Level-set method

meaning the interface moves through the domain. The initial position of the interface is given, and the future position is then part of the solution. In this report, the cracks appearing in the 3-dimensional concrete beam are xed interfaces. However, the crack propagates and one would assume the interface to be moving. This is not the case, since the crack propagation speed is unknown and not part of the solution in the displacement variational principle. For this reason, the propagation of the crack is treated as a quasi-static process and thus, as a xed interface.

A nal distinction is made between so-called strong and weak discontinuities. Strong discontinuities refer to a jump in a eld quantity across an interface, whereas a weak discontinuity refers to a jump in the gradient of the eld quantity across an interface. A discontinuity in the gradient is also referred to as a kink in the eld variable. An example of a strong and weak discontinuities is given in gure 4.2.

Figure 4.2. Example of (a) strong discontinuity and (b) weak discontinuity in a eld quantity represented as a surface. The interface is represented as a bold line.

With the denition of strong and weak discontinuities, the displacement exhibits a strong discontinuity across a crack and a weak discontinuity across a material interface. An accurate description of the location of the interfaces, e.g. cracks, is necessary in order to enrich the solution appropriately. This issue is addressed in the following section.

4.2 Level-set method

The level-set method is a technique for locating interfaces and is useful in combination with the XFEM, because it facilitates the construction of the enrichment, as will be shown later. However, the method is not a part of the XFEM but is widely used in combination hereof. The level-set method denes interfaces implicitly by the zero-level of a scalar function. The method is restricted by the requirements, that the scalar function must be a continuous function and change sign across the interface. The signed distance function is a particularly useful function in this regard, because it fulls the requirements to the scalar

Group B122b - Spring 2011 4. The eXtended Finite Element Method

function and is easy to implement into a code. Examples of eligible level-set functions for a one-dimensional bar with a discontinuity located at x = 0 are shown in gure 4.3.

Figure 4.3. Eligible level-set functions. The red line is the signed distance function and the black line is an arbitrary level-set function. The interface is located at the red circle on a one-dimensional bar discretized with nodes indicated as blue stars.

The signed distance function is given by equation 4.1. As the name suggests, the signed distance function computes the distance from the discontinuity to a given point and assigns a sign to the distance.

φ(x) = ± min

x^∗∈ Γkx − x^∗k , ∀x ∈ Ω (4.1)

x^∗ The coordinates of a node on the interface.

Γ The set of all nodes x^∗ on the interface.

k.k The Euclidean norm kzk = pz₁²+ z₂²+ ... + z_n². Ω The considered domain.

The sign in equation 4.1 is determined by the sign-equation sign(n · (x − x^∗)), where n is the normal vector to Φ given by n = _k∇Φk^∇Φ , where ∇ is the dierential operator and 26

4.2. Level-set method Master Thesis

k∇Φk = 1 holds for the signed distance function. By convention, n points from the Φ-negative subdomain into the Φ-positive subdomain, which is the reason for the existence of the sign-equation.

Because exact functional representation of discontinuities is often inconvenient, the discontinuities are stored in a discrete way. This is done by evaluating the signed distance function at the nodes, and using standard nite element shape functions to interpolate in-between, see equation 4.2. Note that an error is introduced in this approximation, which decreases with mesh renement.

Φ^h =X

i∈I

Ni(x) · Φ(xi) (4.2)

Φ^h and Φ The approximated level-set function value and exact level-set function value.

x and xi Nodal coordinate and the coordinates of node i.

iand I Node i in the set of all nodes I.

N_i Standard Finite Element shape function belonging to node i.

As previously mentioned, the topological dierence between an open and a closed interface is described by the number of level-set functions needed to describe the discontinuity. In the following two subsections, open and closed interfaces are described using the level-set method.

4.2.1 Description of closed interfaces

Consider a domain Ω ∈ R^d containing an interface. Ω can be decomposed into two subdomains, Ω1 and Ω2, such that Ω = Ω1∪ Ω₂ and the interface Γ12= Ω1∩ Ω₂. Ω1 and Ω₂ may consist of disconnected regions. The interface is then given by the set

Γ₁₂= {x : φ(x) = 0}

This situation is depicted in gure 4.4, where the signed-distance function has been used as the level-set function.

Group B122b - Spring 2011 4. The eXtended Finite Element Method

Figure 4.4. (a) The domain Ω is decomposed into two subdomains Ω1 and Ω2. Γ12 describes the interface. The normal vector, n, points from the φ-negative subdomain into the φ-positive subdomain. (b) Contour values of the signed-distance function.

For more than 2 subdomains, 1 level-set function is no longer sucient. In general, for closed interfaces, n level-set functions can separate 2ⁿ subdomains.

4.2.2 Description of open interfaces

Consider a domain Ω ∈ R^d partially cut by an interface. Where one level-set function is able to describe a closed interface, the description of an open interface requires a second level-set function, γ, to describe where the interface ends. The interface is then given by the set

Γ12= {x : φ(x) = 0 and γ(x) ≤ 0}

γ has to fulll the same requirements as those put on φ. However, in computational implementations, γ is often chosen a straight line orthogonal to the tip of the interface and φ is the signed-distance function, which is zero across the interface and extended tangentially from the crack tip. In other words, γ is not necessarily a signed-distance function, but used to dene the end of the open interface, and φ describes a closed interface. In Abaqus 6.10, γ and φ are both signed-distance functions. A crack is a typical open interface, and an example is depicted in gure 4.5.

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