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:

$var element = document.getElementById("moose-equation-cc0ac1bb-a46c-4dc2-86dc-6493fc62f764");katex.render("\\begin{split} \\rho c_p \\frac{\\partial T}{\\partial t} - \\vec \\nabla \\cdot \\left(k\\vec\\nabla T\\right) = 0 \\,\\,\\,&\\mathrm{on}\\,\\, \\Omega \\\\ k \\vec \\nabla T \\cdot \\hat n = h_\\mathrm{htc}\\left(T-T_\\mathrm{inf}\\right) \\,\\,\\, &\\mathrm{on}\\,\\, \\Gamma_\\mathrm{htc} \\\\ T = T_0 \\,\\,\\, &\\mathrm{on}\\,\\, \\Gamma_\\mathrm{D} \\end{split}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-623b3020-b8a6-4893-8f02-6161bb4292d7");katex.render("T_0 = 1.0", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-0da05b8a-d3d7-4d56-9590-befda1c348ff");katex.render("T_\\mathrm{inf} = 0.5", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-c28f09cd-fd27-49e4-8f82-9cb3f06686c4");katex.render("\\rho c_p = 1.0", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-2de019ca-bd44-44e6-9eeb-cc22b5eb6530");katex.render("h_\\mathrm{htc} = 5.0", element, {displayMode:false,throwOnError:false});$ , subject to the initial condition $var element = document.getElementById("moose-equation-2c7c6dcf-c5b1-492d-a071-b9d2922f8380");katex.render("T(t=0)=0.0", element, {displayMode:false,throwOnError:false});$ .

In this example, we solve this using the weak form

var element = document.getElementById("moose-equation-4159c85c-55e8-4511-8119-dda821979df6");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});

where

$var element = document.getElementById("moose-equation-c9de7995-a0e4-4d06-8605-6f70e6cbf20c");katex.render("\\begin{split} T \\in H^1(\\Omega) &: T = T_0 \\,\\,\\, \\mathrm{on}\\,\\, \\Gamma_\\mathrm{D} \\\\ v \\in H^1(\\Omega) &: v = 0 \\,\\,\\, \\mathrm{on}\\,\\, \\Gamma_\\mathrm{D} \\end{split}", element, {displayMode:true,throwOnError:false});$

Example File

[Mesh<<<{"href": "../Mesh/index.html"}>>>]
  type = MFEMMesh
  file = ../mesh/mug.e
  dim = 3
[]

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

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

[Variables<<<{"href": "../Variables/index.html"}>>>]
  [temperature]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[Kernels<<<{"href": "../Kernels/index.html"}>>>]
  [diff]
    type = MFEMDiffusionKernel
    variable = temperature
  []
  [dT_dt]
    type = MFEMTimeDerivativeMassKernel
    variable = 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
    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'
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
  []
  [jacobi]
    type = MFEMOperatorJacobiSmoother
  []
[]

[Solver]
  type = MFEMHypreGMRES
  preconditioner = boomeramg
  l_tol = 1e-16
  l_max_its = 1000
[]

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

[Outputs<<<{"href": "../Outputs/index.html"}>>>]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/HeatTransfer
    vtk_format = ASCII
  []
[]

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

[Mesh]
  type = MFEMMesh
  file = ../mesh/mug.e
  dim = 3
[]

[Problem]
  type = MFEMProblem
[]

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

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

[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'
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
  []
  [jacobi]
    type = MFEMOperatorJacobiSmoother
  []
[]

[Solver]
  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
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/HeatTransfer
    vtk_format = ASCII
  []
[]

Summary
Description
Example File