A machine learning accelerated FE 2 homogenization algorithm for elastic solids
NNoname manuscript No. (will be inserted by the editor)
A machine learning accelerated FE homogenizationalgorithm for elastic solids Saumik Dana · Mary F. Wheeler
Received: date / Accepted: date
Abstract
The FE homogenization algorithm for multiscale modeling iterates be-tween the macroscale and the microscale (represented by a representative volumeelement) till convergence is achieved at every increment of macroscale loading.The information exchange between the two scales occurs at the gauss points ofthe macroscale finite element discretization. The microscale problem is also solvedusing finite elements on-the-fly thus rendering the algorithm computationally ex-pensive for complex microstructures. We invoke machine learning to establish theinput-output causality of the RVE boundary value problem using a neural net-work framework. This renders the RVE as a blackbox which gets the informationfrom the macroscale as an input and gives information back to the macroscaleas output, thereby eliminating the need for on-the-fly finite element solves at theRVE level. This framework has the potential to significantly accelerate the FE algorithm. Keywords
Machine learning · FE homogenization The RVE concept (see [5], [7], [9], [4]) is commonly used in the manufacturingsector to avoid using computationally expensive simulation platforms necessaryto capture microstructural features. In essence, the features are captured in theRVE and averaged out over the RVE before any discretization technique is em-ployed at the macroscale with the averaged properties as parameters. More oftenthan not, a number of simulations are run with different microstructures and thestatistical mean of the results from those simulations on the macroscale are usedas guiding principles for the design of the part. The reason for running multiple
S. DanaPhD, Engineering Mechanics, University of Texas at Austin, TX 78712E-mail: [email protected]. F. WheelerCenter for Subsurface Modeling, Oden Institute of Computational Engineering and Sciences,University of Texas at Austin, TX 78712 a r X i v : . [ c s . C E ] M a r Saumik Dana, Mary F. Wheeler macroscale finite elementgauss point RVE1 234 Γ B Γ R Γ T Γ L N + N − Fig. 1
A 2D depiction of the FE algorithmic framework. The macroscale boundary valueproblem is discretized into finite elements. The gauss point level computations for themacroscale BVP work in conjunction with RVE scale solve corresponding to each gauss point. simulations each with a different microstructure is that the microstructure is onlyknown stochastically and not deterministically. The popular F E numerical ho-mogenization algorithm (see [2], [15], [17]) is commonly employed in which eachgauss point for the finite element calculations at the macroscale is associated witha RVE and the information exchange between the two scales occurs at each ofthose gauss points via the deformation gradient. A 2D depiction of the algorith-mic framework is given in Figure 1. The reason for calling the framework “ F E ”is that both the macroscale and the RVE scale are solved using finite elementmethod.The information exchange between the two scales would need to occure mul-tiple times at every increment of macro load to satisfy the accuracy and precisionrequirements expected of any numerical algorithm designed to solve a set of partialdifferential equations. The number of finite element solves at the RVE scale areproportional to the number of gauss points corresponding to the macroscale finiteelement mesh. Depending on the complexity of the microstructure, the RVE solveitself would entail a lot of finite elements to resolve all the features in the RVE. Thecumbersome computational cost would make the algorithm infeasible for complexmicrostructures. In lieu of that, methods to accelerate the algorithm need to bedevised. One potential feature that can be incorporated in the algorithm is the useof neural network ([16], [14]) to establish the input-output causality of the RVEboundary value problem prior to any finite element solve at the macroscale. Thiswould eliminate the need to solve the RVE boundary value problem on-the-fly asthe neural network can be used as a blackbox which gets the information from themacroscale as an input and gives the information that the macroscale needs as anoutput. The elimination of the on-the-fly RVE solve would substantially reduce the computational burden on the algorithmic framework. In essence, the FE frame-work would effectively be converted to a FE framework since it would require afinite element solve only at the macroscale. It is now important to identify whatinformation is provided to the RVE from the macroscale and what information isprovided back to the macroscale by the RVE. In case of elastic solids, the infor- machine learning accelerated FE homogenization algorithm for elastic solids 3 mation exchange is as followsMacroscale deformation gradient −−−−−−−−−−−−−−→ RVE homogenized stress measure −−−−−−−−−−−−−−−−−−→
MacroscaleThe concepts of deformation gradient and the particular stress measure areexplained in Appendix A. The deformation gradient manifests itself as bound-ary conditions on the RVE. Periodic boundary conditions satisfy the Hill-Mandelcondition (see [8], [6]) of energetic equivalence between the two scales and aregenerally the optimal choice from the standpoint of macroscale accuracy (see [11],[18], [12], [1]). The imposition of periodic boundary conditions are explained inAppendix B. input layer output layerhidden layer F M P M b b targetbias bias P NNM = O ( F M ) Fig. 2
A 1-3-1 neural network with macroscale deformation gradient as input and homoge-nized first P-K stress as the target output.
We follow the outline laid out in [10] to incorporate machine learning in thealgorithmic framework. A neural network is composed of several connected layersof artificial neurons and biases where the data is fed into the input layer andflows through some hidden layers. The output is predicted in the output layer.The neurons from different layers are connected through weights w . In the datacollection phase, the data flows in one way from the input layer to the target. Asimple 1-3-1 neural network with deformation gradient as input and homogenizedfirst P-K stress as target is depicted in Figure 2. At each neuron, an activationfunction is attached. The output of each neuron is computed by multiplying the outputs from the previous layer with the corresponding weights. For the neuron j in layer k , the data from the previous layer k − j in layer k is computed as o kj = F ( N (cid:88) i =1 w ij o k − i + b k − i ) Saumik Dana, Mary F. Wheeler where N is the number of neurons in the previous layer k − w ij is the weightconnecting neurons i and j , o k − i is the output of neuron i in layer k − b k − i is its bias. A common choice for the activation function is the sigmoid F = 21 + e − x − – The RVE boundary value problem is solved using finite elements for a myriadof deformation gradients with the non-linear neo-Hookean model as the stress-strain relation σ = 12 λJ ( J − I + µJ ( b − I )The deformation gradient is fed to the RVE problem via periodic boundaryconditions as explained in Appendix B. The homogenized first P-K stress isobtained for each of these deformation gradients using the relationship (7). – This data is then used to build the input-output causality as follows P NNM = O ( F M ) (1)where O is the map between the deformation gradient and the output of theneural network. 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 internalforces at macroscale finite element nodesIf these internal forces are in balance with the prescribed macro load, incrementalconvergence has been achieved and steps 1 − − machine learning accelerated FE homogenization algorithm for elastic solids 5 Algorithm 1
Machine learning based FE homogenization for elastic solids Use machine learning to establish the relationship (1) F M ← I (cid:46) Initialize deformation gradient for E ∈ T h do (cid:46) loop over macroscale finite elements for g ∈ G do (cid:46) loop over gauss pointsRVE ↔ g (cid:46) Assign a RVE to each gauss pointDiscretize the RVECalculate homogenized macroscopic tangent stiffness in accordance with (10)Assemble macroscopic tangent stiffness over gauss pointsAssemble macroscopic tangent stiffness over finite elements while t ≤ T do Apply increment of macro load while (Internal force-Macro load > TOL) do (cid:46) at macroscale finite element nodesSolve macroscale problem for δ F M F M ← F M + δ F M (cid:46) Update deformation gradient for E ∈ T h do (cid:46) loop over macroscale finite elements for g ∈ G do (cid:46) loop over gauss pointsPrescribe periodic BCs in accordance with (2) Solve RVE problem (cid:46)
Not needed as the relationship (1) has been estalishedusing machine learning
Calculate first P-K stress in accordance with (1)Compute internal forces at finite element nodes
Conflict of interest
The authors declare that they have no conflict of interest.
A The deformation gradient and first P-K stress xyz
X xu undeformed configuration deformed configuration dA n n d f dA Fig. 3 X is position vector of point in reference configuration and x = X + u is the positionvector the same point in the deformed configuration. Meanwhile, an elemental area dA withunit normal n deforms to dA with unit normal n under the transformation.As shown in Figure 3, let u be the macroscale deformation field. The macroscale defor-mation gradient F M is the macroscale spatial derivative of x in the reference configuration as Saumik Dana, Mary F. Wheelerfollows F = x ⊗ ∇ X ≡ I + u ⊗ ∇ X An incremental force d f is defined with respect to the Cauchy stress σ and the first Piola-Kirchoff stress P in the deformed and reference configurations respectively as follows d f = σ n dA = Pn dA B Periodic boundary conditions on RVE B Γ R Γ T Γ L N + N − Fig. 4
Typical 2D RVE with pertinent microstructural features. Γ L and Γ R are mirror imagesso are Γ T and Γ B . This helps in easy implementation of periodic boundary conditions on theRVE in accordance with [18].The typical RVE for imposition of periodic boundary conditions is shown in Figure 4. Aftereach macroscale BVP solve, the deformation gradient is updated and the new position vectorsof the vertices of the RVE are obtained using x = F M X (2)where X represents position vector in the reference configuration. This alongwith the shapeperiodicity 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 boundaryconditions. C Computation of homogenized first P-K stress at the macroscale
The linear momentum balance for the macroscale BVP in the reference configuration is givenby ∇ X · P M + b = where b is the body force vector. The macroscale incremental constitutive law is δ P M = C M δ F M (3)where C M is the fourth order macroscale material property tensor. The determination of C M proceeds as follows: First, the RVE scale linear momemtum balance is expressed in the indicialnotation as P ik,k + b i = 0 i, k = 1 , , homogenization algorithm for elastic solids 7where the notation ( · ) ,k ≡ ∂ ( · ) ∂X k is used to denote the spatial derivative in the referenceconfiguration. Before we proceed, we assume that the body force is zero, and obtain thefollowing using chain rule for differentiation( P ik X j ) ,k = P ik,k X j + P ik δ jk = − (cid:0)(cid:18) b i X j + P ij (4)We express the macroscale first P-K stress in indicial notation as follows P M ij = 1 V (cid:90) V P ij dV = 1 V (cid:90) V ( P ik X j ) ,k dV = 1 V (cid:90) Γ P ik n k X j dΓ (5)where the third equality follows from (4) and the fourth equality follows from divergencetheorem. We then write (5) in tensorial notation as P M = 1 V (cid:90) Γ t ⊗ X dΓ (6)We know that the RVE level BVP is also solved using finite elements. Let N p be the number ofboundary nodes for the RVE scale discretized domain and let f ( i ) p be the force on i th boundarynode. We can rewrite (6) as P M = 1 V (cid:90) Γ t ⊗ X dΓ = 1 V N p (cid:88) i =1 f ( i ) p ⊗ X ( i ) (7) D Computation of homogenized tangent stiffness at macroscale
Let u f represent the displacement DOFs corresponding to the interior nodes and u p representthe displacement DOFs corresponding to the boundary nodes. The force displacement relationfor the RVE scale problem is (cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:42) K RV E (cid:20) K pp K pf K fp K ff (cid:21) (cid:26) δ u p δ u f (cid:27) = (cid:26) δ f p (cid:27) where the matrix K RV E is dictated by the microstructure and is known apriori. We knock offDOFs corresponding to internal nodes to obtain[ (cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:58) KK pp − K pf ( K ff ) − K fp ] { δ u p } = { δ f p } (8)The incremental macroscopic first PK stress is obtained as δ P M = 1 V N p (cid:88) i =1 δ f ( i ) p ⊗ X ( i ) ( from (7))= 1 V N p (cid:88) i =1 N p (cid:88) j =1 K ( ij ) δ u ( j ) p ⊗ X ( i ) ( from (8))= 1 V N p (cid:88) i =1 N p (cid:88) j =1 K ( ij ) δ F M X ( j ) ⊗ X ( i ) ( δ u = δ F M X ) (9)Comparing (9) with (3), we get C M abcd = 1 V N p (cid:88) i =1 N p (cid:88) j =1 K ( ij ) ac X ( i ) b X ( i ) d a, b, c, d = 1 , , References
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