38 research outputs found
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
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
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
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
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
An Isogeometric Analysis Approach to Fluid Flow in a Fractured Porous Medium
In this paper we present an isogeometric approach for calculating fluid flow in a fractured porous medium. The fluid flow away from the crack is modelled using Darcy's relation. A similar relation is assumed for the fluid flow inside the crack. Here, the higher porosity is modelled using a different permeability. An isogeometric analysis approach with B-splines is used for both the parametrisation of the geometry and the discretisation of the weak form of the equilibrium equations. The crack is described in a discrete manner by using the continuity reduction property of the B-splines. To ease the integration into existing finite element technology, a finite element data structure based on B'ezier extraction is used. The B'ezier extraction operator decomposes the B-spline based elements to B'ezier elements which bear a close resemblance to Lagrange elements. The global smoothness of B-splines is localized to an element level similar to finite element analysis, making isogeometric analysis compatible with existing finite element codes while still exploiting the excellent properties of the spline basis functions. The results of the new model are demonstrated in an example in which the fluid flow is considered around a crack in a specimen under mode-I loading
Stochastic Finite Element Method for analyzing static and dynamic pull-in of microsystems
Electro–mechanical sensors and actuators are a specific type of microsystems. The electrostatic pull-in value is one of the defining characteristics for these devices. Because the material and geometrical properties of micro fabricated systems are often very uncertain, this pull-in value can be subject to considerable variations. Therefore it is important to be able to estimate how uncertainty of mechanical properties propagates to the uncertainty of pull-in values. In this work the required design sensitivities of static and dynamic pull-in are derived. These sensitivities are used to perform a perturbation–based stochastic FEM analysis of an electromechanical device. This stochastic analysis consists of an uncertainty analysis and a reliability analysis. This stochastic analysis is validated by an expensive crude Monte Carlo computation.Mechanical, Maritime and Materials Engineerin