层合圆柱三维温度场分析的半解析-精细积分法

针对圆柱体的三维温度场分析,提出了一种高效的半解析-精细积分法。将温度场展开为环向坐标的Fourier级数,并对径向坐标进行差分离散,从而把三维热传导方程简化为一系列二阶常微分方程;将这些二阶常微分方程转化为哈密顿体系下的一阶状态方程,并利用两点边值问题的精细积分法求解。由于该方法仅对径向坐标进行差分离散,故相对于传统的数值方法离散规模大幅度减少,不仅提高了计算效率、降低了存贮量,而且缓解了代数方程的病态问题。此外,针对Fourier半解析解,根据热平衡原理推导出了两种材料衔接面的半解析差分方程,从而为求解复合材料层合柱问题打下了基础。算例结果表明,即使对于细长比高达400的圆柱杆件,此方法仍然可以给出精度较高的解答。

The semi-analytical precise integration method for the analysis of three dimensional temperature field of laminated cylinders

A highly effective semi-analytical precise integration method for the analysis of the three dimensional temperature field of cylinders is proposed.The temperature field is firstly expanded to Fourier series with ring coordinates.Then the differential equations are discretized into difference equations of the radial coordinates.As a result,the three dimensional heat transfer equations are simplified into a series of second-order ordinary differential equations.These ordinary differential equations are subsequently transformed into the first-order state equations under the Hamilton system,which are then solved through the precise integration method for the two-point boundary-value problems.Due to the fact that this method only discretizes the differential equations into difference equations of the radial coordinates,the discrete scale is much smaller than that of the traditional computational methods.By applying this novel method,not only the computational efficiency is improved and more computational resource is saved,but also the ill-conditioned problems of the algebraic equations can be mitigated.Furthermore,according to the result of the Fourier semi-analytical method and the principle of thermal equilibrium,a semi-analytical differential equation for the interface between two different materials in a cylinder is derived.This can be potentially applied to the investigation of cylindrical members laminated with composite materials.Even dealing with the cylindrical member whose length is 400 times longer than the radius,the numerical examples still demonstrate high accuracy and efficiency.