Computing Line Integrals
The Ray Tracing Module can be utilized to integrate fields along a line throughout the domain. Example fields include Variables, AuxVariables, Material properties, and Functions.
The discussion that follows will describe how to integrate a variable across a line in a simple diffusion problem. To integrate other fields, the process remains the same except for the definition of the RayKernel.
Problem Definition
We begin with the standard "simple diffusion" problem:
[Mesh]
[gmg]
type = GeneratedMeshGenerator
dim = 2
nx = 5
ny = 5
xmax = 5
ymax = 5
[]
[]
[Variables/u]
[]
[Kernels/diff]
type = Diffusion
variable = u
[]
[BCs]
[left]
type = DirichletBC
variable = u
boundary = left
value = 0
[]
[right]
type = DirichletBC
variable = u
boundary = right
value = 1
[]
[]
[Executioner]
type = Steady
solve_type = 'PJFNK'
petsc_options_iname = '-pc_type -pc_hypre_type'
petsc_options_value = 'hypre boomeramg'
[]
For this problem, we seek the value of the integrals (where is the finite-element solution)
and
in which we will denote the first line, , as diag and the second, , as right_up for simplicity.
Note that the integral along the second line, , is trivial due to the Dirichlet boundary condition,
which implies
Defining the Study
A RepeatableRayStudy is defined that generates and executes the rays that compute the variable line integral:
[UserObjects/study]
type = RepeatableRayStudy
names = 'diag
right_up'
start_points = '0 0 0
5 0 0'
end_points = '5 5 0
5 5 0'
execute_on = TIMESTEP_END
# Needed to cache trace information for RayTracingMeshOutput
# always_cache_traces = true
# Needed to cache Ray data for RayTracingMeshOutput
# data_on_cache_traces = true
[]
The study object defines two rays to be exectued on TIMESTEP_END:
diagfrom toright_upfrom to
Defining the RayKernel
RayKernels are objects that are executed on the segments of the rays. In this case, we wish to compute the integral of a variable so we will define a VariableIntegralRayKernel:
[RayKernels/u_integral]
type = VariableIntegralRayKernel
variable = u
[]
The u_integral VariableIntegralRayKernel will accumulate the variable line integral of the u Variable for our defined rays, diag and right_up.
Other commonly used IntegralRayKernels are the FunctionIntegralRayKernel and the MaterialIntegralRayKernel.
Integral Output
Lastly, we need to obtain the accumulated integrals from the study. We will utilize a RayIntegralValue Postprocessor to achieve this:
[Postprocessors]
[diag_line_integral]
type = RayIntegralValue
ray_kernel = u_integral
ray = diag
[]
[right_up_line_integral]
type = RayIntegralValue
ray_kernel = u_integral
ray = right_up
[]
[]
The accumulated integrals are seen in output:
Postprocessor Values:
+----------------+--------------------+------------------------+
| time | diag_line_integral | right_up_line_integral |
+----------------+--------------------+------------------------+
| 0.000000e+00 | 0.000000e+00 | 0.000000e+00 |
| 1.000000e+00 | 3.535534e+00 | 5.000000e+00 |
+----------------+--------------------+------------------------+
Segment-wise Integral Output
The segment-wise accumulated integral can also be outputted in a mesh format using the RayTracingMeshOutput. For more information, see Example.