Abstract
Broken cells in the finite cell method—especially those with a small volume fraction—lead to a high condition number of the global system of equations. To overcome this problem, in this paper, we apply and adapt an eigenvalue stabilization technique to improve the ill-conditioned matrices of the finite cells and to enhance the robustness for large deformation analysis. In this approach, the modes causing high condition numbers are identified for each cell, based on the eigenvalues of the cell stiffness matrix. Then, those modes are supported directly by adding extra stiffness to the cell stiffness matrix in order to improve the condition number. Furthermore, the same extra stiffness is considered on the right-hand side of the system—which leads to a stabilization scheme that does not modify the solution. The performance of the eigenvalue stabilization technique is demonstrated using different numerical examples.
Highlights
In fictitious domain or immersed boundary methods, such as the finite cell method (FCM) [13,32], CutFEM [6,7,8], or CutIGA [15], the cells/elements generally do not conform to the boundary of the geometry of interest
We proposed an eigenvalue stabilization technique for the finite cell method in order to improve its robustness for nonlinear analysis at finite strains
The approach, originally proposed by Loehnert [29], is based on the eigenvalue decomposition of the stiffness matrix for cells that are cut by the boundary of the geometry
Summary
In fictitious domain or immersed boundary methods, such as the finite cell method (FCM) [13,32], CutFEM [6,7,8], or CutIGA [15], the cells/elements generally do not conform to the boundary of the geometry of interest. In nonlinear finite strain analysis, badly broken To this end, different stabilization techniques have been developed for fictitious domain methods to overcome the problem of ill-conditioning. We present an eigenvalue stabilization technique, based on the approach of Loehnert [29], to improve the robustness of the FCM for nonlinear problems by reducing the condition number of broken cells without affecting the solution significantly. To this end, an eigenvalue decomposition of the cell stiffness matrix is computed for every broken cell.
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