Condition number analysis and preconditioning of the finite cell method

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

Phase-field models for brittle and cohesive fracture

In this paper we first recapitulate some basic notions of brittle and cohesive fracture models, as well as the phase-field approximation to fracture. Next, a critical assessment is made of the sensitivity of the phase-field approach to brittle fracture, in particular the degradation function, and the use of monolithic versus partitioned solution schemes. The last part of the paper makes extensions to a recently developed phase-field model for cohesive fracture, in particular for propagating cracks. Using some simple examples the current state of the cohesive phase-field model is shown

Error-estimate-based adaptive integration for immersed isogeometric analysis

The Finite Cell Method (FCM) together with Isogeometric analysis (IGA) has been applied successfully in various problems in solid mechanics, in image-based analysis, fluid–structure interaction and in many other applications. A challenging aspect of the isogeometric finite cell method is the integration of cut cells. In particular in three-dimensional simulations the computational effort associated with integration can be the critical component of a simulation. A myriad of integration strategies has been proposed over the past years to ameliorate the difficulties associated with integration, but a general optimal integration framework that suits a broad class of engineering problems is not yet available. In this contribution we provide a thorough investigation of the accuracy and computational effort of the octree integration scheme. We quantify the contribution of the integration error using the theoretical basis provided by Strang's first lemma. Based on this study we propose an error-estimate-based adaptive integration procedure for immersed isogeometric analysis. Additionally, we present a detailed numerical investigation of the proposed optimal integration algorithm and its application to immersed isogeometric analysis using two- and three-dimensional linear elasticity problems

Critical time-step size analysis and mass scaling by ghost-penalty for immersogeometric explicit dynamics

In this article, we study the effect of small-cut elements on the critical time-step size in an immersogeometric context. We analyze different formulations for second-order (membrane) and fourth-order (shell-type) equations, and derive scaling relations between the critical time-step size and the cut-element size for various types of cuts. In particular, we focus on different approaches for the weak imposition of Dirichlet conditions: by penalty enforcement and with Nitsche's method. The stability requirement for Nitsche's method necessitates either a cut-size dependent penalty parameter, or an additional ghost-penalty stabilization term is necessary. Our findings show that both techniques suffer from cut-size dependent critical time-step sizes, but the addition of a ghost-penalty term to the mass matrix serves to mitigate this issue. We confirm that this form of `mass-scaling' does not adversely affect error and convergence characteristics for a transient membrane example, and has the potential to increase the critical time-step size by orders of magnitude. Finally, for a prototypical simulation of a Kirchhoff-Love shell, our stabilized Nitsche formulation reduces the solution error by well over an order of magnitude compared to a penalty formulation at equal time-step size

Residual-based error estimation and adaptivity for stabilized immersed isogeometric analysis using truncated hierarchical B-splines

We propose an adaptive mesh refinement strategy for immersed isogeometric analysis, with application to steady heat conduction and viscous flow problems. The proposed strategy is based on residual-based error estimation, which has been tailored to the immersed setting by the incorporation of appropriately scaled stabilization and boundary terms. Element-wise error indicators are elaborated for the Laplace and Stokes problems, and a THB-spline-based local mesh refinement strategy is proposed. The error estimation .and adaptivity procedure is applied to a series of benchmark problems, demonstrating the suitability of the technique for a range of smooth and non-smooth problems. The adaptivity strategy is also integrated in a scan-based analysis workflow, capable of generating reliable, error-controlled, results from scan data, without the need for extensive user interactions or interventions.Comment: Submitted to Journal of Mechanic

Error-estimate-based Adaptive Integration For Immersed Isogeometric Analysis

The Finite Cell Method (FCM) together with Isogeometric analysis (IGA) has been applied successfully in various problems in solid mechanics, in image-based analysis, fluid-structure interaction and in many other applications. A challenging aspect of the isogeometric finite cell method is the integration of cut cells. In particular in three-dimensional simulations the computational effort associated with integration can be the critical component of a simulation. A myriad of integration strategies has been proposed over the past years to ameliorate the difficulties associated with integration, but a general optimal integration framework that suits a broad class of engineering problems is not yet available. In this contribution we provide a thorough investigation of the accuracy and computational effort of the octree integration scheme. We quantify the contribution of the integration error using the theoretical basis provided by Strang's first lemma. Based on this study we propose an error-estimate-based adaptive integration procedure for immersed isogeometric analysis. Additionally, we present a detailed numerical investigation of the proposed optimal integration algorithm and its application to immersed isogeometric analysis using two- and three-dimensional linear elasticity problems.Comment: To CAMW

A molecular signature of epithelial host defense: comparative gene expression analysis of cultured bronchial epithelial cells and keratinocytes

BACKGROUND: Epithelia are barrier-forming tissues that protect the organism against external noxious stimuli. Despite the similarity in function of epithelia, only few common protective mechanisms that are employed by these tissues have been systematically studied. Comparative analysis of genome-wide expression profiles generated by means of Serial Analysis of Gene Expression (SAGE) is a powerful approach to yield further insight into epithelial host defense mechanisms. We performed an extensive comparative analysis of previously published SAGE data sets of two types of epithelial cells, namely bronchial epithelial cells and keratinocytes, in which the response to pro-inflammatory cytokines was assessed. These data sets were used to elucidate a common denominator in epithelial host defense. RESULTS: Bronchial epithelial cells and keratinocytes were found to have a high degree of overlap in gene expression. Using an in silico approach, an epithelial-specific molecular signature of gene expression was identified in bronchial epithelial cells and keratinocytes comprising of family members of keratins, small proline-rich proteins and proteinase inhibitors. Whereas some of the identified genes were known to be involved in inflammation, the majority of the signature represented genes that were previously not associated with host defense. Using polymerase chain reaction, presence of expression of selected tissue-specific genes was validated. CONCLUSION: Our comparative analysis of gene transcription reveals that bronchial epithelial cells and keratinocytes both express a subset of genes that is likely to be essential in epithelial barrier formation in these cell types. The expression of these genes is specific for bronchial epithelial cells and keratinocytes and is not seen in non-epithelial cells. We show that bronchial epithelial cells, similar to keratinocytes, express components that are able to form a cross-linked protein envelope that may contribute to an effective barrier against noxious stimuli and pathogens

The cohesive band model: A cohesive surface formulation with stress triaxiality

In the cohesive surface model cohesive tractions are transmitted across a two-dimensional surface, which is embedded in a three-dimensional continuum. The relevant kinematic quantities are the local crack opening displacement and the crack sliding displacement, but there is no kinematic quantity that represents the stretching of the fracture plane. As a consequence, in-plane stresses are absent, and fracture phenomena as splitting cracks in concrete and masonry, or crazing in polymers, which are governed by stress triaxiality, cannot be represented properly. In this paper we extend the cohesive surface model to include in-plane kinematic quantities. Since the full strain tensor is now available, a three-dimensional stress state can be computed in a straightforward manner. The cohesive band model is regarded as a subgrid scale fracture model, which has a small, yet finite thickness at the subgrid scale, but can be considered as having a zero thickness in the discretisation method that is used at the macroscopic scale. The standard cohesive surface formulation is obtained when the cohesive band width goes to zero. In principle, any discretisation method that can capture a discontinuity can be used, but partition-of-unity based finite element methods and isogeometric finite element analysis seem to have an advantage since they can naturally incorporate the continuum mechanics. When using interface finite elements, traction oscillations that can occur prior to the opening of a cohesive crack, persist for the cohesive band model. Example calculations show that Poisson contraction influences the results, since there is a coupling between the crack opening and the in-plane normal strain in the cohesive band. This coupling holds promise for capturing a variety of fracture phenomena, such as delamination buckling and splitting cracks, that are difficult, if not impossible, to describe within a conventional cohesive surface model. © 2013 Springer Science+Business Media Dordrecht