Closed-form and numerical solutions to the laser heating process

1998 ◽  
Vol 212 (2) ◽  
pp. 141-151 ◽  
Author(s):  
B S Yilbas ◽  
M Sami ◽  
A Al-Farayedhi
Keyword(s):  
Closed Form ◽  
Laser Heating ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Heat Of Fusion ◽  
Heating Process ◽  
Temperature Profiles ◽  
Depth Analysis ◽  
Analytical Approaches

The laser processing of engineering materials requires an in-depth analysis of the applicable heating mechanism. The modelling of the laser heating process offers improved understanding of the machining mechanism. In the present study, a closed-form solution for a step input laser heating pulse is obtained and a numerical scheme solving a three-dimensional heat transfer equation is introduced. The numerical solution provides a comparison of temperature profiles with those obtained from the analytical approach. To validate the analytical and numerical solutions, an experiment is conducted to measure the surface temperature and evaporating front velocity during the Nd—YAG laser heating process. It is found that the temperature profiles resulting from both theory and experiment are in a good agreement. However, a small discrepancy in temperatures at the upper end of the profiles occurs. This may be due to the assumptions made in both the numerical and the analytical approaches. In addition, the equilibrium time, based on the energy balance among the internal energy gain, conduction losses and latent heat of fusion, is introduced.

Exact solution for time exponentially varying pulsed laser heating: Convective boundary condition case

2001 ◽  
Vol 215 (5) ◽  
pp. 591-606 ◽  
Author(s):  
M Kalyon ◽  
B S Yilbas
Keyword(s):  
Boundary Condition ◽  
Exact Solution ◽  
Closed Form ◽  
Temperature Rise ◽  
Laser Heating ◽  
Closed Form Solution ◽  
Convective Boundary Condition ◽  
Form Solution ◽  
Heating Process ◽  
Convective Boundary

Laser heating offers considerable advantages over conventional methods. The closed-form solution for the temperature rise in the substrate during the laser heating process gives insight into the physical phenomena involving during the heating process and the material response to a laser heating pulse. In the present study, the exact solution for the temperature rise due to a time exponentially varying pulse and convective boundary condition at the surface is obtained. The closed-form solution to the solutions available in the literature for a step input intensity pulse with a convective boundary condition at the surface as well as a time exponentially varying pulse with a non-convective boundary condition at the surface is deduced. A Laplace transformation method is used in the analysis. In order to account for a pulse resembling a typical laser pulse, an intensity function resulting in exponentially increasing and decaying intensity distribution is employed in the source term in the governing transport equation. The effects of the pulse parameters β′, β′/γ′ and Biot number Bi on the resulting temperature profiles are presented and the material response to a pulse profile resembling a typical actual laser pulse is discussed. It is found that the closed-form solution obtained in the present study becomes identical with those presented in the previous studies for different pulse and boundary conditions. Moreover, the coupling effect of pulse parameter β and Bi is significant for the temperature rise at the surface.

scholarly journals Analysis of hyperbolic Pennes bioheat equation in perfused homogeneous biological tissue subject to the instantaneous moving heat source

2021 ◽  
Vol 3 (4) ◽  
Author(s):  
Ali Kabiri ◽  
Mohammad Reza Talaee
Keyword(s):  
Heat Source ◽  
Closed Form ◽  
Method Comparison ◽  
Closed Form Solution ◽  
Form Solution ◽  
Temperature Profiles ◽  
Moving Heat Source ◽  
Bioheat Equation ◽  
Perfusion Rate

AbstractThe one-dimensional hyperbolic Pennes bioheat equation under instantaneous moving heat source is solved analytically based on the Eigenvalue method. Comparison with results of in vivo experiments performed earlier by other authors shows the excellent prediction of the presented closed-form solution. We present three examples for calculating the Arrhenius equation to predict the tissue thermal damage analysis with our solution, i.e., characteristics of skin, liver, and kidney are modeled by using their thermophysical properties. Furthermore, the effects of moving velocity and perfusion rate on temperature profiles and thermal tissue damage are investigated. Results illustrate that the perfusion rate plays the cooling role in the heating source moving path. Also, increasing the moving velocity leads to a decrease in absorbed heat and temperature profiles. The closed-form analytical solution could be applied to verify the numerical heating model and optimize surgery planning parameters.

Modeling Controlled Articulated Flexible Systems Using Assumed Modes: Part I — Theory

2001 ◽  
Author(s):  
Theodore G. Mordfin ◽  
Sivakumar S. K. Tadikonda
Keyword(s):  
Closed Form ◽  
Structural Model ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Flexible Multibody Dynamics ◽  
Companion Paper ◽  
Form Solution ◽  
Control Laws ◽  
Comparison Functions ◽  
Flexible Behavior

Abstract Guidelines are sought for generating component body models for use in controlled, articulated, flexible multibody dynamics system simulations. In support of this effort, exact closed-form and numerical solutions are developed for the small elastic motions of a planar, flexible, single link system, in which the link is represented as an Euler-Bernoulli bar in transverse vibration. The link is connected to ground by a pin joint, and the articulation is controlled by proportional and proprotional/derivative (PD) feedback control laws. The characteristics of the closed-form solution are shown to consist of combinations of the characteristic expressions associated with classical end conditions. A large-articulation flexible body model of a controlled-articulation flexible link is then developed and linearized about an arbitrary reference angle. This model uses the method of assumed modes to represent the flexible behavior of the link. It is shown the model is analytically equivalent to a purely structural model which uses a hybrid set of assumed modes, and that numerical convergence can be investigated in terms of admissible functions and quasi-comparison functions. Numerical evaluation of the use of various types of assumed modes is presented in a companion paper.

Three-dimensional stress state in inclined backfilled stopes obtained from numerical simulations and new closed-form solution

2018 ◽  
Vol 55 (6) ◽  
pp. 810-828 ◽  
Author(s):  
Abtin Jahanbakhshzadeh ◽  
Michel Aubertin ◽  
Li Li
Keyword(s):  
Stress State ◽  
Closed Form ◽  
Numerical Solutions ◽  
Hanging Wall ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Form Solution ◽  
Solid Waste Disposal ◽  
Inclination Angles ◽  
Backfilled Stopes

Backfill is commonly used world-wide in underground mines to improve ground stability and reduce solid waste disposal on the surface. Practical solutions are required to assess the stress state in the backfilled stopes, as the stress state is influenced by the fill settlement that produces a stress transfer to the adjacent rock walls. The majority of existing analytical and numerical solutions for the stresses in backfilled openings were developed for two-dimensional (plane strain) conditions. In reality, mine stopes have a limited extension in the horizontal plane so the stresses are influenced by the four walls. This paper presents recent three-dimensional (3D) simulations results and a new 3D closed-form solution for the vertical and horizontal stresses in inclined backfilled stopes with parallel walls. This solution takes into account the variation of the stresses along the opening width and height, for various inclination angles and fills properties. The numerical results are used to validate the analytical solution and illustrate how the stress state varies along the opening height, length, and width, for different opening sizes and inclination angles of the footwall and hanging wall. Experimental results are also used to assess the validity of the proposed solution.

Squeeze Film Characteristics of Circular Contacts at Constant Squeeze Velocity Motion with Non-Newtonian Lubricants

2011 ◽  
Vol 63-64 ◽  
pp. 147-151
Author(s):  
Li Ming Chu ◽  
Wang Long Li ◽  
Hsiang Chen Hsu
Keyword(s):  
Closed Form ◽  
Numerical Solutions ◽  
Hydrodynamic Lubrication ◽  
Closed Form Solution ◽  
Elastohydrodynamic Lubrication ◽  
Form Solution ◽  
Squeeze Film ◽  
High Pressure Stage ◽  
Time Calculation ◽  
Film Characteristics

In this paper, the numerical solutions in pure squeeze motion are explored by using hydrodynamic lubrication (HL) and elastohydrodynamic lubrication (EHL) models at constant squeeze velocity with power law lubricants. This paper also proposes a closed form solution to calculate the relationship between central pressure and central film thickness under HL condition. In order to save time calculation, the present closed form solution can be used as the initial condition for analysis of EHL at the high-pressure stage. In addition, this paper also discussed the HL and EHL squeeze film characteristics.

Formulation of surface temperature for laser evaporative pulse heating: The time exponentially decaying pulse case

2002 ◽  
Vol 216 (3) ◽  
pp. 289-299 ◽  
Author(s):  
B S Yilbas ◽  
M Kalyon
Keyword(s):  
Analytical Solution ◽  
Closed Form ◽  
Closed Form Solution ◽  
Solid Substrate ◽  
Form Solution ◽  
Solution Temperature ◽  
Heating Process ◽  
Pulse Parameter ◽  
Pulse Profile ◽  
Transform Method

Modelling of the laser heating process is fruitful, since it enhances the understanding of the physical processes involved and minimizes the experimental cost. In the present study, an analytical solution for the temperature distribution inside the solid substrate is obtained using a Laplace transform method. A time exponentially decaying laser pulse profile is introduced in the analysis. The phase change process and recession velocity are accommodated to account for the evaporation at the surface. The closed-form solution obtained is compared with the analytical solution obtained previously for a conduction limited heating case. It is found that the closed-form solution obtained from the present study reduces to a previously obtained analytical solution when the pulse parameter, β∗, is set to zero in the closed-form solution. Temperature predictions from simulations agree well with the results obtained from the closed-form solution.

scholarly journals Closed-Form Solution of Large Deflected Cantilever Beam a gainst Follower Loading Using Complex Analysis

2017 ◽  
Vol 11 (12) ◽  
pp. 12 ◽  
Author(s):  
Ibrahim Mousa Abu-Alshaikh
Keyword(s):  
Numerical Methods ◽  
Closed Form ◽  
Cantilever Beam ◽  
Complex Analysis ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Form Solution ◽  
Beam Deflection ◽  
Integral Approach ◽  
Loading Conditions

The literature reveals that the non-conservative deflection of an elastic cantilever beam caused by applying follower tip loading was investigated and solved by various numerical methods like: Runge Kutta, iterative shooting, finite element, finite difference, direct iterative and non-iterative numerical methods. This is due to the fact that the Euler–Bernoulli nonlinear differential equation governing the problem contains the “slope at the free end”, this slope however needs special numerical treatment. On the other hand, some of these methods fail to find numerical solutions for extremely large loading conditions. Hence, this paper is aimed to obtain a closed-form solution for solving the large deflection of a cantilever beam opposed to a concentrated point follower load at its free end. This closed-form solution when compared with other conventional numerical approaches is characterized by simplicity, stability and straightforwardness in getting the beam deflection and slopes even for extremely large loading conditions. The closed-form solution is obtained by applying complex analysis along with elliptic-integral approach. Very good results were obtained when the elastica of the beam compared with that of various numerical methods which are used in analyzing similar problem.

Numerical Solutions of Generalized Oscillator Equations

2013 ◽  
Author(s):  
Yufeng Xu ◽  
Om P. Agrawal
Keyword(s):  
Closed Form ◽  
Numerical Scheme ◽  
Fractional Integral ◽  
Numerical Solutions ◽  
Lagrange Equation ◽  
Closed Form Solution ◽  
Fractional Power ◽  
Form Solution ◽  
Harmonic Oscillators ◽  
General Nature

Harmonic oscillators play a fundamental role in many areas of science and engineering, such as classical mechanics, electronics, quantum physics, and others. As a result, harmonic oscillators have been studied extensively. Classical harmonic oscillators are defined using integer order derivatives. In recent years, fractional derivatives have been used to model the behaviors of damped systems more accurately. In this paper, we use three operators called K-, A- and B-operators to define the equation of motion of an oscillator. In contrast to fractional integral and derivative operators which use fractional power kernels or their variations in their definitions, the K-, A- and B-operators allow the kernel to be arbitrary. In the case when the kernel is a power kernel, these operators reduce to fractional integral and derivative operators. Thus, they are more general than the fractional integral and derivative operators. Because of the general nature of the K-, A- and B-operators, the harmonic oscillators are called the generalized harmonic oscillators. The equations of motion of a generalized harmonic oscillator are obtained using a generalized Euler-Lagrange equation presented recently. In general, the resulting equations cannot be solved in closed form. A numerical scheme is presented to solve these equations. To verify the effectiveness of the numerical scheme, a problem is considered for which a closed form solution could be found. Numerical solution for the problem is compared with the analytical solution. It is demonstrated that the numerical scheme is convergent, and the order of convergence is 2. For a special kernel, this scheme reduces to a scheme presented recently in the literature.

The Impact of Arbitrary Oriented Ellipsoidal Shell With a Barrier: Analytical Study

2015 ◽  
Vol 82 (4) ◽  
Author(s):  
Shahab Mansoor-Baghaei ◽  
Ali M. Sadegh
Keyword(s):  
Closed Form ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Linearization Method ◽  
Form Solution ◽  
Ellipsoidal Shell ◽  
Explicit Finite Element ◽  
Closed Form Solutions ◽  
Transmitted Force ◽  
The Impact

In this paper, a closed form solution of an arbitrary oriented hollow elastic ellipsoidal shell impacting with an elastic flat barrier is presented. It is assumed that the shell is thin under the low speed impact. Due to the arbitrary orientation of the shell, while the pre-impact having a linear speed, the postimpact involves rotational and translational speed. Analytical solution for this problem is based on Hertzian theory (Johnson, W., 1972, Impact Strength of Materials, University of Manchester Institute of Science and Technology, Edward Arnold Publication, London) and the Vella’s analysis (Vella et al., 2012, “Indentation of Ellipsoidal and Cylindrical Elastic Shells,” Phys. Rev. Lett., 109, p. 144302) in conjunction with Newtonian method. Due to the nonlinearity and complexity of the impact equation, classical numerical solutions cannot be employed. Therefore, a linearization method is proposed and a closed form solution for this problem is accomplished. The closed form solution facilitates a parametric study of this type of problems. The closed form solution was validated by an explicit finite element method (FEM). Good agreement between the closed form solution and the FE results is observed. Based on the analytical method the maximum total deformation of the shell, the maximum transmitted force, the duration of the contact, and the rotation of the shell after the impact were determined. Finally, it was concluded that the closed form solutions were trustworthy and appropriate to investigate the impact of inclined elastic ellipsoidal shells with an elastic barrier.

scholarly journals A Closed Form Solution for Nonlinear Oscillators Frequencies Using Amplitude-Frequency Formulation

2012 ◽  
Vol 19 (6) ◽  
pp. 1415-1426 ◽  
Author(s):  
A. Barari ◽  
A. Kimiaeifar ◽  
M.G. Nejad ◽  
M. Motevalli ◽  
M.G. Sfahani
Keyword(s):  
Closed Form ◽  
Numerical Solutions ◽  
Closed Form Solution ◽  
Analytical Techniques ◽  
Nonlinear Oscillators ◽  
Form Solution ◽  
Closed Form Expression ◽  
System Response ◽  
Classical Dynamics ◽  
Analytical Technique

Many nonlinear systems in industry including oscillators can be simulated as a mass-spring system. In reality, all kinds of oscillators are nonlinear due to the nonlinear nature of springs. Due to this nonlinearity, most of the studies on oscillation systems are numerically carried out while an analytical approach with a closed form expression for system response would be very useful in different applications. Some analytical techniques have been presented in the literature for the solution of strong nonlinear oscillators as well as approximate and numerical solutions. In this paper, Amplitude-Frequency Formulation (AFF) approach is applied to analyze some periodic problems arising in classical dynamics. Results are compared with another approximate analytical technique called Energy Balance Method developed by the authors (EBM) and also numerical solutions. Close agreement of the obtained results reveal the accuracy of the employed method for several practical problems in engineering.

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