Large deformation modelling in geomechanics
MPM and LBM-DEM
Krishna Kumar, kks32@cam.ac.uk
University of Cambridge
The University of Texas at Austin, USA
5th February 2018
Cambridge-Berkeley Computational Geomechanics
- Material Point Method
- Lattice-Boltzmann + Discrete Element Method
- Finite Element Method - Thermo-Hydro Mechanical Coupling
- Lattice Element Method
Lattice Element Method
The LEM code is capable of modelling 30 million beam elements.
LEM fracturing
Wong et al., (2016)
Agent Based Modelling of cities
Soga et al., (2017)
Network analysis of Herpes Simplex Virus (HSV2)
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Global landslide hazard
Oso landslide (2014)
Multiscale modelling in geomechanics
Discrete Element Method
- Particle level interaction based on Newton's equation of motion
- The contact force is computed as:
- The Newton's equation of motion
$F_n=\left\{ \begin{matrix} \text{ }0\text{ },\text{ }{{\delta }_{n}}>0 \\
-{{k}_{n}}{{\delta }_{n}}-{{\gamma }_{n}}\frac{d{{\delta }_{n}}}{dt},\text{ }{{\delta }_{n}}<0 \\
\end{matrix} \right.$
$F_n =m \times a $
Mesh-based vs Mesh-free techniques
Material Point Method
Material Point Method
Granular column collapse
Experimental results (Lube et al 2005)
Micro to Macro
MPM v DEM column collapse
MPM v DEM column collapse
DEM column collapse
MPM v DEM column collapse
Collisional dissipation mechanism is missing in the continuum approach.
MPM slope failure
Horizontal velocity (m/s)
MPM v DEM uniform impact (200 J)
MPM v DEM runout slope v collapse
MPM slope failure: pore pressure changes
Selborne case study of a 9 m high cut-slope slope (Soga et al., 2016)
Possible boundary conditions of submarine run‐out
- Presence of ambient water (larger drag force & less gravity).
- Water entrainment.
- Pore pressure does not dissipate.
Submarine landslides
MPM submarine landslide
Depth-averaged Material Point Method (Taka et al., 2012)
Mechanism of submarine landslides
Modelling Test at 1g Condition
- Material type influences the mode of the flow.
- Target: Clay‐rich flow (Less diffusive, Hydroplaning).
Mechanism of submarine runout
MPM submarine landslide: Water entrainment
Run-out for different water entrainment (Taka et al., 2012)
LBM - DEM simulation of granular collapse in a fluid
aspect ratio 'a' of 6
Lattice Boltzmann - MRT
\[f_{i}(x + dx, t +\Delta t) - f_{i}(x, t) = -S_{\alpha i}(
f_{i}(x, t) - f_{i} ^ {eq}(x, t))\]
- $S_{\alpha i}$ is the collisional matrix.
- Probability density of finding a particle : $f(x,\varepsilon, t) $, where, x is position, $\varepsilon$ is velocity, and t is time.
LBM-DEM fluid-solid coupling
$$\Delta t_{s}=\frac{\Delta t}{\mathit{n}_{s}} \qquad (\mathit{n}_{s}=[\Delta t/ \Delta t_{D}]+1) $$
- At every fluid iteration, $\mathit{n}_{s}$ sub-steps of DEM iterations are performed using the time step $\Delta t_{s}$.
- The hydrodynamic force is unchanged during the sub-cycling.
LBM laminar & turbulent flows
Smagorinsky model (LES):
Karman Vortex Street
Collapse in a fluid
Granular collapse in a fluid: Effect of aspect ratio
aspect ratio 'a' of 0.4
aspect ratio 'a' of 4
Collapse in a fluid: Runout evolution
Critical time $\tau_c=\sqrt{H/g}$ (Staron and Hinch, 2005)
where, H = Height of the granular pile.
LBM - DEM simulation of granular collapse in a fluid
aspect ratio 'a' of 8
Runout: dry vs fluid
Dry collapse flowed further than the underwater collapse
Collapse in a fluid: Effect of permeability
Dirichlet boundary conditions constrain the pressure/density at the boundaries (Zou and He, 1997)
$\rho_0=\sum_{a}f_{a} \mbox{ and } \textbf{u}=\frac{1}{\rho_0}\sum_{a}f_{a}$
Reduction in radius
Collapse in a fluid: Effect of permeability
Reduction ‘r’=0.7R (High permeability)
Reduction ‘r’=0.9R (Low permeability)
Effect of permeability: runout
Effect of permeability: runout
Effect of permeability: kinetic energy
Effect of permeability: runout
Collapse in a fluid: Effect of permeability
Hydrodynamic force (x-dir)
Low permeability condition has large fluctuations in hydrodynamic forces.
Collapse in a fluid: Effect of permeability
Hydrodynamic force (x-dir)
250 - particle at the bottom of the flow;
872 - particle at middle of the flow; 1007 - particle at the surface of the flow
Effect of permeability: stress
Collapse in a fluid: Effect of permeability
Effect of permeability: pore-water pressure
Effect of permeability: effective stress
Low permeability - no effective stresses at the flow front (hydroplaning)
Effect of permeability: drag vs hydroplaning
High permeable flow front experiences drag, while low permeable flow experiences hydroplaning.
Effect of permeability: runout (loose)
Collapse on an inclined plane
aspect ratio 'a' of 6 on a slope of 5*
Collapse of a dense column on an inclined plane
Collapse of a dense column on an inclined plane
Collapse of a dense column on slopes: runout
Collapse of a loose column on slopes: runout
Loose v dense: Initiation phase
Loose v dense: Initiation phase
Pore-pressure distribution along the failure plane during initiation.
Loose v dense: Runout phase
Loose v dense: Runout phase
Water entrainment front (~15d length) at a slope of 5*
Loose v dense: Runout phase
Collapse on slopes: loose v dense
Kinetic energy evolution
Loose v dense: Settlement phase
Collapse on slopes: loose v dense
LBM-DEM Multi-GPU implementation
LBM - DEM a = 0.8 & 10,000 particles
- LBM Nodes = 50 Million : DEM grains = 10000 discs
- Run-time = 4 hours
- Speedup = 125x on a Pascal P100
2D to 3D
