For SiC single crystals grown on-axis by the modified Lely method, thermoelastic stresses are computed. The crystal boundary temperatures are obtained by an axisymmetric heat transfer analysis of the ...whole growth furnace. By adding a small azimuthally varying temperature disturbance to the axisymmetric background temperature field, three-dimensional thermal stresses are generated. This problem is solved by decomposing all field variables into Fourier series whereby the Fourier coefficient fields are still two-dimensional ones. The azimuthal integration of the weak form of the finite element method (FEM) is performed analytically, while all coefficient fields as well as the intentional axisymmetric reference problem are solved numerically in the domain of radius and
z-axis only. For several growth stadiums, the total three-dimensional shear stresses are compared to the axisymmetric background shear stresses.
A numerical study of hydrodynamic instabilities of melt flows in Czochralski growth has been carried out. The background is that high temperature melting rare-earth scandates show often undesirable ...spiral growth and cork screw instabilities. The flow is driven by buoyancy, thermocapillarity and rotation forces. Using a simple numerical model, bifurcation analysis has been performed in order to separate the unstable from stable parameter regions. By applying an extended bifurcation analysis it is possible to find that solutions can be ambiguous, steady state and oscillatory. The numerical approach is based on a finite element discretisation using a fast solver for steady flow state calculation and bifurcation analysis. Results which confirm the hypothesis of initiating undesired spiral instabilities by heat and momentum disturbances are presented in order to help crystal growers to choose parameters which do not lead to spiral instability.
The kinetics of the dissolution of carbon dioxide in water and subsequent chemical reactions through to the formation of calcium carbonate, a system of reactions integral to carbon sequestration and ...anthropogenic ocean acidification, is mathematically modelled using the mass action law. This group of reactions is expressed as a system of five coupled nonlinear ordinary differential equations, with 14 independent parameters. The evolution of this system to equilibrium at 25°C and 1 atm, following an instantaneous injection of gaseous carbon dioxide, is simulated. An asymptotic analysis captures the leading-order behaviour of the system over six disparate time scales, yielding expressions for all species in each time scale. These approximations show excellent agreement with simulations of the full system, and give remarkably simple formulae for the equilibrium concentrations.
Three-dimensional (3D) thermoelastic stresses are computed for SiC single crystals which are grown with an off-axis orientation of the seed. The geometry is axisymmetric. The results are presented in ...terms of the resolved shear stress acting on the slip system. While in the on-axis case the resolved shear stress consists only of the component
σ
rz
, in the off-axis case it consists of all stress components, whereas the dominant components are
σ
rz
,
σ
rr
and
σ
zz
.
In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated ...with the incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard dual-weighted-residual approach, originally developed for the estimation of target functionals of the solution, to eigenvalue/stability problems. The underlying analysis consists of constructing both a dual eigenvalue problem and a dual problem for the original base solution. In this way, errors stemming from both the numerical approximation of the original nonlinear flow problem and the underlying linear eigenvalue problem are correctly controlled. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.
In this article we consider the
a posteriori
error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the ...steady incompressible Navier–Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork or Hopf bifurcation occurs when the underlying physical system possesses reflectional or
Z
2
symmetry. Here, computable
a posteriori
error bounds are derived based on employing the generalization of the standard Dual–Weighted–Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed
a posteriori
error indicator on adaptively refined computational meshes are presented.
Throughout much of condensed matter science, correlated disorder is a key to material function. While structural and compositional defects are known to exist within a variety of metal-organic ...frameworks (MOFs), the prevailing understanding is that these defects are only ever included in a random manner. Here we show--using a combination of diffuse scattering, electron microscopy, anomalous X-ray scattering and pair distribution function measurements--that correlations between defects can in fact be introduced and controlled within a hafnium terephthalate MOF. The nanoscale defect structures that emerge are an analogue of correlated Schottky vacancies in rocksalt-structured transition metal monoxides and have implications for storage, transport, optical and mechanical responses. Our results suggest how the diffraction behaviour of some MOFs might be reinterpreted, and establish a strategy of exploiting correlated nanoscale disorder as a targetable and desirable motif in MOF design.
In this article we consider the
a posteriori
error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the ...steady incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork bifurcation occurs when the underlying physical system possesses rotational and reflectional or
O
(2) symmetry. Here, computable
a posteriori
error bounds are derived based on employing the generalization of the standard Dual Weighted Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed
a posteriori
error indicator on adaptively refined computational meshes are presented. Here, particular attention is devoted to the problem of flow through a cylindrical pipe with a sudden expansion, which represents a notoriously difficult computational problem.
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