, the matrix will remain the same for all elements. The shape functions of an element can be defined as:
Ni = pP−1 (3.1.4)
Where p is a vector of polynomials also dependent on the Pascal’s triangle, in the example of a 4 noded element it defined as:
p =n1 ξ η ξηo (3.1.5)
To obtain the derivatives of the shape function in order to ξ and η for each node, the vector p must be derived in order of the dimensions stated before and multiplied by the inverse of P. ∂Ni ∂ξ = ∂p ∂ξP −1 (3.1.6) ∂Ni ∂η = ∂p ∂ηP −1 (3.1.7)
Knowing these derivatives it is needed to obtain the derivatives of the shape func-tions in order to global coordinates. For that to be possible, the Jacobian matrix for each Gauss point has to be calculated [33]:
JI = "∂N1 ∂ξ · · · ∂Ni ∂ξ ∂N1 ∂η · · · ∂Ni ∂η # x1 y1 .. . ... xi yi (3.1.8)
In order to obtain the shape function derivated in order of the global coordinates, the following equation is obtained:
"∂N1 ∂x · · · ∂Ni ∂x ∂N1 ∂y · · · ∂Ni ∂y # = J−1I "∂N1 ∂ξ · · · ∂Ni ∂ξ ∂N1 ∂η · · · ∂Ni ∂η # (3.1.9)
3.1.4 Stiffness Matrix
In FEM, for every element there is its own local stiffness which will be assembled in the global stiffness matrix. The global stiffness matrix is the result of the assembly of all local stiffness matrices. In order to calculate the stiffness matrix for one element, the deformation matrix for a given problem has to be created. The deformation matrix for a 2D quadrilateral element subjected to plane stress is given by [21]:
B = ∂N1 ∂x 0 · · · ∂Ni ∂x 0 0 ∂N1 ∂y · · · 0 ∂Ni ∂y ∂N1 ∂y ∂N1 ∂x · · · ∂Ni ∂y ∂Ni ∂x (3.1.10)
Knowing the deformation matrix for a Gauss Point, it is now possible to calculate the stiffness matrix for that same Gauss Point and, therefore for an element. In order to calculate this matrix, the constitutive matrix presented before is needed. The stiffness matrix for an element is given by:
K(e)= ni X i=1 nj X j=1 BTi · c · Bj· WiWj (3.1.11)
Where i and j represent the number of Gauss Points in the x and y directions, respectively, and Wi and Wj represent the weights of Gauss Points in those directions.
The global stiffness matrix is obtained by assembling all element stiffness matrices [33].
3.1.5 Force Vector
To apply a force along a boundary, the nodes regarding that boundary need to be identified in order to create a new set of integration points along it to discretize that boundary and integrate a function along that line.
The process to obtain the shape functions is the same as described before. The shape functions will be used to create a matrix known as "Shape Function Matrix",
N, given by [21]: N = " N1 0 · · · Ni 0 0 N1 · · · 0 Ni # (3.1.12)
Having defined the Shape Function Matrix, it is now possible to write the external force vector, fe= ni X i=1 nj X j=1 BTi · NT¯tW iWj (3.1.13)
3.1.6 Displacements, Strains and Stresses
The goal of applying a numerical method to a given problem is to obtain results of the behaviour of a solid with a certain load applied to it. Therefore, to analyse its behaviour it is important to obtain its displacement, strain and stresses, all dependant on each other.
The displacement vector can be obtained by:
u = K−1f (3.1.14)
Where K represents the stiffness matrix with all imposed constraints and f the force vector of the applied nodal forces.
The strain is calculated for each Gauss Point of an element and it is obtained using the following formula:
εi = Bu(e) (3.1.15)
Where B is the deformation matrix of an element and u(e) is the displacement vector for a given element.
The stress vector can also be obtained for each Gauss Point from the relationships between stress and strain, indicated in equation 2.1.12. The stress vector in a Gauss point is given by:
σi = cε (3.1.16)
The stresses and strains are originally obtained for the Gauss Points but can be extrapolated to the nodes in several ways. Since it is not important for the outline of this work, it will not be approached but it can be consulted in [33].
3.2 Meshless Methods
Like the majority of other nodal dependent numerical methods, most meshless methods approaches respect a common outline. First, the given problem is studied and the domain and boundaries are established. Then, its conditions are identified and, finally, the problem domain is numerically discretized by a set of nodes. This set of nodes can follow a regular or irregular distribution, since in the case of meshless methods such procedure does not form a mesh. In truly meshless methods, the only information required to implement the method is the spatial coordinates of the set of nodes discretizing the domain [1].
3.2.1 Nodal Generation and Connectivity
Node generation in meshless methods is similar to the mesh creation in FEM but in this case, no elements are created. So, the domain is divided along its directions, and nodes are created according to those divisions.
While in FEM the nodal connectivity is assured with the predefined mesh and the nodes belonging to the same element interacting between each other and with the boundary nodes of its neighbour element, in RPIM the nodal connectivity is deter-mined after the nodal distribution and is made by overlapping influence domains. In order to maximise performance of this method, the best way is to have all influence domains contain approximately the same number of nodes [1].
In NNRPIM, the nodal connectivity is imposed through the overlap of influence cells. Influence-cells are the equivalent of the influence domain in other meshless methods [1].
Two types of influence-cells are used in NNRPIM [1]:
• First degree influence-cell: composed by the first degree natural neighbours of an interest point.
• Second degree influence-cell: composed by the first and second degree neighbours of an interest point. The second degree natural neighbours are the natural neighbours of the the first degree natural neighbours of an interest point. In figure 3.2 a Voronoï diagram and an example of each type of influence-cell is presented.
1st degree influence-cell 2nd degree influence-cell
Node
Figure 3.2: Example of a Voronoï diagram.
3.2.2 Influence-Domains
As mentioned before, in meshless methods, unlike FEM, there are no elements and the nodal connectivity is achieved by inforcing the overlap of the influence domains of
each interest point.
Commonly, most meshless methods use a background mesh to generate the inte-gration points. This mesh can be dependent or independent from the nodes. Meshless methods using independent background integration meshes are not truly meshless methods, since a meshless method should construct all its numerical entities from the nodal mesh, without any other additional source of information [2].
These influence domains can be defined by searching a certain number of nodes inside a fixed area or volume around an integration point. However, its size and shape variation along the same problem can affect the performance of the method and its final result. It is very important that all influence domains contain approximately the same number of nodes.
There are various ways to define an influence domain. It can be defined by a fixed area with a rectangular or circular shape, and, therefore all influence domains will have the same area but might have a different number of nodes inside it, which can lead to less accurate results. Or it can be defined by its number of nodes, in which a radial search around the integration point is made to find the nodes closer to it. This can make the size of the influence-domain of each point different, but with the same number of nodes in it. For two dimensional space, three types of influence domains are shown in figure 3.3.
(a) (b) (c) ni = 9 nj = 7 ni = 7 nj = 5 ni = 8 nj = 8
Figure 3.3: (a) Fixed rectangular shaped influence-domain; (b) Fixed circular shaped
influence-domain; (c) Flexible circular shaped influence-domain.
Due to the simplicity of its application, influence-domains defined by area are commonly applied. Several works, such as [34], [10], and [7], suggest that the number of nodes inside an influence domain should be between 9 and 16.
3.2.3 Influence-Cells and Natural Neighbours
A new approach to define influence domains in Meshless Methods was proposed by Dinis, Natal and Belinha [2]. It consists on defining the influence-domains using the concept of Voronoï diagrams and Delaunay triangulation, and is the basis for the Natural Neighbour Radial Point Interpolation Method (NNRPIM).
This theory is applicable to a D-dimensional space. However, in this work, only two-dimensional problems will be analysed, therefore, this theory will be presented for a two-dimensional Euclidian space.
Considering a set of N nodes discretizing a domain Ω ⊂ R2:
N = {n1, n2, · · · , nN} ∈ R2 (3.2.1)
The Voronoï diagram of N is the partition of the domain defined by N in sub-regions Vi, closed and convex [1]. Vi is the geometric place associated to the node ni
in a way that all points are closer to ni than any other node. Sub-regions V are the
set of "Voronoï Cells" which define the Voronoï diagram, with k = {1, · · · , N } cells. Mathematically, a Voronoï cell is defined by [1]:
Vi = {x ∈ R2 : kx − xik < kx − xjk, ∀j 6= i} (3.2.2)
Where x represents an interest point of the domain and k · k the Euclidean metric norm, which is the distance between the interest point and the points xi and xj [1].
Voronoï diagrams have several possible applications, from natural sciences to engi-neering, as it is only a mathematical concept. To help understand how a Voronoï cell is built, a nodal set will be discretized in figure 3.4. In this figure, the Voronoï cell,V0, for the node n0 is built. Then, node n4 is assumed as a possible neighbour and vector
u40 is determined [1]:
u40=
(x0− x4) kx0− x4k
(3.2.3)
Being u40 = {u40, v40, w40}. Using this vector, the plane π40 can be defined and the following condition can be established [1]:
u40x + v40y + w40z ≥ (u40x4+ v40y4+ w40z4) (3.2.4)
Where the nodes that do not respect this condition are eliminated as natural neigh-bours of the node n0. As seen in figure 3.4, n8 was excluded, as it did not respect this condition. This process is repeated for each node until all natural neighbours of n0 and, the condition created for them, is respected. Only the nodes of the perimeter V0∗ are considered neighbour nodes. As it is possible to observe, the V0∗ is the homothetic of the cell V0.
n9 n8 n7 n6 n5 n4 n3 n2 n1 n0 n9 n0 n1 n2 n3 n4 n5 n6 n7 n8 V0∗ n6 n5 n4 n3 n2 n1 n0 V0 n0 n1 n2 n3 n4 n5 n6 n7 n8 n9 (a) (b) (c) (d)
Figure 3.4: Voronoï cell construction: (a) initial set of potential neighbour nodes of
n0; (b) First trial plane; (c) Final trial cell containing only the natural neighbours of
node n0; (d) Voronoï cell V0
Besides creating influence domains, the natural neighbours are also used to create a background mesh dependent on nodes, which revolves around the application of the concept of the Delaunay triangles. This concept is the geometrical dual of the Voronoï diagram and it implies that a Delaunay edge only exists between two nodes in the pane if their Voronoï cells share a common edge and connects the nodes belonging to those cells [1]. Delaunay triangles are used to create integration points and its construction can be observed in figure 3.5.
(a) (b)
Figure 3.5: Construction of the Delaunay triangles: (a) initial Voronoï diagram; (b)
respective Delaunay triangulation.
As observed in figure 3.6, it is important to notice that, in the case of regular meshes, some middle points MIi are coincident with some intersection corners PIi,
therefore, it creates triangular instead of quadrilateral Voronoï sub-cells as it happens in figure 3.7 [1]. n6 nI n2 n1 n3 n8 n7 n5 n4 MI7 PI8= MI8 nI n6 nI n2 n1 n3 n8 n7 n5 n4 MI5 MI3 MI1 MI1 PI3= PI4 PI5= PI6 PI7= PI8 PI1= PI2 PI5= PI6= MI6 PI3= PI4= MI4 PI1= PI2= MI2 PI7= PI8= MI8 (c) (b) (a)
Figure 3.6: (a) Voronoï cell and its respective PIi intersection points. (b) Middle
points MIi and the respective generated triangles. (c) Generated triangle.
nI MI4 PI4 MI5 PI5 PI3 PI4 PI1 PI2 PI6 nI n6 n5 n4 n3 n2 n1 n1 n2 n3 n4 n5 n6 nI PI6 PI2 PI1 PI4 PI3 PI5 (a) (b) (c)
Figure 3.7: (a) Voronoï cell and its respective PIi intersection points. (b) Middle
points MIi and the respective generated quadrilaterals. (c) Generated quadrilateral.
sub-cells, where [2]: AVI = n X i=1 ASIi, ∀ASIi ≥ 0 (3.2.5)
Where AVI represents the size of the Voronoï cell VI and ASI represents the size of
the Voronoï sub-cell [2]. Notice that, if the Voronoï cells are a partition without gaps of the global domain, then the set of sub-cells are also a partition of the global domain. Starting with the geometrical shapes seen before, numerous integration schemes can be constructed [1]. In this work, an integration scheme based on Gauss-Legendre numerical integration is shown.
The simplest integration scheme can be established by using the sub-cells, and is obtained by inserting only one integration point on the barycentre of those sub-cells. The coordinates of each integration point are calculated for each sub cell, as indicated in figures 3.7 and 3.6 where xi = {xi, yi} and the weight of the integration point has
an equal value as the area of each sub-cell. The areas of a triangle shape sub-cell is given by: A∆I = 1 2abs det " x2− x1 y2 − y1 x3− x1 y3 − y1 # ! (3.2.6)
And for the quadrilateral shape, the area is given by [2]
A∆I = 1 2abs det " x2− x1 y2− y1 x3− x1 y3− y1 # + " x4 − x1 y4− y1 x3 − x1 y3− y1 # ! (3.2.7) The coordinates (x1+ x2)/2 (x1+ x3)/2 x2 x3 (x2+ x3)/2 x1 xI xI x2 x1 x3 x4 (x1+ x2)/2 (x 1+ x4)/2 (x3+ x4)/2 (x2+ x3)/2 (a) (b)
Figure 3.8: Integration Points’ generation: (a) Triangular sub-cell; (b) Quadrilateral
sub-cell.
3.2.4 Shape Functions
The RPIM and NNRPIM shape functions are more complex than shape functions used for FEM. NNRPIM and RPIM use Radial Point Interpolators, where the PIM generic functions are replaced with radial basis functions [2].
