44 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
Stabilized immersed isogeometric analysis for the Navier-Stokes-Cahn-Hilliard equations, with applications to binary-fluid flow through porous media
Binary-fluid flows can be modeled using the Navier-Stokes-Cahn-Hilliard
equations, which represent the boundary between the fluid constituents by a
diffuse interface. The diffuse-interface model allows for complex geometries
and topological changes of the binary-fluid interface. In this work, we propose
an immersed isogeometric analysis framework to solve the
Navier-Stokes-Cahn-Hilliard equations on domains with geometrically complex
external binary-fluid boundaries. The use of optimal-regularity B-splines
results in a computationally efficient higher-order method. The key features of
the proposed framework are a generalized Navier-slip boundary condition for the
tangential velocity components, Nitsche's method for the convective
impermeability boundary condition, and skeleton- and ghost-penalties to
guarantee stability. A binary-fluid Taylor-Couette flow is considered for
benchmarking. Porous medium simulations demonstrate the ability of the immersed
isogeometric analysis framework to model complex binary-fluid flow phenomena
such as break-up and coalescence in complex geometries
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