Time-domain Radial Integration Boundary Element Method For Solving Unsteady Heat Conduction Problems

The object of boundary element method (BEM) analysis focuses on the linear problems at the initial stage. In subsequent several decades, computational efficiency, application field and other problems of BEM are researched by the domestic and foreign researcher and many outstanding achievements are obtained. Because of the special advantage of BEM, BEM is one of the most popular numerical method in the field of engineering and scientific computation and plays an important role in the numerical simulation of many actual engineering filed. Generally, there are two difficult problems when the complex time-domain problem is solved by using BEM. One is that because of lack of fundamental solution to the complex problem, especially to the nolinear problems, appeared domain integrals are unavoidable in the integral equation. The method of deal with domain integrals decide whether the BEM can be applied efficiently. Another one is that the processing technique of derivative term with respect to time impact directly on the stability of algorithm and the computational efficiency for the general time-domain problems. The two difficult problems play a crucial role in the application limitations of BEM. Up to now, radial integration method (RIM) is considered one of the best tools to transform the domain integral into the boundary integral. In this study, radial integration BEM (RJBEM) and different time-domain methods are combined to research the numerical method of unsteady heat conduction problems. The main contents of the present study are concluded as follows:(1) For the first time, extended application of RIBEM is developed to solve one-phase solidification problem. In this part, the transient heat conduction equation with constant coefficient of the two-dimensional one-phase solidification problem is analyzed by existing RIBEM. A scheme of variable time step is determined by the obtained heat flux of nodes on the moving boundary. Meanwhile, the precision of result is assured. Lastly, the position of moving interface is determined by front-tracking method at the different iteration time. The results of numerical examples verify the validity of present method. The present study not only extends the applied field of RIBEM but also establishes foundation for application of RIBEM to solve more complex phase change problems.(2) The precise integration BEM is proposed to analyze the unsteady Fourier heat conduction problem. For the unsteady Fourier heat conduction problem with constant or variable thermal conductivity and heat source, firstly, the space domain of problem is discretized by RIBEM and the system of differential equations about the temperature of nodes is obtained. Then, the system of fist order ordinary differential equations of temperature is obtained by eliminating the unknown heat flux of boundary node so that the discrete system of ordinary differential equations can be solved by using the adaptive precise integration method. Lastly, numerical results are obtained at different time. The numerical results show that the present method can obtain highly precise results even if the time step is relatively large and can efficiently avoid the numerically unstable phenomenon by the traditional time difference. Particularly, the precision of results is independent of the time step when the load term can be expressed analytically and can be integrated analytically.(3) The precise time-domain expanding BEM is proposed to solve the unsteady Fourier and non Fourier heat conduction problems. Firstly, the recursive and time-independent formulations of the governing equation and boundary conditions are derived by the precise time-domain expanding method for the unsteady Fourier heat conduction problems with heat source, constant or variable thermal conductivity and the non Fourier heat conduction problems with constant thermal conductivity. Then, for the two and three-dimensional problems, the integral equation of recursive form is established uniformly based on the fundamental solution of potential problem. In addition, the radial integration method is used to transform all domain integrals into the boundary integral and the recursive discrete equation is formed by BEM. So far a new numerical method is presented for the related problem. Comparing to the traditional time difference method, numerical results show that the present method can still obtain highly precise results even for the relatively large time step and has a better numerical stability.The time-domain radial integration BEM is presented in this thesis. It can be applied to solve many different types of unsteady heat conduction problems. The present study not only enrich the application field of BEM but also has a better numerical stability. And the present method can give highly precise numerical results. Also, the present study has a reference value for numerical analysis of other time-dependent problems.