Some Analytical and Numerical Solutions to Inverse Problems Applied to Optimizing Phase-Transformation Tracking in Gas Quenching

2003 ◽  
Vol 125 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Michael Vynnycky ◽  
Je´ro^me Ferrari ◽  
Noam Lior
Keyword(s):  
Phase Transformation ◽  
Numerical Solutions ◽  
Asymptotic Methods ◽  
Convective Heat Transfer Coefficient ◽  
Asymptotic Series ◽  
Cooling Curve ◽  
Nonlinear Heat Equation ◽  
Attractive Alternative ◽  
Fourier Number ◽  
Gas Quenching

A transient inverse heat conduction problem focused on gas quenching of steel plates and rings is posed and solved, both analytically and numerically. The quenching objective is to calculate the transient convective heat transfer coefficient which would produce an optimized phase transformation cooling curve. The governing nonlinear heat equation is nondimensionalised, and a small parameter, the reciprocal of the Fourier number, is identified. This allows the construction of an analytic solution in the form of an asymptotic series. For higher values of the reciprocal Fourier number, a numerical scheme incorporating the function specification and Keller Box methods is used to generate solutions. Comparison of the results proves favorable, and suggests that for this inverse problem asymptotic methods provide an attractive alternative to solely numerical ones.

scholarly journals Data Driven Solutions and Discoveries in Mechanics Using Physics Informed Neural Network

2020 ◽  
Author(s):  
Qi Zhang ◽  
Yilin Chen ◽  
Ziyi Yang
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Automatic Differentiation ◽  
Numerical Solutions ◽  
Data Driven ◽  
Attractive Alternative ◽  
The Finite Element Method ◽  
The Neural Network ◽  
Engineering Problems ◽  
Free Parameters

Deep learning has achieved remarkable success in diverse computer science applications, however, its use in other traditional engineering fields has emerged only recently. In this project, we solved several mechanics problems governed by differential equations, using physics informed neural networks (PINN). The PINN embeds the differential equations into the loss of the neural network using automatic differentiation. We present our developments in the context of solving two main classes of problems: data-driven solutions and data-driven discoveries, and we compare the results with either analytical solutions or numerical solutions using the finite element method. The remarkable achievements of the PINN model shown in this report suggest the bright prospect of the physics-informed surrogate models that are fully differentiable with respect to all input coordinates and free parameters. More broadly, this study shows that PINN provides an attractive alternative to solve traditional engineering problems.

Rough Contacts Modeled by Nonlinear Coatings

2007 ◽  
Author(s):  
I. I. Kudish
Keyword(s):  
Contact Problem ◽  
Numerical Solutions ◽  
Asymptotic Methods ◽  
Experimental Studies ◽  
Nonlinear Function ◽  
Normal Displacement ◽  
Experimental Fact ◽  
Axially Symmetric ◽  
Elastic Solids ◽  
Local Pressure

A number of experimental studies [1–3] revealed that the normal displacement in a contact of rough surfaces due to asperities presence is a nonlinear function of local pressure and it can be approximated by a power function of pressure. Originally, a linear mathematical model accounting for surface roughness of elastic solids in contact was introduced by I. Shtaerman [4]. He assumed that the effect of asperities present in a contact of elastic solids can be essentially replaced by the presence of a thin coating simulated by an additional normal displacement of solids’ surfaces proportional to a local pressure. Later, a similar but nonlinear problem formulation that accounted for the above mentioned experimental fact was proposed by L. Galin. In a series of papers this problem was studied by numerical and asymptotic methods [5–9]. The present paper has a dual purpose: to analyze the problem analytically and to provide some asymptotic and numerical solutions. The results presented below provide an overview of the results obtained on the topic and published by the author earlier in the journals hardly accessible to the international tribological community (such as Russian and mathematical journals) and, therefore, mostly unknown by tribologists. A number of recent publications on contacts of rough elastic solids supports the view that these results are still of value to the specialists involved in nanotribology. The existence and uniqueness of a solution of a contact problem for elastic bodies with rough (coated) surfaces is established based on the variational inequalities approach. Four different equivalent formulations of the problem including three variational ones were considered. A comparative analysis of solutions of the contact problem for different values of initial parameters (such as the indenter shape, parameters characterizing roughness, elastic parameters of the substrate material) is done with the help of calculus of variations and the Zaremba-Giraud principle of maximum for harmonic functions [10,11]. The results include the relations between the pressure and displacement distributions for rough and smooth solids as well as the relationships for solutions of the problems for rough solids with fixed and free contact boundaries. For plane and axially symmetric cases some asymptotic and numerical solutions are presented.

On the weakly nonlinear development of Tollmien-Schlichting wavetrains in boundary layers

1996 ◽  
Vol 323 ◽  
pp. 133-171 ◽  
Author(s):  
Xuesong Wu ◽  
Philip A. Stewart ◽  
Stephen J. Cowley
Keyword(s):  
Numerical Solutions ◽  
Asymptotic Methods ◽  
Nonlinear Effects ◽  
Mean Flow ◽  
Flow Distortion ◽  
Weakly Nonlinear ◽  
Nonlinear Development ◽  
Finite Distance ◽  
Tollmien Schlichting ◽  
High Reynolds

The nonlinear development of a weakly modulated Tollmien-Schlichting wavetrain in a boundary layer is studied theoretically using high-Reynolds-number asymptotic methods. The ‘carrier’ wave is taken to be two-dimensional, and the envelope is assumed to be a slowly varying function of time and of the streamwise and spanwise variables. Attention is focused on the scalings appropriate to the so-called ‘upper branch’ and ‘high-frequency lower branch’. The dominant nonlinear effects are found to arise in the critical layer and the surrounding ‘diffusion layer’: nonlinear interactions in these regions can influence the development of the wavetrain by producing a spanwise-dependent mean-flow distortion. The amplitude evolution is governed by an integro-partial-differential equation, whose nonlinear term is history-dependent and involves the highest derivative with respect to the spanwise variable. Numerical solutions show that a localized singularity can develop at a finite distance downstream. This singularity seems consistent with the experimentally observed focusing of vorticity at certain spanwise locations, although quantitative comparisons have not been attempted.

Similarity solutions and the computation formulas of a nonlinear fractional-order generalized heat equation

2019 ◽  
Vol 33 (10) ◽  
pp. 1950122 ◽  
Author(s):  
Yufeng Zhang
Keyword(s):  
Heat Equation ◽  
Fractional Derivative ◽  
Numerical Solutions ◽  
Similarity Solutions ◽  
Nonlinear Heat Equation ◽  
Order Ordinary Differential Equation ◽  
Nonlinear Heat Conduction Equation ◽  
Liouville Fractional Derivative ◽  
Generalized Heat Equation ◽  
Two Parameters

A generalized nonlinear heat equation with the fractional derivative is proposed, whose similarity solutions are derived from a type of special scalar transformation with two parameters. With the help of separated variable method, two special series solutions of the standard heat equation are obtained. Finally, through computation of the left Riemann–Liouville fractional derivative, we obtain two approximated computation formulas of the factional-order ordinary differential equation which could be used to calculate the numerical solutions of the generalized nonlinear heat conduction equation.

Heat Transfer Between Colliding Surfaces and Particles

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Like Li ◽  
Renwei Mei ◽  
James F. Klausner ◽  
David W. Hahn
Keyword(s):  
Heat Transfer ◽  
Contact Area ◽  
Thermal Field ◽  
Initial Period ◽  
Asymptotic Methods ◽  
Hertzian Contact ◽  
Thermal Conductivity Ratio ◽  
Contact Radius ◽  
Two Dimensional ◽  
Fourier Number

Collisional heat transfer between two contacting curved surfaces is investigated computationally using a finite difference method and analytically using various asymptotic methods. Transformed coordinates that scale with the contact radius and the diffusion length are used for the computations. Hertzian contact theory of elasticity is used to characterize the contact area as a function of time. For an axisymmetric contact area, a two-dimensional self-similar solution for the thermal field during the initial period of contact is obtained, and it serves as an initial condition for the heat transfer simulation throughout the entire duration of collision. A two-dimensional asymptotic heat transfer result is obtained for small Fourier number. For finite Fourier numbers, local analytical solutions are presented to elucidate the nature of the singularity of the thermal field and heat flux near the contact point. From the computationally determined heat transfer during the collision, a closed-form formula is developed to predict the heat transfer as a function of the Fourier number, the thermal diffusivity ratio, and the thermal conductivity ratio of the impacting particles.

Freight Car Lateral Dynamics—An Asymptotic Sketch

1984 ◽  
Vol 106 (1) ◽  
pp. 107-113 ◽  
Author(s):  
Alan M. Whitman ◽  
Abdul-Rahman M. Khaskia
Keyword(s):  
Critical Speed ◽  
Numerical Solutions ◽  
Asymptotic Methods ◽  
The Other ◽  
Lateral Dynamics ◽  
Wide Range ◽  
Freight Car

We analyze a model of a freight car using asymptotic methods and obtain a formula for the critical speed that is valid over a wide range of warp stiffnesses. We also discuss corrections to this formula that are engendered by the other parameters that influence the motion. The results of the analysis are compared with numerical solutions of the model in order to assess their accuracy.

scholarly journals Long wavelength vortices in time-periodic flows

1998 ◽  
Vol 39 (4) ◽  
pp. 498-512
Author(s):  
Andrew P. Bassom ◽  
P. J. Blennerhassett
Keyword(s):  
Viscous Fluid ◽  
Numerical Solutions ◽  
Asymptotic Methods ◽  
Critical Value ◽  
Infinite Layer ◽  
Periodic Flows ◽  
Long Wavelength ◽  
Common Features ◽  
Neutral Curves ◽  
Time Periodic

AbstractThe linear stability properties are examined of long wavelength vortex modes in two time-periodic flows. These flows are the motion which is induced by a torsionally oscillating cylinder within a viscous fluid and, second, the flow which results from the sinusoidal heating of an infinite layer of fluid. Previous studies concerning these particular configurations have shown that they are susceptible to vortex motions and linear neutral curves have been computed for wavenumbers near their critical value. These computations become increasingly difficult for long wavelength motions and here we consider such modes using asymptotic methods. These yield simple results which are formally valid for small wavenumbers and we show that the agreement between these asymptotes and numerical solutions is good for surprisingly large wavenumbers. The two problems studied share a number of common features but also have important differences and, between them, our methods and results provide a basis which can be extended for use with other time-periodic flows.

Experimental Study on Nitrogen-Spray Water Quenching for 9SiCr

2012 ◽  
Vol 430-432 ◽  
pp. 365-368
Author(s):  
Bao Dong Shao ◽  
Li Feng Wang ◽  
He Ming Cheng ◽  
Rui Jie Wang ◽  
Jian Yun Li ◽  
...  
Keyword(s):  
Residual Stress ◽  
Experimental Study ◽  
Phase Transformation ◽  
Alloy Steel ◽  
Martensite Phase ◽  
The Other ◽  
Water Quenching ◽  
Cooling Curve ◽  
Quenching Medium ◽  
The One

Alloy steel 9SiCr is quenched by a certain proportion of mixture gas of nitrogen and spray water at atmosphere, and the continual cooling curve during quenching is determined, the rigidity of quenched workpiece is measured and the metallographic map is captured. The results show that it can improve the cooling effect of quenching medium on the one hand; on the other hand, it can decrease the deformation and residual stress of workpieces. The results show that the rigidity of quenched workpiece is about 62 (HRC), which is corresponding with the rigidity after oil quenching. Martensite phase transformation occurs at the end of quenching that can be seen from the metallographic map.

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