Condition number analysis and preconditioning of the finite cell method

TL;DR

This paper analyzes the conditioning issues in the Finite Cell Method caused by small cell volumes and introduces the SIPIC preconditioner to improve solver performance and accuracy.

Contribution

It establishes a scaling relation for condition numbers and develops the SIPIC preconditioner to address ill-conditioning in FCM.

Findings

SIPIC significantly improves condition numbers.

Preconditioned solvers show faster convergence.

Enhanced accuracy enables detailed mesh convergence studies.

Abstract

The (Isogeometric) Finite Cell Method - in which a domain is immersed in a structured background mesh - suffers from conditioning problems when cells with small volume fractions occur. In this contribution, we establish a rigorous scaling relation between the condition number of (I)FCM system matrices and the smallest cell volume fraction. Ill-conditioning stems either from basis functions being small on cells with small volume fractions, or from basis functions being nearly linearly dependent on such cells. Based on these two sources of ill-conditioning, an algebraic preconditioning technique is developed, which is referred to as Symmetric Incomplete Permuted Inverse Cholesky (SIPIC). A detailed numerical investigation of the effectivity of the SIPIC preconditioner in improving (I)FCM condition numbers and in improving the convergence speed and accuracy of iterative solvers is presented for the Poisson problem and for two- and three-dimensional problems in linear elasticity, in which Nitche's method is applied in either the normal or tangential direction. The accuracy of the preconditioned iterative solver enables mesh convergence studies of the finite cell method.