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Department of Precision and Microsystems Engineering

The Moving Node Approach in Topology Optimization

An Exploration to a Flow-inspired Meshless Method-based Topology Optimization

Method

Johannes. T. B. Overvelde

Report no : EM 12.006

Coach : Dr. ir. Matthijs Langelaar

Professor : Prof. dr. ir. Fred van Keulen

Specialization : Solid and Fluid Mechanics

Structural Optimization and Computational Mechanics

Type of report : Master thesis

Date : 18 April 2012

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1

Faculty of Mechanical, Maritime and Materials Engineering

The Moving Node Approach in Topology

Optimization

An Exploration to a Flow-inspired Meshless Method-based Topology

Optimization Method

A Master’s Thesis in

Mechanical Engineering

by

Johannes T.B. Overvelde

April 18, 2012

Faculty of Mechanical, Maritime and Materials Engineering

The Moving Node Approach in Topology

Optimization

An Exploration to a Flow-inspired Meshless Method-based Topology

Optimization Method

A Master’s Thesis in

Mechanical Engineering

Track: Solid and Fluid Mechanics (SFM)

Specialization: Structural Optimization and Computational Mechanics

Author Exam Committee

Johannes T.B. Overvelde Prof.dr.ir. Fred van Keulen

Student no. 1228145 Dr.ir. Matthijs Langelaar

www.overvelde.com Prof.dr.ir. Miguel A. Guti´errez De La Merced

Sergio R. Turteltaub Ph.D.

c

!Johannes T.B. Overvelde

April 18, 2012

Preface

This Master’s thesis is submitted in fulﬁllment of the requirements for the Degree of

Master of Science in the subject of mechanical engineering at the Faculty of Mechanical,

Maritime and Materials Engineering at Delft University of Technology.

The subject of the thesis ﬂows naturally from the composition of my Master’s track

Solid and Fluid Mechanics, with a specialization in Optimization and Computational

Engineering. The problem deﬁnition and proposed ideas arise from a combination of

practices and methods employed in solid mechanics, optimization and ﬂuid mechanics.

The ﬁeld of solid mechanics and optimization are often associated with each other

and the combination of the two proves to be very fruitful, i.e. topology optimization. In

this thesis, ﬂuid dynamics is thrown into the mix and together the ﬁelds create a trinity

that shows great potential. I am excited to present the results.

Haarlem April 18, 2012

Johannes T.B. Overvelde

i

Acknowledgments

This Master’s thesis is submitted in fulﬁllment of the requirements for the De-

gree of Master of Science in the subject of mechanical engineering at the Faculty

of Mechanical, Maritime and Materials Engineering at Delft University of Tech-

nology. I wish here to express my sincere appreciation to Prof. dr. ir. Fred van

Keulen, professor and chair of the research group Structural Optimization and

Computational Mechanics - Applied Mechanics at the Department of Precision

and Microsystems Engineering (PME) at the Faculty of Mechanical, Maritime

and Materials Engineering at Delft University of Technology, for his constructive

criticism and for allowing and supporting me to pursue my own research ideas.

I am grateful to Dr. ir. Matthijs Langelaar, assistant professor at the research

group Structural Optimization and Computational Mechanics - Applied Mechan-

ics at the Department of Precision and Microsystems Engineering (PME) at the

Faculty of Mechanical, Maritime and Materials Engineering of Delft University of

Technology for the numerous fruitful and candid discussions. I highly appreciate

his enthusiasm and input. My colleagues Nico van Dijk, Hans Goosen, Miquel

Guti´errez De La Merced, Evert Hooijkamp, Kelvin Ng Wei Siang, Hugo Peters,

Samee-ur Rehman, Mohammad Samimi, Saputra, Kostiantyn Vandyshev, Alexan-

der Verbart, Tim van Wageningen, Jeroen Wolfs, Stanley Wong and Marco Zocca

at the research group Structural Optimization and Computational Mechanics -

Applied Mechanics at the Department of Precision and Microsystems Engineer-

ing (PME) at the Faculty of Mechanical, Maritime and Materials Engineering of

Delft University of Technology, all of whom I gratefully acknowledge here with

my sincere thanks for their critical contributions during the weekly CHARLES

meetings.

I would also like to seize this opportunity to thank Katia Bertoldi, Ph.D.,

assistant professor in Applied Mechanics at the School of Engineering and Applied

Sciences at Harvard University, for her kind encouragements, trust and interest.

Katia has been an incredible source of inspiration during my research internship

at Harvard University in 2010, and remained so unremittingly in the succeeding

ii

years. I am glad we continued working together and maintain in touch through

boundless email and Skype communication.

I wish to thank my employers Ir. Tom Santegoeds and Dr. ir. Ronald van Dijk,

proprietors of Femto Engineering for their kind-heartedness and ﬂexibility.

Of course I thank all family and friends for their love and support. Finally,

I am most grateful to my ﬁanc´ee Sanne Slagman, for all her support throughout

my studies. She has encouraged and taught me to reach for the sky. Without her,

I would not have been able to ﬁnish my studies so excellently. Above all, I thank

her for her inﬁnite trust and love.

iii

Contents

Preface i

Acknowledgments ii

Contents iv

1 Introduction 1

1.1 FEM-based topology optimization . . . . . . . . . . . . . . . . . . 1

1.2 Meshless method-based topology optimization . . . . . . . . . . . 3

1.3 Flow-inspired meshless method-based topology optimization . . . 5

1.4 Thesisstructure............................ 7

2 Meshless Methods 9

2.1 Approximation methods . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 FEM approximation . . . . . . . . . . . . . . . . . . . . . 11

2.1.2 Kernel approximation . . . . . . . . . . . . . . . . . . . . . 14

2.1.3 MLS approximation . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Discretization of the linear elasticity problem . . . . . . . . . . . . 23

2.2.1 EFGmethod ......................... 25

2.2.2 MLPG mixed collocation method . . . . . . . . . . . . . . 33

2.3 Exemplaryproblem.......................... 39

2.3.1 Two-dimensional cantilever beam . . . . . . . . . . . . . . 39

2.3.2 EFG and MLPG mixed collocation solutions . . . . . . . . 40

2.3.3 Inﬂuence of discretization parameters . . . . . . . . . . . . 43

2.4 Characteristics of meshless methods for the MNA . . . . . . . . . 47

3 Eﬀect of the nodal distribution in the EFG method 48

3.1 Eﬀect of the nodal distributions on accuracy . . . . . . . . . . . . 49

3.1.1 Global eﬀect of nodal distribution . . . . . . . . . . . . . . 49

3.1.2 Local eﬀect of the nodal position in the MLS approximation 58

iv

3.2 Eﬀect of the nodal position on the material distribution . . . . . . 65

3.2.1 Complexly shaped material distribution . . . . . . . . . . . 65

3.2.2 Material distribution exemplary problem . . . . . . . . . . 68

3.3 Requirements for the MNA . . . . . . . . . . . . . . . . . . . . . . 72

4 Material distribution deﬁned by nodal compaction 74

4.1 Material density based on the nodal compaction . . . . . . . . . . 75

4.2 Characteristics of regular nodal distributions . . . . . . . . . . . . 78

4.2.1 General characteristics . . . . . . . . . . . . . . . . . . . . 79

4.2.2 Changes in the background mesh position . . . . . . . . . 82

4.2.3 Relative displacement of nodes . . . . . . . . . . . . . . . . 87

4.3 Compliance sensitivity . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3.1 Finite Diﬀerence compliance sensitivity . . . . . . . . . . . 91

4.3.2 Analytical compliance sensitivity . . . . . . . . . . . . . . 93

4.4 Nodal compaction material distribution in the MNA . . . . . . . . 97

5 The MNA in topology optimization 99

5.1 An MNA topology optimization algorithm . . . . . . . . . . . . . 100

5.1.1 Nodalmovement ....................... 101

5.1.2 Decoupling of discretization and mass distribution functions 102

5.1.3 Asymptotic density function . . . . . . . . . . . . . . . . . 103

5.2 MNA topology optimization exemplary problems . . . . . . . . . 105

5.3 Characteristics of the MNA . . . . . . . . . . . . . . . . . . . . . 112

Summary 115

Outlook 117

Bibliography 119

Appendix 124

A Gaussquadrature........................... 124

B Displacement and stress results . . . . . . . . . . . . . . . . . . . 126

v

Chapter 1

Introduction

1.1 FEM-based topology optimization

This Master’s thesis is situated in the ﬁeld of topology optimization. Topology

optimization is concerned with ﬁnding the optimal material layout, in which the

optimal layout is determined according to speciﬁc performance targets. The ma-

terial layout in various ﬁelds, such as solid mechanics, ﬂuid dynamics and thermo-

dynamics can be optimized using topology optimization. Applications of topology

optimization are for example: satellite support structure designs that weigh less

than a speciﬁed mass, yet are strong enough to carry the instruments during

launch [1]; optimization of the mixing performance of laminar static mixers [2];

two-phase composites design having maximum thermal expansion, zero thermal

expansion, or negative thermal expansion [3]; optimization of the crashworthiness

of cars, where during impact the front of the car keeps a certain acceleration and

only a certain intrusion is reached [4].

The diﬀerence between the various topology optimization problems is the phys-

ical problem, i.e. the governing diﬀerential equations and boundary conditions.

The mathematical algorithms underlying the optimization of the layout are uni-

versal. Thus, two main components of topology optimization can be distinguished:

the physical problem and the optimization problem. The interaction between both

problems is illustrated in Fig. 1.1.

Although the mathematical optimization algorithms are universal and can be

applied to various physical ﬁelds, a variety of optimization methods has been

proposed in literature. At least three main methods can be identiﬁed: the density-

based homogenization method, the Evolutionary Structural Optimization (ESO)

method and the level-set method. These three methods are discussed next.

In 1988 Bendsøe and Kikuchi [5] introduced one of the ﬁrst topology opti-

1

Figure 1.1: The two components of topology optimization. The physical problem is solved

and forms the input for the optimization problem. In the optimization problem the shape is

adapted to better meet the performance targets. This cycle is repeated until the optimal design

is reached.

mization methods: the homogenization method. In the homogenization method

the design space1is covered with cells that contain a microstructure. The shape

of this microstructure is parameterized such that the cell can represent various

compositions, from completely ﬁlled with material to completely empty. With

these cells the rather diﬃcult topology optimization problem is simpliﬁed to a

sizing problem of the microstructural parameters, in which the parameters can

vary from cell to cell. However, for some objectives the homogenization method

may not result in plausible designs. The homogenization method often produces

layouts with inﬁnitesimal holes, which make the design diﬃcult to fabricate.

Some variations on the homogenization method have been proposed. Espe-

cially the Solid Isotropic Material with Penalization (SIMP) method has gained

wide-spread attention [6], [7]. The power of the SIMP method lies in its con-

ceptual and computational simplicity. In this method the presence of material is

given by the density of the cells, instead of the shape of the microstructure. By

varying the normalized cell density between zero and one, material can locally

be added or removed. Penalization of intermediate density is introduced in this

formulation in order to obtain manufacturable designs. However, numerical insta-

bilities, such as checkerboard patterns, mesh-dependencies and local minima can

occur in this method [8]. In order to prevent these problems, additional measures

should be incorporated in the method. For instance, checkerboard patterns can

be prevented by minimum length scale ﬁlters.

Another topology optimization method, called the ESO method, has been

proposed in 1993 by Xie and Steven [9]. This method is based on the gradual

removal of material to achieve an optimal design. In contrast to the SIMP method,

the design space contains only material with a normalized cell density of either

1The domain in which material can be present. Note that the ﬁnal layout will always be contained

inside the design space.

2

zero or one, and therefore intermediate density does not exist. However, the ﬁnal

design is highly sensitive to the initial design space and the size of the underlying

cell structure [10].

The level-set method, proposed in 2000 by Sethian and Weigmann [11], tries

to overcome some of the aforementioned problems such as numerical instability

and intermediate density. This method describes the varying layout of the struc-

ture with a level-set function. Material can therefore also be gradually added

in the level-set method. The removal or addition of the material occurs near the

boundaries of the design, in which the tracking of boundaries is done by a level-set

algorithm. The addition of material makes the ﬁnal design less sensitive to the

initial design and the underlying cell structure. However, intermediate density

may still exist near the boundary, depending on the numerical implementation.

The physical problem can be described in terms of a partial diﬀerential equa-

tion, or a set of partial diﬀerential equations. The solution to such equations is

often found by a numerical method, when the problem is too complex to solve

analytically. All aforementioned topology optimization methods have in common

that the governing equations of the physical problem are typically solved by the

same numerical technique, called the Finite Element Method (FEM). By use of

this method the complex partial diﬀerential equations are reduced to a set of

simple linear algebraic equations.

FEM is based on the discretization of the domain Ωand its boundary Γby

a mesh. To create the mesh, nodes are distributed in the domain Ωand on its

boundary Γ. These nodes are connected with each other through elements. Thus,

the domain is discretized with a ﬁxed interconnection between the nodes. As an

example, a triangular mesh of a two-dimensional domain is shown in Fig. 1.2a.

However, some diﬃculties arise when using a mesh. It is for instance diﬃcult

to determine a good robust mesh, especially in three-dimensional domains [12].

Also, in large deformation and displacement problems, the mesh will deform and

become less reliable. Remeshing is used to solve this problem, although it is

computationally time consuming [13], [14]. Another problem for which standard

FEM is not well suited, is the propagation of cracks in solids [15].

1.2 Meshless method-based topology optimization

To overcome some diﬃculties that arise when using FEM, meshless discretization

techniques have been applied. These techniques are also called meshless methods.

3

Ω

Γ

(a) Typical FEM discretization

Ω

Γ

(b) Typical meshless method discretization

Figure 1.2: Typical FEM and meshless method discretization of a rectangular domain Ωand

its boundary Γ(black line). The domain is discretized using nodes (blue circles). For the FEM

discretization the element boundaries are in red. For the meshless method discretization the

boundary of the inﬂuence or local domain is given in red.

In these methods the simple linear algebraic equations are constructed entirely

in terms of nodes.2These nodes have no ﬁxed elements connecting each other,

in contrast the connections are simply formed between neighboring nodes. In

Fig. 1.2b a discretized rectangular domain Ωand its boundary Γare shown. The

inﬂuence domain of nodes, which determines the connections between neighboring

nodes, is also shown.

One of the ﬁrst meshless methods, called the Smoothed Particle Hydrodynam-

ics (SPH) method, was constructed in 1977 by Gingold and Monaghan [17] and

independently by Lucy [18]. This method was ﬁrst created for modeling astrophys-

ical phenomena, and later to model ﬂuid dynamic problems. The ﬁrst attempt

to model solid mechanics (impact) problems was by Libersky [19] in 1993. The

SPH method is based on the strong form notation of partial diﬀerential equations

(hereafter: strong form) [20], [21]. The strong form in solid mechanics contains

second order derivatives, and therefore the discretization becomes troublesome.

Other methods were developed in the early 1990’s. These methods are based on

the weak form notation of partial diﬀerential equations (hereafter: weak form) [20],

[21]. For the weak form there is no necessity to discretize second order derivatives.

The ﬁrst developed method based on the weak form, and mainly applied to solid

mechanics, is the Element-free Galerkin (EFG) method [12]. Although the EFG

method does not require a mesh to give relations between nodes, a background

mesh is necessary for the evaluation of the integrals present in the weak form.

2The term ‘node’ will be used throughout this thesis, although in some literature the terminology

‘particle’ is used instead [16], [17].

4

More recently, in 1998, Atluri and Zhu proposed a diﬀerent method called

the Meshless Local Petrov-Galerkin (MLPG) method, which is based on the local

weak form [22]. In this method, the integrals present in the local weak form are

evaluated on sub-domains, therefore no background mesh is needed.

The EFG and MLPG methods are some of the most common methods. Ob-

viously, these are not the only available meshless methods. A more complete

overview of meshless methods is provided in the work of Nguyen et al [14]. How-

ever, to limit the scope, this thesis focusses on the EFG and MLPG mixed collo-

cation method.

During the last decade, meshless methods have also been applied to discretize

and solve the physical problem in topology optimization. As a proof of concept,

various meshless methods have been applied to discretize the governing equations

of linear physical problems with the use of the SIMP method [23], [24], [25], [26],

ESO method [13] and the level-set method [27], [28], [29], [30], [31], [32]. The mesh-

less SIMP method has also been applied to intricate non-linear problems [33], [34],

[35], [36], elucidating the capabilities of meshless methods in topology optimiza-

tion. In principle, these meshless method-based topology optimization methods

have been introduced to increase the capabilities of topology optimization for use

in non-linear physical problems. In these topology optimization applications, the

meshless methods have been used as a direct replacement of FEM, in which the

nodal distribution remained unchanged during the optimization process.

1.3 Flow-inspired meshless method-based topology opti-

mization

A possible alternative to topology optimization methods based on regular nodal

distributions, might be generated from the ﬁeld of ﬂuid dynamics. In ﬂuid dy-

namics, the governing partial diﬀerential equations are usually expressed in the

Eulerian form, i.e. at each coordinate in the problem domain Ωthe ﬂuid velocity

and density are tracked [37]. This approach works well for problems in which

the complete problem domain Ωconsists of one type of ﬂuid. However, for free

surface, multi-phase and mixing ﬂows the Eulerian description of the ﬂuid is un-

suitable. A Lagrangian form of the governing partial diﬀerential equations is more

compatible with these types of boundary ﬂows [16].

New developments in discretization techniques depart from a Lagrangian for-

mulation of the governing equations. In the Lagrangian form, the ﬂuid is modeled

5

Figure 1.3: Fluid ﬂow around a moving propeller, found with the SPH method. The results

have been obtained by Femto Engineering [38].

as mass-containing nodes, which can move through the problem domain Ω. Since

the mass is ﬁxed to these nodes, it is much simpler to combine ﬂuids and multiple

phases. The meshless SPH method is often used, in order to solve the governing

partial diﬀerential equations in the Lagrangian form. Fig. 1.3 shows an instanta-

neous SPH image of a ﬂuid ﬂow.

Comparable to the Eulerian form in ﬂuid dynamics, are the topology optimiza-

tion methods based on ﬁxed and regular discretizations of the problem domain. In

these approaches, the presence of material is determined at every coordinate in the

design space. For the SIMP, ESO and level-set methods the amount of material

at every coordinate depends on the coordinate density, which can either take on

continuous or discrete values. In the Eulerian approach, this coordinate density

is mostly used as the design variable in the topology optimization problem.

This leaves open the possibility to investigate the opportunities of a ﬂow-

inspired Lagrangian approach in topology optimization. Such a ﬂow-inspired ap-

proach could result in a topology optimization method in which the density is

determined from the position of the mass-containing nodes. The design variable

for the Lagrangian approach is then not the density, but the nodal position. The

layout of the problem domain Ωcould possibly be transformed by moving the

nodes through the design space. Although this approach has not been the subject

of discussion in literature, it could possibly provide in exciting new openings in

topology optimization.

6

Figure 1.4: Normalized density of two constructed nodal distributions illustrating the MNA

concept, where the nodes are depicted as the blue circles. Material (white area) exist around

the nodes, and no material is present at the black areas. Grey areas contain intermediate density.

The question is then:

What are the opportunities for a ﬂow-inspired moving node approach in topology

optimization?

This thesis explores the possibility of a ﬂow-inspired meshless method-based ap-

proach in topology optimization. By employing the position of the nodes as de-

sign variables in the topology optimization method, this approach could possibly

provide for exciting new opportunities in topology optimization. The topology

optimization problem then transforms into a ﬂow-like problem, in which the ma-

terial moves to a more optimal distribution. An example of a change in layout

by the redistribution of nodes is shown in Fig. 1.4. This concept is comparable

to replacing the mesh in CFD by mass-containing nodes in an SPH setting. The

featuring characteristic of this newly proposed meshless method-based topology

optimization method is its moving node approach. Therefore, in this thesis this

newly proposed topology optimization method will be referred to as the Moving

Node Approach (hereafter: MNA).

1.4 Thesis structure

The focus of this thesis is the exploration of a ﬂow-inspired topology optimization

method and the assessment of its potential. To highlight this newly proposed

method, only the optimization of the layout of well established and relatively sim-

ple physical problems is considered in this thesis. More speciﬁcally, the employed

physical problem is the two-dimensional linear elastic solid under force loading

7

(hereafter: linear elasticity problem).

Since the MNA method will be based on meshless methods, it is of key impor-

tance to dissect relevant characteristics of widely used meshless methods. To gain

better insight in meshless methods, two diﬀerent meshless methods, respectively

the EFG and MLPG mixed collocation method, are investigated. Therefore, in

Chap. 2 the linear elasticity problem is discretized and an exemplary problem is

solved using these two meshless methods. Furthermore, a convergence study on

the accuracy of both meshless methods is performed.

In Chap. 3 the meshless method with the highest potential is tested further in

order to explore the eﬀect of the nodal distribution on the accuracy of the solution

and on the material distribution. The favorable properties of the meshless method

that can be utilized in the formulation of the MNA in topology optimization are

determined from exemplary problems. Moreover, additional conditions on the

nodal distribution are determined.

Chap. 4 will choose and adapt an existing meshless method in order to solve the

physical problem and eﬀectively be able to move the material layout in MNA. The

material distribution in the meshless method is altered by introducing a material

density deﬁned by the nodal compaction. Moreover, in this chapter the favorable

characteristics of this material distribution are determined.

Finally, in Chap. 5 the observations and alterations from previous chapters are

applied to propose an MNA topology optimization algorithm. This algorithm is

then used to determine the optimal shape of some exemplary problems. Based on

these ﬁndings favorable characteristics of the MNA are derived.

8

Chapter 2

Meshless Methods

In this chapter meshless method-based discretization techniques are discussed.

To gain better insight in meshless methods, the workings of two distinct meshless

methods, respectively the EFG and MLPG mixed collocation method, are investi-

gated. Sec. 2.1 discusses the approximations methods used in meshless methods.

In Sec. 2.2 the linear elasticity problem is discretized using both the EFG and

MLPG mixed collocation method. Next, in Sec. 2.3 the aforementioned meshless

methods are applied to the exemplary problem of a two-dimensional cantilever

beam and the speciﬁc characteristics of both methods are discussed. Finally, in

Sec. 2.4 the favorable characteristics of meshless methods for the MNA in topology

optimization are discussed.

2.1 Approximation methods

Sec. 2.1 introduces approximation methods, which are at the base of most dis-

cretization techniques. It is essential to have a clear understanding of approxima-

tion methods, in order to be able to use meshless methods. Here, three diﬀerent

approximation methods are discussed: FEM approximations (Sec. 2.1.1), kernel

approximations (Sec. 2.1.2) and Moving Least Squares (MLS) approximations

(Sec. 2.1.3). In order to provide insight in their workings, a general notation is

introduced as well as an exemplary problem in order to demonstrate how the

approximation methods function.

General notation

The three aforementioned approximation methods share a general notation which

is also used in FEM. A continuous scalar function u(x) and its derivative u,j(x) can

9

be approximated with a ﬁnite number of known scalar function values u(xI)=uI,

where I=1...n and nis the number of coordinates xI. These coordinates xIfor

which the scalar function is known are called the nodes. Here, xis the vector

containing the coordinates xifor i=1...l for a l-dimensional domain, and the

subscript ,j is the notation for the derivative with respect to coordinate xj.At

each node Ia shape function φI(x) is deﬁned, in order to interpolate between

the known scalar values at each node uIand to ﬁnd the approximated scalar

value and its derivative for all x. The weight of each shape function is given by

uI=u(xI), where the weight determines the contribution of each shape function

to the total approximation. The approximation of the continuous scalar function

u(x) is written as

u(x)≈uh(x)=

n

!

I=1

φI(x)uI(2.1)

and the approximation of the derivative u,j(x) equals

u,j(x)≈uh

,j(x)=

n

!

I=1

φI

,j(x)uI,(2.2)

where φI(x) is the value of the shape function belonging to node Iat coordinate

x,andφI

,j(x) is the value of the derivative in direction xjof the shape function

from node Iat coordinate x.

One-dimensional exemplary problem

Although the three approximation methods share a general notation, they dif-

fer in their shape functions. Sec. 2.1.1, Sec. 2.1.2 and Sec. 2.1.3 elaborate on

these distinctions. To exemplify these dissimilarities in the shape functions of the

approximations, the following one-dimensional problem is used.

The problem domain Ωis given by a line of length L= 10. The coordinates of

the domain are given by x. The boundary Γis given by the outer left coordinate

x= 0 and the outer right coordinate x=L. At each coordinate a scalar function

u(x) and its derivative u,x(x) are respectively deﬁned by

u(x)=−[L−x]2+L2+L2

4cos "6πx

L#(2.3)

10

0 L

0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Nodes

uI

u(x)

(a) Scalar function u(x) and scalar nodal val-

ues uI

0 L

−50

0

50

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Nodes

u,x(x)

(b) Derivative of the scalar function u(x),x

Figure 2.1: Scalar function u(x) and derivative of scalar function u,x(x) from Eq. 2.3 and Eq. 2.4.

The nodal values uIare also shown. The blue integers indicate the node numbers.

xI−1xIxI+1

φI(x)

Figure 2.2: Shape function φI(red line) for node I. Node Iand its neighboring nodes are

depicted as blue circles.

and

u,x(x)=2[L−x]−6πL

4sin "6πx

L#.(2.4)

The scalar function u(x) and its derivative u,x(x), can be approximated using the

values at a ﬁnite number of nodes. In the introduced one-dimensional problem,

the domain Ωis discretized by sixteen nodes, which are equally spaced. The scalar

function at each node is known, and is given by u(xI)=uIfor I=1...16. The

nodes, u(x)andu,x(x) are shown in Fig. 2.1.

2.1.1 FEM approximation

In order to create the shape function in FEM, connections between nodes need to

be assigned. The total collection of connections is also called the mesh. To mesh

the domain in the one-dimensional problem (shown in Fig. 2.1), neighboring nodes

need to be connected. This results in ﬁfteen elements, where a single element is a

connection between two nodes.

11

0 L

0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Nodes

uI

u(x)

uh(x)

(a) uh(x)

0 L

−50

0

50

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Nodes

u,x(x)

u,x

h(x)

(b) uh

,x(x)

Figure 2.3: Approximation of the scalar function uh(x) and the derivative of the scalar function

uh

,x(x) from (2.3) and (2.4) with FEM.

Now that the mesh is assigned, the shape function of each node can be con-

stituted. The simplest shape function in FEM is the linear shape function. The

value of this shape function equals one at the corresponding node and zero at

other nodes. Linear interpolation is applied between the corresponding node and

the nodes that are connected through elements [21], [20]. Fig. 2.2 shows the linear

shape function used in FEM. Once the shape function for each node is known,

the derivative of each shape function can be determined by simple taking the

derivatives of the linear interpolation functions.

At this point, the scalar values uI, the shape functions φI(x) and the derivative

of the shape functions φI

,x(x) are known for all nodes (I=1...n). From Eq. 2.1

and Eq. 2.2 the approximation of uh(x)anduh

,x(x) can be determined and are

shown in Fig. 2.4. The ﬁgure displays uh(xI)=uI, which is called the Kronecker

Delta property [39], [40], [41].1The approximation of the scalar function uh(x)

closely follows the scalar function u(x). However, because the derivative of the

shape function is discontinuous, the approximation of the derivative of the scalar

function uh

,x(x) is not very accurate. By increasing the number of nodes in the

mesh, the approximation of both uh(x)anduh

,x(x) will become more accurate,

although discontinuities in the derivative will always exist. A solution to these

discontinuities is the use of higher order shape functions.

1Although this property applies to FEM, meshless methods do in general not share this characteristic.

This will be further discussed in Sec. 2.1.2 and Sec. 2.1.3.

12

ΩIΩx

x

(a) Inﬂuence and support domain in FEM

ΩIΩx

x

(b) Inﬂuence and support domain in meshless

methods

Figure 2.4: One-dimensional nodal distribution (circles) and their typical FEM and meshless

method shape functions. For the node indicated by the black dot, the inﬂuence domain ΩIis

shown. The support domain Ωxof coordinate xis also shown.

Local characteristic of FEM

A characteristic of the shape functions in FEM is their local inﬂuence. The shape

function φI(x) is only unequal to zero if xlies on an element connected to node

I. Fig. 2.2 shows this local inﬂuence for a one-dimensional case. For each node I

the inﬂuence domain ΩIis a sub-domain of the total domain Ω, where the shape

function φI(x) is unequal to zero. For FEM this domain ΩIis the area covered

by the elements connected to node I. Thus, there are ndeﬁned inﬂuence domains

ΩIin the total domain Ω. For each coordinate x, a support (or support domain)

is given by Ωx. The support Ωxof coordinate xconsists of all nodes for which

the shape function is unequal to zero. Fig.2.4a shows the inﬂuence domain and

support domain for a one-dimensional FEM approximation.

Although the deﬁnition of these domains seems rather thorough for FEM,

meshless methods require a proper understanding of the inﬂuence domain ΩIof

node Iand support domain Ωxof coordinate x. With this characteristic con-

cerning locality, the summation in Eq. 2.1 and Eq. 2.2 can in FEM be reduced

to

u(x)≈uh(x)=

k

!

J=1

φJ(x)uJ(2.5)

and

u,j(x)≈uh

,j(x)=

k

!

J=1

φJ

,j(x)uJ,(2.6)

where J=1...k are the nodes in the support Ωxof coordinate x.

13

2.1.2 Kernel approximation

In the previous discussed FEM approximation (Sec. 2.1.1) a mesh needs to be

assigned in order to be able to calculate the shape functions for each node. The

kernel approximation is an approximation method in which there is no mesh

required. The kernel approximation is used in the meshless method called the

SPH method, which was the ﬁrst existing meshless method [17], [18]. A clear

description of the kernel approximation is given in the work of Liu and Liu [16].

Their work gives a thorough and clear description of the SPH method. Therefore

it is used as a guideline to explain the kernel approximation.

Basic approximation

A scalar function u(x) can be represented by the following integral notation:

u(x)=$Ω

u(x!)δ(x−x!)dx!with δ(x−x!)=%1x=x!

0x$=x!.(2.7)

Here δ(x−x!) is called the Dirac delta function. Eq. 2.7 is an exact representation

of u(x). An approximation of the scalar function uh(x) can be found by smoothing

a ﬁnite number of known values u(xI)=uI, where I=1...n.Tobeableto

smooth the known values in Eq. 2.7, the Dirac delta function is replaced by a

kernel function W(x−x!,d), and the integral is replaced by a summation of the

known scalar values uI. The approximation of u(x) is then

u(x)≈uh(x)=

n

!

I=1

W(x−xI,d)mI

ρ(x)uI,(2.8)

where mIis the a speciﬁed weight of each node, ρ(x) is the density at coordinate

xand dis a parameter specifying the size of the inﬂuence domain ΩIof the kernel

function. The properties of the kernel function will be discussed below. The

density ρ(x) is determined using the summation density approach, which equals

ρ(x)=

n

!

J=1

mJW(x−xJ,d).(2.9)

This approach speciﬁes that the density at each coordinate is the weighted average

of mI. Eq. 2.8 can be rewritten in the general form (Eq. 2.1) that is used in

14

approximation methods:

uh(x)=

n

!

I=1

φI(x)uIwith φI(x)= mIW(x−xI,d)

&n

J=1 mJW(x−xJ,d).(2.10)

The derivative of the approximation then becomes

uh

,j(x)=

n

!

I=1

φI

,j(x)uI(2.11)

with

φI

,j(x)= mIW,j (x−xI,d)

&n

J=1 mJW(x−xJ,d)−φI(x)&n

K=1 mKW,j(x−xK,d)

&n

J=1 mJW(x−xJ,d).(2.12)

Properties of the kernel function

For a accurate approximation uh(x) of the scalar function u(x), the kernel function

W(x−x!,d) should have three properties. The ﬁrst property is deﬁned by

$Ω

W(x−x!,d)dx!=1,(2.13)

which is called the unity condition. It implies that integration of the kernel func-

tion should produce unity. The second property is called the compact condition.

For each node Ian inﬂuence domain ΩIis deﬁned. The weight function of node

Iis zero outside this domain. This condition can be written as

W(x−xI,d) = 0 outside ΩI.(2.14)

Only for coordinates xthat lie within the inﬂuence domain of node I, the scalar

value uIhas an inﬂuence on the approximation. In accordance with this condition,

the summation in Eq. 2.11 is reduced to a summation over the nodes in the support

Ωxof coordinate x. The third and ﬁnal property is the delta function condition:

lim

d→0W(x−xI,d)=δ(x−x!).(2.15)

This condition ensures that for coordinates xcloser to a node, the value of the

kernel function is higher. Moreover, for smaller inﬂuence domains ΩI, the kernel

function resembles the Dirac delta function more closely. According to Eq. 2.7 this

results in a better approximation. However, reducing the inﬂuence domain size d

is only possible if there are enough nodes in the support domain Ωxof coordinate

15

x.

A normalized kernel function, that satisﬁes these aforementioned three condi-

tions, is the cubic spline weight function [14], [42], [43]:

W(x−xI,d)=W(r)=α

2

3−4r2+4r3if 0 ≤r≤1

2

4

3−4r+4r2−4

3r3if 1

2≤r≤1

0 otherwise

.(2.16)

The derivative of the cubic spline weight function equals

W,r(r)=α

−8r+12r2if 0 ≤r≤1

2

−4+8r−4r2if 1

2≤r≤1

0 otherwise

,(2.17)

where ris the normalized distance and αis a constant depending on the dimension

and shape of the kernel function. In general, αis chosen in such a way that the

unity condition from Eq. 2.13 is satisﬁed. However, because the summation den-

sity approach from Eq. 2.9 is used, the ﬁnal approximation becomes independent

of α.

In a one-dimensional domain, the normalized distance ris chosen such that

r=|x−xi|

d.(2.18)

In accordance with this equation, the inﬂuence domain ΩIof node Iis given by

−d≤xI≤d. A one-dimensional example of the cubic spline weight function

and its derivative, is shown in Fig. 2.5. Moreover, Fig. 2.4b shows the inﬂuence

domain and support domain for a one-dimensional Kernel approximation.

In a two-dimensional domain, the inﬂuence domain of a node can adopt any

shape. However, commonly used inﬂuence domains are circular or rectangular.

The kernel function for a circular inﬂuence domain ΩIis

W(x−xI,d)=W(r)(2.19)

and the derivative is

W,j(x−xI,d)=W,j (r)=w,r (r)r,j ,(2.20)

where ,j is the derivative with respect to the xjdirection and r=||x−xI||

d.Fora

16

W(r)

xI−1xIxI+1

d

(a) W(r)

W,r(r)

xI−1xIxI+1

d

(b) W,r(r)

Figure 2.5: A one-dimensional example of the spline kernel function and its derivative. The size

of the support domain is equals 2d.

rectangular inﬂuence domain ΩIthe kernel function equals

W(x−xI,d)=W(r1,d

1d)W(r2,d

2d)(2.21)

and the derivatives equal

W,1(x−xI,d)=W,1(r1)W(r2)=W,r1(r1)r1,1W(r2)and

W,2(x−xI,d)=W(r1)W,2(r2)=W(r1)W,r2(r2)r2,2,(2.22)

where rj=|xj−xI

j|

djdfor j=1,2andd1dand d2dare respectively the sizes of the

domain in the x1and x2direction. d1and d2are values determined from the

average node distance in de x1and x2direction. The size of the inﬂuence domain

ΩIalters when varying the parameter d.

An example of a problem domain Ωis shown in Fig. 2.6. In this ﬁgure both

circular and rectangular inﬂuence domains are shown. The size dof the rectangu-

lar domain should be chosen such that the total domain Ωis completely covered

by the sum of the local inﬂuence domains.

Solution to the one-dimensional exemplary problem

The kernel approximation is applied to the one-dimensional problem as shown

in Fig. 2.1. Because the kernel approximation is a meshless method, there is no

need for a mesh. Instead, the size dof the inﬂuence domain is provided for each

node. For simplicity, the inﬂuence domain is chosen to be similar for each node,

where d= 2. According to the approximation in Eq. 2.10, a weight mneeds to be

17

Ω

Γ

(a) Circular inﬂuence domains

Ω

Γ

(b) Rectangular inﬂuence domains

Figure 2.6: Example of a rectangular problem domain Ωand boundary Γ. The domain is

discretized with nodes (blue circles), each having an inﬂuence domain depicted in red.

assigned to each node. This weight is chosen to be the same for each node. The

cubic spline function from Eq. 2.16 is used as the weight function.

Next, the scalar function u(x) from Eq. 2.10 and its derivative u,x(x) from

Eq. 2.11 are approximated and shown in Fig. 2.7. This ﬁgure clearly shows that

the approximation of the scalar function u(x) is continuous, but does not satisfy

the Kronecker delta criterion i.e., uh(xI)$=uI. Although the approximation does

not satisfy this criterion, the approximation is still reasonably good. However,

near the boundaries of the domain Ω, the derivative of the approximation u,x(x)

is inaccurate. Since the derivative is often used in meshless methods, the accu-

racy of the approximation needs to be improved. In order to do so, an other

approximation method that reduces these inconsistencies near the boundaries is

explained in Sec. 2.1.3.

2.1.3 MLS approximation

The previously explained kernel approximation (Sec. 2.1.2) does not require a

mesh, however the derivative of the approximation shows inconsistencies near

the boundaries of the domain Ω. The MLS approximation is in fact, a more

general formulation of the kernel approximation, in which these inconsistencies

are reduced. Originally, the MLS approximation was created for ﬁtting a smooth

curve through a set of points, but nowadays the most common application is

probably the meshless EFG method [14], [15], [12].

18

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0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Nodes

uI

u(x)

uh(x)

(a) uh(x)

0 L

−50

0

50

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Nodes

u,x(x)

u,x

h(x)

(b) uh

,x(x)

Figure 2.7: Approximation of a scalar function u(x) and the derivative of the scalar function

u,x(x) with the kernel approximation. The size of the inﬂuence domain ΩIaround each node is

speciﬁed by d= 2.

Basic approximation

A scalar function u(x) can be approximated with a ﬁnite number of nnodes, for

which the scalar value u(xI)=uIis known (in which I=1...n). The approxima-

tion is deﬁned by

u(x)≈uh(x)=

m

!

l=1

pl(x)al(x)=pT(x)a(x),(2.23)

where p(x) is a vector containing monomials, mis the number of monomials in

p(x)anda(x) is a vector containing mcoeﬃcients. Two examples of a monomial

basis p(x) are the linear basis

p(x)=+1x,T1Dm=2

p(x)=+1xy

,T2Dm=3 (2.24)

and the quadratic basis

p(x)=+1xx

2,T1Dm=3

p(x)=+1xyx

2y2xy,T2Dm=6.(2.25)

To determine the coeﬃcients in a(x), a weighted least square ﬁt is performed

around coordinate x. The weighted least square ﬁt is found by minimizing a

19

weighted discrete L2norm, in which the norm equals

J(x)=

k

!

I=1

W(x−xI,d)-pT(xI)a(x)−uI.2.(2.26)

Here, xIare the coordinates of the nodes and W(x−xI,d) is a kernel function.

The properties of the kernel function are discussed in detail in Sec. 2.1.2. Note

that the summation over all nnodes in domain Ω, can be reduced to a summation

over the knodes in the support Ωxof coordinate x. This can be assumed since

the kernel function W(x−xI,d) has compact support. Eq. 2.26 is rewritten as

follows:

J(x)=[Pa(x)−u]TW(x)[Pa(x)−u],(2.27)

with

u=

u1

.

.

.

un

,P=

p1(x1)··· pm(x1)

.

.

.....

.

.

p1(xn)··· pm(xn)

and

W(x)=

W(x−x1,d) 0 ··· 0

0W(x−x2,d)··· 0

.

.

..

.

.....

.

.

0 0 ··· W(x−xn,d)

.(2.28)

The values of the coeﬃcients a(x) are found by minimizing the discrete L2

norm. The minimum is determined by taking the derivative of the functional

J(x) with respect to each term in a(x) and equal to zero i.e., ∂J(x)

∂al(x)= 0 for

l=1...m:

k

!

I=1

W(x−xI,d)2p(xI)-pT(xI)a(x)−uI.=0 (2.29)

or

k

!

I=1

W(x−xI,d)p(xI)pT(xI)a(x)=

k

!

I=1

W(x−xI,d)p(xI)uI.(2.30)

Eq. 2.30 can be rewritten in matrix notation as

A(x)a(x)=B(x)u,(2.31)

20

with

A(x)=PTW(x)P=

k

!

I=1

W(x−xI,d)p(xI)pT(xI)(2.32)

and

B(x)=PTW(x)=

k

!

I=1

W(x−xI,d)p(xI).(2.33)

The coeﬃcients a(x) are found by solving Eq. 2.31 according to

a(x)=A−1(x)B(x)u.(2.34)

The approximation from Eq. 2.23 can be rewritten by substituting the vector

a(x) with Eq. 2.34. With the use of the general formulation for the approximation

(Eq. 2.1), Eq. 2.23 becomes

uh(x)=pT(x)A−1(x)B(x)u=

k

!

I=1

φI(x)uI,(2.35)

where the shape function φI(x) equals

φI(x)=pT(x)A−1(x)W(x−xI,d)p(xI)

=cT(x)W(x−xI,d)p(xI)(2.36)

with

c(x)=A−1(x)p(x).(2.37)

The derivative u,j(x) of a scalar function u(x) is found according to Eq. 2.2.

Thus, in the MLS approximation it is only necessary to take the derivative of the

shape function from Eq. 2.36, which equals

φI

,j(x)=cT

,j(x)W(x−xI,d)p(xI)+cT(x)W,j (x−xI,d)p(xI) (2.38)

with

c,j(x)=-A−1.,j (x)p(x)+A−1(x)p,j (x)

=−A−1(x)A,j(x)A−1(x)p(x)+A−1(x)p,j (x)

=−A−1(x)[A,j (x)c(x)+p,j (x)] ,(2.39)

21

in which the derivative of the kernel function can be calculated analytically (see

Sec. 2.1.2) and the derivative of p(x) is simply a vector containing the derivative

of each individual term.

Similarity between MLS and kernel approximation

As mentioned before, the MLS approximation is a more general formulation of

the kernel approximation. However, the formulations of the shape functions given

in Eq. 2.10 and Eq. 2.36 apparently do not match. In one particular case, the

formulation of the shape function in the MLS and kernel approximation coincide.

This is explained below, starting from the MLS shape function.

Consider the monomial vector p(x) to contain a constant i.e., p(x)=1.

Eq. 2.36 is then written as

φI(x)=pT(x)A−1(x)W(x−xI,d)p(xI)=A−1(x)W(x−xI,d).(2.40)

Next, Eq. 2.32 can be written as

A(x)=

k

!

J=1

W(x−xJ,d)p(xJ)pT(xJ)=

k

!

J=1

W(x−xJ,d).(2.41)

When combining these two equations, this results in

φI(x)= W(x−xI,d)

&k

J=1 W(x−xJ,d).(2.42)

The shape function in the kernel approximation (Eq. 2.10) has previously been

deﬁned as

φI(x)= mIW(x−xI,d)

&k

J=1 mJW(x−xJ,d).(2.43)

The formulation of the MLS shape function and kernel shape function in Eq. 2.42

and Eq. 2.43 coincide when the weight mfor each node equals one. In conclusion,

the simplest form of the MLS approximation is the kernel approximation. The

MLS approximation provides more consistent results when increasing the number

of terms in the monomial vector p(x).

Solution to the one-dimensional exemplary problem

The MLS approximation is used to approximate the scalar function u(x) and its

derivative u,x(x) from Fig. 2.1. Although no mesh is required, the size dof the

22

φ8(x,d=2)

φ8(x,d=3)

φ8(x,d=4)

3 4 5 6 7 8 9 10 11 12 13

(a) φ8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

(b) All shape functions φIfor I=1...16

Figure 2.8: MLS shape function φIfor the center node I= 8 for d=2,3,4 and shape functions

φIfor all nodes, i.e. I=1...16, and d= 2.

inﬂuence domain of the kernel function has to be given. This size will aﬀect the

accuracy of the approximation. Therefore, the approximation from Eq. 2.3 is

compared for diﬀerent sizes d=2,3,4. The choice of monomials in p(x) also

aﬀects the approximation, therefore a linear and quadratic basis are compared

as well. The cubic spline weight function is chosen as the kernel function (see

Sec. 2.1.2).

First, the shape function corresponding to the center node I= 8 is shown in

Fig. 2.8 for d=2,3,4 and a linear monomial basis. The shape function for each

node is also given, with d= 2. Since the number of nodes decreases near the

boundary, the values of the shape function increase near the boundary.

By the use of the shape functions and their derivatives, the scalar function

Eq. 2.3 and its derivative Eq. 2.4 are approximated with a discrete number of

nodes according to Eq. 2.35. The approximations of the scalar function and its

derivative are shown in Fig. 2.9a and Fig. 2.9b, with d=2,3,4 and a linear

basis. In Fig. 2.9c and Fig. 2.9d the approximations of the scalar function and

its derivative are shown, with d= 3 and a linear or quadratic basis. Note that

for the MLS approximation, the Kronecker delta criterion uh(xi)=uidoes not

hold. This can be seen in Fig. 2.9a and Fig. 2.9c, where the approximation does

not intersect the scalar values at the nodes.

2.2 Discretization of the linear elasticity problem

This section introduces two meshless methods, which are based on the MLS ap-

proximation (Sec. 2.1.3). The MLS approximation is utilized to discretize the

governing equations for a linear elastic two-dimensional solid. However, there are

23

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0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

d=2

d=3

d=4

Nodes

uI

u(x)

uh(x)

(a) uh(x) linear basis

0 L

−50

0

50

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

d=2

d=3

d=4

Nodes

u,x(x)

u,x

h(x)

(b) uh

,x(x) with linear basis

0 L

0

20

40

60

80

100

120

140

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

p=linear

p=quadratic

Nodes

uI

u(x)

uh(x)

(c) uh(x) with d=3

0 L

−50

0

50

100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

p=linear

p=quadratic

Nodes

u,x(x)

u,x

h(x)

(d) uh

,x(x) with d=3

Figure 2.9: The approximation of the scalar function u(x) and its derivative uh

,x(x) with the

MLS approximation. Both functions are given in Eq. 2.3 and Eq. 2.4. The domain is discretized

with sixteen equally spaced nodes, and the approximation is calculated for diﬀerent meshless

parameters dand monomial basis p.

24

several forms of these linear elastic equations, and this is exactly where the two

meshless methods diﬀer from each other. Sec. 2.2.1 discusses the EFG method,

developed by Belytschko et al. [12]. This method is based on the weak form

of the linear elastic equations and requires a background mesh for the integra-

tion. Sec. 2.2.2 discusses the more recent MLPG method, developed by Atluri

and Zhu [40], [22]. The MLPG framework is based on a local weak form of

the linear elastic equations. To avoid integration, the mixed collocation form is

adopted [41], [39].

2.2.1 EFG method

The EFG method was proposed in 1994 by Belytschko et al. [12]. The formulation

of the EFG method has undergone some slight changes in order to be suitable for

multiple applications [14], [15], [13], [44]. In the EFG method, the linear elastic

equations are discretized by utilizing the MLS approximation. In this section, ﬁrst

the linear elastic equations and the weak form on which the EFG method will be

applied are recalled. Second, the linear elastic equations are discretized using

the EFG method. The enforcement of essential boundary conditions is discussed

next. Finally, the evaluation of the integrals present in the discretized equations

is discussed.

Linear elastic equations

The governing equations for a linear elastic solid, which occupies a domain Ω

bounded by Γ, are given by

LTσ+b=0 in Ω

σn=¯

ton Γt

u=¯

uon Γu

,(2.44)

where Lis the diﬀerential operator, σis the stress vector, bis the body force

vector, nis the normal vector on boundary Γt,¯

tis the prescribed traction on

boundary Γt,uis the displacement vector and ¯

uis the prescribed displacement

on boundary Γu. In two dimensions the vectors and matrices are

u=5u1(x)

u2(x)6,σ=

σ11(x)

σ22(x)

σ12(x)

,b=5b1(x)

b2(x)6,¯

u=5¯u1(x)

¯u2(x)6,¯

t=5¯

t1(x)

¯

t2(x)6,

25

n=5¯n1(x)

¯n2(x)6and L=

∂/∂x10

0∂/∂x2

∂/∂x2∂/∂x1,

(2.45)

where x=+x1x2,T. For a two-dimensional linear isotropic plain stress material,

the constitutive equations are

σ=D"with D=E

1−ν2

1ν0

ν10

001

and "=Lu,(2.46)

where "is the strain in the material, Dis a matrix containing the material

properties, Eis the Young’s modulus and νis the Poisson’s ratio.2

The EFG method is based on the weak form of Eq. 2.44, which can be found

by applying the principle of minimal potential energy [20]. The weak form is

determined by multiplying the linear elasticity equations with a test function v

and integrating this multiplication over the problem domain Ω:

$Ω{vTLTσ+vTb}dΩ=0 (2.47)

or

$Ω

[Lδu]TD[Lu]dΩ−$Ω

δuTbdΩ−$Γt

δuT¯

tdΓ=0,(2.48)

where in the EFG method the test function vis chosen to be similar to the

displacement uand δucorresponds to inﬁnitesimal variations of the displacement.

Discretization of the weak form

An approximation of the displacement uis found by discretizing Eq. 2.47 with

the MLS approximation (Sec. 2.1.3). With the use of the general formulation for

the shape functions (Eq. 2.1), the approximation of the displacement is written

as

u=5u1(x)

u2(x)6≈5uh

1(x)

uh

2(x)6=5&k

I=1 φI(x)ˆuhI

1

&k

I=1 φI(x)ˆuhI

26=

k

!

I=1

ΦIˆ

uhI

2Further in the text the two-dimensional linear isotropic plain stress linear elasticity equations will be

referred to as the linear elasticity equations.

26

=5φ1(x)0... φ

k(x)0

0φ1(x)... 0φk(x)6

ˆuh1

1

ˆuh1

2

.

.

.

ˆuhk

1

ˆuhk

2

=Φˆ

uh,(2.49)

with

ΦI=5φI(x)0

0φI(x)6and ˆ

uhI =5ˆuhI

1

ˆuhI

26.(2.50)

Here, ˆ

uhis the vector containing the approximated nodal displacement ˆuhI

i(for

i=1,2andI=1...k)andΦis the matrix containing all shape functions at xof

the nodes within the support domain Ωx. In the MLS approximation the shape

functions are given by Eq. 2.36.3Similarly, the inﬁnitesimal displacement δuis

approximated according to

δu≈

k

!

I=1

ΦIδˆ

uhI with δˆ

uhI =

δˆuh1

1

δˆuh1

2

.

.

.

δˆuhk

1

δˆuhk

2

.(2.51)

The derivative of the displacement Lu is approximated as

Lu ≈LΦˆ

uh=

∂/∂x10

0∂/∂x2

∂/∂x2∂/∂x1

5φ1(x)0... φ

k(x)0

0φ1(x)... 0φk(x)6

ˆuh1

1

ˆuh1

2

.

.

.

ˆuhk

1

ˆuhk

2

=

φ1

,1(x)0... φ

k

,1(x)0

0φ1

,2(x)... 0φk

,2(x)

φ1

,2(x)φ1

,1(x)... φ

k

,2(x)φk

,1(x)

ˆuh1

1

ˆuh1

2

.

.

.

ˆuhk

1

ˆuhk

2

3The formulation of the MLS approximation from Eq. 2.36 is slightly altered, i.e. uI

iis replaced with

ˆuhI

i. The approximation in the EFG method is not based on smoothing the exact nodal displacement,

but it smoothes an approximated nodal displacement, which is found by solving a set of equations.

Since the Kronecker delta criterion uI

i=uhI

idoes not hold, the approximated displacement values uhI

i

are replaced with approximated virtual nodal values ˆuhI

i.

27

=Bˆ

uh=

k

!

I=1

BIˆ

uhI ,(2.52)

with

BI=

φI

,1(x)0

0φI

,2(x)

φI

,2(x)φI

,1(x)

,(2.53)

where BIis the strain-displacement matrix of node Iat coordinate x.

The approximation of the individual elements from Eq. 2.47 is now known,

and the next step is to combine these elements. Using Eq. 2.52 and Eq. 2.54 the

ﬁrst term in Eq. 2.48 becomes

$Ω

δ[Lu]TD[Lu]dΩ=$Ω5k

!

I=1

BIδˆ

uhI 6T

D5k

!

J=1

BJˆ

uhJ 6dΩ

=$Ω

k

!</