The second-order strain gradient wave equation, which is based on the generalized continuum mechanics theory, can enrich the content of classical continuum mechanics theory with the incorporation of the second-order spatial derivative term of displacement. Furthermore, it incorporates scale parameters of the media characteristics to bridge the gap between a micromodel and classical continuum mechanics, thus reflecting the microstructure characteristics within the media. Wang derived a constitutive equation of the single-parameter second-order strain gradient theory using a nonlocal theory and provided a mathematical expression for the second-order strain gradient asymmetric elastic wave equation combined with the geometric equation and differential equation of motion. The second-order strain gradient theory can reflect the smaller-scale effect of seismic wave propagation within the neighborhood of average particle diameter l of the medium. Considering the relatively weak spatial scale effect, the numerical dispersion generated when the difference operator approximates the differential operator can suppress the scale effect. Thus, to accurately describe and analyze the scale effects, the finite-difference operator must be optimized. This paper proposes an improved black window optimization algorithm to obtain optimized finite-difference coefficients and perform numerical modelling based on the second-order strain gradient wave equation. Numerical modelling results reveal that the numerical dispersion is well suppressed, and the smaller spatial scale effects can be clearly observed and extracted from the seismograms.

基于广义连续介质力学理论的二阶应变梯度波动方程，通过引入位移二阶空间导数项，丰富了经典连续介质力学理论的内容。此外，它还结合了介质特性的尺度参数，以弥合微观模型和经典连续介质力学之间的差距，从而反映介质内的微观结构特性。Wang利用非局部理论推导了单参数二阶应变梯度理论的本构方程，并结合几何方程和运动微分方程给出了二阶应变梯度非对称弹性波方程的数学表达式。二阶应变梯度理论可以反映地震波在介质平均粒径l附近传播的小尺度效应。考虑到空间尺度效应相对较弱，差分算子逼近微分算子时产生的数值弥散可以抑制尺度效应。因此，为了准确地描述和分析尺度效应，必须对有限差分算子进行优化。本文提出一种改进的黑窗优化算法，以获得优化的有限差分系数，并基于二阶应变梯度波动方程进行数值建模。数值模拟结果表明数值色散得到了很好的抑制，