Internal Volume

Overview

An important feature in a fuel performance code is the ability to compute the internal or void volume enclosed by a finite element mesh. This is necessary for computing the plenum/gap volume, for example.

The procedure is as follows. $var element = document.getElementById("moose-equation-7b28f289-bf89-4d6c-834a-c03835763de6");katex.render("\\int_V \\text{div}(F) \\text{d}V = \\int_{dS} F \\cdot \\hat{n} \\text{d}S", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-8970ec02-b676-47f7-ba25-4721242825dc");katex.render("F", element, {displayMode:false,throwOnError:false});$ is a field, $var element = document.getElementById("moose-equation-900e3650-6bd3-4c6d-8f09-4f98b5e4b778");katex.render("\\hat{n}", element, {displayMode:false,throwOnError:false});$ is the normal of the surface ( $var element = document.getElementById("moose-equation-49307701-d86b-42a0-8c75-9686f1c4198b");katex.render("[n_x, n_y, n_z]^T", element, {displayMode:false,throwOnError:false});$ ), $var element = document.getElementById("moose-equation-790b2668-0535-4f72-8471-f94521a1d721");katex.render("V", element, {displayMode:false,throwOnError:false});$ is the volume of the domain, and $var element = document.getElementById("moose-equation-f30c2cc1-e4f7-4e8e-8a48-09b29a39319b");katex.render("S", element, {displayMode:false,throwOnError:false});$ is the surface of the domain.

We choose $var element = document.getElementById("moose-equation-12de8912-669a-4618-8704-d54cfc5a74d4");katex.render("F = \\left[ x, 0, 0 \\right]^T", element, {displayMode:true,throwOnError:false});$ giving $var element = document.getElementById("moose-equation-ff643e06-492e-4c42-ad6a-71ef4c94e694");katex.render("\\text{div}(F) = 1", element, {displayMode:true,throwOnError:false});$

The integral becomes

$var element = document.getElementById("moose-equation-903eeebb-5d71-4f18-888f-e5ef67008d14");katex.render("\\int_V \\text{d}V = \\int_{dS} x \\cdot n_x \\int{d}S", element, {displayMode:true,throwOnError:false});$

Computing the volume reduces to a surface integral of the x-coordinate times the x-component of the normal vector.

Demonstration

As a demonstration, we take an axisymmetric mesh composed of two elements along the centerline, one above the other. The height of each element, $var element = document.getElementById("moose-equation-dbee795e-20cf-44b0-88ba-864276270734");katex.render("h", element, {displayMode:false,throwOnError:false});$ , is $var element = document.getElementById("moose-equation-90ae5000-ef67-4d08-9cb7-2bf8703a3f60");katex.render("1/(2\\pi)", element, {displayMode:false,throwOnError:false});$ . The initial radius, $var element = document.getElementById("moose-equation-bedd130c-bf86-4377-b72b-3d8f2172adc5");katex.render("r_1", element, {displayMode:false,throwOnError:false});$ , is 1. This gives $var element = document.getElementById("moose-equation-98637ba5-1e09-4245-91e1-db2f34d2ca6e");katex.render("V = \\pi h r_1^2 = \\pi 2/(2\\pi) 1^2 = 1", element, {displayMode:true,throwOnError:false});$

In a second step, the right face of one of the elements is translated to $var element = document.getElementById("moose-equation-bc6c7c36-1263-43b3-9ae2-af4f020bac7f");katex.render("r_2", element, {displayMode:false,throwOnError:false});$ = 1.5380168369562588. The other node on the right side is left unchanged. This volume is then $var element = document.getElementById("moose-equation-fe116719-93db-4f94-baa7-398c935d27b7");katex.render("V = \\pi h/2 (r_2^2 + 1/3 (r_1^2+r_1 r_2+r_2^2)) = 2", element, {displayMode:true,throwOnError:false});$

Table 1: Comparison of the Bison computed internal volume to the analytical solution.

Step	Analytical	Bison	Error (%)
1	1.0	1.0	0.0
2	2.0	2.0	0.0

Input Syntax

note:Sign Convention

The InternalVolume feature is written to compute internal or void volumes as positive. In the example shown here, where the volume of a solid object is computed, the volume will be reported as a negative value. We use a scale_factor of -1 in the input file shown below to have the volume reported as a positive value.

#
# Volume Test
#
# This test is designed to compute the volume of a space when displacements
#   are imposed.
#
# The mesh is composed of one block (1) with two elements.  The mesh is
#   such that the initial volume is 1.  One element face is displaced to
#   produce a final volume of 2.
#
#     r1
#   +----+   -
#   |    |   |
#   +----+   h    V1 = pi * h * r1^2
#   |    |   |
#   +----+   -
#
#   becomes
#
#   +----+
#   |     \
#   +------+      v2 = pi * h/2 * ( r2^2 + 1/3 * ( r2^2 + r2*r1 + r1^2 ) )
#   |      |
#   +------+
#      r2
#
#   r1 = 1
#   r2 = 1.5380168369562588
#   h  = 1/pi
#
#  Note:  Because the InternalVolume PP computes cavity volumes as positive,
#         the volumes reported are negative.
#

[GlobalParams<<<{"href": "../../../syntax/GlobalParams/index.html"}>>>]
  displacements = 'disp_x disp_y'
[]

[Mesh<<<{"href": "../../../syntax/Mesh/index.html"}>>>]
  coord_type = RZ
  [mesh]
    type = GeneratedMeshGenerator<<<{"description": "Create a line, square, or cube mesh with uniformly spaced or biased elements.", "href": "../../../source/meshgenerators/GeneratedMeshGenerator.html"}>>>
    dim<<<{"description": "The dimension of the mesh to be generated"}>>> = 2
    nx<<<{"description": "Number of elements in the X direction"}>>> = 1
    ny<<<{"description": "Number of elements in the Y direction"}>>> = 2
    xmin<<<{"description": "Lower X Coordinate of the generated mesh"}>>> = 0
    ymin<<<{"description": "Lower Y Coordinate of the generated mesh"}>>> = 0
    xmax<<<{"description": "Upper X Coordinate of the generated mesh"}>>> = 1
    ymax<<<{"description": "Upper Y Coordinate of the generated mesh"}>>> = 0.3183098861837907
  []
  [surface]
    type = SideSetsFromBoundingBoxGenerator<<<{"description": "Defines new sidesets using currently-defined sideset IDs inside or outside of a bounding box.", "href": "../../../source/meshgenerators/SideSetsFromBoundingBoxGenerator.html"}>>>
    input<<<{"description": "The mesh we want to modify"}>>> = mesh
    bottom_left<<<{"description": "The bottom left point (in x,y,z with spaces in-between)."}>>> = '-0.1 -0.1 0.0'
    top_right<<<{"description": "The bottom left point (in x,y,z with spaces in-between)."}>>> = '1.1 0.16 0.0'
    included_boundaries<<<{"description": "A set of boundary names or ids whose sides will be included in the new sidesets.  A side is only added if it also belongs to one of these boundaries."}>>> = right
    boundary_new<<<{"description": "Boundary on specified block within the bounding box to assign"}>>> = pull
  []
  # This test uses ElementalVariableValue postprocessors on specific
  # elements, so element numbering needs to stay unchanged
  allow_renumbering = false
[]

[Functions<<<{"href": "../../../syntax/Functions/index.html"}>>>]
  [disp_x]
    type = PiecewiseLinear<<<{"description": "Linearly interpolates between pairs of x-y data", "href": "../../../source/functions/PiecewiseLinear.html"}>>>
    x<<<{"description": "The abscissa values"}>>> = '0. 1.'
    y<<<{"description": "The ordinate values"}>>> = '0. 0.5380168369562588'
  []
[]

[Variables<<<{"href": "../../../syntax/Variables/index.html"}>>>]
  [disp_x]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
  []

  [disp_y]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = FIRST
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = LAGRANGE
  []
[]

[AuxVariables<<<{"href": "../../../syntax/AuxVariables/index.html"}>>>]
  [volumetric_strain]
    order<<<{"description": "Specifies the order of the FE shape function to use for this variable (additional orders not listed are allowed)"}>>> = CONSTANT
    family<<<{"description": "Specifies the family of FE shape functions to use for this variable"}>>> = MONOMIAL
  []
[]

[Physics<<<{"href": "../../../syntax/Physics/index.html"}>>>/SolidMechanics<<<{"href": "../../../syntax/Physics/SolidMechanics/index.html"}>>>/QuasiStatic<<<{"href": "../../../syntax/Physics/SolidMechanics/QuasiStatic/index.html"}>>>]
  [all]
    volumetric_locking_correction<<<{"description": "Flag to correct volumetric locking"}>>> = false
    decomposition_method<<<{"description": "Methods to calculate the finite strain and rotation increments"}>>> = EigenSolution
    incremental<<<{"description": "Use incremental or total strain (if not explicitly specified this defaults to incremental for finite strain and total for small strain)"}>>> = true
    strain<<<{"description": "Strain formulation"}>>> = FINITE
  []
[]

[AuxKernels<<<{"href": "../../../syntax/AuxKernels/index.html"}>>>]
  [fred]
    type = RankTwoScalarAux<<<{"description": "Compute a scalar property of a RankTwoTensor", "href": "../../../source/auxkernels/RankTwoScalarAux.html"}>>>
    rank_two_tensor<<<{"description": "The rank two material tensor name"}>>> = total_strain
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = volumetric_strain
    scalar_type<<<{"description": "Type of scalar output"}>>> = VolumetricStrain
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = timestep_end
  []
[]

[BCs<<<{"href": "../../../syntax/NuclearMaterials/BCs/index.html"}>>>]
  [no_x]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = disp_x
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'left top'
    value<<<{"description": "Value of the BC"}>>> = 0.0
  []

  [no_y]
    type = DirichletBC<<<{"description": "Imposes the essential boundary condition $u=g$, where $g$ is a constant, controllable value.", "href": "../../../source/bcs/DirichletBC.html"}>>>
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = disp_y
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'left top pull'
    value<<<{"description": "Value of the BC"}>>> = 0.0
  []

  [x]
    type = FunctionDirichletBC<<<{"description": "Imposes the essential boundary condition $u=g(t,\\vec{x})$, where $g$ is a (possibly) time and space-dependent MOOSE Function.", "href": "../../../source/bcs/FunctionDirichletBC.html"}>>>
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = pull
    variable<<<{"description": "The name of the variable that this residual object operates on"}>>> = disp_x
    function<<<{"description": "The forcing function."}>>> = disp_x
  []
[]

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [elasticity_tensor]
    type = ComputeIsotropicElasticityTensor<<<{"description": "Compute a constant isotropic elasticity tensor.", "href": "../../../source/materials/ComputeIsotropicElasticityTensor.html"}>>>
    youngs_modulus<<<{"description": "Young's modulus of the material."}>>> = 1e6
    poissons_ratio<<<{"description": "Poisson's ratio for the material."}>>> = 0.3
  []

  [stress]
    type = ComputeFiniteStrainElasticStress<<<{"description": "Compute stress using elasticity for finite strains", "href": "../../../source/materials/ComputeFiniteStrainElasticStress.html"}>>>
  []
[]

[Executioner<<<{"href": "../../../syntax/Executioner/index.html"}>>>]
  type = Transient
  solve_type = PJFNK

  start_time = 0.0
  dt = 1.0
  end_time = 1.0
[]

[Postprocessors<<<{"href": "../../../syntax/Postprocessors/index.html"}>>>]
  [bison_internalVolume]
    type = InternalVolume<<<{"description": "Computes the volume of an enclosed area by performing an integral over a user-supplied boundary.", "href": "../../../source/postprocessors/InternalVolume.html"}>>>
    boundary<<<{"description": "The list of boundary IDs from the mesh where this object applies"}>>> = 'bottom right top'
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'initial timestep_end'
    scale_factor<<<{"description": "A scale factor to be applied to the internal volume calculation"}>>> = -1
  []
  [analytical_internalVolume]
    type = ParsedPostprocessor<<<{"description": "Computes a parsed expression with post-processors", "href": "../../../source/postprocessors/ParsedPostprocessor.html"}>>>
    pp_names<<<{"description": "Post-processors arguments"}>>> = ''
    use_t<<<{"description": "Make time (t) variable available in the function expression."}>>> = true
    expression<<<{"description": "function expression"}>>> = 'if(t>0,2,1)'
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'initial timestep_end'
  []
  [error]
    type = ParsedPostprocessor<<<{"description": "Computes a parsed expression with post-processors", "href": "../../../source/postprocessors/ParsedPostprocessor.html"}>>>
    expression<<<{"description": "function expression"}>>> = '(bison_internalVolume-analytical_internalVolume)/analytical_internalVolume*100'
    pp_names<<<{"description": "Post-processors arguments"}>>> = 'bison_internalVolume analytical_internalVolume'
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'initial timestep_end'
  []

  [volStrain0]
    type = ElementalVariableValue<<<{"description": "Outputs an elemental variable value at a particular location", "href": "../../../source/postprocessors/ElementalVariableValue.html"}>>>
    elementid<<<{"description": "The ID of the element where we monitor"}>>> = 0
    variable<<<{"description": "The variable to be monitored"}>>> = volumetric_strain
  []
  [volStrain1]
    type = ElementalVariableValue<<<{"description": "Outputs an elemental variable value at a particular location", "href": "../../../source/postprocessors/ElementalVariableValue.html"}>>>
    elementid<<<{"description": "The ID of the element where we monitor"}>>> = 1
    variable<<<{"description": "The variable to be monitored"}>>> = volumetric_strain
  []
[]

[Outputs<<<{"href": "../../../syntax/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
[]

(assessment/verification/Other/internal_volume/rz.i)

#
# Volume Test
#
# This test is designed to compute the volume of a space when displacements
#   are imposed.
#
# The mesh is composed of one block (1) with two elements.  The mesh is
#   such that the initial volume is 1.  One element face is displaced to
#   produce a final volume of 2.
#
#     r1
#   +----+   -
#   |    |   |
#   +----+   h    V1 = pi * h * r1^2
#   |    |   |
#   +----+   -
#
#   becomes
#
#   +----+
#   |     \
#   +------+      v2 = pi * h/2 * ( r2^2 + 1/3 * ( r2^2 + r2*r1 + r1^2 ) )
#   |      |
#   +------+
#      r2
#
#   r1 = 1
#   r2 = 1.5380168369562588
#   h  = 1/pi
#
#  Note:  Because the InternalVolume PP computes cavity volumes as positive,
#         the volumes reported are negative.
#

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  coord_type = RZ
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 1
    ny = 2
    xmin = 0
    ymin = 0
    xmax = 1
    ymax = 0.3183098861837907
  []
  [surface]
    type = SideSetsFromBoundingBoxGenerator
    input = mesh
    bottom_left = '-0.1 -0.1 0.0'
    top_right = '1.1 0.16 0.0'
    included_boundaries = right
    boundary_new = pull
  []
  # This test uses ElementalVariableValue postprocessors on specific
  # elements, so element numbering needs to stay unchanged
  allow_renumbering = false
[]

[Functions]
  [disp_x]
    type = PiecewiseLinear
    x = '0. 1.'
    y = '0. 0.5380168369562588'
  []
[]

[Variables]
  [disp_x]
    order = FIRST
    family = LAGRANGE
  []

  [disp_y]
    order = FIRST
    family = LAGRANGE
  []
[]

[AuxVariables]
  [volumetric_strain]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    volumetric_locking_correction = false
    decomposition_method = EigenSolution
    incremental = true
    strain = FINITE
  []
[]

[AuxKernels]
  [fred]
    type = RankTwoScalarAux
    rank_two_tensor = total_strain
    variable = volumetric_strain
    scalar_type = VolumetricStrain
    execute_on = timestep_end
  []
[]

[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_x
    boundary = 'left top'
    value = 0.0
  []

  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = 'left top pull'
    value = 0.0
  []

  [x]
    type = FunctionDirichletBC
    boundary = pull
    variable = disp_x
    function = disp_x
  []
[]

[Materials]
  [elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 1e6
    poissons_ratio = 0.3
  []

  [stress]
    type = ComputeFiniteStrainElasticStress
  []
[]

[Executioner]
  type = Transient
  solve_type = PJFNK

  start_time = 0.0
  dt = 1.0
  end_time = 1.0
[]

[Postprocessors]
  [bison_internalVolume]
    type = InternalVolume
    boundary = 'bottom right top'
    execute_on = 'initial timestep_end'
    scale_factor = -1
  []
  [analytical_internalVolume]
    type = ParsedPostprocessor
    pp_names = ''
    use_t = true
    expression = 'if(t>0,2,1)'
    execute_on = 'initial timestep_end'
  []
  [error]
    type = ParsedPostprocessor
    expression = '(bison_internalVolume-analytical_internalVolume)/analytical_internalVolume*100'
    pp_names = 'bison_internalVolume analytical_internalVolume'
    execute_on = 'initial timestep_end'
  []

  [volStrain0]
    type = ElementalVariableValue
    elementid = 0
    variable = volumetric_strain
  []
  [volStrain1]
    type = ElementalVariableValue
    elementid = 1
    variable = volumetric_strain
  []
[]

[Outputs]
  csv = true
[]

Overview
Demonstration
Input Syntax