Computer simulation of the surface heating process by the movable laser

The model of heating the target surface in air environment with moving laser beam is created. The equation was solved for the case of dynamically changing laser beam energy and constant parameters of the air flow.


Introduction
The task of heating the surface of the target moving laser beam is relevant, as is used in many industrial processes.
Practical implementation of this operation to ensure optimal heating and cooling parameters of the target is impossible without building a mathematical model that takes into account the specific conditions of the heating laser beam and cooling the surface by convection. The complexity and urgency of this problem increases when the beam heats the surface is moving.
The mechanism of target surface heating with a pulsating-periodic laser beam, moves over the surface of the plate in a predetermined manneris considered. [1][2][3].
The target is a flat plate with predetermined thickness δ. Spatial coordinate system, in which the problem is solved, is stationary. The laser beam moves over the surface of the plate in a predetermined manner with the speed v(x,y,t) ( fig.1).   The system of equations describing the process of heating-cooling the target surface is as follows: ,

Mathematical model
where (1) -is the differential equation of the heat conductivity; (2) -is the boundary conditions on the laser hot spot surface Sare of follows; (3) -is the boundary conditions of heat exchange with ambience on the surface Sare the next; (4) -is the initial conditions for t = 0; T -is the target surface temperature; t -is the time; x, y, z -are spatial coordinates; λ х , λ y ,λ z -are heat conductivity coefficientsin the direction of axisx, yand z; V -target volume; q l = f(x,y,t,e) -theheat flux from the laser beam; αis the heat exchange coefficient; T g -ambient temperature; S -target surface; T 0 -initial temperature.
To solve this problem a method of finite elements [4,5] is used. The decision comes down to the definition of steady-state value of the next functional: -the heat flux, conditioned by the heat capacity of the material, index " * " corresponds to a temperature distribution in the space fixed at the moment.
The problem reduces to the definition of change of the temperature distribution over the volume of the target as a function of time: As a finite element we use the cubic hybrid element, which is formed by the union of several tetrahedrons. A function of the temperature distribution is a linear function and the heat capacity of the element is concentrated in the nodes [4,5,6]  ( ) where l N -shape coefficient of tetrahedral finite element; l T -the temperature of the nodal points of a finite element; l =1, 2, 3, 4.
The linear function of a temperature approximation (7) in this element is as follows: where α 1 , α 2 , α 3 , α 4 -are the coefficients of the approximating function.
Vector temperature values in nodes is as follows: where T V -volume of the tetrahedron; x l , y l , z l -coordinates of the nodal points; indexes a l , b l , c l , d l -the coefficients of the approximating function of temperature in the amount of finite element; l∈ {i, j, k, m}; i, j, k, m = 1, 2, 3, 4 are obtained by cyclic permutation in the sequence.
The heat flow, which is absorbed by the finite element due to the heat capacity of the material expressed through heat flows in nodes: Let us consider the original functional in the class of linear functions of temperature (9). Minimization of this functional can be carried out over the nodal values [4], as they uniquely define the temperature at any point in the field of research.
Let us introduce into the original functional (5) conditions (7) -(9) and differentiate it respect to the nodal temperature of the tetrahedron T l for the field of the finite element: To get the matrix form one should combine equations: where λ -is the heat conduction matrix of the finite elements; С 4 -is the heat capacity diagonal matrix of size 4×4; T -vector of temperature in all nodal points [4].
The final equation of the process of the functional minimization on the temperature at the nodal points are obtained by combining all the derivatives (11) over all finite elements on which the field of research is sampled [4].
Heat conduction matrix, heat capacity matrix and vectors of the nodal heat flow for the structure can be obtained by adding the respective members of heat conduction matrix, heat capacity matrix and finite elements heat flows. Resulting equation of the finite element method for this case takes the form: where Λ , С n -is the global heat conductivity matrix and heat capacity matrix correspondingly of size where p -is the index corresponding to the time; The vector T of size n at the initial time is determined from the initial conditions. The solution of resulting linear system of algebraic equations is carried out by the method of adjoint gradient [4,8].
The algorithm for solving system (1) -(4) was worked out. The results are shown on fig.3 as isotherms.
Computer experiment was held for target of 10mm thick, the material has the conductivity 0.5 W/(mK), grid of size 300×100×20 vas used.  Numerical simulation of the process is carried out by means of theoretical and computational complex of computer modeling and visualization of the heat exchange processes [9].

Conclusion
The developed method of numerical solution process of the surface heating by the movable laser beam allows to carry out a computer simulation of the target surface heating and cooling, and investigate the dynamics of target surface heating by pulsating-periodic laser.