INTEGRATED NURBS-BASED XFEM FOR MODELING CRACK PROPAGATION
3.2. 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
( ) ( ) *( ) ( )
standard FE enrichment
h
i i j j
i I j J
N N
∈ ∈
=∑ +∑ Ψ
u x x u x x a
, (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
( ) ( ) 4 ( ( ) )
1
h
i i j j k k
i I j J k K
N N S∗ N∗ F
∈ ∈ ∈ =
=∑ +∑ +∑ ∑
u x u b x cℓ ℓ x
ℓ
, (3.6)
where uh(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).
( ) ( )
( )
1, for , 0 1, for , 0
S t
t
ψ ψ
>
=
− <
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 Fℓ( )x , also known as branch functions in many literatures.
( ) ( )
( ) ( )
1 2
3 4
, sin , , cos ,
2 2
, sin sin , , cos sin ,
2 2
F r r F r r
F r r F r r
θ θ
θ θ
θ θ
θ θ θ θ
= =
= =
(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
F r θ = r θ2
b) 2( ), cos
F rθ = r θ2
c) 3( ), sin sin
F r θ = r θ2 θ
d) 4( ), cos sin
F r θ = r θ2 θ
Figure 3.2 Branch functions
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 xΓ , (3.9)
where xΓis the normal projection of x on Γc, i.e. the closest point of x on Γc. 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 xΓ.
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 φi(x,t) for each crack tip of the cracks. For example, an edge-crack which has only one tip needs one function
Tip 2
Tip 1
( ),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 ith crack tip xtipi .
Figure 3.4. Signed distance function
Figure 3.5 Construction of initial level set functions
ϕ1 > 0
ϕ 1< 0
ϕ 2> 0 ϕ 2< 0
ψ > 0
ψ < 0
x
d
ψ < 0 ψ = 0
n xΓ
Γ ψ > 0
As sketched in Figure 3.5, the crack is considered to be the zero level set of ψ, where both ϕ1≤0 and ϕ2 ≤0 in case of an interior crack, or where ϕ1≤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 φi
( ), max( )i
t i
ϕ x = ϕ . (3.12)
The function φi(x,t) enables the ability to define the location of crack surface in a more generalized and simpler form:
( ) ( )
{x:ψ x,t =0 and ϕ x,t ≤0}. (3.13)
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, φi 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 φi, then reconstructing the φ function. In each step, the incremental length and the angle of the propagating direction, θc, are known. The displacement of the crack tip is given by the vector
(F Fx, y) tip ni , +1 tipi ,n
= = −
F x x , (3.14)
where xtip ni , +1 is the current crack tip (step n+1) and xtipi ,n =(x yi, i) is the crack tip at step n. Let the values of ψ and φi at step n be ψn and φin
. The updated values of ψ and φi, i.e. ψn+1 and φin+1
, are determined by the following algorithm [Stolarska et al. (2001)]:
(1) φin+1
is updated using equation (3.11).
(2) F is not necessarily orthogonal to the zero level set of φin
. Hence, φin
is rotated to become ˆi
ϕ so that F is orthogonal
( ) ( )
ˆi i Fx i Fy
x x y y
ϕ = − + −
F F (3.15)
(3) The crack is extended by computing new values of ψn+1 only where ϕˆi >0, which is referred to as Ωupdate. Let the region where ϕˆi ≤0 be Ωno update.
(4) Once all φin+1
corresponding to a crack are updated, φn+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, φi 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:
( )
1
Ncp
i i i
ψ Nψ
=
=∑
x , (3.16)
where Ncp 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).
Figure 3.6 Level set functions
The level set functions can also be used to identify the enriched elements. In a given element, let ψmax and ψmin are the maximum and minimum nodal values of ψ on the nodes of that element, respectively. Similarly, let φmax and φmin 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 <0, (3.17a)
then the crack cuts through the element completely and the control points support that element is enriched by the discontinuous enrichment (3.7); and if
max min 0
ψ ⋅ψ ≤ and ϕmax⋅ϕmin ≤0, (3.17b)
then the crack tip lies within the element and the control points support that element are enriched by the branch functions (3.8).
Figure 3.7 Selection of enriched elements and enriched nodes by using level set functions
It should be clarified that for the NURBS-based XFEM, an element is determined to be split element, i.e. discontinuous-enriched, or tip element, i.e. tip-enriched, from the nodal values of ψ and φ. However, the degrees of freedom are associated with the control points, not with nodes.
Thus, the enrichment has to be chosen for the control points. The values of enrichment function in (3.6) are computed at the control points. The level set method models the crack growth independently of the finite element mesh, thus no re-meshing is needed when the crack propagates.
Figure 3.7 demonstrates an example of enriched elements and thus enriched nodes detected using the level set method with the criteria (3.17). The nodes of split elements are denoted by a star (*) and the nodes of tip element are denoted by a square.
However, it is found that the criteria (3.17) given by [Storlaska et al. (2001)] seem not to perform well for all the configurations to identify split elements and tip elements correctly. Examples are given in Figure 3.8.
In both cases given in Figure 3.8, the element 2 is truly split element. However, it is not identified by criteria (3.17) as a split element, because it has a node at which the value of ϕ function is positive.
Figure 3.8 Examples of identifying split elements not correctly
Figure 3.9 gives even a worse case than in Figure 3.8. In Figure 3.9, some split elements are identified as tip elements and some none – enriched elements which are far from the crack are identified as split elements.
From all the mentioned observations, it is necessary to extend the criteria (3.17) to make them work properly. In the framework of this thesis, the above issue is solved by using geometric characteristics. Using geometry calculation, the split elements and tip elements identified by criteria (3.17) are double checked to ensure that they are truly split elements and truly tip elements.
Figure 3.9 Wrong identification of split elements and tip elements