Advanced FEM for free surface flow with application to landslides

Abstract

An advanced space-time finite element method is presented to investigate movements of landslides and their interaction with flexible structures. The mechanics of liquefied soil is described by Navier-Stokes-equations for visco-plastic non-newtonian fluid. Likewise the fluid the kinematics of the structure is described by velocities, taking large rotations into account. This leads to a monolithic fluid-structure interaction approach considering the multi-field problem as a whole. The discretized model equations are assembled in a single set of algebraic equations, which are solved by applying Newton-Raphson scheme. Free surface motion of landslide is described by the level-set method. To reduce computational effort the fragmented finite element method is used, where only active finite elements are evaluated. A pde-based extrapolation of the velocity-field is applied to ensure an accurate transport of distance function, which defines the profile in space and time of the free surfaces. PhenomenologyThe ground moves a few millimeters or even centimeters each year. But sometimes the ground moves meters or even kilometers at a time. This is called landslide, mudslide or avalanche. During a landslide sediment, rock and debits move down-slope by the influence of gravity. They can be natural or man-made, usually a landslide requires a trigger mechanism like earthquake, rain, or just time. Depending on time and place a landslide may affect lives, dwellings, goods and chattels and also natural habitat of animals and plants. Governing EquationsThe state of motion is characterized by displacements u and velocities v =u. The non-linear Green-Lagrangian strain rate tensorĖ = sym F TḞ describes the rate of deformation, where F = I + ∇ 0 u denotes the deformation gradient. The balance equation of momentum is given bywhere ρ 0 indicates structural density and f 0 the body forces. An linear elastic material behaviour is assumed, which is described by the second Piola-Kirchhoff stress tensor S, the Green-Lagrangian strain tensor and the tensor C of elasticity. C is deduced from the strain-energy function Ψ of the St. Venant-Kirchhoff modelThe motion of viscous and compressible fluids is governed by the Navier-Stokes equations ∂ρv ∂tPressure can rise to infinity at large volumetric strain ε v = ρ0 ρ − 1 ensured by the state equation p = −K ln (1 − ε v ), in terms of bulk modulus K. The flow characteristics of the granular material is modelled by a Bingham fluid [1,6]. Hence the deviator stress is given bywith the deviatoric part of the strain rate tensor D = sym (∇v) .The model parameters angle of friction ϕ and cohesion c characterize the limit state between material at rest and at flow. The regularisation parameter 1 ensures the evaluation of stress states for disappearing strain rates. Fluid viscosity is given by η. Between fluid and structure the shear stress t = −t n µ 1 − e −α∆v is taken into account, which depends on normal stress t n , sliding friction coefficient µ, regulation parameter α and relative velocity ∆v . Since a...