Quadractic basis functions for an open, non-uniform knot vector.. The isogeometric concept also has many great advantages such as it allows the control of the order and the continuity of
INTRODUCTION
Isogeometric Analysis (IGA)
The concept of Isogeometric Analysis was originally proposed by [Hughes et al (2005)] In this procedure, Non-Uniform Rational B-Spline (NURBS) functions, widely used in the Computer Aided Design (CAD) are adopted to be used as shape functions for Computer Aided Engineering (CAE), in a fashion similar to the iso-parametric concept of the standard FEM The use of NURBS in CAE was motivated by its good characteristics such as exactness in reproducing the geometry Unlike the conventional Lagrangian basis functions used in the standard FEM, the NURBS-based approach essentially provides a better connection between the CAD and CAE and reduces the source of errors caused by the inaccurate approximation of the geometry The isogeometric concept also has many great advantages such as it allows the control of the order and the continuity of the basis functions due to the methods of order elevation and knots refinement In case of second order or higher, the IGA can yield a number of degrees of freedom smaller than that of the standard FEM, provided the same number of elements NURBS also possesses the capability to approximate the discontinuous characteristics better than the classical FEM, which enables us to model the discontinuities (e.g cracks) by means of IGA, e.g see [Verhoosel et al (2011)] and [de Luycker et al (2011)] Recently, the development of T-Spline ([Sederberg et al (2003)] and [Sederberg et al (2004)]) have introduced the ability of local refinement into IA, instead of global refinement as experienced by NURBS The T-Spline approach seems to be more flexible and gives higher accuracy in comparison to the NURBS on conditions that multiple patches are needed (see [Bazilevs et al (2008)]).
The Extended Finite Element Method (XFEM)
The Extended Finite Element Method (see [Moởs et al (1999)] and [Fries and Belytschko (2010)]), is a popular non-remeshing method that is highly suitable for modeling the discontinuities and singularity problems (cracks in the scope of this thesis) In order to capture the singular fields at the crack tips, enrichment functions are incorporated into the standard finite element approximation space The enrichment functions somehow must be determined appropriately in advances; especially they fully depend upon the materials and the configuration of the problems yet to be solved Thus, different types of discontinuities such as cracks, voids, inclusions, biomaterials, etc have different sets of enrichment functions
In order to model the crack geometry, a level set method is employed In this method, the crack is modeled as the zero level set of function ψ(x) Another type of level set function, i.e φ(x), is used to keep trace of the crack tip Function ψ(x) is defined as the signed distance from the point x to the crack surface and function φ(x) is defined as the signed distance from the point x to the line contains crack tip and orthogonal to the crack surfae The level set functions are updated accordingly to the growth of crack.
The scope of the thesis
Because of the great advantages of IGA and XFEM in modeling discontinuities and the exactness of the geometry, the main objective of the present thesis is a possible combination of the NURBS basis functions and XFEM (named as NURBS-based XFEM) to model the crack growth problems in 2D solids To be able to simulate the crack growth, the propagating angle of crack path is determined by employing the maximum circumferential stress criterion
The remainder of this thesis is outlined as follows Chapter 2 presents a short review of IGA based on the NURBS Next, Chapter 3 describes the formulation of the NURBS-based XFEM in the framework of fracture mechanics Several benchmark problems are numerically investigated by the proposed approach are given in Chapter 4 Finally, the conclusions and future works in the last chapter
FUNDAMENTALS OF B-SPLINES AND NURBS
Univariate B-Spline basis functions
Univariate B-Spline basis functions are constructed from a given knot vector A knot vector, defined by Ξ = { ξ ξ 1 , 2 , … , ξ n p + + 1 }, is a non-decreasing sequence of the coordinates within the parametric space In the knot vector Ξ, ξ i ∈ ℝ, represents the i-th knot, , i = 1, 2, …, n+p+1; p stands for the polynomial order, while n is the number of basis functions Note that the polynomial order and the polynomial degree have the same meaning in our approach The knot values can be repeatable, e.g ξ i = ξ i+1 The number of appearance of a knot value in the knot vector is called multiplicity of that value A knot vector is called open knot vector if the multiplicity of both of its first and last knots is p+1 For example, given polynomial order p = 2, Ξ = {0,0,0,2,3,4,4,5,5,5} is an open knot vector In this thesis, only the open knot vectors are used
Generally, the B-Spline basis functions of order p = 0 are defined by:
For p = 1, 2, 3, …, the basis functions are defined by Cox-de Boor recursion formula:
An efficient algorithm to compute the basis functions and their derivatives is presented in “The NURBS Book” [Piegl and Tiller (1999)]
The B-Spline basis functions possess the following properties:
(5) Control of continuity : If a knot has multiplicity k, the basis functions are C p-k – continuous at that location If p = k, the basis functions are interpolatory If p < k, the basis functions are discontinuous
Figure 2.1 Basis functions of order 1, 2 for the uniform knot vector Ξ={0,1, 2, 3, 4,5, }
Figure 2.2 Basis functions for the knot vector Ξ={0, 0, 0,1, 2, 2, 2}
Figure 2.3 Quadractic basis functions for the open, non-uniform knot vector
{0, 0, 0,1, 2, 3, 4, 4,5, 5,5} ΞA representative example of quadratic B-Spline basis is shown in Figure 2.3 The basis function is interpolatory at the first and last knot values, due to the use of open knot vector, and also at ξ =4, where the multiplicity is k = 2 and thus, the continuity is C 0 Across element boundaries, the basis is C p-1 = C 1 continuous.
B-Spline curve
Figure 2.4 A quadratic B-Spline curve in R 2 , associated with the knot vector
{0, 0, 0,1, 2,3, 4, 4,5,5,5} ΞB-Spline curves in R d are mathematically defined by a linear combination of B-Spline basis functions (see [Hughes et al (2005)], [Cottrell et al (2009)] and [Piegl and Tiller (1999)]) The vector-valued coefficients associated with the basis functions are a set of points called control points These points are similar to the nodal coordinates in the classical Finite Element Analysis
However, the control points are in general not interpolated by the curve The polygon created from these points is usually named by control polygon Given n basis functions N i,p (ξ) corresponding to the knot vector Ξ 1 = { ξ ξ 1 , 2 , … , ξ n p + + 1 , } and a set of control points { } B i , i = 1, 2,
…, n, the B-Spline curve in R d is given by
A representative of the B-Spline curve, is given in Figure 2.4, where the curve is colored red and the control points are colored black Coordinates of the control points are given in Table 2.1
It is noted that the curve is interpolatory at the first and last control points, because an open knot vector is used The curve is also interpolatory at the sixth control point, due to the fact that the multiplicity of the knot value ξ = 4 is equal to the polynomial order It is also noted that curve is tangent to the control polygon at the first, last and sixth control points
Table 2.1 Control points for the quadratic B-Spline curve depicted in Figure 2.4
Multivariate B-Spline functions
Multivariate B-Spline functions are obtained by a tensor product of the univariate B-Spline basis functions
For each parametric direction ℓ, a knot vector is given by Ξ ℓ = { ξ ξ 1 ℓ , 2 ℓ , … , ξ n ℓ ℓ + + p ℓ 1 , } The corresponding set of the basis functions is { N i ℓ ℓ , p } Then the d p – dimensional B-Spline functions are defined as:
B-spline surface
Figure 2.5 The control net and mesh for the biquadratic B-Spline surface with
Table 2.2 Control points for the biquadratic B-
Spline surface depicted in Figure 2.5 i j B ij
Given a set of control points in the physical space, called a control net { } B ij , where i = 1, 2, …, n and j = 1, 2, …, m, polynomial orders p and q, knot vectors Ξ 1 = { ξ ξ 1 , 2 , … , ξ n p + + 1 , } and
2 1, 2, , m q 1, η η η + + Ξ = … , the B-Spline surface is thus defined by:
S B (2.5) where N i,p (ξ) and M j,q (η) are the univariate basis functions corresponding to the knot vectors Ξ 1 and Ξ 2 , respectively
A representative example of a B-Spline surface is depicted in Figure 2.5.
Refinement
There are three types of refinement in the B-Spline approach: knot insertion (h-refinement), order elevation (p-refinement) and k-refinement
Knot insertion: in this mechanism, knots are inserted without changing a curve geometrically or parametrically Existing knot values can be repeated by knot insertion, thereby increasing their multiplicity, which means the continuity of the basis will be reduced However, the continuity of the curve is preserved Knot insertion splits existing elements into new ones Therefore, it can be compared with the h-refinement strategy as in tandard Finite Element Analysis (FEA) An example of knot insertion is given in Figure 2.6
The reverse process of the knot insertion is called knot removal
Order elevation: this process involves raising the order of the basis functions used to describe the geometrical characteristics to one order higher It is comparable with the p-refinement strategy as in the standard FEA Similar to the knot insertion, the curve does not change both geometrically and parametrically after order elevation The discontinuity in the various derivatives exist in the original curve must be preserved, which means the multiplicity of every knot value must increase by one during order elevation The process can be repeated as much as necessary to reach the desired order An example of order elevation is given in Figure 2.7
The reverse process of the order elevation is called order reduction
Figure 2.6 Example of knot insertion k-refinement: this type of the refinement does not exist in the standard FEA Its main purpose is to obtain both, higher polynomial order and higher continuity of the basis functions This process involves order elevation to get higher order and knot removal to reduce the multiplicity of each knot value to get higher continuity
The algorithms used for all the refinement techniques pointed out above can be found in “The NURBS book” [Piegl and Tiller (1997)])
The above techniques of refinement are essential in automatically meshing and mesh refinement in the isogemetric analysis They also allow the ability to control the polynomial degree of the basis functions and the order of continuity, which is impossible with the standard Finite Element Analysis
Figure 2.7 Elevate the order of the curve in Figure 2.4 by one
Non Uniform Rational B-splines (NURBS)
By assigning each control point a weight value, NURBS basis is then given as follows: univariate: ( ) ( )
The NURBS curve is obtained by using (2.4) with the univariate NURBS basis functions given by (2.6) The NURBS surface is obtained by using (2.5) associated with the bivariate NURBS basis functions given by (2.7)
2.7 Derivatives of NURBS basis functions The derivatives of univariate NURBS basis are presented in the following
Denote the weight function by
=∑ , (2.8) the univariate NURBS basis in (2.6) can then be expressed as
The first-order derivative is given by
For the calculation of higher-order derivatives of the NURBS basis functions, the notation is simplified by defining
The higher-order derivatives are finally expressed recursively as
Multiple patches
In some practical cases, it is necessary to describe the domain not only by one patch but by multiple patches For instance, when different materials are to be used in different parts of the domain, describing each sub-domain by one patch is a recommended option For complicated geometries, it is also recommended to use multiple patches to divide and simplify the work
These patches must be compatible, meaning that on the coarsest mesh, mappings and parameterizations on the adjoining patch faces are identical [Cottrell et al (2009)] Each control point on a face must be one-to-one correspondent with a control point on the adjoining face This relationship is preserved during refinement
A representative for multiple patches is presented in Figure 2.8 Figure 2.8 a) models an L- shaped panel by one patch and Figure 2.8 b) models the panel by three patches The patch boundaries are colored red
(b) Figure 2.8 An L-shaped panel modeled by a) one patch and b) three patches
Incoporating NURBS into Finite Element Analysis
The geometry is first described by NURBS A set of control points and the knot vectors are then pre-defined for this purpose Knot vectors are accordingly given in the parametric space, in which the elements are partitioned through the knots Using a mapping from the parametric space to the physical space, the finite elements and the nodes in the physical space are then obtained (see [Cottrell et al (2009)] and [Hughes et al (2005)]) After that, the element matrices and relevant vectors are computed and assembled into global ones in a fashion similarly to the standard FEM It is necessary to emphasize that in isogeometric analysis, degrees of freedom are
00.050.10.150.20.250.30.350.40.450.5 associated with the control points, not with the nodes Although nodes exist in the isogeometric analysis, their importance is much smaller than that of control points The shape functions are associated with the control points and hence, the elements are support by the control points, not nodes Since the number of control points in most of the cases is smaller than the number of nodes, the number of unknowns can be reduced in the isogeometric approach In isogeometric analysis, the control points play a role similarly to that of nodes in standard FEM, but control points and nodes are totally different
Figure 2.9 Mapping of an element from physical space (Ω e ) to parametric space ( ˆΩ e ) and to the parent element space (Ωɶ e )
Meshing is done by the following scheme: a very coarse mesh is initially created, and then the mesh is refined using knot insertion The refined meshes of the one in Figure 2.5 are depicted in Figure 2.10 If order elevation and k-refinement are necessary, they should be done on the coarsest mesh, before doing the procedure of knot insertion ξ η Ωˆ e ξɶ ηɶ
(d) Figure 2.10 Initial mesh (a) and Refined mesh ((b), (c) and (d))
For the sake of completeness, the connectivity arrays required for the construction of element and for the assembly procedure are additionally presented here For convenience, notation is taken the same as in [Cottrell et al (2005)]
Firstly, the concept of the NURBS coordinates is introduced Given knot vector
1 1, 2, , n p 1, ξ ξ ξ + + Ξ = … and knot vector Ξ 2 = { η η 1 , 2 , … , η m q + + 1 , } The knots of these two knot vectors define a two dimensional space called the index space (see Figure 2.11 as an example)
Figure 2.11 The index space view of the mesh in Figure 2.4 Ω 1 is the support region of one shape function Ω 2 is the support region of another shape function The purple area is the overlapping area of Ω 1 and Ω 2
The NURBS coordinates of any vertex in the index space are simply the indices of the knots that define the vertex For example, the vertex created by the intersection of the knot lines ξ 3 and η 2 ξ7
Ω2 η4 ξ2 ξ3 ξ4 ξ5 ξ6 η1 η2 η3 ξ1 η5 η6 has NURBS coordinates (3, 2) Note that this is the vertex where the support of the blue function begins Actually, the NURBS coordinates are mostly used to identify the knots at which the support of a shape function begins
The INC (for “NURBS coordinates”) array is defined as an array that takes a global shape function number and a parametric direction as input and returns the index of the one-dimensional basis function of that parametric direction as output Let n = 4, p = 2, m = 3 and q = 2, we have the INC array as in Table 2.3
Table 2.3 An example of INC array
Given the index A of the global function Nɶ A ( )ξ η, =N i ( )ξ ⋅M j ( )η , i and j are obtained by
By using the NURBS coordinates, the one-dimensional basis functions N i and M j , which are established to construct the global function Nɶ A , are obtained Because the compact support of N i and M j are already known (see Section 2.1), the support of Nɶ A is defined correspondingly
Knowing the NURBS coordinates of Nɶ A means knowing the support of Nɶ A and hence, knowing if a given point is within the support of Nɶ A or not
The NURBS coordinates essentially provide a means to determine which functions having the support in a given element Let us continue using the denotation about the knot vectors in Section 2.9.1 Given an element e, the total number of the basis functions that support the element e is expressed as
The IEN array connects each local (elemental) number of a basis function with its global number For example, Table 2.4 shows the IEN array of the mesh depicted in Figure 2.5 It is noted that, the elements can have overlapping area in index space, but they do not have overlapping area in physical space
IEN (for “element nodes”) array is similar to the concept of the element nodes in the standard finite elements In the isogeometric analysis, however, each basis function does not associate with a node, but with a control point and each control point is associated with a shape function
Given the element number e, and the local basis function number b, the corresponding global basis function number B is obtained by
Figure 2.12 Global numbering of control points for Figure 2.5 in parametric space Each control point is associated with a shape function The red and blue lines denote element 1 and element 2, respectively
Table 2.4 An example of IEN array
Imposing essential boundary conditions
(c) Figure 2.13 Location of interpolating points for the linear constraints for NURBS case
In the isogeometric approach, the degrees of freedom are associated with the control points, not with the nodes like the standard FEM Therefore, the essential boundary conditions must be applied at the control points Besides, the NURBS functions do not satisfy the Kronecker delta property, and it thus leads to a difficulty in imposing the essential boundary conditions In cases the boundaries are fixed, the essential boundary conditions can be imposed directly, especially degree 2 [0 0 0 1 1 1]
[0 0 1 1] [0 0 1/2 1 1] [0 0 1/3 2/3 1 1] [0 0 1/4 1/2 3/4 1 1] when the open knot vectors are used, because of the interpolation property at the boundaries
Nevertheless, to be able to deal with complicated domains, e.g curved boundaries, where the physical boundaries do not pass the control points (see Figure 2.5 as a representative example), some specific techniques are required to apply boundary conditions A technique frequently used to impose arbitrary essential boundary conditions is Interpolating points (see [de Luycker et al
(2011)] and [Wang and Xuan (2010)] The main idea of this technique is to map the prescribed values of points located at the physical essential boundary to the control points associated with that boundary The procedure is briefly presented in the following
Given a point at the boundary of a domain, the displacement field of this point is generally approximated by
, (2.19) where the first term is associated with the interior control points whereas the second term is associated with the boundary control points Due to the characteristics of the open knot vectors, the shape functions associated with the interior control points vanish identically at the boundary
=∑ ∈Γ u ξ ξ d ξ , (2.20) where ξ B denote the coordinates in the parameter space Applying (2.20) to m points at the physical boundary and rewritten the total number of the control points on boundaries as n for simplicity, the following set of linear equations is obtained
, (2.21) where N B ( ) ξ m is denoted by N B m In matrix form
Vector d, which is the “real” boundary conditions that have to be applied, is obtained by solving (2.22) Hence, matrix N should be invertible, which means that the number of points chosen on the physical boundary must be equal to the number of the control points involved at the boundary, i.e m=n Figure 2.13 illustrates the method used to pick the constrained points [de Luycker et al (2011)]
INTEGRATED NURBS-BASED XFEM FOR MODELING CRACK PROPAGATION
Governing equations
Consider a domain Ω bounded by its boundary Γ as depicted in Figure 3.1
Figure 3.1 Body with internal boundaries subject to loads
In Figure 3.1, the boundary Γ is subjected, respectively, to the essential boundary conditions prescribed by the displacement at Γ u , and to the natural boundary conditions imposed by tractions at Γ t so that Γ u ∪Γ = Γ t The crack surface Γ c is assumed to be traction-free Γ u Γ t Γ c y x z Ω
The equilibrium equations and the boundary conditions are given as
= Γ u u (3.1e) where σ is the Cauchy stress tensor, n denotes the unit outward normal while b, t being the body force and traction, respectively, and u stands for the nodal displacements
For linear elastic fracture mechanics, the constitutive relation is given by the Hooke’s law: σ Cε, (3.2) where C is the matrix of the material properties and ε is the linear strain tensor
The weak form of the equilibrium equation, considering the constitutive relation, is given by
∫ ε u C ε u ∫ b u ∫ t u , (3.4) where δu is the test function.
Extended Finite Element method (XFEM)
Literally, XFEM is considered as a local extrinsic partition of unity method In order to handle the discontinuities, e.g cracks, the standard FEM approximation space is “extended” or
“enriched” by adding functions which satisfy the partition of unity in a certain sub-domain close to the discontinuities, in order to capture the special knowledge of the solution when discontinuities exist (e.g jumps, kinks, singularities, etc.) into the approximation space
Enrichment functions should be chosen appropriately dependent on each type of considered discontinuities, e.g cracks, voids, inclusions, etc Following is the general form of XFEM formulation
, (3.5) where the function Ψ(x) defines the enrichment functions Enrichment part is added independently to the standard part; hence it is called “extrinsic” enrichment Standard approximation space does not change by adding “extrinsic” enrichment However, more unknowns are added correspondingly to the added enrichment functions This concept is essentially opposite to the “intrinsic” approach, which replaces the standard shape functions by embedding some special shape functions to be able to capture the singular fields of discontinuities The number of shape functions and the number of unknowns remain unchanged in the intrinsic method
In particular, the shape functions of N and N * are usually chosen to be identical, but in general the shape function of N * can be different from the N functions, as long as N * satisfies the property of partition of unity In the present work, N * is identical to N because of the convenience in the implementation
Due to the advantages of utilizing the enrichment functions in the finite element approximation, the cracks are thus independent of the finite element mesh The versatile XFEM enables the ability to model the evolution of cracks without re-meshing, which is a cumbersome task in the conventional FEM For modeling a crack, two different types of the enrichment functions are used The generalized Heaviside function, which is usually chosen as the sign function, is employed to model the discontinuity of the crack part far away from the crack tip; while a set of four-fold span enrichment functions known as branch functions (see [Moởs et al (1999)] and [Fries and Belytschko (2010)]) are used to enrich the crack tips to describe their singularities
General form of the XFEM for modeling the crack is given by
, (3.6) where u h (x) is the approximated function of the displacement field; N is the shape function computed at the control points; u, b, c, respectively, are the unknown degrees of freedom corresponding to the sets named as I, J and K The term I is the set of total nodes in the problem domain, whereas J is the set of nodes enriched by the sign function S( x)
x x x (3.7) where ψ(x,t) is the level set function discussed in Section 3.3
Set K is the set of nodes enriched by the tip enrichments F ℓ ( ) x , also known as branch functions in many literatures
(3.8) where ( ) r , θ are the local polar coordinates defined at the crack tip
In NURBS-based XFEM, the NURBS basis functions are used as the shape functions, and the unknown degrees of freedom are associated with the control points instead of nodes in the standard XFEM Hence, the sets I, J and K are also associated with control points, correspondingly
In cases where two or more than two crack tips exist in the system, e.g center crack, or double edge crack problems, each crack tip requires its own set of asymptotic near tip enrichment functions For example, the crack in Figure 3.3 has two tips Thus, two sets of branch functions are required, one set for tip 1 and one set for tip 2 a) 1 ( ) , sin
Figure 3.3 An arbitrary crack placed on a mesh 3.3 The level set method
To be able to detect the discontinuous surfaces, the level set method has been recommended by ([Stolarska et al (2001)], [Ventura et al (2002)], [Ventura et al (2003)]) The heart of this method is to measure the signed distance function of an arbitrary point x to the crack surface Γc The distance from a point x to the crack surface is given by d = −x x Γ , (3.9) where x Γ is the normal projection of x on Γ c , i.e the closest point of x on Γ c Hence, the signed distance function φ ( ) x is then defined by
( ) , t min sign ( ( ) ) , c ψ x = x x− ⋅ n x x⋅ − x∈Γ , (3.10) where n is the unit normal of Γc at x Γ
An endpoint of the crack, i.e crack tip is represented as the intersection of ψ(x,t) with an orthogonal zero level set function φ(x,t) It is necessary to define one function φ i (x,t) for each crack tip of the cracks For example, an edge-crack which has only one tip needs one function
( ) ,t ϕ x only, while cracks that are entirely in the interior of the problem domain needs two functions, ϕ1 ( ) x,t and ϕ2 ( ) x,t
( ) , ( tip ) ˆ i t i ϕ x = −x x ⋅t, (3.11) where ˆt is a unit vector tangent to the crack at the i th crack tip x tip i
Figure 3.5 Construction of initial level set functions ϕ 1 > 0 ϕ 1 < 0 ϕ 2 > 0 ϕ 2 < 0 ψ > 0 ψ < 0 x d ψ < 0 ψ = 0 n x Γ Γ ψ > 0
As sketched in Figure 3.5, the crack is considered to be the zero level set of ψ, where both ϕ 1 ≤0 and ϕ 2 ≤0 in case of an interior crack, or where ϕ 1 ≤0 in case of an edge crack In cases that more than one crack tip exists, it is convenient to define a single function ϕ ( ) x ,t to unify all the functions φ i
The function φ i (x,t) enables the ability to define the location of crack surface in a more generalized and simpler form:
Within the framework of crack growth problems, the level set must be updated appropriately, but only nodes locally close to the crack are updated In addition, it is assumed that once a part of a crack has formed, that part will be fixed, i.e no longer change shape or move [Stolarska et al
(2001)] Therefore, the level set functions ψ, φ i and φ need only to be updated on a small region of elements surrounding each crack tip Thus, the level set representation is confined to a narrow band of elements around the crack
The evolution of the crack is modeled by appropriately updating the functions ψ and φ i , then reconstructing the φ function In each step, the incremental length and the angle of the propagating direction, θ c , are known The displacement of the crack tip is given by the vector
F x x , (3.14) where x tip n i , + 1 is the current crack tip (step n+1) and x tip i ,n = ( x y i , i ) is the crack tip at step n Let the values of ψ and φ i at step n be ψ n and φ i n
The updated values of ψ and φ i , i.e ψ n+1 and φ i n+1
, are determined by the following algorithm [Stolarska et al (2001)]:
(2) F is not necessarily orthogonal to the zero level set of φ i n
Hence, φ i n is rotated to become ˆ i ϕ so that F is orthogonal
(3) The crack is extended by computing new values of ψ n+1 only where ϕˆ i >0, which is referred to as Ω update Let the region where ϕˆ i ≤0 be Ω no update
(4) Once all φ i n+1 corresponding to a crack are updated, φ n+1 is updated using (3.12)
The level set method is coupled with the XFEM to model the evolution of cracks The values of ψ, φ i and φ at the nodes are stored For the NURBS-based XFEM, the values at control points are also stored The values at other points within a given element are approximated from the values of the control points that support the given element as follows:
=∑ x , (3.16) where N cp is the number of control points which support the element The other level set functions are also approximated by using the same form of (3.16)
The level set functions can also be used to identify the enriched elements In a given element, let ψ max and ψ min are the maximum and minimum nodal values of ψ on the nodes of that element, respectively Similarly, let φ max and φ min are the maximum and minimum nodal values of φ on the nodes of that element, respectively There are two types of enriched elements: split element and tip element Split elements are elements that are completely cut by the crack, while the element that contains the crack tip is called tip element According to [Stolarska et al (2001)], if max min 0 ψ ⋅ψ ≤ and ϕmax