Integrated use of finite element analysis and gaussian process regression in the structural analysis of AISI 316 stainless steel chimney systems

In this study, the Gaussian Process Regression (GPR) method, a statistical-based machine learning approach, was employed to process the datasets obtained from the structural analysis results generated using the ANSYS academic version.

Parametric analysis

In the GPR method, 50 results were generated at randomly varying pressures and loads to serve as the dataset.

To obtain these results more efficiently, rather than individually, the parametric analysis feature of the ANSYS Academic Structural Analysis module was utilized.

The input variables of the dataset were defined as the applied pressure (P₁) and the Z-directional force (P₂). In contrast, the target variables were selected as the maximum total deformation (P₃) and the maximum equivalent (Von Mises) stress (P₄).

The program was then executed to generate a dataset of 50 samples. The parametric analysis dataset is summarized in Table 2.

Gaussian process regression (GPR)

In regression problems, the GPR method, which provides high accuracy, adopts a probabilistic approach to identify patterns within the dataset. This characteristic enables not only the prediction itself but also the assessment of the reliability of that prediction in engineering problems³⁴.

GPR is based on a Gaussian process, defined as an infinite-dimensional joint distribution of random variables. The fundamental assumption is that a normal distribution can represent the model output for any input point, and that all points jointly follow a Gaussian distribution³⁵.

A regression problem can be defined as follows³⁶.

Where:

\(\:\text{X}\in\:{\mathbb{R}}^{n\times\:d}\) : Input matrix (nnn samples, ddd features).

\(\:\text{y}\in\:{\mathbb{R}}^{n}\) : Observed output vector.

\(\:f(\cdot\:)\) : Function to be learned.

\(\:\epsilon\:\sim\:\mathcal{N}\left(0,{\sigma\:}_{n}^{2}\right)\) : Independently and identically distributed Gaussian noise.

GPR assumes that the function \(\:f(\cdot\:)\) It is modeled by a Gaussian process, which is expressed in the following Eq. 

Where;

\(\:m\left(\mathbf{x}\right)\) : The mean function is generally taken to be zero, \(\:m\left(\mathbf{x}\right)=0\))

\(\:k\left(\mathbf{x},{\mathbf{x}}^{{\prime\:}}\right)\) : The most used kernel function, the Squared Exponential (RBF).

The covariance (kernel) function is expressed as follows.

Where:

\(\mathcal{l}\) : Characteristic length scale.

\({\sigma\:}_{f}^{2}\) : Signal variance.

Let the \(\:\mathbf{y}\) be observed for the inputs \(\:\mathbf{X}\) In the dataset. For a new input point \(\:{\mathbf{x}}_{\text{z\:}}\)The output prediction is expressed as a normal distribution, as given in the following Eq. 

Where;

\(\:K\) : Kernel matrix of the training data, \(\:{K}_{ij}=k\left({\mathbf{x}}_{i},{\mathbf{x}}_{j}\right)\)

\(\:{\mathbf{k}}_{z}\) : Covariance vector between the new point and the training data

\(\:{\mu\:}_{\text{z}}\) : Predicted meaning (model output)

\(\:{\sigma\:}_{\text{*}}^{2}\) : Predicted variance (uncertainty)

Through this formulation, it is possible to compute not only the prediction but also the confidence intervals. This capability makes the GPR model highly valuable in engineering applications for decision support systems, reliability analyses, and uncertainty propagation.

In this study, the GPR model was developed in MATLAB using 50 parametric analysis results obtained from ANSYS. The applied pressure (P1) and Z-directional force (P2) were used as the input variables, while the total deformation (P3) and equivalent (Von Mises) stress (P4) were selected as the target variables. The model was trained in MATLAB using the fitrgp function, and various kernel functions were tested to determine the most suitable configuration. The GPR flowchart is presented in Fig. 8.

The GPR algorithm developed in MATLAB was executed with two additional external inputs that were not included in the initial dataset to predict the corresponding results. When the algorithm is run, it prompts the user to input new values. In this case, the algorithm was executed with input values of^3,5 (MPa) and [−500, −1300] (N), yielding the corresponding results. The MATLAB output screen of the GPR algorithm is shown in Fig. 9.

The GPR MATLAB Result Screen.

First, the algorithm was executed using an input of 3 MPa pressure and − 500 N load. According to the results, the predicted Von Mises stress was 72.07 MPa, and the total deformation was 0.27 mm. Second, the algorithm was run with an input of 5 MPa pressure and − 1300 N load, yielding a predicted Von Mises stress of 120.07 MPa and a total deformation of 0.46 mm.

Based on these results, the GPR model achieved exceptionally high accuracy in predicting both deformation and stress values, with an R² value greater than 0.999, consistent with the literature^37,38. This demonstrates that the model is highly suitable for producing rapid and reliable predictions, particularly when based on numerical results obtained from finite element analyses.

To compare the points predicted by the GPR algorithm with the structural analysis results from the ANSYS academic module, structural analyses were performed for the same input values. The structural analysis results for an input pressure of 3 MPa and a load of − 500 N are shown in Fig. 10.

Structural Analysis a) Von Mises b) Total Deformation.

In the structural analysis calculations for the condition of 3 MPa pressure and − 500 N load, the Von Mises stress was found to be 74.34 MPa, and the total deformation was 0.075 mm.

The structural analysis results for an input pressure of 5 MPa and a load of − 1300 N are presented in Fig. 11.

Structural Analysis (a) Von Mises (b) Total Deformation.

In the structural analysis calculations for the condition of 5 MPa pressure and − 1300 N load, the Von Mises stress was found to be 123.84 MPa, and the total deformation was 0.12 mm.

To better understand the relationship between the GPR algorithm model and the ANSYS structural analysis results, an error analysis was performed. The error rates can be determined using the equations given below.

Based on this equation, an error analysis was conducted to evaluate the uncertainties between the experimental and simulation results. According to the study, the comparison of the GPR results with the ANSYS simulation results, along with a summary of the total error rate, is presented in Table 3.

Accordingly, the prediction results obtained using the Gaussian Process Regression (GPR) method were compared with the ANSYS simulation outputs. For both analysis points, the error rate of the GPR model for Von Mises stress was below 3%, indicating a high level of predictive accuracy. However, the error rates for total deformation values were found to exceed 70%. This suggests that the GPR model is not sufficiently sensitive in predicting low-amplitude deformations. Therefore, it can be observed that, in chimney design, the GPR model produces reliable results for Von Mises stress, considered a more critical parameter than total deformation in structural analysis.

Due to the high error percentages observed in predicting the displacement target (P3) at the design points between the GPR model (ARD-SE, standardize = on) and the ANSYS structural analysis results, the applied methods were re-evaluated.

First, the total deformation output from ANSYS structural analysis, which was used for training the model and comparing the GPR results, was carefully reviewed. Since the boundary conditions remained unchanged, the primary factor influencing changes in total deformation was the mesh configuration. A convergence study was conducted until all results became independent of the mesh settings, thereby confirming the numerical reliability of the ANSYS results. Furthermore, as the total deformation value is not given at a single point but rather at any location along the entire body, its magnitude varies with different load cases and amplitudes. Consequently, predicting this value accurately is inherently challenging.

Secondly, the possibility of achieving lower error percentages in the GPR model through different settings was investigated. For P3, both z-score and log(1 + P3) transformations were tested, while the SE-ARD and Matérn-5/2 kernels were compared using Bayesian hyperparameter optimization combined with 5-fold cross-validation. Although these trials reduced the in-model absolute error to the micron level, the error rate at two independent validation points remained above 70%, similar to the initial case. The primary reasons are (i) the inflation of percentage error due to the denominator effect at very small displacement amplitudes, and (ii) a global scaling bias between the training labels and the ANSYS reference values.

In addition, for structural analyses performed using finite element software such as ANSYS, the standards specify that the maximum allowable total deformation for a 3 m-high steel chimney should be at least 3 mm (0.1% of the chimney height)³⁹. Because the deformation is under 3 mm and von Mises stress predictions are accurate, these error percentages are acceptable for this design.

Below are shown both the length scales in standardized space and their corresponding values ​​converted to raw physical units (transformation: \(\:{\mathcal{l}}_{\text{d}}^{\text{r}\text{a}\text{w}}={\mathcal{l}}_{\text{d}}^{\text{s}\text{t}\text{d}}\cdot\:\text{s}\text{t}\text{d}\left({\text{X}}_{\text{d}}\right)\)).

P3 – Total Deformation (mm).

• Kernel: Squared Exponential (ARD).

• Length scales (std): \(\:{\mathcal{l}}_{\text{P}1}=3.04268,{\mathcal{l}}_{\text{P}2}=3.02382\)

• Length scales (raw): \(\:{\mathcal{l}}_{\text{P}1}^{\text{r}\text{a}\text{w}}=4.04271\text{M}\text{P}\text{a},{\mathcal{l}}_{\text{P}2}^{\text{r}\text{a}\text{w}}=894.907\text{}\text{N}\)

• Signal variance: \(\:{{\upsigma\:}}_{\text{f}}^{2}=2.57537\times\:{10}^{-14}{\text{}\text{m}\text{m}}^{2}\)

• Noise variance (for information): \(\:{{\upsigma\:}}_{\text{n}}^{2}=1.51524\times\:{10}^{-6}{\text{}\text{m}\text{m}}^{2}\)

P4 – Equivalent (von Mises) Stress (MPa).

• Kernel: Squared Exponential (ARD).

• Length scales (std): \(\:{\mathcal{l}}_{\text{P}1}=0.112238,{\mathcal{l}}_{\text{P}2}=0.0919009\)

• Length scales (raw): \(\:{\mathcal{l}}_{\text{P}1}^{\text{r}\text{a}\text{w}}=0.149127\text{M}\text{P}\text{a},{\mathcal{l}}_{\text{P}2}^{\text{r}\text{a}\text{w}}=27.1983\text{}\text{N}\)

• Signal variance: \(\:{{\upsigma\:}}_{\text{f}}^{2}=4.17694\times\:{10}^{-9}{\text{M}\text{P}\text{a}}^{2}\)

• Noise variance (for information): \(\:{{\upsigma\:}}_{\text{n}}^{2}=0.107867{\text{M}\text{P}\text{a}}^{2}\)

The training protocol involved model selection through 5-fold cross-validation with a random seed of 42. Alternative kernels, including Matérn-5/2, as well as target scaling (z-score and log(1 + P3)), were tested. The final selection was made for SE-ARD using the hyperparameters specified above.