# The Finite Cell Method

Thanks to its versatility, the Finite Element Method (FEM) has become the most frequently applied numerical method in Computational Mechanics. Although the FEM is in a mature state there are still problems where its application is difficult. These problems arise, for example, when considering heterogeneous materials or more generally when discretizing structures which have a very complex geometry which might even change during the computation. In such cases mesh generation can become very involved. To overcome these problems we propose to apply the Finite Cell Method (FCM) which can be considered as a combination of a fictitious domain method with high-order finite elements. The main idea is to embed the physical domain into an extended domain which can be easily discretized with a structured mesh consisting of finite cells. In this way, mesh generation is dramatically simplified and the burden is shifted towards the integration of the stiffness matrices. However, the integration can be performed adaptively in a fully automatic way. During the integration the geometry and varying material properties are taken into account while the discretization error is controlled by adjusting the polynomial degree of the cells. To illustrate the basic idea, Figure 1 shows a perforated square plate which is discretized by four cells. Note that the geometry of the circular hole is taking into account during the integration of the stiffness matrices. When performing a p-extension, the error in energy norm decreases exponentially, yielding a very efficient discretization.

Figure 1. Error in the energy norm for a perforated plate .

## PhD students

• Alireza Abedian (Isfahan University of Technology, currently: IUT);
Subject: Adaptive integrations schemes for the FCM, application of the FCM to finite elastoplasticity.
• Maedeh Ranjbar (Isfahan University of Technology, currently: TUHH);
Subject: Damage Mechanics using FCM.
• Yaser Mirbagheri (Isfahan University of Technology, currently: IUT);
Subject: Dynamic problems using FCM.
• Aliakbar Taghipour (Isfahan University of Technology, currently: IUT);
Subject: Application of the FCM to finite elastoplasticity.

The joint research project of the above mentioned partners is funded by the Alexander von Humboldt Foundation in the framework of the alumni/research group linkage programm.
Period of funding: July 2010 – September 2014

## Publications

• A. Abedian, J. Parvizian, A. Düster, E. Rank.
The FCM compared to the h-version FEM for elasto-plastic problems
Applied Mathematics and Mechanics (English Edition), accepted for publication, 2014.
• N. Zander, T. Bog, M. Elhaddad, R. Espinoza, H. Hu, A.F. Joly, C. Wu, P. Zerbe, A. Düster, S. Kollmannsberger, J. Parvizian, M. Ruess, D. Schillinger, E. Rank.
FCMLab: A Finite Cell Research Toolbox for MATLAB
Advances in Engineering Software, accepted for publication, 2014.
• M. Ranjbar, M. Mashayekhi, J. Parvizian, A. Düster, E. Rank.
Using the finite cell method to predict crack initiation in ductile
materials
Computational Materials Science, 82:427-434, 2014.
• A. Abedian, J. Parvizian, A. Düster, E. Rank.
The finite cell method for the J2 flow theory of plasticity
Finite Elements in Analysis and Design, 69:37-47, 2013.
Performance of different integration schemes in facing discontinuities in the finite cell method
International Journal of Computational Methods, 10: 1350002/1-24, 2013.
• J. Parvizian, A. Düster, E. Rank.
Topology Optimization Using the Finite Cell Method.
Optimization and Engineering, 13: 57-78, 2012.
• A. Düster, J. Parvizian, E. Rank.
Topology optimization based on the finite cell method.
PAMM. Proc. Appl. Math. Mech. 10, 151- 152, 2010.
Finite Cell Method for Elasto-Plastic Problems.
Proceedings of the Tenth International Conference on Computational Structures Techology, Valencia, Spain, September 14-17, 2010.
• E. Rank, A. Düster, D. Schillinger, Z. Yang.
The Finite Cell Method: High order simulation of complex structures without meshing.
Proceedings of the International Symposium on Computational Structural Engineering (CSE), Shanghai, China, June 22-24, 2009.
• A. Düster, J. Parvizian, Z. Yang, E. Rank.
The Finite Cell Method for 3D problems of solid mechanics.
Computer Methods in Applied Mechanics and Engineering, 197:3768-3782, 2008.
• A. Düster, J. Parvizian, Z. Yang, E. Rank.
A high order fictitious domain method for patient specific surgery planning.
Proceedings of APCOM’07 in conjunction with EPMESC XI, Japan, December 3-6, 2007.
• J. Parvizian, A. Düster, E. Rank.
Finite Cell Method: h- and p-extension for embedded domain problems in Solid Mechanics.
Computational Mechanics, 41: 121-133, 2007.