Friday, April 26, 2013

MAFELAP 2013 presentation

Stéphane P. A. Bordas

http://people.brunel.ac.uk/~icsrsss/bicom/mafelap2013/
 

"Advances in extended finite element methods for fracture and heterogeneous materials"

This presentation will address recent advances in enriched numerical methods to simplify the treatment of evolving discontinuities in the field variables or their derivatives: cracks or material interfaces; and to treat geometrically intricate domains and their evolution.

The presentation will be composed of three parts:

(1) advances in numerical methods aiming at simplifying the treatment of complex geometries;

Two competing approaches coexist in the literature to simplify the solution of partial differential equations over domains of complex and/or evolving geometries. One focuses on streamlining the transition between computer aided design (CAD) data and the solution of problems over the corresponding domains. An example of this is isogeometric analysis (Hughes et al. 2005) where the geometry description and the approximation of the field variables are tied, thus enabling an exact treatment of the boundary as well as simplifying eventual geometric design iterations.
The second approach follows an orthogonal direction, where the geometry is uncoupled from the field variable discretisation, e.g. embedded boundary methods such as the structured extended finite element method of (Belytschko et al. 2003).
We will present results emanating from both lines of thought: isogeometric analysis of three-dimensional structures and embedded interfaces for complex geometries including sharp edges and vertices.

(2) advances in enriched formulations for evolving discontinuities such as cracks

The extended finite element method (XFEM) was introduced in (Belytschko and Black, 1999), with, as a basis, the partition of unity enrichment method of (Melenk and Babuška, 1996). The XFEM enables the simulation of evolving discontinuities without or with minimal remeshing.
We present recent developments in the area and focus on tackling difficulties associated with the control of the conditioning number of the system, the control of the error in quantities of interest, element distortion and blending errors between different partitions of unity.

(3) applications of those methods to problems in Silicon wafer manufacturing and brain surgery simulation.

Finally, as an application, we will present a simple method to grow several hundreds of cracks in two-dimensions in order to predict their growth and coalescence in brittle materials using the XFEM.
We will also present recent results permitting the simulation of cutting and contact during brain surgery simulation at 30 frames per second using an implicit time integration method and a hybrid, asynchronous CPU/GPU solver.

We finish the presentation by conclusions and propositions for future work.

Acknowledgements: Stéphane P.A. Bordas wishes to thank the organisers of MAFELAP for their kind invitation. He is grateful to the European Research Council for funding the research presented (ERC Stg grant agreement No. 279578: "RealTCut Towards real time multiscale simulation of cut- ting in non-linear materials with applications to surgical simulation and computer guided surgery").