**1. Introduction**

Nanoporous gold (np-Au) made by dealloying can be produced as macroscopic objects that exhibit a bi-continuous network of nanoscale pores and solid "ligaments" connected in nodes. An overview of the fascinating morphologies and mechanical properties of this material is provided in [1–3]. The skeleton of the structure is formed by ligaments, which can be controlled in their average diameter by altering the dealloying conditions, thus allowing one to examine the impact of the ligament size on the macroscopic mechanical properties [4,5]. It has been recently demonstrated that the dealloying process can be applied sequentially and allows one to produce hierarchically organized nanoporous metals with superior macroscopic properties compared to similar materials with only one hierarchy level [6].

So far, even for one hierarchy level, no model exists that allows for the prediction of the macroscopic mechanical properties based on the parameters used in the sample preparation. Recently, the evolution of the ligament size and the network connectivity during thermal treatment was modeled with kinetic Monte Carlo simulations [7] for a large range of solid fractions, but the connection to the macroscopic mechanical properties is still missing. For a selected microstructure, this is realized by conventional meshing and finite

**Citation:** Huber, N. A Strategy for Dimensionality Reduction and Data Analysis Applied to Microstructure–Property Relationships of Nanoporous Metals. *Materials* **2021**, *14*, 1822. https:// doi.org/10.3390/ma14081822

Academic Editors: Francisco Chinesta, Elías Cueto and Benjamin Klusemann

Received: 12 March 2021 Accepted: 5 April 2021 Published: 7 April 2021

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**Copyright:** © 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

element (FE) simulation, e.g., as shown in [8], but from this point, it is still a long way to go towards an all-inclusive process–microstructure–property model that handles all required steps along a fully automated work flow and at the required efficiency. An overview of the elements needed for such a work flow based on efficient simulation models, data mining, and AI is presented in [9].

A key element represents the relationship that efficiently translates a set of microstructural parameters and material properties of the solid phase into macroscopic properties. Together with the structural information from, e.g., high-resolution 3D tomography and image analysis [10,11], all relevant aspects are currently under development. As pointed out in [9], they altogether will allow for an efficient scan of large multidimensional parameter spaces of descriptors and reliably predict the macroscopic mechanical properties for any assumed constitutive law on the level of a single ligament. Moving from scarce data to rich data allows for data mining of the fundamental structure–property relationships. The objective is to derive robust approximations that generalize the available data and support our understanding of the underlying physics well beyond the application of machine learning as a black box method.

In this work, we concentrate on the relationship that allows predicting mechanical properties based on microstructural information or, formulated as an inverse problem, enables us to determine microstructural descriptors from macroscopic test data. Due to the complex morphology of this material, FE modeling of np-Au with all its structural complexity is highly challenging. Two general paths exist, which are summarized in [9]. One route uses random structures (spinodal decomposition, leveled waves); the second is based on unit cells (Gibson–Ashby, gyroid, diamond). The mechanical behavior of random structures is usually predicted with molecular dynamics (MD) simulations [12–14] or with continuum mechanics using FE-solid or voxel models [8,15,16]. In combination with plasticity, also the FE-models lead to large computing times and allow only for a very limited number of simulations. Furthermore, the limited model size makes it extremely difficult to simulate a nanoindentation test that averages over sufficient features, such that it can be analyzed like an experiment. One of the rare examples that goes in this direction is the work of Farkas et al. [14], which presents a MD simulation of nanoindentation in a single crystal with a relative density of 0.67 and ligament diameter of 2 nm.

As pointed out in [9], FE-beam models provide the efficiency and flexibility needed for the generation of larger data sets and, at the same time, allow for an independent variation of all structural and material parameters of interest. This modeling technique has been successfully applied in studying the mechanical behavior of foams [17–22] and nanporous metals [10,23–27]. Research in this field concentrated mainly on the anisotropy of the macroscopic elastic properties as well as aspects of the structure–property relationships for elastic–plastic macroscopic compression. Until recently, the quantitative correct prediction of materials with relative densities >10% was limited to cylindrical ligament shapes [26]. The nodal correction proposed by Odermatt et al. enables us to expand FE simulations towards variations of the ligament shape from concave to convex [27]. The advantage of this approach is that the computational efficiency of FE-beam models is maintained. This paves the way for handling hundreds of simulations with a predictive model that is at the same time large enough for the simulation of nanoindentation.

The scope of this work is to study the influence of microstructure and material parameters on the macroscopic response of a porous metal. We will investigate the macroscopic behavior under compression as well as nanoindentation. For scanning the multidimensional parameter space, a highly efficient simulation model is required. Furthermore, the model set-up should allow for the independent variation of all important structural inputs. To this end, we use a representative volume element (RVE) that approximates the complex morphology of an open pore material by a diamond structure [23,24]. Using this unit cell, it is possible to define the degree of randomization and connectivity of the structure [28]. Together with the material parameters defining the mechanical behavior of the solid phase, this generates a highly dimensional parameter space that is hard to scan in

a dense manner by numerical simulations. The dimensionality of the problem and limited number of simulations makes analysis of the underlying structure–property relationships very challenging. If we limit the number of parameters to five (two material and three microstructural parameters) and the number of variations per parameter to three, a systematic variation with one parameter at a time would end up with 243 simulations, which is already at the limit of the computer's capacity. Adding more parameters or increasing the number of increments is almost impossible. Therefore, this investigation requires a strategy that exploits all available methods that contribute to reduce the dimensionality of the problem.

In this sense, the present work also serves as a guide, demonstrating how such a problem can be tackled systematically by means of a dimensional analysis, inclusion of a priori knowledge about the physical problem at hand, data generation strategies, principal component analysis, machine learning, and visualization. Along this path, Section 2 describes the FE-beam models used for generation of the data for macroscopic compression and nanoindentation. Sections 3 and 4 deal with dimensionality reduction of the macroscopic compression and nanoindentation problem, respectively, where both sections follow the same methodology. Finally, it is shown that for important dependencies, simple mathematical formulations can be derived that relate the major influences of microstructure and mechanical properties to the macroscopic response.

#### **2. FE-Model and Data Generation**

FE-beam modeling is used to predict the macroscopic response of nanoporous metals during macroscopic compression and nanoindentation. The generation of the representative volume element (RVE) is established in the literature and is described only briefly in Section 2.1. In contrast, the simulation of nanoindentation is novel. The incorporation of a conical indenter and strategies for achieving an efficient simulation model that copes with the nonlinearities arising from the contact problem is described in Section 2.2.
