习强+傅卓佳+蔡加正

摘要： 采用基本解方法结合扩展精度技术和Kirchhoff变换求解功能梯度材料的二维热传导问题.在求解瞬态热传导问题时运用Laplace变换处理时间变量，将时域问题转化为频域问题求解；采用基本解方法计算得到高精度的频域数值解，再分别采用Stehfest和Talbot这2种数值Laplace逆变换恢复原瞬态热传导问题的计算结果.通过3个非线性功能梯度材料的稳态和瞬态热传导基准算例，分析结合扩展精度技术的基本解方法的计算精度与扩展精度位数、边界布点数和虚拟边界参数三者之间的关系.比较Stehfest和Talbot这2种数值Laplace逆变换算法的优劣.采用结合扩展精度技术的基本解方法数值研究热传导系数随位置剧烈变化的功能梯度材料热传导行为.数值结果表明该方法具有求解精度高、适用性好等特点，能高效模拟非线性功能梯度材料的二维稳态与瞬态热传导行为.

关键词： 基本解方法； 扩展精度技术； 数值Laplace逆变换； 功能梯度材料； 热传导

中图分类号： O241.82；O343.6文献标志码： A

傅卓佳（1985—），男，浙江绍兴人，副教授，博士，研究方向为径向基函数、软物质力学建模、流固耦合、工程反问题、偏微分方程数值解、波传播及结构振动，（Email） paul212063@hhu.edu.cn

Abstract： The method of fundamental solution method in conjunction with the extended precision technique and Kirchhoff transformation is used to solve 2D heat conduction problem of nonlinear functionally graded materials. For transient heat conduction analysis， Laplace transform technique is applied to handle the time variable and the corresponding time domain problem can be transformed frequency domain problem. The numerical solution with highprecision in frequency domain is calculated by fundamental solution method. Two numerical Laplace inversion techniques Stehfest and Talbot are employed to retrieve the calculation results of original transient heat conduction. The effects of the extended precision digit， the boundary node number and fictitious boundary parameter on the calculation accuracy of the fundamental solution method in conjunction with the extended precision arithmetic are analyzed under three benchmark examples about steadystate and transient heat conduction problems in nonlinear functionally graded materials. The two numerical Laplace inversion techniques Stehfest and Talbot are compared. The thermal behavior of functionally graded materials with drasticchanged thermal conductivity is numerically studied by the fundamental solution method in conjunction with the extended precision arithmetic. The numerical results show that the method is with high accuracy and good applicability， and can effectively simulate 2D steadystate and transient heat conduction behavior of nonlinear functionally graded materials.

Key words： fundamental solution method； extended precision arithmetic； numerical Laplace inverse transform； functionally graded material； heat conduction

0引言

功能梯度材料是新一代复合材料，其导热系数、比热容和密度等物理参数沿着材料位置逐渐改变.[1]由于良好的热传导特性，功能梯度材料的使用环境多为高温或超高温[2-4]，比如：热障涂层、返回舱的热保护等，因此很有必要了解功能梯度材……