Stabilization techniques and adaptive conjugate gradient solver tolerances for the finite cell method

Roman Sartorti; Christian Kühne; Lars Radtke; Alexander Düster

doi:10.3934/acse.2025003

Article Contents

March 2025, Volume 3, 24-45. Doi: 10.3934/acse.2025003

This volume Previous Article Reduced order models for fluid flows at various Mach number solved using discontinuous Galerkin method Next Article Fourier spectral method for nonlocal equations on bounded domains

Stabilization techniques and adaptive conjugate gradient solver tolerances for the finite cell method

1.
Numerical Structural Analysis with Application in Ship Technology, Institute for Ship Structural Design and Analysis, Hamburg University of Technology, Germany
2.
Chair of Structural Mechanics, Faculty of Mechanical and Naval Engineering, University of Rostock, Albert-Einstein-Str. 2, 18059 Rostock, Germany

^*Corresponding author: Roman Sartorti
^*Corresponding author: Roman Sartorti

^† Contributing authors: Christian Kühne, Lars Radtke, Alexander Düster

Received: December 2024

Revised: March 2025

Early access: March 25, 2025

Published online: March 25, 2025

Roman Sartorti is supported by the Deutsche Forschungsgemeinschaft (DFG) under the grant number DU 405/21-1 and the project number 505137962.
Lars Radtke is supported by the Deutsche Forschungsgemeinschaft (DFG) under the grant number DU 405/20-1 and the project number 503865803.

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Abstract

Immersed boundary methods such as the finite cell method provide a versatile tool for the analysis of structures, which are difficult to discretize in a boundary conforming manner due to their complex geometries. Using Cartesian grids and a fictitious domain approach, the effort is shifted from meshing toward the quadrature, which needs to be adapted to discontinuous integrands for cut cells, i.e., elements cut by the immersed boundary. Further, the condition number of the stiffness matrix of such discretizations is typically much larger compared to boundary fitted discretizations, making the use of iterative solvers challenging. In order to restore the performance of iterative solvers, i.e., lower the condition number of the stiffness matrix, several stabilization methods are available. In this work, we compare the performance of two stabilization methods. The classical $ \alpha $-stabilization method uses a material with low (but non-zero) stiffness in the fictitious domain. The so called eigenvalue- or $ \epsilon $-stabilization is based on an eigendecomposition of the stiffness matrix and selectively stabilizes modes, which are associated with very low eigenvalues. Generally, both stabilization methods introduce an error, which, however, can be corrected. This work includes an investigation of such correction mechanisms for linear and nonlinear problems.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

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Figure 1. Illustration of the concept of a fictitious domain method

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Figure 2. Schematic of the eigenvalue stabilization procedure for a single cell (from left to right). First, the eigenvalues are computed. Second, the critical non-zero eigenmodes are extracted. Third, the eigenmodes are orthonormalized, and, finally, the eigenvalues are scaled and the stabilization system is constructed

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Figure 3. Plate with hole geometry and boundary conditions

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Figure 4. High-order boundary fitted FEM mesh (A) and a Cartesian FCM grid (B)

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Figure 5. Volume fraction of the cells within the finite cell mesh

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Figure 6. Influence of the stabilization parameters on the error in the energy norm (left) and the number of CG-iterations (right)

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Figure 7. Condition number of the cut cells for $ p = 3 $ without stabilization (left), with $ \alpha $-stabilization $ \alpha = 10^{-6} $ (center), and with $ \epsilon $-stabilization $ \epsilon = 10^{-6} $ (right)

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Figure 8. Combination of the data from Fig. 6 (left) and the same data after one corrector iteration (right). For each $ \alpha $-graph $ \alpha = 10^{-4-2\, k} $ and $ \epsilon = 0 $, and for each $ \epsilon $-graph $ \epsilon = 10^{-2-2\, k} $ and $ \alpha = 0 $, where $ k = 0, 1, 2, 3 $

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Figure 9. Progress of the error and deviation in energy norm for $ \alpha $-stabilization (top) and $ \epsilon $-stabilization (bottom) for $ p = 3 $

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Figure 10. Relative error in energy norm for FCM simulation ansatz order $ p = 5 $ for varying stabilization parameters

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Figure 11. Convergence in energy norm for $ p = 5 $ and varying stabilization tolerances. Each mark corresponds to the values at the end of one Newton-Raphson iteration

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Figure 12. Convergence for different right-hand side modifications for $ \epsilon = {10^{{-2}}} $ and $ \alpha = 0.0 $

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Figure 13. Convergence for different right-hand side modifications for $ \epsilon = 0.0 $ and $ \alpha = {10^{{-2}}} $

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Figure 14. Evolution of the CG solver convergence indicator over the CG-iteration (left) and the relative error in energy norm for each Newton-Raphson iteration (right)

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