Introduction to the Extended Finite Element Method

1 Konstantinos Agathos Dipl. Civil Eng. Aristotle University of Thessaloniki Prof. Dr. Eleni Chatzi, Chair of Structural Mechanics, IBK, D-BAUG Institute of Structural Engineering 1 Introduction to the Extended Finite Element Method Method of Finite elements II The Extended Finite Element Method (XFEM) is a numerical Method , based on the Finite Element Method (FEM), that is especially designed for treating discontinuities. Discontinuities are generally divided in strong and weak discontinuities. Institute of Structural Engineering 2 Introduction Method of Finite elements II Strong discontinuities are discontinuities in the solution variable of a problem. In structures, the solution variable is usually the displacements so strong discontinuities are displacement jumps, cracks and holes.

2 Institute of Structural Engineering 3 Strong and Weak discontinuities Displacement jumpCracked Bar Method of Finite elements II Weak discontinuities are discontinuities in the derivatives of the solution variable. In structures such discontinuities would invole kinks in the displacements (jump in the strains), as for example in bimaterial problems. Institute of Structural Engineering 4 Strong and Weak discontinuities Str ain jumpE1E2 KinkBimaterial Bar Method of Finite elements II The biggest part of this presentation will be dealing with the modeling of strong discontinuities and more specificaly with cracks. All formulations will be derived for the 2D cracked domain case and in the end the corresponding formulations for weak discontinuities will be given.

3 So some basic concepts of fracture mechanics will be briefly mentioned Institute of Structural Engineering 5 Fracture Mechanics Method of Finite elements II Problem Statement Determine the stress, strain and displacement distribution in structures in the presence of flaws such as cracks and small holes. Institute of Structural Engineering 6 Fracture Mechanics SmallHoleCrackMethod of Finite elements II Problem geometry (cracked domain case) Institute of Structural Engineering 7 Fracture Mechanics YXCrack tipCrack tipCrack planeCrack fr on tyzx2D Crack 3D Crack Method of Finite elements II Crack opening modes ( how can the two crack surfaces deform) Institute of Structural Engineering 8 Fracture Mechanics Mode I Mode II Mode III Method of Finite elements II Institute of Structural Engineering 9 Fracture Mechanics Method of Finite elements II Institute of Structural Engineering 10 Fracture Mechanics Method of Finite elements II Institute of Structural Engineering 11 Fracture Mechanics Method of Finite elements II Westergaard solution parameters Institute of Structural Engineering 12 Fracture Mechanics XYr Method of Finite elements II Institute of Structural Engineering 13 Fracture Mechanics Stresses approach infinity near the crack tip Method of Finite elements II In order to model the crack with FEM, the geometry has to be explicitly represented by the mesh, nodes have to be placed across the crack and on the crack tip.

4 Example: Institute of Structural Engineering 14 FEM Solution Crack tip 1 Crack tip 2 CrackCracked domain FEM mesh Crack tip Mesh refinement Crack tip 1 Crack tip 2 CrackCrack tip 3 Crack propagation Remeshing Method of Finite elements II Remarks: Mesh refinement is usually necessary near the crack tips in order to represent the asymptotic fields asociated with the crack tips. As the crack propagates remeshing is needed which is computationally expensive especially in complex geometries and 3D domains. In some cases when remeshing, results need to be projected from one mesh to the other which further increases the computational cost. Institute of Structural Engineering 15 FEM Solution Method of Finite elements II Institute of Structural Engineering 16 Partition of Unity Method of Finite elements II Institute of Structural Engineering 17 Partition of Unity Method of Finite elements II In the above equation two factors have to be determined: type of enrichment functions used (next section).

5 Parts of the approximation that are going to be enriched. In the case of cracks, the nature of the discontinuity is local since stress, strain and displacement fields are discontinuous or singular only near the crack tips or along the crack, so enrichment should be local too, only nodes near the crack are enriched. This matter is going to be addressed in more detail in a following section. Institute of Structural Engineering 18 Partition of Unity Method of Finite elements II Institute of Structural Engineering 19 Near Tip Enrichment Method of Finite elements II Institute of Structural Engineering 20 Near Tip Enrichment Method of Finite elements II A simple example is considered first in order to demostrate the concept, the results will the be generalized to more complex cases.

6 The objective is to represent mesh 1 using mesh 2 plus some enrichment terms Institute of Structural Engineering 21 Jump Enrichment 12394567810YX1234567811 YXMesh 1 Mesh 2 Method of Finite elements II Institute of Structural Engineering 22 Jump Enrichment Method of Finite elements II Institute of Structural Engineering 23 Jump Enrichment Method of Finite elements II Institute of Structural Engineering 24 Jump Enrichment Method of Finite elements II Institute of Structural Engineering 25 Jump Enrichment Jump enrichment terms for arbitrary crack orientation Method of Finite elements II Institute of Structural Engineering 26 Signed Distance Function XndX Bimater ial in ter fac eMethod of Finite elements II Absolute value of the signed distance function: Institute of Structural Engineering 27 Signed Distance Function 1D absolute value of the signed distance function 2D absolute value of the signed distance function for arbitrary discontinuity Method of Finite elements II Institute of Structural Engineering 28 XFEM Displacement Approximation Method of Finite elements II In order to select the nodes to be enriched the following definition is necessary: The support of a node is the set of elements that contain that node.

7 Institute of Structural Engineering 29 Selection of enriched nodes IINodal support of external and internal node Method of Finite elements II Institute of Structural Engineering 30 Selection of enriched nodes Method of Finite elements II Node enrichment examples: Institute of Structural Engineering 31 Selection of enriched nodes tip enrichment jump enrichment Method of Finite elements II In order to facilitate the evaluation of the enrichment functions and their derivatives, which is necessary for the calculation of the stiffness matrices, the level set Method (LSM) is employed in most XFEM implementations. The level set Method is also a powerful tool for tracking moving interfaces, which makes it s use very common in problems such as crack propagation. Institute of Structural Engineering 32 Level Set Method Method of Finite elements II Institute of Structural Engineering 33 Level Set Method Method of Finite elements II Institute of Structural Engineering 34 Level Set Method Method of Finite elements II Institute of Structural Engineering 35 Level Set Method Circular level set function Elliptical level set function of Finite elements II Non-regular discontinuity Non-regular level set function Institute of Structural Engineering 36 FE Approximation of Level Sets zer o lev el setFE appr oxima tionFE approximation of level sets Method of Finite elements II Institute of Structural Engineering 37 Crack Representation with Level Sets Xr фCrack tip2D crack 3D crack Method of Finite elements II Institute of Structural Engineering 38 Weak Form and

8 Discretization Method of Finite elements II Institute of Structural Engineering 39 Weak Form and Discretization Method of Finite elements II Gauss quadrature is not apropriate for the numerical integration of the discontinuous enrichment functions, so usually one of the following approaches is employed: Institute of Structural Engineering 40 Numerical Integration Division into subtriangles Division into subquads Method of Finite elements II Apart from stresses strains and displacements, one quantity of interest when post processing XFEM results is the stress intensity factors. Their calculation is based on the evaluation of an integral (interaction integral) over an area around the crack tip. The procedure is similar to the one for the FE case. Stress intensity factors are necessary for the calculation of the stress fields around the crack tip as well as for determining the direction of crack propagation.

9 Institute of Structural Engineering 41 Post processing Method of Finite elements II The Method can be Extended in a very straightforward manner to more general and complex problems such as: Crack propagation Branched and intersecting cracks Plastic enrichment Nonlinear Finite elements Dynamic problems So a wide variety of applications can be treated. Institute of Structural Engineering 42 Extentions Method of Finite elements II References/recomended reading: Extended Finite Element Method for Fracture Analysis of Structures by Soheil Mohammadi, Blackwell Publishing, 2008 Extended Finite Element Method for Crack Propagation by Sylvie Pommier, John Wiley & Sons, 2011 Elastic Crack Growth in Finite elements by T. Belytschko and T. Black, International Journal for Numerical methods in Engineering, 1999 A Finite Element Method for Crack Growth without Remeshing by N.

10 Mo s, J. Dolbow and T. Belytschko, International Journal for Numerical methods in Engineering, 1999 Institute of Structural Engineering 43 References Method of Finite elements II