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Content of this file 
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1) What's this
2) How to use
3) Bug reporting
4) License information
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1)     WHAT'S THIS
==============================
This Reduced Basis (RB) tool is an efficient solution to the problem of
hyperelastic homogenization under finite strains. It takes the macroscopic
stretch tensor "Ubar" as input and outputs the corresponding homogenized first
Piola-Kirchhoff stress "Pbar". It does so by means of a linear combination of
pre-computed F-fluctuations such that the overall hyperelastic energy is
minimized. When the coefficients are found, Pbar is computed as the volume
average of the microscopic stress field P.
The RB matrix B contains the RB elements as columns. All matrices are stored in
row-major format (in contrast to Matlab/Octave default, hence some
transpositions near the reshape commands). The columns of the RB matrix B
contain deformation gradient fluctuations (9 components) at each quadrature
point (N_qp points) for each basis element (N elements), thus
                       size(B) = [ 9*N_qp , N ].
The example data contains multiple sets of approximately uniformly distributed
directions in 5 dimensions, the RB of an example micro-structure, and the
quadrature weights. The microstructure is the unit cube with an off-center
cubical inclusion. Due to the nature of the method, no mesh and not even point
coordinates are necessary. For the sake of brevity, such information is not
contained in this repository.
Since the original mesh is very simple, all quadrature weights are equal
in this example. The code, however, is general and does not exploit this.

This software package is related to the open access research article
Authors: Oliver Kunc and Felix Fritzen
Title  : Finite strain homogenization using a reduced basis and efficient
         sampling
Journal: Mathematical and Computational Applications
         Special Issue "Machine Learning, Low-Rank Approximations and Reduced
         Order Modeling in Computational Mechanics"
Year   : 2019
Volume : 24
Number : 2
DOI    : 10.3390/mca24020056
URL    : https://dx.doi.org/10.3390/mca24020056

2)     HOW TO USE
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2.1)   REQUIREMENTS
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This program was tested with Matlab R2018a. As only basic functions are 
utilized, it is expected to be compatible with most earlier and later releases.

2.2)   DOCUMENTATION
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As the main code consists of 50 lines of code, only inline comments are provided
for documentation purposes. All main steps contain references to the
corresponding formulas in the open access paper.

2.3)   EXECUTION
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Start the program by running ReducedBasisDemonstrator.m
You can vary the loading direction (e.g. by picking another row of the matrix
"directions"), the deviatoric amplitude "t", and the dilatational amplitude "J".
See the paper for more information or contact the authors.

You may create your own set of equidistributed directions by using the program
provided in the repository https://github.com/EMMA-Group/MinimumEnergyPoints

2.4)   HOW TO CITE
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When referring to this software, please cite the article mentioned above in
Section 1) "WHAT'S THIS".

3) BUG REPORTING
==============================
Please report bugs and suggestions to the administrators of the corresponding
github repository:
        https://github.com/EMMA-Group/ReducedBasisDemonstrator
or directly e-mail
        kunc@mechbau.uni-stuttgart.de

4) LICENSE INFORMATION
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See file LICENSE