Applied Materials Today
Volume 20, September 2020, 100674
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Analytical transport network theory & structural mechanics: Towards computationally inexpensive multi-functional materials design

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Highlights

  • Transport and mechanical analyses of complex, 3D network structures.

  • Computationally inexpensive, interpretable, and hierarchical framework.

  • Opportunity for future integration into topology optimization/machine learning.

Abstract

Analytical transport network (ATN) theory is extended to account for static, structural mechanics. The extension provides a consistent, computationally economical, and interpretable formalism for examining morphological/topological trade-offs relevant to the design of multi-functional materials. The approach is demonstrated through application to over 100,000 computationally-generated lattice structures consisting of non-uniform channels. It is orders of magnitude faster and less memory intensive than finite element analysis. This, along with its compact mathematical formalism, present the opportunity for future integration into multi-functional characterization/design frameworks (e.g., topology optimization and machine learning).

Introduction

Materials for emerging advanced technologies must increasingly serve multiple functions during operation. As a result, there is a desire for synthetic systems that mimic the multi-functionality common in hierarchically-structured natural systems, such as bone or mammalian lungs [1]. Consider, for example, battery and fuel cell electrode materials.

Electrode materials must (1) provide a sufficient density of active sites; (2) facilitate the transport of reactants/products to/from those sites; (3) act to dissipate local heating; (4) minimize parasitic reactions; (5) offer mechanical toughness and durability; and (6) readily integrate into the broader device architecture. An electrode material's capacity to achieve this multi-functionality is dictated by its microstructure.

In order to illustrate the preceding point, let us take the solid oxide fuel cell anode material, Ni-YSZ (yttria-stablilized zirconia), as a specific example. Ni-YSZ consists of three phases: nickel, YSZ, and pore. Each facilitates the transport of specific reactants/products. Junctions where the phases intersect, “triple-phase boundaries,” are the material's active sites.

There is thus a trade-off between efficient transport and activity in Ni-YSZ. Electrodes with straighter, less tortuous transport pathways offer greater transport efficiency at the cost of eliminating triple phase boundary density [2]. Similarly, increasing the weight percentage of YSZ increases the cermet's structural integrity and leads to better CTE match with the electrolyte, but comes at the cost of fewer active sites and reduced gas and/or electron transport efficiency. Similar trade-offs exist in battery electrode materials [3,4].

The question is: Which characteristics of a material's microstructure dictate its capacity for multi-functionality and, in turn, performance? And, furthermore, how can microstructure be designed to optimize multi-functionality? These questions are two sides of the same coin. The first is posed from the perspective of materials characterization, the second, from that of materials design. In order to design a material's microstructure, it is first necessary to understand which microstructural characteristics determine the properties of interest.

Methods for characterizing microstructure-property relationships often rely on numerical simulation techniques to interpret data obtained from materials characterization methods, e.g., 3-D imaging [5]. Finite element analysis (FEA), for example, can be used to calculate a 3-D structure's tortuosity factor or to estimate its mechanical stiffness [6,7]. Machine learning models for certain structure-property relationships have also been developed [8]. Numerical and machine learning approaches benefit from high accuracy and the ability to account for complex physical phenomena occurring in geometrically intricate domains.

The primary obstacle for numerical techniques is computational cost. Slow, memory intensive simulations act as bottlenecks in the analysis of larger sample sets, and they increase the cost of materials characterization. A secondary obstacle is low interpretability. By discretizing a structural domain into millions of elements or by feeding an image into a 3D convolution neural network, the structure-property link is quickly obscured, and additional, expert analysis may be required. The limitations of high fidelity techniques are exacerbated by complexity in the modeling domain's geometry as well as in the physical phenomena being considered. Nonlinearity, coupling, and multi-physics, for example, significantly increase computational costs and decrease interpretability.

Moving from materials characterization to design, we find a similar picture. Progress in designing and fabricating advanced, 3-D material structures is accelerating. A combination of emerging advanced manufacturing techniques, additive methods in particular, and advances in topology optimization (sometimes referred to as generative design) are fueling these advances [9]. Structural optimization methods, however, often rely on numerical simulation techniques such as FEA and thus come with significant computational expense and low interpretability [10].

In response to these materials characterization/design challenges, a variety of reduced order and approximate analytical/semi-analytical models have been developed, e.g., [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. They offer desirable complements to higher fidelity techniques as (1) screening tools; (2) interpretation aides; and as (3) means for speeding up numerical calculations (e.g., by providing a better initial guess of the solution). In certain contexts, approximate methods have been shown to accelerate topology optimization [10]. They have also been incorporated into “physics-informed neural networks,” e.g., as in [20].

The computational economy and interpretability of approximate/reduced order models are particularly desirable for the characterization/design of multifunctional materials where the complexity of numerical simulation can be immense. Early characterization/design stages in particular stand to benefit from replacing a single, multi-physics, multi-scale numerical model with a set of approximate, computationally inexpensive models, which permit examining a larger initial sample set. The best subset of the initial sample set can then be passed on to higher fidelity techniques.

Ideally the set of approximate models used in a multifunctional characterization/design context would share common nomenclature and form. Having a consistent mathematical formalism across phenomena facilitates the interpretable quantification of design trade-offs. Furthermore, it aides in coordinating efforts to design portions of a multi-functional material system. Currently, few approximate modeling approaches that treat different physical phenomena, e.g., transport and mechanics, share a consistent formalism. Similarly, few approaches offer a consistent formalism that lends itself to treating hierarchical structures, which are known to give rise to emergent material properties [21].

Analytical transport network (ATN) theory is an approximate, lumped element approach for modeling transport phenomena in geometrically complex network structures [13,14,22,23]. It consists of a channel- and a network-scale model. The channel-scale model accounts for the particular physics being considered and the local morphology of each of the network's channels. The graph-theory-based network-scale model relates the network's aggregate behavior to the relative arrangement of the channels, i.e., to the network's topology.

In addition to local potential and flow distributions, the network-scale model provides effective material properties. These properties can be used to express the network as an effective channel. Networks of networks can then be treated as networks of effective channels and the analysis can be repeated in a consistent manner through the length scales of a hierarchical structure.

To date, ATN theory exists for diffusive flow [13], diffusive flow with linear surface exchange [14], Hagen-Poiseuille flow, and linearly coupled flow (e.g., linear electrokinetic flow of a Newtonian fluid) [22,23]. Relative to finite element analysis, ATN can be implemented up to 6 orders of magnitude faster using down to 10 times less memory [14]. Furthermore, it is readily automated [23].

In this work, we extend the ATN approach to static structural mechanics. In doing so, we make it possible to characterize/design a complex network structure with respect to transport and mechanics in a single framework. Like ATN's transport formalism [13,24], the mechanics formalism is designed for future integration into established optimization approaches [10]. This opens up the future opportunity to accelerate multi-functional material structure optimization in a manner that also increases the optimization results’ interpretability.

In the next section, we present the channel-scale mechanics model. It is an ATN adaptation of classic Timoshenko beam theory. We discuss the impact of channel morphology on its stiffness to various loading conditions. Exploiting the mechanics model's shared formalism with ATN transport models, we examine trade-offs between mechanical and transport performance at the channel scale.

In Section 3, we present the network-scale model and define a number of “mechanical tortuosity factors.” Analogous to the tortuosity factor common to diffusive transport analysis, the mechanical factors quantify the influence of a structure's morphology and topology on its stiffness to loading. In addition, we introduce a method for normalizing a network's stiffness matrix and explore how the eigenvalues of the normalized matrix can be used in conjunction with those of the normalized Laplacian matrix (for transport analysis) to characterize the network's structure.

In Section 4, we apply the results of Sections 2 and 3 to a series of 100,000 randomly-generated network structures and demonstrate ATN's application to a large sample set at low computational cost. We then take a subset of the 100,000 structures and randomly combine them to generate 100 “superstructures,” which we subject to ATN analysis that includes 10 individual models. Each iteration of ATN analysis’ 10 models require ~1s. This is orders of magnitude faster than FEA. Furthermore, the analyses directly provide a variety of local morphological and topological properties, which aid in interpreting the estimates and which are not readily available from FEA.

To close we extend a previously-developed algorithm for extracting ATN geometric inputs from a voxel-based image to mechanical analyses. Compared to meshing, the algorithm is fully automated and runs in a matter of seconds. We show that the approach is robust to artifacts introduced into the structure from the extraction process. Through these examples, we demonstrate ATN's advantages as a computationally efficient, approximate modeling approach for early-stage materials characterization and design.

Section snippets

Channel-scale model

Here, we derive the 12-degree-of-freedom stiffness matrix for a straight channel with non-uniform cross-sectional area distribution. All geometrical properties are consistent with Analytical Transport Network Theory [13,14,22,23]. This makes it possible to consider the influence of channel morphology on both transport and structural mechanics within the same framework.

Network-scale model

Once local stiffness matrices for each of the network's members have been calculated and expressed in the global coordinate frame (kg,i), the network's stiffness matrix, Kg, can be assembled [36]. We provide an overview of assembly's established underlying theory from the ATN perspective in the supplemental materials. Here, we note our assumptions and move directly to solving the system.

Results & discussion

Here, we apply the ATN approach to characterize the mechanical and transport properties of 100,000 random variations of an octet unit cell. We begin with a description of the “baseline structure,” which is an octet unit cell. Then we describe our procedure for generating variations on that cell and examine how the harmonic means of the normalized Laplacian and stiffness matrices relate to the structures’ transport and mechanical properties. We neglect internal shear for these analyses.

Conclusions

In this work, we extended the framework of analytical transport network (ATN) theory to include static, structural mechanics (including thermal expansion). We exploited the shared formalisms of the ATN mechanics and transport models to examine morphological and topological trade-offs relevant to the design of multi-functional materials.

We demonstrated the technique's computational economy and ready supply of local property information by applying it to over 100,000 example structures. For each

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations. The method for generating/re-producing the data used for this analysis is documented in the main text and in further detail in the supplementary materials. The analyses methods themselves are documented in full in the text, supplementary materials, and in cited references.

CRediT authorship contribution statement

Alex P. Cocco: Methodology, Software, Validation, Formal analysis, Writing - original draft, Visualization. Kyle N. Grew: Validation, Writing - review & editing, Supervision.

Declaration of Competing Interest

None.

Acknowledgements

The authors gratefully acknowledge financial support for this work from the Army Research Laboratory under a Director's Research Initiative Early Career Award (KG). AC acknowledges support through Oak Ridge Associated Universities (ORAU) postdoctoral fellowship program at the Army Research Laboratory under the Cooperative Agreement Number W911NF-17-2-0038.

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