The Dual Interpolation Boundary Face Method For Heat Conduction Problems

The heat conduction is a common engineering problem in the field of production technology.The current numerical integration methods for analyzing heat conduction problems include finite difference method,finite volume method and finite element method.Among these methods,the finite element method,which is more adaptable to irregular areas,stands out.Because the finite element method needs to derive the trial function when calculating the stiffness matrix and requires the trial function to satisfy C~0 continuity requirement at least,so the calculation accuracy is dependent on the shape of the units and the heat flux is one order lower than temperature.The dual interpolation boundary face method based on the boundary integral equation can maximize the advantage of discontinuous requirement for the test function in the boundary integral equation under the premise of ensuring the accuracy of the numerical calculation,thereby reducing the dependence on the quality of the grid,as well as the requirements for mesh continuity,so the dual interpolation boundary face method will not require geometric repair of complex CAD models with"noise".In addition,the method,which retains the complete geometric information of the actual model and avoids geometric errors,can automatically analyze the CAD model by using the boundary representation(B-rep)data structure in the CAD solid modeling system,thus it naturally integrates CAD and CAE.This paper focuses on the dual interpolation boundary face method,and is dedicated to the analysis of heat conduction problems with this method.The main contents are as follows:(1)Combined with the Galerkin method,the dual interpolation boundary face method is applied to solve the homogeneous steady-state heat conduction problem.This paper studies the combinability of the dual interpolation method and the symmetric Galerkin boundary element method,and derives the general formula of the dual interpolation Galerkin boundary face method in detail.The new method will not need hypersingular boundary integral equations,and the symmetric coefficient matrix is obtained through a series of matrix operations.Numerical results show that,compared with the traditional Galerkin boundary face method,the algorithm has higher calculation accuracy and faster convergence speed.(2)The dual interpolation boundary face method is applied to solve the non-homogeneous steady-state heat conduction problem.Since the basic solution of the non-homogeneous heat conduction problem does not consider the non-homogeneous term,a volume component will appear in the derived boundary integral equation.This article does not make any conversion to the domain integral.After the grid is generated in the domain,the domain integrals brought by the known source items are directly calculated through the defined dual interpolation unit in the domain.Since the domain integral term is only placed at the right end of the equation as a known quantity when solving the equation,and does not participate in the inversion of the coefficient matrix,it does not change the solution dimension of the boundary integral equation essentially.Introducing intra-domain integration,so that the dual interpolation boundary face method can be applied to a wider range of heat conduction problems.Numerical results show that,compared with the traditional boundary face method,the algorithm has higher accuracy and efficiency,and is suitable for solving heat conduction problems with domain integral terms.(3)Based on the study of the dual interpolation boundary face method for transient heat conduction problems,the dual interpolation boundary face method is applied to the two-dimensional transient heat conduction analysis for the first time.Numerical results show that,compared with the traditional pseudo-initial condition boundary face method and precise integration boundary face method,the algorithm has higher accuracy and efficiency,which further proves the generality of the algorithm.(4)Based on the study of the dual interpolation boundary face method for the three-dimensional steady-state heat conduction problem,the three-dimensional steady-state heat conduction problem is solved for the first time in combination with the fully-automatically divided binary tree mesh.This algorithm truly realizes the integration of CAD and CAE,verifies the feasibility of using discontinuous grids for CAE analysis,and lays a solid foundation for fully automated CAE analysis.Numerical results show that,compared with the finite element method,the algorithm does not require any model simplification,retains all geometric information of the actual model,and uses fewer source points to achieve higher accuracy.(5)Based on the research of the precise integration method for transient heat conduction problems,the dual interpolation boundary face method is extended to the three-dimensional transient heat conduction problem for the first time,and the dual interpolation precise integration boundary face method for the three-dimensional transient heat conduction problem is developed.Numerical results show that,compared with the traditional precise integration boundary face method,this new algorithm has higher accuracy and efficiency;the algorithm is suitable for numerical calculations with small time step,and it effectively solves the numerical instability problem in the three-dimensional transient heat conduction analysis when the time step length becoming smaller.