1 Introduction
The RVE concept (see hashin1962 , hill1963 , hill1972 , hashin1983 ) is commonly used in the manufacturing sector to avoid using computationally expensive simulation platforms necessary to capture microstructural features. In essence, the features are captured in the RVE and averaged out over the RVE before any discretization technique is employed at the macroscale with the averaged properties as parameters. More often than not, a number of simulations are run with different microstructures and the statistical mean of the results from those simulations on the macroscale are used as guiding principles for the design of the part. The reason for running multiple simulations each with a different microstructure is that the microstructure is only known stochastically and not deterministically. The popular numerical homogenization algorithm (see geers , ozdemir2008 , schroder2014 ) is commonly employed in which each gauss point for the finite element calculations at the macroscale is associated with a RVE and the information exchange between the two scales occurs at each of those gauss points via the deformation gradient. A 2D depiction of the algorithmic framework is given in Figure 1. The reason for calling the framework “” is that both the macroscale and the RVE scale are solved using finite element method.
The information exchange between the two scales would need to occure multiple times at every increment of macro load to satisfy the accuracy and precision requirements expected of any numerical algorithm designed to solve a set of partial differential equations. The number of finite element solves at the RVE scale are proportional to the number of gauss points corresponding to the macroscale finite element mesh. Depending on the complexity of the microstructure, the RVE solve itself would entail a lot of finite elements to resolve all the features in the RVE. The cumbersome computational cost would make the algorithm infeasible for complex microstructures. In lieu of that, methods to accelerate the algorithm need to be devised.
One potential feature that can be incorporated in the algorithm is the use of neural network (rumelhart , nguyen1990neural ) to establish the inputoutput causality of the RVE boundary value problem prior to any finite element solve at the macroscale. This would eliminate the need to solve the RVE boundary value problem onthefly as the neural network can be used as a blackbox which gets the information from the macroscale as an input and gives the information that the macroscale needs as an output. The elimination of the onthefly RVE solve would substantially reduce the computational burden on the algorithmic framework. In essence, the FE framework would effectively be converted to a FE framework since it would require a finite element solve only at the macroscale. It is now important to identify what information is provided to the RVE from the macroscale and what information is provided back to the macroscale by the RVE. In case of elastic solids, the information exchange is as followsThe concepts of deformation gradient and the particular stress measure are explained in Appendix A. The deformation gradient manifests itself as boundary conditions on the RVE. Periodic boundary conditions satisfy the HillMandel condition (see hillmandel1 , hazanov1998 ) of energetic equivalence between the two scales and are generally the optimal choice from the standpoint of macroscale accuracy (see michel1999 , kouznetsova2001 , miehe2003 , dijk2015 ). The imposition of periodic boundary conditions are explained in Appendix B.
2 The machine learning aspect
We follow the outline laid out in mlpeter
to incorporate machine learning in the algorithmic framework. A neural network is composed of several connected layers of artificial neurons and biases where the data is fed into the input layer and flows through some hidden layers. The output is predicted in the output layer. The neurons from different layers are connected through weights
. In the data collection phase, the data flows in one way from the input layer to the target. A simple 131 neural network with deformation gradient as input and homogenized first PK stress as target is depicted in Figure 2. At each neuron, an activation function is attached. The output of each neuron is computed by multiplying the outputs from the previous layer with the corresponding weights. For the neuron
in layer , the data from the previous layer is summed up and then altered by an activation function. The output of neuron in layer is computed aswhere is the number of neurons in the previous layer , is the weight connecting neurons and , is the output of neuron in layer and is its bias. A common choice for the activation function is the sigmoid
In the training phase, the weights of neural network will be initialized firstly, (see nguyen
), which is followed by the weights updating using a training algorithm such that the global error is minimized. The global error, also named as loss function or network performance, is defined according to the difference between the network prediction and the target data. To minimize the global error, the LevenbergMarquardt algorithm (see
hagan ) is applied to update the weights. The steps in the machine learning addendum to the algorithm are
The RVE boundary value problem is solved using finite elements for a myriad of deformation gradients with the nonlinear neoHookean model as the stressstrain relation
The deformation gradient is fed to the RVE problem via periodic boundary conditions as explained in Appendix B. The homogenized first PK stress is obtained for each of these deformation gradients using the relationship (7).

This data is then used to build the inputoutput causality as follows
(1) where is the map between the deformation gradient and the output of the neural network.
3 Algorithmic framework in a nutshell
In the initialization stage,

Establish the inputoutput causality of the RVE boundary value problem as in (1)
Once the initalization phase is complete,

An increment of macro load is applied

Macroscale BVP is solved using the macroscale stiffness computed in (10)

The macroscale deformation gradient is updated

Periodic boundary conditions are imposed on RVE in accordance with (2)

Homogenized first PK stress is obtained in accordance with (1)

The gauss point level homogenized first PK stress is used to compute internal forces at macroscale finite element nodes
If these internal forces are in balance with the prescribed macro load, incremental convergence has been achieved and steps are repeated. If that is not the case, steps are repeated.
Conflict of interest
The authors declare that they have no conflict of interest.
Appendix A The deformation gradient and first PK stress
As shown in Figure 3, let be the macroscale deformation field. The macroscale deformation gradient is the macroscale spatial derivative of in the reference configuration as follows
An incremental force is defined with respect to the Cauchy stress and the first PiolaKirchoff stress in the deformed and reference configurations respectively as follows
Appendix B Periodic boundary conditions on RVE
The typical RVE for imposition of periodic boundary conditions is shown in Figure 4. After each macroscale BVP solve, the deformation gradient is updated and the new position vectors of the vertices of the RVE are obtained using
(2) 
where represents position vector in the reference configuration. This alongwith the shape periodicity of the RVE enables the implementation of periodic boundary conditions on RVE. It is easy to see that the prescribed periodic boundary conditions are Dirichlet boundary conditions.
Appendix C Computation of homogenized first PK stress at the macroscale
The linear momentum balance for the macroscale BVP in the reference configuration is given by
where is the body force vector. The macroscale incremental constitutive law is
(3) 
where
is the fourth order macroscale material property tensor. The determination of
proceeds as follows: First, the RVE scale linear momemtum balance is expressed in the indicial notation aswhere the notation
is used to denote the spatial derivative in the reference configuration. Before we proceed, we assume that the body force is zero, and obtain the following using chain rule for differentiation
(4) 
We express the macroscale first PK stress in indicial notation as follows
(5) 
where the third equality follows from (4) and the fourth equality follows from divergence theorem. We then write (5) in tensorial notation as
(6) 
We know that the RVE level BVP is also solved using finite elements. Let be the number of boundary nodes for the RVE scale discretized domain and let be the force on boundary node. We can rewrite (6) as
(7) 
Appendix D Computation of homogenized tangent stiffness at macroscale
Let represent the displacement DOFs corresponding to the interior nodes and represent the displacement DOFs corresponding to the boundary nodes. The force displacement relation for the RVE scale problem is
where the matrix is dictated by the microstructure and is known apriori. We knock off DOFs corresponding to internal nodes to obtain
(8) 
The incremental macroscopic first PK stress is obtained as
(9) 
Comparing (9) with (3), we get
(10) 
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