158 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
Discontinuities without discontinuity: The Weakly-enforced Slip Method
Tectonic faults are commonly modelled as Volterra or Somigliana dislocations
in an elastic medium. Various solution methods exist for this problem. However,
the methods used in practice are often limiting, motivated by reasons of
computational efficiency rather than geophysical accuracy. A typical
geophysical application involves inverse problems for which many different
fault configurations need to be examined, each adding to the computational
load. In practice, this precludes conventional finite-element methods, which
suffer a large computational overhead on account of geometric changes. This
paper presents a new non-conforming finite-element method based on weak
imposition of the displacement discontinuity. The weak imposition of the
discontinuity enables the application of approximation spaces that are
independent of the dislocation geometry, thus enabling optimal reuse of
computational components. Such reuse of computational components renders
finite-element modeling a viable option for inverse problems in geophysical
applications. A detailed analysis of the approximation properties of the new
formulation is provided. The analysis is supported by numerical experiments in
2D and 3D.Comment: Submitted for publication in CMAM
Duality-based two-level error estimation for time-dependent PDEs: application to linear and nonlinear parabolic equations
We introduce a duality-based two-level error estimator for linear and nonlinear time-dependent problems. The error measure can be a space-time norm, energy norm, final-time error or other error related functional. The general methodology is developed for an abstract nonlinear parabolic PDE and subsequently applied to linear heat and nonlinear Cahn-Hilliard equations. The error due to finite element approximations is estimated with a residual weighted approximate-dual solution which is computed with two primal approximations at nested levels. We prove that the exact error is estimated by our estimator up to higher-order remainder terms. Numerical experiments confirm the theory regarding consistency of the dual-based two-level estimator. We also present a novel space-time adaptive strategy to control errors based on the new estimator
Inverting elastic dislocations using the Weakly-enforced Slip Method
Earthquakes cause lasting changes in static equilibrium, resulting in global
deformation fields that can be observed. Consequently, deformation measurements
such as those provided by satellite based InSAR monitoring can be used to infer
an earthquake's faulting mechanism. This inverse problem requires a numerical
forward model that is both accurate and fast, as typical inverse procedures
require many evaluations. The Weakly-enforced Slip Method (WSM) was developed
to meet these needs, but it was not before applied in an inverse problem
setting. Consequently, it was unknown what effect particular properties of the
WSM, notably its inherent continuity, have on the inversion process. Here we
show that the WSM is able to accurately recover slip distributions in a
Bayesian-inference setting, provided that data points in the vicinity of the
fault are removed. In a representative scenario, an element size of 2 km was
found to be sufficiently fine to generate a posterior probability distribution
that is close to the theoretical optimum. For rupturing faults a masking zone
of 20 km sufficed to avoid numerical disturbances that would otherwise be
induced by the discretization error. These results demonstrate that the WSM is
a viable forward method for earthquake inversion problems. While our
synthesized scenario is basic for reasons of validation, our results are
expected to generalize to the wider gamut of scenarios that finite element
methods are able to capture. This has the potential to bring modeling
flexibility to a field that if often forced to impose model restrictions in a
concession to computability.Comment: The associated software implementation is openly available in zenodo
at https://doi.org/10.5281/zenodo.507179
A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations
While many methods exist to discretize nonlinear time-dependent partial differential equations (PDEs), the rigorous estimation and adaptive control of their discretization errors remains challenging. In this paper, we present a methodology for duality-based a posteriori error estimation for nonlinear parabolic PDEs, where the full discretization of the PDE relies on the use of an implicit-explicit (IMEX) time-stepping scheme and the finite element method in space. The main result in our work is a decomposition of the error estimate that allows to separate the effects of spatial and temporal discretization error, and which can be used to drive adaptive mesh refinement and adaptive time-step selection. The decomposition hinges on a specially-tailored IMEX discretization of the dual problem. The performance of the error estimates and the proposed adaptive algorithm is demonstrated on two canonical applications: the elementary heat equation and the nonlinear Allen-Cahn phase-field model
Connecting species’ geographical distributions to environmental variables: range maps versus observed points of occurrence
Connecting the geographical occurrence of a species with underlying environmental variables is fundamental for many analyses of life history evolution and for modeling species distributions for both basic and practical ends. However, raw distributional information comes principally in two forms: points of occurrence (specific geographical coordinates where a species has been observed), and expert-prepared range maps. Each form has potential short-comings: range maps tend to overestimate the true occurrence of a species, whereas occurrence points (because of their frequent non-random spatial distribution) tend to underestimate it. Whereas previous comparisons of the two forms have focused on how they may differ when estimating species richness, less attention has been paid to the extent to which the two forms actually differ in their representation of a species’ environmental associations. We assess such differences using the globally distributed avian order Galliformes (294 species). For each species we overlaid range maps obtained from IUCN and point-of-occurrence data obtained from GBIF on global maps of four climate variables and elevation. Over all species, the median difference in distribution centroids was 234 km, and median values of all five environmental variables were highly correlated, although there were a few species outliers for each variable. We also acquired species’ elevational distribution mid-points (mid-point between minimum and maximum elevational extent) from the literature; median elevations from point occurrences and ranges were consistently lower (median −420 m) than mid-points. We concluded that in most cases occurrence points were likely to produce better estimates of underlying environmental variables than range maps, although differences were often slight. We also concluded that elevational range mid-points were biased high, and that elevation distributions based on either points or range maps provided better estimates
On the singular nature of the elastocapillary ridge
The functionality of soft interfaces is crucial to many applications in
biology and surface science. Recent studies have used liquid drops to probe the
surface mechanics of elastomeric networks. Experiments suggest an intricate
surface elasticity, also known as the Shuttleworth effect, where surface
tension is not constant but depends on substrate deformation. However,
interpretations have remained controversial due to singular elastic
deformations, induced exactly at the point where the droplet pulls the network.
Here we reveal the nature of the elastocapillary singularity on a hyperelastic
substrate with various constitutive relations for the interfacial energy.
First, we finely resolve the vicinity of the singularity using goal-adaptive
finite element simulations. This confirms the universal validity, also at large
elastic deformations, of the previously disputed Neumann's law for the contact
angles. Subsequently, we derive exact solutions of nonlinear elasticity that
describe the singularity analytically. These solutions are in perfect agreement
with numerics, and show that the stretch at the contact line, as previously
measured experimentally, consistently points to a strong Shuttleworth effect.
Finally, using Noether's theorem we provide a quantitative link between wetting
hysteresis and Eshelby-like forces, and thereby offer a complete framework for
soft wetting in the presence of the Shuttleworth effect.Comment: 17 Pages, 7 figure
A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations
While many methods exist to discretize nonlinear time-dependent partial differential equations (PDEs), the rigorous estimation and adaptive control of their discretization errors remains challenging. In this paper, we present a methodology for duality-based a posteriori error estimation for nonlinear parabolic PDEs, where the full discretization of the PDE relies on the use of an implicit-explicit (IMEX) time-stepping scheme and the finite element method in space. The main result in our work is a decomposition of the error estimate that allows to separate the effects of spatial and temporal discretization error, and which can be used to drive adaptive mesh refinement and adaptive time-step selection. The decomposition hinges on a specially-tailored IMEX discretization of the dual problem. The performance of the error estimates and the proposed adaptive algorithm is demonstrated on two canonical applications: the elementary heat equation and the nonlinear Allen-Cahn phase-field model
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
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