Verification of ANSYS and Matlab Heat Conduction Results Using an 'Intrinsically' Verified Exact Analytical Solution

2021 ◽  
Author(s):  
Robert McMasters ◽  
Filippo de Monte ◽  
Giampaolo D'Alessandro ◽  
James Beck
Keyword(s):  
Analytical Solution ◽  
Numerical Solutions ◽  
Exact Analytical Solution ◽  
The Body ◽  
Transient Temperature ◽  
Dimensionless Time ◽  
Fe Method ◽  
Time Step ◽  
First Case ◽  
Constant Heating

Abstract A two-dimensional transient thermal conduction problem is examined and numerical solutions to the problem generated by ANSYS and Matlab, employing the finite element (FE) method, are compared against an 'intrinsically' verified analytical solution. Various grid densities and time-step combinations are used in the numerical solutions, including some as recommended by default in the ANSYS software, including coarse, medium and fine spatial grids. The transient temperature solutions from the analytical and numerical schemes are compared at four specific locations on the body and time-dependent error curves are generated for each point. Additionally, tabular values of each solution are presented for a more detailed comparison. Two different test cases are examined for the various numerical solutions using selected grid densities. The first case involves uniform constant heating on a portion of one surface for a long duration, up to a dimensionless time of 30. The second test case still involves uniform constant heating but for a dimensionless time of one, immediately followed by an insulated condition on that same surface for another duration of one dimensionless time unit. Although the errors at large times for both ANSYS and Matlab are extremely small, the errors found within the short-duration test are more significant, in particular when the heating is suddenly set 'on'. Surprisingly, very small errors occur when the heating is suddenly set 'off'.

Verification of ANSYS and Matlab Conduction Results Using Analytical Solutions

2020 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
Giampaolo D'Alessandro ◽  
James V. Beck
Keyword(s):  
Analytical Solution ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Early Time ◽  
The Body ◽  
Numerical Code ◽  
Transient Temperature ◽  
Dimensionless Time ◽  
Time Step ◽  
First Case

Abstract A two-dimensional transient thermal conduction problem is examined and numerical solutions to the problem generated by ANSYS and Matlab, employing the finite element method, are compared against an analytical solution. Various different grid densities and time-step combinations are used in the numerical solutions, including some as recommended by default in the ANSYS software, including coarse, medium and fine spatial grids. The transient temperature solutions from the analytical and numerical schemes are compared at four specific locations on the body and time-dependent error curves are generated for each point. Additionally, tabular values of each solution are presented for a more detailed comparison. The errors found in the numerical solutions by comparing them directly with the analytical solution vary depending primarily on the time step size used. The errors are much larger if calculated using the analytical solution at a given time as a basis of the comparison between the two solutions as opposed to using the steady-state temperature as a basis. The largest errors appear in the early time steps of the problem, which is typically the regime wherein the largest errors occur in mathematical solutions to transient conduction problems. Conversely, errors at larger values of dimensionless time are extremely small and the numerical solutions agree within one tenth of one percent of the analytical solutions at even the worst locations. In addition to difficulties during the early time values of the problem, temperatures calculated on convective boundaries or prescribed-heat-flux boundaries are locations generating larger-magnitude errors. Corners are particularly difficult locations and require finer gridding and finer time steps in order to generate a very precise solution from a numerical code. These regions are compared, using several grid densities, against the analytical solutions. The analytical solutions are, in turn, intrinsically verified to eight significant digits by comparing similar analytical solutions against one another at very small values of dimensionless time. The solution developed using the Matlab differential equation solver was found to have errors of a similar magnitude to those generated using ANSYS. Two different test cases are examined for the various numerical solutions using the selected grid densities. The first case involves steady heating on a portion of one surface for a long duration, up to a dimensionless time of 30. The second test case involves constant heating for a dimensionless time of one, immediately followed by an insulated condition on that same surface for another duration of one dimensionless time unit. Although the errors at large times were extremely small, the errors found within the short duration test were more significant.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Analytical-Solution Based Corner Correction for Transient Thermal Measurement

2015 ◽  
Vol 137 (11) ◽  
Author(s):  
H. Jiang ◽  
W. Chen ◽  
Q. Zhang ◽  
L. He
Keyword(s):  
Heat Transfer ◽  
Analytical Solution ◽  
Data Processing ◽  
Numerical Solutions ◽  
Thermal Measurement ◽  
Time Step ◽  
Experimental Data Processing ◽  
Numerical Solution Methods ◽  
Transient Thermal ◽  
Analytical Approaches

The one-dimensional (1D) conduction analytical approaches for a semi-infinite domain, widely adopted in the data processing of transient thermal experiments, can lead to large errors, especially near a corner of solid domain. The problems could be addressed by adopting 2D/3D numerical solutions (finite element analysis (FEA) or computational fluid dynamics (CFD)) of the solid field. In addition to needing the access to a conduction solver and extra computing effort, the numerical field solution based processing methods often require extra experimental efforts to obtain full thermal boundary conditions around corners. On a more fundamental note, it would be highly preferable that the experimental data processing is completely free of any numerical solutions and associated discretization errors, not least because it is often the case that the main purposes of many experimental measurements are exactly to validate the numerical solution methods themselves. In the present work, an analytical-solution based method is developed to enable the correction of the 2D conduction errors in a corner region without using any conduction solvers. The new approach is based on the recognition that a temperature time trace in a 2D corner situation is the result of the accumulated heat conductions in both the normal and lateral directions. An equivalent semi-infinite 1D conduction temperature trace for a correct heat transfer coefficient (HTC) can then be generated by reconstructing and removing the lateral conduction component at each time step. It is demonstrated that this simple correction technique enables the use of the standard 1D conduction analysis to get the correct HTC completely analytically without any aid of CFD or FEA solutions. In addition to a transient infrared (IR) thermal measurement case, two numerical test cases of practical interest with turbine blade tip heat transfer and film cooling are used for validation and demonstration. It has been consistently shown that the errors of the conventional 1D conduction analysis in the near corner regions can be greatly reduced by the new corner correction method.

scholarly journals Numerical Solution Strategies in Permeation Processes

2021 ◽  
Vol 413 ◽  
pp. 29-46
Author(s):  
Axel von der Weth ◽  
Daniela Piccioni Koch ◽  
Frederik Arbeiter ◽  
Till Glage ◽  
Dmitry Klimenko ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Ad Hoc ◽  
Numerical Solutions ◽  
Matrix Multiplication ◽  
Transport Processes ◽  
Time Step ◽  
Stability Range ◽  
The Stability ◽  
The One ◽  
Crank Nicolson

In this work, the strategy for numerical solutions in transport processes is investigated. Permeation problems can be solved analytically or numerically by means of the Finite Difference Method (FDM), while choosing the Euler forward explicit or Euler backwards implicit formalism. The first method is the easiest and most commonly used, while the Euler backwards implicit is not yet well established and needs further development. Hereafter, a possible solution of the Crank-Nicolson algorithm is presented, which makes use of matrix multiplication and inversion, instead of the step-by-step FDM formalism. If one considers the one-dimensional diffusion case, the concentration of the elements can be expressed as a time dependent vector, which also contains the boundary conditions. The numerically stable matrix inversion is performed by the Branch and Bound (B&B) algorithm [2]. Furthermore, the paper will investigate, whether a larger time step can be used for speeding up the simulations. The stability range is investigated by eigenvalue estimation of the Euler forward and Euler backward. In addition, a third solver is considered, referred to as Combined Solver, that is made up of the last two ones. Finally, the Crank-Nicolson solver [9] is investigated. All these results are compared with the analytical solution. The solver stability is analyzed by means of the Steady State Eigenvector (SSEV), a mathematical entity which was developed ad hoc in the present work. In addition, the obtained results will be compared with the analytical solution by Daynes [6,7].

scholarly journals Shape Recovery of Deformed Biomolecular Droplets: Dependence on Condensate Viscoelasticity

2021 ◽  
Author(s):  
Huan-Xiang Zhou
Keyword(s):  
Analytical Solution ◽  
Interfacial Tension ◽  
Shape Recovery ◽  
Numerical Solutions ◽  
Exact Analytical Solution ◽  
Viscous Fluids ◽  
Spherical Droplet ◽  
Time Constants ◽  
Shape Dynamics ◽  
Shear Relaxation

ABSTRACTPhase-separated biomolecular condensates often appear as micron-sized droplets. Due to interfacial tension, the droplets usually have a spherical shape and, upon deformation, tend to recover their original shape. Likewise, interfacial tension drives the fusion of two droplets into a single spherical droplet. In all previous studies on shape dynamics, biomolecular condensates have been modeled as purely viscous. However, recent work has shown that biomolecular condensates are viscoelastic, with shear relaxation occurring not instantaneously as would in purely viscous fluids. Here we present an exact analytical solution for the shape recovery dynamics of biomolecular droplets, which exhibits rich time dependence due to viscoelasticity. For condensates modeled as purely viscous, shape recovery is an exponential function of time, with the time constant given by the “viscocapillary” ratio, i.e., viscosity over interfacial tension. For viscoelastic droplets, shape recovery becomes multi-exponential, with shear relaxation yielding additional time constants. The longest of these time constants can be dictated by shear relaxation and independent of interfacial tension, thereby challenging the currently prevailing viscocapillarity-centric view derived from purely viscous fluids. These results highlight the importance of viscoelasticity in condensate shape dynamics and expand our understanding of how material properties affect condensate dynamics in general, including aging. The analytical solution presented here can also be used for validating numerical solutions of fluid-dynamics problems and for fitting experimental and molecular simulation data.

EXACT ANALYTICAL SOLUTION OF BIOHEAT EQUATION SUBJECTED TO INTENSIVE MOVING HEAT SOURCE

2017 ◽  
Vol 17 (05) ◽  
pp. 1750081 ◽  
Author(s):  
MOHAMMAD REZA TALAEE ◽  
ALI KABIRI
Keyword(s):  
Analytical Solution ◽  
Heat Source ◽  
Numerical Solutions ◽  
Exact Analytical Solution ◽  
Blood Perfusion ◽  
Temperature Profiles ◽  
Moving Heat Source ◽  
Bioheat Equation ◽  
Line Heat Source ◽  
Moving Line

Presented is the analytical solution of Pennes bio-heat equation, under localized moving heat source. The thermal behavior of one-dimensional (1D) nonhomogeneous layer of biological tissue is considered with blood perfusion term and modeled under the effect of concentric moving line heat source. The procedure of the solution is Eigen function expansion. The temperature profiles are calculated for three tissues of liver, kidney, and skin. Behavior of temperature profiles are studied parametrically due to the different moving speeds. The analytical solution can be used as a verification branch for studying the practical operations such as scanning laser treatment and other numerical solutions.

scholarly journals Analytical treatment of transient temperature and thermal stress distribution in CW end pumped laser rod: Thermal response optimization study

2014 ◽  
Vol 18 (2) ◽  
pp. 399-408 ◽  
Author(s):  
Khalid Shibib ◽  
Mayada Tahir ◽  
Mohammad Mahdi
Keyword(s):  
Temperature Distribution ◽  
Analytical Solution ◽  
Numerical Solutions ◽  
Integral Transform ◽  
Beam Quality ◽  
Thermal Response ◽  
Optical Path Difference ◽  
Transient Temperature ◽  
Analytical Treatment ◽  
Transform Method

The analytical solution of transient temperature distribution and Tresca failure stress in CW end- pumped laser rod has been derived using integral transform method. The analytical result is compared with numerical solutions presented by other works and good agreement has been found. Analytical solution with its clear physical meaning and its explicit form permits to predict the influence of various factors on the solution. The optical path difference which gives a valuable means to quantify the optical properties of laser material such as designed beam quality, will converge to a constant value as steady state temperature distribution is reached. One can obtain the dominate factors which affect the laser response to bring the laser rod to the thermal equilibrium; it has been found that fast response can be achieved by reducing pumping power, increasing extracted heat from the rod , choosing a crystal having high thermal diffusivity and decreasing laser rod radius while its volume remains constant. One final advantage of the analytical solution is that a fast result can be obtained where the numerical solution usually is a time consuming technique.

scholarly journals The exact analytical solution of the dual-phase-lag two-temperature bioheat transfer of a skin tissue subjected to constant heat flux

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Hamdy M. Youssef ◽  
Najat A. Alghamdi
Keyword(s):  
Heat Flux ◽  
Analytical Solution ◽  
Exact Analytical Solution ◽  
Constant Heat Flux ◽  
Bioheat Transfer ◽  
Skin Tissue ◽  
Phase Lag ◽  
Dual Phase ◽  
Constant Heat ◽  
Two Temperature

Abstract This work is dealing with the temperature reaction and response of skin tissue due to constant surface heat flux. The exact analytical solution has been obtained for the two-temperature dual-phase-lag (TTDPL) of bioheat transfer. We assumed that the skin tissue is subjected to a constant heat flux on the bounding plane of the skin surface. The separation of variables for the governing equations as a finite domain is employed. The transition temperature responses have been obtained and discussed. The results represent that the dual-phase-lag time parameter, heat flux value, and two-temperature parameter have significant effects on the dynamical and conductive temperature increment of the skin tissue. The Two-temperature dual-phase-lag (TTDPL) bioheat transfer model is a successful model to describe the behavior of the thermal wave through the skin tissue.

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