ThesisPDF Available

The Moving Node Approach in Topology Optimization: An Exploration to a Flow-inspired Meshless Method-based Topology Optimization Method

Authors:
  • AMOLF & Eindhoven University of Technology

Abstract

In this thesis the possibility of a flow-inspired Meshless Method based topology optimization method is explored. More specific, an investigation is carried out to the possibility of using the position of the nodes from the meshless EFG method as design variables in topology optimization. In order to determine the possibility of using the nodal positions as design variables, the influence of variations in the nodal distribution on the EFG method accuracy is investigated. It turned out that the solution to the governing equations obtained with the EFG method is highly sensitive to nodal displacements. The most accurate results are obtained for regularly spaced nodal distributions. Moreover, the relation between the nodal distribution and the material distribution is investigated. The discretization of irregular shapes, often encountered in topology optimization, proves difficult. The difficulties arise from discontinuities in the material distribution and the occurrence of large material domains without nodes. These complications are mainly caused by the interaction between the nodal influence domains and the fixed background mesh. The above-mentioned difficulties are avoided by the introduction of a material density that depends on the nodal compaction. Although the fixed background mesh is still present in this material distribution, the shape of the structure can be altered continuously. Moreover, with this material description the layout of the structure becomes sensitive to changes in the nodal position. This sensitivity can be utilized to alter the shape towards an optimal layout. However, the influence of the nodal position on the EFG method accuracy remains a problem. Finally, a fairly simple Moving Node Approach (hereafter: MNA) topology optimization algorithm is proposed. In this proposed algorithm the influence of the nodal position on the EFG method is circumvented by using two types of nodes. One node type deals with the discretization of the linear elasticity equations and the other node type deals with the distribution of mass. In order to reach the optimal layout, each mass node is accelerated along the compliance sensitivity.
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 Eect of the nodal distribution in the EFG method 48
3.1 Eect of the nodal distributions on accuracy . . . . . . . . . . . . 49
3.1.1 Global eect of nodal distribution . . . . . . . . . . . . . . 49
3.1.2 Local eect of the nodal position in the MLS approximation 58
iv
3.2 Eect 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 Dierence 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 dierence between the various topology optimization problems is the phys-
ical problem, i.e. the governing dierential 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 dicult 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 dicult 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 dierential equa-
tion, or a set of partial dierential 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 dierential equations are reduced to a set of
simple linear algebraic equations.
FEM is based on the discretization of the domain and its boundary Γby
a mesh. To create the mesh, nodes are distributed in the domain and 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 diculties arise when using a mesh. It is for instance dicult
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 diculties that arise when using FEM, meshless discretization
techniques have been applied. These techniques are also called meshless methods.
3
(a) Typical FEM discretization
Ω
Γ
(b) Typical meshless method discretization
Figure 1.2: Typical FEM and meshless method discretization of a rectangular domain and
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 and 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 dierential 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 dierential 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 dierent 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 dierential equations are usually expressed in the
Eulerian form, i.e. at each coordinate in the problem domain the fluid velocity
and density are tracked [37]. This approach works well for problems in which
the complete problem domain consists 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 dierential 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 . 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 dierential 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 could 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 dierent 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 eect 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 eectively 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 dierent
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 is 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)=[Lx]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.
xI1xIxI+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[Lx]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 is 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
(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 Iis
shown. The support domain xof 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 Iis a sub-domain of the total domain , where the shape
function φI(x) is unequal to zero. For FEM this domain Iis the area covered
by the elements connected to node I. Thus, there are ndefined influence domains
Iin the total domain . For each coordinate x, a support (or support domain)
is given by x. The support xof 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 Iof
node Iand support domain xof 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 xof 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)=$
u(x!)δ(xx!)dx!with δ(xx!)=%1x=x!
0x$=x!.(2.7)
Here δ(xx!) 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(xx!,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(xxI,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 Iof 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(xxJ,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(xxI,d)
&n
J=1 mJW(xxJ,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 (xxI,d)
&n
J=1 mJW(xxJ,d)φI(x)&n
K=1 mKW,j(xxK,d)
&n
J=1 mJW(xxJ,d).(2.12)
Properties of the kernel function
For a accurate approximation uh(x) of the scalar function u(x), the kernel function
W(xx!,d) should have three properties. The first property is defined by
$
W(xx!,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 Iis defined. The weight function of node
Iis zero outside this domain. This condition can be written as
W(xxI,d) = 0 outside I.(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
xof coordinate x. The third and final property is the delta function condition:
lim
d0W(xxI,d)=δ(xx!).(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 I, 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 xof coordinate
15
x.
A normalized kernel function, that satisfies these aforementioned three condi-
tions, is the cubic spline weight function [14], [42], [43]:
W(xxI,d)=W(r)=α
2
34r2+4r3if 0 r1
2
4
34r+4r24
3r3if 1
2r1
0 otherwise
.(2.16)
The derivative of the cubic spline weight function equals
W,r(r)=α
8r+12r2if 0 r1
2
4+8r4r2if 1
2r1
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=|xxi|
d.(2.18)
In accordance with this equation, the influence domain Iof node Iis given by
dxId. 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 Iis
W(xxI,d)=W(r)(2.19)
and the derivative is
W,j(xxI,d)=W,j (r)=w,r (r)r,j ,(2.20)
where ,j is the derivative with respect to the xjdirection and r=||xxI||
d.Fora
16
(a) W(r)
W,r(r)
xI1xIxI+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 Ithe kernel function equals
W(xxI,d)=W(r1,d
1d)W(r2,d
2d)(2.21)
and the derivatives equal
W,1(xxI,d)=W,1(r1)W(r2)=W,r1(r1)r1,1W(r2)and
W,2(xxI,d)=W(r1)W,2(r2)=W(r1)W,r2(r2)r2,2,(2.22)
where rj=|xjxI
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
Ialters when varying the parameter d.
An example of a problem domain is 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 is 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
(a) Circular influence domains
Ω
Γ
(b) Rectangular influence domains
Figure 2.6: Example of a rectangular problem domain and 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 , 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 . 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
(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 Iaround 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 mcoecients. 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 coecients 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(xxI,d)-pT(xI)a(x)uI.2.(2.26)
Here, xIare the coordinates of the nodes and W(xxI,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 , can be reduced to a summation
over the knodes in the support xof coordinate x. This can be assumed since
the kernel function W(xxI,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(xx1,d) 0 ··· 0
0W(xx2,d)··· 0
.
.
..
.
.....
.
.
0 0 ··· W(xxn,d)
.(2.28)
The values of the coecients 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(xxI,d)2p(xI)-pT(xI)a(x)uI.=0 (2.29)
or
k
!
I=1
W(xxI,d)p(xI)pT(xI)a(x)=
k
!
I=1
W(xxI,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(xxI,d)p(xI)pT(xI)(2.32)
and
B(x)=PTW(x)=
k
!
I=1
W(xxI,d)p(xI).(2.33)
The coecients a(x) are found by solving Eq. 2.31 according to
a(x)=A1(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)A1(x)B(x)u=
k
!
I=1
φI(x)uI,(2.35)
where the shape function φI(x) equals
φI(x)=pT(x)A1(x)W(xxI,d)p(xI)
=cT(x)W(xxI,d)p(xI)(2.36)
with
c(x)=A1(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(xxI,d)p(xI)+cT(x)W,j (xxI,d)p(xI) (2.38)
with
c,j(x)=-A1.,j (x)p(x)+A1(x)p,j (x)
=A1(x)A,j(x)A1(x)p(x)+A1(x)p,j (x)
=A1(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)A1(x)W(xxI,d)p(xI)=A1(x)W(xxI,d).(2.40)
Next, Eq. 2.32 can be written as
A(x)=
k
!
J=1
W(xxJ,d)p(xJ)pT(xJ)=
k
!
J=1
W(xxJ,d).(2.41)
When combining these two equations, this results in
φI(x)= W(xxI,d)
&k
J=1 W(xxJ,d).(2.42)
The shape function in the kernel approximation (Eq. 2.10) has previously been
defined as
φI(x)= mIW(xxI,d)
&k
J=1 mJW(xxJ,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 aect the
accuracy of the approximation. Therefore, the approximation from Eq. 2.3 is
compared for dierent sizes d=2,3,4. The choice of monomials in p(x) also
aects 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
(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
(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 dierent meshless
parameters dand monomial basis p.
24
several forms of these linear elastic equations, and this is exactly where the two
meshless methods dier 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
bounded by Γ, are given by
LTσ+b=0 in
σn=¯
ton Γt
u=¯
uon Γu
,(2.44)
where Lis the dierential 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
1ν2
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 :
${vTLTσ+vTb}d=0 (2.47)
or
$
[Lδu]TD[Lu]d$
δuTbd$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)65uh
1(x)
uh
2(x)6=5&k
I=1 φI(xuhI
1
&k
I=1 φI(xuhI
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 x. 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
$
δ[Lu]TD[Lu]d=$5k
!
I=1
BIδˆ
uhI 6T
D5k
!
J=1
BJˆ
uhJ 6d
=$
k
!</