A machine learning accelerated FE^2 homogenization algorithm for elastic solids

03/21/2020 ∙ by Saumik Dana, et al. ∙ The University of Texas at Austin 0

The FE^2 homogenization algorithm for multiscale modeling iterates between the macroscale and the microscale (represented by a representative volume element) till convergence is achieved at every increment of macroscale loading. The information exchange between the two scales occurs at the gauss points of the macroscale finite element discretization. The microscale problem is also solved using finite elements on-the-fly thus rendering the algorithm computationally expensive for complex microstructures. We invoke machine learning to establish the input-output causality of the RVE boundary value problem using a neural network framework. This renders the RVE as a blackbox which gets the information from the macroscale as an input and gives information back to the macroscale as output, thereby eliminating the need for on-the-fly finite element solves at the RVE level. This framework has the potential to significantly accelerate the FE^2 algorithm.



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1 Introduction

Figure 1: A 2D depiction of the FE algorithmic framework. The macroscale boundary value problem is discretized into finite elements. The gauss point level computations for the macroscale BVP work in conjunction with RVE scale solve corresponding to each gauss point.

The RVE concept (see hashin-1962 , hill-1963 , hill-1972 , hashin-1983 ) 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 , ozdemir-2008 , schroder-2014 ) 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 input-output 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 on-the-fly 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 on-the-fly 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 follows

The 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 Hill-Mandel condition (see hillmandel1 , hazanov-1998 ) of energetic equivalence between the two scales and are generally the optimal choice from the standpoint of macroscale accuracy (see michel-1999 , kouznetsova2001 , miehe-2003 , dijk-2015 ). The imposition of periodic boundary conditions are explained in Appendix B.

2 The machine learning aspect

Figure 2:

A 1-3-1 neural network with macroscale deformation gradient as input and homogenized first P-K stress as the target output.

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 1-3-1 neural network with deformation gradient as input and homogenized first P-K 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 as

where 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 Levenberg-Marquardt 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 non-linear neo-Hookean model as the stress-strain relation

    The deformation gradient is fed to the RVE problem via periodic boundary conditions as explained in Appendix B. The homogenized first P-K stress is obtained for each of these deformation gradients using the relationship (7).

  • This data is then used to build the input-output causality as follows


    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 input-output 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 P-K stress is obtained in accordance with (1)

  • The gauss point level homogenized first P-K 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.

Use machine learning to establish the relationship (1)
Initialize deformation gradient
for  do loop over macroscale finite elements
     for  do loop over gauss points
         RVE Assign a RVE to each gauss point
         Discretize the RVE
         Calculate homogenized macroscopic tangent stiffness in accordance with (10)      
     Assemble macroscopic tangent stiffness over gauss points
Assemble macroscopic tangent stiffness over finite elements
while  do
     Apply increment of macro load
     while (Internal force-Macro load TOL) do at macroscale finite element nodes
         Solve macroscale problem for
          Update deformation gradient
         for  do loop over macroscale finite elements
              for  do loop over gauss points
                  Prescribe periodic BCs in accordance with (2)
                  Solve RVE problem Not needed as the relationship (1) has been estalished using machine learning
                  Calculate first P-K stress in accordance with (1)                        
         Compute internal forces at finite element nodes      
Algorithm 1 Machine learning based FE homogenization for elastic solids

Conflict of interest

The authors declare that they have no conflict of interest.

Appendix A The deformation gradient and first P-K stress

Figure 3:

is position vector of point in reference configuration and

is the position vector the same point in the deformed configuration. Meanwhile, an elemental area with unit normal deforms to with unit normal under the transformation.

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 Piola-Kirchoff stress in the deformed and reference configurations respectively as follows

Appendix B Periodic boundary conditions on RVE

Figure 4: Typical 2D RVE with pertinent microstructural features. and are mirror images so are and . This helps in easy implementation of periodic boundary conditions on the RVE in accordance with kouznetsova2001 .

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


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 P-K 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



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 as

where 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


We express the macroscale first P-K stress in indicial notation as follows


where the third equality follows from (4) and the fourth equality follows from divergence theorem. We then write (5) in tensorial notation as


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


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


The incremental macroscopic first PK stress is obtained as


Comparing (9) with (3), we get



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