Small and large deformation analysis with the p- and B-spline versions of the Finite Cell Method

• Published in: Computational mechanics Vol. 50; no. 4; pp. 445 - 478
• Main Authors: Schillinger, Dominik, Ruess, Martin, Zander, Nils, Bazilevs, Yuri, Düster, Alexander, Rank, Ernst
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.10.2012; Springer; Springer Nature B.V
• Subjects: Analysis; Basis functions; Boundary conditions; Classical and Continuum Physics; Complexity; Computational Science and Engineering; Convergence; Coordination compounds; Deformation analysis; Dirichlet problem; Elasticity; Engineering; Femur; Finite element method; Iterative methods; Mathematical analysis; Mesh generation; Metal foams; Methods; Nonlinear analysis; Original Paper; Robustness (mathematics); Splines; Theoretical and Applied Mechanics; Embedded domain methods; Version of the Finite Cell Method; B-spline version of the Finite Cell Method; Large deformation solid mechanics; Weak boundary conditions; Immersed boundary methods; Fictitious domain methods
• ISSN: 0178-7675; 1432-0924
• DOI: 10.1007/s00466-012-0684-z

The Finite Cell Method (FCM) is an embedded domain method, which combines the fictitious domain approach with high-order finite elements, adaptive integration, and weak imposition of unfitted Dirichlet boundary conditions. For smooth problems, FCM has been shown to achieve exponential rates of convergence in energy norm, while its structured cell grid guarantees simple mesh generation irrespective of the geometric complexity involved. The present contribution first unhinges the FCM concept from a special high-order basis. Several benchmarks of linear elasticity and a complex proximal femur bone with inhomogeneous material demonstrate that for small deformation analysis, FCM works equally well with basis functions of the p -version of the finite element method or high-order B-splines. Turning to large deformation analysis, it is then illustrated that a straightforward geometrically nonlinear FCM formulation leads to the loss of uniqueness of the deformation map in the fictitious domain. Therefore, a modified FCM formulation is introduced, based on repeated deformation resetting, which assumes for the fictitious domain the deformation-free reference configuration after each Newton iteration. Numerical experiments show that this intervention allows for stable nonlinear FCM analysis, preserving the full range of advantages of linear elastic FCM, in particular exponential rates of convergence. Finally, the weak imposition of unfitted Dirichlet boundary conditions via the penalty method, the robustness of FCM under severe mesh distortion, and the large deformation analysis of a complex voxel-based metal foam are addressed.