The importance of accurate time-integration in the numerical modelling of P-wave propagation

Abstract

The numerical dissipation characteristics of the Newmark and generalised-α time-integration schemes are investigated for P-wave propagation in a fully saturated level-ground sand deposit, where higher frequencies than those for S-waves are of concern. The study focuses on resonance, which has been shown to be of utmost importance for triggering liquefaction due to P-waves alone. The generalised-α scheme performs well, provided that the time-step has been carefully selected. Conversely, the dissipative Newmark method can excessively damp the response, changing radically the computed results. This implies that a computationally prohibiting small time-step would be required for Newmark to provide an accurate solution

A case study of liquefaction: demonstrating the application of an advanced model and understanding the pitfalls of the simplified procedure

“…In all analyses the non-linear solver is based on a modified Newton-Raphson scheme with a substepping stress point algorithm (Potts & Zdravković, 1999), while the generalised α-method of Chung & Hulbert (1993) is used as the time-integration scheme with a spectral radius at infinity, ρ∞, of 0.818 (Chung & Hulbert, 1993;Kontoe, 2006;Kontoe et al, 2008;Han et al 2015a). The suitability of the CH scheme in analyses modelling the higher frequency vertical ground motion has been demonstrated in Tsaparli et al (2017b). A time step of 0.004 s and 0.005 s was found adequate to ensure accuracy when the horizontal component of LPCC and RHSC, respectively, was used as input alone, but this had to be decreased to 0.003 s for vertical motion and bi-directional analyses, due to the wider frequency content of the input motions in the vertical direction.…”

Temporal and null‐space filter for the material point method

“…This approach is shown to be able to damp out the high frequency modes only without causing excessive numerical dissipation for structural engineering . Kontoe has shown that the generalized‐α algorithm is more accurate and has better numerical dissipation characteristics than other dissipated time integration schemes . Moreover, the generalized‐α algorithm can mitigate the spurious high frequency mode but does not affect the lower modes strongly.…”

“…29 Kontoe 30 has shown that the generalized-α algorithm is more accurate and has better numerical dissipation characteristics than other dissipated time integration schemes. 31,32 Moreover, the generalized-α algorithm can mitigate the spurious high frequency mode but does not affect the lower modes strongly. Additionally, it allows controlling the numerical dissipation with a single parameter, which is more appropriate than damping related to the numerical time step size.…”

“…Hence, the mapping in Equation (31) may result in a nonunique solution. In other words, there possibly exists a set of stress vectors null p at material points, which leads to zero internal forces after mapping with Equation (31). Because the balance equations are solved at the grid, the null p in the null-space of ∇S cannot be detected on the background grid and represents the null-space noise.…”

“…The rank of the gradient mapping matrix ∇S, written as rank(∇S), is the set of linearly independent column vector spaces. Equation (31) and Equation (32) give the gradient mapping based on the ∇S and its transpose ∇S T , which transfer the data between material points and grid nodes and vice versa. Therefore, the following four fundamental subspaces are considered:…”