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mainpage.h
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/**
@{
\brief Implementation of transformation into the logarithmic strain space in deal.ii
\author jfriedlein
@tableofcontents
@mainpage Transformation into the logarithmic strain space
@section intro Introduction
@todo refer to papers by Miehe
@subsection subsec_overview Overview
This overview shall give you a first impression what to expect ...
@subsection subsec_interface Interface - How to incorporate the transformation into your code
@todo add a simple example code that calls the pre and post with e.g. an elastic matmod as coding example in section code
@subsection subsec_more_basics Some more basics
@subsection subsec_resources Some resources/links
@section code The commented program
\code
#ifndef ln_space_H
#define ln_space_H
\endcode
@section includes Include Files
The data type SymmetricTensor and some related operations, such as trace, symmetrize, deviator, ... for tensor calculus
\code
#include <deal.II/base/symmetric_tensor.h>
#include <deal.II/base/exceptions.h>
\endcode
@todo Check whether the following three headers are needed at all
\code
#include <iostream>
#include <fstream>
#include <cmath>
#include "functions.h"
using namespace dealii;
namespace ln_space
{
// template<int dim>
// void transform ( /*input->*/ Tensor<2, dim> &F, double &alpha_n, SymmetricTensor<2,dim> &eps_p_n, void (*func_pointer)(auto),
// /*output->*/ SymmetricTensor<2,dim> &stress_S, SymmetricTensor<4,dim> &C, SymmetricTensor<4,dim> &Lambda, double &stress_vM )
// {
// Vector<double> eigenvalues(dim);
// std::vector< Tensor<1, dim> > eigenvector(dim);
// std::vector< SymmetricTensor<2, dim> > eigenbasis(dim);
// Vector<double> ea(dim);
// Vector<double> da(dim);
// Vector<double> fa(dim);
// SymmetricTensor<2, dim> hencky_strain;
\endcode
\code
// pre_ln ( /*input->*/ F,
// /*output->*/ hencky_strain, ea, da, fa, eigenvector, eigenvalues, eigenbasis );
\endcode
\code
// // We only create this variable to emphasise the difference between the stress \a T from the
// // small strain material modell and the 2nd Piola Kirchhoff stress \a S transformed back by the post processing
// SymmetricTensor<2,dim> T_stress;
\endcode
\code
// func_pointer( /*input->*/ hencky_strain, alpha_n, eps_p_n, /*output->*/ T_stress, stress_vM, C, Lambda );
\endcode
\code
// // ToDo: We also have to transform Lambda, stress_vM
\endcode
\code
// // also transforms the T_stress to the 2nd Piola-Kirchhoff stress
// post_ln ( /*input->*/ ea, da, fa, eigenvalues, eigenbasis,
// /*output->*/ T_stress, C );
\endcode
\code
// stress_S = T_stress;
// }
template<int dim>
void pre_ln ( /*input->*/ Tensor<2, dim> &F,
/*output->*/ SymmetricTensor<2,dim> &hencky_strain, Vector<double> &ea, Vector<double> &da, Vector<double> &fa,
std::vector<Tensor<1,dim> > &eigenvector, Vector<double> &eigenvalues, std::vector< SymmetricTensor<2,dim> > &eigenbasis )
{
\endcode
Following "Algorithms for computation of stresses and elasticity moduli in terms of Seth–Hill’s family of generalized strain tensors" by Miehe&Lambrecht \n
Table I. Algorithm A
\code
/*
* 1. Eigenvalues, eigenvalue bases and diagonal functions:
*/
\endcode
Get the symmetric right cauchy green tensor
\code
SymmetricTensor<2, dim> right_cauchy_green_sym = symmetrize( transpose(F) * F);//Physics::Elasticity::Kinematics::C(F);
\endcode
Compute Eigenvalues, Eigenvectors and Eigenbasis
\code
{
\endcode
Get Eigenvalues and Eigenvectors from the deal.ii function \a eigenvectors(*)
\code
for (unsigned int i = 0; i < dim; ++i) {
eigenvalues[i] = eigenvectors(right_cauchy_green_sym)[i].first;
eigenvector[i] = eigenvectors(right_cauchy_green_sym)[i].second;
}
\endcode
Check if the found eigenvectors are perpendicular to each other
\code
if ((std::fabs(eigenvalues(0) - 1) > 1e-10)
&& (std::fabs(eigenvalues(1) - 1) > 1e-10)
&& (std::fabs(eigenvalues(2) - 1) > 1e-10)) {
for (unsigned int i = 0; i < dim; ++i) {
for (unsigned int j = i + 1; j < dim; ++j) {
AssertThrow( (std::fabs(eigenvector[i] * eigenvector[j]) < 1e-12),
ExcMessage("ln-space<< Eigenvectors are not perpendicular to each other.") );
}
}
}
\endcode
Compute eigenbasis Ma: eigenbasis = eigenvector \otimes eigenvector
\code
for (unsigned int i = 0; i < dim; ++i) {
eigenbasis[i] = symmetrize( outer_product(eigenvector[i], eigenvector[i]) );
AssertThrow( eigenvalues(i) >= 0.0,
ExcMessage("ln-space<< Eigenvalue is negativ. Check update_qph.") );
}
}
\endcode
Compute diagonal function \a ea and its first and second derivate \a da and \a fa \n
\code
for (unsigned int i = 0; i < dim; ++i)
{
ea(i) = 0.5 * std::log( std::abs(eigenvalues(i)) ); // diagonal function
da(i) = std::pow(eigenvalues(i), -1.0); // first derivative of diagonal function ea
fa(i) = -2.0 * std::pow(eigenvalues(i), -2.0); // second derivative of diagonal function ea
AssertThrow( ea(i) == ea(i),
ExcMessage( "ln-space<< Ea is nan due to logarithm of negativ eigenvalue. Check update_qph.") );
AssertThrow( da(i) > 0.0,
ExcMessage( "ln-space<< First derivative da of diagonal function is "+std::to_string(da(i))+" < 0.0 . Check update_qph.") );
}
\endcode
Compute the Hencky strain
\code
for (unsigned int a = 0; a < dim; ++a)
hencky_strain += ea(a) * eigenbasis[a];
\endcode
Output-> SymmetricTensor<2, dim> hencky_strain, Vector<double> ea, da, fa,
\code
// std::vector<Tensor<1, dim>> eigenvector, Vector<double> eigenvalues, std::vector< SymmetricTensor<2, dim> > eigenbasis
}
template<int dim>
void post_ln ( /*input->*/ Vector<double> &ea, Vector<double> &da, Vector<double> &fa,
Vector<double> &eigenvalues, std::vector< SymmetricTensor<2,dim> > &eigenbasis,
/*output->*/ SymmetricTensor<2,dim> &second_piola_stress_S, SymmetricTensor<4,dim> &elasto_plastic_tangent )
{
const double comp_tolerance = 1e-8;
SymmetricTensor<2,dim> stress_measure_T_sym = second_piola_stress_S; // the output argument is abused as an input argument
/*
* 3. Set up coefficients \a theta, \a xi and \a eta
*/
\endcode
Compute the coefficients based on the eigenvalues, eigenvectors and ea,da,fa
\code
Tensor<2, dim> theta;
Tensor<2, dim> xi;
double eta = 999999999.0;
\endcode
For three different eigenvalues \f$ \lambda_a \neq \lambda_b \neq \lambda_c \f$
\code
if (
( !(std::fabs(eigenvalues(0) - eigenvalues(1)) < comp_tolerance) )
&&
( !(std::fabs(eigenvalues(0) - eigenvalues(2)) < comp_tolerance) )
&&
( !(std::fabs(eigenvalues(1) - eigenvalues(2)) < comp_tolerance) )
)
{
eta = 0.0;
for (unsigned int a = 0; a < dim; ++a)
for (unsigned int b = 0; b < dim; ++b)
if (a != b)
{
theta[a][b] = (ea(a) - ea(b))
/ (eigenvalues(a) - eigenvalues(b));
xi[a][b] = (theta[a][b] - 0.5 * da(b))
/ (eigenvalues(a) - eigenvalues(b));
for (unsigned int c = 0; c < dim; ++c)
if ((c != a) && (c != b))
{
eta +=
ea(a)
/ (2.0
* (eigenvalues(a)
- eigenvalues(b))
* (eigenvalues(a)
- eigenvalues(c)));
}
}
}
// For three equal eigenvalues \f$ \lambda_a = \lambda_b = \lambda_c \f$
else if ( (std::fabs(eigenvalues(0) - eigenvalues(1)) < comp_tolerance)
&&
(std::fabs(eigenvalues(1) - eigenvalues(2)) < comp_tolerance) )
{
eta = 0.0;
for (unsigned int a = 0; a < dim; ++a)
{
for (unsigned int b = 0; b < dim; ++b)
if (a != b)
{
theta[a][b] = 0.5 * da(0);
xi[a][b] = (1.0 / 8.0) * fa(0);
}
}
eta = (1.0 / 8.0) * fa(0);
}
\endcode
For two equal eigenvalues a and b: \f$ \lambda_a = \lambda_b \neq \lambda_c \f$
\code
else if ( (std::fabs(eigenvalues(0) - eigenvalues(1)) < comp_tolerance)
&&
( !(std::fabs(eigenvalues(1) - eigenvalues(2)) < comp_tolerance) ) )
{
eta = 0.0;
for (unsigned int a = 0; a < dim; ++a)
for (unsigned int b = 0; b < dim; ++b)
if ((a != b) && ((a == 2) || (b == 2)))
{
theta[a][b] = (ea(a) - ea(b))
/ (eigenvalues(a) - eigenvalues(b));
xi[a][b] = (theta[a][b] - 0.5 * da(b))
/ (eigenvalues(a) - eigenvalues(b));
}
theta[0][1] = 0.5 * da(0);
theta[1][0] = theta[0][1];
xi[0][1] = (1.0 / 8.0) * fa(0);
xi[1][0] = xi[0][1];
eta = xi[2][0];
}
\endcode
For two equal eigenvalues a and c: \f$ \lambda_a = \lambda_c \neq \lambda_b \f$
\code
else if ( (std::fabs(eigenvalues(0) - eigenvalues(2)) < comp_tolerance)
&&
(!(std::fabs(eigenvalues(1) - eigenvalues(2)) < comp_tolerance)) )
{
eta = 0.0;
for (unsigned int a = 0; a < dim; ++a)
for (unsigned int b = 0; b < dim; ++b)
if ( (a != b) && ((a == 1) || (b == 1)) )
{
theta[a][b] = (ea(a) - ea(b))
/ (eigenvalues(a) - eigenvalues(b));
xi[a][b] = (theta[a][b] - 0.5 * da(b))
/ (eigenvalues(a) - eigenvalues(b));
}
theta[0][2] = 0.5 * da(0);
theta[2][0] = theta[0][2];
xi[0][2] = (1.0 / 8.0) * fa(0);
xi[2][0] = xi[0][2];
eta = xi[1][0];
}
\endcode
For two equal eigenvalues b and c: \f$ \lambda_b = \lambda_c \neq \lambda_a \f$
\code
else if ( (std::fabs(eigenvalues(1) - eigenvalues(2)) < comp_tolerance)
&&
(!(std::fabs(eigenvalues(0) - eigenvalues(1)) < comp_tolerance)) )
{
eta = 0.0;
for (unsigned int a = 0; a < dim; ++a)
for (unsigned int b = 0; b < dim; ++b)
if ( (a != b) && ((a == 0) || (b == 0)) )
{
theta[a][b] = (ea(a) - ea(b))
/ (eigenvalues(a) - eigenvalues(b));
xi[a][b] = (theta[a][b] - 0.5 * da(b))
/ (eigenvalues(a) - eigenvalues(b));
}
theta[1][2] = 0.5 * da(1);
theta[2][1] = theta[1][2];
xi[1][2] = (1.0 / 8.0) * fa(1);
xi[2][1] = xi[1][2];
eta = xi[0][1];
}
else
{
deallog << "ln-space<< eigenvalues:0: " << eigenvalues[0] << std::endl;
deallog << "ln-space<< eigenvalues:1: " << eigenvalues[1] << std::endl;
deallog << "ln-space<< eigenvalues:2: " << eigenvalues[2] << std::endl;
AssertThrow( false,
ExcMessage("ln-space<< Eigenvalue case not possible, check update_qph!") );
}
\endcode
Ensure that \a eta was initialised in one of the cases
\code
AssertThrow( (eta < 9999999), ExcMessage("Eta in update_qph not initialised") );
/*
* 4. Lagrangian stresses and elasticity moduli
*/
\endcode
Compute projection tensor P
\code
Tensor<4, dim> projection_tensor_P;
for (unsigned int a = 0; a < dim; ++a)
{
projection_tensor_P += da(a) * (Tensor<4,dim> ) outer_product(eigenbasis[a],eigenbasis[a]);
for (unsigned int b = 0; b < dim; ++b)
if (b != a)
projection_tensor_P += theta[a][b] * get_tensor_operator_G(eigenbasis[a],eigenbasis[b]);
}
\endcode
Check whether the projecton tensor is symmetric and store it into a \a SymmetricTensor
\code
AssertThrow( symmetry_check(projection_tensor_P), ExcMessage( "ln-space<< Projection tensor P is not symmetric") );
SymmetricTensor<4,dim> projection_tensor_P_sym = symmetrize(projection_tensor_P);
\endcode
Compute the double contraction of T and L
\code
Tensor<4, dim> projection_tensor_T_doublecon_L;
for (unsigned int a = 0; a < dim; ++a)
{
projection_tensor_T_doublecon_L += fa(a)
* (stress_measure_T_sym * eigenbasis[a])
* (Tensor<4, dim> ) (outer_product(eigenbasis[a], eigenbasis[a]));
for (unsigned int b = 0; b < dim; ++b)
if (b != a)
{
projection_tensor_T_doublecon_L += 2.0 * xi[a][b]
* (
get_tensor_operator_F_right( eigenbasis[a], eigenbasis[b], eigenbasis[b], stress_measure_T_sym )
+ get_tensor_operator_F_left( eigenbasis[a], eigenbasis[b], eigenbasis[b], stress_measure_T_sym )
+ get_tensor_operator_F_right( eigenbasis[b], eigenbasis[b], eigenbasis[a], stress_measure_T_sym )
+ get_tensor_operator_F_left( eigenbasis[b], eigenbasis[b], eigenbasis[a], stress_measure_T_sym )
+ get_tensor_operator_F_right( eigenbasis[b], eigenbasis[a], eigenbasis[b], stress_measure_T_sym )
+ get_tensor_operator_F_left( eigenbasis[b], eigenbasis[a], eigenbasis[b], stress_measure_T_sym )
);
for (unsigned int c = 0; c < dim; ++c)
if ( (c != a) && (c != b) )
{
projection_tensor_T_doublecon_L += 2.0 * eta
* (
get_tensor_operator_F_right( eigenbasis[a], eigenbasis[b], eigenbasis[c], stress_measure_T_sym )
+ get_tensor_operator_F_left( eigenbasis[b], eigenbasis[c], eigenbasis[a], stress_measure_T_sym )
);
}
}
}
\endcode
Check whether the tensor is symmetric and store it into a \a SymmetricTensor
\code
AssertThrow( symmetry_check(projection_tensor_T_doublecon_L),
ExcMessage("ln-space<< Projection tensor T:L is not symmetric") );
SymmetricTensor<4,dim> projection_tensor_T_doublecon_L_sym = symmetrize(projection_tensor_T_doublecon_L);
\endcode
Compute the retransformed values
\code
second_piola_stress_S = stress_measure_T_sym * projection_tensor_P_sym;
elasto_plastic_tangent = projection_tensor_P_sym * elasto_plastic_tangent * projection_tensor_P_sym // Note: we work on the input argument \a elasto_plastic_tangent
+ projection_tensor_T_doublecon_L_sym;
}
}
#endif // ln_space_H
\endcode
@section END The End
Hosted via GitHub according to https://goseeky.wordpress.com/2017/07/22/documentation-101-doxygen-with-github-pages/ \n
Design of the documentation inspired by the deal.ii tutorial programs.
@}
*/