金属纤维烧结过程的分数阶模型

基于金属纤维烧结结点的几何模型，运用Caputo分数阶微分方程，建立时间分数阶表面扩散模型，使用有限差分法求数值解，实现金属纤维烧结过程的数值模拟.对不同的阶数进行模拟，得到0~1阶烧结过程数值结果及颈长变化规律；在阶数为0.9阶时，模拟初始夹角为0°、30°、60°、90°时的烧结过程.结果表明：阶数等于1时的结果与整数阶扩散模型一致；烧结的初始阶段，整数阶与分数阶模拟的颈半径迅速生长，随着烧结的进行，分数阶模拟的烧结颈长出现局部波动，最后以大于整数阶的增长速率增长；阶数固定时，初始夹角越小，增长速率越大.分数阶表面扩散模型比整数阶表面扩散模型能更好地描述纤维烧结过程中烧结结点的复杂变化.

A Fractional Model of Metal Fiber Sintering Process

Based on geometric model of metal fiber sintering nodes, with Caputo fractional differential equations, a time fractional surface diffusion model is established. Numerical solution by finite difference method is made. Numerical simulation of metal fiber sintering process is realized. Numerical simulation of sintering process and variation of neck length as fractional order varies from 0 to 1 are obtained. As the order is fixed at 0.9, sintering process at initial included angles of 0°, 30°, 60° and 90° are simulated. It shows that as the order is equal to 1 the result is consistent with integer order diffusion model. Neck radius with integer order and fractional order grows rapidly in initial stage of sintering. With progress of sintering, fractional simulation of sintering neck length appears local fluctuation, and finally grows at an increase rate greater than the integer order. As the order is fixed, the smaller the initial angle, the greater the rate of growth. The fractional order surface diffusion model describes well the complex change of sintering node during fiber sintering process than the integer order surface diffusion model.