scholarly journals Modeling near-surface temperatures of airless bodies with application to the Moon

2019 ◽  
Vol 627 ◽  
pp. A129
Author(s):  
P. Gläser ◽  
D. Gläser
Keyword(s):  
Window Size ◽  
Solar Limb ◽  
Heat Diffusion ◽  
Integration Time ◽  
Integration Step ◽  
Step Size ◽  
Time Step ◽  
Laser Altimeter ◽  
Near Surface ◽  
Surface Temperatures

In this study we present a model to determine surface and sub-surface temperatures of airless bodies in the solar system. To precisely model direct sunlight we incorporated the solar limb darkening effect of the solar disk. Scattered sunlight and thermal re-radiation from nearby planets is also considered in our model. We further consider multiple scattering of reflected sunlight and thermal re-radiation on the modeled object itself. The finite volume method is applied to solve the model for which we present full derivations for the governing equations that control scattering and heat diffusion into the sub-surface. We assessed errors stemming from the chosen discretization of the depth profile, the window size from which scattering is considered, as well as from the chosen integration step-size and the spatial resolution of the Digital Terrain Model (DTM). Exemplarily, we determine surface and sub-surface (2 m depth) temperatures for the lunar polar areas. Topography of the lunar poles is modeled by measurements of the Lunar Orbiter Laser Altimeter (LOLA). We integrated temperatures over a 18.6-year time frame using 180 m pixel−1 LOLA DTMs of the poles, a 60 × 60 km window, and a 12 h integration time-step. The resulting preliminary temperature maps for the lunar poles are presented. Further, we show that our model agrees with temperatures obtained by the Diviner lunar radiometer experiment.

Newmark-Beta Method in Discrete Elastic Rods Algorithm to Avoid Energy Dissipation

2019 ◽  
Vol 86 (8) ◽  
Author(s):  
Weicheng Huang ◽  
Mohammad Khalid Jawed
Keyword(s):  
Energy Dissipation ◽  
Time Integration ◽  
Integration Scheme ◽  
Integration Step ◽  
Elastic Rods ◽  
Computationally Efficient ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Integration Scheme

Discrete elastic rods (DER) algorithm presents a computationally efficient means of simulating the geometrically nonlinear dynamics of elastic rods. However, it can suffer from artificial energy loss during the time integration step. Our approach extends the existing DER technique by using a different time integration scheme—we consider a second-order, implicit Newmark-beta method to avoid energy dissipation. This treatment shows better convergence with time step size, specially when the damping forces are negligible and the structure undergoes vibratory motion. Two demonstrations—a cantilever beam and a helical rod hanging under gravity—are used to show the effectiveness of the modified discrete elastic rods simulator.

scholarly journals Precise-Integration Time-Domain Formulation for Optical Periodic Media

Materials ◽  
2021 ◽  
Vol 14 (24) ◽  
pp. 7896
Author(s):  
Joan Josep Sirvent-Verdú ◽  
Jorge Francés ◽  
Andrés Márquez ◽  
Cristian Neipp ◽  
Mariela Álvarez ◽  
...  
Keyword(s):  
Time Domain ◽  
Fdtd Method ◽  
Fdtd Simulation ◽  
Integration Time ◽  
Periodic Media ◽  
Step Size ◽  
Time Step ◽  
Precise Integration ◽  
Time Step Size ◽  
Wide Range

A numerical formulation based on the precise-integration time-domain (PITD) method for simulating periodic media is extended for overcoming the Courant-Friedrich-Levy (CFL) limit on the time-step size in a finite-difference time-domain (FDTD) simulation. In this new method, the periodic boundary conditions are implemented, permitting the simulation of a wide range of periodic optical media, i.e., gratings, or thin-film filters. Furthermore, the complete tensorial derivation for the permittivity also allows simulating anisotropic periodic media. Numerical results demonstrate that PITD is reliable and even considering anisotropic media can be competitive compared to traditional FDTD solutions. Furthermore, the maximum allowable time-step size has been demonstrated to be much larger than that of the CFL limit of the FDTD method, being a valuable tool in cases in which the steady-state requires a large number of time-steps.

scholarly journals Mathematical Modeling of the Dynamics of Linear Electrical Systems with Parallel Calculations

Energies ◽  
2021 ◽  
Vol 14 (10) ◽  
pp. 2930
Author(s):  
Sławomir Cieślik
Keyword(s):  
Mathematical Modeling ◽  
Power Systems ◽  
Real Time ◽  
Integration Time ◽  
Integration Step ◽  
Time Step ◽  
Parallel Calculations ◽  
Electrical Systems ◽  
Backward Euler ◽  
Real Time Simulations

The dynamics of power systems is often analyzed using real-time simulators. The basic requirements of these simulators are the speed of obtaining the results and their accuracy. Known algorithms (backward Euler or trapezoidal rule) used in real-time simulations force the integration time step to be reduced to obtain the appropriate accuracy, which extends the time of obtaining the results. The acceleration of obtaining the results is achieved by using parallel calculations. The paper presents an algorithm for mathematical modeling of the dynamics of linear electrical systems, which works stably with a relatively large integration time step and with accuracy much better than other algorithms widely described in the literature. The algorithm takes into account the possibility of using parallel calculations. The proposed algorithm combines the advantages of known methods used in the analysis of electrical circuits, such as nodal analysis, multi-terminal electrical component theory, and transient states analysis methods. However, the main advantage over other algorithms is the use of the method based on average voltages in the integration step (AVIS method). The attention was focused on the presentation of the scientifically acceptable general principle offered to mathematical modeling of dynamics of linear electrical systems with parallel computations. However, the evidence of its effective application in the analysis of the dynamics of electric power and electromechanical systems was indicated in the works carried out by the team of authors from the Institute of Electrical Engineering UTP University of Science and Technology in Bydgoszcz (Poland).

scholarly journals Quantifying and attributing time step sensitivities in present-day climate simulations conducted with EAMv1

2020 ◽  
Author(s):  
Hui Wan ◽  
Shixuan Zhang ◽  
Philip J. Rasch ◽  
Vincent E. Larson ◽  
Xubin Zeng ◽  
...  
Keyword(s):  
Time Integration ◽  
Cloud Fraction ◽  
Coarse Grained ◽  
Step Size ◽  
Time Step ◽  
Substantial Impact ◽  
Near Surface ◽  
Upper Troposphere ◽  
Truncation Errors ◽  
Climate Simulations

Abstract. This study assesses the relative importance of time integration error in present-day climate simulations conducted with the atmosphere component of the Energy Exascale Earth System Model version 1 (EAMv1) at 1-degree horizontal resolution. We show that a factor-of-6 reduction of time step size in all major parts of the model leads to significant changes in the long-term mean climate. Examples of such changes include warming in the lower troposphere, cooling in the tropical and subtropical upper troposphere, as well as decreases of relative humidity throughout the troposphere accompanied by cloud fraction decreases. These changes imply that the reduction of temporal truncation errors leads to a notable although unsurprising degradation of agreement between the simulated and observed present-day climate; the model would require retuning to regain optimal climate fidelity in the absence of those truncation errors. A coarse-grained attribution of the time step sensitivities is carried out by separately shortening time steps used in various components of EAM or by revising the numerical coupling between some processes. Our analysis leads to the counter-intuitive finding that the marked decreases in the subtropical low-cloud fraction and total cloud radiative effect are caused not by the step size used for the collectively subcycled turbulence, shallow convection and stratiform cloud macro- and microphysics parameterizations but by the step sizes used outside the subcycles. Further analysis suggests that the coupling frequency between the subcycles and the rest of EAM has a substantial impact on the marine stratocumulus decks while the deep convection parameterization has a significant impact on trade cumulus. The step size of the cloud macro- and microphysics subcycles appears to have a primary impact on cloud fraction at most latitudes in the upper troposphere as well as in the mid-latitude near-surface layers. Impacts of step sizes used by the dynamical core and radiation appear to be relatively small. These results provide useful clues to help better understand the root causes of time step sensitivities in EAM. The experimentation strategy used here can also provide a pathway for other models to identify and reduce time integration errors.

Numerical oscillation in seepage analysis of unsaturated soils

2001 ◽  
Vol 38 (3) ◽  
pp. 639-651 ◽  
Author(s):  
Muthusamy Karthikeyan ◽  
Thiam-Soon Tan ◽  
Kok-Kwang Phoon
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Unsaturated Soil ◽  
Unsaturated Soils ◽  
Heat Diffusion ◽  
Step Size ◽  
Time Step ◽  
Seepage Analysis ◽  
Time Step Size ◽  
Element Method

The finite element method provides a popular means of analyzing groundwater flow in an unsaturated soil. In such problems, oscillatory results are often observed in the finite element solution. Such a phenomenon is observed, for example, when a typical finite element program such as Seep/w is used to model water infiltration into unsaturated soils. Numerical oscillations are often found near the wetting front where the hydraulic gradient is the steepest. These oscillations do not always reduce with decreasing or increasing time-step size alone; rather, an appropriate ratio between time-step size and element size is required. As the pore-water pressures predicted from a transient seepage analysis are used as input groundwater conditions for other types of analysis such as slope stability, contaminant transport, and capillary barrier, these oscillations may have important practical ramifications. Since seepage analysis is common in engineering practice, it is important that appropriate criteria are identified to minimize, if not to remove, the oscillations. In this paper, numerical examples are provided to demonstrate that a simple set of criteria, developed in heat diffusion problems with constant properties to control oscillation, is also applicable to one- and two-dimensional unsaturated seepage analyses, for a range of material nonlinearities that are frequently encountered in unsaturated soils.Key words: unsaturated soil, soil-water characteristic curve, seepage analysis, finite element method, numerical oscillation.

scholarly journals A method for solving heat transfer with phase change in ice or soil that allows for large time steps while guaranteeing energy conservation

2021 ◽  
Vol 15 (6) ◽  
pp. 2541-2568
Author(s):  
Niccolò Tubini ◽  
Stephan Gruber ◽  
Riccardo Rigon
Keyword(s):  
Heat Transfer ◽  
Phase Change ◽  
Model Performance ◽  
Grid Method ◽  
Frozen Soil ◽  
Integration Time ◽  
Step Size ◽  
Time Step ◽  
Conservation Of Energy ◽  
Time Step Size

Abstract. The accurate simulation of heat transfer with phase change is a central problem in cryosphere studies. This is because the non-linear behaviour of enthalpy as function of temperature can prevent thermal models of snow, ice, and frozen soil from converging to the correct solution. Existing numerical techniques rely on increased temporal resolution in trying to keep corresponding errors within acceptable bounds. Here, we propose an algorithm, originally applied to solve water flow in soils, as a method to solve these integration issues with guaranteed convergence and conservation of energy for any time step size. We review common modelling approaches, focusing on the fixed-grid method and on frozen soil. Based on this, we develop a conservative formulation of the governing equation and outline problems of alternative formulations in discretized form. Then, we apply the nested Newton–Casulli–Zanolli (NCZ) algorithm to a one-dimensional finite-volume discretization of the energy–enthalpy formulation. Model performance is demonstrated against the Neumann and Lunardini analytical solutions and by comparing results from numerical experiments with integration time steps of 1 h, 1 d, and 10 d. Using our formulation and the NCZ algorithm, the convergence of the solver is guaranteed for any time step size. With this approach, the integration time step can be chosen to match the timescale of the processes investigated.

scholarly journals On the numerical stability of surface-atmosphere coupling in weather and climate models

2016 ◽  
Author(s):  
Anton Beljaars ◽  
Emanuel Dutra ◽  
Gianpaolo Balsamo
Keyword(s):  
Heat Flux ◽  
Surface Temperature ◽  
Numerical Stability ◽  
Surface Heat ◽  
Heat Diffusion ◽  
Integration Time ◽  
Time Level ◽  
Time Step ◽  
Flux Coupling ◽  
Surface Atmosphere

Abstract. Coupling the atmosphere with the underlying surface presents numerical stability challenges in cost-effective model integrations used for operational weather prediction or climate simulations. These are due to the choice of large integration time-step, aiming at reducing computational burden, and to an explicit flux coupling formulation, often preferred for its simplicity and modularity. The atmospheric models therefore use the surface-layer temperatures (representative of the uppermost soil, snow, ice, water, etc.,) at previous integration time-step in all surface-atmosphere heat-flux calculations and prescribe fluxes to be used in the surface models' integrations. Although both models may use implicit formulations for the time stepping, the explicit flux coupling can still lead to instabilities. In this study, idealized simulations with a fully coupled implicit system are performed to derive an empirical relation between surface heat flux and surface temperature at the new time level. Such a relation mimics the fully implicit formulation by allowing to estimate the surface temperature at the new time level without solving the surface heat diffusion problem. It is based on similarity reasoning and applies to any medium with constant heat diffusion and heat capacity parameters. The advantage is that modularity of the code is maintained and that the heat flux can be computed in the atmospheric model in such a way that instabilities in the snow or ice code are avoided. Applicability to snow/ice/soil models with variable density is discussed, and the loss of accuracy turns out to be small.

scholarly journals A method for solving heat transfer with phase change in ice or soil that allows for large time steps while guaranteeing energy conservation

2020 ◽  
Author(s):  
Niccolò Tubini ◽  
Stephan Gruber ◽  
Riccardo Rigon
Keyword(s):  
Heat Transfer ◽  
Phase Change ◽  
Model Performance ◽  
Frozen Soil ◽  
Integration Time ◽  
Nonlinear Behaviour ◽  
Step Size ◽  
Time Step ◽  
Conservation Of Energy ◽  
Time Step Size

Abstract. The accurate simulation of heat transfer with phase change is a central problem in cryosphere studies. This is because the nonlinear behaviour of enthalpy as function of temperature can prevent thermal models of snow, ice and frozen soil from converging to the correct solution. Existing numerical techniques rely on increased temporal resolution in trying to keep corresponding errors withing acceptable bounds. Here, we propose an algorithm, originally applied to solve water flow in soils, as a method to solve these integration issues with guaranteed convergence and conservation of energy for any time step size. We review common modeling approaches, focusing on the fixed-grid method and on frozen soil. Based on this, we develop a conservative formulation of the governing equation and outline problems of alternative formulations in discretized form. Then, we apply the nested Newton-Casulli-Zanolli (NCZ) algorithm to a one-dimensional finite-volume discretization of the energy-enthalpy formulation. Model performance is demonstrated against the Neumann and Lunardini analytical solutions and by comparing results from numerical experiments with integration time steps of one hour, one day, and ten days. Using our formulation and the NCZ algorithm, the convergence of the solver is guaranteed for any time step size. With this approach, the integration time step can be chosen to match the time scale of the processes investigated.

scholarly journals The Influence of an Integration Time Step on Dynamic Calculation of a Vehicle-Track-Bridge under High-Speed Railway

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 431
Author(s):  
Junjie Ye ◽  
Hao Sun
Keyword(s):  
High Speed ◽  
Coupled Model ◽  
Short Wave ◽  
Integration Time ◽  
Vertical Acceleration ◽  
Dynamic Calculation ◽  
Time Step ◽  
High Speed Railway ◽  
Track Irregularity ◽  
Track Bridge

In order to study the influence of an integration time step on dynamic calculation of a vehicle-track-bridge under high-speed railway, a vehicle-track-bridge (VTB) coupled model is established. The influence of the integration time step on calculation accuracy and calculation stability under different speeds or different track regularity states is studied. The influence of the track irregularity on the integration time step is further analyzed by using the spectral characteristic of sensitive wavelength. According to the results, the disparity among the effect of the integration time step on the calculation accuracy of the VTB coupled model at different speeds is very small. Higher speed requires a smaller integration time step to keep the calculation results stable. The effect of the integration time step on the calculation stability of the maximum vertical acceleration of each component at different speeds is somewhat different, and the mechanism of the effect of the integration time step on the calculation stability of the vehicle-track-bridge coupled system is that corresponding displacement at the integration time step is different. The calculation deviation of the maximum vertical acceleration of the car body, wheel-sets and bridge under the track short wave irregularity state are greatly increased compared with that without track irregularity. The maximum vertical acceleration of wheel-sets, rails, track slabs and the bridge under the track short wave irregularity state all show a significant declining trend. The larger the vibration frequency is, the smaller the range of integration time step is for dynamic calculation.

scholarly journals Thermal Submeso Motions in the Nocturnal Stable Boundary Layer. Part 2: Generating Mechanisms and Implications

2021 ◽  
Author(s):  
Lena Pfister ◽  
Karl Lapo ◽  
Larry Mahrt ◽  
Christoph K. Thomas
Keyword(s):  
Boundary Layer ◽  
Stable Boundary Layer ◽  
Side Wall ◽  
Cold Pool ◽  
Continuous Data ◽  
Valley Bottom ◽  
Near Surface ◽  
Cold Air ◽  
Stable Boundary ◽  
Surface Temperatures

AbstractIn the stable boundary layer, thermal submesofronts (TSFs) are detected during the Shallow Cold Pool experiment in the Colorado plains, Colorado, USA in 2012. The topography induces TSFs by forming two different air layers converging on the valley-side wall while being stacked vertically above the valley bottom. The warm-air layer is mechanically generated by lee turbulence that consistently elevates near-surface temperatures, while the cold-air layer is thermodynamically driven by radiative cooling and the corresponding cold-air drainage decreases near-surface temperatures. The semi-stationary TSFs can only be detected, tracked, and investigated in detail when using fibre-optic distributed sensing (FODS), as point observations miss TSFs most of the time. Neither the occurrence of TSFs nor the characteristics of each air layer are connected to a specific wind or thermal regime. However, each air layer is characterized by a specific relationship between the wind speed and the friction velocity. Accordingly, a single threshold separating different flow regimes within the boundary layer is an oversimplification, especially during the occurrence of TSFs. No local forcings or their combination could predict the occurrence of TSFs except that they are less likely to occur during stronger near-surface or synoptic-scale flow. While classical conceptualizations and techniques of the boundary layer fail in describing the formation of TSFs, the use of spatially continuous data obtained from FODS provide new insights. Future studies need to incorporate spatially continuous data in the horizontal and vertical planes, in addition to classic sensor networks of sonic anemometry and thermohygrometers to fully characterize and describe boundary-layer phenomena.

Sign in / Sign up

Export Citation Format

Share Document