A unified Lagrangian formulation for solid and fluid dynamics and its possibility for modelling submarine landslides and their consequences

Abstract

Consequences of submarine landslides include both their direct impact on offshore infrastructure, such as subsea electric cables and gas/oil pipelines, and their indirect impact via the generated tsunami. The simulation of submarine landslides and their consequences has been a long-standing challenge majorly due to the strong coupling among sliding sediments, seawater and infrastructure as well as the induced extreme material deformation during the complete process. In this paper, we propose a unified finite element formulation for solid and fluid dynamics based on a generalised Hellinger-Reissner variational principle so that the coupling of fluid and solid to be achieved naturally in a monolithic fashion. In order to tackle extreme deformation problems, the resulting formulation is implemented within the framework of the particle finite element method. The correctness and robustness of the proposed unified formulation for single-phase problems (e.g. fluid dynamics problems involving Newtonian/Non-Newtonian flows and solid dynamics problems) as well as for multi-phase problems (e.g. two-phase flows) are verified against benchmarks. Comparisons are carried out against numerical and analytical solutions or experimental data that are available in literatures. Last but not least, the possibility of the proposed approach for modelling submarine landslides and their consequences is demonstrated via a numerical experiment of an underwater slope stability problem. It is shown that the failure and postfailure process of the underwater slope can be predicted in a single simulation with its direct 2 threat to a nearby pipeline and indirect threat by generating tsunami being estimated as well.

A case study and implication: particle finite element modelling of the 2010 Saint-Jude sensitive clay landslide

“…A key feature of the PFEM lies in its inheritance not only of the capability of a particle approach for handling general large deformation problems but also of the solid mathematical foundation of the FEM. To date, the PFEM has been used successfully for modelling landslides and their relevant large deformation problems such as the granular flow (Dávalos et al 2015;Zhang et al 2014;Cante et al 2014;Zhang et al 2016), the debris flow and its interaction with barriers (Franci and Zhang 2018), a flow-like landslide (Zhang et al 2015), landslide-generated waves (Salazar et al 2016;Cremonesi et al 2011), fluid-soil-structure interactions (Oñate et al 2011), submarine landslides and their impact to nearby subsea pipelines (Zhang et al 2019), just to mention a few. The possibility of the used PFEM for modelling retrogressive landslides has been explored in (Zhang et al 2017).…”

Mathematical Optimization Problems for Particle Finite Element Analysis Applied to 2D Landslide Modeling

“…Indeed, efforts have been devoted to the analysis of the convergence properties of the utilized IPM for solving nonlinear optimization problems; For example, the global and local convergence properties of the IPM for nonlinear programming were demonstrated by Tits et al (2003), while its stability and convergence rates were analyzed mathematically by Alizadeh et al (1998). Recently, the possibility of using the optimization-based PFEM for simulating challenging landslide problems has been explored through modeling a flow-like landslide (Zhang et al 2015), progressive failure of landslides in sensitive clays (Zhang et al 2017), as well as submarine landslides and their consequences such as induced tsunamis and the impact on ocean pipelines (Zhang et al 2019).…”

Reply to Referee #1

“…We will improve the texts and figures in the revision. As for the accuracy, the whole computational configuration is similar to the experiment of the collapse of aluminum bars [ 9,10].…”

“…SPH [8] and MPM [2]. The whole computational configuration is similar to the experiment of the collapse of aluminum bars [9,10]. As for the question on remeshing and mapping, Figure 5 in the literature [10] (concerning the dam break problem) shows that the frequency of mapping doesn't play a significant role in such case.…”