analysis of crack propagation in an elastic bar using meshfree methods
TRANSCRIPT
ANALYSIS OF CRACK PROPAGATION IN AN ELASTIC BAR USING MESHFREE
METHODS
Introduction
Mesh free methods is used to establish a system of equations for the whole problem domain
without the use of predefined mesh. Mesh free methods use a set of nodes scattered within the
problem domain as well as the sets of scattered nodes on boundary domain to represent the
domain or the boundaries. These sets of nodes do not form a mesh, which means that there is no
relationship between nodes is required, at least for field variable interpolation.
Aim of the thesis
The aim was to implement the Element Free Galerkin method to compare the
method and its accuracy with the exact and FEM solution for crack propagation in an elastic bar.
The understanding of the basic procedure for meshfree method is developed and MATLAB code
is developed to solve the system of equations for calculating the increase in the length of crack in
the bar under the influence of point load and body force. The increase in the length due to
temperature is included in the MATLAB code. The Element Free Galerkin method was chosen
for this Master Thesis, to investigate its applicability and accuracy.
Features of mesh free methods
The some of the most important features of meshfree methods, often in comparison to
analogous properties of mesh-based methods are :-
(a) Absence of mesh
In meshfree methods the connectivity of the nodes is determined at run-
time
No mesh alignment sensitivity, this is a serious problem in mesh-based
calculations.
h-adaptivity is comparably simple with meshfree methods as only nodes
have to be added, and the connectivity is then computed at run-time
automatically. p-adaptivity is also conceptionally simpler than in mesh
based methods.
No mesh generation at the beginning of the calculations is necessary. This
is still not fully automatic may require human intervention especially in
complex domains.
No re-meshing during the calculations. Especially in problems with large
deformations of domains or moving discontinuities a frequent re-meshing
is needed in mesh based methods, which is difficult.
(b) Continuity of shape functions: The shape functions of meshfree methods may be
easily constructed to have the desired order of continuity.
Meshfree methods readily fulfill the requirement on the continuity arising
from the order of the problem under consideration.
No post-processing is required in order to determine the smooth
derivatives of the unknown functions.
(c) Convergence: For the same order of consistency numerical experiments suggest that
the convergence results of the meshfree methods are often considerably better than the results
obtained by the mesh-based shape functions.
(d) Computational effort: In practice, for a given reasonable accuracy, meshfree methods
are often considerably more time-consuming than their mesh-based counter parts.
Meshfree shape functions are of a more complex nature than the
polynomial-like shape functions of mesh-based methods. Consequently,
the number of integration points for a sufficiently accurate evaluation of
the integrals of the weak form is considerably larger in meshfree methods
than in mesh-based methods.
At each integration point the following steps are necessary to evaluate
the meshfree shape function: Neighbor search, solution of small system
of equations and small matrix-matrix and matrix –vector operations to
determine the derivatives.
The resulting global system of equations has in general a larger
bandwidth for meshfree methods than for comparable mesh-based
methods.
(e) Essential boundary conditions: Most of the meshfree methods lack the Kronecker
delta property. This is in contrast to the mesh-based methods which often posses this property.
Consequentially the imposition of the essential boundary conditions requires certain attention in
meshfree methods and may degrade the convergence of the method.
Overview to Meshfree Methods
2.1 Introduction
The Finite Element Methods used for modeling and analysis of practical problems in
many fields of engineering are well-established. With the development of the computer
technology, the Finite Element Methods have become the most popular tool due to its advantages
in computer implementation. This is a robust and thoroughly developed technique.
However, the finite element methods have some shortcomings. It is not always
advantageous in handling some problems, consider the modeling of large deformation process
(remarkable loss in accuracy happens when the elements in the mesh become extremely skewed
or compressed), the growth of cracks with arbitrary and complex paths, moving discontinuities,
fracture mechanics and phase change problems. These kinds of problems are difficult to be
solved efficiently by finite element methods because of its reliance on the pre-defined mesh.
The solution was found to re-mesh the domain of the problem at each step during the
simulation allowing the mesh-lines to remain distortion free. Mesh generation each time is a far
more time consuming and expensive task than the assembly and the solution.
Therefore, a new method was needed. Meshless methods have been developed to handle
these difficulties. These do not require a mesh to discretize the problem domain. All meshfree
methods share the common feature that the partial differential equation model can be discretized
without any structured and pre-defined meshing. Mesh-free approximations are constructed
using a set of scattered particles or nodes that have no particular pre-defined relationship or
connection among them.
The meshfree methods eliminate the difficulties experienced by finite element methods
by approximating it entirely in terms of nodes which may or may not be uniformly distributed in
the domain of interest. The connectivity between nodes is completely defined by the overlap of
the nodal domains of influence. This methodology uses only the nodes.
2.2 Literature review
Mesh free methods go back to the seventies. The major difference to finite element
methods is that the domain of interest is discretized only with nodes, often called particles. These
particles interact via meshfree shape functions in a continuum framework similar as finite
elements do although particle “connectivity” can change over the course of a simulation. The
most important advantages of mesh free methods compared to finite element methods are their
higher order continuous shape functions that can be exploited for higher smoothness; simpler
incorporation of p and h-adaptivity and certain advantages in crack problems. The most
important drawback of meshfree methods is probably their higher computational cost, regardless
of some instability that certain meshfree methods have.
One of the oldest meshfree methods is the Smooth Particle Hydrodynamics (SPH)
developed by Lucy and Gingold and Monaghan in 1977. SPH was first applied in astrophysics to
model phenomena such as supernova and was later employed in fluid dynamics. In 1993,
Petschek and Libersky extended SPH to solid mechanics. Early SPH formulations suffered from
spurious instabilities and inconsistencies that were a hot topic of investigations. Many corrected
SPH versions were developed that improved either the stability behavior of SPH or its
consistency. Consistency, often referred to as completeness in a Galerkin framework, means the
ability to reproduce exactly a polynomial of certain order.
Based on the idea of Lancaster and Salkauskas and probably motivated by the purpose
to model arbitrary crack propagation without computational expensive re-meshing, the group of
Prof. Ted Belytschko developed the Element Free Galerkin (EFG) method in 1994. The EFG
method is based on an MLS approximation and avoids inconsistencies inherent of some SPH
formulations. In 1995, the group of Prof. W.K. Liu proposed a similar method, the Reproducing
Kernel Particle Method (RKPM). Though the method is very similar to the EFG method, it
originates from wavelets rather than from curve-fitting. The first method that employed an
extrinsic basis was the hp-cloud method of Duarte and Oden. In contrast to the EFG and RKPM
method, the hp-cloud method increases the order of consistency (or completeness) by an
extrinsic basis. In other words, additional unknowns were introduced into the variational
formulation to increase the order of completeness. This idea was later adopted (and modified) in
the XFEM context through the extrinsic basis (or extrinsic enrichment) and used to describe the
crack kinematics rather than to increase the order of completeness in a p-refinement sense. The
group of Prof. Ivo Babuska discovered certain similarities between finite element methods &
meshfree methods and formulated a general framework, the Partition of Unity Finite Element
Method (PUFEM) that is similar to the generalized Finite Element Method (GFEM) of
Strouboulis and colleagues.
Another very popular meshfree method worth mentioning is the Meshless Local Petrov
Galerkin (MLPG) method developed by the group of Prof. S.N. Atluri in 1998. The main
difference of the MLPG method to all other methods mentioned above is that local weak forms
are generated over overlapping sub-domains rather than using global weak forms. The
integration of the weak form is then carried out in these local sub-domains. In this context, Atluri
introduced the notion of “truly” meshfree methods since truly meshfree methods do not need
construction of any background mesh that is needed for integration.
The issue of integration in meshfree methods was a topic of investigations since its early
times. Methods that are based on a global weak form may use different types of integration
schemes: nodal integration, stress-point integration and Gauss quadrature based on a background
mesh that does not necessarily need to be aligned with the particles. Nodal integration is from the
computational point of view, the easiest and cheapest way to build the discrete equations but
similar to finite elements, meshfree methods based on nodal integration suffer from instability
due to rank deficiency. Adding stress points to the nodes can eliminate (or at least alleviate) this
instability. The term stress-point integration comes from the fact that additional nodes were
added to the particles where only stresses are evaluated. All kinematics values are obtained from
the "original" particles. The concept of stress points was actually first introduced in one
dimension in an SPH setting by Dyka. This concept was introduced into higher order dimensions
by Randles and Libersky and the group of Prof. Belytschko. There is a subtle difference between
the stress point integration of Belytschko and Randles and Libersky. While Randles and Libersky
evaluate stresses only at the stress points, Belytschko and colleagues evaluate stresses also at the
nodes. Meanwhile, many different versions of stress point integration were developed. The most
accurate way to obtain the governing equations is Gauss quadrature. In contrast to finite
elements, integration in meshfree methods is not exact. A background mesh has to be constructed
and usually a larger number of quadrature points as in finite elements are used. For example,
while usually 4 quadrature points are used in linear quadrilateral finite elements, Belytschko
recommends use of 16 quadrature points in the EFG method.
Another important issue regarding the stability of mesh free methods is related to the kernel
function, often called window or weighting function. The weighting function is somehow related
to the meshfree shape function. The weight function can be expressed in terms of material
coordinates or spatial coordinates. We then refer to Lagrangian or Eulerian kernels, respectively.
Early meshfree methods such as SPH use an Eulerian kernel. Many meshfree methods that are
based on Eulerian kernels have a so-called tensile instability, meaning the method gets unstable
when tensile stresses occur. In a sequence of papers by Belytschko, it was shown that the tensile
instability is caused by the use of an Eulerian kernel. Meshfree methods based on Lagrangian
kernels do not show this type of instability. Moreover, it was demonstrated that for some given
strain softening constitutive models, methods based on Eulerian kernels were not able to detect
the onset of material instability correctly while methods that use Lagrangian kernels were able to
detect the onset of material instability correctly. This is a striking drawback of Eulerian kernels
when one wishes to model fracture.
However, a general stability analysis is difficult to perform and will of course also depend on
the underlying constitutive model. Note also, that Libersky proposed a method based on Eulerian
kernels and showed stability in the tension region though he did not consider strain softening
materials. For too large deformations, methods based on Lagrangian kernels tend to get unstable
as well since the domain of influence in the current configuration can become extremely
distorted. Some recent methods to model fracture try to combine Lagrangian and Eulerian
kernels though certain aspects still have to be studied, e.g. what happens in the transition area or
how are additional unknowns treated (in case an enrichment is used).
In mesh free methods, we talk about approximation rather than interpolation since the
meshfree shape functions do not satisfy the Kronecker-delta property. This entails certain
difficulties in imposing essential boundary conditions. Probably the simplest way to impose
essential boundary conditions is by boundary collocations. Another opportunity is to use the
penalty method, Lagrange multipliers or Nitsche’s method.
Coupling to finite elements is one more alternative that was extensively pursued in the
literature-in this case; the essential boundary conditions are imposed in the finite element
domain. In the first coupling method by Belytschko, the meshfree nodes have to be located at the
finite element nodes and a blending domain is constructed such that the shape functions are zero
at the finite element boundary. In this first approach, discontinuous strains were obtained at the
meshfree-finite element interface.
Many improvements were made and methods were developed that exploit the advantage of
both meshfree methods and finite elements, e.g. the Moving Particle Finite Element Method
(MPFEM) by Su Hao et al. or the Reproducing Kernel Element Method (RKEM) developed by
the group of Prof. W.K. Liu. Meanwhile, several textbooks on mesh free methods have been
published, WK Liu and S Li, Prof T Belytschko, SN Atluri and some books by Prof. GR Liu.
Over the last years a number of different mesh free methods have been developed. Some of the
most well known methods are:
2.2.1 Smooth particle hydrodynamics (SPH) was introduced by Lucy (1977) and further
developed by Monaghan (1982). It is the simplest method, partly because it is a point collocation
method and also because the shape functions are very simple with no special cases at the
boundary. It has some problems with both stability and accuracy, but many corrections have
been made to improve the method, for example better integration scheme and correction term in
the shape functions.
2.2.2 Element free galerkin (EFG) was developed by Dr, Belytschko et al. (1994). The shape
functions are constructed with MLS approximation, and the test function is chosen as the shape
function. Boundary conditions are enforced by Lagrange multiplier and in general a lot of gauss
integration points are needed to get accurate results.
2.2.3 Reproducing kernel particle method (RKPM) was created by Liu et al. (1995). It is a
particle method, but instead of point collocation it uses a Galerkin formulation. Also the shape
function has a correction term to improve the accuracy at the boundary.
2.2.4 Truly meshfree method or Mesh less local petrov-galerkin (MLPG) was originated by
SN Atluri and Zhu (1998). Instead of a global weak form, it has a local weak form. Therefore no
integration over the domain is necessary, so no background mesh is needed as for example EFG
and RKPM. MLPG can have different shape functions and test functions and is then named as
MLPG1, MLPG2 and so on.
2.2.5 Natural element method (NEM) was developed by Prof N Sukumar et al. (1998). NEM
solves a Galerkin formulation of the problem; here the shape functions are constructed in a
different fashion. The domain is divided in Voroni cells so the NEM fulfills the Kronecker-delta
property and therefore it is straightforward to implement essential boundary conditions.
There are uncountable methods that exist in the literature. During the last two decades,
several meshfree methods for seeking approximate solutions of partial differential equations
have been proposed; we can have a look on the Table 2.1. All of these methods, except for the
MLPG, the SPH and the collocation method are not truly meshfree since the use of shadow
elements are required for evaluating integrals appearing in the governing weak formulations.
TABLE 2.1
Some Meshfree methods developed and their features
S.
no.
Method Reference Approximation Function
1 Diffuse element method Nayroles et al., 1992 MLS approximation,
Galerkin method
2 Element free Galerkin
Method (EFG)
Belytschko et al.
(1994)
MLS approximation,
Galerkin method
3 Meshless local Petrov-
Galerkin Method
(MLPG)
Atlury and Zhu, 1998 MLS approximation,
Petrov-Galerkin method
4 Finite Point Method Onate et al., 1996;
Liszka and
Orkisz,1980;Jensen,19
80
Finite Differential
Representation (taylor
series), MLS
approximation
5 Smooth particle
hydrodynamics
Lucy,1977; Gingold
and Monagan,1977
Integral representation
6 Reproducing kernel
particle method
Liu WK et al.1993 Integral representation
(RKPM)
7 hp-clouds Odan and Abani, 1994;
Armando and
Oden,1995
Partition of unity, MLS
8 Partition of unity FEM Babuska and Melenk, Partition of unity, MLS
1995
9 Point interpolation
method
Liu GR and Gu, 1999,
2000, 2001
Point interpolation
10 Boundary node method Mukherjee and
Mukherjee 1997
MLS
11 Boundary point
interpolation method
Liu GR and Gu, 2000;
Gu and Liu GR, 2001
Point interpolation
2.3 Classification of the meshfree methods
The meshfree methods may be classified in three classes shown by flowchart 2.1. These
are classified by the construction procedure for partition of unity, based upon this the
approximation function is selected. The choice of the test function forms the last step in the
characterization of the meshfree methods:
(a) Construction of the partition of unity
(b) Choice of an approximation function
(c) Choice of the test unction
FEMMesh-based methods
MLS RKPM Meshfree methods
PU
Construction ofPartition of Unity
Intrinsic basis only Additional extrinsic basis
Choice of approximation
Choice of test function
1. SPH2. EFG3. LSMM4. MLPG 15. MLPG 26. MLPG 3 etc.
1. PUM2. XFEM3. hp-clouds
1. FDM2. FVM3. Least square FEM
Flowchart 2.1 Classification of meshfree methods
2.4 Closure
In this chapter, the historical background and the classification regarding the meshfree
methods has been presented in brief. The most of referred papers considered the elastic bar with
uniform cross-section subjected to body force and Timoshenko beam as an example to verify and
compare the results of meshfree methodology.
Fundamentals of Meshfree Methods
3.1 Introduction
One of the most important things engineers and designers do is to model physical
phenomena. Virtually every phenomena in nature, whether mechanical, chemical or biological
can be described with the aid of laws of physics in terms of algebraic, differential or integral
equations relating the various quantities of interest.
This analytical description of physical phenomena and process is called mathematical
model. The use of a numerical method and a computer to evaluate the mathematical model of a
process and estimate its characteristics is called numerical simulation. There are various
numerical simulation methods, such as finite difference method (FDM), finite element method
(FEM) and variational methods etc. for analysis of any physical phenomenon occurring in
nature. The development of various methods of analysis is presented by the flowchart 3.1 in a
chronological order. In this chapter, we will discuss finite element method and compare the
meshfree methodology with help of the flowchart 3.2. The various terms in meshfree methods
are also defined.
3.2 Meshfree method in comparison to FEM
A brief introduction to the differences between finite element method and meshfree
methods is as discussed below.
3.2.1 Finite element method
The finite element method is a powerful numerical method for solving engineering
problems and can analyze complex, structural, mechanical and electrical system. The finite
element method is used to analyze both linear and nonlinear system and to solve problems in
static and dynamic systems. Because of its various applications such as heat transfer,
Methods of Analysis
Experimentation on actual model
Experimentation on Prototype
Mathematical model
Numerical Simulationor
Computer simulation
Classical Variational methods
1. Rayleigh-Ritz method2. Galerkin method
3. Collocation method, etc
Approximation methods
1. Finite difference method (FDM)2. Finite volume method (FVM)3. Finite element method (FEM)
4. Meshfree methods
electrostatic potential, fluid mechanics, vibration analysis and so on, the finite element method
has become very popular.
Flowchart 3.1 Development of methods of analysis
GEOMETRY GENERATION
Element Mesh Generation Nodal Mesh Generation
FEM MeshfreeFEM
Shape Function CreationBased on Element Predefined
Shape Function Creation BasedOn Nodes in a Local Domain
System Equation for Elements System Equation for Nodes
Global Matrix Assembly
Essential Boundary Condition
Support Specification
Solution for displacements
Computation of Strains andStresses from Displacements
Results Assessment
Flowchart 3.2 Flowchart for FEM and meshfree method procedures
In many physical problems, it is very difficult to find an analytical solution for complex
geometry with complex boundary conditions. In the finite element method, a complex region
defining a continuum is discretized into simple geometric shapes called finite elements that are
connected at specified node points as shown in Figure 3-1. The shapes of the elements are
intentionally made as simple as possible, such as triangles and quadrilaterals in two-dimensional
domains, and tetrahedral, pentahedral and hexahedra in three dimensions. The entire mosaic-like
pattern of elements is called a mesh.
Figure 3.1 Boundary representations in FEM and meshfree methods
The material properties and the governing relationships are considered over these
elements and expressed in terms of unknown field values at element corners. These equations
describe the physical problem that is to be analyzed and generally can be expressed using the
weak form based on variational principle.
3.2.2 Meshfree method
Creation of a mesh for the problem domain is a prerequisite in FEM packages and the
analysts spend the majority of their time in creating the mesh. The stresses obtained using FEM
packages are discontinuous and less accurate if the mesh density and quality is poor. When
handling large deformation, considerable accuracy is lost because of the element distortion. The
root of these difficulties was the element mesh and the idea of eliminating the mesh has evolved
to counter this problem.
The meshfree method is used to establish a system of algebraic equations for the whole
problem domain without using a predefined mesh. Meshfree methods use a set of nodes scattered
within the problem domain as well as node scattered on the boundaries of the domain to
represent the problem domain and its boundaries. These sets of scattered nodes do not form a
mesh, which means that no information on the relationship between the nodes is required, at least
for field variable interpolation. In Mesh Free method (Liu 2003) [2], adaptive schemes can be
easily developed, as there is no mesh, no connectivity is involved. This provides flexibility in
adding or deleting points/ nodes whenever and wherever needed. The analyst can save the time
spend on conventional mesh generation because there is no mesh, and the nodes can be created
by a computer in a fully automated manner. This can translate into major cost and time savings
in modeling and simulation projects.
The fundamental difference between FEM and Meshfree method is the construction of
the shape functions. In FEM, the shape functions are constructed using elements. These shape
functions are therefore predetermined for different types of elements before the finite element
analysis starts. In Meshfree method, the shape functions constructed are usually only for a
particular point of interest, and the shape function changes as the location of the point of interest
changes. The construction of the element free shape function is performed during the analysis.
Modeling the geometry is also a difference between these two methods. In FEM, curved parts of
the geometry and its boundary can be modeled using curves and curved surfaces using high-
order elements. If linear elements are used, these curves and surfaces are straight lines or flat
surfaces. Figure 3-1 shows an example of smooth boundary approximated in the finite element
model by the straight line edges of triangular elements. The accuracy of representation of the
curved parts is controlled by the number of elements and the order of the approximation.
In Mesh Free methods, the boundary is represented by nodes as shown in Figure 3-1. At
any point between two nodes on the boundary, one can interpolate using Meshfree shape
functions. Because the Meshfree shape functions are created using nodes in a moving local
domain, the curved boundary can be approximated very accurately even if linear polynomial
basis functions are used.
3.2.2.1 Meshfree terminology
The common terms used in the meshfree methodology are concisely defined and
presented for easy assimilation.
3.2.2.2 Support domain
The support domain is defined as the domain/area or field that is affected or influenced
by any point of interest xQ, in the problem domain Ω, this point of interest may or may not be a
node. The influence domain is defined as the domain that is affected or influenced by any node
of interest xi in the domain. The influence domain is defined for each node in the problem
domain, and it can be different from node to node to represent the area of influence of any node.
Figure 3.2 Circular and rectangular domains of Influence
The concept of influence domain is clarified by the figure 3.2; node 1 has an influence
domain of radius r1, and node 2 has an influence of radius r2. These domains of influence may
have different shapes and dimensions. Most commonly used shapes are circular or rectangular
ones. The figure 3.3 depicts the circular domain of influence of a node represented by Ω I,
whereas the problem domain is shown by Ω.
Figure 3.3 Circular domains of influence
3.2.2.3 Node connectivity
In case of finite element methods the nodes are connected by the elements whereas in
case of meshfree methods the node connectivity is established by the overlapping domains of
influence. Figure 3.4 shows
the overlapping domains of influence and local node numbering at point of interest xQ.
Figure 3.4 Overlapping domains of influence and node numbering at point xQ
3.2.2.4 Dimension of support domain
The dimensions of domain of influence affects the accuracy of the interpolation at the
point of interest, therefore the selection of suitable dimension of support domain is very
important.[2,8] To define the support domain for a point xQ, the dimension of the support domain
ds is determined by
d s=α s dc 3.1
Where αs is the dimensionless size of the support domain. And dc is the characteristic
length that relates to the nodal spacing near the point xQ. If the nodes are uniformly distributed dc
is simply the distance between the two neighboring nodes.
3.2.2.5 Weight function
The weight function plays an important role in the performance of approximate solution.
These are monotonically decreasing functions with respect to distance from x to xi which implies
that the magnitude decreases as the distance from x to xi increases. It is obvious that weight
function should be continuous and positive in its domain of support. Different types of weight
functions are used in EFG method. In this work quartic spline weight function has been used. [2,
8, 9] The function is expressed as:
W ( x−x i )=1−6 r 2+8 r3−3 r4 ,0
r≤1r>1 3.2
Where r is given by r=
||x−x i||d i , di, is the size of the domain of support of node i.
The plot of quartic weight function and its derivative is shown in figure 3.5. The value of
radius of support domain of influence is taken to be equal to 2.0, the domain is given by Ω= (0,
5). Only one curve has been plotted each for weight function and the derivative of weight
function for the central node.
Figure 3.5 Quartic spline weight function and its derivative
3.2.2.6 Effect of dimension of support domain
The smoothening of the weighting function, because of the selection of the support
domain of influence for the node i (central node in our plots) can be visualized by the plot in the
figure 3.6. The figure plots the Quartic weight function and its derivatives for different values of
support domain i.e. di= 1.5, 2.0 and 2.4. With the increase of the dimension of the support
domain the curves are smoother which produces the smooth shape functions.
Figure 3.6 Effect of support domain on weight function and its derivative
3.3 Closure
The primary difference between finite element method and meshfree method were
discussed in brief in this chapter. The terminology concerned to meshfree methods is defined.
The effect of the support domain on the weighting function is presented by the plot with different
values.
Element Free Galerkin Method
4.1 Introduction
The Element Free Galerkin method is one of the most used meshfree methods
based on the diffuse elements method proposed by Nayroles et al. (1992) and
developed further by Belytscho et al (1994)[7]. The major features of it are as follows:
Moving least square approximation (MLS) is used for the construction of the
shape function.
Galerkin procedure is employed to derive the discrete equations from the
weak form.
Integration is performed with background cells or mesh for calculation of
system matrices.
4.2 Basic procedure of meshfree analysis
The problem of elastic bar with uniform cross-sectional area will be used to
illustrate the basic steps involved in the meshfree analysis of any solid mechanics
problem. The elastic bar with loading conditions is represented by figure 4.1. The
displacement of any point ‘x’ on the bar is given by u(x). The governing equilibrium
equation in terms of displacement u(x) as given by Reddy [4] is:
ddx (EA
dudx )+b( x )=0
, 0<x<1 4.1
Subject to boundary conditions
u(0) =0
and
Representation,Number of nodes
Weight function
Approximation &Shape Function
Weak Form
Material property,Loading
Nodal Discrete Equations
Global Matrix Assembly
Solve for Nodal Displacement Parameters
Basis function
[EAdudx ]
x=1=P
Flowchart 4.1 Meshfree solution procedure
L
x
P
dx
Body force b(x)
Point Load
Figure 4.1 One-dimensional bar subjected to point-load and body force
The equation 4.1 can be written as, using the equilibrium of forces acting
on a small element dx of the bar, Reddy [4]:
dσdx
+b( x )=0 4.2
where
Stress = σ = εE 4.3
Strain = ε=du
dx ¿¿ 4.4
Considering the same problem for elastic bar of length L=1, with uniform cross-
sectional Area A, subjected to body force b due to density ρ, Young’s modulus E and a
point load P with consistent units. The exact analytical solution of the governing
equation 4.1 subject to boundary conditions is given by:
u( x )= 1EA (Px+b( x− x2
2 )) 4.5
4.2.1 Domain representation
In meshfree method we model the problem and represent it by the set of nodes
scattered in the problem domain and its boundary. Specify the boundary and loading
conditions. The density and distribution of the nodes depends upon the accuracy
requirement and the resource availability. These nodes are often called field nodes as
these will carry the value of the field variables in the meshfree formulation. Figure 4.2
represents the One-dimensional meshfree representation along with the boundary
conditions of elastic bar using 11 nodes.
.
Figure 4.2 Meshfree re-presentation of one-dimensional elastic bar
4.2.2 Displacement interpolation
The field variable, say a component of displacement u at any point at x within the
problem domain is interpolated using the displacements at its nodes within the support
domain of the point at xi, and is given by:
4.6
Where n is the number of nodes included in the support domain of the point x
and is the shape function of the ith node determined using the nodes that are
included in the small support domain of x.
4.2.3 Formation of system equations
The discrete equations of a meshfree method are formulated using the shape
function and weak form of governing differential equations. These equations are written
in nodal matrix form and are assembled into the global system matrices for the entire
problem domain. The derivation of meshfree discrete equations or the algebraic
equations of meshfree approximation involves following:
a) Approximate solution over the domain
b) Weak form
c) Derivation of discrete equations
4.2.3.1 Approximate solution over the domain
The approximate solutions are sought over the problem domain
Ω = (0, L) at once in meshfree methods the approximate solution is given by the form of
approximation u(x) [2, 7]
4.7
Where p(x) is the complete polynomial of order m and a(x) is given by:
4.8
4.9
The order of polynomial is defined as the order of basis function, in one
dimension the complete linear basis function is given as
pT ( x )=[1 x ] 4.10
On putting equation 4.9 and 4.10 in equation 4.7 for approximate solution we get
u( x )=a0 ( x )+a1 ( x )x 4.11
The unknown coefficient ai of a(x) varies with x. These approximations are
known as Moving Least Square (MLS) approximations in curve and surface fitting and
were first described by Lancaster and Salkauskas [2, 9]. The MLS method is now
widely used as an alternative for constructing meshfree shape functions for
approximation. It was used in Element Free Galerkin method by Belytschko et al.
(1994). MLS approximation has two major features that make it popular: 1) the
approximated field function is continuous and smooth in the entire problem domain and
2) it is capable of producing an approximation with the desired order of consistency.
Figure 4.3 Data fitting using least square method
The unknown coefficients ai of a(x) at any point are determined by minimizing the
difference between the local approximation at that point and nodal parameters ui.
4.12
Where
is the quartic spline weight function, and solving equation 4.12 to minimize the
functional J
and 4.13
We get 4.14
This is further rearranged and can be written compactly as:
4.15
4.16
We get the reduced form of Equation-4.14 as
A( x )a ( x )=B( x )u 4.17
On manipulation for evaluating a(x)
a (x )=A−1 ( x )B( x )u 4.18
Recalling the Equation-4.7 of approximate solution and substituting the values
we get this equation for approximation:
4.19
Hence, the approximate solution can be written concisely as
. 4.20
Where the moving least square shape function or is defined by
4.21
Thus, we get the equation of shape function to be used for substituting in the
weak form to establish the discrete equations. The obtained plots of shape function for a
set of 13 nodes in the problem domain Ω = (0, 1) are shown in the figure 4.4.
Figure 4.4 MLS shape functions at the gauss points
4.2.3.2 Weak form
The partial differential equation that governs the solid mechanics problem can be
represented as equilibrium equation, Liu GR [2]
LT σ+b=0 4.22
where b is the vector of external body forces in the x, y, and z directions
and σ is the stress tensor.
b=bx
b y
bz
4.23
and L is a differential operator matrix:
L=[d /dx
00
d /dy00
00
0d /dz
d /dzd / dy
d /dzd /dy
0d /dx
d /dx0
] 4.24
The constrained Galerkin weak form with the implementation of Lagrange
multipliers λ is used, which is developed & given by Liu GR [2] and can be written as:
∫0
L
δ( Lu )T (cLu )dx−∫0
L
δuT b dx−∫Γt
δuT P dΓ−∫Γu
δλT (u−u )dΓ−∫Γu
δuT λ dΓ=0
……….4.25
where
Stress tensor
L= differential operator
u= displacement
b= body force vector
P= point load on the natural boundary
∫0
L
δ( Lu )T (cL u )d x−∫0
L
δu T b d x−∫Γt
δu T P dΓ −∫Γu
δ λT (u−u )dΓ −∫Γu
δu T λ dΓ =0 = prescribed displacement on essential boundary
The imposition of Lagrange multiplier λ is necessary for satisfying the boundary
conditions and implementation of solution for meshfree methods. The Lagrange
multiplier λ can be interpreted as the reaction forces needed to fulfill the displacement
conditions at the essential boundary.
4.2.3.3 Derivation of discrete equations
The discrete nodal equations are obtained by using the weak form of equilibrium
equation and imposing the boundary conditions. Weak form, of problem under
consideration is given by equation-4.25, is recalled and approximate solution equation
4.20 is substituted for u, which yields the following system of linear algebraic equations
in matrix form:
[ K GGT 0 ][uλ ]=[ fq ]
4.26
Where
K ij=∫0
L
EAdφ
iT
dx
dφ j
dxdx
Gik=−∫Γu
L
φk dΓui
f i=∫Γ t
φi P dΓ t +∫0
L
φ ibdx
q i=−∫Γu
u dΓ u
These discrete nodal equations are assembled into global matrix, to solve for the
nodal displacement parameter values.
4.3 Imposition of boundary conditions
The Element Free Galerkin (EFG) shape functions obtained using moving least
square approximation do not satisfy the Kronecker delta property:
4.27
Therefore, one can conclude that they are not truly interpolants
that is why these are called approximation shape functions. This means that the found
values of are not the nodal values. The approximation at th node is dependent
upon the nodal parameters as well as the nodal parameters through ,
corresponding to all other nodes within the domain of influence of node . This makes
the imposition of essential boundary conditions difficult. In this thesis work, Lagrange
multiplier technique, Belytscho et al. [2, 7] is used to enforce the essential boundary
conditions.
4.4 Solutions of discrete equations for nodal parameters
The major difficulty in the solution of meshfree discrete equations is the
numerical integration of the weak form. This is due to the non-polynomial form of
meshfree shape functions including Moving least square approximation (MLS).
Therefore, exact integration is the most difficult to perform for meshfree methods.
Many techniques have been developed, in this work Gauss-Legendre
Quadrature technique has been used to perform the integration over the
Domain Ω = (0, 1), GR Liu [2], Reddy [4]. The Gauss Legendre quadrature formula is
given by:
∫a
b
F ( x )dx=∫ F (ξ )dξ≈∑I=1
r
F (ξ I )w I 4.28
Where wI are the weight factors and ξI are the base points.
Gauss-Legendre Quadrature technique utilizes a background mesh; the method
is ideally applicable to small and moderate deformations. The considered problem has
been solved by developing the MATLAB language code, using the uniformly distributed
set of 13 nodes. MATLAB is the trade mark of math-works, software developers. The
obtained nodal displacement parameters have been compared to exact analytical
solution. The plot obtained considering the all the variable material properties to be unity
is shown in figure 4.5. The effect of the dimensionless size of the support domain is
studied by plotting the curves with different values, figure 4.6 & 4.7. A portion of figure
4.6 has been enlarged to bring forward the clarity between the curves. It is decided to
have further programming with this value equal to 2.0, which gives better approximation.
The material properties have been considered here only as the representative values
for the development of the MATLAB program code.
Figure 4.5 Displacement vs length for bar of uniform area of cross-section
Figure 4.6 Effect of dimensionless size of support domain αS
Figure 4.7 Effect of dimensionless size of support domain αS (enlarged)
4.5 Closure
This chapter described the basic formulation of Element Free Galerkin method
and approximation of field variable (displacement). Application of Lagrange multipliers is
briefly described for imposing the essential boundary conditions. For evaluating the
global matrices Gauss-Legendre Quadrature technique with background cell integration
is used. The results for exact and meshfree method are plotted. In the next chapter the
application of Element Free Galerkin Method to elastic tapering bar is presented.