Efficient simulation of heterogeneous structures with the Finite Cell Method

Project members:
Mahan Gorji, Prof. Dr.-Ing. habil. Alexander Düster

Project description:
Numerical methods play an important role for solving engineering problems. Especially in structural mechanics, the Finite Element Method (FEM) is widely used. When the geometry becomes complex, the mesh generation takes a long time during the overall design process. To address this problem the Finite Cell Method (FCM) is applied, which is a combination of high-order finite elements with fictitious domain methods. It has been shown, that the FCM performs well as for linear elastic problems as well for nonlinear problems, e.g. for large deformations or elastoplasticity [1, 4, 5].

When material interfaces occur, the convergence rate of the standard FCM deteriorates due to the related discontinuities. Such material interfaces appear for example in heterogeneous structures. To overcome this issue, the FCM is extended by the local enrichment, which in our case is based on the hp-d method combined with a high-order partition of unity (PUM) approach; also known as the hp-d/PUM-FCM [3]. With the aid of the local enrichment, physical effects caused by the material interface such as kinks in the displacements or jumps in the stresses and strains, can be captured very precisely. Furthermore, the local enrichment is utilized in XFEM, where it is used for applications in fracture mechanics [2].

In Figure 1, a plate with heterogenous material is analyzed. The white circles and ellipsoids denote the holes, while the darkgreen ones denote the inclusions (stiffer material). In Figure 2, the von Mises stress is plotted along the cutline AA’. It is clear that the proposed method (hp-d/PUM-FCM) shows a good agreement as compared to the reference solution performed by ABAQUS [3].

Figure 1: A heterogenous plate under compression [3].
Figure 2: The von Mises stress along the cutline AA’ [3].

References:

[1] A. Abedian, J. Parvizian, A. Düster, and E. Rank. Finite cell method
compared to h-version finite element method for elasto-plastic problems. Applied Mathematics and Mechanics, 35(10):1239–1248, 2014.

[2] T.-P. Fries and T. Belytschko. The extended/generalized finite element
method: An overview of the method and its applications. International
Journal for Numerical Methods in Engineering, 84(3):253–304, 2010.

[3] M. Joulaian and A. Düster. Local enrichment of the finite cell method for problems with material interfaces. Computational Mechanics, 52:741–762, 2013.

[4] J. Parvizian, A. Düster, and E. Rank. Finite cell method: h- and p-extension for embedded domain problems in solid mechanics. Computational Mechanics, 41:121–133, 2007.

[5] D. Schillinger, M. Ruess, N. Zander, Y. Bazilevs, A. Düster, and E. Rank.
Small and large deformation analysis with the p- and B-spline versions of the finite cell method. Computational Mechanics, 50:445–478, 2012.