Data-driven physics-constrained recurrent neural networks for multiscale damage modeling of metallic alloys with process-induced porosity

Abstract

Computational modeling of heterogeneous materials is increasingly relying on multiscale simulations which typically leverage the homogenization theory for scale coupling. Such simulations are prohibitively expensive and memory-intensive especially when modeling damage and fracture in large 3D components such as cast metallic alloys. To address these challenges, we develop a physics-constrained deep learning model that surrogates the microscale simulations. We build this model within a mechanistic data-driven framework such that it accurately predicts the effective microstructural responses under irreversible elasto-plastic hardening and softening deformations. To achieve high accuracy while reducing the reliance on labeled data, we design the architecture of our deep learning model based on damage mechanics and introduce a new loss component that increases the thermodynamical consistency of the model. We use mechanistic reduced-order models to generate the training data of the deep learning model and demonstrate that, in addition to achieving high accuracy on unseen deformation paths that include severe softening, our model can be embedded in 3D multiscale simulations with fracture. With this embedding, we also demonstrate that state-of-the-art techniques such as teacher forcing result in deep learning models that cause divergence in multiscale simulations. Our numerical experiments indicate that our model is more accurate than pure data-driven models and is much more efficient than mechanistic reduced-order models.

Appendices

Appendices

A Multiscale Continuum Damage Model

There are two popular approaches for modeling damage [69, 70]: (1) a discrete approach based on fracture mechanics which models displacement discontinuity at discontinuous interfaces, and (2) a continuous approach based on continuum mechanics which models damage as strain softening by inelastic strains. In our work, we adopt the latter approach to generate a database of microstructural responses. The continuous approach, however, often suffers from non-objectivity where localized damage bands become narrower and narrower upon mesh refinement, i.e., they are mesh dependent (this dependency is caused by ill-posed governing equations due to damage-induced non-positive definiteness). One way to address this issue is to modify the continuum constitutive law by introducing crack bandwidth via nonlocal functions.

However, integrating nonlocal functions within a multiscale damage model is quite difficult [18, 60]. This is because, on the one hand, a material characteristic length $l_m$ is needed to stabilize ill-posed governing equations (caused by non-positive definite stiffness matrix) to address the mesh dependency at microscale. On the other hand, a macroscale characteristic length $l_M$ is required to provide nonlocality for distant macro elements. Simultaneously imposing $l_m$ and $l_M$ helps to stabilize the multiscale damage model but is not physically realistic. If only $l_m$ is imposed and $l_M$ is neglected, the multiscale model merely transfers the damage-induced localization to the finer scale which is equivalent to a single scale model with $l_m$ and contradicts with the purpose of multiscale modeling without properly transmitting microscale damage to macroscale [60].

One way to properly impose nonlocal functions in multiscale damage models is to introduce a material characteristic length in each RVE that accounts for the influence of damage on neighboring RVEs (and associated macro elements) [71]. We adopt another approach to properly impose both the micro and macro damage regularizations [26]: our micro damage model is regularized by a predefined material fracture energy and its resulting effective damage parameter is subsequently regularized by a macro damage model via a nonlocal function for neighboring RVEs. We provide more details on our approach below.

We adopt an continuum damage model to simulate strain softening in ductile metals whose load-carrying capacity drops due to the degradation of yield stress and stiffness. To simulate the onset of softening, we choose ductile damage initiation criteria which models the effective strain at damage initiation, i.e., ${\bar{E}}_{d}^{pl}$, as a function of stress and strain states. We presume ${\bar{E}}_{d}^{pl}$ is a constant and that damage begins when the equivalent plastic strain is equal or greater than it, i.e., ${\bar{E}}^{pl} \geqslant {\bar{E}}_{d}^{pl}$.

A major challenge of continuum damage models is the softening-induced non-positive definite stiffness matrix that results in slow solution convergence and negative wave speeds [60]. Specifically, the ill-posed problem causes equilibrium equations to lose objectivity with respect to mesh sizes by exhibiting spurious mesh sensitivity. In microstructural simulations, to address the lack of objectivity to mesh choices, we convert the stress–strain relation in the constitutive equation to a stress-displacement relation to formulate the micro-damage evolution after damage initiation as:

$$\begin{aligned} G_f = \int _{{\bar{E}}_0^{pl}}^{{\bar{E}}_f^{pl}} l_e S_y \,d{{\bar{E}}^{pl}} = \int _{0}^{{\bar{u}}_f^{pl}} S_y \,d{{\bar{u}}^{pl}} \end{aligned}$$

(A-1)

where $l_e$ indicates the element’s characteristic length in an arbitrary RVE, and $G_f$ represents the dissipated energy that opens a unit area of crack after damage initiation. The equivalent plastic displacement ${\bar{u}}^{pl}$ is the fracture work conjugate to the yield stress $S_y$ from the onset of damage (with the effective plastic strain ${\bar{E}}_0^{pl}$ and zero plastic displacement ${\bar{u}}^{pl}$) until the final failure (with the effective fracture strain ${\bar{E}}_f^{pl}$ and the fracture displacement ${\bar{u}}_{f}^{pl}$). Using Equation (A-1) we define the microscale damage evolution based on an exponential form of the released energy [72] as:

$$\begin{aligned} D_m = 1 - exp\left( -\frac{1}{G_f} \int _{0}^{{\bar{u}}_f^{pl}} S_y \,d{{\bar{u}}^{pl}}\right) \end{aligned}$$

(A-2)

where $D_m$ represents the microscale damage parameter that monotonically increases in the range of [0.0, 1.0]. We note that in our context of isotropic continuum damage, $D_m$ is a scalar and it becomes a tensor in anisotropic damage models. In addition, we note that $D_m$ approaches 1.0 asymptotically with infinitely large ${\bar{u}}^{pl}$ in Equation (A-2). In practice, we set $D_m$ as 1.0 when the dissipated energy exceeds $0.99G_f$. In our continuum damage model, we formulate a softening response of a micro point with an elasto-plastic behavior as:

$$\begin{aligned} {\varvec{S}}_m = (1-D_m){\varvec{S}}^0_m; \hspace{0.3cm} {\varvec{S}}^0_m = {\mathbb {C}}^{el}:{\varvec{E}}^{el}_m = {\mathbb {C}}^{el}:({\varvec{E}}_m-{\varvec{E}}^{pl}_m) \nonumber \\ \end{aligned}$$

(A-3)

where ${\varvec{S}}_m$ and ${\varvec{S}}^0_m$ are, respectively, the damaged stress and the reference stress that undergoes the same deformation path but in the absence of damage at a micro material point. ${\mathbb {C}}^{el}$ represents the fourth-order elasticity tensor. ${\varvec{E}}_m$, ${\varvec{E}}^{el}_m$ and ${\varvec{E}}^{pl}_m$ are the microscale total strain, elastic strain, and plastic strain, respectively.

From the microscale stress, we can compute the effective stress via Equation (2). Additionally, we compute the RVE’s effective damage parameter [71] by:

$$\begin{aligned} D_M = 1 - \frac{\Vert {\varvec{S}}_M: {\varvec{S}}_M^0 \Vert }{\Vert {\varvec{S}}_M^0:{\varvec{S}}_M^0 \Vert } \end{aligned}$$

(A-4)

where the homogenized damage parameter $D_M$ indicates the damage status of the RVE. Its value depends on the values of the effective stress ${\varvec{S}}_M$ and the effective reference stress ${\varvec{S}}_M^0$ without damage; thus, it is clear that the effective damage parameter is not a function of homogenized plastic strains.

We proceed to constrain the effective damage parameter $D_M$ via an integral-type non-local damage model to mitigate the spurious mesh dependency on the macroscale as:

$$\begin{aligned} {\hat{D}}_M({\varvec{P}}, {\varvec{P}}') = \int _{B} {\omega (\Vert {\varvec{P}}-{\varvec{P}}'\Vert )}D_M({\varvec{P}}') \,d{\varvec{P}}' \end{aligned}$$

(A-5)

where ${\hat{D}}_M({\varvec{P}}, {\varvec{P}}')$ is the non-local damage parameter at the macroscopic point ${\varvec{P}}$ surrounded by points ${\varvec{P}}'$ in the compact neighborhood B. $D_M({\varvec{P}}')$ represents the local damage parameter at ${\varvec{P}}'$ and $\omega$ indicates the non-local weighting function which depends on the distance $\Vert {\varvec{P}}-{\varvec{P}}'\Vert$ between the studied point and its supporting points. In this work, we define $\omega$ by a polynomial bell-shape function as:

$$\begin{aligned} {\omega (\Vert {\varvec{P}}-{\varvec{P}}'\Vert )} = \frac{\omega _\infty (\Vert {\varvec{P}}-{\varvec{P}}'\Vert )}{\int _{B} {\omega _\infty (\Vert {\varvec{P}}-{\varvec{P}}'\Vert )} \,d{\varvec{P}}'} \end{aligned}$$

(A-6a)

$$\begin{aligned} {\omega _\infty (\Vert {\varvec{P}}-{\varvec{P}}'\Vert )} = \left\langle 1 - \frac{4(\Vert {\varvec{P}}-{\varvec{P}}'\Vert )^2}{l_d^2} \right\rangle ^2 \end{aligned}$$

(A-6b)

where $\langle \dots \rangle$ is the Macauley bracket defined as $\langle x \rangle = max(0,x)$, $l_d$ denotes the strain localization bandwidth whose value represents the non-local interacting radius, and the support domain B is a sphere with a radius of $l_d/2$ in 3D models.

In our multiscale damage model, we compute the regularized macro stress, ${\varvec{S}}$, as:

$$\begin{aligned} {\varvec{S}}=(1-{\hat{D}}_M){\varvec{S}}^0_M \end{aligned}$$

(A-7)

where we assume the effective reference stress ${\varvec{S}}^0_M$ can be closely approximated by the effective damaged stress (${\varvec{S}}_M$) and the effective damage parameter ($D_M$) from microstructural analyses. That is:

$$\begin{aligned} {\varvec{S}}^0_M={\varvec{S}}_M/(1-D_M) \end{aligned}$$

(A-8)

We can directly relate ${\varvec{S}}$ to the RVEs’ effective damaged stress (${\varvec{S}}_M$) via ${\varvec{S}}={\varvec{S}}_M(1-{\hat{D}}_M)/(1-D_M)$. These two stresses are identical if there are no macro nonlocal functions, i.e., ${\hat{D}}_M=D_M$. However, as discussed before, it is important to properly impose damage regularization on each scale to address the mesh dependency issue of continuum damage mechanics [18, 60]. In our model, we regularize micro damage by material fracture energy in Equation (A-2) and macro damage by nonlocal functions in Equation (A-5). Hence, the two stresses (${\varvec{S}}$ and ${\varvec{S}}_M$) in our case are directly related via a coefficient of $(1-{\hat{D}}_M)/(1-D_M)$.

We now demonstrate that the relation in Equation (A-8) is directly related to the definition of the effective damage parameter in Equation (A-4):

$$\begin{aligned}{} & {} {{\varvec{S}}_M} = (1-D_M) {\varvec{S}}_M^0 \end{aligned}$$

(A-9a)

$$\begin{aligned}{} & {} {{\varvec{S}}_M: {\varvec{S}}_M^0} = (1-D_M) {\varvec{S}}_M^0: {\varvec{S}}_M^0 \end{aligned}$$

(A-9b)

$$\begin{aligned}{} & {} {({\varvec{S}}_M:{\varvec{S}}_M^0)/({\varvec{S}}_M^0:{\varvec{S}}_M^0)} {=} (1{-}D_M) ({\varvec{S}}_M^0:{\varvec{S}}_M^0)/({\varvec{S}}_M^0:{\varvec{S}}_M^0) \nonumber \\ \end{aligned}$$

(A-9c)

$$\begin{aligned}{} & {} {({\varvec{S}}_M:{\varvec{S}}_M^0)/({\varvec{S}}_M^0:{\varvec{S}}_M^0)} = (1-D_M) \end{aligned}$$

(A-9d)

where we can obtain the definition of the effective damage parameter as in Equation (A-4) since $0 \le 1-D_M \le 1$.

B Hybrid Constitutive Integration

The non-positive definiteness of the stiffness matrix is the primary reason for the slow convergence of classic implicit time integration schemes that are used in continuum damage simulations. For illustration, consider the constitutive equation of an isotropic damage model integrated by an implicit backward-Euler integration scheme. Its algorithmic tangent operator at an arbitrary macroscopic IP can be written as:

$$\begin{aligned} {\mathbb {C}}_{n+1}^{alg}= & {} \frac{\partial {\varvec{S}}_{n+1}}{\partial {\varvec{E}}_{n+1}} = (1-D_{n+1}){\mathbb {C}}^{el} \nonumber \\{} & {} - \frac{S_{n+1} - H_{n}{\bar{E}}_{n+1}^{pl}}{({\bar{E}}_{n+1}^{pl})^3} {\varvec{S}}_{n+1}^{0} \otimes {\varvec{S}}_{n+1}^{0} \end{aligned}$$

(B-1)

where ${\mathbb {C}}_{n+1}^{alg}$, ${\bar{E}}_{n+1}^{pl}$, $S_{n+1}$, ${\varvec{S}}_{n+1}^{0}$ and $H_{n}$ represent the fourth-order algorithmic tangent operator, equivalent plastic strain, equivalent stress, referenced stress tensor, and softening modulus, respectively. The subscripts denote time steps and the symbol $\otimes$ represents the cross product between tensors. Softening causes negative values for $H_{n}$ which can render ${\mathbb {C}}_{n+1}^{alg}$ indefinite. A non-positive ${\mathbb {C}}_{n+1}^{alg}$ leads to an ill-conditioned elemental stiffness matrix with near-zero or negative eigenvalues, and further deteriorates the global stiffness matrix in the element assembly process. Such ill-posed matrices dramatically reduce the efficiency of iterative solvers (e.g., Newton–Raphson methods) and often cause job abortion before final convergence.

To fundamentally resolve the convergence issue, we adopt a hybrid time integration scheme [26, 73] to integrate the governing equations of elasto-plastic and softening equations explicitly-implicitly. The basic idea of the hybrid integration is to maintain the positive definiteness of the system’s algebraic tangent operator by separately integrating constitutive equations in two consecutive stages via explicit and implicit schemes. At the first stage, we explicitly extrapolate internal material state variables at time step $n+1$ from step n to compute the explicit stress state $\tilde{{\varvec{S}}}_{n+1}$ that balances the equilibrium equation between internal and external forces. At the second stage, we compute the implicit stress state ${\varvec{S}}_{n+1}$ based on the current strain state ${\varvec{E}}_{n+1}$ using the classic backward Euler method to update the trial stress tensor and yield functions for the next time step where the tangent operator between $\tilde{{\varvec{S}}}_{n+1}$ and ${\varvec{E}}_{n+1}$ is kept positive definite.

For the elasto-plastic model, we choose the material state variable as the incremental plastic strain tensor $\triangle \tilde{{\varvec{E}}}_{n+1}^{pl}$ such that $\tilde{{\varvec{S}}}_{n+1}$ can be computed as:

$$\begin{aligned}{} & {} \tilde{{\varvec{S}}}_{n+1}(\triangle \tilde{{\varvec{E}}}_{n+1}^{pl}) = \tilde{{\varvec{S}}}_{n+1}^{trial} - {\mathbb {C}}^{el}:\triangle \tilde{{\varvec{E}}}_{n+1}^{pl} = {\mathbb {C}}^{el}:{\varvec{E}}_{n+1} \nonumber \\{} & {} \quad - {\mathbb {C}}^{el}:{\varvec{E}}_{n}^{pl} - {\mathbb {C}}^{el}:\triangle \tilde{{\varvec{E}}}_{n+1}^{pl} \nonumber \\{} & {} \quad \triangle \tilde{{\varvec{E}}}_{n+1}^{pl} = \frac{\triangle t_{n+1}}{\triangle t_n} \triangle {\varvec{E}}_n^{pl} \end{aligned}$$

(B-2)

where ${\varvec{E}}_n^{pl}$ represents the implicit incremental plastic strain tensor at time step n, $\triangle t_n$ and $\triangle t_{n+1}$ indicate the lengths of time steps at two consecutive steps. The algorithmic tangent operator (under loading⁵) is therefore computed as:

$$\begin{aligned}{} & {} \tilde{{\mathbb {C}}}_{n+1}^{alg} = \frac{\partial {\tilde{{\varvec{S}}}_{n+1}(\triangle \tilde{{\varvec{E}}}_{n+1}^{pl})}}{\partial {{\varvec{E}}_{n+1}}} \nonumber \\{} & {} \quad = \frac{\partial ({\mathbb {C}}^{el}:{\varvec{E}}_{n+1} - {\mathbb {C}}^{el}:{\varvec{E}}_{n}^{pl} - {\mathbb {C}}^{el}:\triangle \tilde{{\varvec{E}}}_{n+1}^{pl})}{\partial {{\varvec{E}}_{n+1}}} = {\mathbb {C}}^{el} \nonumber \\ \end{aligned}$$

(B-3)

In a similar manner, for isotropic continuum damage models, we choose the explicitly interpolated material state variable in the hybrid integration as the incremental plastic multiplier $\triangle {\tilde{\lambda }}_{n+1}$, i.e., $\triangle {\tilde{\lambda }}_{n+1} = (\triangle t_{n+1} / \triangle t_n) \triangle \lambda _n$. We can then write its explicit damaged stress and algorithmic tangent operator (under loading⁶) as:

$$\begin{aligned}{} & {} \tilde{{\varvec{S}}}_{n+1} = (1-{\tilde{D}}_{n+1}) {\varvec{S}}_{n+1}^0 = (1-{\tilde{D}}_{n+1}) {\mathbb {C}}^{el}:{\varvec{E}}_{n+1};\nonumber \\{} & {} {\tilde{D}}_{n+1} = {\tilde{D}}_{n+1} (D_n, \triangle {\tilde{\lambda }}_{n+1}) \end{aligned}$$

(B-4)

$$\begin{aligned}{} & {} \tilde{{\mathbb {C}}}_{n+1}^{alg} = \frac{\partial {\tilde{{\varvec{S}}}_{n+1}}}{\partial {{\varvec{E}}_{n+1}}} = (1-{\tilde{D}}_{n+1}) {\mathbb {C}}^{el} \end{aligned}$$

(B-5)

where ${\varvec{S}}_{n+1}^0$ is the effective stress tensor, and ${\tilde{D}}_{n+1}$ represents the explicit state of the damage variable which is a function of its previous implicit state $D_n$ and the current explicit incremental plastic multiplier $\triangle {\tilde{\lambda }}_{n+1}$.

In the hybrid integration scheme, the loading tangent operators of the elasto-plastic model in Equation (B-3) and the damage model in Equation (B-5) are trivially equal to the elastic modulus ${\mathbb {C}}^{el}$ and $(1-{\tilde{D}}_{n+1}) {\mathbb {C}}^{el}$. Hence, the hybrid integration scheme preserves the positive-definiteness of the governing equations and also allows to assemble the global stiffness matrix only once before online simulations. The global stiffness matrix remains constant for the elasto-plastic regime and only needs partial updates on matrix entries associated with the softening IPs by Equation (B-5). As softening is often highly localized in small regions, the global stiffness can be incrementally updated during the entire elasto-plastic-hardening-softening process [26]; saving significant memory footprints with robust convergence performance.

C Deflated Clustering Analysis

Simulation of microstructural softening via the classic FE² method involves demanding computational costs, which is prohibitive for generating big training data for machine learning models. To accelerate the database generation, we adopt our previously developed mechanistic ROM, i.e., deflated clustering analysis (DCA) [25, 26]. Its high efficiency comes from two facts: (1) the number of unknown variables in the system is dramatically reduced from a large number of finite elements to a few clusters by agglomerating elements via clustering as shown in Fig. 23, and (2) the algebraic equations of the reduced system contains much fewer close-to-zero eigenvalues that results in better convergence comparing to the classic FE system.

Our DCA utilizes k-means clustering, i.e., an unsupervised machine learning technique for data interpretation and grou**, to agglomerate neighboring elements into a set of interactive irregular-shape clusters. The clustering begins with feeding the coordinates of element centroids into a feature space where randomly scattered cluster seeds serve as initial cluster means. Clusters accept or reject elements by iteratively minimizing the within-cluster variance until all elements are assigned to a cluster. The clustering procedure can be mathematically stated as a minimization problem as:

$$\begin{aligned} {\varvec{C}} = \min \limits _{{\varvec{C}}}\sum \limits _{I = 1}^k \sum \limits _{n \in C^I} \Vert \pmb {\varphi }_n - \bar{\pmb {\varphi }}_I \Vert ^2 \end{aligned}$$

(C-1)

where ${\varvec{C}}$ represents the k clusters with ${\varvec{C}} = \{C^1, C^2, \dots , C^k\}$. $\pmb {\varphi }_n$ and $\bar{\pmb {\varphi }}_I$ indicate the coordinates of the centroid of the $n^{th}$ element and the mean of the coordinates of the $I^{th}$ cluster, respectively. A clustering example is illustrated in Fig. 23 where the discrete domain of a 2D generic RVE with 5, 000 elements is decomposed into 100 clusters.

Fig. 19

Demonstration of clustering in ROM: The domain of a generic 2D RVE with 5, 000 elements in (a) are decomposed into 100 clusters in (b) where elements in the same cluster are assigned with the same color

We construct clustering-based reduced mesh via Delaunay triangularization by connecting cluster centroids where the topological relations between clusters are preserved from the original FE mesh. By assuming the motions of cluster centroids are directly related to clustering nodes, we can compute the nodal displacements via polynomial augmented radian point interpolation [74] as:

$$\begin{aligned} {\varvec{u}}_c = {\varvec{R}}{\varvec{a}} + {\varvec{Z}}{\varvec{b}} \end{aligned}$$

(C-2)

where ${\varvec{u}}_c$ represents the displacements of cluster centroids. ${\varvec{a}}$ is the coefficient vector of the radial basis function matrix ${\varvec{R}}$, and ${\varvec{b}}$ is the coefficient vector of the polynomial basis matrix ${\varvec{Z}}$. Meanwhile, the radial coefficient and the polynomial basis need to satisfy the following equation for every node per cluster and every polynomial basis function to ensure solution uniqueness [74] as:

$$\begin{aligned} {\varvec{Z}}{\varvec{a}} = {\varvec{0}} \end{aligned}$$

(C-3)

Fig. 20

Multiscale cube model: a Every integration point of the macro-cube model is associated with a porous RVE; and b The RVE domain is discretized by different numbers of clusters

The displacements of cluster centroids are augmented with rotational degrees of freedom to represent the six rigid body motions in a 3D deflation space [75], including three translations and three rotations. Upon the completion of a non-linear analysis on the reduced mesh, the displacement solutions can be projected back to the original FE mesh by:

$$\begin{aligned} {\varvec{u}}_{i}^{j} = {\varvec{W}}_{i}^{j} \pmb {\lambda }_{j} \end{aligned}$$

(C-4)

where ${\varvec{u}}_{i}^{j}$ represents the displacement vector at the $i^{th}$ node in the $j^{th}$ cluster. In addition, $\pmb {\lambda }_{j}$ is the rigid body motion of the centroid of the $j^{th}$ cluster, while the ${\varvec{W}}_{i}^{j}$ indicates the deflation matrix for the $i^{th}$ node in the $j^{th}$ cluster as:

$$\begin{aligned} \pmb {\lambda }_{j}= & {} [u_{jx}, u_{jy}, u_{jz}, \theta _{jx}, \theta _{jy}, \theta _{jz}]^{T}; \nonumber \\ {\varvec{W}}_{i}^{j}= & {} \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0 &{} z_{i}^{j} &{} -y_{i}^{j} \\ 0 &{} 1 &{} 0 &{} -z_{i}^{j} &{} 0 &{} x_{i}^{j} \\ 0 &{} 0 &{} 1 &{} y_{i}^{j} &{} -x_{i}^{j} &{} 0 \end{bmatrix} \end{aligned}$$

(C-5)

where $u_{jx}$ and $\theta _{jx}$ are the displacement and rotation of the $j^{th}$ cluster along x axis, and the ($x_{i}^{j}$, $y_{i}^{j}$, $z_{i}^{j}$) are the relative 3D coordinates of the $i^{th}$ node with respect to the centroid of the $j^{th}$ cluster. By assuming all elements in the same cluster share identical stress and strain fields, microstructural effective responses can be reproduced in a highly efficient manner such that the unknown variables are dramatically decreased from FE system that accounts for distinct field variables per element to the reduced system with much fewer distinct solutions per cluster.

To demonstrate the efficacy of our DCA, we compare its simulation results on a 3D multiscale cube against the classic FE² method in Fig. 20. The macro-cube is fully constrained at its bottom surface, and it is subject to an upward extension on the top surface with $d = 7$ mm. The cube is meshed with 12 tetrahedral elements of reduced-integration (one IP at the center of each tetrahedron). We assume each macro-IP is associated with the same porous RVE containing one spherical pore in the middle as shown in Fig. 20a.

Fig. 21

Results of the multiscale cube model: a Comparison of the softening load–displacement curves between FE² and FE-ROM with different clusters; and b Comparison of computational time

To determine the number of clusters for a given problem (for any clustering-based ROM, e.g., DCA, SCA, or SCA’s variants), we can perform a quick preliminary convergence study where we gradually increase the number of clusters and determine the minimum number of clusters above which the results insignificantly change. This convergence study can also be done by comparing the results of the ROM to that of direction numerical solutions (DNS). Yet another method is to formulate a data-driven inverse optimization problem [59] where the cluster number is considered as an optimization variable. In this work, we carry out a convergence study to ensure our ROM’s solutions do not change as the number of clusters increase and that they are consistent with the DNS, i.e., FE². Specifically, we apply four clustering levels (k) of 400, 800, 1, 200 and 1, 600 to an RVE meshed with 15, 000 elements and investigate the effects of k on the RVE’s effective softening behaviors, see Fig. 20.

We compare the reaction force-displacement curves from FE² and FE-ROM in Fig. 21a. By considering the FE² solutions as the benchmark, we observe that: (1) the FE-ROM solutions with $k = 400$ slightly overestimate the component’s strength as insufficient clustering in the RVE artificially strengthens the material [23, 25]; and (2) as k increases, the FE-ROM responses (especially the post-failure behaviors) become closer and closer to the benchmark. Specifically, we observe that when k increases to 1, 200 and 1, 600, FE-ROMs achieve sufficiently accurate results compared to FE².

Fig. 22

Architectures of GRU layer and cells: The internal structure and mathematical operations are demonstrated in the GRU cell at time step t

Fig. 23

Experimental characterization of our aluminum alloy A356: a A356 ingots are melted in a high-temperature furnace with degassing to remove porosity; b Heat treatment of cast tensile bars; c Composition analysis; and d Tensile tests of the cast alloys

We compare the computational costs of the different solvers in Fig. 21b. While all experiments are performed on an HPC by paralleling 60 CPU cores with 360 GB RAM, the clock time of FE² is the longest (about 24.9 hours). The clock time of the ROM with 1, 200 and 1, 600 clusters is about 2.5 and 3.2 hours, resulting in the acceleration factors of 9.9 and 7.8, respectively. Considering the fact that the ROM with $k = 1,200$ is about $28\%$ faster than its counterpart with $k = 1,600$ while achieving similar accuracy, we adopt $k = 1,200$ while building the training dataset in Sect. 4.

For efficient generation of (micro)structure-performance datasets, we note that many other ROMs can also be used for porous microstructural analyses. For example, self-consistent analysis (SCA) [23, 76, 77] and virtual clustering analysis (VCA) [24] can achieve highly efficient and accurate microstructural homogenization results by treating pores as a soft material with the $0.1\%$ modulus of matrix materials [78]. Another method is the FEM-cluster-based analysis (FCA) [79] where the Hill-Mandel theorem is replaced with the energy equivalence theorem without filling pores with reference material properties. As our focus in this paper is on building the deep learning model that can faithfully surrogate microstructural analyses, we use our in-house DCA package and plan to leverage other methods such as SCA in our future works.

D Gated Recurrent Unit

To alleviate vanishing and exploding gradient issues of RNNs in processing long sequential data, long short term memory (LSTM) and gated recurrent unit (GRU) are typically used. GRU is a variant of the LSTM that, while providing similar accuracy, is more parsimonious and hence computationally more efficient. It is for this reason that we choose GRU as the memory cell in our proposed RNN architecture as in Fig. 4.

To demonstrate the working mechanism of GRUs, we three interconnected cells of a GRU layer in Fig. 22. In a GRU layer, a typical cell at an arbitrary time step t generates predictions $\hat{{\varvec{y}}}_t$ and internal memory-like hidden variables ${\varvec{h}}_t$ after reading in the current inputs ${\varvec{x}}_t$ and the hidden variables ${\varvec{h}}_{t-1}$ from the previous cell. Compared to the RNN cell in Fig. 3b, the GRU cell uses reset and update gates to regulate its internal information flow. The reset gate ${\varvec{r}}_t$ reads ${\varvec{x}}_t$ and ${\varvec{h}}_{t-1}$ to determine the candidate hidden state $\hat{{\varvec{h}}}_t$ by filtering out less important information passing from the previous cell. Its operations include:

$$\begin{aligned} {\varvec{r}}_t&=\sigma \left( {\varvec{W}}_{h r} {\varvec{h}}_{t-1}+{\varvec{W}}_{x r} {\varvec{x}}_t+{\varvec{b}}_r\right) \end{aligned}$$

(D-1a)

$$\begin{aligned} \tilde{{\varvec{h}}}_t&={\text {tanh}}\left( {\varvec{r}}_t \odot {\varvec{W}}_{h {\tilde{h}}} {\varvec{h}}_{t-1}+{\varvec{W}}_{x {\tilde{h}}} {\varvec{x}}_t+{\varvec{b}}_{{\tilde{h}}}\right) \end{aligned}$$

(D-1b)

where $\sigma$ is the sigmoid activation function that returns a value in the range of [0, 1], tanh is the hyperbolic tangent function, and $\odot$ represents the Hadamard product. ${\varvec{W}}_{hr}$, ${\varvec{W}}_{xr}$, ${\varvec{W}}_{h {\tilde{h}}}$, ${\varvec{W}}_{x {\tilde{h}}}$ are the weight matrices associated with the hidden state, the input state, the hidden-to-candidate hidden state and the input-to-candidate hidden state, respectively. ${\varvec{b}}_r$ and ${\varvec{b}}_{{\tilde{h}}}$ are the biases applied to the sigmoid function in the reset gate and the hyperbolic tangent function, respectively.

The update gate (which has its weights and biases) similarly operates on ${\varvec{x}}_t$ and ${\varvec{h}}_{t-1}$: it linearly interpolates the previous hidden state ${\varvec{h}}_{t-1}$ and the candidate hidden state $\tilde{{\varvec{h}}}_t$ to update the memory-like hidden state ${\varvec{h}}_t$ which is then passed to the next cell:

$$\begin{aligned} {\varvec{u}}_t&=\sigma \left( {\varvec{W}}_{hu} {\varvec{h}}_{t-1}+{\varvec{W}}_{x u} {\varvec{x}}_t+{\varvec{b}}_u\right) \end{aligned}$$

(D-2a)

$$\begin{aligned} {\varvec{h}}_t&={\varvec{u}}_t \odot {\varvec{h}}_{t-1}+\left( 1-{\varvec{u}}_t\right) \odot \tilde{{\varvec{h}}}_t+{\varvec{b}}_h \end{aligned}$$

(D-2b)

where ${\varvec{W}}_{hu}$ and ${\varvec{W}}_{xu}$ are the weights applied onto the hidden state and input state in the update gate. ${\varvec{b}}_u$ and ${\varvec{b}}_h$ are the two biases associated to the sigmoid function and the generation of the current hidden state. The cell output at the current time step $\hat{{\varvec{y}}}_t$ is then obtained by linearly transforming the hidden state:

$$\begin{aligned} \hat{{\varvec{y}}}_t ={\varvec{W}}_{h y} {\varvec{h}}_t+{\varvec{b}}_y \end{aligned}$$

(D-3)

where ${\varvec{W}}_{hy}$ and ${\varvec{b}}_y$ are the weights and biases associated with the current output state $\hat{{\varvec{y}}}_t$. We note that all the weights and biases of the GRU networks are iteratively updated by BPTT during training.

E Experimental Material Characterization

For the microstructural simulations in Appendix C we assume the microstructure only contains porosity and the matrix material (i.e., aluminum alloy A356). So, in this section, we briefly discuss the experimental characterization process that can be used to obtain the effective elastoplastic and damage properties of the matrix material, see Fig. 23. Our experiment consists of several steps. In the first step, we melt aluminum A356 ingots in a furnace which is pre-heated to about $800^{\circ }\,\hbox {C}$. During the melting process, we apply degassing [80] to remove gases (e.g., hydrogen contents) and gas-induced porosity before casting as tensile coupons. In the second step, we apply a standard T6 heat treatment to improve the A356 alloy’s strength and toughness. The heat treatment involves a high temperature treatment at $540^{\circ }\,\hbox {C}$ for 8 hours to dissolve alloy elements into aluminum matrix, a quenching process to freeze alloy elements within the solid solution, and an artificial aging process at about $155^{\circ }\,\hbox {C}$ for 3.5 hours to precipitate alloy elements and form grain structures. We also perform composition analysis and find that our A356 alloy contains about $92.05\%$ aluminum (weight fraction), $6.72\%$ silicon, $0.09\%$ steel, $0.0028\%$ magnesium, and other alloy elements. In the third step, we use X-ray computed tomography (CT) to inspect the porosity defect in tensile coupons to ensure the cast alloy is free of pores. Finally, we perform the tensile test on the tensile coupons and measure their averaged elastoplastic and damage parameters (which are provided in Sect. 4.1).