Numerical Modelling and Optimization of the Second-Order Strain Gradient Wave Equation

In the conventional continuum mechanics theory, the strain energy density function at the point of the medium material only depends on the classical strain, and the medium is modeled as an ideal continuum. For the isotropic medium, has no internal structure, no additional internal degrees of freedom, and no internal characteristic scales, therefore, applying the law of conservation of moment of momentum will inevitably lead to the conclusion that shear stress is equivalent. However, all media have extremely complicated microstructures, it is not suitable to describe the more complex interactions occurring in generalized continua by means of the sole Cauchy stress tensor. Wang derived and gave the mathematical expression of the second-order strain gradient asymmetric elastic wave equation from the nonlocal theory. The derived wave equation can describe the smaller spatial scale effects with the particle diameter of the constituent medium as the neighborhood of seismic wave propagation, caused by the heterogeneity from the micro-structure interactions. Considering that the smaller spatial scale effect is weak, the numerical dispersion generated when the difference operator approximates the differential operator can suppress the scale effect. Therefore, in order to better describe and analyze the scale effects, the finite difference operator needs to be optimized. This paper proposes an improved BWOA algorithm to obtain the optimized finite difference coefficients and conducts numerical modelling based on the second-order strain gradient asymmetric elastic wave equation. Through numerical modelling analysis, it is shown that the numerical dispersion is well suppressed, and the smaller spatial scale effects can be obviously observed and extracted from the seismograms.