scholarly journals Numerical Gradient Schemes for Heat Equations Based on the Collocation Polynomial and Hermite Interpolation

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 93 ◽  
Author(s):  
Hou-Biao Li ◽  
Ming-Yan Song ◽  
Er-Jie Zhong ◽  
Xian-Ming Gu
Keyword(s):  
Hermite Interpolation ◽  
Mesh Size ◽  
Dirichlet Boundary Conditions ◽  
Maximum Norm ◽  
Heat Equations ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
The One ◽  
Delay Partial Differential Equations

As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is that it is unconditionally stable and convergent on the order O ( τ 2 + h 4 ) (where τ is the time step size and h is the mesh size), under the maximum norm for a class of nonlinear delay partial differential equations with initial and Dirichlet boundary conditions. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. The convergence order of this kind of method is also O ( τ 2 + h 4 ) under the discrete maximum norm when the spatial step size is twice the one of H-OCD, which accelerates the computational process. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in the time direction. The results of numerical experiments are consistent with the theoretical analysis.

The Dispersion in One- and Two-Dimensional Finite Element Solutions to the Heat Equation

2002 ◽  
Author(s):  
W. Dauksher ◽  
A. F. Emery
Keyword(s):  
Finite Element ◽  
Heat Equations ◽  
Two Dimensional ◽  
Step Size ◽  
Matrix Formulation ◽  
Time Step ◽  
Time Step Size ◽  
Time Stepping ◽  
Dimensional Finite Element ◽  
The One

The dispersive errors in the finite element solution to the one- and two-dimensional heat equations are examined as a function of element type and size, capacitance matrix formulation, time stepping scheme and time step size.

scholarly journals SHAKTI: Subglacial Hydrology and Kinetic, Transient Interactions v1.0

2018 ◽  
Vol 11 (7) ◽  
pp. 2955-2974 ◽  
Author(s):  
Aleah Sommers ◽  
Harihar Rajaram ◽  
Mathieu Morlighem
Keyword(s):  
Water Pressure ◽  
Water Flux ◽  
Ice Sheet ◽  
Mesh Size ◽  
Governing Equations ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Transient Interactions ◽  
Ice Sheet Dynamics

Abstract. Subglacial hydrology has a strong influence on glacier and ice sheet dynamics, particularly through the dependence of sliding velocity on subglacial water pressure. Significant challenges are involved in modeling subglacial hydrology, as the drainage geometry and flow mechanics are constantly changing, with complex feedbacks that play out between water and ice. A clear tradition has been established in the subglacial hydrology modeling literature of distinguishing between channelized (efficient) and sheetlike (inefficient or distributed) drainage systems or components and using slightly different forms of the governing equations in each subsystem to represent the dominant physics. Specifically, many previous subglacial hydrology models disregard opening by melt in the sheetlike system or redistribute it to adjacent channel elements in order to avoid runaway growth that occurs when it is included in the sheetlike system. We present a new subglacial hydrology model, SHAKTI (Subglacial Hydrology and Kinetic, Transient Interactions), in which a single set of governing equations is used everywhere, including opening by melt in the entire domain. SHAKTI employs a generalized relationship between the subglacial water flux and the hydraulic gradient that allows for the representation of laminar, turbulent, and transitional regimes depending on the local Reynolds number. This formulation allows for the coexistence of these flow regimes in different regions, and the configuration and geometry of the subglacial system evolves naturally to represent sheetlike drainage as well as systematic channelized drainage under appropriate conditions. We present steady and transient example simulations to illustrate the features and capabilities of the model and to examine sensitivity to mesh size and time step size. The model is implemented as part of the Ice Sheet System Model (ISSM).

An Aerodynamic Investigation of an Isolated Rotating Formula 1 Wheel Assembly

2012 ◽  
Vol 134 (12) ◽  
Author(s):  
John Axerio-Cilies ◽  
Gianluca Iaccarino
Keyword(s):  
Large Scale ◽  
Cross Flow ◽  
Reynolds Stress Model ◽  
Navier Stokes ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Turbulence Closures ◽  
Patch Boundary ◽  
The One

The flowfield around a 60% scale rotating Formula 1 tire in contact with the ground in a closed wind tunnel at a Reynolds number of 500,000 was examined computationally and experimentally. The goal of this study was to assess the accuracy of unsteady Reynolds-averaged Navier–Stokes (URANS) equations and confirm the existence of large scale vortical and flow recirculating features. A replica deformable F1 tire model that includes four tire treads and all brake components was used to determine the sensitivity of the wake to cross flow within the tire hub as well as the flow blockage caused by the brake assembly. Several turbulence closures were employed and the one that matched closest to the experimental PIV data was the Reynolds stress model. The variability between the six turbulence closures is shown by comparing velocity profiles, pressure distributions, and vortex eccentricity. The sensitivity of the wake to four different hub geometries, contact patch boundary conditions, multiple reference frame (MRF) rotor and spoke treatment, and time step size are also discussed.

Study on Numerical Oscillations of Forming Load Occurring in Forward Open Cold Extrusion

2020 ◽  
Vol 311 ◽  
pp. 48-55
Author(s):  
Akash Meena ◽  
Jae Dong Yoo ◽  
Kwang Min Kim ◽  
Wan Jin Chung ◽  
Man Soo Joun
Keyword(s):  
Degrees Of Freedom ◽  
Mesh Size ◽  
Extrusion Process ◽  
Oscillatory Behavior ◽  
Cold Extrusion ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Forming Load ◽  
Reduction Of Area

A forward open cold extrusion process is evaluated to reveal the sensitivity of forming load to numerical factors including number of total degrees of freedom, time step size and mesh size. The allowable reduction of area was examined by varying number of finite elements and the total solution steps to determine the reliability, in which the results are affected significantly by the positions of the lowest and highest nodal points in contact with the die. An investigation of the oscillations of the forming load in forward open extrusion revealed a set of critical numerical conditions that minimize unwanted oscillatory behavior. The limits of the material initial radii were obtained, suggesting that rigid plasticity with rigid dies is inappropriate for simulations of forward open cold extrusion with extreme reduction of area. Due to artificial numerical deformation of the rigid zone and die elastic deformation thus each of which has a non-negligible influence on the maximum reduction of area.

scholarly journals Analysis of a Decoupled Time-Stepping Scheme for Evolutionary Micropolar Fluid Flows

2016 ◽  
Vol 2016 ◽  
pp. 1-13 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Micropolar Fluid ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Optimal Order ◽  
Step Size ◽  
Time Step ◽  
Order Error ◽  
Time Step Size ◽  
Time Stepping

Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction.

On the Influence of Inflow Model Selection for Time-Domain Tiltrotor Aeroelastic Analysis

2021 ◽  
Author(s):  
Ethan Corle ◽  
Matthew Floros ◽  
Sven Schmitz
Keyword(s):  
Experimental Data ◽  
Particle Method ◽  
Comprehensive Analysis ◽  
Solution Stability ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Vortex Particle Method ◽  
Whirl Flutter ◽  
Vortex Particle

The methods of using the viscous vortex particle method, dynamic inflow, and uniform inflow to conduct whirl-flutter stability analysis are evaluated on a four-bladed, soft-inplane tiltrotor model using the Rotorcraft Comprehensive Analysis System. For the first time, coupled transient simulations between comprehensive analysis and a vortex particle method inflow model are used to predict whirl-flutter stability. Resolution studies are performed for both spatial and temporal resolution in the transient solution. Stability in transient analysis is noted to be influenced by both. As the particle resolution is refined, a reduction in simulation time-step size must also be performed. An azimuthal time step size of 0.3 deg is used to consider a range of particle resolutions to understand the influence on whirl-flutter stability predictions. Comparisons are made between uniform inflow, dynamic inflow, and the vortex particle method with respect to prediction capabilities when compared to wing beam-bending frequency and damping experimental data. Challenges in assessing the most accurate inflow model are noted due to uncertainty in experimental data; however, a consistent trend of increasing damping with additional levels of fidelity in the inflow model is observed. Excellent correlation is observed between the dynamic inflow predictions and the vortex particle method predictions in which the wing is not part of the inflow model, indicating that the dynamic inflow model is adequate for capturing damping due to the induced velocity on the rotor disk. Additional damping is noted in the full vortex particle method model, with the wing included, which is attributed to either an interactional aerodynamic effect between the rotor and the wing or a more accurate representation of the unsteady loading on the wing due to induced velocities.

A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

1997 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Stiff Systems ◽  
Step Size ◽  
Step Method ◽  
Integral Of Motion ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

Numerical Simulation of Proppant Transport in Hydraulically Fractured Reservoirs

2021 ◽  
Author(s):  
Seyhan Emre Gorucu ◽  
Vijay Shrivastava ◽  
Long X. Nghiem
Keyword(s):  
Hydraulic Fracturing ◽  
Hydraulic Fracture ◽  
Fractured Reservoirs ◽  
Injection Timing ◽  
Fracture Permeability ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Proppant Transport ◽  
Fracture Closure

Abstract An existing equation-of-state compositional simulator is extended to include proppant transport. The simulator determines the final location of the proppant after fracture closure, which allows the computation of the permeability along the hydraulic fracture. The simulation then continues until the end of the production. During hydraulic fracturing, proppant is injected in the reservoir along with water and additives like polymers. Hydraulic fracture gets created due to change in stress caused by the high injection pressure. Once the fracture opens, the bulk slurry moves along the hydraulic fracture. Proppant moves at a different speed than the bulk slurry and sinks down by gravity. While the proppant flows along the fracture, some of the slurry leaks off into the matrix. As the fracture closes after injection stops, the proppant becomes immobile. The immobilized proppant prevents the fracture from closing and thus keeps the permeability of the fracture high. All the above phenomena are modelled effectively in this new implementation. Coupled geomechanics simulation is used to model opening and closure of the fracture following geomechanics criteria. Proppant retardation, gravitational settling and fluid leak-off are modeled with the appropriate equations. The propped fracture permeability is a function of the concentration of immobilized proppant. The developed proppant simulation feature is computationally stable and efficient. The time step size during the settling adapts to the settling velocity of the proppants. It is found that the final location of the proppants is highly dependent on its volumetric concentration and slurry viscosity due to retardation and settling effects. As the location and the concentration of the proppants determine the final fracture permeability, the additional feature is expected to correctly identify the stimulated region. In this paper, the theory and the model formulation are presented along with a few key examples. The simulation can be used to design and optimize the amount of proppant and additives, injection timing, pressure, and well parameters required for successful hydraulic fracturing.

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