АЛГОРИТМ ПОСТРОЕНИЯ РАЦИОНАЛЬНЫХ АППРОКСИМАЦИЙ ФУНКЦИЙ В ИЗОБРАЖЕНИЯХ ЛАПЛАСА С ПРИМЕНЕНИЕМ К ДИНАМИЧЕСКИМ ЗАДАЧАМ ПОРОУПРУГОСТИ

Abstract

In this paper, the Vector Fitting method is used for fitting functions in Laplace domain by pole relocating. The used version of the method is described for one function and for vector functions. An algorithm based on the Vector Fitting method for obtaining strictly proper rational approximations of vector functions is presented. Algorithm produces approximations with increasing orders until the specified convergence condition is satisfied. Specific recommendations for selecting algorithm parameters are given. The proposed algorithm allows to adaptively select frequency samples that are used for obtaining rational approximations, without any previously known information about the nature of functions in Laplace domain. Time-domain solutions are obtained by analytical inversion of the rational approximations. The developed algorithm is tested in detail on the problem of wave propagation in one-dimensional poroelastic column with finite length. The model of linear isotropic fully saturated poroelasticity proposed by Biot is employed. For the general case, analytical solutions of the problem are presented in Laplace domain for displacements, stresses, pore pressures and flux. For the limiting value of the permeability of a poroelastic material, the corresponding analytical solutions are also given in time domain. Rational approximations of solutions in Laplace domain obtained with the proposed algorithm are presented in details for two values of the permeability. The convergence of the algorithm is investigated depending on the number of complex frequencies. It was found that with an increase in the order of approximation, the relative error changes similarly for all considered functions. It is shown that when parameters of the algorithm are chosen within the recommended ranges, the solutions in time domain are stable and the amplitude of the oscillations caused by truncation of the high frequency parts of the solutions remains moderate over the entire specified time interval. For the limiting value of the permeability, high accuracy of the obtained approximate results in comparison with analytical solutions in time domain is clearly demonstrated.