Tutorial 6: Drag anchor¶

Introduction¶

This is the most ambitious tutorial - a 3D dynamic simulation of an AC-14 drag anchor being pulled through a bed of very loose, submerged sand.

The problem combines almost everything in AMPSSIE: the Willam-Warnke sand with depth-dependent stiffness from Tutorial 3, the dynamic time integration and moving boundaries from Tutorial 5, and an articulated anchor built from three points joined by a hinge, an angle stop and a pull line.

This tutorial has four sections:

Problem description
Input setup
Deploying and running the problem
Viewing the results

Problem description¶

You will define the soil, the anchor, the three loading stages, the solver and the output in the input file. All the inputs to the simulation are defined using the input_data.json file format.

Only half of the anchor and soil is modelled, using the vertical \(xz\) plane of symmetry through the anchor at \(y = 0\). Instead of solving the whole length of soil the anchor will pass through, a boundary track keeps only a window of soil around the anchor active.

Figure 1: Schematic of the half-symmetric anchor problem, with the soil block and the AC-14 anchor positioned at the surface.

Figure reproduced from ¹.

Soil block: One layer \(100\) m long, \(10\) m wide and \(18.123\) m deep, with \(2 \times 2 \times 2\) material points in each element. The base element size is given as \(4.0\) m, which is rounded to \(0.1 \times 2^5 = 3.2\) m because the mesh around the anchor is refined to \(0.1\) m. The footprint is then rounded up to whole elements (\(102.4 \times 12.8\) m) and the soil surface up to a whole row of material points, at \(19.2\) m (see how element sizes set the mesh).

Boundary conditions: The side at \(y = 0\) is the symmetry plane and carries rollers, as does the opposite side. The base is fixed. While the soil settles, the two ends are rollers; once the anchor is in place they become fixed planes that move with the anchor.

Material: Willam-Warnke elasto-plastic sand at a relative density of \(D_r = 4\%\), below the water table, with the parameters from the Brinkgreve correlations ²:

Property	Correlation	Value at \(D_r = 4\%\)	Input key
Reference Young's modulus	\(E^{ref} = 60\,000\; D_r/100\) kPa	\(2.4\) MPa	`"E": 2.4e6`
Friction angle	\(\phi = 28 + 12.5\; D_r/100\)	\(28.5^\circ\)	`"friction angle": 28.5`
Dilation angle	\(\psi = -2 + 12.5\; D_r/100\), not below 0	\(0^\circ\)	`"dilation angle": 0.0`
Buoyant unit weight	\(\gamma' = 19 + 1.6\; D_r/100 - \gamma_w\) kN/m\(^3\)	\(9.254\) kN/m\(^3\)	`"density": 925.4`
Stiffness exponent	\(m = 0.7 - D_r/320\)	\(0.6875\)	`"exponent": 0.6875`
Earth pressure at rest	\(K_0 = 1 - \sin\phi\)	\(0.5228\)	`"reference stress"` (below)
Poisson's ratio	-	\(0.25\)	`"nu": 0.25`
Cohesion	-	\(0.3\) kPa	`"cohesion": 300.0`

\(\gamma_w = 9.81\) kN/m\(^3\) is the unit weight of water. Because the sand is submerged, its buoyant density is used, so that gravity produces effective stresses; it is entered as the unit weight divided by \(10\) m/s\(^2\). As in Tutorial 3, the reference stress for the depth-dependent stiffness is \(p^{ref}/K_0 = 100/0.5228 = 191.3\) kPa.

Anchor: The AC-14 anchor (see Figure 2) has two parts, a fluke and a shank, joined by a hinge. Each part is a rigid body welded to a point at its centre of mass. A third point with no geometry is the end of the pull line; moving it drags the anchor. Because only half the anchor is modelled, the masses are halved (the full anchor is \(8700\) kg).

Figure 2: Anchor design: top-down schematic of the AC-14 anchor in (a), the frame abstraction with pivot point, centres of mass and the opening angle in (b), and the extended profile including the pull wire in (c).

Figure reproduced from ³.

Point	Represents	Position as drawn (m)	Mass (kg)	Rotational inertia (kg\(\cdot\)m\(^2\))	Geometry
1	Fluke centre of mass	\((0.62, 0, 0)\)	\(3291.6\)	\(275\)	`fluke_round_nose.stl`
2	Shank centre of mass	\((1.7592, 0, 0)\)	\(1058.4\)	\(337.5\)	`shank_scv2.stl`
3	Pull point	\((12.5294, 0, 0)\)	\(1\)	\(10^{-6}\)	none

Three constraints join the points:

Constraint	Type	Points	Details
Hinge	`"rotation lock"`	1, 2	Pin at \((0.489, 0, 0)\) with its axis along \(y\); pin stiffness \(10^9\) N/m, axis stiffness \(10^8\) N\(\cdot\)m.
Opening-angle stop	`"distance stop"`	1, 2	Goes taut when the hinge has opened \(35^\circ\); stiffness \(8.58 \times 10^9\) N/m.
Pull line	`"distance lock"`	2, 3	A \(9.5\) m rod from the shank's eyelet at \((3.0294, 0, 0)\) to the pull point; stiffness \(EA/L\) with \(EA = 10^{10}\) N.

Frictional contact between the anchor and the sand uses \(\mu = 0.45\).

Loading: Three stages:

Stage 1 - initial stresses: static; the anchor is switched off and gravity is ramped up over 5 load steps.
Stage 2 - settling: dynamic, \(1\) s in steps of \(0.005\) s; the anchor is placed on the soil surface and settles under its own weight with the pull point held still.
Stage 3 - drag: dynamic, in steps of \(0.08\) s; the pull point moves in \(+x\) at \(0.1\) m/s. The tutorial file runs for \(3\) s, a drag of \(0.3\) m; for a longer drag increase "final time" - for example \(190\) s drags the pull point \(19\) m.

Solver: Newton-Raphson, pseudo-static for stage 1 and implicit dynamic for stages 2 and 3.

Input setup¶

The input file is a single JSON object - a human-readable, editable text file. The complete file for this problem can be found here, and every key is described on the input_data.json file format page.

Description

input_data.json

Machine and domain¶

"GPU": "on" runs the analysis on an NVIDIA GPU; a problem of this size is intended for a GPU or an HPC node.

The domain "size" is only a lower bound. The background grid is enlarged automatically to hold the soil block, which here makes it a cube of side \(102.4\) m.

"GPU": "on",

"domain": {
    "size": 2.0,
    "gravity": [0.0, 0.0, -9.81]
}

Material points¶

One layer of Willam-Warnke sand with the properties from the Problem description, over a \(100 \times 10\) m footprint whose \(y = 0\) side is the plane of symmetry.

"material points": {
    "extra capacity": 1.2,
    "element size": 4.0,
    "number of material points per element 1": 2,
    "material size": { "min": [0.0, 0.0], "max": [100, 10] },
    "layers": [
        {
            "thickness": 18.123,
            "material": { "type": "willam warnke",
                          "E": 2.4e6, "nu": 0.25,
                          "friction angle": 28.5, "dilation angle": 0.0, "cohesion": 300.0,
                          "density": 925.4,
                          "E overburden": { "reference stress": 1.913e5, "exponent": 0.6875 } }
        }
    ]
}

Rigid body: points¶

The anchor is one system of three points. Points 1 and 2 are the centres of mass of the fluke and the shank. Their boundary conditions leave them free to move in the \(xz\) plane and to rotate about \(y\); the other three degrees of freedom are fixed by the symmetry.

Point 3, the end of the pull line, carries no geometry and only a nominal mass, so all six of its degrees of freedom are held; stage 3 replaces its \(x\) condition to pull.

The anchor is drawn lying along the \(x\) axis, and "offset" moves the whole system \(1\) m in \(+x\).

"rigid bodies": [
    {
        "offset": [1.0, 0.0, 0.0],
        "points": [
            { "position": [0.62, 0.0, 0.0], "mass": 3291.60, "rotational inertia": [275.0, 275.0, 275.0],
              "boundary conditions": ["free", "fixed", "free", "fixed", "free", "fixed"] },
            { "position": [1.7592, 0.0, 0.0], "mass": 1058.40, "rotational inertia": [337.5, 337.5, 337.5],
              "boundary conditions": ["free", "fixed", "free", "fixed", "free", "fixed"] },
            { "position": [12.5294, 0.0, 0.0], "mass": 1.0, "rotational inertia": [1.0e-6, 1.0e-6, 1.0e-6],
              "boundary conditions": ["fixed", "fixed", "fixed", "fixed", "fixed", "fixed"] }
        ],

Rigid body: constraints¶

The "rotation lock" is the hinge between the fluke and the shank, pinned at \((0.489, 0, 0)\) with both axes along \(y\).
The "distance stop" limits how far the hinge opens. Its two anchors start at the same place, \((0.62, 0, 0)\), one carried by the fluke and one by the shank. As the hinge opens they move apart, and when the opening about the rotation centre reaches \(35^\circ\) the stop goes taut.
The "distance lock" is the pull line: a \(9.5\) m rod from the shank's eyelet at \((3.0294, 0, 0)\) to the pull point at \((12.5294, 0, 0)\).

        "constraints": [
            { "type": "rotation lock", "points": [1, 2],
              "rotation centre": [0.489, 0.0, 0.0],
              "direction 1": [0.0, 1.0, 0.0],
              "direction 2": [0.0, 1.0, 0.0],
              "stiffness": 1.0e9, "rotational stiffness": 1.0e8 },

            { "type": "distance stop", "points": [1, 2],
              "rotation centre": [0.489, 0.0, 0.0],
              "position 1": [0.62, 0.0, 0.0],
              "position 2": [0.62, 0.0, 0.0],
              "angle limit": 35, "stiffness": 8.5814e9 },

            { "type": "distance lock", "points": [2, 3],
              "position 1": [3.0294, 0.0, 0.0],
              "position 2": [12.5294, 0.0, 0.0],
              "distance": 9.5, "stiffness": 1.0526e9 }
        ],

Rigid body: geometry¶

The fluke surface is welded to point 1 and the shank surface to point 2. Both STL files are drawn in global coordinates, in the same frame as the point positions, and their tetrahedral meshes are stored in the mesh-cache files.

        "stl files": [
            { "name": "fluke", "stl": "fluke_round_nose.stl",
              "mesh cache": "fluke_round_nose_mesh.txt", "point": 1 },
            { "name": "shank", "stl": "shank_scv2.stl",
              "mesh cache": "shank_scv2_mesh.txt", "point": 2 }
        ]
    }
]

Contact¶

Coulomb friction with \(\mu = 0.45\) between the anchor and the sand.

"contact": {
    "friction coefficient": 0.45
}

Stage 1: initial stresses¶

A static stage with the anchor switched off. Gravity is ramped from zero over 5 load steps on a uniform \(3.2\) m mesh, with rollers on the four vertical faces and a fixed base. The top face lies in the empty space above the soil.

{
    "type": "static",
    "load steps": 5,
    "load": "gravity ramp",
    "rigid bodies": "off",
    "adaptivity": { "type": "uniform" },
    "boundary conditions": { "min": ["roller", "roller", "fixed"], "max": ["roller", "roller", "fixed"] }
}

Stage 2: settling¶

A dynamic stage of \(1\) s in steps of \(0.005\) s, in which the anchor settles into the sand under its own weight while the pull point stays where it is. "rigid body surface placement": "on" first moves the anchor vertically so that it rests on the soil surface. \(\beta = 0.5\) and \(\gamma = 1.0\) add numerical damping.

The mesh is refined to \(0.1\) m wherever the fluke and shank surfaces are. "boundary track" replaces the two \(x\) ends with fixed planes that follow the anchor: one \(0.5\) m behind it, which deletes the soil the anchor leaves behind, and one \(2.5\) m ahead of it. The soil beyond the leading plane stays inactive until the plane reaches it.

{
    "type": "dynamic",
    "dt": 5.0e-3,
    "final time": 1.0,
    "beta": 0.5,
    "gamma": 1.0,
    "rigid bodies": "on",
    "rigid body surface placement": "on",
    "adaptivity": {
        "type": "rigid body surface",
        "element size": 0.1
    },
    "boundary conditions": { "min": ["fixed", "roller", "fixed"], "max": ["roller", "roller", "fixed"] },
    "boundary track": {
        "min": { "type": ["fixed", "off", "off"], "distance": [0.5, 0.0, 0.0],
                 "advance": ["positive", "positive", "positive"], "remove": ["yes", "no", "no"] },
        "max": { "type": ["fixed", "off", "off"], "distance": [2.5, 0.0, 0.0],
                 "advance": ["positive", "positive", "positive"], "remove": ["no", "no", "no"] }
    }
}

Stage 3: drag¶

The drag itself. "rigid body boundary conditions" replaces the conditions of the pull point (system 1, point 3) for this stage, prescribing a movement of \(0.008\) m in \(+x\) every \(0.08\) s time step - \(0.1\) m/s. The anchor is already in place, so "rigid body surface placement" is "off", and the leading plane of the boundary track moves \(5\) m ahead of the anchor.

"final time": 3.0 drags the pull point \(0.3\) m. Increase it for a longer drag.

{
    "type": "dynamic",
    "dt": 8.0e-2,
    "final time": 3.0,
    "beta": 0.5,
    "gamma": 1.0,
    "rigid bodies": "on",
    "rigid body surface placement": "off",
    "adaptivity": {
        "type": "rigid body surface",
        "element size": 0.1
    },
    "rigid body boundary conditions": [
        { "system": 1, "point": 3, "boundary conditions": [{ "prescribed": 0.008 }, "fixed", "fixed", "fixed", "fixed", "fixed"] }
    ],
    "boundary conditions": { "min": ["fixed", "roller", "fixed"], "max": ["roller", "roller", "fixed"] },
    "boundary track": {
        "min": { "type": ["fixed", "off", "off"], "distance": [0.5, 0.0, 0.0],
                 "advance": ["positive", "positive", "positive"], "remove": ["yes", "no", "no"] },
        "max": { "type": ["fixed", "off", "off"], "distance": [5.0, 0.0, 0.0],
                 "advance": ["positive", "positive", "positive"], "remove": ["no", "no", "no"] }
    }
}

Solver¶

Newton-Raphson iterations to a tolerance of \(10^{-6}\) in each step, with at most 20 iterations before the step is retried with half the increment, and the same ghost stabilisation settings as Tutorial 3.

"solver": {
    "tolerance": 1.0e-6,
    "max newton iterations": 20,
    "poor factor": 0.25,
    "ghost factor": 0.025,
    "ghost factor mass": 0.25
}

Output¶

VTK files are written to vtk_AC14 and CSV files to csv_AC14, each every \(0.25\%\) of a stage's steps. rigid_body.csv records the position, velocity, reaction force and reaction moment of all three points, which is what the anchor's trajectory and holding force are calculated from.

"output": {
    "vtk": "on",
    "vtk directory": "vtk_AC14",
    "vtk percent": 0.25,
    "vtk material point fields": ["displacement", "velocity", "stress", "strain", "volume"],
    "vtk rigid body fields": ["position", "velocity", "reaction force"],

    "csv": "on",
    "csv directory": "csv_AC14",
    "csv percent": 0.25,
    "csv material point fields": ["initial position", "position", "stress"],
    "csv rigid body fields": ["position", "velocity", "reaction force", "reaction moment"]
}

Deploying and running the problem¶

AMPSSIE is written in the Julia programming language. See the installation guide for installing Julia and AMPSSIE, and Tutorial 1 for a first, small run.

Deployment instructions

terminal

Setting up the run folder¶

Create a folder for the run containing:

input_data.json, copied from the complete input file;
fluke_round_nose.stl, fluke_round_nose_mesh.txt, shank_scv2.stl and shank_scv2_mesh.txt, from the top level of the AMPSSIE repository.

The STL and mesh-cache paths in the input file are relative to the input file, and the output folders vtk_AC14 and csv_AC14 are created in the folder Julia is started from.

Running the problem¶

Start Julia in the run folder with the AMPSSIE project active (--project), then load AMPSSIE and run the input file. "GPU": "on" requires an NVIDIA GPU with CUDA; set it to "off" to run on the CPU, which will be slow for this problem.

The progress line counts through stage 1/3, stage 2/3 and stage 3/3. Its fields are explained in Tutorial 1.

$ cd path/to/run_folder
$ julia --project=path/to/AMPSSIE -t auto

julia> using S3MPM

julia> S3MPM.non_linear_solve("input_data.json");

Viewing the results¶

Visualising the output in ParaView¶

The walkthrough below uses the same ParaView controls as Tutorial 1. Only active material points are written, so once the anchor is on the soil the output shows the window of soil around it rather than the whole block.

1. Open the soil and the anchor. File → Open, go to vtk_AC14, hold Ctrl and select the mps_1_..vtu (soil) and surface_..vtu (fluke and shank) series, click OK and then Apply.