Isogeometric Analysis For Solving Transient Heat Conduction Problems

Heat transfer is one of the most common physical phenomena in the nature,it is found in many fields such as industrial casting,aerospace,electronic chips and health care.The essence of the problem is to solve the differential equation with boundary condition,in which the transient heat conduction has a time derivative term.The simplest and most effective way to deal with this problem is numerical methods.Isogeometric analysis is a new numerical calculation method which has been popular in recent years.The method is based on the idea of finite element moderate parameter and takes the spline function as a shape function.In this paper,isogeometric analysis and two different time-domain methods are combined respectively to develop a numerical method for solving transient heat conduction.The main contents of the paper are concluded as follows:(1)Based on the isogeometric analysis and Precise Integration Method,an accurate and efficient numerical algorithm for solving transient heat conduction is proposed.Firstly,the space domain of problem is discretized by isogeometric analysis by using Non-uniform rational B-spline(NURBS),the natural NURBS mesh and the system of ordinary differential equations in time domain are obtained.In the time domain,Precise Integration Method is used to deal with the system of ordinary differential equations.For large scale structures,by using the physical nature of heart conduction and sparse property of matrix,the special structure of matrix exponential is given and a fast precise integration method for solving differential equations is obtained.Lastly,the numerical examples show that the present method can obtain highly precise results even for the less control nodes and larger time step.(2)Based on the isogeometric analysis and Time-Domain Expanding Algorithm,an accurate and steady numerical algorithm for solving linear and nonlinear transient heat conduction is proposed.In this method,time dependent variables are expanded by Taylor formula in the time domain and the time independent transient heat conduction constitutive equation and boundary conditions are given.Next,by using isogeometric analysis,the system of linear equations in a recursion format are obtained.Lastly,the numerical examples show that the accuracy of the present method is not sensitive to the time step and the results have high precision and good stability.