Transient Heat Transfer

Summary

Solves a transient heat conduction problem with a boundary parameterized by a heat transfer coefficient that exchanges heat with a thermal reservoir.

Description

This problem solves the transient heat equation with strong form:

$var element = document.getElementById("moose-equation-6c0c626f-dbc0-417b-ad13-9ed5c0625780");katex.render("\\begin{split} \\rho c_p \\frac{\\partial T}{\\partial t} - \\vec \\nabla \\cdot \\left(k\\vec\\nabla T\\right) = 0 \\,\\,\\,&\\mathrm{on}\\,\\, \\Omega \\\\ k \\vec \\nabla T \\cdot \\hat n = h_\\mathrm{htc}\\left(T-T_\\mathrm{inf}\\right) \\,\\,\\, &\\mathrm{on}\\,\\, \\Gamma_\\mathrm{htc} \\\\ T = T_0 \\,\\,\\, &\\mathrm{on}\\,\\, \\Gamma_\\mathrm{D} \\end{split}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where $var element = document.getElementById("moose-equation-d6b33da2-cda9-4510-b3fa-ce0020b71512");katex.render("T_0 = 1.0", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-f6fac8e5-df9d-4dd1-8c32-f9b4fcc9b8a7");katex.render("T_\\mathrm{inf} = 0.5", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-f9bc5bb5-5850-45a1-bb00-99d3bb217e0d");katex.render("\\rho c_p = 1.0", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , and $var element = document.getElementById("moose-equation-77d2d4c5-fbec-4a86-8c30-bacb238b4c4d");katex.render("h_\\mathrm{htc} = 5.0", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , subject to the initial condition $var element = document.getElementById("moose-equation-f1a34386-190e-4bda-8d08-4204c62e87d1");katex.render("T(t=0)=0.0", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ .

In this example, we solve this using the weak form

var element = document.getElementById("moose-equation-37eaffdc-fd08-4a2e-9c3e-3d9efcb0cc7b");katex.render("\\left(\\rho c_p \\frac{\\partial T}{\\partial t}, v\\right)_\\Omega + (k \\vec \\nabla T, \\vec \\nabla v)_\\Omega - (k \\vec \\nabla T \\cdot \\hat n, v)_{\\partial \\Omega} = 0 \\,\\,\\, \\forall v \\in V", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where

$var element = document.getElementById("moose-equation-29209351-5779-426d-86c7-e96bb97c5fce");katex.render("\\begin{split} T \\in H^1(\\Omega) &: T = T_0 \\,\\,\\, \\mathrm{on}\\,\\, \\Gamma_\\mathrm{D} \\\\ v \\in H^1(\\Omega) &: v = 0 \\,\\,\\, \\mathrm{on}\\,\\, \\Gamma_\\mathrm{D} \\end{split}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

Example File

[Mesh<<<{"href": "../Mesh/index.html"}>>>]
  type = MFEMFileMesh
  file = ../mesh/mug.e
[]

[Problem<<<{"href": "../Problem/index.html"}>>>]
  type = MFEMProblem
[]

[FESpaces<<<{"href": "../FESpaces/index.html"}>>>]
  [H1FESpace]
    type = MFEMScalarFESpace<<<{"description": "Convenience class to construct scalar finite element spaces.", "href": "../../source/mfem/fespaces/MFEMScalarFESpace.html"}>>>
    fec_type<<<{"description": "Specifies the family of FE shape functions."}>>> = H1
    fec_order<<<{"description": "Order of the FE shape function to use."}>>> = FIRST
  []
[]

[Variables<<<{"href": "../Variables/index.html"}>>>]
  [temperature]
    type = MFEMVariable<<<{"description": "Class for adding MFEM variables to the problem (`mfem::ParGridFunction`s).", "href": "../../source/mfem/variables/MFEMVariable.html"}>>>
    fespace<<<{"description": "The finite element space this variable is defined on."}>>> = H1FESpace
  []
[]

[AuxVariables<<<{"href": "../AuxVariables/index.html"}>>>]
  inactive<<<{"description": "If specified blocks matching these identifiers will be skipped."}>>> = average_temperature
  [average_temperature]
    type = MFEMVariable<<<{"description": "Class for adding MFEM variables to the problem (`mfem::ParGridFunction`s).", "href": "../../source/mfem/variables/MFEMVariable.html"}>>>
    fespace<<<{"description": "The finite element space this variable is defined on."}>>> = H1FESpace
  []
[]

[AuxKernels<<<{"href": "../AuxKernels/index.html"}>>>]
  inactive<<<{"description": "If specified blocks matching these identifiers will be skipped."}>>> = average_field
  [average_field]
    type = MFEMScalarTimeAverageAux<<<{"description": "Calculates a running time average of a scalar coefficient projected onto an auxvariable", "href": "../../source/mfem/auxkernels/MFEMScalarTimeAverageAux.html"}>>>
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = average_temperature
    source<<<{"description": "Scalar coefficient to average. A functor is any of the following: a variable, an MFEM material property, a function, a postprocessor or a number."}>>> = temperature
  []
[]

[Kernels<<<{"href": "../Kernels/index.html"}>>>]
  [diff]
    type = MFEMDiffusionKernel<<<{"description": "Adds the domain integrator to an MFEM problem for the bilinear form $(k\\vec\\nabla u, \\vec\\nabla v)_\\Omega$ arising from the weak form of the Laplacian operator $- \\vec\\nabla \\cdot \\left( k \\vec \\nabla u \\right)$.", "href": "../../source/mfem/kernels/MFEMDiffusionKernel.html"}>>>
    variable<<<{"description": "Variable labelling the weak form this kernel is added to"}>>> = temperature
  []
  [dT_dt]
    type = MFEMTimeDerivativeMassKernel<<<{"description": "Adds the domain integrator to an MFEM problem for the bilinear form $(k \\dot{u}, v)_\\Omega$ arising from the weak form of the operator $k \\dot{u}$.", "href": "../../source/mfem/kernels/MFEMTimeDerivativeMassKernel.html"}>>>
    variable<<<{"description": "Variable labelling the weak form this kernel is added to"}>>> = temperature
  []
[]

[BCs<<<{"href": "../BCs/index.html"}>>>]
  active<<<{"description": "If specified only the blocks named will be visited and made active"}>>> = 'bottom top_convective'
  [bottom]
    type = MFEMScalarDirichletBC<<<{"description": "Applies a Dirichlet condition to a scalar variable.", "href": "../../source/mfem/bcs/MFEMScalarDirichletBC.html"}>>>
    variable<<<{"description": "Variable on which to apply the boundary condition"}>>> = temperature
    boundary<<<{"description": "The list of boundaries (ids or names) from the mesh where this object applies. Defaults to all boundaries."}>>> = '1'
    coefficient<<<{"description": "The coefficient setting the values on the essential boundary. A functor is any of the following: a variable, an MFEM material property, a function, a postprocessor or a number."}>>> = 1.0
  []
  [top_convective]
    type = MFEMConvectiveHeatFluxBC<<<{"description": "Convective heat transfer boundary condition with temperature and heat transfer coefficent given by material properties to add to MFEM problems.", "href": "../../source/mfem/bcs/MFEMConvectiveHeatFluxBC.html"}>>>
    variable<<<{"description": "Variable on which to apply the boundary condition"}>>> = temperature
    boundary<<<{"description": "The list of boundaries (ids or names) from the mesh where this object applies. Defaults to all boundaries."}>>> = '2'
    T_infinity<<<{"description": "Name of a coefficient specifying the far-field temperature. A functor is any of the following: a variable, an MFEM material property, a function, a postprocessor or a number."}>>> = .5
    heat_transfer_coefficient<<<{"description": "Name of the coefficient specifying the heat transfer coefficient. A functor is any of the following: a variable, an MFEM material property, a function, a postprocessor or a number."}>>> = 5
  []
  [top_dirichlet]
    type = MFEMScalarDirichletBC<<<{"description": "Applies a Dirichlet condition to a scalar variable.", "href": "../../source/mfem/bcs/MFEMScalarDirichletBC.html"}>>>
    variable<<<{"description": "Variable on which to apply the boundary condition"}>>> = temperature
    boundary<<<{"description": "The list of boundaries (ids or names) from the mesh where this object applies. Defaults to all boundaries."}>>> = '2'
  []
[]

[Solvers<<<{"href": "../Solvers/index.html"}>>>]
  inactive<<<{"description": "If specified blocks matching these identifiers will be skipped."}>>> = 'jacobi'
  [boomeramg]
    type = MFEMHypreBoomerAMG<<<{"description": "Hypre BoomerAMG solver and preconditioner for the iterative solution of MFEM equation systems.", "href": "../../source/mfem/solvers/MFEMHypreBoomerAMG.html"}>>>
  []
  [jacobi]
    type = MFEMOperatorJacobiSmoother<<<{"description": "MFEM solver for performing Jacobi smoothing of the equation system.", "href": "../../source/mfem/solvers/MFEMOperatorJacobiSmoother.html"}>>>
  []
  [main]
    type = MFEMHypreGMRES<<<{"description": "Hypre solver for the iterative solution of MFEM equation systems using the generalized minimal residual method.", "href": "../../source/mfem/solvers/MFEMHypreGMRES.html"}>>>
    preconditioner<<<{"description": "Optional choice of preconditioner to use."}>>> = boomeramg
    l_tol<<<{"description": "Set the relative tolerance."}>>> = 1e-16
    l_max_its<<<{"description": "Set the maximum number of iterations."}>>> = 1000
  []
[]

[Executioner<<<{"href": "../Executioner/index.html"}>>>]
  type = MFEMTransient
  device = cpu
  assembly_level = legacy
  dt = 2.0
  start_time = 0.0
  end_time = 6.0
[]

[VectorPostprocessors<<<{"href": "../VectorPostprocessors/index.html"}>>>]
  [line_sample]
    type = MFEMVariableLineValueSampler<<<{"description": "Sample an MFEM variable along a specified line.", "href": "../../source/mfem/vectorpostprocessors/MFEMVariableLineValueSampler.html"}>>>
    variable<<<{"description": "The variable that this VectorPostprocessor samples"}>>> = 'temperature'
    start_point<<<{"description": "The beginning of the line"}>>> = '2.125 0 -2.375'
    end_point<<<{"description": "The ending of the line"}>>> = '2.125 0 2.625'
    num_points<<<{"description": "The number of points to sample along the line"}>>> = 101
  []
[]

[Outputs<<<{"href": "../Outputs/index.html"}>>>]
  [CSV]
    type = CSV<<<{"description": "Output for postprocessors, vector postprocessors, and scalar variables using comma seperated values (CSV).", "href": "../../source/outputs/CSV.html"}>>>
    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'
    file_base<<<{"description": "The desired solution output name without an extension. If not provided, MOOSE sets it with Outputs/file_base when available. Otherwise, MOOSE uses input file name and this object name for a master input or uses master file_base, the subapp name and this object name for a subapp input to set it."}>>> = OutputData/HeatTransfer/heattransfer
  []
[]

(test/tests/mfem/kernels/heattransfer.i)

[Mesh]
  type = MFEMFileMesh
  file = ../mesh/mug.e
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
  []
[]

[Variables]
  [temperature]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[AuxVariables]
  inactive = average_temperature
  [average_temperature]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[AuxKernels]
  inactive = average_field
  [average_field]
    type = MFEMScalarTimeAverageAux
    variable = average_temperature
    source = temperature
  []
[]

[Kernels]
  [diff]
    type = MFEMDiffusionKernel
    variable = temperature
  []
  [dT_dt]
    type = MFEMTimeDerivativeMassKernel
    variable = temperature
  []
[]

[BCs]
  active = 'bottom top_convective'
  [bottom]
    type = MFEMScalarDirichletBC
    variable = temperature
    boundary = '1'
    coefficient = 1.0
  []
  [top_convective]
    type = MFEMConvectiveHeatFluxBC
    variable = temperature
    boundary = '2'
    T_infinity = .5
    heat_transfer_coefficient = 5
  []
  [top_dirichlet]
    type = MFEMScalarDirichletBC
    variable = temperature
    boundary = '2'
  []
[]


[Solvers]
  inactive = 'jacobi'
  [boomeramg]
    type = MFEMHypreBoomerAMG
  []
  [jacobi]
    type = MFEMOperatorJacobiSmoother
  []
  [main]
    type = MFEMHypreGMRES
    preconditioner = boomeramg
    l_tol = 1e-16
    l_max_its = 1000
  []
[]

[Executioner]
  type = MFEMTransient
  device = cpu
  assembly_level = legacy
  dt = 2.0
  start_time = 0.0
  end_time = 6.0
[]

[VectorPostprocessors]
  [line_sample]
    type = MFEMVariableLineValueSampler
    variable = 'temperature'
    start_point = '2.125 0 -2.375'
    end_point = '2.125 0 2.625'
    num_points = 101
  []
[]


[Outputs]
  [CSV]
    type = CSV
    execute_on = 'timestep_end'
    file_base = OutputData/HeatTransfer/heattransfer
  []
[]

Summary
Description
Example File