Introduction

The progress of advanced technologies is critically dependent on the development of new materials satisfying ever increasing service requirements. Accordingly, the acceleration of materials development is widely recognized as a strategic research direction. The key challenge in the development of new structural metallic materials is that, in addition to optimizing the chemical composition of each material system, significant consideration is required for the hierarchical internal structure of the material, i.e., its microstructure. The microstructure of materials decides their properties and, at the same time, is highly dependent on the processing history (which typically involves thermo-mechanical treatment) and its variables. These strong interconnections emphasize the importance of establishing processing–structure–property (PSP) relationships for efficient materials design and optimization of the processing routes to provide for materials with targeted performance characteristics specific for each technological application.

Given the high cost and significant labor requirements of empirical trial-and-error approaches to establishing PSP relationships, increasing emphasis is placed on computational methods. The adoption of computational approaches is fueled by the progress in the development of high-performance computing infrastructures as well as significant advances in physics-based modeling at different length scales. Example applications of computational tools for materials development include prediction of properties depending on the chemical composition [1] or microstructure [2, 3], simulations of microstructure evolution during deformation [4,5,6,7,8] or heating/cooling [9]. Integration of these models and tools into an integrated computational framework that forms a digital twin of a manufacturing route is a promising approach to accelerating materials development and processing optimization. Such integration of computational tools and models is the central idea behind the Integrated Computational Materials Engineering (ICME) strategy.

More recently, the emergence of cloud computing, and ever increasing rates of data generation combined with improving data storage capacities has led to a growing interest in data science and machine learning approaches in many scientific domains, including materials science. A paradigm shift towards data-driven materials research is occurring [10, 11] to leverage the potential of these new approaches. Perhaps, one of the best known examples is the Materials Genome Initiative [12], which prioritized acceleration of materials development through harnessing the data-driven revolution in science.

Accelerated materials development and implementation, in part, could clearly benefit from widely available platforms that (i) provide cyberinfrastructure for storage and management of materials data that facilitates sharing and collaborations across facilities and among research groups and (ii) integrate computational tools for essential tasks of materials research, such as microstructure analysis and establishing PSP relationships. Early work on such platforms has already demonstrated their utility for various materials systems [13,14,15,16,17,18,19]. Notable examples of research and development in this direction include Materials Commons (materials collaboration platform and information repository) [13], Integrated Collaborative Environment (cyberinfrastructure for collaborative materials research) [19], MiCloud (commercial server-client based ICME) [17], Composite Hub (platform for composites community) [16], and Citrination Platform (materials data infrastructure) [18]. In this contribution, we present BisQue as an interdisciplinary, cloud-based platform aimed to equip materials researchers with web-based tools that cover tasks ranging from raw data storage and sharing, to microstructure analysis and microstructure-sensitive predictions of engineering properties with an emphasis on 3D analysis. The paper is organized such that we briefly describe the essentials of the overall BisQue infrastructure as well as the current state of the BisQue/Materials part of the platform in “BisQue for Cloud-Based Materials Research.” We then present integration of a new data-driven model into BisQue/Materials along with its example use cases in “Implementation of Microstructure–Property Linkages” and “Case Studies” and close with a summary and outlook for the BisQue/Materials platform in “Summary and Outlook”.

BisQue for Cloud-Based Materials Research

BisQue overview

BisQue, which stands for Bio-Image Semantic Query User Environment, is a platform developed at the UCSB and made available for researchers as a Software-as-a-Service.Footnote 1 Originally, BisQue was developed to provide online resources for biological image management and analysis [20, 21]. The motivation is that, in such image-centered disciplines as biology, it is common for researchers to deal with thousands of images, which requires non-trivial solutions concerning data storage and provenance, efficient processing, and data sharing to facilitate collaboration. The development of BisQue enables researchers to store terabytes of image data online, share it instantly with collaborators, and perform analysis on the uploaded datasets in the cloud. The system facilitates image processing, annotation, batch analysis, and parametric studies as well as efficient web-based rendering and visualization. Analysis and processing of images are typically supported through modules that implement algorithms for image classification, quantification methods, or even deep learning. The main advantage of the availability of these tools and algorithms in the BisQue platform is that users (with minimal computational resources) can run computationally intensive jobs in the cloud using the distributed high-performance computing resources of BisQue without the need to download large datasets to their local computers.

To support such rich and scalable functionality, the BisQue platform has a multilevel interconnected architecture shown in Fig. 1. The architecture is built upon four data types distinguished and supported in the platform—images, metadata, tables, and generic blobs with the four Core Services centered around each of these data types. The Core Services have the highest level of authentication with read and write permissions on data stored on the server and thus are usually curated by the BisQue administrators. The major function of the Core Services is to provide communication between the data and other elements of the platform through interoperable data formats such as XML and JSON and accessible binary formats. Other elements of the architecture that communicate with the data through the Core Services are web pages and visualization tools that represent data in a human-interpretable way, API tools, and Modules. Modules allow for analysis on (parts of) the data communicated between the server and the module through the Core Services. Modules that can be developed by users have access to the data provided by the Core Services and can write new data to the server derived (if any) as a result of an analysis. At the same time, with the original data is sacred philosophy, BisQue does not allow modules to edit or overwrite data already stored in the cloud. Furthermore, to promote reproducibility and help users keep track of research, BisQue writes every action and analysis carried out on each file in the metadata associated with the file. For example, metadata of an image (or any other file) contains the date and time when the image was uploaded to BisQue as well as details on which modules were used for the analysis and processing of the image accompanied by information on the input and output for each module execution.

Fig. 1
figure 1

Multilevel architecture of BisQue, including data, services, and modules

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In addition to this comprehensive server-side infrastructure for analysis that includes Services and Modules, BisQue allows for external analysis on data stored in BisQue. For these purposes, any environment that can fetch web resources and parse XML can be used to access the data. External interaction with data stored in BisQue service can be further simplified by using the Bisque API available for most of the popular programming languages: MATLAB, Python, Java, and C++. While external analysis relies on the computational resources of the user, it offers the flexibility of custom and rapid analysis without the need in module deployment in the BisQue platform. Compared to completely offline analysis (i.e., without BisQue), external analysis with BisQue offers efficient fetching of only parts of the data stored in the cloud, which is especially useful when a large dataset is shared among collaborators and analysis only on a small portion of the dataset is required. More details on the philosophy behind the BisQue infrastructure can be found elsewhere [20, 21].

Current Materials Tools in BisQue

Given prior success and utility of BisQue for image-based research in the cloud for researchers in biology as well as in other image-rich disciplines (e.g., life sciences, environmental and marine sciences), we recently began extending BisQue to the materials science domain. Our strategy for allowing cloud-based materials research was to significantly enhance the Table Service available in BisQue from simple tabular data (i.e., csv files) to comprehensive support of the hierarchical HDF5 file format with subsequent implementation of modules for this data type. The specific choice of introducing HDF5 as the “materials format” was made given the abundance of materials data with hierarchical and multimodal structure, which is especially true for microstructure data, central to many materials science efforts. For example, electron back-scattered diffraction (EBSD) is a widely used characterization technique, which yields multimodal maps that contain crystal orientations, back-scattered gray scale, image quality and pattern indexing measures at each pixel [22]. The HDF5 format, which has a hierarchical structure similar to a computer file system, was recognized as a suitable file format for such data [23]. The hierarchy allowed by the HDF5 format is especially useful to fuse data on the same region of the material from different sources (e.g., chemistry from energy dispersive spectroscopy or strain fields from digital image correlation and orientations from EBSD) or to efficiently store 3D microstructure data often obtained as slices by serial sectioning of a material sample [23, 24].

The added support of the hierarchical HDF5 file format alone allowed materials scientists to (i) store and organize rich materials data such as spatially resolved data on chemistry, morphology, and crystallographic orientation in the cloud and (ii) share them with collaborators, (iii) quickly explore the contents of HDF5 files—all online in a web browser. This functionality is further enhanced by data provenance tracking offered throughout the BisQue infrastructure.

As the next step in the development of the BisQue platform for the materials science community, we integrated the Dream.3D software package [25] into BisQue as a new internal module. Dream.3D uses HDF5 as a native file format which facilitated seamless integration of Dream.3D with the Table Service supporting HDF5 format. Dream.3D is a powerful open-source software package for 3D microstructure reconstruction and characterization [26] as well as virtual microstructure generation [27]. Integration of Dream.3D into BisQue allowed materials scientists to perform all these tasks in a web browser using the high-performance computing resources of the BisQue server. The availability of Dream.3D in the cloud is advantageous because modern 3D microstructure datasets (such as those obtained using the TriBeam platform [28, 29]) can easily reach terabytes in size and require significant computational resources for their reconstruction and analysis. Dream.3D analysis via BisQue can be done essentially from any (even portable) device with very modest memory and CPU requirements.

One of the successful use cases of the existing BisQue/Materials infrastructure featuring Dream.3D includes data curation and 3D microstructure analysis based on datasets obtained from the TriBeam system [30]. Specifically, the BisQue/Materials platform allowed researchers to (i) store online raw experimental datasets acquired in TriBeam; (ii) share them with collaborators worldwide; (iii) perform parametric studies of 3D microstructure reconstruction; (iv) store, share, and characterize the reconstructed volumes (i.e., processed data); and (v) capture optimized reconstruction and analysis workflows as Dream.3D pipelines also stored in BisQue as resources, shareable with other researchers. This application of the BisQue/Materials platform is presented in detail by Polonsky et al. [30].

Implementation of Microstructure–Property Linkages

Motivation and Approach

Our experience with the BisQue/Materials platform for 3D materials research (discussed above) showed tremendous advantages of having tools for materials science in the cloud. At the same time, prior work focused solely on microstructures and specifically 3D microstructure reconstruction and analysis [30]. In “Introduction”, we underlined that PSP relationships are key to efficient design and development of materials and central to most efforts in materials science and engineering. Consequently, the natural next step in the development of the BisQue/Materials platform is introducing tools that help establish the PSP relationships for materials and leverage the 3D microstructure information for prediction of structure-sensitive properties.

As mentioned in “BisQue overview,” BisQue allows for two types of tool integrations: external (through the BisQue API) and internal (through modules). We envision that the external interaction with BisQue (through an API, e.g., using local Python/MATLAB scripts) is most useful for exploratory research or custom analysis very specific to an individual researcher or research group. On the other hand, it is sensible to implement internally (i.e., as BisQue modules) those computational tools that can be useful to a broader materials science community, which is expected to be the case for PSP relationships.

Our approach, therefore, is to take advantage of the modular and flexible architecture of BisQue and integrate new tools for PSP relationships as modules into the BisQue/Materials infrastructure utilizing the Table Service support of the HDF5 format. This specific choice ensures the interoperability with Dream.3D, which is useful for providing microstructure datasets for which PSP relationships can be sought. Indeed, it can be envisioned that users can upload raw microstructure data (e.g., slices from serial sectioning), use Dream.3D to reconstruct 3D microstructure volumes, and store them into HDF5 files in BisQue. These reconstructed volumes can be then directly fed into the tools for PSP relationships, which will access the necessary 3D microstructure information from the corresponding arrays stored in the HDF5 file, without having to download or move any data.

Integration of models for PSP relationships with Dream.3D can be also beneficial in combination with the capabilities of Dream.3D to generate virtual (“synthetic”) microstructures. This combination is useful, for example, for design efforts focused on the microstructure exploration and optimization of a targeted property. With these design endeavors in mind and to illustrate the development and integration of tools in the BisQue/Materials platform for cloud-based computing of PSP relationships, we present an implementation of a data-driven model in BisQue for estimation of macroscopic yield strength in two-phase materials given their 3D microstructure information and individual strength characteristics of the constituent phases.

Model for Two-Phase Composites

Given our targeted application—cloud-based computing of PSP relationships—the models to be employed should offer a combination of accuracy and low computational cost to keep the user experience in the BisQue/Materials platform responsive and efficient. In this study, we consider one of the classical problems of micromechanics—homogenization, i.e., prediction of macroscopic (or effective) properties of a composite (or more generally, any heterogeneous material) given the properties of the microscale constituents and the 3D information of their spatial distribution (i.e., topology) which we refer to as microstructure. Despite homogenization being a long-standing problem and the development of numerous models to address it, the trade-off is clearly apparent between the accuracy and the computational cost in most of the widely used approaches.

For example, classical bounds for effective properties (Voigt [31], Reuss [32], Hashin–Shtrikman [33]) as well as classical mean-field approaches (Mori–Tanaka [34], self-consistent [35,36,37]) require very modest computational resources and an estimate of the effective property (or its bounds) can be obtained almost instantaneously with modern computers. At the same time, most of these approaches are based on simplifying microstructure descriptions, such as volume fractions. These simplistic first-order metrics overlook a number of important microstructural features such as morphology and alignment of the constituents as well as their relative spatial arrangement, which can also affect the effective properties of the material. The limitation of these approaches to first-order metrics of the structure of the material (i.e., volume fractions) leads to the predictions of these approaches being limited to first-order estimates of the properties. On the other end of the spectrum of the existing modeling strategies are numerical full-field approaches, such as finite element (FE) methods [3, 38,39,40,41,42,43]. In this case, the microstructure of the material at the given length scale can be explicitly taken into account through consideration of a representative volume element (RVE) of the microstructure (i.e., the representative microstructural volume element, MVE [44]). A numerical FE simulation can be carried out on the selected RVE to accurately determine the response of the microstructure and obtain the targeted effective property of the material. While powerful, the numerical approaches have high computational cost, especially in the context of the targeted interactive web-based application. Numerical approaches, and the FE method in particular, also often require dedicated and commercial software (e.g., ABAQUS or ANSYS), which presents an additional obstacle for implementation in open-source platforms such as BisQue.

Recently, the potential to overcome existing trade-offs between accuracy and computational cost has been achieved through data-driven and machine learning approaches. Specifically, a new physics-based framework employing data science strategies—Materials Knowledge Systems (MKS)—has demonstrated a combination of high accuracy and modest computation cost in addressing micromechanics problems, including homogenization [39, 45,46,47,48]. The central idea underlying the MKS approach is the calibration of physics-based functions (inspired by statistical continuum theories [49,50,51,52,53]) to numerical datasets produced by micromechanical FE simulations on ensembles of digitally generated (synthetic) MVEs [39, 45, 48, 54]. MKS showed success in addressing both linear (elastic) [48, 54] and non-linear (plastic [39, 46, 47, 55], damage [56]) properties and phenomena. In the context of two-phase materials, MKS has shown its utility over a wide range of volume fractions, phase topologies, and even property contrasts (i.e., ratio of the individual properties of the constituents) [39, 46]. An additional advantage of the MKS approach in the context of cloud-based computing is that the models developed within the framework have analytical forms that facilitate their implementation into such platforms as BisQue. For these reasons, the MKS approach to the model development was chosen for implementation into BisQue.

The MKS approach to micromechanics is based on statistical continuum theories that allow for incorporation of the effects of higher-order microstructure statistics on effective properties [49,50,51,52,53]. In these theories, higher-order microstructure statistics are captured through the formalism of n-point correlation functions. The central impediment of the classical statistical theories that limited their wide use lies in difficulties in analytical derivation of the high-order terms (Green’s function kernels) that capture the underlying physics of the microstructure–property relationship. In the MKS approach, these difficulties are overcome with the aid of modern statistical learning methods. A typical workflow of the model development within the MKS approach includes the following steps: (i) constitution of a training ensemble of digital microstructures, (ii) their quantification by n-point statistics and principal component analysis (PCA), (iii) estimation of their micromechanical responses by FE simulations, and (iv) calibration and verification of the model. In this work, we employ the model developed in a prior study [39], where more details on the data generation as well as model calibration and verification can be found. The essentials of the model development are outlined below.

Training Ensemble

The training ensemble was constituted of 456 MVEs of two-phase microstructures (measuring 27 voxels in all three dimensions) synthetically generated in Dream.3D. These MVEs were generated to cover a wide range of volume fractions of the phases and different sizes and shapes of the particles or grains constituting the phases. The choice of the MVE size of 27 × 27 × 27 voxels was guided by the conflicting requirements that the MVEs need to be (i) sufficiently large to capture the heterogeneity of the microstructure and to cover the length scale of the micromechanical interaction between the constituents and, at the same time, (ii) as small as possible to minimize the computational cost involved in the generation of the training dataset. The first requirement is usually fulfilled when the Green’s function underlying the sought microstructure–property relationship decays to values close to zero over a half of the MVE size. Previous work on micromechanical localization [48, 54] showed that 213 MVEs are sufficient to calibrate model coefficients related to the Green’s function for composites with moderate contrasts. In this and previous related studies [39, 46], slightly larger MVEs were adopted and successfully used for the model development.

Microstructure Quantification

Upon generation, microstructures were quantified through the calculation of n-point statistics. While the model development framework allows incorporating spatial statistics up to very high orders, n, in this study, we considered 2-point statistics, i.e., n = 2. The models based on 2-point statistics have been shown successful for material systems with low to moderate property contrasts [39, 45, 57]. Furthermore, the microstructure quantification step in the case of 2-point statistics can take advantage of efficient algorithms based on fast Fourier transforms (FFT) [58]. These algorithms were used for obtaining 2-point autocorrelation functions for the second (hard) phase. One autocorrelation function was considered because only one 2-point correlation is independent in two-phase materials [59, 60]. To enable efficient model development, the high dimensionality of 2-point statistics was reduced using PCA. PCA provides for patterns (principal component bases) in 2-point statistics sorted according to the explained variance in the ensemble (with the first basis capturing the most variance). Establishing these patterns (i.e., bases, ϕ) allows us to represent any k th microstructure or, more precisely, its 2-point statistics, f(k), using the weights of these patterns—principal component scores (or coordinates in the new basis), α(k):

$$ \mathbf{f}^{(k)} \approx \sum\limits_{j = 1}^{R} \alpha_{j}^{(k)} \boldsymbol{\phi}_{j} + \langle \mathbf{f} \rangle, $$
(1)

where the number of the first principal components, R, is a hyper-parameter of the model development chosen to satisfy Eq. 1 to a good approximation. In the MKS framework, the set of the first R principal component scores, α, obtained in PCA serves as a compact representation of 2-point statistics in reduced order models capturing the microstructure–property linkage of interest.

Micromechanical FE Simulations

The property of interest of the MVEs—effective yield strength—was obtained in micromechanical FE simulations carried out in ABAQUS software [61]. Isochoric tensile strain was considered with periodic boundary conditions. The constitutive response of both phases was described by isotropic von Mises plasticity. Effective yield strength was obtained by volume-averaging the stress tensor components over all integration points in the MVE followed by calculation of the effective von Mises stress, which is equal to effective yield strength for rate-independent plasticity with no strain hardening considered here. Running hundreds of micromechanical FE simulations for generation of training data involves considerable one-time computational cost and may require high-performance computing clusters for parallel calculations. For the present work, a single FE simulation typically takes 5–15 min to complete. These simulations are the most computationally demanding step, all other steps in the model development (e.g., microstructure quantification, polynomial regression) can be performed on a typical desktop.

Model Development

The dataset obtained in steps (i)–(iii), which included microstructure data and their efficient representation by principal component scores of 2-point statistics and their properties from FE simulations, allowed for establishing a data-driven model capturing the quantitative microstructure–property relationship. The model for effective yield strength, \(\bar {s}\) (normalized by the yield strength of the soft phase, s1), was sought in the form of an analytical polynomial function of principal component scores of 2-point statistics, α:

$$ \bar{s}/s_{1} = \sum\limits_{q} A_{q} \alpha^{q}, $$
(2)

where \(\alpha ^{q} = \alpha _{1}^{q_{1}}\alpha _{2}^{q_{2}} {\ldots } \alpha _{R}^{q_{R}}\) are the polynomial terms (monomials) of the first R principal component scores of 2-point statistics, α = {α1,α2,…,αR}, and A = {A0,A1,…,AN} are the influence coefficients capturing the microstructure–property relationship. Multi-index q is an array of exponents, q = {q1,q2,…,qR}, each element of which is a non-negative integer that varies from 0 to the selected maximum degree, Q (i.e., q ∈ [0,Q]). As an example, a polynomial function of the second degree (Q = 2) with three PC scores (R = 3) would be written as \(\bar {s}/s_{1} = A_{000} + A_{100}\alpha _{1} + A_{010}\alpha _{2} + A_{001}\alpha _{3} + A_{200}{\alpha _{1}^{2}} + A_{020}{\alpha _{2}^{2}} + A_{002}{\alpha _{3}^{2}} + A_{110}\alpha _{1}\alpha _{2} + A_{101}\alpha _{1}\alpha _{3} + A_{011}\alpha _{2}\alpha _{3}\). The unknown influence coefficients, A, were calibrated to the training data. The polynomial degree, Q, and the number of principal component scores, R, were the hyper-parameters of the model which were selected such that the calibrated model has shown a good fit to the training data and at the same time generalized well to new MVEs “unseen” by the model during the training step.

The developed reduced order model for effective yield strength of two-phase materials in the form of a calibrated polynomial function was used in the present work for implementation into the BisQue/Materials platform as described in the next section.

Model Implementation into BisQue

One of the key advantages of the MKS approach in the present context is that the microstructure-sensitive models can be obtained in the form of efficient analytical functions (Eq. 2), which facilitates their integration into open-source platforms such as BisQue. Indeed, the analytical form of the models allows for a straightforward implementation in any of the popular scripting languages (Python, MATLAB, C++) supported in BisQue without requiring any commercial software. In this work, the specific polynomial function for effective yield strength developed and calibrated as described above was implemented in Python, which ensured its seamless integration with the other key elements of the BisQue architecture also written in Python.

Prediction of the effective yield strength for a new two-phase MVE using the calibrated homogenization model requires the following steps: (i) calculation of 2-point statistics, f∗, for the new MVE;Footnote 2 (ii) transformation of 2-point statistics into the principal component space using the basis vectors, ϕ, obtained during PCA on the training data, which yields principal component scores for the new MVE, α∗; (iii) calculation of the effective yield strength, \(\bar {s}_{\ast }\), using the principal component scores of the MVE, α∗, as the input variables representing the microstructure in the polynomial function (Eq. 2) with the coefficients, A, calibrated during the model development. To allow the execution of these steps in the BisQue platform, first, all the necessary libraries were installed on the server, e.g., PyMKS [62], to support efficient FFT-based calculations of 2-point statistics [58]. Second, the basis vectors, ϕ, and the calibrated coefficients of the polynomial function, A, obtained during the model development were stored in binary files and uploaded to BisQue as generic resources. Rather than hard-coding the calibrated polynomial coefficients and basis vector into the module, this choice was made to allow the users to select which model (i.e., combination of basis vectors and calibrated polynomial coefficients) to utilize for their specific material system as more models become available on the platform.

In addition to making available the files containing the calibrated function and the principal component basis, we developed a Python code—the central element of the module—to perform the actual property calculations using these basis vectors and the polynomial. The function takes as input the following: (i) path to HDF5 file(s) with 3D microstructure data of two-phase MVE for which the property is sought, (ii) path to the file with the calibrated polynomial, (iii) path to the file with the principal component basis. Upon execution, the Python code outputs effective yield strength value(s) obtained for the provided HDF5 file(s) with microstructure data.

During the call, the Python script loads the basis vectors, ϕ, and the polynomial with the calibrated coefficients, A, calculates 2-point statistics for the given microstructure, f∗, transforms it into the loaded principal component basis, and uses the principal component scores obtained by transformation, α∗, as input variables for the loaded polynomial function, which finally returns the calculated effective yield strength. It should be recognized that a particular MVE size (27 × 27 × 27 voxels) of training microstructures was used during the model development. A new microstructure, for which the property prediction is sought, can be naturally of a different size. To ensure the functionality of the model for MVEs of an arbitrary size, the Python script checks the dimensions of the given MVE and either (i) pads 2-point statistics with zeros to the training MVE size for MVEs smaller than 27 × 27 × 27 or (ii) truncates 2-point statistics to the training MVE size for MVEs larger than 27 × 27 × 27. Unless pushed to extremes (i.e., extremely small or extremely large MVEs), the model can be expected to provide for reasonable estimates of the effective yield strength in both cases (truncation and padding). Truncation and padding can be used here because 2-point autocorrelation functions have a peak at the center (which is equal to the volume fraction of the corresponding constituent) and asymptotically decay to values close to zero from the center to the edges. It is these near-zero values, which correspond to long-range spatial correlations and have relatively minor effect on the sought property, that become omitted during truncation. For the same reason, 2-point autocorrelations obtained from smaller MVEs can be padded with zeros. We emphasize, that it is much more advantageous to truncate or pad 2-point statistics rather than raw microstructure volumes. Indeed, in the case of truncation of larger volumes, 2-point statistics are still sampled from the whole volume provided and only long-range correlations are omitted. Padding raw microstructures, unlike 2-point statistics, is simply impractical. We finally note that, while no specific physical dimensions were attached to the training MVEs, the implied characteristic length scale of the microstructural heterogeneity captured in the MVE was in the range of tens of micrometers (typical to dual-phase steel or W-Cu composites considered below). If a new microstructure dataset is obtained at a significantly different grid, its resolution can be adjusted to the voxel size of the training MVEs and then truncated or padded as discussed above.

In addition to the data-driven homogenization model developed using the MKS approach, we also included calculation of the classical Voigt [31] and Reuss [32] bounds of effective properties in the Python code of our module. Given the yield strength of the soft phase, s1, and the hard phase, s2, as well as their volume fractions, f1 and f2, Voigt and Reuss bounds of the effective (macroscopic) yield strength, \(\bar {s}_{V}\) and \(\bar {s}_{R}\), can be obtained as follows:

$$ \bar{s}_{V} = f_{1}s_{1} + f_{2}s_{2} $$
(3)
$$ \frac{1}{\bar{s}_{R}} = \frac{f_{1}}{s_{1}} + \frac{f_{2}}{s_{2}} $$
(4)

These bounds of effective yield strength are returned as a result of the module execution alongside the prediction obtained with the MKS homogenization model.

figure a

To allow efficient and intuitive interaction between the BisQue user and this function (as any other module), BisQue offers tools for the development of web-based graphical user interfaces of modules. The elements (buttons, input fields, drop-down menu, etc.) of the user interface can be defined in an XML file accompanying the Python code. In this work, the main input for the module is 3D microstructure data (spatially resolved phase labels) stored in an HDF5 file. Accordingly, the web elements of the interface are defined in XML to allow browsing and selecting an HDF5 resource in BisQue containing the microstructure information. Another option for input is a BisQue dataset, which is useful for calculations on a series of HDF5 files organized in a BisQue dataset. Calculations on datasets are advantageous for parallel property calculations on a number of microstructures at once. In addition, the interface features selection of the file containing the calibrated model and the PC basis as described above. The model (basis vectors and calibrated polynomial coefficients) described in this study is uploaded to BisQue as an example and openly available to the community. Once the files containing the model and the microstructure data are selected, calculation on either individual microstructures or microstructure datasets can be invoked by a Run button. The results of the calculation are displayed in the same web page in the Results field. The developed user interface with all these web elements is shown in Fig. 2 after execution on an example microstructure.

Fig. 2
figure 2

Web user interface of the BisQue module for prediction of effective yield strength of a two-phase composite developed in this work available at (https://bisque.ece.ucsb.edu/module_service/predict_strength/)

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The next section presents example use cases of the new module now available in BisQue for rapid property screening of MVEs (i) generated synthetically and (ii) subsampled from large experimentally measured and reconstructed volumes of a W–Cu composite.

Case Studies

Screening Virtual Microstructures

In this section, we present a case study for cloud-based property screening of 3D digital microstructures as an application for the new module implemented into the BisQue/Materials platform. The goal of the case study is rapid evaluation of the effective yield strength of two-phase materials over a large range of potential 3D topologies of the phases. In this case study, we focus on two-phase materials with a contrast in yield strength equal to 5 (i.e., s2/s1 = 5), which is typical of the strength contrast in widely used dual-phase steels containing ferrite (soft phase) and martensite (hard phase) [63]. The case study included the following steps: (i) composition of input parameters controlling 3D topologies of phases in two-phase microstructures (detailed below), (ii) generation of an ensemble of MVEs for the composed sets of topology parameters, (iii) calculation of effective yield strength of the MVEs using the MKS homogenization model (as well as Voigt and Reuss bounds). Dream.3D was employed for steps (i)–(ii), whereas the new module with the MKS homogenization model (and classical bounds) was used for step (iii). Furthermore, relatively computationally intensive steps (ii)–(iii) were completely carried out in BisQue/Materials platform from a web browser.

The first step—composition of input topology parameters—was carried out locally, on a personal computer. The StatsGenerator utility of Dream.3D was employed for the generation of the topology parameters, which included relative size (quantified by mean and deviation of log-normal distribution [27]) and shape (quantified by ellipsoid aspect ratios) characteristics of the grains constituting the phases and the phase volume fractions (more details on the adopted microstructure generation workflow can be found in [39]). First, a master HDF5 file with the parameters and a fixed volume fraction was manually created using StatsGenerator. Then, this set of parameters was duplicated 90 times with varying volume fractions ranging from 5 to 95% with a 1% step by using a Python script. The set of all generated input HDF5 files was then uploaded into BisQue as a dataset to carry out the second, more computationally intensive, step—microstructure generation.

Specifically, the uploaded dataset was used as input for a Dream.3D pipeline that instantiates 3D MVEs based on the parameters defining the targeted topology of the phases. The algorithm of Dream.3D for microstructure generation randomly places ellipsoids as precursors of the microstructure constituents and then iteratively fills the allocated 3D volume matching the user-defined size and shape parameters as closely as possible [27]. Since it is a more computationally demanding procedure than step (i), especially for a large set of MVEs, it was performed in the BisQue platform without any computational burden on the user’s personal computer. Execution of step (ii), i.e., microstructure generation, resulted in an ensemble of MVEs stored as individual HDF5 resources that contained 3D spatial information on the phases specified by their phase labels in each spatial voxel of the MVE. These MVEs can be visualized directly in BisQue with the aid of the web-based 3D rendering tools, without the need for downloading the dataset. An example MVE generated and visualized in BisQue is shown in Fig. 3a.

Fig. 3
figure 3

Case study on synthetic two-phase MVEs: a example MVE generated and visualized in BisQue, in which the hard phase is semi-transparent; b low-dimensional representation of the MVEs in the principal component space shown as a 2D projection in terms of the first two principal components, training MVEs are shown in gray, new MVEs are shown in red; c effective yield strength predicted by the classical bounds (Voigt and Reuss) and the MKS homogenization model, yield strength is plotted against the volume fraction of the hard phase, f2

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These MVEs, as described in “Implementation of Microstructure–Property Linkages,” were directly used as input for the MKS homogenization model for prediction of effective yield strength. Specifically, 2-point statistics were computed for each MVE and transformed into the principal component basis established during the model development. A 2D projection of the reduced order representation of the generated mirostructures in the principal component space is shown in Fig. 3b. In this particular ensemble of MVEs, the first principal component score (PC1) is highly correlated with the volume fraction of the hard phase [39], while the second principal component score (PC2) roughly corresponds to the size of the continuous regions of the hard phase. Since the MVEs were generated in Dream.3D with a particular targeted grain size (different from that used for the generation of training MVEs) and with a volume fraction of the hard phase varying from 5 to 95%, their PC scores appear as falling along a curve in the 2D projection of the PC space shown in Fig. 3b. The principal component scores obtained by this transformation were used as microstructure variables in the calibrated polynomial function (Eq. 2) as described in “Model for Two-Phase Composites.” Voigt and Reuss models were also used to obtain bounds of effective yield strength for the same microstructures based on their volume fractions only (Eqs. 3 and 4). All these calculations of the effective yield strength using the three models were carried out in the BisQue platform from the web user interface shown in Fig. 2. The output was displayed on the same web page as a list of values. The calculated values were parsed using Python and are visualized in Fig. 3c . It is seen that the classical bounds are quite widely separated (especially for MVEs with volume fractions of both phases close to 50%), which leads to significant uncertainty in the effective yield strength. At the same time, the MKS homogenization model provides for a prediction taking into account higher-order microstructure information beyond volume fractions, which presents potential benefits for microstructure-sensitive design and optimization efforts [50]. The use of the MKS homogenization model implemented in BisQue is also advantageous in terms of the computational cost: predictions for 90 MVEs takes only 6 min even if calculations are performed sequentially using a BisQue installation on a local server at UCSB. In comparison, approximately the same CPU time (5–10 min) would be required for a traditional FE simulation for only one MVE on a workstation.

Effective Yield Strength of W–Cu Composite

We next demonstrate the use of the presented module for the prediction of effective yield strength based on 3D microstructure data obtained experimentally for a W–Cu composite [64]. W–Cu composites with moderate volume fractions of copper (up to 40%) and continuous tungsten phase (which are considered here) are used in engineering components for high-temperature applications [64, 65]. The data analyzed in this study was obtained by Echlin et al. [64], who collected large 3D microstructure datasets (e.g., 615 × 525 × 250 μm3) from W–Cu samples with the aid of the TriBeam system. In this study, we consider several sets of MVEs of varying sizes. The analysis is relevant to the practical problem of the selection of a representative volume element (RVE) of a microstructure, i.e., a minimal volume characterized by the same property of interest as the microstructure as a whole [44, 66]. Concerning the selection of the RVE for W–Cu, Echlin et al. analyzed RVEs in terms of thermal and elastic properties using the Reuss/Voigt and Hashin–Shtrikman bounds based on volume fractions of the phases. Mignone et al. [67] analyzed effective elastic properties using FE simulations on experimentally measured 3D microstructures. The MKS model and its implementation in the BisQue/Materials platform allows for rapid screening MVEs of various sizes to determine a reasonable RVE size while accounting for microstructure beyond volume fraction and without the need of computationally demanding FE simulations. The MKS model calibrated to a yield strength contrast of s2/s1 = 5 can be applied to W–Cu composite assuming yield strength of copper, s1 ≈ 150 MPa, and of tungsten, s2 ≈ 750 MPa [68]. For the present case study, we considered 13 sets of 20 MVEs randomly sampled from large volumes obtained by 3D reconstruction of experimentally measured microstructure data. The 13 sets corresponded to different MVE sizes with the MVE edge length varying from 12.5 to 65 μm. After sampling, the datasets were uploaded to BisQue and analyzed with the developed module as described in the previous section. The analysis results are shown in Fig. 4b and c, which includes the results obtained using both the MKS model and the classical bounds implemented in the module. Figure 4b presents distributions of the effective yield strength values over 20 MVEs of different sizes. It is seen that the values are scattered when the MVE size is relatively small (especially in the range of 10–30 μm). As the MVE size increases, the scatter becomes less significant and the predicted effective yield strength values become more narrowly distributed within the dataset. The same conclusion is clearly seen in the dependence of the mean effective yield strength on the MVE size with error bars indicating standard deviation (Fig. 4c). Indeed, in the case of MKS model predictions, the standard deviation drops from 16.7% for the MVE edge length of 12.5 μm to 1.7% for the MVE edge length of 30 μm, while the mean value converges to \(\bar {s}/s_{1} \approx 3.84\). The convergence of the effective yield strength corresponds to increase in the statistical microstructure equivalence of MVEs within each dataset as the MVE size increases. Accordingly, classical bounds converge because of converging volume fractions [44], while MKS model predictions converge because of converging 2-point statistics [66].

Fig. 4
figure 4

Case study on experimental microstructure volume elements (MVEs) sampled from microstructure data on W–Cu composite: a MVEs of different sizes visualized as interfaces between the phases, b distribution of effective yield strength values for different MVE sizes predicted by the MKS model implemented in BisQue and by classical Voigt/Reuss bounds, c mean effective yield strength values (MKS prediction and bounds) as a function of the MVE size with error bars indicating standard deviation

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This case study highlights that rapid screening of effective properties enabled by computationally efficient MKS model can be used for the determination of the RVE for heterogeneous materials. The implementation of the MKS model in the BisQue platform offers additional advantages. First, the computational infrastructure and parallelization of calculations available through the BisQue platform allow screening very large datasets. To illustrate the computational advantages, property predictions for sets of 20 MVEs took only 1.5 min equally for smallest and largest MVEs studied here. Second, MKS model implemented in BisQue simplifies the workflow (in this case of RVE determination), which includes minimal software requirements (only web browser is needed) and minimal domain knowledge of the user in modeling of materials. Once the optimal RVE size is efficiently identified using these new tools, more detailed physics-based calculations (aiming, for example, strain localization [3, 63], damage [56], or crack initiation [69]) can be performed on the selected RVE keeping the computational cost of the physics-based simulation to a minimum. Finally, while the current study focused on predicting effective yield strength of two-phase materials based on microstructure quantification through 2-point statistics, the same data-driven model and module development workflows can be used to build models based on higher-order spatial statistics (e.g., 3-point correlations and beyond [57]) and/or for capturing other PSP relationships [46, 54, 56, 70].

Summary and Outlook

In this work, we presented the BisQue/Materials platform for efficient web-based 3D materials research. The platform can be used by materials scientists not only for data storage and sharing (with provenance) but also as cyber-infrastructure integrating materials science tools for tasks ranging from 3D microstructure reconstruction and analysis to prediction of properties based on 3D microstructure information. Towards this end, we described our philosophy on the development of such tools as internal BisQue modules that can be made available to the community and researchers worldwide to carry out analyses and calculations on their own datasets. The process of module development was exemplified through implementation of a data-driven model for prediction of effective yield strength of two-phase materials made of isotropically plastic constituents. A Python function implementing the model was integrated into BisQue along with a web-based user interface that allows users to perform calculations from a web browser of any personal computer or other (even portable) device with minimal requirements to software and computational resources on the user end.

The use of the new web-based tool for prediction of strength was demonstrated in two case studies. The first case study focused on screening potential microstructures in terms of their effective yield strength. The case study showed that both generation of microstructure volume elements and calculation of the effective yield strength can be carried out efficiently in the BisQue/Materials platform. This effort is a first step towards the development of a cloud computing platform for automated exploration of microstructures for performance optimization. The second case study was on prediction of effective yields strength for W–Cu composites based on experimentally measured 3D microstructures. The case study demonstrated that the developed module in the BisQue platform can be used for efficient determination of the representative volume element, i.e., a minimal microstructure volume element with the same property as the whole microstructure. The modular and flexible architecture of the BisQue platform provides for an excellent venue for the development of computational tools for efficient materials research. We invite researchers in the community to use the existing capabilities as well as contribute models to make the platform even more useful and versatile.