Research On Fast Boundary Element Method For Coupled Heat Radiation-conduction Problems In Porous Material

The boundary element method(BEM)is one of the most commonly used numerical methods for solving engineering and scientific problems,and compared with finite element method,the advantage of BEM is only the discretization of boundaries,which reduces the dimension of the problems.This advantage makes BEM very suitable for simulating structures with complex boundaries or interfaces,such as random porous structures.However,the coefficient matrix arising from BEM is always full and asymmetric,and then the traditional solution techniques are inefficient,which makes BEM not available for large scale problems.The fast boundary element method based on the analytic expanding of kernel function reduces the complexity of operation to O(N).However,it is very hard to find the appropriate expanding because of the complex expressions of kernel function.In this thesis,we study the kernel independent fast boundary element algorithm and corresponding high efficient and precision calculation method for solving coupled radiation-conduction heat transfer problem of porous and random porous structures.And this method is applicable to the complex problems,which is impossible to the appropriate analytic expanding of their kernel functions.In the first part,a kernel independent fast boundary element method is developed for solving the coupled radiation-conduction heat transfer problem of porous structures.Firstly,the boundary integral equation of the coupled radiation-conduction heat transfer problem is derived and corresponding boundary element discrete schemes are given.And then,based on the Chebyshev interpolation method,the numerical expansion and shifting formulation are constructed,including the multipole expansion and local expansion.Moreover,a new tree structure based on multi-layer bucket is established to store the coefficient matrix information and implement the product of matrix and vector quickly.Finally,the algorithm procedure of the fast FEM for coupled radiation-conduction heat transfer problem is presented and the computation efficiency of the algorithm is also analyzed.Numerical results show the efficiency and accuracy of this method.In the second part,a multi-scale boundary element method based on the stochastic asymptotic homogenization theory is developed for the thermal conduction problem of random porous structures.There exist complicated microstructures and multiscale characters in random porous structures and it is difficult or impossible to evaluate their physical or mechanical behaviors by the traditional numerical methods.Firstly,based on the scale separation,the thermal conduction problem is decomposed into the homogenized problem and cell problems.And based on the radial basis function,the boundary integral equations of homogenized and cell problems and corresponding discrete schemes are derived then.Finally,the stochastic multiscale boundary element algorithm procedure combining the Monte Carlo technique is presented.Numerical results show that the proposed stochastic multiscale boundary element method is feasible for solving the thermal conduction problem of random porous structures and can greatly improve the computational efficiency.In the third part,a new stochastic multi-scale boundary element method based on is developed for predicting the coupled radiation-conduction heat transfer performance in random porous structures.Firstly,the radiation boundary condition in random pores and the coupled radiation-conduction heat transfer model are given.And based on the scale separation and multiscale asymptotic expansion,the nonlinear homogenized problem and cell problems are established and corresponding discrete schemes are also derived.Finally,the stochastic multiscale boundary element algorithm procedure combining the Monte Carlo technique is presented.Numerical results show that the stochastic multiscale boundary element method is not only feasible,but also accurate for predicting the coupled radiation-conduction heat transfer performance in random porous structures.In the fourth part,we study the topology optimization algorithm of heat-dissipating structure based on the level set method and fast boundary element method.Firstly,the structural boundaries of design domain are defined as zero level set,and the change and shape of structural boundaries can be captured accurately according to evolution of zero level set.Then,the kernel independent fast boundary element method is introduced to solve the thermal conduction problem and its dual problem,which avoids reconstruction of three-dimensional finite element mesh.Finally,the topology optimization iterative algorithm coupling the level set method and fast boundary element method is presented.Numerical results show that the method greatly improve the efficiency of calculation and optimization,which is very useful for the topology optimization of large heat-dissipating structures.