# Step 5: Time-Dependent Shear Displacement and Tractions
% Meatadata extracted from parameter files
```{include} step05_sheardisptractrate-synopsis.md
```
## Simulation parameters
In this example we build on Step 3 and make the Dirichlet (displacement) and Neumann (traction) boundary conditions a bit more complicated by adding variation in time.
The simulation has a duration of 5 years with a time step of 1 year.
The time-dependent boundary conditions use the same initial amplitude values for the first time step before adding in a constant rate increase at a time of 1 year.
{numref}`fig:example:box:3d:step05:diagram` shows the boundary conditions on the domain.
The parameters specific to this example are in `step05_sheardisptractrate.cfg`.
:::{figure-md} fig:example:box:3d:step05:diagram
Boundary conditions for shear deformation.
We constrain the x and y displacements on the +x and -x boundaries.
We apply tangential (shear) tractions on the +y and -y boundaries.
At a time of 1 year we increase the amplitude at a constrant rate $b$ ($H(t)$ corresponds to the heavyside step function).
:::
This is a time-dependent problem, so we must specify the start and end times of the simulation along with the initial time step.
With an initial time step of 1 year, we start the simulation at -1 year so that the first solve will advance the simulation to a time of 0.
We also specify a relaxation time on the order of the time scale of the simulation to allow for reasonable nondimensionalization of time.
```{code-block} cfg
---
caption: Time stepping parameters for Step 5.
---
[pylithapp.problem]
initial_dt = 1.0*year
start_time = -1.0*year
end_time = 5.0*year
[pylithapp.problem.scales]
relaxation_time = 10.0*year
```
For the time-dependent Dirichlet and Neumann boundary conditions, we specify both the initial displacement and a constant rate; the constant rate begins at t=1 year.
```{code-block} cfg
---
caption: Time-dependent boundary conditions for Step 5. We show the details for the -x and -y boundaries.
---
bc = [bc_xneg, bc_xpos, bc_yneg, bc_ypos, bc_zneg]
bc.bc_xneg = pylith.bc.DirichletTimeDependent
bc.bc_xpos = pylith.bc.DirichletTimeDependent
bc.bc_yneg = pylith.bc.NeumannTimeDependent
bc.bc_ypos = pylith.bc.NeumannTimeDependent
bc.bc_zneg = pylith.bc.DirichletTimeDependent
[pylithapp.problem.bc.bc_xneg]
label = boundary_xneg
label_value = 10
constrained_dof = [0, 1]
use_initial = True
use_rate = True
db_auxiliary_field = spatialdata.spatialdb.SimpleDB
db_auxiliary_field.description = Dirichlet BC -x edge
db_auxiliary_field.iohandler.filename = sheardisprate_bc_xneg.spatialdb
db_auxiliary_field.query_type = linear
[pylithapp.problem.bc.bc_yneg]
label = boundary_yneg
label_value = 12
use_initial = True
use_rate = True
db_auxiliary_field = spatialdata.spatialdb.UniformDB
db_auxiliary_field.description = Neumann BC +x edge
db_auxiliary_field.values = [initial_amplitude_tangential_1, initial_amplitude_tangential_2, initial_amplitude_normal, rate_start_time, rate_amplitude_tangential_1, rate_amplitude_tangential_2, rate_amplitude_normal]
db_auxiliary_field.data = [-9.0*MPa, 0.0*MPa, 0.0*MPa, 1.0*year, -2.25*MPa/year, 0.0*MPa/year, 0.0*MPa/year]
```
## Running the simulation
```{code-block} console
---
caption: Run Step 5 simulation
---
$ pylith step05_sheardisptractrate.cfg
# The output should look something like the following.
>> /Users/baagaard/software/unix/micromamba/envs/pylith-debug/lib/python3.12/site-packages/pylith/apps/PyLithApp.py:79:main
-- info (application-flow)
-- Running on 1 process(es).
# -- many lines omitted --
>> /Users/baagaard/src/cig/pylith/libsrc/pylith/problems/TimeDependent.cc:473:void pylith::problems::TimeDependent::solve()
-- info (application-flow)
-- Component 'timedependent.problem': Solving equations.
0 TS dt 0.1 time -0.1
0 SNES Function norm 8.562738880728e-01
Linear solve converged due to CONVERGED_ATOL iterations 4
1 SNES Function norm 3.584100837616e-09
Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
# -- many lines omitted --
5 TS dt 0.1 time 0.4
0 SNES Function norm 2.140684720394e-01
Linear solve converged due to CONVERGED_ATOL iterations 0
1 SNES Function norm 4.123676062187e-10
Nonlinear solve converged due to CONVERGED_FNORM_ABS iterations 1
6 TS dt 0.1 time 0.5
>> /Users/baagaard/software/unix/micromamba/envs/pylith-debug/lib/python3.12/site-packages/pylith/problems/Problem.py:222:finalize
-- info (application-flow)
-- Finalizing problem.
```
The output written to the terminal now contains multiple time steps.
The PETSc TS (time stepping) monitor shows the time step and time in nondimensional units.
## Visualizing the results
In {numref}`fig:example:box:3d:step05:solution` we use the `pylith_viz` utility to visualize the x displacement field.
```{code-block} console
---
caption: Visualize PyLith output using `pylith_viz`.
---
pylith_viz --filenames=output/step05_sheardisptractrate-domain.h5 warp_grid --component=x
```
You can move the slider or use the `p` and `n` keys to change the increment or decrement time.
:::{figure-md} fig:example:box:3d:step05:solution
Solution for Step 5 at a time of 5.0 years.
The colors of the shaded surface indicate the x displacement, and the deformation is exaggerated by a factor of 1000.
The undeformed configuration is shown by the gray wireframe.
:::