1]Department of Mechanical Engineering, The University of British Columbia,
Vancouver, BC, Canada

2]Department of Civil Engineering, The University of British Columbia,
Vancouver, BC, Canada

# Graph Neural Network for Stress Predictions in Stiffened Panels Under
Uniform Loading

Yuecheng Cai  Jasmin Jelovica  [  [

###### Abstract

Machine learning (ML) and deep learning (DL) techniques have gained
significant attention as reduced order models (ROMs) to computationally
expensive structural analysis methods, such as finite element analysis (FEA).
Graph neural network (GNN) is a particular type of neural network which
processes data that can be represented as graphs. This allows for efficient
representation of complex geometries that can change during conceptual design
of a structure or a product. In this study, we propose a novel graph embedding
technique for efficient representation of 3D stiffened panels by considering
separate plate domains as vertices. This approach is considered using Graph
Sampling and Aggregation (GraphSAGE) to predict stress distributions in
stiffened panels with varying geometries. A comparison between a finite-
element-vertex graph representation is conducted to demonstrate the
effectiveness of the proposed approach. A comprehensive parametric study is
performed to examine the effect of structural geometry on the prediction
performance. Our results demonstrate the immense potential of graph neural
networks with the proposed graph embedding method as robust reduced-order
models for 3D structures.

###### keywords:

Machine learning, Deep learning, Reduced order models, Graph embedding, Graph
Neural Networks, Stiffened panels, Stress prediction, Geometric variations

##  1 Introduction

###  1.1 Motivation

The progress in developing efficient modern structures is significantly driven
by advancements in structural analysis methods such as Finite Element Analysis
(FEA) [clough1990original]. With the adoption of this powerful tool, engineers
can improve design of engineering structures. People can now routinely analyze
structures with diverse geometries without the need for intricate mathematical
or analytical solutions to governing differential equations. Consequently,
there is a growing emphasis on pursuing cost-effective structural solutions,
as they not only enhance performance but also lead to reductions in weight and
manufacturing costs [sullivan2018effect].

Many thin-walled structures, such as bridges, ships, and aircraft incorporate
stiffened panels in their design, which are generally effective in carrying a
diverse range of loads and relatively easy to manufacture. The stiffened panel
consists of a plate and discrete stiffeners welded onto it. The geometry of
both stiffeners and face plates used in the construction is often rectangular
because this is relatively easy to fabricate and can always provide good
performance. The panels often need to be optimized in design.

The optimization of stiffened panels and large structures incorporating the is
mostly performed by using sizing optimization [samanipour2020adaptive,
jelovica2022improved] and sometimes topology optimization [chu2021design].
Structural optimization often relies on FEA to estimate stresses under loads.
Although FEA can be successfully applied to solve numerous complex problems,
several challenges remain in structural optimization:

  * •

The computational costs increase as the complexity of the model rises,
including the refinement of the mesh;

  * •

Evaluation of each design requires independent simulations;

  * •

Re-meshing is often necessary when the structure changes geometrically.

With these challenges, the use of FEM in structural optimization is expensive.
In order to overcome this problem, traditional reduced-order models (ROMs)
have been used, such as Multivariate Adaptive Regression Splines (MARS),
Kriging (KRG), Radial Basis Functions (RBF), and Response Surface Method (RSM)
[chen2006review]. These methods aim to maintain the precision of high-fidelity
models while having a relatively low computational cost.

However, applying the ROMs above to complicated engineering problems is
limited due to their inherent assumptions. Moreover, they may lose fidelity
when the structure changes geometrically. More recently, with the advancements
in machine learning (ML) and in particular deep learning (DL) methodologies,
there is a tendency to adopt ML/DL models as ROMs, taking advantage of their
versatility in capturing various data properties. Particularly in mechanical
engineering, ML/DL techniques are effective modelling tools and approximators
that often exceed conventional ROM techniques in accuracy and capacity to
represent even nonlinearities of engineering problems [mai2022robust,
shojaeefard2013modelling, kabir2021failure].

###  1.2 Deep learning-based reduced order models for structural analysis

Typical ROMs require a structure to be represented parametrically, where the
structural variables are identified as inputs to the ML model. The typical
technique for reduced-order modelling is artificial neural networks (ANNs).
One of the most commonly used is multi-layer perceptron, also known as MLP,
which has been widely implemented in various fields, as evidenced by numerous
scholarly works. This is because MLP with one hidden layer is a universal
approximator, which can accurately predict any continuous function arbitrarily
well if there are enough hidden units, as stated in Ref
[hornik1989multilayer].

An early study of MLP-based ROM for structural optimization can be found in
Ref [papadrakakis1998structural], where MLP with one hidden layer was employed
as ROM to replace the structural analysis in optimization. With the demand for
higher prediction accuracy, deeper neural networks have been employed by
researchers to accomplish more complex tasks such as the prediction of axial
load carrying capacity [bisagni2002post], buckling load [sun2021prediction],
shear [limbachiya2021application] and lateral-torsional [ferreira2022lateral]
resistance, etc. More recent advancements in ANNs for structural analysis can
be found in Ref. [mandal2021application, zarringol2023artificial,
zhu2023artificial].

However, limited by the monolithic structure of traditional neural networks,
even if additional layers can be added to increase the architecture’s
complexity, MLPs are inefficient at accurately describing complicated
structural behaviour since they demand a large amount of training data and
computational resources. In addition, MLPs tend to overfit, and they are less
interpretable than other types of neural network (NN) approaches, therefore
its capabilities are limited and not suited for advanced problems.

Therefore, to capture more complex features, some researchers have used more
advanced NNs such as convolutional neural networks (CNNs) as ROMs for the
structures that can be represented as images (2D matrices) or composition of
images (3D matrices). A few examples can be found from Refs.
[ramkumar2021unconventional, banga20183d], where different composite
materials’ modulus, strength, and toughness are predicted by feeding enough
composite configurations (around 1,000,000). In addition, researchers utilized
CNN for stress predictions in different structures, for instance, the maximum
stress of brittle materials [wang2021stressnet], and stress contour in
components of civil engineering structures [bolandi2022deep].

Instead of approximating the mechanical properties and structural responses of
structures, some researchers also train CNNs or generative adversarial
networks (GANs) to predict the optimal structural design directly; interested
readers could check Refs. [nie2021topologygan, achour2020development,
mao2020designing, shu20203d] for more information.

However, modelling engineering structures using fixed-size vectors or
matrices, such as images, proves to be challenging, considering that one of
the variables is structural geometry. It introduces a dimensional change in
the design input, which is primarily handled by re-training for NN approaches
such as MLP and CNN. Furthermore, stiffened panels are discrete structures,
which consist of repeated structural units, e.g., stiffeners and plates, whose
connection can not be neglected as they affect the mechanical response of the
structure. Motivated by these two reasons, this paper proposes an approach
that transforms structural models into graphs. This transformation permits
flexibility in varying the dimension of design inputs, a characteristic that
aligns well with the capabilities of graph neural networks (GNNs).

GNNs have made significant strides across various fields, such as computer
vision [xu2017scene], traffic prediction [yao2018deep], chemistry
[gilmer2017neural], biology [fout2017protein], and recommender systems
[ying2018graph], etc. In fluid mechanics, researchers have employed GNNs in
replacing the expensive CFD simulations using dynamic graphs in recent years
[lino2022multi, shao2023pignn, pfaff2020learning], where they converted mesh
to the vertex and edge features, consequently applying GNNs to learn the
temporal evolution of the properties in a fluid system. However, since GNN
just gained significant attention in recent years, not many advancements have
been made in the structural engineering field. In a recent study by Zheng et
al. [zheng2023tso], a graph embedding approach was employed to represent 2D
and 3D trusses as graphs, with vertices denoting the joints and edges
representing the bars. This method aimed to optimize the arrangement of bars
within trusses while adhering to stress and displacement constraints.
Similarly, other recent investigations applying GNNs to truss-related problems
can be found in references such as [bacciu2020gentle] and [whalen2022toward].
These studies employed GNNs in conjunction with various techniques, including
Q-learning and transfer learning, to address distinct objectives.
Nevertheless, there has been a notable absence of research dedicated to the
application of GNNs to more intricate 3D structures beyond the scope of truss
problems.

In this paper, we extend the previous research by introducing an innovative
graph embedding technique tailored for 3D stiffened panels. We use this novel
graph embedding method for predicting stresses in stiffened panels, employing
a Graph Sampling and Aggregation network (GraphSAGE). We compare the proposed
graph embedding with the conventional finite-element-vertex embedding that has
been used in fluid mechanics. Additionally, we conduct a comprehensive
parametric study for various structural geometry parameters including various
boundary conditions and complex geometric variations.

##  2 Methodolgy

###  2.1 Basis of Graph Neural Networks

Graph Neural Networks (GNNs) represent a specialized subset of neural networks
(NNs) renowned for their capacity to handle data with graph embeddings. Unlike
the CNN, which is typically used for tasks involving data that is defined on a
regular grid, such as images, GNNs are designed to process and analyze data
represented in the form of graphs, such as data with complex relationships
between entities or data that has a natural representation as a network.

In general, a graph can be defined as $G=(V,E,A)$, where $V$ represents the
set of vertices, $E$ indicates the set of edges between these vertices, and
$A$ is the adjacency matrix. We denote the edge going from vertex $v_{i}$ to
vertex $v_{j}$ as $e_{ij}=(v_{i},v_{j})\in E$. If a graph is undirected, every
two vertices contain two edges $e_{ij}=(v_{i},v_{j})\in E$, and
$e_{ji}=(v_{j},v_{i})\in E$. The adjacency matrix $A\in\mathbb{R}^{N\times N}$
is a convenient way to represent the graph structure, where $N=|V|$ is the
number of vertices, $A_{ij}=1$ if $e_{ij}\in E$. For an undirected graph,
$A_{ij}=A_{ji}$. Therefore, a graph is associated with graph attributes
$X\in\mathbb{R}^{N\times D}$ and $A\in\mathbb{R}^{N\times N}$, where $D$ is
the number of input features of each vertex. It is worth mentioning that in
this study, all graphs are defaulted as undirected graphs.

To conduct convolution on a graph, researchers developed techniques such as
the graph spectral method, which leverages the eigenvalues and eigenvectors of
the graph Laplacian matrix to define graph convolutions, few of the most
famous networks are Spectral CNN [bruna2013spectral], Chebyshev Spectral CNN
(ChebNet) [defferrard2016convolutional], and Graph Convolution Network (GCN)
[kipf2016semi], etc. However, they require a large memory to conduct the graph
convolution, which is not efficient in handling large graphs. Hence, the
emergence of the graph spatial method, which operates by leveraging the
spatial interconnections among vertices and their adjacent neighbours, for
instance, original Graph Neural Network (GNN) [scarselli2008graph], Graph
Sampling and Aggregation (GraphSAGE) [hamilton2017inductive], and Graph
Isomorphism Network (GIN) [xu2018powerful], etc.

Figure 1: Architecture of the GraphSAGE Network employed in this study. Each
vertex represents a structural unit, including plate span, stiffener web and
flange. In the input graph, vertices hold the geometric details of their
respective structural units. Meanwhile, in the output graph, vertices contain
the stress information associated with their corresponding structural
components.

All the above-mentioned techniques can be summarized into message-passing
neural networks (MPNNs) [battaglia2018relational], which represent a general
framework of graph convolution. Under this framework, we specifically employ
the GraphSAGE network [hamilton2017inductive], which is a prominent and
benchmark method for many graph-related tasks. To preserve the maximum
information of each vertex, the ‘sum’ operator is determined as the
aggregation function [xu2018powerful]. Utilizing the MPNN framework, the
GraphSAGE operator with a ‘sum’ aggregator is defined as:

$\textbf{h}_{v}^{l}=\sigma(\textbf{W}^{l}(\textbf{h}_{v}^{l-1}+sum_{u\in\mathcal{N}(v)}\textbf{h}_{u}^{l-1}))$
(1)

where $\textbf{h}_{v}^{l-1}$ and $\textbf{h}_{v}^{l}$ are the embedding for
vertex $v$ at layer $l-1$ and $l$, respectively. $\textbf{W}^{l}$ is the
trainable parameter at the current layer $l$. Message propagation is conducted
through vector concatenation, followed by the message update phase through the
$tanh$ activation function $\sigma$.

A general structure of a GraphSAGE network can be found in Fig. 1. Structural
geometric data and external loading are initially transformed into the
proposed graph representation, as detailed in Section 2.2. At each layer, the
GraphSAGE operator is employed to process and learn the features of the graph.
Batch normalization has been utilized after each GraphSAGE convolution layer
to stabilize the training process of the GNN. Mean square error (MSE) is
adopted as the loss function in this study. The hyperparameters for the
utilized model have been fine-tuned and are detailed in Table 1. Note that the
hyperparameter settings are consistent throughout the investigation in Section
4.

Table 1: Hyperparameter setting of the Employed GraphSAGE Network Category | Value
---|---
Number of layers | 32
Number of hidden neurons for each layer | 64
Activation function | tanh
Optimizer | adam
Learning rate | 0.02
Batch size | 512
L2 regularization factor | 1e-4
|

###  2.2 Proposed graph embedding for stiffened panels

As indicated in the previous section, graph embedding is the prerequisite for
a structure being handled by a GNN. In this study, we propose a simple but
efficient method to graphically represent the stiffened panel.

Figure 2: Proposed graph representation versus finite element-vertex graph
representation. The representation of the finite element-vertex graph is
displayed on the top right. Each element is substituted by a vertex, and the
connections between elements are denoted as edges. The proposed graph layout
is presented at the bottom. Every vertex stands for a structural component,
such as the plate span, stiffener web, and flange, with their
interrelationships defined by edges.

Fig. 2 illustrates a conventional finite-element-vertex graph representation
where each element is a vertex. Our approach is to represent each stiffener or
plate between stiffeners as a vertex. The conventional method maps each finite
element to a corresponding vertex, while connections between these elements
are determined as edges. This transformation allows the direct representation
of the finite element (FE) model of the structure as a graph. Consequently, as
the mesh refinement progresses, the graph naturally scales in size in a
proportional manner.

However, it is essential to consider that the time complexity of GNN training
is proportional to the number of vertices $|V|$. The employed graphs in this
study are sparse graphs, with the following assumptions:

  * •

The GNN architecture is fixed, with a depth of $L$, and a transformed graph
embedding size of $F_{i}$ at layer $i$;

  * •

The average number of neighbours $k$ of vertices in the graph is about the
same for different graph representations;

  * •

The time complexity of feature transformation and aggregation dominates other
operators.

The time complexity for feature transformation is
$\mathcal{O}(N\times\Sigma_{i=1}^{L}F_{i-1}\times F_{i})$, and
$\mathcal{O}(N\times k\times\Sigma_{i=1}^{L}F_{i})$ for aggregation. Combining
the time complexity for both feature transformation and aggregation, we can
get the total training complexity:

$\mathcal{O}(N\times\Sigma_{i=1}^{L}F_{i-1}\times F_{i})+\mathcal{O}(N\times
k\times\Sigma_{i=1}^{L}F_{i})$ (2)

Given constants $C_{1}=\Sigma_{i=1}^{L}F_{i-1}\times F_{i}$ and
$C_{2}=k\times\Sigma_{i=1}^{L}F_{i}$, the combined complexity can be written
as: $\mathcal{O}(N\times(C_{1}+C_{2}))$. We can clearly see that
$\mathcal{O}(N\times(C_{1}+C_{2}))\propto\mathcal{O}(N)$, where the primary
factor is the number of vertices $N$ in a graph. Therefore, adopting the
conventional finite-element-vertex graph representation can lead to a
significant increase in computational resources as the level of structural
discretization grows.

In this paper, we introduce an efficient graph embedding technique designed
for modelling stiffened panels with a reduced number of vertices, as indicated
in Figure 2. More specifically, we represent each structural unit, including
plates between stiffeners, stiffener webs, and flanges, as individual vertices
within the graph. Each vertex incorporates both geometrical information of the
physical entity and its boundary conditions. Our objective is to reduce the
total number of vertices from $N$ to a smaller number $N^{\prime}$, thereby
expediting the GNN’s training process by approximately $N/N^{\prime}$ than
with the conventional approach. For each vertex, we employ eight variables,
including structural unit width, length, thickness, and boundary conditions
for each edge, together with the value of the applied pressure. The
connectivity between each structural unit is encoded by the adjacency matrix
and is not reflected in the vertex input embedding.

The objective of this study is to predict the von Mises stress distribution
across stiffened panels, which is crucial for structural design. Thus, it is
necessary to obtain the grid space (mesh) stress information of each
structural unit. Utilizing the proposed graph embedding, and given that the
vertex output dimensions do not affect the GNN training time, we have
allocated a grid space of $10\times 20$ to each structural unit. For each
vertex, the von Mises stress information of each node on the grid space is
determined as output. Since vertices in the graph can only accept vector
inputs, the number of input features $D$ for each vertex has been reshaped to
a dimension of $1\times 200$.

##  3 Data preparation

The case study are stiffened panels because they represent the basic repeating
unit of many large-scale structures. To approximate the stiffened panel of
real-life structures, the span of the panel has been set as $3m\times 3m$.
Main plate thickness ranges from $10mm$ to $20mm$. Each panel contains $2$ to
$8$ stiffeners, with a random height from $0.1m$ to $0.4m$. All stiffeners
have a T-shaped cross-section, with a rectangular flange whose width ranges
from $0.05m$ to $0.15m$. The thickness for both stiffener web and flange can
change from $5mm$ to $20mm$. We allow the stiffener/flange height and
thickness to continuously vary in this range, which allows a wider design
space. We assume that the plate of the panel is subjected to a uniformly
distributed pressure, which ranges from $0.05MPa$ to $0.1MPa$. The summary of
the upper and lower limits for the structure’s geometrical details can be
found in Table 2.

Table 2: Lower and upper limits of stiffened panel geometrical variables Category | Lower limit | Upper limit | Unit
---|---|---|---
Plate thickness | 10 | 20 | mm
Stiffener thickness | 5 | 20 | mm
Stiffener height | 100 | 400 | mm
Flange thickness | 5 | 20 | mm
Flange width | 50 | 150 | mm
Number of stiffeners | 2 | 8 | $-$
|  |  |

In this study, we have systematically considered two key variables for a
comprehensive parametric study: structural boundary conditions and geometric
variations. In both cases, we maintain the remaining variables fixed to
carefully assess the effect of introducing a specific variable under
investigation. For each case separately we prepare a total of 2000 randomly
generated design configurations, allocating 80% of them for training and 20%
for validation. Detailed analysis for each parametric test case is exhibited
and discussed in Section 4.

The dataset is obtained using parametric models prepared in MATLAB and
executed through ABAQUS FEM software. The static analysis is performed on a
model discretized using the ‘SR4’ element. The training procedure of GNN is
executed using Pytorch Geometric and carried out on a computing device with a
GTX 3090 GPU.

##  4 Results and discussion

In this section, we first compare the proposed entity-vertex graph embedding
with the finite element-vertex embedding, to demonstrate the efficiency of the
proposed approach. Afterwards, we present the results of our investigation
into the influence of various parameters on the accuracy of the trained GNN
model. The parameters under examination include boundary conditions and
geometric complexity. Understanding how these parameters affect the neural
network’s performance is crucial in optimizing its accuracy and robustness for
real-world applications. For each parameter, we first discuss its influence on
neural network accuracy, followed by a detailed analysis and comparison of
predictions with the ground truth data.

###  4.1 Impact of graph representation on GNN computational resources

To quantitatively study the influence of graph embedding on GNN training time
complexity as discussed in 2.2, we compare our proposed stiffened panel graph
embedding with a finite-element-vertex graph representation through a simple
test case. This comparison primarily focuses on the training time difference
between the two graph embedding techniques, the test case presented here omits
the stiffener flange for simplicity. All other structural variables remain
consistent with those described in Table 2. Two graph neural networks have
been trained independently based on the same dataset, GraphSAGE architecture,
and hyperparameter setting, differing only in graph embedding. An example for
both embeddings can be found in Fig. 2.

Figure 3 illustrates the comparison of stress contour predictions using
GraphSAGE with two graph embeddings, both of which yield comparable
performance levels. The training computational resources for both approaches
are presented in Table 3. Utilizing the same batch size, the GNN trained with
the proposed graph embedding achieves approximately 27 times faster training
times per epoch compared to the conventional method. The GPU memory
requirements also differ significantly: the proposed approach consumes 0.5 GB
for a batch size of 64, while the finite element-vertex embedding requires
23.4 GB. These results demonstrate the benefits of the proposed approach,
which is useful across a broader spectrum of computer devices.

Table 3: Graph embedding computational resources comparison Category | Entity-vertex embedding | Element-vertex embedding
---|---|---
Running time per epoch | 0.2553 seconds | 6.9375 seconds
GPU memory | 0.5 GB | 23.4 GB
|  |

(a) Element-vertex embedding

(b) Entity-vertex embedding

(c) Ground truth

Figure 3: Comparison between finite element-vertex embedding and the proposed
entity-vertex embedding

###  4.2 Parametric study

####  4.2.1 The effect of boundary conditions

In this subsection, we examine the impact of structural boundary conditions on
the performance of the GraphSAGE model. Entity-vertex embedding is used, where
the entity is plate between the stiffeners (or edge and stiffener), stiffener
web and stiffener flange. While considering all the variables outlined in
Table 2, we also incorporate the structural boundary conditions as an
additional variable. Specifically, plate edges and stiffeners are assigned
with random boundary conditions separately. All edges of the plate are either
simply supported or fixed. In addition, the edges of the stiffener web and
flange are free, simply supported, or fixed.

Table 4: Comparison of GraphSAGE predictions and FEA results for stiffened panel with varying structural boundary conditions Test examples | Model | 3D Stress Distribution | Stiffener webs and Flanges | Stress Range
---|---|---|---|---
1 | GNN |  |  |
FEA |  |
2 | GNN |  |  |
FEA |  |
Table 5: Structural geometry details in parametric study for boundary conditions. Category | Test Example 1 | Test Example 2 | Unit
---|---|---|---
Plate thickness | 14.51 | 12.58 | mm
Stiffener web thickness | 9.08 | 16.62 | mm
Stiffener web height | 308.37 | 223.98 | mm
Flange thickness | 16.83 | 15.61 | mm
Flange width | 111.12 | 86.52 | mm
Number of stiffeners | 8 | 3 | $-$
Uniform pressure | 0.071 | 0.072 | MPa
BCs @ Plate edges | Simply support | Fixed | $-$
BCs @ Stiffener web and Flange edges | Free | Simply support | $-$
|  |  |

(a) Test example 1

(b) Test example 2

Figure 4: Stress distribution comparison along specified paths in the
stiffened Panel under parametric study with varying boundary conditions Figure
5: GraphSAGE training data versus mean square error for parametric study
considering the effect of boundary conditions.

In Table 4, we present a comparison between the predicted stress distribution
of the stiffened panel and the distribution obtained through finite element
analysis (FEA). We have selected two test examples at random to illustrate the
prediction accuracy of our model. The 3D view of the structure is shown, where
the contour shows the stress distribution and stress intensity accross the
panel. For each test example, the same color for both GraphSAGE prediction and
FEA ground truth represents the same stress value. The geometry of the
structure and the external loading amplitude for both test examples can be
found in Table 5.

For both test examples in Table 4, the prediction of the stress distribution
in the plate and stiffeners shows a high consistency with the FEA. This can
also be verified by the detailed stress comparison in Fig. 4, where we exhibit
the stress distribution at different locations of the stiffened panel. The red
circles denote the FEA results and the blue solid curve denotes the predicted
stress distribution along the specified path with GNN. The location chosen to
be evaluated depends on the geometry and stress distribution of the stiffened
panel. The focus is on the areas that experience relatively higher stress,
such as plate edges and stiffener edges.

We can observe that although both test examples in this test case have
employed different boundary conditions, the GraphSAGE model with the proposed
graph embedding accurately captures the stress distribution of the structure.
The predicted stress distribution exhibits a high consistency with the FEA
results for both test examples at different locations. Although the prediction
shows some slight deviation when predicting the lower stresses such as the
plate center area in test example 1, the trained GraphSAGE network exhibits
good performance in capturing the high-stress distribution. The maximum stress
occurs at the intersection of plate and stiffener web edges in test example 1,
and at the plate center in test example 2. For the high-stress estimation in
both test examples (Fig. 4a (3) and Fig. 4b (1)), the prediction error is less
than 1%.

As is generally known, the amount of data and data quality are paramount for
accurate predictions with neural networks. Fig. 5 depicts the relationship
between training data size and the MSE value for the validation set. The
GraphSAGE model, utilized in this study, demonstrates consistent performance
when the amount of data is 400 and above. This suggests that the proposed
approach holds promise for practical applications in practical situations, as
it demands a relatively small amount of data.

####  4.2.2 The effect of structural geometry

An inherent advantage of graph neural networks (GNNs) over other types of
neural networks lies in their adaptability to structural geometric changes. In
this section, we aim to demonstrate the capacity of GNNs with the proposed
graph embedding technique. Instead of employing uniformly distributed
stiffeners, we introduce randomness to the distribution of stiffeners.
Furthermore, we allow variations in the height of each stiffener separately
and width of each flange separately, aiming at introducing more complexity to
the structural geometries.

Table 6: Comparison of GraphSAGE predictions and FEA results for stiffened panel with complex geometry. Test examples | Model | 3D Stress Distribution | Stiffeners and Flanges | Stress Range
---|---|---|---|---
1 | GNN |  |  |
FEA |  |
2 | GNN |  |  |
FEA |  |
Table 7: Structural geometry details for the two test examples when considering complex geometry. Category | Test Example 1 | Test Example 2 | Unit
---|---|---|---
Plate thickness | 19.85 | 16.14 | mm
Stiffener web thickness | (11.8, 17.9, 12.8, 12.6, 10.8) | (19.8, 17.0, 11.5) | mm
Stiffener web height | (320, 267, 320, 160, 293) | (347, 160, 347) | mm
Flange thickness | (14.9, 19.0, 14.0, 6.7, 6.2) | (14.1, 14.8, 9.8) | mm
Flange width | (74.3, 66.7, 107.8, 146.6, 69.8) | (52.1, 142.2, 104.0) | mm
Number of stiffeners | 5 | 3 | $-$
Uniform pressure | 0.079 | 0.090 | MPa
BCs @ Plate edges | Fixed | Fixed | $-$
BCs @ Stiffener and Flange edges | Fixed | Fixed | $-$
|  |  |

(a) Test example 1

(b) Test example 2

Figure 6: Stress distribution comparison along specified paths in the
stiffened panel under parametric study with complex structural geometry.
Figure 7: GraphSAGE training data versus mean square error for parametric
study with complex structural geometry

Specifically, in this section, the geometry of each stiffener and their
corresponding flanges can be individually changed within the domain as set in
Table 2, and they can be randomly positioned on the stiffened panel. With the
remaining structural geometric parameters the same as in Section 4.2.1, two
test examples are exhibited and their corresponding stress contour and
detailed stress analysis are shown in Table 6 and Fig. 6, respectively. The
corresponding structural geometry details can be found in Table. 7. In each
bracket, the numerical values represent the dimensions of the corresponding
category of structural geometry, with the order of the numbers corresponding
to the sequence of the stiffeners from left to right as shown in the Table. 6.

From Table 6, we can observe that increasing the complexity of the structural
geometry does not hinder the prediction capability of the GraphSAGE model with
the proposed graph embedding. For both test examples, the predicted stress
contour for both plate and stiffeners exhibits a good consistency with the FEA
results. As shown in Fig. 6a, the predicted stress distribution shows a good
alignment with the FEA results for all specified paths including the stiffener
edge and flange edge, where the maximum stress occurs. The GraphSAGE
estimation for the second test example is also accurate and captures all the
stress information of the structure. The maximum prediction error occurs at
the plate edge, which is about 6.18%. The average prediction accuracy for the
other paths is 96.7%.

Following a similar methodology as discussed in Section 4.2.1, we examine the
necessary amount of training data in this particular test case. In contrast to
the previous scenario, the GraphSAGE model requires a larger quantity of
training data to effectively capture stress distributions within more
intricate structures. As depicted in Figure 7, the curve reaches a plateau
after having 1200 training data points. However, it is important to note that
such a large dataset might not be required in practical real-life cases. The
test structures considered in this subsection encompass lots of unrealistic
structures, which can hinder the training process of the GraphSAGE model.

##  5 Conclusion and future work

In this study, we explored the potential of graph neural networks (GNN),
specifically Graph Sampling and Aggregation (GraphSAGE) as a promising avenue
for developing a reduced-order model for stress prediction in stiffened
panels. By proposing a novel graph embedding technique, the research showcased
its superiority over conventional finite element-vertex embedding. A
parametric analysis is conducted to demonstrate the versatility and robustness
of the employed model, emphasizing its ability to handle various boundary
conditions and complex geometric variations. The results indicate that the
integration of GNN and the proposed graph embedding can revolutionize the
structural modelling and analysis, offering both accuracy and efficiency. It
can be concluded that:

  * •

The proposed structural (physical unit) entity-vertex graph embedding is a
viable method to model the 3D structures, such as stiffened panels;

  * •

The proposed structural entity-vertex embedding is more efficient than the
conventional finite element-vertex embedding;

  * •

GraphSAGE with the proposed embedding can handle various variables including
boundary conditions and geometric variations;

  * •

The predicted stress distribution exhibits high consistency with the FEA
simulation.

Compared with the conventional reduced-order models employed in solid and
structural mechanics, the proposed modelling technique in this study first
introduces structural geometry as one of the input variables, which provides a
foundation and potential for further development of reduced-order models for
more complex structures. Even though this paper utilizes the data from the FEA
simulation, obtaining the data in real-life is very difficult. However, this
is a common issue for most data-driven models, which could be potentially
addressed in future works as well.

## Acknowledgments

This research was supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC). This research was supported in part through
computational resources and services provided by Advanced Research Computing
at The University of British Columbia.


# Degenerate Fubini-type polynomials associated with degenerate Apostol-
Bernoulli and Apostol-Euler polynomials of order $\alpha$

Muhammet Cihat DAĞLI
Department of Mathematics, Akdeniz University,
07058-Antalya, Turkey
E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, by introducing the degenerate Fubini-type polynomials, we give
several relations with the help of the Faà di Bruno formula and some
properties of Bell polynomials, and generating function methods. Also, we
derive some new explicit formulas and recurrence relations for Fubini-type
polynomials and numbers. Associating the degenerate Fubini-type polynomials
newly defined here with degenerate Apostol-Bernoulli polynomials and
degenerate Apostol-Euler polynomials of order $\alpha$ enables us to present
additional relations for some degenerate special polynomials and numbers.

Keywords: Apostol-Bernoulli polynomials, Apostol-Euler polynomials, degenerate
Bernoulli polynomials, Generalized Fubini polynomial, Stirling numbers,
explicit formula, recurrence relation, generating function.

Mathematics Subject Classification 2010: 11B68, 11B37, 05A15, 05A19, 11B83,
11Y55.

##  1 Introduction

The higher-order Bernoulli polynomials $B_{n}^{(\alpha)}(x)$ and higher-order
Euler polynomials $E_{n}^{(\alpha)}(x)$, each of degree $n$ in $x$ and in
$\alpha$, are defined by means of the generating functions [23]

$\left(\dfrac{t}{e^{t}-1}\right)^{\alpha}e^{xt}={\displaystyle\sum\limits_{n=0}^{\infty}}B_{n}^{(\alpha)}(x)\dfrac{t^{n}}{n!}$

and

$\text{
}\left(\dfrac{2}{e^{t}+1}\right)^{\alpha}e^{xt}={\displaystyle\sum\limits_{n=0}^{\infty}}E_{n}^{(\alpha)}(x)\dfrac{t^{n}}{n!},$

respectively. For $\alpha=1$, we have the classical Bernoulli polynomials
$B_{n}(x)$ and Euler polynomials $E_{n}(x),$ defined by means of the following
generating functions:

$\dfrac{te^{xt}}{e^{t}-1}={\displaystyle\sum\limits_{n=0}^{\infty}}B_{n}(x)\dfrac{t^{n}}{n!}\text{
}\left(\left|t\right|<2\pi\right)\text{ and
}\dfrac{2e^{xt}}{e^{t}+1}={\displaystyle\sum\limits_{n=0}^{\infty}}E_{n}(x)\dfrac{t^{n}}{n!}\text{
}\left(\left|t\right|<\pi\right).$

In particular, the rational numbers $B_{n}=B_{n}(0)$ and integers
$E_{n}=2^{n}E_{n}(1/2)$ are called classical Bernoulli numbers and Euler
numbers, respectively.

The generalized Apostol-Bernoulli polynomials $B_{n}^{(\alpha)}(x;\gamma)$
were defined by Luo and Srivastava by means of the generating function [19,
21, 22]

$\left(\dfrac{t}{\gamma
e^{t}-1}\right)^{\alpha}e^{xt}={\displaystyle\sum\limits_{n=0}^{\infty}}B_{n}^{(\alpha)}(x;\gamma)\dfrac{t^{n}}{n!}$
$\left(\gamma\in\mathbb{C};\text{ }\left|t\right|<2\pi\text{ if
}\gamma=1\text{; }\left|t\right|<\left|\log\gamma\right|\text{ if }\gamma\neq
1\right),$

and the generalized Apostol-Euler polynomials $E_{n}^{(\alpha)}(x;\gamma)$ by
means of the generating function [20]

$\text{ }\left(\dfrac{2}{\gamma
e^{t}+1}\right)^{\alpha}e^{xt}={\displaystyle\sum\limits_{n=0}^{\infty}}E_{n}^{(\alpha)}(x;\gamma)\dfrac{t^{n}}{n!}$
$\left(\left|t\right|<\pi\text{ if }\gamma=1\text{;
}\left|t\right|<\left|\log(-\gamma)\right|\text{ if }\gamma\neq 1;\text{
}1^{\gamma}=1\right).$

Carlitz [1] defined degenerate Bernoulli polynomials and degenerate Euler
polynomials by

$\frac{t}{\left(1+\lambda t\right)^{1/\lambda}-1}\left(1+\lambda
t\right)^{x/\lambda}={\displaystyle\sum\limits_{n=0}^{\infty}}B_{n}(x;\lambda)\dfrac{t^{n}}{n!},$

and

$\frac{2}{\left(1+\lambda t\right)^{1/\lambda}+1}\left(1+\lambda
t\right)^{x/\lambda}={\displaystyle\sum\limits_{n=0}^{\infty}}E_{n}(x;\lambda)\dfrac{t^{n}}{n!}.$

For $x=0,$ these are called as degenerate Bernoulli and Euler numbers.

The degenerate versions of Apostol-Bernoulli polynomials and Apostol-Euler
polynomials of order $\alpha$ were introduced by [13]

$\left(\frac{t}{\gamma\left(1+\lambda
t\right)^{1/\lambda}-1}\right)^{\alpha}\left(1+\lambda
t\right)^{x/\lambda}={\displaystyle\sum\limits_{n=0}^{\infty}}B_{n}^{\left(\alpha\right)}(x;\lambda;\gamma)\dfrac{t^{n}}{n!}$
(1.1)

and

$\left(\frac{2}{\gamma\left(1+\lambda
t\right)^{1/\lambda}+1}\right)^{\alpha}\left(1+\lambda
t\right)^{x/\lambda}={\displaystyle\sum\limits_{n=0}^{\infty}}E_{n}^{(\alpha)}(x;\lambda;\gamma)\dfrac{t^{n}}{n!},$
(1.2)

respectively. Note that since $\lim_{\lambda\rightarrow 0}\left(1+\lambda
t\right)^{1/\lambda}=e^{t}$, for $\lambda\rightarrow 0,$ $\alpha=\gamma=1,$
the equations (1.1) and (1.2) reduce to the generating functions for classical
Bernoulli and Euler polynomials, respectively.

Let us mention that the above polynomials have been discussed detailed in the
literature. (See for example [6, 9, 10, 18, 20] and related references
therein).

We now focus on Kılar and Simsek’s recent study [14], in which a family of
Fubini-type polynomials $a_{n}^{\left(\alpha\right)}\left(x\right)$ are
introduced as in the following

$\frac{2^{\alpha}}{\left(2-e^{t}\right)^{2\alpha}}e^{xt}={\displaystyle\sum\limits_{n=0}^{\infty}}a_{n}^{\left(\alpha\right)}\left(x\right)\frac{t^{n}}{n!},\text{
\ \ }\alpha\in\mathbb{N}_{0}\text{ and }\left|t\right|<\log 2.$ (1.3)

In particular,
$a_{n}^{\left(\alpha\right)}\left(0\right)=a_{n}^{\left(\alpha\right)}$ are
called Fubini-type numbers. They gave some relationships between these
polynomials and numbers, and other celebrated polynomials and numbers such as
Apostol-Bernoulli numbers, the Frobenius-Euler numbers and the Stirling
numbers via generating function methods and functional equations. Very
recently, Srivastava and Kızılates [38] extended Fubini-type polynomials
$a_{n}^{\left(\alpha\right)}\left(x\right)$ to parametric kind families of the
Fubini-type polynomials by considering the two special generating functions
and obtained many relations concerning these and other parametric special
polynomials and numbers. As emphasized therein, the Fubini-type polynomials
$a_{n}^{\left(\alpha\right)}\left(x\right)$ are special case of generalized
Apostol-Euler polynomials $E_{n}^{(\alpha)}(x;\gamma).$ More concretely,
$E_{n}^{(2\alpha)}(x;-1/2)=2^{3\alpha}a_{n}^{\left(\alpha\right)}\left(x\right).$

Further investigations on Fubini polynomials and numbers can be found in [12,
15, 16, 24, 37, 39], and plenty of references cited therein.

On the other hand, Qi and his colleagues have studied a number of explicit and
recursive formulas, and closed forms for some significant polynomials and
numbers by applying the Faà di Bruno formula (see Eq. (2.1), below), some
properties of the Bell polynomials of the second kind, and a general
derivative formula for a ratio of two differentiable functions. See [3, 4, 5,
8, 11, 25, 26, 27, 28, 29, 30, 31, 33, 34, 36, 40] and related references.

In this paper, we introduce degenerate version of Fubini-type polynomials as

$\frac{2^{\alpha}}{\left(2-\left(1+\lambda
t\right)^{1/\lambda}\right)^{2\alpha}}\left(1+\lambda
t\right)^{x/\lambda}={\displaystyle\sum\limits_{n=0}^{\infty}}a_{n}^{\left(\alpha\right)}\left(x;\lambda\right)\frac{t^{n}}{n!},\text{
\ \ \ }\lambda\in\mathbb{R}.$ (1.4)

Notice that for $x=0,$
$a_{n}^{\left(\alpha\right)}\left(0;\lambda\right)=a_{n}^{\left(\alpha\right)}\left(\lambda\right)$
are called degenerate Fubini-type numbers. Also, for $\lambda\rightarrow 0$,
these reduce to Fubini-type polynomials
$a_{n}^{\left(\alpha\right)}\left(x\right)$ aforementioned above.

In parallel with the conclusion given in [38, Remark 4], we infer a
relationship between degenerate Fubini-type polynomials and degenerate
Apostol-Euler polynomials of order $\alpha,$ i.e.

$a_{n}^{\left(\alpha\right)}\left(x;\lambda\right)=2^{-3\alpha}E_{n}^{(2\alpha)}(x;\lambda;-1/2).$
(1.5)

In this paper, we would like to use Faà di Bruno formula and some properties
of Bell polynomials, and generating function methods in order to obtain some
new explicit formulas, closed forms and recurrence relations for degenerate
Fubini-type polynomials and numbers, and Fubini-type polynomials and numbers.
Moreover, we give a relation between degenerate Fubini-type polynomials and
degenerate Apostol-Bernoulli polynomials of order $\alpha$ and deduce similar
formulas for them.

##  2 Properties of second kind Bell polynomials

The Bell polynomials of the second kind
$B_{n,k}\left(x_{1},x_{2},...,x_{n-k+1}\right)$ for $n\geq k\geq 0$ were
defined by [2, p. 134 and 139]

$B_{n,k}\left(x_{1},x_{2},...,x_{n-k+1}\right)={\displaystyle\sum\limits_{\begin{subarray}{c}1\leq
i\leq n,\text{ }l_{i}\in\left\\{0\right\\}\cup\mathbb{N}\\\
{{\textstyle\sum\nolimits_{i=1}^{n}}}il_{i}=n,\text{
}{{\textstyle\sum\nolimits_{i=1}^{n}}}l_{i}=k\end{subarray}}^{\infty}}\frac{n!}{{\textstyle\prod\nolimits_{i=1}^{l-k+1}}l_{i}!}{\textstyle\prod\limits_{i=1}^{l-k+1}}\left(\frac{x_{i}}{i!}\right)^{l_{i}}.$

The Faà di Bruno formula can be described in terms of the Bell polynomials of
the second kind $B_{n,k}\left(x_{1},x_{2},...,x_{n-k+1}\right)$ by

$\frac{d^{n}}{dt^{n}}f\circ
h\left(t\right)={\displaystyle\sum\limits_{k=0}^{n}}f^{\left(k\right)}\left(h\left(t\right)\right)B_{n,k}\left(h^{\prime}\left(t\right),h^{\prime\prime}\left(t\right),...,h^{(n-k+1)}\left(t\right)\right).$
(2.1)

For $n\geq k\geq 0,$ these polynomials satisfy the following relation [2, p.
135]

$B_{n,k}\left(abx_{1},ab^{2}x_{2},...,ab^{n-k+1}x_{n-k+1}\right)=a^{k}b^{n}B_{n,k}\left(x_{1},x_{2},...,x_{n-k+1}\right),$
(2.2)

where $a$ and $b$ are any complex number. Also, for $n\geq k\geq 0,$ the
following formula is valid for the special case of $B_{n,k}$

$B_{n,k}\left(1,1,...,1\right)=S\left(n,k\right),$ (2.3)

where $S\left(n,k\right)$ denotes the Stirling numbers of the second kind, can
be generated by [2, p. 206]

$\frac{\left(e^{t}-1\right)^{k}}{k!}={\displaystyle\sum\limits_{n=k}^{\infty}}S\left(n,k\right)\frac{t^{n}}{n!}.$

In [32, Remark 1], there existed the formula

$B_{n,k}\left(1,1-\lambda,\left(1-\lambda\right)\left(1-2\lambda\right),...,\prod_{l=0}^{n-k}\left(1-l\lambda\right)\right)=\frac{(-1)^{k}}{k!}{\displaystyle\sum\limits_{l=0}^{k}}\left(-1\right)^{l}\binom{k}{l}\prod_{q=0}^{n-1}\left(l-q\lambda\right),$
(2.4)

which is equivalent to

$B_{n,k}\left(\left\langle\lambda\right\rangle_{1},\left\langle\lambda\right\rangle_{2},...,\left\langle\lambda\right\rangle_{n-k+1}\right)=\frac{(-1)^{k}}{k!}{\displaystyle\sum\limits_{l=0}^{k}}\left(-1\right)^{l}\binom{k}{l}\left\langle\lambda
l\right\rangle_{n},$ (2.5)

established in [35, Theorems 2.1 and 4.1]. In [7, Remark 7.5], the explicit
formulas (2.4) and (2.5) were rearranged as

$\displaystyle
B_{n,k}\left(1,1-\lambda,\left(1-\lambda\right)\left(1-2\lambda\right),...,\prod_{l=0}^{n-k}\left(1-l\lambda\right)\right)$
$\displaystyle=(-1)^{k}\frac{\lambda^{n-1}(n-1)!}{k!}{\displaystyle\sum\limits_{l=1}^{k}}\left(-1\right)^{l}l\binom{k}{l}\binom{l/\lambda-1}{n-1}$

for $\lambda\neq 0$ and

$B_{n,k}\left(\left\langle\lambda\right\rangle_{1},\left\langle\lambda\right\rangle_{2},...,\left\langle\lambda\right\rangle_{n-k+1}\right)=(-1)^{k}\lambda\frac{(n-1)!}{k!}{\displaystyle\sum\limits_{l=1}^{k}}\left(-1\right)^{l}l\binom{k}{l}\binom{\lambda
l-1}{n-1}.$ (2.6)

Here, the generalized binomial coefficient $\binom{z}{w}$ is defined by

$\binom{z}{w}=\begin{cases}\dfrac{\Gamma\left(z+1\right)}{\Gamma\left(w+1\right)\Gamma\left(z-w+1\right)},&\text{
if }z,w,z-w\in\mathbb{C}-\left\\{-1,-2,...\right\\}\text{;}\\\ 0,&\text{if
}z\in\mathbb{C}-\left\\{-1,-2,...\right\\}\text{ and
}w,z-w\in\left\\{-1,-2,...\right\\}\text{.}\end{cases}$

##  3 Results and their proofs

In this section, we give some computational formulas for degenerate Fubini-
type numbers, some explicit formulas and recurrence relations for Fubini-type
polynomials and numbers and consequently, degenerate Apostol-Bernoulli
polynomials and degenerate Apostol-Euler polynomials of order $\alpha.$ Also,
we present further relations for some polynomials, considered here.

###### Theorem 3.1

The degenerate Fubini-type numbers can be computed by the formula:

$a_{n}^{\left(\alpha\right)}\left(\lambda\right)=(n-1)!{\displaystyle\sum\limits_{k=1}^{n}}\frac{\left\langle-2\alpha\right\rangle_{k}}{2^{\alpha+k}}\frac{(-1)^{k}}{\lambda^{k-1}k!}{\displaystyle\sum\limits_{l=1}^{k}}(-1)^{l}l\binom{k}{l}\binom{\lambda
l-1}{n-1},$ (3.1)

where $\left\langle x\right\rangle_{n}$ denotes the falling factorial, defined
for $x\in\mathbb{R}$ by

$\left\langle
x\right\rangle_{n}=\prod\limits_{k=0}^{n-1}\left(x-k\right)=\begin{cases}x\left(x-1\right)...(x-n+1),&\text{if
}n\geq 1\text{;}\\\ 1,&\text{if }n=0.\end{cases}$

Consequently, for the special case of the degenerate Apostol-Euler polynomials
$E_{n}^{(\alpha)}(x;\lambda;\gamma),$ the following relation holds:

$E_{n}^{(2\alpha)}(0;\lambda;-1/2)=(n-1)!{\displaystyle\sum\limits_{k=1}^{n}}\frac{\left\langle-2\alpha\right\rangle_{k}}{2^{k-2\alpha}}\frac{(-1)^{k}}{\lambda^{k-1}k!}{\displaystyle\sum\limits_{l=1}^{k}}(-1)^{l}l\binom{k}{l}\binom{\lambda
l-1}{n-1}.$ (3.2)

Proof. If we apply $f(u)=(2-u)^{-2\alpha}$ and $u=g(t)=\left(1+\lambda
t\right)^{1/\lambda}$ to the Faà di Bruno formula (2.1), and use (2.2) and
(2.6) then, we find that

$\displaystyle\frac{d^{n}}{dt^{n}}\left((2-\left(1+\lambda
t\right)^{1/\lambda})^{-2\alpha}\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\frac{d^{k}}{du^{k}}(2-u)^{-2\alpha}B_{n,k}\left(\frac{\lambda\left(1+t\right)^{\lambda-1}}{\lambda},\frac{\lambda\left(\lambda-1\right)\left(1+t\right)^{\lambda-2}}{\lambda},\right.$
$\displaystyle...,\left.\frac{\lambda\left(\lambda-1\right)...\left(\lambda-\left(n-k\right)\right)\left(1+t\right)^{\lambda-(n-k+1)}}{\lambda}\right),$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\left\langle-2\alpha\right\rangle_{k}(2-u)^{-2\alpha-k}\frac{\left(1+t\right)^{k\lambda-n}}{\lambda^{k}}B_{n,k}\left(\left\langle\alpha\right\rangle_{1},\left\langle\alpha\right\rangle_{2},...,\left\langle\alpha\right\rangle_{n-k+1}\right)$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\left\langle-2\alpha\right\rangle_{k}(2-u)^{-2\alpha-k}\frac{\left(1+t\right)^{k\lambda-n}}{\lambda^{k}}(-1)^{k}\lambda\frac{(n-1)!}{k!}{\displaystyle\sum\limits_{l=1}^{k}}\left(-1\right)^{l}l\binom{k}{l}\binom{\lambda
l-1}{n-1}.$ (3.3)

Now letting $t\rightarrow 0$, which is equivalent to $u\rightarrow 0$ on both
sides of (3.3) and taking into consideration the generating function for
degenerate Fubini-type numbers (for $x=0$ in equation (1.4)) complete the
proof of (3.1). From the relationship (1.5), the identity (3.2) follows
readily.

###### Theorem 3.2

The Fubini-type polynomials $a_{n}^{\left(\alpha\right)}\left(x\right)$
possess the explicit formula

$a_{n}^{\left(\alpha\right)}\left(x\right)=2^{\alpha}{\displaystyle\sum\limits_{k=0}^{n}}\binom{n}{k}{\displaystyle\sum\limits_{i=0}^{k}}\left\langle-2\alpha\right\rangle_{i}\left(-1\right)^{i}S(k,i)x^{n-k},$

where $S(n,k)$ is the Stirling numbers of the second kind. Also, the Fubini-
type numbers $a_{n}^{\left(\alpha\right)}$ can be written in the form

$a_{n}^{\left(\alpha\right)}=2^{\alpha}{\displaystyle\sum\limits_{i=0}^{n}}\left\langle-2\alpha\right\rangle_{i}\left(-1\right)^{i}S(n,i).$
(3.4)

Besides, the generalized Apostol-Euler numbers $E_{n}^{(\alpha)}(x;\gamma)$
can be expressed as

$E_{n}^{(2\alpha)}(x;-1/2)=2^{4\alpha}{\displaystyle\sum\limits_{k=0}^{n}}\binom{n}{k}{\displaystyle\sum\limits_{i=0}^{k}}\left\langle-2\alpha\right\rangle_{i}\left(-1\right)^{i}S(k,i)x^{n-k}.$
(3.5)

Proof. From (2.1), (2.2) and (2.3), we have

$\displaystyle\frac{d^{k}}{dt^{k}}\left(2-e^{t}\right)^{-2\alpha}$
$\displaystyle={\displaystyle\sum\limits_{i=0}^{k}}\left\langle-2\alpha\right\rangle_{i}\left(2-e^{t}\right)^{-2\alpha-i}B_{k,i}\left(-e^{t},-e^{t},...,-e^{t}\right)$
$\displaystyle={\displaystyle\sum\limits_{i=0}^{k}}\left\langle-2\alpha\right\rangle_{i}\left(2-e^{t}\right)^{-2\alpha-i}(-1)^{i}e^{ti}B_{k,i}\left(1,1,...,1\right)$
$\displaystyle\rightarrow{\displaystyle\sum\limits_{i=0}^{k}}\left\langle-2\alpha\right\rangle_{i}(-1)^{i}S\left(k,i\right),\text{
\ \ \ \ as }t\rightarrow 0.$ (3.6)

Also, it is obvious that $\left(e^{xt}\right)^{(k)}=x^{k}e^{xt}\rightarrow
x^{k},$ as $t\rightarrow 0.$ So, by aid of the Leibnitz’s formula for the
$n$th derivative of the product of two functions, we get

$\displaystyle\lim_{t\rightarrow
0}\frac{d^{n}}{dt^{n}}\left[\frac{2^{\alpha}}{\left(2-e^{t}\right)^{2\alpha}}e^{xt}\right]$
$\displaystyle=2^{\alpha}{\displaystyle\sum\limits_{k=0}^{n}}\binom{n}{k}{\displaystyle\sum\limits_{i=0}^{k}}\left\langle-2\alpha\right\rangle_{i}(-1)^{i}S\left(k,i\right)x^{n-k},$

namely, we have $a_{n}^{\left(\alpha\right)}\left(x\right)$ by (1.3). For
$x=0,$ we immediately arrive at the identity (3.4). The equation (3.5) can be
deduced from the relation between the Fubini-type polynomials
$a_{n}^{\left(\alpha\right)}\left(x\right)$ and generalized Apostol-Euler
polynomials $E_{n}^{(\alpha)}(x;\gamma),$ given by (1.5).

###### Theorem 3.3

The Fubini-type polynomials $a_{n}^{\left(\alpha\right)}\left(x\right)$
satisfy the recurrence relation

${\displaystyle\sum\limits_{k=0}^{n}}\binom{n}{k}{\displaystyle\sum\limits_{i=0}^{n-k}}\left\langle
2\alpha\right\rangle_{i}(-1)^{i}S\left(n-k,i\right)a_{k}^{\left(\alpha\right)}\left(x\right)=2^{\alpha}x^{n}$

In particular, the Fubini-type numbers $a_{n}^{\left(\alpha\right)}$ provide
that

${\displaystyle\sum\limits_{k=0}^{n}}\binom{n}{k}{\displaystyle\sum\limits_{i=0}^{n-k}}\left\langle
2\alpha\right\rangle_{i}(-1)^{i}S\left(n-k,i\right)a_{k}^{\left(\alpha\right)}=0.$
(3.7)

In analogy, the generalized Apostol-Euler polynomials
$E_{n}^{(\alpha)}(x;\gamma)$ possess the recurrence relation

${\displaystyle\sum\limits_{k=0}^{n}}\binom{n}{k}{\displaystyle\sum\limits_{i=0}^{n-k}}\left\langle
2\alpha\right\rangle_{i}(-1)^{i}S\left(n-k,i\right)E_{k}^{(2\alpha)}(x;-1/2)=2^{4\alpha}x^{n}.$
(3.8)

Proof. Since

$\left[\left(2-e^{t}\right)^{2\alpha}\right]\left[\frac{2^{\alpha}}{\left(2-e^{t}\right)^{2\alpha}}e^{xt}\right]=2^{\alpha}e^{xt},$

by keeping in mind the generating function of Fubini-type polynomials (1.3)
and by proceeding as in the proof of (3.6), differentiate $n$ times with
respect to $t$ on both sides to deduce that

$\displaystyle{\displaystyle\sum\limits_{k=0}^{n}}\binom{n}{k}\frac{\partial^{n-k}}{\partial
t^{n-k}}\left[\left(2-e^{t}\right)^{2\alpha}\right]\frac{\partial^{k}}{\partial
t^{k}}\left[\frac{2^{\alpha}}{\left(2-e^{t}\right)^{2\alpha}}e^{xt}\right]$
$\displaystyle={\displaystyle\sum\limits_{k=0}^{n}}\binom{n}{k}{\displaystyle\sum\limits_{i=0}^{n-k}}\left\langle
2\alpha\right\rangle_{i}\left(2-e^{t}\right)^{2\alpha-i}(-1)^{i}e^{ti}S\left(n-k,i\right)\frac{\partial^{k}}{\partial
u^{k}}\left[\frac{2^{\alpha}}{\left(2-e^{t}\right)^{2\alpha}}e^{xt}\right]$
$\displaystyle\rightarrow{\displaystyle\sum\limits_{k=0}^{n}}\binom{n}{k}{\displaystyle\sum\limits_{i=0}^{n-k}}\left\langle
2\alpha\right\rangle_{i}(-1)^{i}S\left(n-k,i\right)a_{k}^{\left(\alpha\right)}\left(x\right),\text{
\ \ as }t\rightarrow 0$ $\displaystyle=2^{\alpha}x^{n}.$

Setting $x=0$ yields the equation (3.7) immediately. Formula (3.8) can be
derived by the same motivation stated in the proofs of our previous theorems.

###### Theorem 3.4

The following relationship holds true:

$a_{n-2\alpha}^{\left(\alpha\right)}\left(x;\lambda\right)=\frac{B_{n}^{\left(2\alpha\right)}(x;\lambda;1/2)}{2^{\alpha}\left\langle
n\right\rangle_{2\alpha}},$ (3.9)

where $B_{n}^{\left(\alpha\right)}(x;\lambda;\gamma)$ is the degenerate
Apostol-Bernoulli polynomials of order $\alpha,$ defined by (1.1).

Proof. If we put $\gamma=1/2$ and replace $\alpha$ by $2\alpha$ in (1.1), we
have

$\displaystyle{\displaystyle\sum\limits_{n=0}^{\infty}}B_{n}^{\left(2\alpha\right)}(x;\lambda;1/2)\dfrac{t^{n}}{n!}$
$\displaystyle=\left(\frac{t}{\dfrac{1}{2}\left(1+\lambda
t\right)^{1/\lambda}-1}\right)^{2\alpha}\left(1+\lambda t\right)^{x/\lambda}$
$\displaystyle=2^{\alpha}t^{2\alpha}{\displaystyle\sum\limits_{n=0}^{\infty}}a_{n}^{\left(\alpha\right)}\left(x;\lambda\right)\frac{t^{n}}{n!}$
$\displaystyle=2^{\alpha}{\displaystyle\sum\limits_{n=2\alpha}^{\infty}}a_{n-2\alpha}^{\left(\alpha\right)}\left(x;\lambda\right)\frac{t^{n}}{(n-2\alpha)!}$
$\displaystyle=2^{\alpha}{\displaystyle\sum\limits_{n=2\alpha}^{\infty}}\left\langle
n\right\rangle_{2\alpha}a_{n-2\alpha}^{\left(\alpha\right)}\left(x;\lambda\right)\frac{t^{n}}{n!},$

which concludes the proof.

Let us continue to study degenerate Fubini-type polynomials
$a_{n}^{\left(\alpha\right)}\left(x;\lambda\right)$. Firstly, from the fact

$\displaystyle\left(1+\lambda t\right)^{x/\lambda}$
$\displaystyle={\displaystyle\sum\limits_{n=0}^{\infty}}\left\langle\frac{x}{\lambda}\right\rangle_{n}\lambda^{n}\frac{t^{n}}{n!}$
$\displaystyle={\displaystyle\sum\limits_{n=0}^{\infty}}\left(x\right)_{n,\lambda}\frac{t^{n}}{n!},$

where
$\left(x\right)_{n,\lambda}=x\left(x-\lambda\right)...\left(x-\left(n-1\right)\lambda\right)$
for $n>0$ with $\left(x\right)_{0,\lambda}=1,$ it is easily verify that

$\displaystyle{\displaystyle\sum\limits_{n=0}^{\infty}}a_{n}^{\left(\alpha\right)}\left(x;\lambda\right)\frac{t^{n}}{n!}$
$\displaystyle=\frac{2^{\alpha}}{\left(2-\left(1+\lambda
t\right)^{1/\lambda}\right)^{2\alpha}}\left(1+\lambda t\right)^{x/\lambda}$
$\displaystyle=\left({\displaystyle\sum\limits_{n=0}^{\infty}}a_{n}^{\left(\alpha\right)}\left(\lambda\right)\frac{t^{n}}{n!}\right)\left({\displaystyle\sum\limits_{n=0}^{\infty}}\left(x\right)_{n,\lambda}\frac{t^{n}}{n!}\right)$
$\displaystyle={\displaystyle\sum\limits_{n=0}^{\infty}}\left({\displaystyle\sum\limits_{k=0}^{n}}\binom{n}{k}a_{k}^{\left(\alpha\right)}\left(\lambda\right)\left(x\right)_{n-k,\lambda}\right)\frac{t^{n}}{n!}.$

Comparing the coefficients $\frac{t^{n}}{n!}$ gives the following theorem.

###### Theorem 3.5

For $n\geq 0,$ we have

$a_{n}^{\left(\alpha\right)}\left(x;\lambda\right)={\displaystyle\sum\limits_{k=0}^{n}}\binom{n}{k}a_{k}^{\left(\alpha\right)}\left(\lambda\right)\left(x\right)_{n-k,\lambda}.$

Now, we observe that

$\displaystyle{\displaystyle\sum\limits_{n=0}^{\infty}}\left(a_{n}^{\left(\alpha\right)}\left(x+1;\lambda\right)-a_{n}^{\left(\alpha\right)}\left(x;\lambda\right)\right)\frac{t^{n}}{n!}$
$\displaystyle=\frac{2^{\alpha}\left(1+\lambda
t\right)^{x/\lambda}}{\left(2-\left(1+\lambda
t\right)^{1/\lambda}\right)^{2\alpha}}\left(\left(1+\lambda
t\right)^{1/\lambda}-1\right)$ $\displaystyle=\frac{2^{\alpha}\left(1+\lambda
t\right)^{x/\lambda}}{\left(2-\left(1+\lambda
t\right)^{1/\lambda}\right)^{2\alpha-1}}\left(-1+\frac{1}{2-\left(1+\lambda
t\right)^{1/\lambda}}\right)$ $\displaystyle=\frac{2^{\alpha}\left(1+\lambda
t\right)^{x/\lambda}}{\left(2-\left(1+\lambda
t\right)^{1/\lambda}\right)^{2\alpha}}-\sqrt{2}\frac{2^{\alpha-1/2}\left(1+\lambda
t\right)^{x/\lambda}}{\left(2-\left(1+\lambda
t\right)^{1/\lambda}\right)^{2\left(\alpha-1/2\right)}}$
$\displaystyle={\displaystyle\sum\limits_{n=0}^{\infty}}a_{n}^{\left(\alpha\right)}\left(x;\lambda\right)\frac{t^{n}}{n!}-\sqrt{2}{\displaystyle\sum\limits_{n=0}^{\infty}}a_{n}^{\left(\alpha-1/2\right)}\left(x;\lambda\right)\frac{t^{n}}{n!}$
$\displaystyle={\displaystyle\sum\limits_{n=0}^{\infty}}\left(a_{n}^{\left(\alpha\right)}\left(x;\lambda\right)-\sqrt{2}a_{n}^{\left(\alpha-1/2\right)}\left(x;\lambda\right)\right)\frac{t^{n}}{n!}.$

Comparing the coefficients $\frac{t^{n}}{n!}$ yields the following theorem.

###### Theorem 3.6

For $n\geq 0,$ we have

$a_{n}^{\left(\alpha\right)}\left(x+1;\lambda\right)=2a_{n}^{\left(\alpha\right)}\left(x;\lambda\right)-\sqrt{2}a_{n}^{\left(\alpha-1/2\right)}\left(x;\lambda\right).$

Now, if we differentiate both sides of (1.4) with respect to $t,$ we write

$\displaystyle\frac{d}{dt}\left(\frac{2^{\alpha}\left(1+\lambda
t\right)^{x/\lambda}}{\left(2-\left(1+\lambda
t\right)^{1/\lambda}\right)^{2\alpha}}\right)$
$\displaystyle=2^{\alpha}\left(\frac{x\left(1+\lambda
t\right)^{(x-\lambda)/\lambda}}{\left(2-\left(1+\lambda
t\right)^{1/\lambda}\right)^{2\alpha}}+2\alpha\frac{\left(1+\lambda
t\right)^{(x-\lambda+1)/\lambda}}{\left(2-\left(1+\lambda
t\right)^{1/\lambda}\right)^{2\alpha+1}}\right).$ (3.10)

If we replace $x$ by $x_{1}+x_{2}+\lambda$ and evaluate the terms on both
sides of (3.10), separately, then, we have

$\frac{d}{dt}\left(\frac{2^{\alpha}\left(1+\lambda
t\right)^{\left(x_{1}+x_{2}+\lambda\right)/\lambda}}{\left(2-\left(1+\lambda
t\right)^{1/\lambda}\right)^{2\alpha}}\right)={\displaystyle\sum\limits_{n=0}^{\infty}}a_{n+1}^{\left(\alpha\right)}\left(x_{1}+x_{2}+\lambda;\lambda\right)\frac{t^{n}}{n!},$
(3.11) $\left(x_{1}+x_{2}+\lambda\right)\frac{2^{\alpha}\left(1+\lambda
t\right)^{(x_{1}+x_{2})/\lambda}}{\left(2-\left(1+\lambda
t\right)^{1/\lambda}\right)^{2\alpha}}=\left(x_{1}+x_{2}+\lambda\right){\displaystyle\sum\limits_{n=0}^{\infty}}a_{n}^{\left(\alpha\right)}\left(x_{1}+x_{2};\lambda\right)\frac{t^{n}}{n!}$
(3.12)

and

$\sqrt{2}\alpha\frac{2^{\alpha+1/2}\left(1+\lambda
t\right)^{(x_{1}+x_{2}+1)/\lambda}}{\left(2-\left(1+\lambda
t\right)^{1/\lambda}\right)^{2\left(\alpha+1/2\right)}}=\sqrt{2}\alpha{\displaystyle\sum\limits_{n=0}^{\infty}}a_{n}^{\left(\alpha+1/2\right)}\left(x_{1}+x_{2}+1;\lambda\right)\frac{t^{n}}{n!}.$
(3.13)

Substitute (3.11), (3.12) and (3.13) in (3.10) to reach the following theorem.

###### Theorem 3.7

For $n\geq 0,$ the degenerate Fubini-type polynomials
$a_{n}^{\left(\alpha\right)}\left(x;\lambda\right)$ satisfy the recurrence
relation

$\displaystyle
a_{n+1}^{\left(\alpha\right)}\left(x_{1}+x_{2}+\lambda;\lambda\right)$
$\displaystyle=\left(x_{1}+x_{2}+\lambda\right)a_{n}^{\left(\alpha\right)}\left(x_{1}+x_{2};\lambda\right)+\sqrt{2}\alpha
a_{n}^{\left(\alpha+1/2\right)}\left(x_{1}+x_{2}+1;\lambda\right).$ (3.14)

Letting $\lambda\rightarrow 0$ and taking $x_{1}+x_{2}=y$ in (3.14) allow us
to derive the following formula for Fubini-type polynomials

$a_{n+1}^{\left(\alpha\right)}\left(y\right)=ya_{n}^{\left(\alpha\right)}\left(y\right)+\sqrt{2}\alpha
a_{n}^{\left(\alpha+1/2\right)}\left(y+1\right).$

###### Remark 3.8

From the relationships (3.9) and (1.5), the counterpart identities in Theorems
3.6 and 3.7 can be presented for degenerate Apostol-Bernoulli polynomials and
degenerate Apostol-Euler polynomials of order $\alpha.$

##  4 Conclusion

In our recent study, we have introduced and dealt with degenerate version of
Fubini-type polynomials. Utilizing the Faà di Bruno formula and some
properties of Bell polynomials, and generating function methods, we have
derived some new explicit formulas, closed forms and recurrence relations for
degenerate Fubini-type polynomials and numbers, and Fubini-type polynomials
and numbers, defined by Kılar and Simsek [14]. Furthermore, by associating the
degenerate Fubini-type polynomials with degenerate Apostol-Bernoulli
polynomials and degenerate Apostol-Euler polynomials of order $\alpha,$ we
have presented analog identities for them. As a final note, a relation
involving degenerate Fubini-type polynomials and degenerate Apostol-Genocchi
polynomials of order $\alpha$, defined by [13, Eq. 2.6] can be given and
further relations can be obtained.

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