The photosensitive resin that acts as a temporary matrix for the YSZ acts is hardened via the interaction of the resin polymer and an additive photoinitiator. The photoinitiator is a constituent that is reactive to light exposure. When the photoinitiator is activated by light energy it generates molecular constituents that attract areas of the free floating polymer chains that make up the resin. The photoinitiators selected for these resins bind to them in such a way that multiple polymer chains can bind to a single activated element (Allen, 1996). As free floating polymers in the path of the directed light are bound together in large networks with cross linked elements, the engineering-scale effect of this process is that the material is hardened and fixed in those regions. These photoinitiating additives are known to degrade along with other resin elements over time after the resin mixture is formulated (Nagaoka et al., 2020).

A digital approach to manufacturing often raises the issue of ontologies. An ontology is a set of concepts and classes that define the entities in a domain, their relationships, and their properties. Ontologies in manufacturing have been explored as a method to query information from manufacturing data (Usman et al., 2013), to automate manufacturing design and decision making (Sanfilippo et al., 2019), and to create interoperability among different manufacturing design software applications (Chungoora et al., 2013).

While efforts to create an ontology to link design engineering and manufacturing methods (Chhim et al., 2019) have been explored, the ontology efforts of this project are geared toward making a single model of a manufacturing process that can be be used for experimental design generation (via the response surface method), manufacturing, and simulation, all in the service of sample performance prediction. The ontology is called Advanced Manufacturing Basic Entity Relationships (AMBER) and is based on the ontology Data Integration Aggregated Model and Ontology for Nuclear Deployment (DIAMOND), funded by an award from the Department of Energy’s Office of Nuclear Energy (Browning et al., 2020). Both ontologies make use of the Basic Formal Ontology (BFO) entities (Arp et al., 2015) near the root for future interoperability with other BFO-compatible models. A diagram of the classes in the ontology is shown in Figure 2.

The classes in the ontology are built around representing the properties and relationships that describe factors relevant both to the manufacturing of samples with DLP, and also the data modeling activities and numeric models necessary for making predictions from the relationship between these factors and the resultant performance of the samples.

The AMBER ontology is implemented in the Deep Lynx data lake (Darrington et al., 2020), a DOE-funded open source software utility. Deep Lynx allows ontologies to initialize a database, which can be used for storage, querying, and version control. For this project, the elements of a manufacturing build, decomposed by the definition and constraints of the process that generated it, feedstock material properties, and preform geometry, are all located in the same knowledge graph. The ontology also defines entities to store the resultant characteristics, that may be physical or simulated, and the predictive models themselves that are generated from the relationship between manufacturing inputs and outputs.

The response surface method (RSM) has been used widely in the search for optimal manufacturing parameters (Sik Shim, 2021; Geng et al., 2017; Al-Ahmari et al., 2016; Seo et al., 2017; Jagadish et al., 2016; Öktem et al., 2005). The method uses a constellation of techniques to systematically approach the discovery of optimal resultants in parameterized systems; the components of modern RSM involve generating a design of experiments, carrying out the designs, and using statistical techniques to correlate the relationship between design parameters and experimental resultant values. Typically, for techniques in which design parameters can be set continuously and independently within a set of ranges, the designs are generated using the central composite design or Box-Behnken design (de Oliveira et al., 2019). These methods define efficient designs that explore the contribution of parameters of interest across their available ranges. An alternative experimental design method was used for this study due to the constraints imposed by the manufacturing technique. Although layer height is of interest, batches of samples printed together must have the same basic layer height in order to print together. Because manufacturing was carried out in batches, there are discrete layer heights. As such the design method used in this study is a single parameter design.

Typically, once the experiments are carried out and the results recorded, the response surface is generated using a simple least squares regression. This study opts instead for a Gaussian Process Model (GPM), in order to create surfaces that not only predict regions of results inside the design envelope but also return uncertainty measurements as surfaces that bound the prediction surface. The RSM models were generated using utilities on the stochastic tools module (Slaughter et al., 2023) of the Multiphysics Object-Oriented Simulation Environment (MOOSE) simulation library (Lindsay et al., 2022). The library’s GPM is implemented as follows:

Given a set of input parameters X = { x ⃗ 1 , x ⃗ m } , and a set of corresponding outputs of the form Y = { y ⃗ 1 , y ⃗ m } that result from the function or process Y = f ( X ) the Gaussian model should return a new set of results Y * = { y ⃗ * 1 , y ⃗ * n } given new input parameters X * = { x ⃗ * 1 , x ⃗ * n } without incurring the cost of evaluating Y * = f ( X * ) .

In much the same way a multivariate Gaussian distribution is fully defined by its mean and covariance matrix, the Gaussian Process [shown in Equation 1] is defined by two functions: the mean function and the covariance function, where μ ( x ⃗ ) are the parameters from the training set and μ ( x ′ ⃗ ) are the parameters being evaluated. The mean function μ ( x ⃗ ) and the covariance function k ( x ⃗ , x ′ ⃗ ) are tuned on the training data. G P μ x ⃗ , k x ⃗ , x ′ ⃗ (1)

The mean function returns the mean value for any combination of parameters. The covariance function is meant to describe the relationship that the input parameters in the study have to one another. The output of the covariance function (also called a kernel function) is a covariance matrix that describes the complete set of parameters (every parameter’s relationship to every other parameter). The complete set of covariance matrices available from the kernel function are as follows in Equations 2–4: (2) (3) (4)

With this definition of the covariance function and its output, along with the assumption that the distribution of the training data and the evaluation data will be roughly the same the model can undergo conditioning. The Gaussian Process Model starts with an infinite collection of functions such that any random selection of output variables from the collection will produce a normal distribution. In the conditioning process functions are winnowed down from the infinite set to only those functions that agree with the training and evaluation sets, according to the following in Equations 5, 6: μ ⋆ = μ + K ⋆ K − 1 Y − μ (5) Σ ⋆ = K ⋆ ⋆ − K ⋆ T K − 1 K ⋆ (6)

and the output of the model will be given as shown in Equation 7 Y ⋆ ∼ N μ ⋆ , Σ ⋆ (7)

The particular kernel function used for this study was the radial basis function. It contains the additional terms (or hyperparameters) l , which is a length scale term which correlates loosely to the standard deviation of a given model parameter and affects the smoothness of the fitted model; σ f 2 , which represents the true signal variance and acts as an overall length scale term; and σ n 2 , which acts as a signal noise term which generally improves model fitting even when acting on relatively noiseless data. The radial basis function is shown as follows in Equations 8, 9: k x , x ′ = σ f 2 e x p − r ℓ x , x ′ 2 2 + σ n 2 δ x , x ′ (8) where r ℓ ( x , x ′ ) is expanded as r ℓ x , x ′ = ∑ n x i − x i ′ ℓ i 2 (9)

In this study’s implementation, the Gaussian process pulls its training data from the digital representation of a set of experiments. It prompts the user for initial hyperparameter values, trains the model, and stores the model executable in the graph for future reference and execution. Although the Gaussian process requires more data preparation and more operations than a simple fit using linear or polynomial regression, the digital twin framework uses the GPM in order to deliver models that return uncertainty bounds along with every point of evaluation. In the digital twin’s implementation, the manual data management usually involved with using the Gaussian model is handled by an interface that automates most onerous data preparation tasks.

Miniature three-point bend bars (25 mm × 4 mm x 1 mm) were printed using an Admatec Admaflex 130 DLP printer with commercially available photocurable YSZ resin (Admatec, Alkmaar, Netherlands). The samples printed in the study were designed using the ASTM C1161-18 design specification (ASTM International, 2018). This is a prismatic geometry optimized for testing the flexural strength of ceramic samples at ambient temperatures. The tesselated geometry was stored in the digital twin architecture as an STL file and also exported as a batch to the printer. Since the Admatec 130 build plate can accommodate 20 samples arranged in a grid pattern, 20 became the batch size wherein the layer height was held constant.

The print settings given in Table 1 were applied to all samples.

The layer height was determined by a depth of cure test in which a small amount of the slurry is placed on the printing surface and exposed to a certain duration of light exposure. The cured material is measured from the printing base After printing, each sample was post-processed with the same thermal treatment that debinds the organic resin matrix and then sinters the remaining ceramic material to its final density. The profile of this heat application is listed in Table 2.

One of the factors impacting the quality of printed samples is exposure to atmospheric humidity. The room in which the printing took place did not have a dehumidifier for the printing area nor for the lab space. Because of this, average humidity values during the time of printing (from 12:00 to 18:00, each day) was considered as a variable in the modeling effort, and as an element of the digital twin framework.

Flexural strength is the ultimate strength (the stress at which a tested sample breaks or cracks) when subjected to a bending load. The flexural strength and the tensile strength of a material, especially a ceramic, are correlated values (Leguillon et al., 2015). The general standard practice for materials that are brittle, such as ceramics, is to use determine material performance for an application from flexural strength, as these measures tend to be the most representative of application performance (Yamaguchi et al., 2019).

Samples were measured for flexural strength using a three-point configuration as illustrated in Figure 3. Conducting three-point bend tests on ceramic materials, as opposed to tensile tests, is driven by several considerations rooted in the inherent characteristics of ceramics. Primarily, ceramics exhibit brittleness, and their tensile strength is notably lower than their compressive strength. Tensile loading can lead to premature and catastrophic failure due to the limited capacity of ceramics to withstand tensile stress. The high friction jaws used in tensile testing lead to many samples cracking, making this technique impractical for this study. In contrast, three-point bend tests provide a controlled environment for studying fracture mechanics, including crack initiation and propagation, which is essential for understanding how ceramics fail. Fracture mechanics in ceramics are better captured through three-point bend tests, as these tests facilitate the controlled study of crack propagation. Tensile tests may not effectively capture critical flaw sizes or stress intensity factors crucial for assessing the fracture process in brittle materials.

Practically, ceramics are often employed in structural applications where they experience bending or flexural loads rather than tension. three-point bend tests simulate these real-world loading conditions, offering more relevant insights into how ceramics behave in practical applications. Additionally, the geometry of ceramic components frequently involves bending, making three-point bend tests more representative of actual service conditions. Moreover, the size and shape of samples for three-point bend tests are more manageable, requiring smaller specimens compared to the larger cross-sectional areas needed for tensile tests. Interpreting data from three-point bend tests is often more straightforward, with a more uniform stress distribution across the sample compared to tensile tests. This simplicity aids in relating the test results to the material’s properties.

Standardization further supports the preference for three-point bend tests in ceramics. Widely recognized testing methods, such as ASTM C1161-18, recommend three-point bend tests as the preferred approach for determining flexural strength and modulus of rupture. This standardization ensures consistency and comparability of results across different studies and laboratories, contributing to the robustness of data obtained from such tests.