MCVolumetricSwellingEigenstrain

Model that calculates and sums the change in fuel pellet volume due to densification and fission product release. This class applies a volumetric strain correction before adding the strain from this class to the diagonal entries of the eigenstrain tensor.

Description

The MCVolumetricSwellingEigenstrain material model computes an eigenstrain tensor that accounts for solid and gaseous swelling and for densification in mixed (U,Pu)C (monocarbides). Built into the model is a reduction in total swelling due to hot pressing that occurs in irradiations where the cladding is much stiffer than the fuel, and fission gas bubbles are present in the fuel.

This model uses correlations from Preusser (1982) to compute the swelling increment for UC and (U,Pu)C as a function of burnup: $var element = document.getElementById("moose-equation-a3d9c4f5-8cff-4866-b04a-76070bb2df55");katex.render("\\frac{\\Delta V}{V}_{UC} = 0.4667 + 1.711 f_{\\phi}\\quad \\text{for } T \\leq 700 \\degree \\text{C}", element, {displayMode:true,throwOnError:false});$ and $var element = document.getElementById("moose-equation-efeb08f3-b521-4981-8abc-aa8bc1ee0b7a");katex.render(" \\frac{\\Delta V}{V}_{UC} = 0.4667 + 1.711 f_{\\phi} + \\left[ \\left(6.412 - 0.0198\\ T + 1.52\\times 10^{-5} T^2 \\right) f_B f_{\\phi} \\right]\\quad \\text{for } T > 700 \\degree \\text{C}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-168aaab0-2dc1-43aa-a8d2-2581cd215d0b");katex.render("T", element, {displayMode:false,throwOnError:false});$ is temperature in Celsius, $var element = document.getElementById("moose-equation-087813ad-2964-467e-ac47-152810e7eb60");katex.render("\\phi", element, {displayMode:false,throwOnError:false});$ is the porosity, and $var element = document.getElementById("moose-equation-8d9f8c51-cd5f-4234-874b-d6692d88fb89");katex.render("\\Delta V/V", element, {displayMode:false,throwOnError:false});$ is given as percent strain per 10 MW-d/kg (which is internally converted to FIMA, but is nearly equal to 1\% FIMA).

The correlations for (U $var element = document.getElementById("moose-equation-cc9399d6-e5d5-4cab-8416-0c501bca458e");katex.render("_{0.8}", element, {displayMode:false,throwOnError:false});$ ,Pu $var element = document.getElementById("moose-equation-baf04306-2588-485e-98c9-7deedb0db37e");katex.render("_{0.2}", element, {displayMode:false,throwOnError:false});$ )C is similar in form the UC, with different coefficients and threshold temperatures: $var element = document.getElementById("moose-equation-f1b89f24-9755-4793-9ad1-68ef68ec8d8d");katex.render("\\frac{\\Delta V}{V}_{(UPu)C} = 0.4667 + 0.9171 f_{\\phi}\\quad \\text{for } T \\leq 900 \\degree \\text{C}", element, {displayMode:true,throwOnError:false});$ and $var element = document.getElementById("moose-equation-b6db5da2-28cd-4ad1-a501-746d40952001");katex.render(" \\frac{\\Delta V}{V}_{(UPu)C} = 0.4667 + 0.9171 f_{\\phi} + \\left[ \\left(11.8062 - 0.0257\\ T + 1.398\\times 10^{-5} T^2 \\right) f_B f_{\\phi} \\right]\\quad \\text{for } T > 900 \\degree \\text{C}", element, {displayMode:true,throwOnError:false});$

For concentrations of plutonium other than 0 or 0.2, a linear correlation is used between $var element = document.getElementById("moose-equation-ffd97427-4a56-4d35-9945-4bebe4203952");katex.render("\\frac{\\Delta V}{V}_{UC}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-6ec9bbf9-0ef9-4e23-acfb-98adbeda054d");katex.render("\\frac{\\Delta V}{V}_{(UPu)C}", element, {displayMode:false,throwOnError:false});$ . In addition, the total swelling is computed from the incremental swelling by $var element = document.getElementById("moose-equation-a704d207-1afb-495b-aba3-497c0d8802c7");katex.render("\\frac{\\Delta V}{V}", element, {displayMode:false,throwOnError:false});$ by the incremental burnup in the given timestep.

The correction factors for porosity $var element = document.getElementById("moose-equation-e5d13189-ca5c-4ed9-8a0a-9cfbbd182cd4");katex.render("f_{\\phi}", element, {displayMode:false,throwOnError:false});$ and for burnup $var element = document.getElementById("moose-equation-5e41c127-21db-436e-b3c9-633d32af31ba");katex.render("f_B", element, {displayMode:false,throwOnError:false});$ are $var element = document.getElementById("moose-equation-8de88324-9a53-47aa-b3f3-9810d7e0b4c7");katex.render(" \\begin{aligned} f_{\\phi}(\\phi, p_{\\text{contact}}) &= \\exp{(0.04 -\\phi)} \\exp{\\left[-\\left( \\frac{p_{\\text{contact}}}{p_0} b \\right) \\right]}, & \\phi -0.04\\geq 0 \\\\ f_B(B) &= \\frac{B}{B_0} - a, & f_B \\geq 0 \\end{aligned}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-4bc925eb-c18c-4a3f-9a20-f3ca8f94bc02");katex.render("p_{\\text{contact}}", element, {displayMode:false,throwOnError:false});$ is the contact pressure (MPa) and $var element = document.getElementById("moose-equation-b8df38c4-e51b-4e15-967d-d7b8480b7361");katex.render("B", element, {displayMode:false,throwOnError:false});$ is the burnup (MW-d/kg). The suggested fit parameters are $var element = document.getElementById("moose-equation-9a7d0cd5-44a9-409e-8cbe-2bbd0d5624e5");katex.render(" \\begin{aligned} a &= 2 \\text{\\ MW-d/kg} \\\\ b &= 0.1 \\\\ p_0 &= 1 \\text{\\ MPa} \\\\ B_0 &= 10 \\text{\\ MW-d/kg} \\end{aligned}", element, {displayMode:true,throwOnError:false});$ The burnup correlation is smoothed via the smootherStep function from MathUtils.h.

An upper limit on swelling of $var element = document.getElementById("moose-equation-a443a57f-df67-47e2-807e-ec341fd3dd1f");katex.render("\\left(\\Delta V/V\\right)_{max} = 4.558", element, {displayMode:false,throwOnError:false});$ for UC and $var element = document.getElementById("moose-equation-2c8c65d4-8e8d-43f3-a49c-116266bb8205");katex.render("\\left(\\Delta V/V\\right)_{max} = 3.406", element, {displayMode:false,throwOnError:false});$ for (U $var element = document.getElementById("moose-equation-8edb6c18-6058-40ca-b2d7-4d46383ffe88");katex.render("_{0.8}", element, {displayMode:false,throwOnError:false});$ ,Pu $var element = document.getElementById("moose-equation-7167866c-f90f-4ae1-b2f3-8cf938c43557");katex.render("_{0.2}", element, {displayMode:false,throwOnError:false});$ )C is suggested by Preusser (1982). For concentrations of plutonium other than 0 or 0.2, a linear correlation is used for $var element = document.getElementById("moose-equation-05cd4e1b-e0bf-421e-9239-47ca41268a5f");katex.render("\\left(\\Delta V/V\\right)_{max}", element, {displayMode:false,throwOnError:false});$ . The swelling is smoothed up to these maximum values using a smootherStep function.

A temperature correction term is available to modify the input temperature value into the correlations here. Such modifications can be used to use this model for other fuel types that exhibit similar behavior, but different temperature dependence, e.g., nitride fuels.

Example Input Syntax

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [swelling]
    type = ADMCVolumetricSwellingEigenstrain<<<{"description": "Model that calculates and sums the change in fuel pellet volume due to densification and fission product release. This class applies a volumetric strain correction before adding the strain from this class to the diagonal entries of the eigenstrain tensor.", "href": "MCVolumetricSwellingEigenstrain.html"}>>>
    eigenstrain_name<<<{"description": "Material property name for the eigenstrain tensor computed by this model. IMPORTANT: The name of this property must also be provided to the strain calculator."}>>> = swelling
    temperature<<<{"description": "Coupled temperature (K)"}>>> = temp
    porosity<<<{"description": "Porosity material property name."}>>> = porosity
    contact_pressure<<<{"description": "Contact pressure (Pa)"}>>> = layered_side_contact_pressure
    burnup<<<{"description": "burnup material property name"}>>> = burnup
    swelling_scalar<<<{"description": "Scalar multiplied against swelling eigenstrain"}>>> = ${scalar}
    temperature_shift<<<{"description": "Calibration value added to the temperature to swelling expressions for different fuel compositions."}>>> = ${temperature_shift}
    X_Pu_to_An_in_MC<<<{"description": "Relative atom fraction of plutonium to actnidies in mono-carbide phase."}>>> = pu
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
[]

The eigenstrain name must also be passed to the strain calculator as shown below:

[Physics<<<{"href": "../../../syntax/Physics/index.html"}>>>]
  [SolidMechanics<<<{"href": "../../../syntax/Physics/SolidMechanics/index.html"}>>>]
    [QuasiStatic<<<{"href": "../../../syntax/Physics/SolidMechanics/QuasiStatic/index.html"}>>>]
      [all]
        strain<<<{"description": "Strain formulation"}>>> = SMALL
        add_variables<<<{"description": "Add the displacement variables"}>>> = true
        eigenstrain_names<<<{"description": "List of eigenstrains to be applied in this strain calculation"}>>> = swelling
        use_automatic_differentiation<<<{"description": "Flag to use automatic differentiation (AD) objects when possible"}>>> = true
      []
    []
  []
[]

Input Parameters

contact_pressureContact pressure (Pa)
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Contact pressure (Pa)
eigenstrain_nameMaterial property name for the eigenstrain tensor computed by this model. IMPORTANT: The name of this property must also be provided to the strain calculator.
C++ Type:std::string
Controllable:No
Description:Material property name for the eigenstrain tensor computed by this model. IMPORTANT: The name of this property must also be provided to the strain calculator.
porosityPorosity material property name.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Porosity material property name.
temperatureCoupled temperature (K)
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled temperature (K)

Required Parameters

X_Pu_to_An_in_MC0Relative atom fraction of plutonium to actnidies in mono-carbide phase.
Default:0
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Relative atom fraction of plutonium to actnidies in mono-carbide phase.
base_nameOptional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases
C++ Type:std::string
Controllable:No
Description:Optional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases
blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
boundaryThe list of boundaries (ids or names) from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundaries (ids or names) from the mesh where this object applies
burnupburnupburnup material property name
Default:burnup
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:burnup material property name
computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
Default:True
C++ Type:bool
Controllable:No
Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
Default:NONE
C++ Type:MooseEnum
Options:NONE, ELEMENT, SUBDOMAIN
Controllable:No
Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
declare_suffixAn optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
execute_onLINEARThe 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.
Default:LINEAR
C++ Type:ExecFlagEnum
Options:XFEM_MARK, NONE, INITIAL, LINEAR, NONLINEAR_CONVERGENCE, NONLINEAR, POSTCHECK, TIMESTEP_END, TIMESTEP_BEGIN, MULTIAPP_FIXED_POINT_END, MULTIAPP_FIXED_POINT_BEGIN, FINAL, CUSTOM
Controllable:No
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.
swelling_scalar1Scalar multiplied against swelling eigenstrain
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scalar multiplied against swelling eigenstrain
temperature_shift0Calibration value added to the temperature to swelling expressions for different fuel compositions.
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Calibration value added to the temperature to swelling expressions for different fuel compositions.
value_range_behaviorEXCEPTIONWhat to do if input value is outside the range of applicability.
Default:EXCEPTION
C++ Type:MooseEnum
Options:ERROR, WARN, IGNORE, EXCEPTION
Controllable:No
Description:What to do if input value is outside the range of applicability.

Optional Parameters

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.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)
C++ Type:std::vector<std::string>
Controllable:No
Description:List of material properties, from this material, to output (outputs must also be defined to an output type)
outputsnone Vector of output names where you would like to restrict the output of variables(s) associated with this object
Default:none
C++ Type:std::vector<OutputName>
Controllable:No
Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

Input Files

(test/tests/solid_mechanics/mc_volumetric_swelling/exact.i)
(assessment/carbide/EBRII/WSA32/analysis/base.i)

References

Timm Preusser. Modeling of carbide fuel rods. Nuclear Technology, 57(3):343–371, 1982. URL: https://doi.org/10.13182/NT82-A26303, arXiv:https://doi.org/10.13182/NT82-A26303, doi:10.13182/NT82-A26303.

@article{preusser:1982,
    author = "Preusser, Timm",
    title = "Modeling of Carbide Fuel Rods",
    journal = "Nuclear Technology",
    volume = "57",
    number = "3",
    pages = "343-371",
    year = "1982",
    publisher = "Taylor \& Francis",
    doi = "10.13182/NT82-A26303",
    URL = "https://doi.org/10.13182/NT82-A26303",
    eprint = "https://doi.org/10.13182/NT82-A26303"
}

(moose/framework/include/utils/MathUtils.h)

// This file is part of the MOOSE framework
// https://mooseframework.inl.gov
//
// All rights reserved, see COPYRIGHT for full restrictions
// https://github.com/idaholab/moose/blob/master/COPYRIGHT
//
// Licensed under LGPL 2.1, please see LICENSE for details
// https://www.gnu.org/licenses/lgpl-2.1.html

#pragma once

#include "Moose.h"
#include "MooseError.h"
#include "MooseTypes.h"
#include "libmesh/libmesh.h"
#include "libmesh/utility.h"
#include "libmesh/numeric_vector.h"
#include "libmesh/compare_types.h"
#include "libmesh/point.h"

namespace MathUtils
{

/// std::sqrt is not constexpr, so we add sqrt(2) as a constant (used in Mandel notation)
static constexpr Real sqrt2 = 1.4142135623730951;

Real poly1Log(Real x, Real tol, unsigned int derivative_order);
Real poly2Log(Real x, Real tol, unsigned int derivative_order);
Real poly3Log(Real x, Real tol, unsigned int derivative_order);
Real poly4Log(Real x, Real tol, unsigned int derivative_order);
Real taylorLog(Real x);
/**
 * Evaluate Cartesian coordinates of any center point of a triangle given Barycentric
 * coordinates of center point and Cartesian coordinates of triangle's vertices
 * @param p0,p1,p2 are the three non-collinear vertices in Cartesian coordinates
 * @param b0,b1,b2 is the center point in barycentric coordinates with b0+b1+b2=1, e.g.
 * (1/3,1/3,1/3) for a centroid
 * @return the center point of triangle in Cartesian coordinates
 */
Point barycentricToCartesian2D(const Point & p0,
                               const Point & p1,
                               const Point & p2,
                               const Real b0,
                               const Real b1,
                               const Real b2);
/**
 * Evaluate Cartesian coordinates of any center point of a tetrahedron given Barycentric
 * coordinates of center point and Cartesian coordinates of tetrahedon's vertices
 * @param p0,p1,p2,p3 are the three non-coplanar vertices in Cartesian coordinates
 * @param b0,b1,b2,b3 is the center point in barycentric coordinates with b0+b1+b2+b3=1, e.g.
 * (1/4,1/4,1/4,1/4) for a centroid.
 * @return the center point of tetrahedron in Cartesian coordinates
 */
Point barycentricToCartesian3D(const Point & p0,
                               const Point & p1,
                               const Point & p2,
                               const Point & p3,
                               const Real b0,
                               const Real b1,
                               const Real b2,
                               const Real b3);
/**
 * Evaluate circumcenter of a triangle given three arbitrary points
 * @param p0,p1,p2 are the three non-collinear vertices in Cartesian coordinates
 * @return the circumcenter in Cartesian coordinates
 */
Point circumcenter2D(const Point & p0, const Point & p1, const Point & p2);
/**
 * Evaluate circumcenter of a tetrahedrom given four arbitrary points
 * @param p0,p1,p2,p3 are the four non-coplanar vertices in Cartesian coordinates
 * @return the circumcenter in Cartesian coordinates
 */
Point circumcenter3D(const Point & p0, const Point & p1, const Point & p2, const Point & p3);

template <typename T>
T
round(T x)
{
  return ::round(x); // use round from math.h
}

template <typename T>
T
sign(T x)
{
  return x >= 0.0 ? 1.0 : -1.0;
}

template <typename T>
T
pow(T x, int e)
{
  bool neg = false;
  T result = 1.0;

  if (e < 0)
  {
    neg = true;
    e = -e;
  }

  while (e)
  {
    // if bit 0 is set multiply the current power of two factor of the exponent
    if (e & 1)
      result *= x;

    // x is incrementally set to consecutive powers of powers of two
    x *= x;

    // bit shift the exponent down
    e >>= 1;
  }

  return neg ? 1.0 / result : result;
}

template <typename T>
T
heavyside(T x)
{
  return x < 0.0 ? 0.0 : 1.0;
}

template <typename T>
T
regularizedHeavyside(T x, Real smoothing_length)
{
  if (x <= -smoothing_length)
    return 0.0;
  else if (x < smoothing_length)
    return 0.5 * (1 + std::sin(libMesh::pi * x / 2 / smoothing_length));
  else
    return 1.0;
}

template <typename T>
T
regularizedHeavysideDerivative(T x, Real smoothing_length)
{
  if (x < smoothing_length && x > -smoothing_length)
    return 0.25 * libMesh::pi / smoothing_length *
           (std::cos(libMesh::pi * x / 2 / smoothing_length));
  else
    return 0.0;
}

template <typename T>
T
positivePart(T x)
{
  return x > 0.0 ? x : 0.0;
}

template <typename T>
T
negativePart(T x)
{
  return x < 0.0 ? x : 0.0;
}

template <
    typename T,
    typename T2,
    typename T3,
    typename std::enable_if<libMesh::ScalarTraits<T>::value && libMesh::ScalarTraits<T2>::value &&
                                libMesh::ScalarTraits<T3>::value,
                            int>::type = 0>
void
addScaled(const T & a, const T2 & b, T3 & result)
{
  result += a * b;
}

template <typename T,
          typename T2,
          typename T3,
          typename std::enable_if<libMesh::ScalarTraits<T>::value, int>::type = 0>
void
addScaled(const T & scalar,
          const libMesh::NumericVector<T2> & numeric_vector,
          libMesh::NumericVector<T3> & result)
{
  result.add(scalar, numeric_vector);
}

template <
    typename T,
    typename T2,
    template <typename>
    class W,
    template <typename>
    class W2,
    typename std::enable_if<std::is_same<typename W<T>::index_type, unsigned int>::value &&
                                std::is_same<typename W2<T2>::index_type, unsigned int>::value,
                            int>::type = 0>
typename libMesh::CompareTypes<T, T2>::supertype
dotProduct(const W<T> & a, const W2<T2> & b)
{
  return a * b;
}

template <typename T,
          typename T2,
          template <typename>
          class W,
          template <typename>
          class W2,
          typename std::enable_if<std::is_same<typename W<T>::index_type,
                                               std::tuple<unsigned int, unsigned int>>::value &&
                                      std::is_same<typename W2<T2>::index_type,
                                                   std::tuple<unsigned int, unsigned int>>::value,
                                  int>::type = 0>
typename libMesh::CompareTypes<T, T2>::supertype
dotProduct(const W<T> & a, const W2<T2> & b)
{
  return a.contract(b);
}

/**
 * Evaluate a polynomial with the coefficients c at x. Note that the Polynomial
 * form is
 *   c[0]*x^s + c[1]*x^(s-1) + c[2]*x^(s-2) + ... + c[s-2]*x^2 + c[s-1]*x + c[s]
 * where s = c.size()-1 , which is counter intuitive!
 *
 * This function will be DEPRECATED soon (10/22/2020)
 *
 * The coefficient container type can be any container that provides an index
 * operator [] and a .size() method (e.g. std::vector, std::array). The return
 * type is the supertype of the container value type and the argument x.
 * The supertype is the type that can represent both number types.
 */
template <typename C,
          typename T,
          typename R = typename libMesh::CompareTypes<typename C::value_type, T>::supertype>
R
poly(const C & c, const T x, const bool derivative = false)
{
  const auto size = c.size();
  if (size == 0)
    return 0.0;

  R value = c[0];
  if (derivative)
  {
    value *= size - 1;
    for (std::size_t i = 1; i < size - 1; ++i)
      value = value * x + c[i] * (size - i - 1);
  }
  else
  {
    for (std::size_t i = 1; i < size; ++i)
      value = value * x + c[i];
  }

  return value;
}

/**
 * Evaluate a polynomial with the coefficients c at x. Note that the Polynomial
 * form is
 *   c[0] + c[1] * x + c[2] * x^2 + ...
 * The coefficient container type can be any container that provides an index
 * operator [] and a .size() method (e.g. std::vector, std::array). The return
 * type is the supertype of the container value type and the argument x.
 * The supertype is the type that can represent both number types.
 */
template <typename C,
          typename T,
          typename R = typename libMesh::CompareTypes<typename C::value_type, T>::supertype>
R
polynomial(const C & c, const T x)
{
  auto size = c.size();
  if (size == 0)
    return 0.0;

  size--;
  R value = c[size];
  for (std::size_t i = 1; i <= size; ++i)
    value = value * x + c[size - i];

  return value;
}

/**
 * Returns the derivative of polynomial(c, x) with respect to x
 */
template <typename C,
          typename T,
          typename R = typename libMesh::CompareTypes<typename C::value_type, T>::supertype>
R
polynomialDerivative(const C & c, const T x)
{
  auto size = c.size();
  if (size <= 1)
    return 0.0;

  size--;
  R value = c[size] * size;
  for (std::size_t i = 1; i < size; ++i)
    value = value * x + c[size - i] * (size - i);

  return value;
}

template <typename T, typename T2>
T
clamp(const T & x, T2 lowerlimit, T2 upperlimit)
{
  if (x < lowerlimit)
    return lowerlimit;
  if (x > upperlimit)
    return upperlimit;
  return x;
}

template <typename T, typename T2>
T
smootherStep(T x, T2 start, T2 end, bool derivative = false)
{
  mooseAssert("start < end", "Start value must be lower than end value for smootherStep");
  if (x <= start)
    return 0.0;
  else if (x >= end)
  {
    if (derivative)
      return 0.0;
    else
      return 1.0;
  }
  x = (x - start) / (end - start);
  if (derivative)
    return 30.0 * libMesh::Utility::pow<2>(x) * (x * (x - 2.0) + 1.0) / (end - start);
  return libMesh::Utility::pow<3>(x) * (x * (x * 6.0 - 15.0) + 10.0);
}

enum class ComputeType
{
  value,
  derivative
};

template <ComputeType compute_type, typename X, typename S, typename E>
auto
smootherStep(const X & x, const S & start, const E & end)
{
  mooseAssert("start < end", "Start value must be lower than end value for smootherStep");
  if (x <= start)
    return 0.0;
  else if (x >= end)
  {
    if constexpr (compute_type == ComputeType::derivative)
      return 0.0;
    if constexpr (compute_type == ComputeType::value)
      return 1.0;
  }
  const auto u = (x - start) / (end - start);
  if constexpr (compute_type == ComputeType::derivative)
    return 30.0 * libMesh::Utility::pow<2>(u) * (u * (u - 2.0) + 1.0) / (end - start);
  if constexpr (compute_type == ComputeType::value)
    return libMesh::Utility::pow<3>(u) * (u * (u * 6.0 - 15.0) + 10.0);
}

/**
 * Helper function templates to set a variable to zero.
 * Specializations may have to be implemented (for examples see
 * RankTwoTensor, RankFourTensor).
 */
template <typename T>
inline void
mooseSetToZero(T & v)
{
  /**
   * The default for non-pointer types is to assign zero.
   * This should either do something sensible, or throw a compiler error.
   * Otherwise the T type is designed badly.
   */
  v = 0;
}
template <typename T>
inline void
mooseSetToZero(T *&)
{
  mooseError("mooseSetToZero does not accept pointers");
}

template <>
inline void
mooseSetToZero(std::vector<Real> & vec)
{
  for (auto & v : vec)
    v = 0.;
}

/**
 * generate a complete multi index table for given dimension and order
 * i.e. given dim = 2, order = 2, generated table will have the following content
 * 0 0
 * 1 0
 * 0 1
 * 2 0
 * 1 1
 * 0 2
 * The first number in each entry represents the order of the first variable, i.e. x;
 * The second number in each entry represents the order of the second variable, i.e. y.
 * Multiplication is implied between numbers in each entry, i.e. 1 1 represents x^1 * y^1
 *
 * @param dim dimension of the multi-index, here dim = mesh dimension
 * @param order generate the multi-index up to certain order
 * @return a data structure holding entries representing the complete multi index
 */
std::vector<std::vector<unsigned int>> multiIndex(unsigned int dim, unsigned int order);

template <ComputeType compute_type, typename X, typename X1, typename X2, typename Y1, typename Y2>
auto
linearInterpolation(const X & x, const X1 & x1, const X2 & x2, const Y1 & y1, const Y2 & y2)
{
  const auto m = (y2 - y1) / (x2 - x1);
  if constexpr (compute_type == ComputeType::derivative)
    return m;
  if constexpr (compute_type == ComputeType::value)
    return m * (x - x1) + y1;
}

/**
 * perform modulo operator for Euclidean division that ensures a non-negative result
 * @param dividend dividend of the modulo operation
 * @param divisor divisor of the modulo operation
 * @return the non-negative remainder when the dividend is divided by the divisor
 */
template <typename T1, typename T2>
std::size_t
euclideanMod(T1 dividend, T2 divisor)
{
  return (dividend % divisor + divisor) % divisor;
}

/**
 * automatic prefixing for naming material properties based on gradients of coupled
 * variables/functors
 */
template <typename T>
T
gradName(const T & base_prop_name)
{
  return "grad_" + base_prop_name;
}

/**
 * automatic prefixing for naming material properties based on time derivatives of coupled
 * variables/functors
 */
template <typename T>
T
timeDerivName(const T & base_prop_name)
{
  return "d" + base_prop_name + "_dt";
}

/**
 * Computes the Kronecker product of two matrices.
 * @param product Reference to the product matrix
 * @param mat_A Reference to the first matrix
 * @param mat_B Reference to the other matrix
 */
void kron(RealEigenMatrix & product, const RealEigenMatrix & mat_A, const RealEigenMatrix & mat_B);

} // namespace MathUtils

/// A helper function for MathUtils::multiIndex
std::vector<std::vector<unsigned int>> multiIndexHelper(unsigned int N, unsigned int K);

(test/tests/solid_mechanics/mc_volumetric_swelling/exact.i)

# This tests ADMCVolumetricSwellingEigenstrain due to changing temperature, porosity, burnup, and Pu composition.
# Exact solution is calculated as a function in the input file. The layered side average and user
# object are not strictly necessary -- they just exist to test the method as it will usually appear
# in an input file for a fuel rod.

MWd_per_kgHM_per_FIMA = '${fparse 6.02214076e23 / 0.23803 * 3.41e-11 / 1e6 / 3600 / 24}'
scalar = 0.5
temperature_shift = 4

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  use_displaced_mesh = false
  [gen]
    type = ExamplePatchMeshGenerator
    dim = 3
  []
[]

[AuxVariables]
  [temp]
  []
  [contact_pressure]
  []
  [layered_side_contact_pressure]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [burnup_ramp]
    type = PiecewiseLinear
    x = '0     50   60'
    y = '0.015 0.03 0.4'
  []
  [porosity_ramp]
    type = PiecewiseLinear
    x = '0    10   20'
    y = '0.04 0.04 0.2'
  []
  [contact_pressure_func]
    type = PiecewiseLinear
    x = '0 20 30'
    y = '0 0 1e7'
  []
  [temp_ramp]
    type = PiecewiseLinear
    x = '0   30  40'
    y = '500 500 1500'
  []
  [pu_func]
    type = PiecewiseLinear
    x = '0 40 50 60'
    y = '0 0 0.3 0'
  []
[]

[AuxKernels]
  [temp_aux]
    type = FunctionAux
    variable = temp
    function = temp_ramp
  []
  [contact_pressure_aux]
    type = FunctionAux
    variable = contact_pressure
    function = contact_pressure_func
  []
  [layered_side_contact_pressure]
    type = SpatialUserObjectAux
    variable = layered_side_contact_pressure
    user_object = layered_side_contact_pressure
  []
[]

[UserObjects]
  [layered_side_contact_pressure]
    # This reads the contact pressure on the right side and propagates it leftward so that it can
    # be used in ADMCVolumetricSwellingEigenstrain for inner elements as well.
    type = LayeredSideAverage
    direction = y
    num_layers = 1
    variable = contact_pressure
    execute_on = linear
    boundary = right
  []
[]

[Physics/SolidMechanics/QuasiStatic/all]
  strain = SMALL
  add_variables = true
  eigenstrain_names = swelling
  use_automatic_differentiation = true
[]

[BCs]
  [left]
    type = ADDirichletBC
    variable = disp_x
    value = 0
    boundary = left
  []
  [bottom]
    type = ADDirichletBC
    variable = disp_y
    value = 0
    boundary = bottom
  []
  [back]
    type = ADDirichletBC
    variable = disp_z
    value = 0
    boundary = back
  []
[]

[Materials]
  [mats]
    type = ADGenericFunctionMaterial
    prop_names = 'porosity      pu      burnup'
    prop_values = 'porosity_ramp pu_func burnup_ramp'
    outputs = all
  []
  [elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 1
    poissons_ratio = 0.3
  []
  [swelling]
    type = ADMCVolumetricSwellingEigenstrain
    eigenstrain_name = swelling
    temperature = temp
    porosity = porosity
    contact_pressure = layered_side_contact_pressure
    burnup = burnup
    swelling_scalar = ${scalar}
    temperature_shift = ${temperature_shift}
    X_Pu_to_An_in_MC = pu
    outputs = all
  []
  [stress]
    type = ADComputeLinearElasticStress
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  nl_abs_tol = 1e-10
  end_time = 60
[]

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
  []
  [porosity]
    type = ElementAverageValue
    variable = porosity
  []
  [contact_pressure]
    type = ElementAverageValue
    variable = contact_pressure
  []
  [pu]
    type = ElementAverageValue
    variable = pu
  []
  [burnup]
    type = ElementAverageValue
    variable = burnup
  []
  [burnup_old]
    type = ElementAverageValue
    variable = burnup
    execute_on = 'TIMESTEP_BEGIN'
  []
  [volumetric_strain]
    type = ElementAverageValue
    variable = volumetric_swelling_strain
  []

  [burnup_correction]
    type = ParsedPostprocessor
    pp_names = burnup
    expression = 'burnup_mwd_kg := burnup * ${MWd_per_kgHM_per_FIMA};
                x := (burnup_mwd_kg - 20.0) / (20.2 - 20.0);
                clamped_x := if(x < 0, 0, if(x > 1, 1, x));
                mix := (clamped_x)^3 * (clamped_x * (clamped_x * 6.0 - 15.0) + 10.0);
                (burnup_mwd_kg / 10.0 - 2.0) * mix'
  []
  [porosity_correction]
    type = ParsedPostprocessor
    pp_names = 'porosity contact_pressure'
    expression = 'offset := max(porosity - 0.04, 0);
                if(offset > 0.0, exp(-offset) * exp(-contact_pressure / 1.0e6 * 0.1), 1.0)'
  []
  [swelling_old]
    type = ElementAverageValue
    variable = volumetric_swelling_strain
    execute_on = TIMESTEP_BEGIN
  []
  [strain_inc]
    type = ParsedPostprocessor
    pp_names = 'burnup burnup_old porosity_correction burnup_correction temp swelling_old pu'
    expression = 'T := temp - 273.15 + ${temperature_shift};
                burnup_mwd_kg := burnup * ${MWd_per_kgHM_per_FIMA};
                solid_swelling := 0.4667;
                athermal_swelling := (0.9171 - 1.711) / 0.2 * pu + 1.711;
                u_thermal_swelling := if(T > 700, 6.412 - 0.0198 * T + 1.52e-5 * T^2, 0);
                pu_thermal_swelling := if(T > 900, 11.8062 - 0.0257 * T + 1.398e-5 * T^2, 0);
                thermal_swelling := (pu_thermal_swelling - u_thermal_swelling) / 0.2 * pu + u_thermal_swelling;
                solid_swelling + (athermal_swelling + thermal_swelling * burnup_correction) * porosity_correction'
  []
  [strain_inc_max]
    type = ParsedPostprocessor
    pp_names = 'pu'
    expression = '(3.406 - 4.558) / 0.2 * pu + 4.558'
  []
  [swelling_exact]
    type = ParsedPostprocessor
    pp_names = 'burnup burnup_old strain_inc swelling_old pu strain_inc_max'
    expression = 'x := (strain_inc - strain_inc_max * 0.9) / (strain_inc_max - strain_inc_max * 0.9);
                clamped_x := if(x < 0, 0, if(x > 1, 1, x));
                mix := (clamped_x)^3 * (clamped_x * (clamped_x * 6.0 - 15.0) + 10.0);
                total_inc := strain_inc * (1 - mix) + strain_inc_max * mix;
                ${scalar} * total_inc * (burnup - burnup_old) * ${MWd_per_kgHM_per_FIMA} / 1000 + swelling_old'
  []

  [swelling_diff]
    type = ParsedPostprocessor
    pp_names = 'volumetric_strain swelling_exact'
    expression = '(volumetric_strain - swelling_exact) / swelling_exact'
    outputs = none
  []
  [swelling_diff_max]
    type = TimeExtremeValue
    postprocessor = swelling_diff
    value_type = abs_max
  []
[]

[Outputs]
  csv = true
[]

(test/tests/solid_mechanics/mc_volumetric_swelling/exact.i)

# This tests ADMCVolumetricSwellingEigenstrain due to changing temperature, porosity, burnup, and Pu composition.
# Exact solution is calculated as a function in the input file. The layered side average and user
# object are not strictly necessary -- they just exist to test the method as it will usually appear
# in an input file for a fuel rod.

MWd_per_kgHM_per_FIMA = '${fparse 6.02214076e23 / 0.23803 * 3.41e-11 / 1e6 / 3600 / 24}'
scalar = 0.5
temperature_shift = 4

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  use_displaced_mesh = false
  [gen]
    type = ExamplePatchMeshGenerator
    dim = 3
  []
[]

[AuxVariables]
  [temp]
  []
  [contact_pressure]
  []
  [layered_side_contact_pressure]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [burnup_ramp]
    type = PiecewiseLinear
    x = '0     50   60'
    y = '0.015 0.03 0.4'
  []
  [porosity_ramp]
    type = PiecewiseLinear
    x = '0    10   20'
    y = '0.04 0.04 0.2'
  []
  [contact_pressure_func]
    type = PiecewiseLinear
    x = '0 20 30'
    y = '0 0 1e7'
  []
  [temp_ramp]
    type = PiecewiseLinear
    x = '0   30  40'
    y = '500 500 1500'
  []
  [pu_func]
    type = PiecewiseLinear
    x = '0 40 50 60'
    y = '0 0 0.3 0'
  []
[]

[AuxKernels]
  [temp_aux]
    type = FunctionAux
    variable = temp
    function = temp_ramp
  []
  [contact_pressure_aux]
    type = FunctionAux
    variable = contact_pressure
    function = contact_pressure_func
  []
  [layered_side_contact_pressure]
    type = SpatialUserObjectAux
    variable = layered_side_contact_pressure
    user_object = layered_side_contact_pressure
  []
[]

[UserObjects]
  [layered_side_contact_pressure]
    # This reads the contact pressure on the right side and propagates it leftward so that it can
    # be used in ADMCVolumetricSwellingEigenstrain for inner elements as well.
    type = LayeredSideAverage
    direction = y
    num_layers = 1
    variable = contact_pressure
    execute_on = linear
    boundary = right
  []
[]

[Physics/SolidMechanics/QuasiStatic/all]
  strain = SMALL
  add_variables = true
  eigenstrain_names = swelling
  use_automatic_differentiation = true
[]

[BCs]
  [left]
    type = ADDirichletBC
    variable = disp_x
    value = 0
    boundary = left
  []
  [bottom]
    type = ADDirichletBC
    variable = disp_y
    value = 0
    boundary = bottom
  []
  [back]
    type = ADDirichletBC
    variable = disp_z
    value = 0
    boundary = back
  []
[]

[Materials]
  [mats]
    type = ADGenericFunctionMaterial
    prop_names = 'porosity      pu      burnup'
    prop_values = 'porosity_ramp pu_func burnup_ramp'
    outputs = all
  []
  [elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 1
    poissons_ratio = 0.3
  []
  [swelling]
    type = ADMCVolumetricSwellingEigenstrain
    eigenstrain_name = swelling
    temperature = temp
    porosity = porosity
    contact_pressure = layered_side_contact_pressure
    burnup = burnup
    swelling_scalar = ${scalar}
    temperature_shift = ${temperature_shift}
    X_Pu_to_An_in_MC = pu
    outputs = all
  []
  [stress]
    type = ADComputeLinearElasticStress
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  nl_abs_tol = 1e-10
  end_time = 60
[]

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
  []
  [porosity]
    type = ElementAverageValue
    variable = porosity
  []
  [contact_pressure]
    type = ElementAverageValue
    variable = contact_pressure
  []
  [pu]
    type = ElementAverageValue
    variable = pu
  []
  [burnup]
    type = ElementAverageValue
    variable = burnup
  []
  [burnup_old]
    type = ElementAverageValue
    variable = burnup
    execute_on = 'TIMESTEP_BEGIN'
  []
  [volumetric_strain]
    type = ElementAverageValue
    variable = volumetric_swelling_strain
  []

  [burnup_correction]
    type = ParsedPostprocessor
    pp_names = burnup
    expression = 'burnup_mwd_kg := burnup * ${MWd_per_kgHM_per_FIMA};
                x := (burnup_mwd_kg - 20.0) / (20.2 - 20.0);
                clamped_x := if(x < 0, 0, if(x > 1, 1, x));
                mix := (clamped_x)^3 * (clamped_x * (clamped_x * 6.0 - 15.0) + 10.0);
                (burnup_mwd_kg / 10.0 - 2.0) * mix'
  []
  [porosity_correction]
    type = ParsedPostprocessor
    pp_names = 'porosity contact_pressure'
    expression = 'offset := max(porosity - 0.04, 0);
                if(offset > 0.0, exp(-offset) * exp(-contact_pressure / 1.0e6 * 0.1), 1.0)'
  []
  [swelling_old]
    type = ElementAverageValue
    variable = volumetric_swelling_strain
    execute_on = TIMESTEP_BEGIN
  []
  [strain_inc]
    type = ParsedPostprocessor
    pp_names = 'burnup burnup_old porosity_correction burnup_correction temp swelling_old pu'
    expression = 'T := temp - 273.15 + ${temperature_shift};
                burnup_mwd_kg := burnup * ${MWd_per_kgHM_per_FIMA};
                solid_swelling := 0.4667;
                athermal_swelling := (0.9171 - 1.711) / 0.2 * pu + 1.711;
                u_thermal_swelling := if(T > 700, 6.412 - 0.0198 * T + 1.52e-5 * T^2, 0);
                pu_thermal_swelling := if(T > 900, 11.8062 - 0.0257 * T + 1.398e-5 * T^2, 0);
                thermal_swelling := (pu_thermal_swelling - u_thermal_swelling) / 0.2 * pu + u_thermal_swelling;
                solid_swelling + (athermal_swelling + thermal_swelling * burnup_correction) * porosity_correction'
  []
  [strain_inc_max]
    type = ParsedPostprocessor
    pp_names = 'pu'
    expression = '(3.406 - 4.558) / 0.2 * pu + 4.558'
  []
  [swelling_exact]
    type = ParsedPostprocessor
    pp_names = 'burnup burnup_old strain_inc swelling_old pu strain_inc_max'
    expression = 'x := (strain_inc - strain_inc_max * 0.9) / (strain_inc_max - strain_inc_max * 0.9);
                clamped_x := if(x < 0, 0, if(x > 1, 1, x));
                mix := (clamped_x)^3 * (clamped_x * (clamped_x * 6.0 - 15.0) + 10.0);
                total_inc := strain_inc * (1 - mix) + strain_inc_max * mix;
                ${scalar} * total_inc * (burnup - burnup_old) * ${MWd_per_kgHM_per_FIMA} / 1000 + swelling_old'
  []

  [swelling_diff]
    type = ParsedPostprocessor
    pp_names = 'volumetric_strain swelling_exact'
    expression = '(volumetric_strain - swelling_exact) / swelling_exact'
    outputs = none
  []
  [swelling_diff_max]
    type = TimeExtremeValue
    postprocessor = swelling_diff
    value_type = abs_max
  []
[]

[Outputs]
  csv = true
[]

(test/tests/solid_mechanics/mc_volumetric_swelling/exact.i)

# This tests ADMCVolumetricSwellingEigenstrain due to changing temperature, porosity, burnup, and Pu composition.
# Exact solution is calculated as a function in the input file. The layered side average and user
# object are not strictly necessary -- they just exist to test the method as it will usually appear
# in an input file for a fuel rod.

MWd_per_kgHM_per_FIMA = '${fparse 6.02214076e23 / 0.23803 * 3.41e-11 / 1e6 / 3600 / 24}'
scalar = 0.5
temperature_shift = 4

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  use_displaced_mesh = false
  [gen]
    type = ExamplePatchMeshGenerator
    dim = 3
  []
[]

[AuxVariables]
  [temp]
  []
  [contact_pressure]
  []
  [layered_side_contact_pressure]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [burnup_ramp]
    type = PiecewiseLinear
    x = '0     50   60'
    y = '0.015 0.03 0.4'
  []
  [porosity_ramp]
    type = PiecewiseLinear
    x = '0    10   20'
    y = '0.04 0.04 0.2'
  []
  [contact_pressure_func]
    type = PiecewiseLinear
    x = '0 20 30'
    y = '0 0 1e7'
  []
  [temp_ramp]
    type = PiecewiseLinear
    x = '0   30  40'
    y = '500 500 1500'
  []
  [pu_func]
    type = PiecewiseLinear
    x = '0 40 50 60'
    y = '0 0 0.3 0'
  []
[]

[AuxKernels]
  [temp_aux]
    type = FunctionAux
    variable = temp
    function = temp_ramp
  []
  [contact_pressure_aux]
    type = FunctionAux
    variable = contact_pressure
    function = contact_pressure_func
  []
  [layered_side_contact_pressure]
    type = SpatialUserObjectAux
    variable = layered_side_contact_pressure
    user_object = layered_side_contact_pressure
  []
[]

[UserObjects]
  [layered_side_contact_pressure]
    # This reads the contact pressure on the right side and propagates it leftward so that it can
    # be used in ADMCVolumetricSwellingEigenstrain for inner elements as well.
    type = LayeredSideAverage
    direction = y
    num_layers = 1
    variable = contact_pressure
    execute_on = linear
    boundary = right
  []
[]

[Physics/SolidMechanics/QuasiStatic/all]
  strain = SMALL
  add_variables = true
  eigenstrain_names = swelling
  use_automatic_differentiation = true
[]

[BCs]
  [left]
    type = ADDirichletBC
    variable = disp_x
    value = 0
    boundary = left
  []
  [bottom]
    type = ADDirichletBC
    variable = disp_y
    value = 0
    boundary = bottom
  []
  [back]
    type = ADDirichletBC
    variable = disp_z
    value = 0
    boundary = back
  []
[]

[Materials]
  [mats]
    type = ADGenericFunctionMaterial
    prop_names = 'porosity      pu      burnup'
    prop_values = 'porosity_ramp pu_func burnup_ramp'
    outputs = all
  []
  [elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 1
    poissons_ratio = 0.3
  []
  [swelling]
    type = ADMCVolumetricSwellingEigenstrain
    eigenstrain_name = swelling
    temperature = temp
    porosity = porosity
    contact_pressure = layered_side_contact_pressure
    burnup = burnup
    swelling_scalar = ${scalar}
    temperature_shift = ${temperature_shift}
    X_Pu_to_An_in_MC = pu
    outputs = all
  []
  [stress]
    type = ADComputeLinearElasticStress
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  nl_abs_tol = 1e-10
  end_time = 60
[]

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
  []
  [porosity]
    type = ElementAverageValue
    variable = porosity
  []
  [contact_pressure]
    type = ElementAverageValue
    variable = contact_pressure
  []
  [pu]
    type = ElementAverageValue
    variable = pu
  []
  [burnup]
    type = ElementAverageValue
    variable = burnup
  []
  [burnup_old]
    type = ElementAverageValue
    variable = burnup
    execute_on = 'TIMESTEP_BEGIN'
  []
  [volumetric_strain]
    type = ElementAverageValue
    variable = volumetric_swelling_strain
  []

  [burnup_correction]
    type = ParsedPostprocessor
    pp_names = burnup
    expression = 'burnup_mwd_kg := burnup * ${MWd_per_kgHM_per_FIMA};
                x := (burnup_mwd_kg - 20.0) / (20.2 - 20.0);
                clamped_x := if(x < 0, 0, if(x > 1, 1, x));
                mix := (clamped_x)^3 * (clamped_x * (clamped_x * 6.0 - 15.0) + 10.0);
                (burnup_mwd_kg / 10.0 - 2.0) * mix'
  []
  [porosity_correction]
    type = ParsedPostprocessor
    pp_names = 'porosity contact_pressure'
    expression = 'offset := max(porosity - 0.04, 0);
                if(offset > 0.0, exp(-offset) * exp(-contact_pressure / 1.0e6 * 0.1), 1.0)'
  []
  [swelling_old]
    type = ElementAverageValue
    variable = volumetric_swelling_strain
    execute_on = TIMESTEP_BEGIN
  []
  [strain_inc]
    type = ParsedPostprocessor
    pp_names = 'burnup burnup_old porosity_correction burnup_correction temp swelling_old pu'
    expression = 'T := temp - 273.15 + ${temperature_shift};
                burnup_mwd_kg := burnup * ${MWd_per_kgHM_per_FIMA};
                solid_swelling := 0.4667;
                athermal_swelling := (0.9171 - 1.711) / 0.2 * pu + 1.711;
                u_thermal_swelling := if(T > 700, 6.412 - 0.0198 * T + 1.52e-5 * T^2, 0);
                pu_thermal_swelling := if(T > 900, 11.8062 - 0.0257 * T + 1.398e-5 * T^2, 0);
                thermal_swelling := (pu_thermal_swelling - u_thermal_swelling) / 0.2 * pu + u_thermal_swelling;
                solid_swelling + (athermal_swelling + thermal_swelling * burnup_correction) * porosity_correction'
  []
  [strain_inc_max]
    type = ParsedPostprocessor
    pp_names = 'pu'
    expression = '(3.406 - 4.558) / 0.2 * pu + 4.558'
  []
  [swelling_exact]
    type = ParsedPostprocessor
    pp_names = 'burnup burnup_old strain_inc swelling_old pu strain_inc_max'
    expression = 'x := (strain_inc - strain_inc_max * 0.9) / (strain_inc_max - strain_inc_max * 0.9);
                clamped_x := if(x < 0, 0, if(x > 1, 1, x));
                mix := (clamped_x)^3 * (clamped_x * (clamped_x * 6.0 - 15.0) + 10.0);
                total_inc := strain_inc * (1 - mix) + strain_inc_max * mix;
                ${scalar} * total_inc * (burnup - burnup_old) * ${MWd_per_kgHM_per_FIMA} / 1000 + swelling_old'
  []

  [swelling_diff]
    type = ParsedPostprocessor
    pp_names = 'volumetric_strain swelling_exact'
    expression = '(volumetric_strain - swelling_exact) / swelling_exact'
    outputs = none
  []
  [swelling_diff_max]
    type = TimeExtremeValue
    postprocessor = swelling_diff
    value_type = abs_max
  []
[]

[Outputs]
  csv = true
[]

(assessment/carbide/EBRII/WSA32/analysis/base.i)

initial_fuel_density = 13630.0

[GlobalParams]
  density = ${initial_fuel_density}
  order = SECOND
  family = LAGRANGE
  energy_per_fission = 3.2e-11 # J/fission
  volumetric_locking_correction = false
  displacements = 'disp_x disp_y'
  value_range_behavior = IGNORE
[]

[Problem]
  type = ReferenceResidualProblem
  reference_vector = 'ref'
  extra_tag_vectors = 'ref'
  group_variables = 'disp_x disp_y'
[]

[Mesh]
  coord_type = RZ
  use_displaced_mesh = false
  [smeared_pellet_mesh]
    type = FuelPinMeshGenerator
    clad_thickness = 0.508e-3
    pellet_outer_radius = 4.301e-3
    pellet_height = 343e-3
    clad_top_gap_height = 43.5e-3
    clad_gap_width = 0.145e-3
    bottom_clad_height = 8e-3
    top_clad_height = 8e-3
    clad_bot_gap_height = 0.2e-3 # unknown
    clad_mesh_density = customize
    pellet_mesh_density = customize
    nx_p = 6
    ny_p = 260
    nx_c = 4
    ny_c = 260
    ny_cu = 3
    ny_cl = 3
    pellet_quantity = 1
    elem_type = QUAD8
  []
  patch_size = 30
  patch_update_strategy = auto
  partitioner = centroid
  centroid_partitioner_direction = y
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
  [temp]
    initial_condition = 350
  []
[]

[AuxVariables]
  [contact_pressure]
  []
  [layered_side_contact_pressure]
    block = pellet
    order = CONSTANT
    family = MONOMIAL
  []
  [pellet_wall_temp]
    order = CONSTANT
    family = MONOMIAL
  []
  [radial_heat_flux]
    order = CONSTANT
    family = MONOMIAL
  []
  [clad_inner_wall_temp]
    order = CONSTANT
    family = MONOMIAL
  []
  [volumetric_strain]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [power_history]
    type = PiecewiseLinear
    x = '0 1e5   70e6 70100000' # unknown, final burnup was 12.2 %
    y = '0 1     1    0'
  []
  [axial_peaking_factors]
    type = PiecewiseLinear
    axis = y
    x = '0    46e-3 111e-3 200e-3 248e-3 313e-3 352e-3'
    y = '86e3 86e3  97e3   100e3  96e3   87e3   87e3'
  []
  [fission_func]
    type = ParsedFunction
    symbol_names = 'axial_peaking_factors power_history'
    symbol_values = 'axial_peaking_factors power_history'
    expression = 'linear_heating_rate := axial_peaking_factors * power_history;
             pellet_cross_sectional_area := 3.14159 * pow( 4.301e-3, 2 );
             volumetric_power := linear_heating_rate / pellet_cross_sectional_area;
             energy_per_fission := 3.2e-11;
             volumetric_power / energy_per_fission'
  []
  [burnup_func] # Burnup func is needed because BurnupAux doesn't currentl export an ADMaterialProperty.
    type = PiecewiseLinear
    x = '0 1e5   70e6  70100000'
    y = '0 0     0.122 0.122'
  []

  [coolant_press_ramp]
    type = PiecewiseLinear
    x = '0       70100000'
    y = '0.151e6 0.151e6' # unknown, Na coolant
  []
  [coolant_inlet_temp]
    type = PiecewiseLinear
    x = '0   1e5 70e6 70100000'
    y = '350 644 644  350'
  []
  [coolant_outlet_temp]
    type = PiecewiseLinear
    x = '0   1e5 70e6 70100000'
    y = '350 746 746  350'
  []
  [coolant_wall_temp]
    type = ParsedFunction
    symbol_values = 'coolant_inlet_temp coolant_outlet_temp'
    symbol_names = 'coolant_inlet_temp coolant_outlet_temp'
    expression = 'rod_length := 343e-3 + 43.5e-3 + 2 * 8e-3;
             delta_temp := coolant_outlet_temp - coolant_inlet_temp;
             interpolated_temp := coolant_inlet_temp + y * delta_temp / rod_length;
             max( min( interpolated_temp, coolant_outlet_temp ), coolant_inlet_temp )'
  []
  [radial_heat_flux]
    type = ParsedFunction
    symbol_values = 'coolant_wall_temp power_history axial_peaking_factors'
    symbol_names = 'coolant_wall_temp power_history axial_peaking_factors'
    expression = 'linear_heating_rate := axial_peaking_factors * power_history;
             pellet_radius := 4.301e-3;
             pellet_circumference := 3.14159 * pellet_radius * 2;
             linear_heating_rate / pellet_circumference'
  []
  [clad_inner_wall_temp]
    type = ParsedFunction
    symbol_values = 'radial_heat_flux coolant_wall_temp'
    symbol_names = 'radial_heat_flux coolant_wall_temp'
    expression = 'conductivity_316ss := 22;
             clad_thickness := 0.51e-3;
             coolant_wall_temp + radial_heat_flux / conductivity_316ss * clad_thickness'
  []
  [pellet_wall_temp]
    type = ParsedFunction
    symbol_values = 'radial_heat_flux coolant_wall_temp clad_inner_wall_temp'
    symbol_names = 'radial_heat_flux coolant_wall_temp clad_inner_wall_temp'
    expression = 'conductivity_gap := 1;
             gap_thickness := 0.145e-3;
             clad_inner_wall_temp + radial_heat_flux / conductivity_gap * gap_thickness'
  []

  [contact_pressure_func]
    type = ConstantFunction
    value = 12e6
  []
  [plenum_pressure]
    type = ConstantFunction
    value = 12.4e6 # initial fill
  []

  [porosity_ramp]
    type = PiecewiseLinear
    x = '0    1e5  70e6  70100000'
    y = '0.14 0.14 0.145 0.145' # Pellets start at 88.1% TD, but no data on final porosity.
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [fuel]
    block = pellet
    strain = SMALL
    incremental = true
    extra_vector_tags = 'ref'
    eigenstrain_names = 'fuel_thermal_expansion fuel_swelling'
    use_automatic_differentiation = true
  []
  [clad]
    block = clad
    strain = SMALL
    incremental = true
    extra_vector_tags = 'ref'
    use_automatic_differentiation = true
  []
[]

[Kernels]
  [gravity]
    type = ADGravity
    variable = disp_y
    value = -9.81
    extra_vector_tags = 'ref'
  []
  [heat]
    type = ADHeatConduction
    variable = temp
    extra_vector_tags = 'ref'
  []
  [heat_ie]
    type = ADHeatConductionTimeDerivative
    variable = temp
    extra_vector_tags = 'ref'
  []
  [heat_source]
    type = ADFissionRateHeatSource
    variable = temp
    block = pellet
    fission_rate = fission_rate
    extra_vector_tags = 'ref'
  []
[]

[AuxKernels]
  [contact_pressure_aux]
    type = FunctionAux
    variable = contact_pressure
    function = contact_pressure_func
  []
  [layered_side_contact_pressure]
    type = SpatialUserObjectAux
    variable = layered_side_contact_pressure
    user_object = layered_side_contact_pressure
    block = pellet
  []

  [pellet_wall_temp]
    type = FunctionAux
    variable = pellet_wall_temp
    function = pellet_wall_temp
  []
  [radial_heat_flux]
    type = FunctionAux
    variable = radial_heat_flux
    function = radial_heat_flux
  []
  [clad_inner_wall_temp]
    type = FunctionAux
    variable = clad_inner_wall_temp
    function = clad_inner_wall_temp
  []
  [volumetric_strain]
    type = ADRankTwoScalarAux
    rank_two_tensor = total_strain
    variable = volumetric_strain
    scalar_type = VolumetricStrain
  []
[]

[BCs]
  [no_x_all]
    type = DirichletBC
    variable = disp_x
    boundary = 12
    value = 0.0
  []
  [no_y_fuel]
    type = DirichletBC
    variable = disp_y
    boundary = 20
    value = 0.0
  []
  [no_y_clad]
    type = DirichletBC
    variable = disp_y
    boundary = 1
    value = 0.0
  []
  [Pressure]
    [coolantPressure]
      boundary = '1 2 3'
      function = coolant_press_ramp
    []
    [plenumPressure]
      boundary = 9
      function = plenum_pressure
    []
  []
  [clad_outer_temp]
    type = FunctionDirichletBC
    boundary = '1 2 3'
    function = coolant_wall_temp
    variable = temp
  []
  [pellet_to_clad_cooling]
    type = FunctionDirichletBC
    boundary = 10
    function = pellet_wall_temp
    variable = temp
  []
[]

[Materials]
  [fission]
    type = ADGenericFunctionMaterial
    prop_names = fission_rate
    prop_values = fission_func
    outputs = all
  []
  [burnup]
    type = ADGenericFunctionMaterial
    block = pellet
    prop_names = burnup
    prop_values = burnup_func
    outputs = all
  []
  [fuel_thermal]
    block = pellet
    type = ADMCThermal
    temperature = temp
    porosity = porosity
    X_Pu = 0.0
    outputs = all
  []
  [fuel_porosity]
    block = pellet
    type = ADGenericFunctionMaterial
    prop_names = porosity
    prop_values = porosity_ramp
    outputs = all
  []
  [fuel_elasticity_tensor]
    block = pellet
    type = ADMCElasticityTensor
    temperature = temp
    porosity = porosity
    output_properties = 'youngs_modulus poissons_ratio'
    outputs = all
  []
  [fuel_thermal_expansion]
    block = pellet
    type = ADMCThermalExpansionEigenstrain
    eigenstrain_name = fuel_thermal_expansion
    stress_free_temperature = 350
    temperature = temp
  []
  [fuel_creep]
    block = pellet
    type = ADMCCreepUpdate
    temperature = temp
    fission_rate = fission_rate
    outputs = all
  []
  [fuel_swelling]
    block = pellet
    type = ADMCVolumetricSwellingEigenstrain
    eigenstrain_name = fuel_swelling
    temperature = temp
    porosity = porosity
    contact_pressure = layered_side_contact_pressure
    burnup = burnup
    outputs = all
  []
  [fuel_radial_return_stress]
    block = pellet
    type = ADComputeMultipleInelasticStress
    inelastic_models = 'fuel_creep'
  []
  [fuel_density]
    block = pellet
    type = ADStrainAdjustedDensity
    strain_free_density = ${initial_fuel_density}
  []

  [clad_elasticity_tensor]
    block = clad
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 190e9
    poissons_ratio = 0.265
  []
  [clad_stress]
    block = clad
    type = ADComputeLinearElasticStress
  []
  [clad_thermal]
    block = clad
    type = ADSS316Thermal
    temperature = temp
  []
  [clad_density]
    block = clad
    type = ADStrainAdjustedDensity
    strain_free_density = 8000
  []
[]

[UserObjects]
  [layered_side_contact_pressure]
    # This reads the contact pressure at the pellet OD and propagates it inward so that it can
    # be used in ADMCVolumetricSwellingEigenstrain for inner elements as well.
    type = LayeredSideAverage
    direction = y
    num_layers = 100
    variable = contact_pressure
    execute_on = linear
    boundary = 10
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]

[Dampers]
  [limitT]
    type = MaxIncrement
    max_increment = 100.0
    variable = temp
  []
  [limitX]
    type = MaxIncrement
    max_increment = 1e-5
    variable = disp_x
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 50
  l_tol = 8e-3
  nl_max_its = 15
  nl_rel_tol = 1e-4
  nl_abs_tol = 1e-10

  start_time = -200
  n_startup_steps = 1
  end_time = 70100000
  dtmin = 1
  dtmax = 2e6

  [Quadrature]
    order = fifth
    side_order = seventh
  []

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 2e2
    time_t = ' 0   1e5 70e6 70100000'
    time_dt = '1e2 1e2 1e2  1e2'
    optimal_iterations = 10
    growth_factor = 2
    cutback_factor = 0.5
    iteration_window = 1
    linear_iteration_ratio = 100
  []
[]

[Postprocessors]
  [_dt]
    type = TimestepSize
  []
  [num_lin_it]
    type = NumLinearIterations
  []
  [num_nonlin_it]
    type = NumNonlinearIterations
  []
  [burnup]
    block = pellet
    type = ADElementAverageMaterialProperty
    mat_prop = burnup
  []
  [FCT]
    type = NodalExtremeValue
    boundary = 12
    variable = temp
    execute_on = 'initial timestep_end'
  []
  [volumetric_strain]
    type = ElementAverageValue
    block = pellet
    variable = volumetric_strain
  []
  [pellet_outer_disp_x]
    type = NodalExtremeValue
    boundary = 10
    variable = disp_x
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  color = true
  exodus = true
  perf_graph = true
  csv = true
  [console]
    type = Console
    max_rows = 25
    time_step_interval =  1
    output_linear = true
  []
  [chkfile]
    type = CSV
    hide = 'num_nonlin_it num_lin_it _dt'
  []
[]

Description
Example Input Syntax
Input Parameters
Input Files
References