Auxiliary Boundary Value Problem Method For Stable Temperature Field Problem And Elasticity Problem

As we all know,the finite element method is the dominant numerical simulation technology and it has been widely used in the fields of scientific calculation and engineering analysis.However,the finite element method needs to discretize its computational domain into certain regularized grids that are called regional elements,a certain degree of correlation is generally required between these elements.For three-dimensional problems,the time spent on data processing is much longer than the calculation time,especially for some problems with complex geometric calculation areas,it may pay considerable computational cost,and sometimes it even cause problems in mathematical theory.Moreover,as a domain discretization technique,it is difficult for the finite element method to solve those inverse problems that only measure boundary data.In addition,when using the finite element method to simulate coating structure and ultra-thin structure problems,it will cause deformed elements.As a numerical simulation technology with strong competitiveness in recent years,the boundary element method makes up for the deficiency of the finite element method and it has the advantage of reducing the dimension of the problem.The boundary element method has been widely used in steady-state and transient mechanical problems,linear problems such as infinite sound field,magnetic field simulation,and various non-uniform materials and nonlinear problems.In particular,the gradient calculation formula of the physical quantity obtained by the boundary element method can be analytically derived from the basic integral equation.Therefore,the obtained gradient value of the physical quantity has the same level of accuracy as the physical quantity itself,which is difficult for the numerical method that are based on the internal grid(such as finite element method)to achieve.However,unlike the finite difference and finite element methods,the boundary element method involves singular kernel integrals,which are caused by the basic solutions(and their derivatives)of the governing differential equations of the problem under study.The key to the implementation of the boundary element method is how to deal with singular kernel integrals.Today,the analysis and research of singular integrals are still the difficulty and focus of the boundary element family,which is determined by the properties of the method itself.Although many techniques and methods for dealing with singular integrals have been developed in the boundary element method,they all have the disadvantages of complicated theoretical derivation,large amount of calculation,and difficulty in programming.Different from the existing research,this paper proposes the auxiliary boundary value problem method for stable temperature field problem and elasticity problem.Constructing an auxiliary boundary value problem with the same solution domain as the original boundary value problem,and the auxiliary boundary value problem has a known solution,so the system matrix of the boundary integral equation can be obtained by solving this auxiliary boundary value problem,then it can be used to solve the original boundary value problem.It is worth noting that there is no need to recalculate the system matrix when solving the original boundary value problem,so the efficiency of the auxiliary boundary problem method is not bad.The auxiliary boundary value problem method avoids the calculation of strong singular integrals,and it is suitable for numerical implementation under any geometric unit and higher order interpolation.It has the advantages of simple mathematical theory,easy program design,high calculation accuracy,etc.It provides a new way to solve the gradient boundary integral equation of coordinate variables.