scholarly journals Improved convergence and stability properties in a three-dimensional higher-order ice sheet model

2011 ◽  
Vol 4 (3) ◽  
pp. 1569-1610
Author(s):  
J. J. Fürst ◽  
O. Rybak ◽  
H. Goelzer ◽  
B. De Smedt ◽  
P. de Groen ◽  
...  
Keyword(s):  
Velocity Field ◽  
Finite Difference ◽  
Three Dimensional ◽  
Ice Sheet ◽  
Higher Order ◽  
Velocity Fields ◽  
Force Balance ◽  
Staggered Grid ◽  
Comparison Results ◽  
Resultant Velocity

Abstract. We present a novel finite difference implementation of a three-dimensional higher-order ice sheet model that performs well both in terms of convergence rate and numerical stability. In order to achieve these benefits the discretisation of the governing force balance equation makes extensive use of information on staggered grid points. Using the same iterative solver, an existing discretisation that operates exclusively on the regular grid serves as a reference. Participation in the ISMIP-HOM benchmark indicates that both discretisations are capable of reproducing the higher-order model inter-comparison results. This allows a direct comparison not only of the resultant velocity fields but also of the solver's convergence behaviour which holds main differences. First and foremost, the new finite difference scheme facilitates convergence by a factor of up to 7 and 2.6 in average. In addition to this decrease in computational costs, the precision for the resultant velocity field can be chosen higher in the novel finite difference implementation. For high precisions, the old discretisation experiences difficulties to converge due to large variation in the velocity fields of consecutive Picard iterations. Finally, changing discretisation prevents build-up of local field irregularites that occasionally cause divergence of the solution for the reference discretisation. The improved behaviour makes the new discretisation more reliable for extensive application to real ice geometries. Higher precision and robust numerics are crucial in time dependent applications since numerical oscillations in the velocity field of subsequent time steps are attenuated and divergence of the solution is prevented. Transient applications also benefit from the increased computational efficiency.

scholarly journals Improved convergence and stability properties in a three-dimensional higher-order ice sheet model

2011 ◽  
Vol 4 (4) ◽  
pp. 1133-1149 ◽  
Author(s):  
J. J. Fürst ◽  
O. Rybak ◽  
H. Goelzer ◽  
B. De Smedt ◽  
P. de Groen ◽  
...  
Keyword(s):  
Velocity Field ◽  
Finite Difference ◽  
Three Dimensional ◽  
Ice Sheet ◽  
Higher Order ◽  
Force Balance ◽  
Stability And Convergence ◽  
Staggered Grid ◽  
Comparison Results ◽  
Resultant Velocity

Abstract. We present a finite difference implementation of a three-dimensional higher-order ice sheet model. In comparison to a conventional centred difference discretisation it enhances both numerical stability and convergence. In order to achieve these benefits the discretisation of the governing force balance equation makes extensive use of information on staggered grid points. Using the same iterative solver, a centred difference discretisation that operates exclusively on the regular grid serves as a reference. The reprise of the ISMIP-HOM experiments indicates that both discretisations are capable of reproducing the higher-order model inter-comparison results. This setup allows a direct comparison of the two numerical implementations also with respect to their convergence behaviour. First and foremost, the new finite difference scheme facilitates convergence by a factor of up to 7 and 2.6 in average. In addition to this decrease in computational costs, the accuracy for the resultant velocity field can be chosen higher in the novel finite difference implementation. Changing the discretisation also prevents build-up of local field irregularites that occasionally cause divergence of the solution for the reference discretisation. The improved behaviour makes the new discretisation more reliable for extensive application to real ice geometries. Higher accuracy and robust numerics are crucial in time dependent applications since numerical oscillations in the velocity field of subsequent time steps are attenuated and divergence of the solution is prevented.

scholarly journals Experimental Validation of Falling Liquid Film Models: Velocity Assumption and Velocity Field Comparison

Polymers ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1205
Author(s):  
Ruiqi Wang ◽  
Riqiang Duan ◽  
Haijun Jia
Keyword(s):  
Experimental Data ◽  
Particle Image Velocimetry ◽  
Velocity Field ◽  
Experimental Validation ◽  
Particle Image ◽  
Velocity Fields ◽  
Number Range ◽  
Comparison Results ◽  
Image Velocimetry ◽  
Experimental Particle

This publication focuses on the experimental validation of film models by comparing constructed and experimental velocity fields based on model and elementary experimental data. The film experiment covers Kapitza numbers Ka = 278.8 and Ka = 4538.6, a Reynolds number range of 1.6–52, and disturbance frequencies of 0, 2, 5, and 7 Hz. Compared to previous publications, the applied methodology has boundary identification procedures that are more refined and provide additional adaptive particle image velocimetry (PIV) method access to synthetic particle images. The experimental method was validated with a comparison with experimental particle image velocimetry and planar laser induced fluorescence (PIV/PLIF) results, Nusselt’s theoretical prediction, and experimental particle tracking velocimetry (PTV) results of flat steady cases, and a good continuity equation reproduction of transient cases proves the method’s fidelity. The velocity fields are reconstructed based on different film flow model velocity profile assumptions such as experimental film thickness, flow rates, and their derivatives, providing a validation method of film model by comparison between reconstructed velocity experimental data and experimental velocity data. The comparison results show that the first-order weighted residual model (WRM) and regularized model (RM) are very similar, although they may fail to predict the velocity field in rapidly changing zones such as the front of the main hump and the first capillary wave troughs.

Three‐dimensional induction logging problems, Part 2: A finite‐difference solution

Geophysics ◽  
2002 ◽  
Vol 67 (2) ◽  
pp. 484-491 ◽  
Author(s):  
Gregory A. Newman ◽  
David L. Alumbaugh
Keyword(s):  
Finite Difference ◽  
Electric Field ◽  
Krylov Subspace ◽  
Inverse Operator ◽  
Three Dimensional ◽  
Staggered Grid ◽  
Linear System Of Equations ◽  
Order Of Magnitude ◽  
Finite Difference Solution ◽  
Difference Solution

A 3‐D finite‐difference solution is implemented for simulating induction log responses in the quasi‐static limit that include the wellbore and bedding that exhibits transverse anisotropy. The finite‐difference code uses a staggered grid to approximate a vector equation for the electric field. The resulting linear system of equations is solved to a predetermined error level using iterative Krylov subspace methods. To accelerate the solution at low induction numbers (LINs), a new preconditioner is developed. This new preconditioner splits the electric field into curl‐free and divergence‐free projections, which allows for the construction of an approximate inverse operator. Test examples show up to an order of magnitude increase in speed compared to a simple Jacobi preconditioner. Comparisons with analytical and mode matching solutions demonstrate the accuracy of the algorithm.

scholarly journals Effect of higher-order stress gradients on the centennial mass evolution of the Greenland ice sheet

2012 ◽  
Vol 6 (4) ◽  
pp. 2961-3010
Author(s):  
J. J. Fürst ◽  
H. Goelzer ◽  
P. Huybrechts
Keyword(s):  
Standard Model ◽  
Ice Sheet ◽  
Coupled Model ◽  
Greenland Ice Sheet ◽  
Signal Propagation ◽  
Higher Order ◽  
Force Balance ◽  
Appreciable Effect ◽  
Model Version ◽  
The Standard Model

Abstract. We use a three-dimensional thermo-mechanically coupled model of the Greenland ice sheet to assess the effects of marginal perturbations on volume changes on centennial time scales. The model is designed to allow for five ice dynamic formulations using different approximations to the force balance. The standard model is based on the shallow ice approximation for both ice deformation and basal sliding. A second model version relies on a higher-order Blatter/Pattyn type of core that resolves effects from gradients in longitudinal stresses and transverse horizontal shearing, i.e. membrane-like stresses. Together with three intermediate model versions, these five versions allow for gradually more dynamic feedbacks from membrane stresses. Idealised experiments were conducted on various resolutions to compare the time-dependent response to imposed accelerations at the marine ice front. If such marginal accelerations are to have an appreciable effect on total mass loss on a century time scale, a fast mechanism to transmit such perturbations inland is required. While the forcing is independent of the model version, inclusion of direct horizontal coupling allows the initial speedup to reach several tens of kilometres inland. Within one century, effects from gradients in membrane stress alter the inland signal propagation and transmit additional dynamic thinning to the ice sheet interior. But the centennial overall volume loss differs only by some percents from the standard model as the dominant response is a diffusive inland propagation of geometric changes. In our experiments, the volume response is even attenuated by direct horizontal coupling. The reason is a faster adjustment of the sliding regime by instant stress transmission in models that account for the effect of membrane stresses. Ultimately, horizontal coupling decreases the reaction time to perturbations at the ice sheet margin.

A Semi-Staggered Numerical Procedure for the Incompressible Navier-Stokes Equations on Curvilinear Grids

2011 ◽  
Author(s):  
Hessam Babaee ◽  
Sumanta Acharya
Keyword(s):  
Finite Difference ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Second Order ◽  
Staggered Grid ◽  
Navier Stokes ◽  
Benchmark Problems ◽  
Navier Stokes Equations ◽  
Curvilinear Grids

An accurate and efficient finite difference method for solving the three dimensional incompressible Navier-Stokes equations on curvilinear grids is developed. The semi-staggered grid layout has been used in which all three components of velocity are stored on the corner vertices of the cell facilitating a consistent discretization of the momentum equations as the boundaries are approached. Pressure is stored at the cell-center, resulting in the exact satisfaction the discrete continuity. The diffusive terms are discretized using a second-order central finite difference. A third-order biased upwind scheme is used to discretize the convective terms. The momentum equations are integrated in time using a semi-implicit fractional step methodology. The convective and diffusive terms are advanced in time using the second-order Adams-Bashforth method and Crank-Nicolson method respectively. The Pressure-Poisson is discretized in a similar approach to the staggered gird layout and thus leading to the elimination of the spurious pressure eigen-modes. The validity of the method is demonstrated by two standard benchmark problems. The flow in driven cavity is used to show the second-order spatial convergence on an intentionally distorted grid. Finally, the results for flow past a cylinder for several Reynolds numbers in the range of 50–150 are compared with the existing experimental data in the literature.

scholarly journals Greenland under changing climates: sensitivity experiments with a new three-dimensional ice-sheet model

1995 ◽  
Vol 21 ◽  
pp. 1-7 ◽  
Author(s):  
Adeline Fabre ◽  
Anne Letréguilly ◽  
Catherine Ritz ◽  
Anne Mangeney
Keyword(s):  
Ice Core ◽  
Three Dimensional ◽  
Ice Sheet ◽  
Greenland Ice Sheet ◽  
Velocity Fields ◽  
Geothermal Heat ◽  
Climate History ◽  
Isostatic Adjustment ◽  
Temperature And Velocity Fields ◽  
Different Climates

A new three-dimensional, time-dependent ice-sheet model, including the calculation of the coupled temperature and velocity fields, isostatic adjustment of the bedrock and a mass-balance parameterization, was used to reconstruct the evolution of the Greenland ice sheet in response to a climate history derived from the oxygen-18 measured in the GRIP ice core. Steady-state experiments were done to test the sensitivity of the model, first to variations of poorly known parameters, secondly to different climates. These experiments show that the modelled ice sheet is not very sensitive to variations in the geothermal heat flux, but very sensitive to changes in the accumulation.

scholarly journals A higher order perfectly matched layer formulation for finite-difference time-domain seismic wave modeling

Geophysics ◽  
2015 ◽  
Vol 80 (1) ◽  
pp. T1-T16 ◽  
Author(s):  
David P. Connolly ◽  
Antonios Giannopoulos ◽  
Michael C. Forde
Keyword(s):  
Finite Difference ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
Degrees Of Freedom ◽  
Higher Order ◽  
Staggered Grid ◽  
First Order ◽  
Enhanced Absorption ◽  
Absorption Performance ◽  
Difference Time

We have developed a higher order perfectly matched layer (PML) formulation to improve the absorption performance for finite-difference time-domain seismic modeling. First, we outlined a new unsplit “correction” approach, which allowed for traditional, first-order PMLs to be added directly to existing codes in a straightforward manner. Then, using this framework, we constructed a PML formulation that can be used to construct higher order PMLs of arbitrary order. The greater number of degrees of freedom associated with the higher order PML allow for enhanced flexibility of the PML stretching functions, thus potentially facilitating enhanced absorption performance. We found that the new approach can offer increased elastodynamic absorption, particularly for evanescent waves. We also discovered that the extra degrees of freedom associated with the higher order PML required careful optimization if enhanced absorption was to be achieved. Furthermore, these extra degrees of freedom increased the computational requirements in comparison with first-order schemes. We reached our formulations using one compact equation thus increasing the ease of implementation. Additionally, the formulations are based on a recursive integration approach that reduce PML memory requirements, and do not require special consideration for corner regions. We tested the new formulations to determine their ability to absorb body waves and surface waves. We also tested standard staggered grid stencils and rotated staggered grid stencils.

A Three-Dimensional Upper Bound Elemental Technique for Forging Analysis

1996 ◽  
Vol 210 (4) ◽  
pp. 353-359 ◽  
Author(s):  
R S Lee ◽  
C T Kwan
Keyword(s):  
Velocity Field ◽  
Upper Bound ◽  
Metal Flow ◽  
Three Dimensional ◽  
Velocity Fields ◽  
Connecting Rod ◽  
Total Energy Consumption ◽  
Pure Lead ◽  
Forming Load ◽  
Theoretical Predictions

In this paper, two kinematically admissible velocity fields are derived for the proposed three-dimensional arbitrarily triangular and trapezoidal prismatic upper bound elemental technique (UBET) elements. These elements are applied to the portions between the circular shaped part and the straight rod part with three-dimensional metal flow in connecting rod forging, and then the capability of the proposed elements are shown. From the derived velocity fields, the upper bound loads on the upper die and the velocity field are determined by minimizing the total energy consumption with respect to some chosen parameters. Experiments with connecting rod forging were carried out with commercial pure lead billets at ambient temperature. The theoretical predictions of the forming load is in good agreement with the experimental results. It is shown that the proposed UBET elements in this work can effectively be used for the prediction of the forming load and velocity field in connecting rod forging.

Sign in / Sign up

Export Citation Format

Share Document