Solution And Simulation For 2D Heat Conduction Process Based On Numerical Calculation

The heat conduction equation is one of the basic equations describing the heat conduction phenomenon.It is widely used in many fields such as engineering thermodynamics,material science,aerospace,and metallurgical engineering.However,most of the heat conduction problems in nature and production are difficult to obtain their exact solutions,and numerical methods are needed to obtain their approximate solutions.Therefore,the high precision and fast speed numerical calculation method is very important to solve the two-dimensional heat conduction problem.At the same time,due to the large amount of data in the numerical solution of the heat conduction problem,it is not easy to analyze the changing trend of the temperature field.The simulation can intuitively show this change,but the domestic-related types of self-developed simulation software are relatively lacking.Therefore,it is of great significance to solve and simulate the two-dimensional heat conduction process based on numerical calculation.Based on the idea of the alternating direction implicit method and the modified Crank-Nicolson method,this paper proposes a time-splitting method for solving the numerical solution of the two-dimensional heat conduction problem.The main work is as follows :(1)Based on the finite difference method to solve the one-dimensional heat conduction problem.Because the two-dimensional heat conduction problem can be decomposed into multiple one-dimensional heat conduction problems by dimensionality reduction,solving onedimensional problems efficiently and accurately is the basis for solving two-dimensional problems.Firstly,the one-dimensional heat conduction equation of the corresponding problem is established,and the orthogonal grid method is used to discretize the heat transfer region.Secondly,the three traditional difference methods and the modified Crank-Nicolson method are used to approximate the difference of the heat conduction equation into algebraic equations.The algebraic equations on each node constitute an algebraic equation group.Finally,the numerical approximate solution of the one-dimensional problem is obtained by solving the algebraic equation group.The experimental results show that the modified Crank-Nicolson method has a smaller error than the traditional difference method,and the convergence speed is faster than the Crank-Nicolson method.With the increase of mesh subdivision degree,The convergence rate of the modified Crank-Nicolson method grows faster.(2)Two-dimensional heat conduction problem is solved based on numerical calculation methods.Based on the idea of the modified Crank-Nicolson method and alternating direction implicit method,this paper proposes a time-splitting method for solving two-dimensional heat conduction problems and uses the Richardson extrapolation method to optimize the proposed method.This method transforms the two-dimensional heat conduction problem into solving two one-dimensional heat conduction problems and uses the modified Crank-Nicolson method to solve the one-dimensional problem,which reduces the computational complexity and error.The experimental results show that compared with the alternating direction implicit method,the error is smaller and the convergence speed is faster when using the proposed method.After using the Richardson extrapolation method to optimize,the solution accuracy and convergence speed are significantly improved,but the CPU time consumed is also increased.Compared with the heat balance method,the error of the proposed method is also smaller.(3)Design and implementation of two-dimensional heat conduction simulation software.Firstly,the graphical visualization interface is constructed by QT,and the temperature field image is drawn by Open GL.The development of two-dimensional heat conduction simulation software,including meshing,data processing,temperature field visualization,and interactive operation,is realized.Then,a series of simulation experiments were carried out under different initial conditions,boundary conditions,spatial steps,and heat transfer objects.The influence of different factors on the simulation results was explored,and the software solution’s accuracy and the simulation’s stability were verified.(4)Analysis of experimental results.The calculation accuracy of the method used in this paper is evaluated.When the grid is divided into 10,20 and 40 segments and the time step is0.002 s,the error of the modified Crank-Nicolson method is reduced by 0.42%,1.69%,and7.09% respectively compared with the Crank-Nicolson method.When the grid is divided into10 × 10 and 20 × 20 segments,and the time step is 0.005 s,the error of this method is reduced by 2.26% and 9.83% respectively compared with the alternating direction implicit method.When the time step is 0.0025 s,the error decreases by 0.59% and 2.40% respectively.