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Geometry and Material Definition

A simple tetrahedral finite element mesh of a cube shown in fig-schematic with side length \(a = 1.0\) mm and resolution \(r = 0.1\) mm as default is generated. The cube is assigned a neo-Hookean material model with shear modulus \(\mu = 10\) kPa and penalty parameter \(K = 1000\) kPa.

Boundary Condition Application

For the experiments I & II, movement of nodes located at \(x = -0.5 a\) is restricted in all directions, i.e., homogeneous Dirichlet boundary conditions are applied (see left and right panels in fig-schematic).

Experiment I - Dirichlet Boundary Conditions

Experiment I provides guidance on the consistent application of inhomogeneous Dirichlet boundary conditions (see left panel in fig-schematic). Nodes located at \(x = 0.5 a\) are translated under application of displacement \(\mathbf{u} = [u_{1},0,0]^{\mathrm{T}}\) utilizing the command line

./run.py
  --experiment dirichlet --sidelength 1.00 --resolution 0.10 --magnitude 0.10

where the parameter experiment selects the correct boundary condition type, parameters sidelength and resolution provide an optional input for the mesh generation and magnitude defines the displacement \(u_{1}^{\mathrm{s}}\) in mm. The application of inhomogeneous Dirichlet boundary conditions is time-dependent and necessitates existence of a specific trace file (.trc), in which the time course for the selected displacement is provided calculated as \(\mathbf{u} = \phi \mathbf{u}^{\mathrm{s}} = \phi [u_{1}^{\mathrm{s}},u_{2}^{\mathrm{s}},u_{3}^{\mathrm{s}}]^{\mathrm{T}}\), where \(\phi\) represents the scale factor (see fig-traces).

Experiment II - Neumann Boundary Conditions

Experiment II provides guidance on the consistent application of Neumann boundary conditions (see right panel in fig-schematic). Nodes located at \(x = 0.5 a\) are translated under application of pressure \(p\) on the corresponding surface triangles utilizing the command line

./run.py
  --experiment neumann --sidelength 1.00 --resolution 0.10 --magnitude 5.00

where the parameter experiment selects the correct boundary condition type, parameters sidelength and resolution provide an optional input for the mesh generation and magnitude defines the applied pressure \(p\) in kPa. Again, the application of Neumann boundary conditions is time-dependent and necessitates existence of a specific trace file (.trc), in which the time course for the selected pressure is provided calculated as \(p = \phi p\), where \(\phi\) represents the scale factor (see fig-traces).

Experimental Results

Upon application of the command lines introduced above, results provided in fig-results will be obtained. While in Experiment I inhomogeneous Dirichlet boundary conditions are consistently stretching the cube, Neumann boundary conditions in Experiment II administer both, compression and tension on the cube (compare boundary condition application traces in fig-traces).

For the experiments III & IV, the plane at \(x = -0.5 a\) is attached to a spring acting in all directions, i.e., homogeneous Robin boundary conditions are applied.

Experiment III - Dirichlet Boundary Conditions

Experiment III provides guidance on the consistent application of inhomogeneous Dirichlet boundary conditions (see left panel in fig-schematic). Nodes located at \(x = 0.5 a\) are translated under application of displacement \(\mathbf{u} = [u_{1},0,0]^{\mathrm{T}}\) utilizing the command line

./run.py
  --experiment dirichlet --fixation weak --sidelength 1.00 --resolution 0.10 --magnitude
  0.10

where the parameter experiment selects the correct boundary condition type, parameters sidelength and resolution provide an optional input for the mesh generation and magnitude defines the displacement \(u_{1}^{\mathrm{s}}\) in mm. The application of inhomogeneous Dirichlet boundary conditions is time-dependent and necessitates existence of a specific trace file (.trc), in which the time course for the selected displacement is provided calculated as \(\mathbf{u} = \phi \mathbf{u}^{\mathrm{s}} = \phi [u_{1}^{\mathrm{s}},u_{2}^{\mathrm{s}},u_{3}^{\mathrm{s}}]^{\mathrm{T}}\), where \(\phi\) represents the scale factor (see fig-traces).

Experiment IV - Neumann Boundary Conditions

Experiment IV provides guidance on the consistent application of Neumann boundary conditions (see right panel in fig-schematic). Nodes located at \(x = 0.5 a\) are translated under application of pressure \(p\) on the corresponding surface triangles utilizing the command line

./run.py
  --experiment neumann --fixation weak --sidelength 1.00 --resolution 0.10 --magnitude
  5.00

where the parameter experiment selects the correct boundary condition type, parameters sidelength and resolution provide an optional input for the mesh generation and magnitude defines the applied pressure \(p\) in kPa. Again, the application of Neumann boundary conditions is time-dependent and necessitates existence of a specific trace file (.trc), in which the time course for the selected pressure is provided calculated as \(p = \phi p\), where \(\phi\) represents the scale factor (see fig-traces).

Experimental Results

Upon application of the command lines introduced above, results provided in fig-results2 will be obtained. While in Experiment III inhomogeneous Dirichlet boundary conditions are consistently stretching the cube, Neumann boundary conditions in Experiment IV administer both, compression and tension on the cube (compare boundary condition application traces in fig-traces).

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