Dynamics with matrices possessing kronecker product structure

In this paper we present an application of Alternating Direction Implicit (ADI) algorithm for solution of non-stationary PDE-s using isogeometric finite element method. We show that ADI algorithm has a linear computational cost at every time step. We illustrate this approach by solving two example non-stationary three-dimensional problems using explicit Euler and Newmark time-stepping scheme: heat equation and linear elasticity equations for a cube. The stability of the simulation is controlled by monitoring the energy of the solution

Time-stepping beyond CFL: a locally one-dimensional scheme for acoustic wave propagation

In this abstract, we present a case study in the application of a time-stepping method, unconstrained by the CFL condition, for computational acoustic wave propagation in the context of full waveform inversion. The numerical scheme is a locally one-dimensional (LOD) variant of alternating dimension implicit (ADI) method. The LOD method has a maximum time step that is restricted only by the Nyquist sampling rate. The advantage over traditional explicit time-stepping methods occurs in the presence of high contrast media, low frequencies, and steep, narrow perfectly matched layers (PML). The main technical point of the note, from a numerical analysis perspective, is that the LOD scheme is adapted to the presence of a PML. A complexity study is presented and an application to full waveform inversion is shown.National Science Foundation (U.S.); Alfred P. Sloan Foundatio

Comparison of Wide and Compact Fourth Order Formulations of the Navier-Stokes Equations

In this study the numerical performances of wide and compact fourth order formulation of the steady 2-D incompressible Navier-Stokes equations will be investigated and compared with each other. The benchmark driven cavity flow problem will be solved using both wide and compact fourth order formulations and the numerical performances of both formulations will be presented and also the advantages and disadvantages of both formulations will be discussed

Wall effects on the transportation of a cylindrical particle in power-law fluids

The present work deals with the numerical calculation of the Stokes-type drag undergone by a cylindrical particle perpendicularly to its axis in a power-law fluid. In unbounded medium, as all data are not available yet, we provide a numerical solution for the pseudoplastic fluid. Indeed, the Stokes-type solution exists because the Stokes’ paradox does not take place anymore. We show a high sensitivity of the solution to the confinement, and the appearance of the inertia in the proximity of the Newtonian case, where the Stokes’ paradox takes place. For unbounded medium, avoiding these traps, we show that the drag is zero for Newtonian and dilatant fluids. But in the bounded one, the Stokes-type regime is recovered for Newtonian and dilatant fluids. We give also a physical explanation of this effect which is due to the reduction of the hydrodynamic screen length, for pseudoplastic fluids. Once the solution of the unbounded medium has been obtained, we give a solution for the confined medium numerically and asymptotically. We also highlight the consequence of the confinement and the backflow on the settling velocity of a fiber perpendicularly to its axis in a slit. Using the dynamic mesh technique, we give the actual transportation velocity in a power-law “Poiseuille flow”, versus the confinement parameter and the fluidity index, induced by the hydrodynamic interactions

Numerical Performance of Compact Fourth Order Formulation of the Navier-Stokes Equations

In this study the numerical performance of the fourth order compact formulation of the steady 2-D incompressible Navier-Stokes equations introduced by Erturk et al. (Int. J. Numer. Methods Fluids, 50, 421-436) will be presented. The benchmark driven cavity flow problem will be solved using the introduced compact fourth order formulation of the Navier-Stokes equations with two different line iterative semi-implicit methods for both second and fourth order spatial accuracy. The extra CPU work needed for increasing the spatial accuracy from second order (O(x2)) to fourth order (O(x4)) formulation will be presented

Treatment of Subclinical Hypothyroidism or Hypothyroxinemia in Pregnancy

Subclinical thyroid disease during pregnancy may be associated with adverse outcomes, including a lower-than-normal IQ in offspring. It is unknown whether levothyroxine treatment of women who are identified as having subclinical hypothyroidism or hypothyroxinemia during pregnancy improves cognitive function in their children

Mathematical Modelling of Transient Thermography and Defect Sizing

The principle employed to obtain an image of a sub-surface defect by transient thermography is deceptively simple. A surface is heated by powerful flash lamps and subsequent thermal transients are recorded by an infrared camera. Defects cause perturbations in heat flow which are revealed by the camera. Whilst there is now a considerable body of practical experience of the application of the technique, there is rather less precise quantitative information about the image formation process that could lead to reliable defect sizing. In earlier papers [1,2] one of the authors considered circular air gap defects by treating them as buried uniformly heated disks. The thermal edge-effect occurring at the tip of a perfect crack-like defect was dealt with analytically by adapting the well established Wiener-Hopf [3] solution for the scattering of light or sound from the edge of a semi-infinite half-plane. The problem was solved in the frequency-domain, i.e. to obtain a thermal wave solution, and then a time-domain solution was obtained by a suitable transformation. The analysis showed an edge-effect amounting to a decay in temperature contrast over a distance of about a thermal diffusion length from the edge of the crack. A crucial feature of the edge-effect was the decay of thermal contrast to zero at the crack tip. This, and the edge-effect as a whole, is caused by the flow of heat around the crack tip from the hot upper surface of the crack to the cold under surface. The symmetry of this process ensures that there is no net flux increase for material in front of the crack tip

Critical number of atoms for attractive Bose-Einstein condensates with cylindrically symmetrical traps

We calculated, within the Gross-Pitaevskii formalism, the critical number of atoms for Bose-Einstein condensates with two-body attractive interactions in cylindrical traps with different frequency ratios. In particular, by using the trap geometries considered by the JILA group [Phys. Rev. Lett. 86, 4211 (2001)], we show that the theoretical maximum critical numbers are given approximately by $N_c = 0.55 ({l_0}/{|a|})$. Our results also show that, by exchanging the frequencies $\omega_z$ and $\omega_\rho$, the geometry with $\omega_\rho < \omega_z$ favors the condensation of larger number of particles. We also simulate the time evolution of the condensate when changing the ground state from $a=0$ to $a<0$ using a 200ms ramp. A conjecture on higher order nonlinear effects is also added in our analysis with an experimental proposal to determine its signal and strength.Comment: (4 pages, 2 figures) To appear in Physical Review