Mechanical Boundary Conditions

Module: tutorials.04_EM_tissue.01_boundary_conditions.run

Section author: Thomas Fastl <thomas.fastl@kcl.ac.uk> and Christoph Augustin <christoph.augustin@medunigraz.at> andMatthias Gsell <matthias.gsell@medunigraz.at>

This tutorial demonstrates the time-dependent application of different boundary condition types, i.e., Dirichlet and Neumann boundary conditions, on a cuboidal geometry. For more details on the definition of mechanical boundary conditions in carpentry see manual section mechanical boundary conditions.

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.

../../_images/MechanicalBoundaryConditions_schematics.png

Schematic representation of the cuboidal geometry including Dirichlet (left panel) and Neumann (right panel) boundary conditions subsequently introduced in Experiment I and Experiment II, respectively.

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).

Note

Node selection for application of Dirichlet boundary conditions is currently based on internal functions, but can alternatively be performed utilizing a vertex file (.vtx).

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).

Note

Element selection for application of Neumann boundary conditions is currently based on internal fuctions, but can alternatively be performed utilizing a surface element file (.neubc).

../../_images/MechanicalBoundaryConditions_traces.png

Dirichlet (blue) and Neumann (red) boundary condition application traces indicating the time-dependent application using a scale factor \phi during simulation specified in corresponding trace files (.trc).

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).

../../_images/MechanicalBoundaryConditions_results.gif

Results obtained upon application of command lines introduced in Experiment I using Dirichlet boundary conditions (left panel) and Experiment II using Neumann boundary conditions (right panel) with the color-code representing displacement in 1-direction (top row) and 1^{\mathrm{st}} principal stress (bottom row).

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).

Note

Node selection for application of Dirichlet boundary conditions is currently based on internal functions, but can alternatively be performed utilizing a vertex file (.vtx).

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).

Note

Element selection for application of Neumann boundary conditions is currently based on internal fuctions, but can alternatively be performed utilizing a surface element file (.neubc).

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).

../../_images/MechanicalBoundaryConditions2_results.gif

Results obtained upon application of command lines introduced in Experiment III using Dirichlet boundary conditions (left panel) and Experiment IV using Neumann boundary conditions (right panel) with the color-code representing displacement in 1-direction (top row) and 1^{\mathrm{st}} principal stress (bottom row).