Generalized Finite Element Methods

1. Generalized Finite Element Methods Discontinuous Approximations Suvranu De

2. This class • Discontinuous approximations • Modification of weight functions • Visibility Criterion • Diffraction Method • Surface Mesh • Enrichment using PUM • X-FEM

3. Why ? Most meshfree approximations are smooth\in both displacements and their derivatives • PROBLEMS • Nonconvex boundaries • Displacement/velocity discontinuity • e.g., crack, shear band, shock • Stress discontinuity • e.g., when the coefficients of the pde are discontinuous such as interface between two materials

4. Solution • Problems with FEM/FVM • Element edges need to be aligned with discontinuities • Mesh refinement close to singularity Types of discontinuities: Strong and weak Solution Strategies • Modification of weight functions • Visibility Criterion • Diffraction Method • Transparency Method • Enrichment using PUM • X-FEM (Belytschko)/ G-FEM (Babuska)

5. Discontinuous approximations • Modification of weight functions • Visibility Criterion • Diffraction Method • Surface Mesh • Enrichment using PUM • X-FEM

6. Visibility Criterion Modification of weight functions Visibility Criterion (Belytschko, Lu, Gu 1994) A ray of light is imagined to be shot from each node (xI) to the point (x) where the weight function is to be evaluated. If the ray encounters a discontinuity, it is terminated and the point (x) is not considered part of the support

7. Visibility Criterion Modification of weight functions For nodes such as J, this method produces the correct discontinuity However, for nodes such as I this method results in artificial discontinuities Significant errors may be introduced if MLS functions are used since the shape functions generated using these weight functions are not even C0 near the tip.

8. Visibility Criterion Modification of weight functions Similar problems at nonconvex boundaries

9. Diffraction Method Modification of weight functions Diffraction Method (Belytschko, et al 1996, Organ, et al 1996) Overcomes the drawback of visibility method Applies to radial weight functions of type In this case, one modifies sI as where

10. Diffraction Method Modification of weight functions Shape functions and weight functions are continuous inside the domain and discontinuous across the line Derivative of shape functions is multiple valued at the tip of the discontinuity. But this is not a problem if no Gauss points are placed aright at the crack tip Works well for cracks and nonconvex boundaries.

11. Boundary mesh Modification of weight functions Problem with every support having its own version of the boundary (Macri, De, 2003)

12. Boundarymesh Modification of weight functions Solved using planar straight line graph (PSLG) at the boundary Can deal with very complex nonconvex boundaries

13. Discontinuous approximations • Modification of weight functions • Visibility Criterion • Diffraction Method • Surface Mesh • Enrichment using PUM • X-FEM

14. XFEM Enrichment using PUM X-FEM uses traditional FEM with enrichments The X-FEM idea is to represent a discontinuity independent of the underlying mesh using enrichment functions

15. X-FEM Enrichment using PUM Consider a crack between two finite elements in R2 The finite element approximation space is

16. X-FEM Enrichment using PUM Enrichment to represent the same crack between two finite elements in R2 Heaviside function The finite element approximation space is Enrichment function

17. Level Sets Enrichment using PUM Discontinuities and interfaces are often represented by level-sets in the X-FEM A level-set function is a scalar function within the domain whose zero-level (i.e., is interpreted as the discontinuity. As a consequence, the domain is divided into two subdomains and on either side of the discontinuity where the level-set function is positive or negative, respectively. e.g.,

18. Level set functions will be typically defined by discrete values at the nodes, They will be then interpolated in the element interiors by standard finite element shape functions Level Sets Enrichment using PUM Often, the signed distance function is used as the level set function:

19. X-FEM Basics Enrichment using PUM X-FEM (or G-FEM) approximation with

20. X-FEM Basics Enrichment using PUM Reproducing elements: Elements all of whose nodes belong to I*. Since the PU property holds the global enrichment is exactly reproduced in these elements Blending elements: Elements some of whose nodes belong to I*. Since the PU property does not hold in these elements, the global enrichment is not exactly reproduced

21. X-FEM Basics Enrichment using PUM Choice of I* The nodal set I* is built from all nodes of elements that are cut by the discontinuity. Whether or not an element is cut by the discontinuity can conveniently be determined on element-level by help of the level-set function

22. X-FEM Basics Enrichment using PUM Choice of y Recall with For weak discontinuities (where the displacement is continuous but its gradient is discontinuous), chose the absolute value of the level set function For strong discontinuities (where the displacement is discontinuous), chose the sign of the level set function which is the Heaviside function

23. X-FEM Enrichment using PUM Special issues • The approximation does, in general, not have the Kronecker-delta property. Shifting of the enrichment functions is usually employed in order to recover this property. • The resulting shape functions of the enrichment involve kinks or jumps in elements cut by a discontinuity. A partitioning of the cut elements into sub-elements for integration purposes is, in general, required. • There are often more than only one enrichment term. Formally, these additional enrichment functions are added to the approximation, so the extension is straightforward.