Verification of ANSYS and Matlab Conduction Results Using Analytical Solutions

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.

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

Journal of Verification Validation and Uncertainty Quantification ◽

10.1115/1.4050610 ◽

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

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NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

The International Journal of Maritime Engineering ◽

10.5750/ijme.v153ia2.851 ◽

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.

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Analytical-Solution Based Corner Correction for Transient Thermal Measurement

Journal of Heat Transfer ◽

10.1115/1.4030980 ◽

2015 ◽

Vol 137 (11) ◽

Cited By ~ 1

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.

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Water wave transients in an ice-covered channel

Canadian Journal of Civil Engineering ◽

10.1139/l11-010 ◽

2011 ◽

Vol 38 (4) ◽

pp. 404-414 ◽

Cited By ~ 4

Author(s):

François Nzokou ◽

Brian Morse ◽

Jean-Loup Robert ◽

Martin Richard ◽

Edmond Tossou

Keyword(s):

Analytical Solutions ◽

Water Waves ◽

Numerical Solutions ◽

Wave Attenuation ◽

Ice Sheet ◽

Ice Cover ◽

Beam Equation ◽

Transverse Cracks ◽

Time Step ◽

The Impact

Many studies show that the propagation of a breakup water surge in impeded rivers (ice cover present) differs from the unimpeded case. Some of the differences are due to ice sheet breaking into pieces as the wave travels downstream while others are due to the effect of a fissured but otherwise intact ice cover’s resistance to motion. This is the subject of this paper: water waves that are sufficiently strong to break the cover away from the banks but not strong enough to create transverse cracks. Although some analytical solutions exist for the propagation of these transients for simple cases, for the first time in the literature, this paper introduces numerical solutions using a FEM model (HYDROBEAM) that simulates this interaction using the one-dimensional Saint-Venant equations appropriately written for rivers having an intact fissured floating ice cover coupled with a classic beam equation subject to hydrostatic loads (often referred to as a beam on an elastic foundation). The governing equations are numerically expressed and are solved using a finite element method (FEM) for the hydrodynamic and ice beam equations separately. A coupling technique is used to converge to a unique solution at each time step (for more information on the numerical characteristics of the model, see companion paper presented by the authors in this issue). The coupled model, gives a first and unique opportunity to compare the simplified analytical solutions to the full numerical solutions. A parametric analysis is herein presented that quantifies the impact of the ice cover's presence and stiffness on wave attenuation and wave celerity as well as to quantify tensile stresses generated in the ice sheet as a function of ice properties (thickness and strength) and channel shape (rectangular and trapezoidal). In general, for rectangular channels, it was found that the simplified analytical solutions are quite representative of the phenomenon namely that short wave transients are affected by the cover’s stiffness but long waves (>400 m) are not.

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Numerical Solution Strategies in Permeation Processes

Defect and Diffusion Forum ◽

10.4028/www.scientific.net/ddf.413.29 ◽

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].

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Progress in Analytical Modeling of Water Hammer

10.1115/fedsm2021-65920 ◽

2021 ◽

Author(s):

Kamil Urbanowicz ◽

Haixiao Jing ◽

Anton Bergant ◽

Michał Stosiak ◽

Marek Lubecki

Keyword(s):

Analytical Solutions ◽

Numerical Solutions ◽

Dynamic Pressure ◽

Complete Solution ◽

Water Hammer ◽

Inverse Transformation ◽

Mathematical Form ◽

Dimensionless Time ◽

Wide Range ◽

Analytical Formulas

Abstract In this paper analytical formulas of water hammer known from the literature are simplified to the shortest possible mathematical form based on dimensionless parameters: dimensionless time, water hammer number, etc. Novel formulas are determined, for example for the flow velocity and wall shear stress in the Muto and Takahashi solution. A complete solution in the Laplace domain is presented and the problem of its inverse transformation is discussed. A series of comparative studies of analytical solutions with numerical solutions and the results of experimental research were carried out. The compared analytical solutions, taking into account the frequency-dependent nature of the hydraulic resistances, show very good agreement with the experimental results in a wide range of water hammer numbers, in particular when Wh ≤ 0.1. On the other hand, it turned out that the analytical model based on the quasi-steady friction in great detail simulates dynamic pressure response in systems characterized by a high value of the water hammer number Wh ≥ 0.5.

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Analytical Solutions of Saint Venant Equations Decomposed in Frequency Domain

Journal of Mechanics ◽

10.1017/s1727719100003403 ◽

2004 ◽

Vol 20 (3) ◽

pp. 187-197

Author(s):

W. H. Chung ◽

Y. L. Kang

Keyword(s):

Analytical Solution ◽

Finite Difference Method ◽

Frequency Domain ◽

Analytical Solutions ◽

Numerical Solutions ◽

Single Equation ◽

Response Functions ◽

Generalized Model ◽

Saint Venant Equations ◽

Flow Domain

AbstractThe Saint Venant equations are often merged into a single equation for being easily solvable. By doing so, the most general form of the single equation is formulated in this study if all terms are preserved. As a result, the generalized model (GM) results and contains several unexpected nonlinear terms that may impose a great limitation on model analyses. In order to identify these redundant terms, this paper discusses the employment of the linearized Saint Venant equations (LSVE) governing subcritical flow in prismatic channels. The LSVE is solved by a new procedure that separates, in the Laplace frequency domain, the governing equation of water depth from that of flow velocity and thus enables us to consider two independent equations rather than two coupled ones. This allows us to obtain analytical solutions in a much easier way. Comparisons of the response functions of LSVE and the linearized generalized model (LGM) show that the two equations provide identical solutions if the redundant terms embedded in LGM are neglected. It then follows that the response function of LGM can be utilized as a replacement for solving the analytical solution of LSVE that is valid for prismatic channels of any shape. Validity of the analytical solution is verified by repeatedly comparing with the corresponding numerical solutions of finite difference method or Crump's algorithm [1], depending on whether the flow domain is finite or semi-infinite. It is clearly demonstrated in this paper that LSVE serves as an excellent substitution for LGM whose variants have been employed for quite a few years.

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Corrigendum

Journal of Plasma Physics ◽

10.1017/s0022377800003020 ◽

1985 ◽

Vol 34 (3) ◽

pp. 481-483

Author(s):

T. G. Forbes ◽

E. R. Priest ◽

A. W. Hood

Keyword(s):

Analytical Solution ◽

Numerical Solution ◽

Current Sheet ◽

Analytical Solutions ◽

Numerical Solutions ◽

Linear Solution ◽

Significant Change ◽

Parameter Values ◽

A Current

Numerical solutions were obtained by Forbes, Priest & Hood (1982) for the resistive decay of a current sheet in an MHD fluid. To check the accuracy of the numerical solutions, a linear, analytical solution was also deived for the regime where diffusion is dominant. In a subsequent reinvestigation of this problem an error in the linear, analytical solution has been discovered. For the parameter values used in the numerical solution this error is too small (≲ 2%) to produce any significant change in the previous test comparison between the numerical and analytical solutions. However, for parameter values much different from those used in the numerical solution, the error in the linear solution can be significant.

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THEORETICAL ANALYTICAL METHOD IN PROBLEM SOLUTION OF LIQUID VACUUM FREEZING IN QUIET STAT

Bulletin of Bryansk state technical university ◽

10.12737/22051 ◽

2016 ◽

Vol 2016 (3) ◽

pp. 123-128

Author(s):

Игорь Лобанов ◽

Igor Lobanov

Keyword(s):

Analytical Solution ◽

Analytical Solutions ◽

Analytical Method ◽

Numerical Solutions ◽

Stationary Process ◽

Problem Solution ◽

Quiet State ◽

Stationary Vacuum ◽

Vacuum Freezing

A generalized closed analytical solution of the problem of a quasi-stationary process in liquid vacuum freezing in a quiet state with regard to the thickness of the frosting layer ξ whereas heretofore numerical solutions of this problem occurred. The advantage of the analytical solutions obtained of the problem of a quasi-stationary vacuum freezing of moisture in a finedispersion state over existing numerical ones consists in the identification of an immanent tie between defining and determined parameters regarding a thickness of the frosting layer ξ. It is also possible to use them directly at the computation without resorting to the help of computers.

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One-dimensional Localization Solutions for Time-dependent Damage

International Journal of Damage Mechanics ◽

10.1177/1056789510395553 ◽

2011 ◽

Vol 20 (8) ◽

pp. 1178-1197

Author(s):

D. Caillerie ◽

C. Dascalu

Keyword(s):

Lower Limit ◽

Analytical Solutions ◽

Numerical Solutions ◽

Damage Model ◽

Creep Damage ◽

The Body ◽

Time Dependent ◽

One Dimensional ◽

Quasistatic Approximation ◽

Uniqueness Of The Solution

This article presents analytical solutions for a class of one-dimensional time-dependent elasto-damage problems. The considered damage evolution law may be seen as a one-dimensional version of the Kachanov–Rabotnov creep damage model with classical loading–unloading conditions. We construct analytical solutions for the quasistatic one-dimensional problem. The evolution consists of a first regime, in which damage and strain grow uniformly, followed by a regime in which localization occurs. In the second regime, the uniqueness of the solution is lost and the deformation of the body is represented by a sequence of arbitrary alternate loading/unloading regions. Complex evolutions with progressive enlargement of the unloading regions in a finite number of steps are also constructed. We study analytically and numerically the features of the obtained bifurcated solutions. It is shown that, at every instant of time, a lower limit exists for the size of the localization zone. This lower limit is actually realized by the solution with successive unloadings constructed in this article. These features help us to understand the behavior of numerical solutions for time-dependent damage in the quasistatic approximation.

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