Research On Transient Heat Transfer Problems By Using The Smoothed Finite Element Methods

Transient heat transfer is a science that studies the change of heat energy over time,which has wide application background and profound significance,from providing more comfort and convenience for human production and life to the development of industrial technology promotes the innovation of productivity.Due to the complexity of the practical research,some numerical methods need to be established to get the solution.In this paper,transient heat transfer problems are analyzed individually using the smoothed finite element methods(S-FEMs)with explicit time integration and Laplace method.When the difference method is used to deal with the time,for a numerical method with the spatial discretization,the computational cost per time step in the explicit method is less than that in the implicit method,thus,the explicit scheme is often adopted in practical engineering problems.However,for explicit method,in order to ensure the stability of the method,there is a certain relationship between time step and mesh size.This work thus studies the stability of S-FEMs,when applied to transient heat transfer problems.Relationships are established between the critical time steps used in S-FEMs with the maximum eigenvalues of the thermal stiffness(conduction)matrix and mass matrix.Because computing the eigenvalues and condition numbers of the stiffness matrices are very expensive but valuable for stability analysis,we proposed a concise and effective algorithm to estimate the maximum eigenvalue and condition number.Intensive numerical examples show that our scheme for computing the critical time step can work accurately and stably for explicit method in FEM and S-FEMs.Further,it is found that the critical time step relates to the “softness” of the model: the numerical method is “softer”,which permits larger critical time step.For example,node-based smoothed finite element method(NS-FEM)is softer than edge-based smoothed finite element method(ES-FEM),which leads to that the critical time step of NS-FEM is larger than that of ES-FEM.The Laplace method is also widely used for solving transient heat transfer problems in engineering technology.It has no error accumulation and which is not possessed by the finite difference method mentioned before.So the author tries to apply Laplace method and S-FEMs to solve transient heat conduction problems,and study the advantages and disadvantages of these two methods in time processing.A number of numerical examples show that the S-FEMs with Laplace method can effectively simulate the transient heat transfer problems in the case of appropriate parameters.