Microstructure-informed reduced modes synthesized with Wang tiles and the Generalized Finite Element Method

Abstract

A recently introduced representation by a set of Wang tiles—a generalization of the traditional Periodic Unit Cell-based approach—serves as a reduced geometrical model for materials with stochastic heterogeneous microstructure, enabling an efficient synthesis of microstructural realizations. To facilitate macroscopic analyses with a fully resolved microstructure generated with Wang tiles, we develop a reduced order modelling scheme utilizing pre-computed characteristic features of the tiles. In the offline phase, inspired by computational homogenization, we extract continuous fluctuation fields from the compressed microstructural representation as responses to generalized loading represented by the first- and second-order macroscopic gradients. In the online phase, using the ansatz of the generalized finite element method, we combine these fields with a coarse finite element discretization to create microstructure-informed reduced modes specific for a given macroscopic problem. Considering a two-dimensional scalar elliptic problem, we demonstrate that our scheme delivers less than 3% error in both the relative $L_2$ and energy norms with only 0.01% of the unknowns when compared to the fully resolved problem. Accuracy can be further improved by locally refining the macroscopic discretization and/or employing more pre-computed fluctuation fields. Finally, unlike standard snapshot-based reduced-order approaches, our scheme handles significant changes in the macroscopic geometry or loading without the need for recalculating the offline phase, because the fluctuation fields are extracted without any prior knowledge of the macroscopic problem.

Appendices

Expressions for stiffness matrix and load vector

Assuming the standard Finite Element approximation of the solution in the form of a piecewise polynomial, recall Eq. (7), individual entries of the stiffness matrix $\mathsf {K}^{\mathcal {T}}$, which corresponds to the Hessian of ${\tilde{\varPi }}\left( {\varvec{G}}^{1}\!,\mathbf {G}^{2}\!,{\hat{\theta }}\right)$ with fixed ${\varvec{G}}^{1}\!$ and $\mathbf {G}^{2}\!$, follow from the standard expression

$$\begin{aligned} K^{\mathcal {T}}_{mn} = \int _{\varOmega ^{\mathcal {T}}} {\varvec{\nabla }} N ^{\mathcal {T}}_{m}({\varvec{x}})\cdot \mathbf {K}({\varvec{x}})\cdot {\varvec{\nabla }}N^{\mathcal {T}}_{n} \,{\mathrm d}{\varvec{x}} \,. \end{aligned}$$

(36)

Analogously, components of load vector $\mathsf {f}^{\mathcal {T}}\left( {\varvec{G}}^{1}\!, \mathbf {G}^{2}\!\right)$ from Eq. (8) are obtained by plugging the assumed solution (5) into the stationary conditions of ${\tilde{\varPi }}\left( {\varvec{G}}^{1}\!,\mathbf {G}^{2}\!,{\hat{\theta }}\right)$ with respect to the sought-for coefficients $u^{\mathcal {T}}_{m}$

$$\begin{aligned} f^{\mathcal {T}}_{m}\left( {\varvec{G}}^{1}\!, \mathbf {G}^{2}\!\right) = -\int _{\varOmega ^{\mathcal {T}}} {\varvec{\nabla }} N ^{\mathcal {T}}_{m}({\varvec{x}})\cdot \mathbf {K}({\varvec{x}})\cdot ({\varvec{G}}^{1}\!+ \mathbf {G}^{2}\!\cdot {\varvec{x}}) \,{\mathrm d}{\varvec{x}} \,.\nonumber \\ \end{aligned}$$

(37)

The integrals are computed element-wise, using the Gauss numerical quadrature rule and the standard finite element procedures [68].

Unique vertex numbering

To generate fluctuation fields that are by construction continuous, boundary degrees of freedom (DOFs) must be enumerated consistently, i.e. corresponding nodes at the edges with the same code within different tiles should be assigned the same number. While the enumeration is straightforward for edge-related DOFs, it does not hold for vertex DOFs. Assigning the same number to all vertex DOFs is not always correct because—depending on the code distribution within the set—it may happen that certain vertices will never coincide during the tile assembly, thus there might be separate vertex group with distinct DOFs. This typically happens for tile sets derived from the vertex-defined Wang tiles [37].

Inspired by the pragmatic question of where to copy a particle that intersects more than one edge during microstructural generation [13], we devised an algorithm that is capable of identifying the separate vertex groups. First, we construct an undirected graph from the provided tile set definition such that (i) each node of the graph corresponds to one half of each edge code (i.e. there is a left and right part for each horizontal and top and bottom part for each vertical code) and (ii) arcs between the graph nodes are obtained from the individual vertices of each tile.

Next, using the Depth-First Search algorithm [56, Section 18.2], we identify the connected components of the graph. If several sub-graphs are identified, the arcs corresponding to this group yield vertices that belong to a distinct group and only these should be assigned the same number. For more details, including illustrations for several tile sets, readers are referred to [13, Section 2.3].

Discrete form of constraints

For completeness, implementation details of the individual constraint matrices from Eq. (24) are provided next. We assume the standard isoparametric finite elements [68]. The boundary of each domain is consistently discretized by restricting planar finite element discretization to the tiles’ boundary, e.g. domain discretization by linear triangles yields linear line elements ${\bar{e}}$ on the boundary. Since the tile domains are square and aligned with coordinate axes, the outer normal ${\varvec{n}} = \begin{bmatrix} n_{x}&n_{y} \end{bmatrix}^\mathsf{T}$ along individual edge elements is constant, which simplifies the following expressions.

First, the average value of $\tilde{ \theta }$ along all boundaries of all tiles, appearing in Eq. (16), is obtained through

$$\begin{aligned} \begin{aligned} \int _{\partial \varOmega ^{\mathcal {S}}} \tilde{ \theta }\,{\mathrm d}{\varvec{x}}&= \sum _{\mathcal {T}\in \mathcal {S}} \int _{\partial \varOmega ^{\mathcal {T}}} \tilde{ \theta }\,{\mathrm d}{\varvec{x}} \\&\approx \sum _{\mathcal {T}\in \mathcal {S}} \sum _{{\bar{e}}} \int _{\varOmega ^{{\bar{e}},\mathcal {T}}} {\bar{\mathsf {N}}}({\varvec{x}})\mathsf {u}^{{\bar{e}},\mathcal {T}} \,{\mathrm d}{\varvec{x}} \\&= \sum _{\mathcal {T}\in \mathcal {S}} \left( \sum _{{\bar{e}}} \int _{\varOmega ^{{\bar{e}},\mathcal {T}}} {\bar{\mathsf {N}}}({\varvec{x}})\,{\mathrm d}{\varvec{x}} \, \mathsf {L}^{{\bar{e}},\mathcal {T}} \right) \mathsf {u}^{\mathcal {T}}_{b} \\&= \underbrace{ \left\{ \sum _{\mathcal {T}\in \mathcal {S}} \left( \sum _{{\bar{e}}} \left\langle {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} \mathsf {L}^{{\bar{e}},\mathcal {T}} \right) \mathsf {L}^{\mathcal {T}} \right\} }_{\mathsf {C}_{0}} \mathsf {u}^{\text {s}} \,, \end{aligned} \end{aligned}$$

(38)

where $\bar{\mathsf {N}}$ is a row vector of shape functions pertinent to the vertices of element ${\bar{e}}$, $\left\langle {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}}$ denotes its integrated counterpart using a suitable quadrature rule, $\mathsf {L}^{{\bar{e}},\mathcal {T}}$ maps boundary DOFs of tile $\mathcal {T}$ onto nodal unknowns of boundary element ${\bar{e}}$ such that $\mathsf {u}^{{\bar{e}},\mathcal {T}} = \mathsf {L}^{{\bar{e}},\mathcal {T}} \mathsf {u}^{\mathcal {T}}_{b}$, and $\mathsf {L}^{\mathcal {T}}$ is the localization matrix introduced in Sect. 3.1.

Proceeding with Eq. (17), the boundary integral related to the average of the first-order gradient of each tile is approximated as

$$\begin{aligned} \begin{aligned} \int _{\partial \varOmega ^{\mathcal {T}}} \tilde{ \theta }{\varvec{n}} \,{\mathrm d}s&\approx \sum _{{\bar{e}}} \int _{\varOmega ^{{\bar{e}},\mathcal {T}}} \begin{bmatrix} n^{{\bar{e}}}_{x} {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\\ n^{{\bar{e}}}_{y} {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\end{bmatrix} \mathsf {u}^{{\bar{e}},\mathcal {T}} \,{\mathrm d}{\varvec{x}} \\&= \left( \sum _{{\bar{e}}} \begin{bmatrix} n^{{\bar{e}}}_{x}\int _{\varOmega ^{{\bar{e}},\mathcal {T}}} {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\,{\mathrm d}{\varvec{x}} \\ n^{{\bar{e}}}_{y}\int _{\varOmega ^{{\bar{e}},\mathcal {T}}} {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\,{\mathrm d}{\varvec{x}} \end{bmatrix} \mathsf {L}^{{\bar{e}},\mathcal {T}} \right) \mathsf {u}^{\mathcal {T}}_{b} \\&= \underbrace{ \left( \sum _{{\bar{e}}} \begin{bmatrix} n^{{\bar{e}}}_{x} \left\langle {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} \\ n^{{\bar{e}}}_{y} \left\langle {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} \end{bmatrix} \mathsf {L}^{{\bar{e}},\mathcal {T}} \right) \mathsf {L}^{\mathcal {T}} }_{\mathsf {C}^{\mathcal {T}}_{\text {I}}} \mathsf {u}^{\mathcal {S}} \,. \end{aligned} \end{aligned}$$

(39)

Depending whether the average first-order gradient is set to vanish for each tile individually, Eq. (19), or for the whole set only, Eq. (20), the corresponding $\mathsf {C}_{\text {I}}$ takes either the form

$$\begin{aligned} \mathsf {C}_{\text {I}} = \begin{bmatrix} \mathsf {C}^{\mathcal {T}_{1}}_{\text {I}} \\ \vdots \\ \mathsf {C}^{\mathcal {T}_{16}}_{\text {I}} \end{bmatrix} \end{aligned}$$

(40)

or

$$\begin{aligned} \mathsf {C}_{\text {I}} = \sum _{\mathcal {T}\in \mathcal {S}} \mathsf {C}^{\mathcal {T}}_{\text {I}} \,. \end{aligned}$$

(41)

Analogously, a similar approximation is used also for the second-order gradient constraints. Note that the second-order gradient is a symmetric $2\times 2$ tensor; hence, it constitutes only three independent scalar constraints, which we represent with a vector using Voigt notation. Starting with the constraints (21) posed on an individual tile and with ${\varvec{x}} = \begin{bmatrix}x&y\end{bmatrix}^\mathsf{T}$, we obtain

$$\begin{aligned} \begin{aligned} \int _{\partial \varOmega ^{\mathcal {T}}}&\tilde{ \theta }\left( {\varvec{x}}\otimes {\varvec{n}} + {\varvec{n}} \otimes {\varvec{x}}\right) \,{\mathrm d}s \\&\approx \sum _{{\bar{e}}} \int _{\varOmega ^{{\bar{e}},\mathcal {T}}} \begin{bmatrix} 2 n^{{\bar{e}}}_{x} x {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\\ 2 n^{{\bar{e}}}_{y} y {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\\ (n^{{\bar{e}}}_{x} y + n^{{\bar{e}}}_{y} x) {\bar{\mathsf {N}}}^\mathsf{T}({\varvec{x}})\end{bmatrix} \mathsf {u}^{{\bar{e}},\mathcal {T}} \,{\mathrm d}{\varvec{x}} \\&= \underbrace{ \left\{ \sum _{{\bar{e}}} \left( \begin{bmatrix} 2 n^{{\bar{e}}}_{x} \left\langle \mathsf {x} {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} \\ 2 n^{{\bar{e}}}_{y} \left\langle \mathsf {y} {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} \\ n^{{\bar{e}}}_{x} \left\langle \mathsf {y} {\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} + n^{{\bar{e}}}_{y} \left\langle \mathsf {y}{\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}} \end{bmatrix} \mathsf {L}^{{\bar{e}},\mathcal {T}} \right) \mathsf {L}^{\mathcal {T}} \right\} }_{\mathsf {C}^{\mathcal {T}}_{\text {II}}} \mathsf {u}^{\mathcal {S}} \,, \end{aligned} \end{aligned}$$

(42)

with $\left\langle \mathsf {x}{\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}}$ and $\left\langle \mathsf {y}{\bar{\mathsf {N}}}\right\rangle _{{\bar{e}}}$ denoting the corresponding quantities integrated over element ${\bar{e}}$. Again, depending whether the constraint is posed on individual tiles, Eq. (22), or the set as a whole, Eq. (23), the tile constraint matrices $\mathsf {C}^{\mathcal {T}}_{\text {II}}$ are either stacked,

$$\begin{aligned} \mathsf {C}_{\text {II}} = \begin{bmatrix} \mathsf {C}^{\mathcal {T}_{1}}_{\text {II}} \\ \vdots \\ \mathsf {C}^{\mathcal {T}_{16}}_{\text {II}} \end{bmatrix} \,, \end{aligned}$$

(43)

or summed,

$$\begin{aligned} \mathsf {C}_{\text {II}} = \sum _{\mathcal {T}\in \mathcal {S}} \mathsf {C}^{\mathcal {T}}_{\text {II}} \,. \end{aligned}$$

(44)