Pressure

Applies a pressure on a given boundary in a given direction

Description

The boundary condition, Pressure applies a force to a mesh boundary in the magnitude specified by the user. A component of the normal vector to the mesh surface (0, 1, or 2 corresponding to the $var element = document.getElementById("moose-equation-fa7d8abd-6004-4166-b33d-f18ae85e2adb");katex.render("\\hat{x}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-011f7865-85d7-4938-82df-616d7e4e2c84");katex.render("\\hat{y}", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-b4954a28-b4f0-4e51-b9bb-d38ce4780725");katex.render("\\hat{z}", element, {displayMode:false,throwOnError:false});$ vector components) is used to determine the direction in which to apply the traction. The boundary condition is typically applied to the displaced mesh.

The magnitude of the Pressure boundary condition can be specified as either a scalar (use the input parameter factor, which defaults to 1.0), a function parameter, or a Postprocessor name. If more than one of these are given, they are multiplied by one another.

note:Can Be Created with the Pressure Action

A set of Pressure boundary conditions applied to multiple variables in multiple components can be defined with the PressureAction.

Jacobian

Cartesian

Let $var element = document.getElementById("moose-equation-aa4769e5-883f-4a10-9308-0a9e5b3484c3");katex.render("N", element, {displayMode:false,throwOnError:false});$ be the number of nodes on a finite element face. Also, let $var element = document.getElementById("moose-equation-fa4c0f67-dbae-485c-b309-829edc194d30");katex.render("h_i", element, {displayMode:false,throwOnError:false});$ be the shape function at node $var element = document.getElementById("moose-equation-d6bce9ba-37dc-4c93-ba7b-700bb12a5621");katex.render("i", element, {displayMode:false,throwOnError:false});$ . Then the vector $var element = document.getElementById("moose-equation-12746220-c75e-4105-97e5-9a3e1d7bc2dc");katex.render("f", element, {displayMode:false,throwOnError:false});$ , which has dimension $var element = document.getElementById("moose-equation-d739ef69-4a01-45fc-9877-9bf784524e63");katex.render("3N\\times1", element, {displayMode:false,throwOnError:false});$ for a 3D model, is defined as $var element = document.getElementById("moose-equation-8400037f-5fc2-45c6-98be-c2007463e00e");katex.render(" f = \\int \\gamma F^T n\\;\\; dA", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-50af92ff-b8d6-4cc7-b7b6-27b41eddc124");katex.render("\\gamma", element, {displayMode:false,throwOnError:false});$ represents a scaling factor, $var element = document.getElementById("moose-equation-ed584c4b-bfa5-4d30-b1a2-0103997cf6b9");katex.render("n", element, {displayMode:false,throwOnError:false});$ is the normal vector, and $var element = document.getElementById("moose-equation-8f26ec7b-2a00-4ed5-9e99-4b278e5ad3ec");katex.render(" F = \\begin{bmatrix} {h_1}I & {h_2}I & ... & {h_N}I \\end{bmatrix}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-01738317-fe54-4223-ab54-e098ce65f736");katex.render("I", element, {displayMode:false,throwOnError:false});$ is the identity tensor.

To find the Jacobian, we take the variation of the term in the integral, $var element = document.getElementById("moose-equation-4c1484c2-273c-4a0a-a27f-42fe00f94f47");katex.render(" \\delta(\\gamma F^T n) = \\gamma(\\delta(F^T)n + F^T\\delta(n))", element, {displayMode:true,throwOnError:false});$

We define $var element = document.getElementById("moose-equation-5daa0c06-5a52-4518-ae5b-d1c43c45a784");katex.render("F_{,\\alpha}", element, {displayMode:false,throwOnError:false});$ as $var element = document.getElementById("moose-equation-dee74bbf-b466-44c1-89d3-94f17463c2b4");katex.render(" F_{,\\alpha} = \\begin{bmatrix} {h_{1,\\alpha}}I & {h_{2,\\alpha}}I & ... & {h_{N,\\alpha}}I \\end{bmatrix}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-3f16cb93-209e-498c-b163-097e8b61cada");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ may be either $var element = document.getElementById("moose-equation-4a645a14-9aa2-4287-bab3-c662aeb87548");katex.render("\\xi", element, {displayMode:false,throwOnError:false});$ or $var element = document.getElementById("moose-equation-1595819f-6da7-4d4f-b945-e616d80f19cf");katex.render("\\eta", element, {displayMode:false,throwOnError:false});$ , the two parametric coordinates associated with the face of an element. $var element = document.getElementById("moose-equation-579c360a-96e6-4907-989c-96f6d3485a52");katex.render("F_{,\\alpha}", element, {displayMode:false,throwOnError:false});$ has dimensions $var element = document.getElementById("moose-equation-c1b02a5c-d8e7-4ce0-ae00-d603765ec10c");katex.render("3 \\times 3n", element, {displayMode:false,throwOnError:false});$ . This allows $var element = document.getElementById("moose-equation-7228d306-73b2-4496-88a8-d3b26fb4b695");katex.render("\\delta F = \\begin{bmatrix} F_{,\\xi}\\delta\\xi & F_{,\\eta}\\delta\\eta \\end{bmatrix}", element, {displayMode:true,throwOnError:false});$ Since the values of the parametric coordinates are fixed according to the integration rule and do not vary, this term becomes zero. We are left with $var element = document.getElementById("moose-equation-c6971049-e305-48b7-a5f5-dfffff74156d");katex.render("\\gamma F^T\\delta n", element, {displayMode:false,throwOnError:false});$ .

The normal vector $var element = document.getElementById("moose-equation-cb71ed3a-98bb-41cf-8985-f23c6830e346");katex.render("n", element, {displayMode:false,throwOnError:false});$ is an outward unit vector at the integration points. We define $var element = document.getElementById("moose-equation-cab4ce67-e183-4ac0-b50f-2a34dad0e835");katex.render("b = q_{,\\xi} \\times q_{,\\eta}", element, {displayMode:false,throwOnError:false});$ where $var element = document.getElementById("moose-equation-c9cb97a5-f7ce-4f3c-bd2f-8fbe2402497b");katex.render("q_{,\\alpha} = F_{,\\alpha}p", element, {displayMode:true,throwOnError:false});$ defined at the integration points with $var element = document.getElementById("moose-equation-5b48d8fd-0fa7-4584-b622-2bfa1e498815");katex.render("p^T = \\begin{bmatrix} x^T_1 & x^T_2 & ... & x^T_n \\end{bmatrix}", element, {displayMode:true,throwOnError:false});$ $var element = document.getElementById("moose-equation-c1c6509e-7dc6-411c-b019-c4fc91bb8647");katex.render("x_i = X_i + u_i", element, {displayMode:false,throwOnError:false});$ where $var element = document.getElementById("moose-equation-7be778a6-ef81-4702-8248-576fe636fd61");katex.render("X_i", element, {displayMode:false,throwOnError:false});$ is the vector of coordinates for node $var element = document.getElementById("moose-equation-f6bc55db-0b32-445d-bfd0-17a32dbb3389");katex.render("i", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-7f33f4b4-b555-4d52-a8fe-f7d98bb20f02");katex.render("u_i", element, {displayMode:false,throwOnError:false});$ is the vector of displacments for node $var element = document.getElementById("moose-equation-259697d9-b83e-4950-8d87-e2efaf81a1c1");katex.render("i", element, {displayMode:false,throwOnError:false});$ .

$var element = document.getElementById("moose-equation-48294a17-c614-4f63-ac49-d9abfe57c18b");katex.render("\\begin{aligned} \\delta b = & \\delta q_{,\\xi} \\times q_{,\\eta} + q_{,\\xi} \\times \\delta q_{,\\eta} \\\\ \\delta q_{,\\xi} = & F_{,\\xi}\\delta p + q_{,\\xi\\eta}\\delta \\eta \\\\ \\delta q_{,\\eta} = & F_{,\\eta}\\delta p + q_{,\\xi\\eta}\\delta \\xi \\end{aligned}", element, {displayMode:true,throwOnError:false});$ If $var element = document.getElementById("moose-equation-d94a11dd-81c2-46f4-bc7d-750ca9c85679");katex.render("\\delta\\xi = \\delta\\eta = 0", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-d312e8d1-557b-45f6-976a-8f10c2cdb928");katex.render("\\begin{aligned} \\delta b = & F_{,\\xi}\\delta p \\times q_{,\\eta} + q_{,\\xi} \\times F_{,\\eta}\\delta p \\\\ = & q_{,\\xi} \\times F_{,\\eta}\\delta p - q_{,\\eta} \\times F_{,\\xi}\\delta p \\\\ = & (q_{,\\xi} \\times F_{,\\eta} - q_{,\\eta} \\times F_{,\\xi}) \\delta p \\end{aligned}", element, {displayMode:true,throwOnError:false});$

We are left with $var element = document.getElementById("moose-equation-6d147494-59e4-42a8-ab67-913de80fb319");katex.render(" \\delta(\\gamma F^T n) = \\frac{\\gamma}{||b||}F^T(q_{,\\xi} \\times F_{,\\eta} - q_{,\\eta} \\times F_{,\\xi}) \\delta p", element, {displayMode:true,throwOnError:false});$

To take the cross product of a vector and a set of vectors in a matrix, we take the cross product of the vector with each vector in the matrix in turn. The $var element = document.getElementById("moose-equation-7d1b9176-8b2b-46e3-948b-9804ed7f49ac");katex.render("j", element, {displayMode:false,throwOnError:false});$ th $var element = document.getElementById("moose-equation-926deb35-8425-4a3d-985d-72adc1d6d43d");katex.render("3\\times3", element, {displayMode:false,throwOnError:false});$ submatrix of $var element = document.getElementById("moose-equation-4c6b338e-0800-4021-b4bc-68c38f7d1714");katex.render("F_{,\\alpha}", element, {displayMode:false,throwOnError:false});$ is $var element = document.getElementById("moose-equation-842ef2b3-0974-4a8b-b01e-09ffaaaedbf8");katex.render("h_{j,\\alpha}I", element, {displayMode:false,throwOnError:false});$ . This gives $var element = document.getElementById("moose-equation-ac0bf1fe-dc8d-49c5-8da9-9272b83a8913");katex.render(" (q_{,\\xi} \\times F_{,\\eta} - q_{,\\eta} \\times F_{,\\xi})_j = \\begin{bmatrix} 0 & -q_{,\\xi(3)}h_{j,\\eta}+q_{,\\eta(3)}h_{j,\\xi} & q_{,\\xi(2)}h_{j,\\eta}-q_{,\\eta(2)}h_{j,\\xi} \\\\ q_{,\\xi(3)}h_{j,\\eta}-q_{,\\eta(3)}h_{j,\\xi} & 0 & -q_{,\\xi(1)}h_{j,\\eta}+q_{,\\eta(1)}h_{j,\\xi} \\\\ -q_{,\\xi(2)}h_{j,\\eta}+q_{,\\eta(2)}h_{j,\\xi} & q_{,\\xi(1)}h_{j,\\eta}-q_{,\\eta(1)}h_{j,\\xi} & 0 \\end{bmatrix}", element, {displayMode:true,throwOnError:false});$ The integrand of the $var element = document.getElementById("moose-equation-3de33c94-7b74-4028-95b4-78a2ff87a39c");katex.render("(i,j)", element, {displayMode:false,throwOnError:false});$ $var element = document.getElementById("moose-equation-7bc09f9d-2915-48a7-bd21-76eff58cf7da");katex.render("3\\times3", element, {displayMode:false,throwOnError:false});$ submatrix of the stiffness is $var element = document.getElementById("moose-equation-4f87a4e5-ab6e-465c-a76a-271a7e943d19");katex.render("\\frac{\\gamma}{||b||}(F^T(q_{,\\xi} \\times F_{,\\eta} - q_{,\\eta} \\times F_{,\\xi}))_{ij} = \\frac{\\gamma}{||b||}h_i \\begin{bmatrix} 0 & -q_{,\\xi(3)}h_{j,\\eta}+q_{,\\eta(3)}h_{j,\\xi} & q_{,\\xi(2)}h_{j,\\eta}-q_{,\\eta(2)}h_{j,\\xi} \\\\ q_{,\\xi(3)}h_{j,\\eta}-q_{,\\eta(3)}h_{j,\\xi} & 0 & -q_{,\\xi(1)}h_{j,\\eta}+q_{,\\eta(1)}h_{j,\\xi} \\\\ -q_{,\\xi(2)}h_{j,\\eta}+q_{,\\eta(2)}h_{j,\\xi} & q_{,\\xi(1)}h_{j,\\eta}-q_{,\\eta(1)}h_{j,\\xi} & 0 \\end{bmatrix}", element, {displayMode:true,throwOnError:false});$

Spherical symmetry

For a problem using spherical symmetry, the Jacobian is much simpler. Here we have $var element = document.getElementById("moose-equation-d5dd917f-fc6e-4862-9ffe-60eacb3f3976");katex.render("f = \\int \\gamma F^Tn \\;\\; dA = \\int\\int \\gamma F^T n r^2 \\sin \\phi \\;\\; d\\phi \\; d\\theta = 4\\pi\\gamma F^T n r^2", element, {displayMode:true,throwOnError:false});$ with $var element = document.getElementById("moose-equation-d99a6235-38fb-40f8-b6c5-e1f2fa2d70a5");katex.render("r = X + u", element, {displayMode:false,throwOnError:false});$ . Here, $var element = document.getElementById("moose-equation-6f1f27fb-b6d5-402d-8e15-304cc52cdbe5");katex.render("n", element, {displayMode:false,throwOnError:false});$ is not a function of the displacements. The variation is $var element = document.getElementById("moose-equation-541d0288-d4a6-4c94-8949-91adc5f9346c");katex.render("8 \\pi \\gamma F^T n r\\delta r", element, {displayMode:true,throwOnError:false});$

Axisymmetry

For 1D axisymmetry, we have $var element = document.getElementById("moose-equation-c045ed78-b5ac-47f0-b623-7418477e68a2");katex.render("f = \\int \\gamma F^Tn \\;\\; dA = \\int \\gamma F^T nr \\;\\; d\\theta = 2\\pi \\gamma F^T n r", element, {displayMode:true,throwOnError:false});$ with $var element = document.getElementById("moose-equation-35b261a0-beb0-45a0-a530-3a521e19eb8d");katex.render("r = X + u", element, {displayMode:false,throwOnError:false});$ and a unit height. Here, $var element = document.getElementById("moose-equation-95375ccb-461c-4dca-acdb-a1281b7d783c");katex.render("n", element, {displayMode:false,throwOnError:false});$ is not a function of the displacements. The variation is $var element = document.getElementById("moose-equation-b7ad112b-e088-4e3d-b939-09ad05e4ca0c");katex.render("2 \\pi \\gamma F^T n \\delta r", element, {displayMode:true,throwOnError:false});$

For 2D axisymmetry, we have $var element = document.getElementById("moose-equation-8758d59e-f877-44a9-b19b-a76247b0885c");katex.render("f = \\int \\gamma F^Tn \\;\\; dA = \\int \\int \\gamma F^T n r \\;\\; d\\theta \\; dz = \\int 2\\pi\\gamma F^T n r \\;\\; dz", element, {displayMode:true,throwOnError:false});$ with $var element = document.getElementById("moose-equation-4652d5cd-9918-4cf1-abe7-ff5b60c72eaa");katex.render("r = X + u", element, {displayMode:false,throwOnError:false});$ . However, both $var element = document.getElementById("moose-equation-40db7313-b9c8-42b6-8081-8878eea7454f");katex.render("n", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-55fa2ac0-6e29-45cb-a0b9-39138d60774d");katex.render("r", element, {displayMode:false,throwOnError:false});$ depend on the displacements. Thus, we have $var element = document.getElementById("moose-equation-5e0d8342-6b41-4af0-86c3-057f5d0a1b0b");katex.render("\\delta(2\\pi\\gamma F^T n r) = 2 \\pi \\gamma F^T (r \\delta n + n \\delta r)", element, {displayMode:true,throwOnError:false});$

Example Input File Syntax

[outerPressure]
  type = Pressure
  boundary = right
  variable = disp_r
  factor = 2
[]

Input Parameters

boundaryThe list of boundary IDs from the mesh where the pressure will be applied
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundary IDs from the mesh where the pressure will be applied
displacementsThe displacements appropriate for the simulation geometry and coordinate system
C++ Type:std::vector<VariableName>
Controllable:No
Description:The displacements appropriate for the simulation geometry and coordinate system

Required Parameters

absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
active__all__ If specified only the blocks named will be visited and made active
Default:__all__
C++ Type:std::vector<std::string>
Controllable:No
Description:If specified only the blocks named will be visited and made active
control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
factor1The factor to use in computing the pressure
Default:1
C++ Type:double
Controllable:No
Description:The factor to use in computing the pressure
functionThe function that describes the pressure
C++ Type:FunctionName
Controllable:No
Description:The function that describes the pressure
hht_alpha0alpha parameter for mass dependent numerical damping induced by HHT time integration scheme
Default:0
C++ Type:double
Controllable:No
Description:alpha parameter for mass dependent numerical damping induced by HHT time integration scheme
inactiveIf specified blocks matching these identifiers will be skipped.
C++ Type:std::vector<std::string>
Controllable:No
Description:If specified blocks matching these identifiers will be skipped.
save_in_disp_xThe save_in variables for x displacement
C++ Type:std::vector<AuxVariableName>
Controllable:No
Description:The save_in variables for x displacement
save_in_disp_yThe save_in variables for y displacement
C++ Type:std::vector<AuxVariableName>
Controllable:No
Description:The save_in variables for y displacement
save_in_disp_zThe save_in variables for z displacement
C++ Type:std::vector<AuxVariableName>
Controllable:No
Description:The save_in variables for z displacement
use_automatic_differentiationFalseFlag to use automatic differentiation (AD) objects when possible
Default:False
C++ Type:bool
Controllable:No
Description:Flag to use automatic differentiation (AD) objects when possible
use_displaced_meshTrueWhether to use the displaced mesh.
Default:True
C++ Type:bool
Controllable:No
Description:Whether to use the displaced mesh.

Optional Parameters

(moose/modules/tensor_mechanics/test/tests/1D_spherical/finiteStrain_1DSphere_hollow.i)

# This simulation models the mechanics solution for a hollow sphere under
# pressure, applied on the outer surfaces, using 1D spherical symmetry
# assumpitions.  The inner radius of the sphere, r = 4mm, is pinned to prevent
# rigid body movement of the sphere.
#
# From Bower (Applied Mechanics of Solids, 2008, available online at
# solidmechanics.org/text/Chapter4_1/Chapter4_1.htm), and applying the outer
# pressure and pinned displacement boundary conditions set in this simulation,
# the radial displacement is given by:
#
# u(r) = \frac{P(1 + v)(1 - 2v)b^3}{E(b^3(1 + v) + 2a^3(1-2v))} * (\frac{a^3}{r^2} - r)
#
# where P is the applied pressure, b is the outer radius, a is the inner radius,
# v is Poisson's ration, E is Young's Modulus, and r is the radial position.
#
# The radial stress is given by:
#
# S(r) = \frac{Pb^3}{b^3(1 + v) + 2a^3(1 - 2v)} * (\frac{2a^3}{r^3}(2v - 1) - (1 + v))
#
# The test assumes an inner radius of 4mm, and outer radius of 9 mm,
# zero displacement at r = 4mm, and an applied outer pressure of 2MPa.
# The radial stress is largest in the inner most element and, at an assumed
# mid element coordinate of 4.5mm, is equal to -2.545MPa.
[Mesh]
  type = GeneratedMesh
  dim = 1
  xmin = 4
  xmax = 9
  nx = 5
[]

[GlobalParams]
  displacements = 'disp_r'
[]

[Problem]
  coord_type = RSPHERICAL
[]

[Modules/TensorMechanics/Master]
  [all]
    strain = FINITE
    add_variables = true
    spherical_center_point = '4.0 0.0 0.0'
    generate_output = 'spherical_radial_stress'
  []
[]

[Postprocessors]
  [stress_rr]
    type = ElementAverageValue
    variable = spherical_radial_stress
  []
[]

[BCs]
  [innerDisp]
    type = DirichletBC
    boundary = left
    variable = disp_r
    value = 0.0
  []
  [outerPressure]
    type = Pressure
    boundary = right
    variable = disp_r
    factor = 2
  []
[]

[Materials]
  [Elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    poissons_ratio = 0.345
    youngs_modulus = 1e4
  []
  [stress]
    type = ComputeFiniteStrainElasticStress
  []
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  line_search = none

  # controls for linear iterations
  l_max_its = 100
  l_tol = 1e-8

  # controls for nonlinear iterations
  nl_max_its = 15
  nl_rel_tol = 1e-10
  nl_abs_tol = 1e-5

  # time control
  start_time = 0.0
  dt = 0.25
  dtmin = 0.0001
  end_time = 0.25
[]

[Outputs]
  exodus = true
[]