Introduction to the Extended Finite Element Method

Transcription

1 Introduction to the Extended Finite Element Method Konstantinos Agathos Dipl. Civil Eng. Aristotle University of Thessaloniki Prof. Dr. Eleni Chatzi, Chair of Structural Mechanics, IBK, D-BAUG 1

2 Introduction 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. 2

3 Strong and Weak discontinuities 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, e.g. cracks and holes. Displacement jump Cracked Bar 3

4 Strong and Weak discontinuities 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. E1 E2 Kink Strain jump Bimaterial Bar 4

5 Fracture Mechanics 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. So some basic concepts of fracture mechanics will be briefly mentioned 5

6 Fracture Mechanics Problem Statement Determine the stress, strain and displacement distribution in structures in the presence of flaws such as cracks and small holes. Small Hole Crack 6

7 Fracture Mechanics Problem geometry (cracked domain case) Y y Crack tip Crack tip X z x Crack plane Crack front 2D Crack 3D Crack 7

8 Fracture Mechanics Crack opening modes (i.e. how can the two crack surfaces deform) Mode I Mode II Mode III 8

9 Fracture Mechanics 9

10 Fracture Mechanics 10

11 Fracture Mechanics 11

12 Fracture Mechanics Westergaard solution parameters Y σ0 α α r θ X 12

13 Fracture Mechanics Stresses approach infinity near the crack tip 13

14 FEM Solution In order to model the crack with FEM, the geometry has to be explicitly represented by the mesh, i.e. nodes have to be placed across the crack and on the crack tip. Example: Crack tip 3 Crack Crack tip 2 Crack tip 1 Cracked propagation domain Remeshing FEM Mesh Crack refinement tip 14

15 FEM Solution 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. 15

16 Partition of Unity 16

17 Partition of Unity 17

18 Partition of Unity In the above equation two factors have to be determined: 1. The type of enrichment functions used (next section). 2. The 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, i.e. only nodes near the crack are enriched. This matter is going to be addressed in more detail in a following section. 18

19 Near Tip Enrichment 19

20 Near Tip Enrichment 20

21 Jump Enrichment A simple example is considered first in order to demostrate the concept, the results will the be generalized to more complex cases: The objective is to represent mesh 1 using mesh 2 plus some enrichment terms Y Y 9 4 X X Mesh 1 Mesh 2 21

22 Jump Enrichment 22

23 Jump Enrichment 23

24 Jump Enrichment 24

25 Jump Enrichment Jump enrichment terms for arbitrary crack orientation 25

26 Signed Distance Function Bimaterial interface n XΓ d X 26

27 Signed Distance Function Absolute value of the signed distance function: 1D absolute value of the signed distance function 2D absolute value of the signed distance function for arbitrary discontinuity 27

28 XFEM Displacement Approximation 28

29 Selection of enriched nodes 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. I I Nodal support of external and internal node 29

30 Selection of enriched nodes 30

31 Selection of enriched nodes Node enrichment examples: tip enrichment jump enrichment 31

32 Level Set Method 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. 32

33 Level Set Method 33

34 Level Set Method 34

35 Level Set Method Circular level set function Elliptical level set function Non-regular discontinuity Non-regular level set function 35

36 FE Approximation of Level Sets FE approximation zero level set FE approximation of level sets 36

37 Crack Representation with Level Sets θ Crack tip ψ r X ф 2D crack 3D crack 37

38 Weak Form and Discretization 38

39 Weak Form and Discretization 39

40 Numerical Integration Gauss quadrature is not apropriate for the numerical integration of the discontinuous enrichment functions, so usually one of the following approaches is employed: Division into subtriangles Division into subquads 40

41 Post processing 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. 41

42 Extentions 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. 42

43 References 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. Moës, J. Dolbow and T. Belytschko, International Journal for Numerical Methods in Engineering,