URBaM: A Novel Surrogate Modelling Method to Determine Design Scaling Rules for Product Families

Abstract

The use of regression-based surrogate models to determine design scaling rules for mechanical product families has proven to be a powerful approach for dimensioning complex geometries. However, there is a broad range of surrogate models in the literature, and each model can be configured in multiple ways. The optimal model election is highly conditioned by the case study nature (e.g., non-linearity level), and consequently, it is mandatory to evaluate different surrogate models. This process can be cumbersome and time consuming, but the election of an inaccurate model may lead to several design–analysis iterations that increase the product cost and development time. Therefore, in this paper, a novel surrogate modelling technique to determine representative design scaling rules for product families—named Univariate Regression-Based Multivariate (URBaM)—is presented. The proposed method seeks to minimize design–analysis iterations while suppressing the process involved in evaluating different surrogate models, independent of the non-linearity level nature of the design problem. The URBaM model is evaluated through six oil and gas valve family design case studies, addressing critical mechanical component design problems across different non-linearity levels (two low, two medium, and two high). The results obtained with the URBaM model in these six cases are compared against 14 configurations of eight widely used techniques in the literature. The obtained results demonstrate that the URBaM model is capable of accurately adapting to different non-linearity levels with a single configuration and presents high stability regarding MAPE, NRMSE, and RMAE metrics from case to case. Consequently, the potential of the URBaM surrogate modelling technique to assist the design process of scalable mechanical product families is proven.

1. Introduction

The use of product families is an extended solution to efficiently develop customized products [1,2]. Two main strategies are followed when adapting the product platform of a family in order to obtain a family member: module-based and scale-based strategies [3,4]. The former, adapts the product platform by combining different functional modules. The latter, adapts the platform by scaling it according to previously defined scaling rules [5]. In particular, scale-based strategies are widespread in the development of mechanical product families, since dimensions play an important role in the structural integrity and functionality of these kinds of products [3].

Nowadays, the evaluation of the functionality and the structural integrity of mechanical products is usually carried out by Finite Element Method (FEM)- or Finite Volume Method (FVM)-based Computer-Aided Engineering (CAE) software [6]. Such verification techniques may become computationally expensive when solving complex problems [7]. This fact becomes especially relevant when poorly defined scaling rules entail an iterative scaling design process where the model must be updated and run in each design iteration, extending the product development time. For this reason, it is mandatory to define representative product platform scaling rules, maintaining structural integrity and functionality conditions in each new scaled product [8].

Scaling rules of mechanical products have been traditionally defined using analytical formulations of mechanics of materials [9]. However, such formulations do not adequately represent the mechanical behavior of complex shape geometries. FEM-FVM CAE models afford representative solutions, but they are sometimes computationally expensive [10]. As an alternative, surrogate models have proven to be a powerful option for dimensioning complex shape geometries by replacing costly to evaluate problems with almost instantly to solve mathematical functions [11,12,13]. The complexity of the geometry can even go further in topological optimization problems, which are also used in the mechanical design field [14].

Surrogate models interrelate user defined input parameters (e.g., dimensions of a product) and an output parameter (e.g., field variable of interest). Thus, surrogate models are built based on phenomenological data of multiple cases within the range of interest called design points, which can be obtained either from an experimental or numerical source. In general, surrogate models for scaling rules are built based on deterministic numerical data [15]. For that purpose, a Design Of Experiment (DOE) to calculate the required field variable for each design point that covers the DOE domain has to be designed [16]. Once the surrogate model is built, the result of any input value configuration within the DOE range can be directly obtained. This allows for optimizing the design variable election process and obtaining reliable and safe product designs in a reduced number of design–analysis iterations.

Multiple surrogate model types exist in the literature [17]. Some of the best known model types are polynomial models [10,18,19]; Radial Basis Function (RBF) models [20,21,22]; Kriging models [23,24,25]; Artificial Neural Network (ANN) models [26,27]; Support Vector Machine Regression (SVMR) models [28,29,30]; Multivariate Adaptive Regression Splines (MARS) models [31,32]; and Random Forest (RF) models [33,34,35]. In addition, there is one further group of modelling, labelled in this paper as “Standalone Weighted Ensemble”, which refers to models that combine different types of independent models such as Kriging, Polynomial models, and SVMR [36,37,38,39,40,41,42]. Such a wide range of options makes it difficult for the user to make the optimum choice. In addition, each model can be configured in multiple ways. Depending on the configuration, the prediction accuracy can be different [24,43]. Thus, the definition of an accurate surrogate model is greatly hindered by the decisions taken at configuration phase. Moreover, several studies in the literature indicate that applying the same surrogate model configuration to problems with different characteristics does not necessarily yield accurate predictions (see

Table 1. Comparative studies of surrogate models reported in the literature.

Ref.	Compared Models	Key Findings
[43]	Ordinary Kriging, Universal Kriging, Linear/Quadratic trend-based Universal Kriging, and Blind Kriging.	Blind Kriging provided most accurate predictions.
[24]	Blind Kriging, Ordinary Kriging, Co-Kriging, and Universal Kriging with pseudo likelihood estimations.	Universal Kriging with maximum likelihood provided most accurate results.
[39]	Polynomial, Kriging, RBF, and Standalone Weighted Ensemble across 8 case studies.	The Standalone Weighted Ensemble model was most accurate in all eight cases. Kriging and RBF showed comparable accuracy in some cases.
[44]	Kriging, RBF, RF, ANN, MARS, and SVMR.	SVMR provided most accurate results.
[38]	Kriging (Matérn basis and linear trend), SVMR (Gaussian kernel), full second order Polynomial, Moving Least Square (MLS), and PPS-Optimal and PPS-GA Standalone Weighted Ensembles across 15 case studies.	PPS-Optimal and PPS-GA were most accurate in 9 out of 15 cases. Kriging ranked the same as PPS models in 3 cases and was the most accurate in 3 of the cases. Kriging outperformed SVMR in 13 of the cases.
[45]	PPS-GA-Standalone Weighted Ensemble and Kriging.	PPS-GA provided most accurate predictions.
[27]	RBF, RF, and ANN.	RBF and RF slightly outperformed ANN.

):

Such contradictory results indicate that the optimal selection of a surrogate model and its configuration is highly conditioned by the case study in which it is employed [36,37]. For this reason, the strategy of using the same type and configuration model for all case studies is inappropriate. In fact, no one specific model independent of the characteristics of a particular problem currently exists in the literature [38,40].

Some authors have defined general rules that associate specific types of surrogate models with the characteristics of particular case studies to facilitate the surrogate model selection [39,46].

Table 2. Guidelines from the literature for selecting surrogate models according to case study nature.

Ref.	Performed Analysis	Main Conclusions
[47]	Compared MARS, RBF, Kriging, and Polynomial models across 14 case studies, classified as linear, slightly non-linear, and highly non-linear cases.	RBF was the most accurate for highly non-linear cases, consistent with findings in [20,41].
		MARS and Kriging require a high number of design points to accurately represent highly non-linear problems, while [41] concludes Kriging is more suitable for slightly non-linear and large problems.
		Polynomial models are particularly suitable for linear or quasi-linear cases, also noted by [20,48].
[49]	Proposed the AutoSM model, which automatically selects the optimal model among Polynomial, Kriging, MARS, and RBF model configurations. It was tested on 10 benchmark functions and 5 test problems, considering problem non-linearity, scale, amount of data, and smoothness.	The work highlights the negative correlation between non-linearity and accuracy, with substantially higher errors in non-linear cases.
[50]	Studied the accuracy of MARS, SVMR, ANN, GP, ALAMO, and RF models, focused on the number of design points in optimization problems.	ANN requires a large number of samples for accurate approximation, and RF and SVMR were less accurate regardless of data amount. However, Ref. [51] recommends RF and SVMR over Kriging and ANN, for cases with a high number of design points.

summarizes general guidelines from various authors, with a focus on the relationship between case study non-linearity and prediction accuracy.

While these conclusions are helpful for selecting a surrogate model, the studies do not associate the proposed guidelines to a standardized metric to evaluate the level of non-linearity of a problem. R. Jin et al. [47] evaluated the level of non-linearity classifying the coefficient of determination R² value of a low order polynomial approach of the DOE. This non-linearity measurement procedure was also used in [49]. However, this metric was not generally adopted by other works, evaluating non-linearity level only qualitatively. Therefore, the guidelines proposed in the literature are sometimes difficult to apply in specific problems. In consequence, authors of papers [26,44] evaluate different types and configurations of surrogate models as a procedure to choose the most appropriate model for each case. Nevertheless, this process could become cumbersome and time-consuming depending on the case of analysis, specifically in an industrial environment.

As an alternative, in this paper a novel surrogate modelling technique capable of adapting to different non-linearity levels with a single configuration—named Univariate Regression-Based Multivariate (URBaM)—is presented. The proposed method was developed with two main objectives. Firstly, the proposed model aims to avoid the cumbersome and time-consuming evaluation process of different surrogate model types and configurations required nowadays to choose the most appropriate model for each case study. Secondly, the method seeks to reduce close to zero design–analysis iterations when scaling a new family member. The accuracy of the proposed technique is evaluated for six case studies of different non-linearity levels (two low, two medium, and two high) and compared against 14 configurations of eight most representative techniques in the literature.

The rest of this article is set out as follows: Section 2 defines the proposed URBaM modelling method. Section 3 details the procedure followed to evaluate the accuracy and the adaptability to different non-linearity level cases of the URBaM models. Here, the URBaM method is compared to the most employed surrogate models in the literature in six case studies classified according to non-linearity levels. Section 4 presents the obtained results with the discussion of this study. Section 5 provides the main conclusions of this study. The fundamentals of the analyzed surrogate models and their configurations are attached in Appendix A.

2. URBaM Modelling Method

The URBaM model is a multilayer surrogate model in which the number of layers corresponds to the amount of design variables p: x_i → L_i, where x_i refers to the design variables, L_i to the layers, and i ∈ {1…p}. In each layer, multiple nodes are defined, N_i,j,t: the number of nodes in a layer is represented by t, while j determines the position of the node in the layer, where j ∈ {1…k_t}. In each of these nodes, input data, x_i,j,t, and the associated layer design variable value, x_i, are transformed into node output data, y_i,j,t, by using selected functions, f_i,j,t. The determined node output is transferred to another node in the concatenated layer above. Therefore, the output of the node, y_i,j,t, becomes the input, x_i−1,j,t, of the node, N_i−1,j,t, in the layer above, where y_i,j,t = x_i−1,j,t. This concatenation of nodes can be understood as a master/slave dependency. A master node in the layer i is fed by the inputs determined by different slave nodes in the layer i + 1. Each slave node transfers only one input to the master node, and the number of slave nodes that each master node can have varies in the model. The number of the associated slave nodes is directly affected by the function f_i,j,t used in the master node. In particular, the function f_i,j,t is composed of k_t unknown parameters, and each master node must have the same number of associated slave nodes.

The master/slave concatenation is repeated throughout all the layers of the model, but the type of function, f_i,j,t, used can be different in each case. The concatenation between nodes can be graphically interpreted by branches, while the global structure takes a tree shape with an initial root node that determines the prediction of y, which is ŷ. Figure 1 shows the global structure of a p design variable URBaM model.

The defined N_i,j,t node structure depends on the used f_i,j,t(x_i) function in each master node. The procedure was followed by the URBaM method for defining f_i,j,t(x_i) functions, thus establishing the global nodal structure, which is defined hereafter; Figure 2 summarizes all necessary steps:

• First, the f_1,1,1(x₁) function is defined in L₁. Here, DOE data is arranged in different data groups. The design points in each of these groups must have different x₁ values, and the rest of design variable values must remain constant (x_z = const., where z ∈ {i + 1…p}) (Step 1.1). In order to create data groups of these characteristics, a full factorial DOE must be defined; not all DOE types in the literature can be used for creating URBaM models.
• After defining data groups, a univariate regression function is selected (Step 1.2) and each of the data groups defined is modelled using it. The applied regressions relate the x₁ design variable by means of simple univariate (2D) regression functions to the output variable y and the unknown coefficients are determined by the least square method (Step 1.3).
• In order to evaluate in future steps how the applied regression fits to the data groups, the coefficient of determination, R², is measured together with the number of unknown coefficients that the applied regression function requires being calculate by the least squares method (Step 1.4).The regressions are applied to each data group (Loop 1.1).
• Multiple simple univariate regression types are available in the literature, and the URBaM model must select one of them prior to defining f_1,1,1(x₁). Table 3 sets out the type of regression functions considered by the URBaM surrogate modelling technique. In order to select the most appropriate function, the model studies all the regression functions in Table 3. To this end, the URBaM model repeats the following procedure: (i) takes one of these functions and applies it to each of the design point groups created in the node, (ii) calculates the R² of each data group and determines the mean value, and (iii) identifies the number of unknown coefficients of the regression function. The same process is repeated with all functions in Table 3 (Loop 1.2). The best function option will be that with a high correlation between regression parameters (and thus, a high R²), while at the same time being a function type without a high number of unknown parameters to avoid overfitting. In other words, the URBaM method selects the function with the fewest number of unknown coefficients, but with the highest R² value, considering 0.95 the minimum R² value (Step 1.5).

  Figure 2. The steps followed by the URBaM modelling technique.

  Figure 2. The steps followed by the URBaM modelling technique.

  Table 3. Type of regression functions used by the URBaM model.

  Model	Regression Function
  Lineal regression	$f\left(x_{\mathrm{i}}\right)=A{x}_{\mathrm{i}}+B$
  Second-order polynomial regression	$f\left(x_{\mathrm{i}}\right)=A{x}_{\mathrm{i}}^2+B{x}_{\mathrm{i}}+C$
  Third order polynomial regression	$f\left(x_{\mathrm{i}}\right)=A{x}_{\mathrm{i}}^3+B{x}_{\mathrm{i}}^2+C{x}_{\mathrm{i}}+D$
  Logarithmic regression	$f\left(x_{\mathrm{i}}\right)=A ln\left(x_{\mathrm{i}}\right)+B$
  Exponential regression	$f\left(x_{\mathrm{i}}\right)=A e^{B{x}_{\mathrm{i}}}$
  Potential regression	$f\left(x_{\mathrm{i}}\right)=A{x}_{\mathrm{i}}^B$

  Table 3. Type of regression functions used by the URBaM model.

  Model	Regression Function
  Lineal regression	$f\left(x_{\mathrm{i}}\right)=A{x}_{\mathrm{i}}+B$
  Second-order polynomial regression	$f\left(x_{\mathrm{i}}\right)=A{x}_{\mathrm{i}}^2+B{x}_{\mathrm{i}}+C$
  Third order polynomial regression	$f\left(x_{\mathrm{i}}\right)=A{x}_{\mathrm{i}}^3+B{x}_{\mathrm{i}}^2+C{x}_{\mathrm{i}}+D$
  Logarithmic regression	$f\left(x_{\mathrm{i}}\right)=A ln\left(x_{\mathrm{i}}\right)+B$
  Exponential regression	$f\left(x_{\mathrm{i}}\right)=A e^{B{x}_{\mathrm{i}}}$
  Potential regression	$f\left(x_{\mathrm{i}}\right)=A{x}_{\mathrm{i}}^B$

• The regression function selected by the URBaM method becomes the f_1,1,1(x₁) function in the root node (N_1,1,1). Figure 3a shows an example of a second-order polynomial regression function selection case of one of the defined design point data groups. Following the same example, Figure 3b shows how the number of slave nodes created from the master node N_1,1,1 is the same as the number of unknown coefficients determined by least square method in f_1,1,1(x₁). In this case, the function has three unknown coefficients, so three slave nodes are created from the master node (Step 1.6).

  Figure 3. (a) Example of a second order polynomial regression fitting in N_1,1,1 and (b) its master/slave node configuration.

  Figure 3. (a) Example of a second order polynomial regression fitting in N_1,1,1 and (b) its master/slave node configuration.
• In each node multiple design point data groups are modelled individually; thus, multiple y_i+1,j,t coefficients will be determined. All of them will comprise the y_i+1,j,t output variable. Following the example in Figure 3a, the A coefficients obtained in the applied regression of each data group will comprise the y_2,1,1 vector, and the B and C coefficients will comprise the y_2,2,1 and y_2,3,1 vectors, respectively.
• The nodal structure in the layer L₂ is defined as a consequence of the selection of f_1,1,1(x₁) function and its unknown coefficients. Thus, the next step consists of defining all the f_2,j,t(x₂) functions of the created nodes in the layer L₂ (Step 2.1). To this end, the obtained y_2,j,t values in L₁ are related to each of the corresponding nodes in L₂ (Figure 3b). In each node, the y_2,j,t values are related to the x₂ design variable by simple univariate regression functions, f_2,j,t(x₂). The procedure that must be followed for defining f_2,j,t(x₂) is the same used in the definition of the f_1,1,1(x₁) function, but defining the new regressions using x₂ and y_2,j,t values (Step 2.2). This process is repeated in all nodes of L₂. As a result, all f_2,j,1(x₂) functions, all the y_i+1,j,t vector values, and the L₃ nodal structure N_3,j,t are determined (Loop 2.1). This process is repeated throughout all the layers, until the last layer L_i is reached, where i = p (Loop 2.2).
• In the last layer L_p, the x_p design variable is related to y_p,j,t by f_p,j,t(x_p) regression function in each N_p,j,t node of the layer, repeating the same procedure. However, in this last layer, the unknown coefficients determined by the least square method in the f_p,j,t(x_p) functions are not used for building new nodes. These last coefficients, termed C_j,t, are constant terms that are used by the URBaM model as inputs for predicting new x test points.

Once all the f_i,j,t(x_i) functions are defined, the C_j,t constant terms and the global structure of the URBaM model shown in Figure 1 are obtained. As a result, the URBaM model is completely defined. In order to predict y values of new x test points, ŷ(x), the x₁, x₂, …, x_p design variable values are introduced progressively in the model from the last layer to the initial one. Thus, the x_p design variable values of the test points and the determined C_j,t coefficient values are entered into the determined f_p,j,t(x_p) functions, obtaining the y_i,j,t values of the nodes in the layer. Following the master/slave concatenation, each slave node will pass the determined y_i,j,t values to its master node, and will be used in f_p−1,j,t(x_p−1) functions. This concatenation is repeated until the root node in the first layer is reached, where the output will be ŷ(x). Figure 4 shows an overview of the concatenation process followed by URBaM for determining ŷ(x).

The URBaM model concatenates multiple 2D univariate regressions, and as a consequence, a global multivariate regression model is obtained. The univariate regressions are implemented progressively for each design variable layer to layer, starting from the last layer and finishing in the root node layer. Thus, in each layer the dimensionality of the nodes changes depending on the layer position. In the last layer nodes, when the x_p design variable value is implemented in the regression functions f_p,j,t(x_p), the model acquires a 2D predictive model form (Figure 5a). When a new design variable, x_p−1, is implemented in the univariate regression functions, f_p−1,j,t(x_p−1), of the L_p−1 layers, the model takes a 3D model shape (Figure 5b). Repeating the same process in the next layer, a 4D regression domain is obtained (Figure 5c). Further repetition results in a 5D regression domain (Figure 5d). Finally, the prediction of the test point, ŷ(x), is obtained in a (p + 1) dimension domain. The dimensionality in each layer can be determined by the following expression: p − (i − 2).

The URBaM method ensembles simple regression functions, similar to ensemble approaches discussed in the literature. However, unlike standard ensemble models, URBaM is specifically designed to adapt to varying degrees of non-linearity. For this purpose, the method progressively applies simple regression functions to each design variable, accounting for how each variable influences the non-linear behavior of the output. Moreover, URBaM accounts for variations within the DOE range by applying different regressions to the same variable in different regions. Since a variable may exhibit near-linear behavior in some parts of the DOE range and highly non-linear behavior in others, this localized treatment allows for URBaM to capture non-linearity changes more effectively. Thus, while URBaM can be classified as an ensemble model, its distinctive strength lies in its ability to adapt to non-linearities.

3. Validation of the URBaM Modelling Method

First, the accuracy and adaptability of the presented URBaM modelling method was evaluated for different non-linearity-level cases. Then, it was compared in each case study against 14 configurations of eight widely used surrogate modelling methods in the literature. For that purpose, six case studies of different non-linearity levels were selected: two high non-linearity-, two medium non-linearity-, and two low non-linearity-level cases. All of them are real engineering problems, focused on the design of valve families in the oil and gas sector.

In all case studies, the design variables that most affect the mechanical performance were selected through a numerical sensitivity study performed in ANSYS^®. The remaining design parameters, which are necessary to geometrically define the studied valve components, were defined by previously set dimensioning rules. These dimensioning rules vary according to the two main parameters that are generally used for defining valves in a family: the valve size (bore diameter Db) and the working pressure (P). Next, a full factorial DOE was assembled considering the identified main design variables together with the valve size and rating within the specified valve family range. The design points of the full factorial DOE were run in ANSYS^®. The surrogate models were defined following the URBaM modelling technique as well as the eight modelling techniques selected for purposes of comparison, in a total of 14 model configurations.

Finally, the accuracy of the surrogate models were evaluated using the split samples strategy, where the test points were randomly defined within the domain of interest. In order to obtain a representative validation sample, the number of test points for each case study was set to approximately 25% of the size of the DOE used to build the models. The validation points were also executed using ANSYS^®.

In this section, first, the metrics used to evaluate the non-linearity level of the selected case studies, as well as the accuracy metrics of the surrogate models are specified. Then, the six case studies are exposed, selected according to problem non-linearity-level criteria. Finally, the configuration of the URBaM model and the eight surrogate model types (their 14 configurations included) selected from the literature for comparison purposes of these six case studies is detailed.

3.1. Problem Classification and Surrogate Model Evaluation Metrics

3.1.1. Problem Non-Linearity Definition Metric

As explained in the introduction, very few works in the literature propose non-linearity evaluation methods, and there is not a standardized and widely accepted criteria for this purpose. In the present work, a non-linearity evaluation criteria that takes as reference the procedure proposed by R. Jin et al. [47] was established. The procedure consists of the following steps: (i) the case study was fitted with a first order polynomial model, (ii) the coefficient of determination was obtained (Equation (1)), and (iii) four non-linearity classification levels were set according to the coefficient of determination, R², value. This classification includes high non-linearity (HNL), medium non-linearity (MNL), low non-linearity (LNL), and quasi-linear (QL).

Table 4. Case study non-linearity level classification.

Non-Linearity Level	R2 Range
High non-linearity (HNL)	0 ≤ R2 ≤ 0.5
Medium non-linearity (MNL)	0.5 < R2 ≤ 0.8
Low non-linearity (LNL)	0.8 < R2 ≤ 0.98
Quasi-linear (QL)	0.98 < R2 ≤ 1

details the R² coefficient ranges considered for each non-linearity level.

$$R^2={\displaystyle \frac{\sum _{i=1}^N{\left(\widehat{y_i}-\overline{y}\right)}^2}{\sum _{i=1}^N{\left(y_i-\overline{y}\right)}^2}}$$

(1)

3.1.2. Surrogate Model Accuracy Metrics

The accuracy of the surrogate model was evaluated by analyzing the Mean Absolute Percentage Error (MAPE), the Normalized Root Mean Square Error (NRMSE), and the Relative Maximum Absolute Error (RMAE). All of them are widely used metrics in the literature for comparing surrogate models [37,45,52].

MAPE (Equation (2)) provides insight into the general error of the model in percentages. However, the error percentage value is highly sensitive for numerical values close to zero, where even cases with small difference between predicted and measured results can lead to high error percentage values. In order to identify when this happens, it is also important to measure the residual difference between the predicted and measured values. The NRMSE and RMAE metrics are used for this. The NRMSE provides insight into the general error of the model, and the RMAE gives information about the maximum error of the model. Although parameters can be normalized in multiple ways according to the literature, H. Chen et al. [52] stated that the method used for normalization does not substantially affect to the accuracy evaluation process. In the present work, NRMSE was normalized with respect to the standard deviation, σ, according to Equation (3). The RMAE metric is shown in Equation (4):

$$MAPE={\displaystyle \frac{100}{N}}\sum _{i=1}^N\left|{\displaystyle \frac{y_{\mathrm{i}}-{\widehat{y}}_{\mathrm{i}}}{y_{\mathrm{i}}}}\right|$$

(2)

$$NRMSE={\displaystyle \frac{\sqrt{\frac{1}{N}\sum _{i=1}^N{\left(y_{\mathrm{i}}-{\widehat{y}}_{\mathrm{i}}\right)}^2}}{\sigma }}$$

(3)

$$\mathrm{R}\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{\mathrm{m}\mathrm{a}\mathrm{x}\left(\left|y_{\mathrm{i}}-{\widehat{y}}_{\mathrm{i}}\right|\right)}{\sigma }}$$

(4)

In order to evaluate the representativeness that each surrogate model configuration analyzed has in the presented engineering design problems, the following limits are considered in the present study: an MAPE limit of 10% is considered as desirable, while an MAPE value of 20% is considered as acceptable (which can be translated into a security factor of 1.1 and 1.2, respectively). In addition, an NRMSE limit of 0.5 and an RMAE limit of 1 were also defined according to user experience. Therefore, the surrogate models that fulfil these three conditions will be considered representative for the analyzed case studies.

3.2. Case Studies

The proposed surrogate modelling method was developed in the context of the oil and gas valve industry to assist product family development. The selected case studies are focused on the dimensioning of critical valve components of different valve families based on FEM calculations. Hereafter, the different case studies are detailed, summarized in Table 5:

• Case study 1: Stem dovetail dimensioning in a downstream slab valve family. Slab valves in downstream applications are used in the transportation of high volumes of fluid. The valve size ranges from 1” to 46” Db, and they may work from 300 lbs up to 2500 lbs pressure ratings. When the valve is in a closed position, the upstream chamber of the valve is pressurized, but not the downstream chamber. The pressure difference generates a normal force between the seat and the gate that must be borne by the valve opening mechanism in a maneuvering. One of the critical areas in the maneuvering mechanism is the stem dovetail, the failure of which would lead to the inoperability of the valve. Therefore, the aim of this case study is to generate a surrogate model to correctly size the stem dovetail. For this purpose the maximum equivalent Von Mises stress in the dovetail was set as the output variable of the analysis, while the dovetail arm thickness (h₁), the dovetail arm cantilever distance (b₁), and the thickness of the dovetail core (t₁) were considered the design variables (see Case 1 in Table 5). Thus, a four-design-variable (h₁, b₁, t₁, and Db) full factorial DOE with 832 design points was set for a single pressure rating, P, of 900 lbs. The non-linearity analysis resulted in R² = 0.33, a high non-linearity (HNL)-level problem.
• Case study 2: Structural bolt dimensioning in a flanged joint of a high-pressure split body ball valve family. The body of this type of valves is divided into two parts that are assembled by means of a flanged joint. The analytical equations correctly size the bolting so as to bear the tensile loads resulting from preloading and the internal high pressure during loading stage. This pressure can range from 5000 psi to 15,000 psi, in valve sizes that range from 1 13/16″ to 16 3/4″. However, an inadequate flange dimensioning may cause it to deflect and consequently couple the tensile loads in the bolting with a bending effect that may compromise the integrity of the joint. Therefore, the aim of this case study is to generate a surrogate model to correctly dimension the flange and the bolting preventing excessive flange deflections. For this purpose the maximum deflection angle in the flange was set as the output variable of the analysis, while the bolt metric (Dm), flange thickness (h₂), and cantilever distance of the flange (b₂) were considered the design variables (see Case 2 in Table 5). Thus, a four-design-variable (Dm, h₂, b₂, and Db) full factorial DOE with 320 design points was set for a single pressure rating, P, of 5000 psi. The non-linearity analysis classified the present problem as an HNL-level case, due to the value of the coefficient of determination R² = 0.43.
• Case study 3: Slab dovetail dimensioning in a downstream slab valve family. This case study corresponds to the same valve family as case study 1. However, in this case, the female dovetail of the slab is dimensioned instead of the stem dovetail. Again, the maximum equivalent Von Mises stress in the dovetail was set as the output variable of the analysis, while the dovetail arm thickness (h₃), the dovetail arm cantilever distance (b₃), and the lateral thickness of the dovetail (t₃) were considered the design variables (see Case 3 in Table 5). Thus, a four-design-variable (h₃, b₃, t₃, and Db) full factorial DOE with 832 design points was set for a single pressure rating P of 300 lbs. The non-linearity analysis classified the present problem as a medium non-linearity (MNL)-level case, due to the value of the coefficient of determination R² = 0.62.
• Case study 4: Stem–ball joint dimensioning for a high-pressure ball valve family. As in case study 2, due to pressure difference between the upstream chamber and downstream chamber, high friction forces appear in this case between the ball and the sealing seats. In addition, the resultant forces may be extremely high as internal pressure may rise up to 5000–15,000 psi. The friction force combined with the ball diameter generates a resistance torque that has to be borne by the maneuvering mechanism. In particular, one of the critical areas in this mechanism is the trunnion stem–ball joint. Therefore, the aim of the present case study is to generate a surrogate model to correctly dimension the stem–ball joint. For this purpose, the maximum equivalent Von Misses stress was set as the output variable of the analysis, while the trunnion diameter (Dt), the groove height (h₄), and groove width (t₄) were considered the design variables (see Case 4 in Table 5). Thus, a three-design-variable (Dt, h₄, and t₄) full factorial DOE with 64 design points was set for a 5000 psi P pressure rating and 2 9/16″ Db valve sizes. The non-linearity analysis classified the present problem as a MNL level case, due to the value of the coefficient of determination, R² = 0.76.
• Case study 5: Flange dimensioning in a bonnet to body joint for a high-pressure ball valve family. This case study is similar to case study 2. However, as commercial valve actuators for valve maneuvering are assembled in the bonnet, bolting diameter and positioning cannot vary in the present case. Therefore, the aim of this case study is to generate a surrogate model to correctly dimension the flange thickness to prevent undesired bending deflections. For this purpose, the deflection angle was set as the output variable of the analysis, while the flange thickness (h₅) and the cantilever distance of the bonnet (b₅), together with the size of the valve Db were considered the design variables (see Case 5 in Table 5). Thus, a three-design-variable (h₅, b₅, and Db) full factorial DOE with 112 design points was set for a single pressure rating P of 5000 psi. The non-linearity analysis classified the present problem as a low non-linearity (LNL)-level case, due to the value of the coefficient of determination R² = 0.85.
• Case study 6: Gate–Seat dimensioning for a high-pressure slab valve family. This case study corresponds to the same valve type defined in Case 1, but for highly pressurized and subsea slab valve families. The present case study analyzes the closure mechanism between the slab and the seat of a valve, where internal sealing must be ensured when the valve is pressurized (up to 5000–15,000 psi) in a closed position for valve sizes that range from 1 13/16″ to 7 1/16″. For this purpose, the contact pressure between the gate and the slab was set as the output variable of the analysis, while the contact thickness of the seat (t₆), together with the size of the valve Db were considered the design variables (see Case 6 in Table 5). Thus, a two-design-variable full factorial DOE with 16 design points was set for a single pressure rating P of 15,000 psi. The non-linearity analysis classified the present problem as an LNL-level case, due to the value of the coefficient of determination R² = 0.96.

  Table 5. Selected case studies: studied area, output variable of interest, design variables, and non-linearity level are detailed.

  Case	Valve Type	Studied Area	Output Variable	R2	Non-Linearity Level
  1	Downstream Slab Valve	Stem dovetail	Dovetail equivalent Von Mises stress	0.33	High non-linearity (HNL)
  2	High-Pressure Ball Valve	Split body flanged joint	Split Body flange bending angle	0.43	HNL
  3	Downstream Slab Valve	Slab dovetail	Dovetail equivalent Von Mises stress	0.62	Medium non-linearity (MNL)
  4	High-Pressure Ball Valve	Stem–ball joint	Von Mises equivalent stress	0.76	MNL
  5	High-Pressure Subsea Ball Valve	Bonnet flanged joint	Bonnet flange bending angle	0.85	Low non-linearity (LNL)
  6	High-Pressure Slab Valve	Gate–Seat closure mechanism	Gate–Seat contact pressure	0.96	LNL

  Table 5. Selected case studies: studied area, output variable of interest, design variables, and non-linearity level are detailed.

  Case	Valve Type	Studied Area	Output Variable	R2	Non-Linearity Level
  1	Downstream Slab Valve	Stem dovetail	Dovetail equivalent Von Mises stress	0.33	High non-linearity (HNL)
  2	High-Pressure Ball Valve	Split body flanged joint	Split Body flange bending angle	0.43	HNL
  3	Downstream Slab Valve	Slab dovetail	Dovetail equivalent Von Mises stress	0.62	Medium non-linearity (MNL)
  4	High-Pressure Ball Valve	Stem–ball joint	Von Mises equivalent stress	0.76	MNL
  5	High-Pressure Subsea Ball Valve	Bonnet flanged joint	Bonnet flange bending angle	0.85	Low non-linearity (LNL)
  6	High-Pressure Slab Valve	Gate–Seat closure mechanism	Gate–Seat contact pressure	0.96	LNL

3.3. Configuration of Surrogate Models

The proposed URBaM modelling method was evaluated and compared with eight of the most studied surrogate modelling techniques in the literature: (i) second-order polynomial, (ii) Artificial Neural Network (ANN), (iii) Radial Basis Function (RBF), (iv) Kriging, (v) Multivariate Adaptive Regression Splines (MARS), (vi) Random Forest (RF), (vii) Support Vector Machine Regression (SVMR), and (viii) Standalone Ensemble model based on the Penalized Predictive Score weighting Genetic Algorithm (PPS-GA). The fundamentals of the selected surrogate modelling techniques for validation purposes are summarized in Appendix A.

Each type of surrogate modelling method presents a wide range of configuration options. Therefore, the URBaM model was compared with a total of 14 recommended model configurations, which are detailed hereafter:

• Second-Order Polynomial: From all the possible polynomial model configurations, the full second-order polynomial model, which considers all the possible interactions between design variables, is the most widely used polynomial model configuration [53]. Therefore, this configuration was selected.
• Artificial Neural Networks (ANNs): One-hidden-layer-based ANN models are the most widely used [48]. However, there is not a specified rule for defining the optimum number of nodes of the hidden layer. Thus, three arbitrary ANN configurations were selected: (i) 3 nodes, (ii) 5 nodes, and (iii) 10 nodes.
• Radial Basis Function (RBF): The RBF model can be configured according to several types of base functions: linear, cubic, multiquadratic, inverse multiquadratic, thin plate spline, and Gaussian. However, several authors highlight the potential of multiquadratic and inverse multiquadratic functions for building accurate RBF models [54,55,56]. Moreover, the effect of the shape parameter c, which governs the RBF functions, was studied by different authors. Thus, E. Acar and M. Rais-Rohani [41] proposed a c = 1 configuration, while M. J. Colaço et al. [57] recommended a c = 1/N configuration, where N corresponds to the number of design points. Therefore, four configurations of the RBF modelling technique were selected: (i) multiquadratic with c = 1, (ii) multiquadratic with c = 1/N, (iii) inverse multiquadratic with c = 1, and (iv) inverse multiquadratic with c = 1/N.
• Kriging: From all possible Kriging model configurations, Universal Kriging is one of the most widely used due to its high potential [24]. In particular, Kriging models are commonly configured using a Gaussian correlation function with a p_u exponent coefficient of 2 [22,48,58]. In addition, the correlation parameter θ_u can be defined as a constant or variable unknown term for each design variable. Simpson et al. [18] reported that fixing constant the correlation parameter sufficiently good results can be obtained. However, defining a variable correlation parameter is more common than fixing it as a constant. Therefore, in the present work, two Universal Kriging configurations were selected: (i) constant correlation parameter and (ii) variable correlation parameter.
• Multivariate Adaptive Regression Splines (MARS): The main characteristic of the MARS modelling method developed by J. H. Friedman [31] is that the modelled domain is divided into different subdomains by means of knots. Each subdomain can be modelled by linear or cubic piecewise functions. According to [59], the use of a cubic function configuration provides more accurate results for non-noisy data. In the current work, the maximum number of subdomains, Max_sub, was limited according to [60]: the minimum value between 200 and max(20;2k), where k is the number of design variables. In addition, as FEM results are considered as non-noisy data, the MARS model was configured based on cubic piecewise functions.
• Random Forest (RF): In the current work, each tree is built with 1/3 of the total design variables (N_DV) selected randomly according to [61], the number of leaves of each tree N_L ≥ 1 [34], the minimum number of design points to split each node in the tree N_S ≥ 5 [61], and a standard deviation threshold of 5% in the model response to split a node [62]. Different numbers of trees were evaluated to ensure that the predictions of the RF models converged to a stable value. Specifically, each case study was checked for 100, 200, and 500 trees. It was found that the results converged over 200 trees in all the studied cases. Therefore, the results for 500 trees were used as reference for validation purposes.
• Support Vector Machine Regression (SVMR): From all the possible SVMR configurations, the ε-insensitive model, derived from the machine learning field, is typically used in numerical regression problems [28,29]. In particular, SVMR models that use linear loss functions and Gaussian kernels for non-linear transformations are among the most widely employed in the literature. Therefore, this configuration was selected.
• Standalone Ensemble model based on Penalized Predictive Score weighting Genetic Algorithm (PPS-GA): According to M. B. Salem and L. Tomaso [38], the ensemble models based on Genetic Algorithm (GA) techniques together with a Penalized Predictive Score (PPS) weighting criterion are among the models that present the highest accuracy. In particular, the weighting parameter values determined by the PPS method are obtained minimizing (i) the RMSE of the design points, (ii) the k-fold cross-validation PRESS value, and (iii) the overfitting penalization. The ensemble models are typically built based on second-order polynomial, Kriging, RBF, Gaussian Process, SVMR, or Moving Least Square [22] standalone models [37,38,41,42]. Therefore, in the present work, different configurations of second-order polynomial, Kriging, SVMR, and Moving Least Square standalone models, detailed in

  Table 6. Summary of the characteristics of the surrogate modelling technique configurations.

  Type	Configuration			Abbreviation	Code Source
  URBaM	Regression function election criteria: R2 ≥ 0.95 and fewest number of unknown coefficients			URBaM	Excel© 365 VBA code
  Second-Order Polynomial	Regression type: Full interaction and quadratic			POL2	ANSYS® 19.2 DesignXplorer™
  ANN	Layer structure: 1 hidden layer		3 nodes	ANN_3	ANSYS® 19.2 DesignXplorer™
  			5 nodes	ANN_5
  			10 nodes	ANN_10
  RBF	Base function type: Multiquadratic		c = 1	RBF_MQ_1	MATLAB® 2023 [63]
  			c = 1/N	RBF_MQ_1/N
  	Base function type: Inverse multiquadratic		c = 1	RBF_IMQ_1
  			c = 1/N	RBF_IMQ_1/N
  Kriging	Trend function: Universal (polynomial)		Constant θu	UKRI_Const	ANSYS® 19.2 DesignXplorer™
  	Kernel type: Gaussian pu = 2		Variable θu	UKRI_Var
  MARS	Domain partition criteria:Maxsub = min (200, max (20, 2k)) Type of function: Cubic spline approximation			MARS	MATLAB® 2023 [59]
  Random Forest	Tree Characteristics:NL ≥ 1; NS ≥ 5; STDerror ≤ 5%; NDV: k/3 Characteristics of the assembly: Nº of trees: 500			RF	MATLAB® 2023 [61]
  SVMR	Loss function type: ε-insensitive with linear loss function			SVMR	ANSYS® 19.2 DesignXplorer™
  	Non-linear transformation kernel: Gaussian
  PPS-GA	Kriging	Trend function: Constant/Polynomial Kernel type: Gaussian/Cubic/Thin Plate Spline Kernel variation θu: Constant/variable		PPS-GA	ANSYS® 19.2 DesignXplorer™
  	Polynomial	Regression types: Linear/quadratic/cross-quadratic
  	SVMR	Loss function type: Laplacian/ε-insensitve Kernel type: Linear/Gaussian/Sigmoidal
  	Moving Least Square (MLS)	Polynomial functions: Constant/linear/quadratic
  		Weight functions: Linear/Gaussian/Wendland

  , are considered to build ensemble models.

The manual modelling of these methods can be cumbersome and error-prone, especially in high-dimensional problems. However, the codes of the most utilized surrogate modelling techniques are available in the literature. In fact, the second-order polynomial, the Kriging, the ANN, the SVMR, and the PPS-GA models are already available in ANSYS^® 19.2 DesignXplorer^™, which was the software package used for building these models. The codes to build and test RBF [63], MARS [59], and RF [61] models were implemented in MATLAB^® 2023. The URBaM modelling method proposed in Section 2 was also programmed in a Visual Basic for Application (VBA) environment of Excel^© 365 software.

Table 6 shows the evaluated modelling techniques, the configurations used, their abbreviation name set for each configured model, and the reference of the code utilized:

4. Results and Discussion

4.1. URBaM Method Evaluation

Figure 6 represents the deviation between the FEM results, used as a reference, and the predictions obtained with the URBaM model for the evaluated validation points in the six case studies. In addition, the ±10% and ±20% deviation error curves are also represented in the charts to identify the error range of the validation points. The following can be observed:

• In Cases 1 (HNL), 3, and 4 (MNL), where the equivalent stress is evaluated, as well as in in Case 6 (LNL), where the contact pressure is evaluated, most of the validation points present an error below ±10%, independent of the non-linearity level. In particular, all points in Case 6 are below ±10% error range. However, multiple validation points present an error in the ±10–20% range in Cases 1, 3, and 4. The 20% error limit was surpassed by a few validation points in these cases. In general, the maximum errors occur for lower output variable magnitudes, as they are more sensitive to any deviation in percentages.
• Most of the validation points in Case 2 (HNL) were below the ±10% error curve, and in Case 5 (LNL) below the ±20% error curve. However, in these cases, the evaluated output variable corresponds to the deflection angle in degrees, which leads to very low-magnitude values (<2°). Therefore, the predictions are more sensitive to any deviation, and they provide higher error percentage values.

  Figure 6. Comparison of the FEM and the URBaM prediction results of the validation points.

  Figure 6. Comparison of the FEM and the URBaM prediction results of the validation points.

As observed, the error percentage is not conditioned to the level of non-linearity of the case. In contrast, it is found that the error sensitivity is influenced by the magnitude of the output variable. In particular, near-zero output values may lead to high errors, even when there are small residual differences. Therefore, analyzing the error percentage metric without taking into account the residual difference between reference and predicted values can lead to incomplete conclusions. Thus, the Normalized Root Mean Square Error (NRMSE) and Relative Maximum Absolute Error (RMAE) metrics were also analyzed to complete this study, together with the Mean Absolute Percentage Error (MAPE).

Figure 7 shows the MAPE, NRMSE, and RMAE values of the validation points predicted with the URBaM modelling technique for the six case studies. The obtained results show the following:

• The MAPE value remained stable near the 10% value in all cases except in Case 2 (HNL). In that case, the error reached the 20% value. This error percentage increase is attributed to the fact that several output values are close to zero.
• The NRMSE metric remained stable close to 0.25 for all case studies.
• The RMAE metric also remained stable for all cases close to 0.7.

  Figure 7. Comparison of the NRMSE, RMAE, and MAPE values of the validation points using the URBaM model.

  Figure 7. Comparison of the NRMSE, RMAE, and MAPE values of the validation points using the URBaM model.

The results show that the average values of MAPE, NRMSE, and RMAE are 10.5%, 0.22, and 0.66, respectively. In addition, the accuracy of the URBaM method remained stable independent of the non-linearity level, as the standard deviation values show: 6% for MAPE, 0.07 for NRMSE, and 0.2 for RMAE. The MAPE observed in Case 2 (HNL) presented the highest deviation with a 21.4% value. However, the same case presented the smallest NRMSE value (0.1).

4.2. Comparison with Well-Known Surrogate Model Configurations

The URBaM modelling technique is compared with eight widely used surrogate modelling techniques (a total of 14 model configurations) in the six case studies with different non-linearity levels.

4.2.1. Case 1 (HNL): Stem Dovetail Dimensioning in a Downstream Slab Valve Family

The MAPE, NRMSE, and RMAE values of Case 1 are shown in Figure 8, Figure 9, and Figure 10, respectively. The highlights of the histograms are set out as follows:

• The models that present an MAPE below 10% are the PPS-GA (5.4%), RBF_MQ_1/N (5.7%), SVMR (7.2%), UKRI_Const (7.6%), POL2 (8.6%), and URBaM (9.5%) models. In addition, the ANN_5 (12.4%), RF (13.7%), ANN_10 (18%), ANN_3 (18.1%), and MARS (19%) models present errors below the 20% criteria.
• The models that present an NRMSE below 0.5 are the PPS-GA (0.2), RBF_MQ_1/N (0.2), UKRI_Const (0.2), URBaM (0.3), SVMR (0.3), POL2 (0.3), RF (0.4), ANN_5 (0.4), and MARS (0.5) models.
• The models that report an RMAE below 1 are the PPS-GA (0.6), RBF_MQ_1/N (0.7), URBaM (1), and UKRI_Const (1) models.

The models that fulfil the established three metric criteria for Case 1 (HNL) are the URBaM, PPS-GA, UKRI_Const, and RBF_MQ_1/N models. Thus, all of them are representative for modelling Case 1. The POL2 and the SVMR models present an MAPE and an NRMSE below the reference limits. However, they surpass the RMAE limit of 1: SVMR (1.6) and POL2 (2). Therefore, in general, these models provide representative prediction values but occasionally report high deviations in the predictions and are considered partially representative models for Case 1.

Figure 8. MAPE values of the different surrogate model configurations analyzed in Case 1 (HNL).

Figure 8. MAPE values of the different surrogate model configurations analyzed in Case 1 (HNL).

Figure 9. NRMSE values of the different surrogate model configurations analyzed in Case 1 (HNL).

Figure 9. NRMSE values of the different surrogate model configurations analyzed in Case 1 (HNL).

Figure 10. RMAE values of the different surrogate model configurations analyzed in Case 1 (HNL).

Figure 10. RMAE values of the different surrogate model configurations analyzed in Case 1 (HNL).

4.2.2. Case 2 (HNL): Structural Bolting Dimensioning in a Flanged Joint of a High-Pressure Split Body Ball Valve Family

The MAPE, NRMSE, and RMAE values of Case 2 are shown in Figure 11, Figure 12, and Figure 13, respectively. The highlights of the histograms are set out as follows:

• All the analyzed models present an MAPE value higher than 20%. As previously stated, this fact is attributed to the magnitude of the output variable of Case 2, which is close to zero. The models that present MAPE values closest to the 20% limit are ANN_10 (20.2%) and URBaM (21.4%).
• The models that present an NRMSE below 0.5 are the URBaM (0.1), ANN_10 (0.1), SVMR (0.2), ANN_5 (0.2), RBF_MQ_1/N (0.2), PPS-GA (0.3), UKRI_Const (0.3), ANN_3 (0.3), MARS (0.3), POL2 (0.4), and RF (0.4) models.
• The models that report an RMAE below 1 are the ANN_10 (0.5), URBaM (0.6), UKRI_Const (0.8), PPS-GA (0.9), and RBF_MQ_1/N (0.9) models.

Figure 11. MAPE values of the different surrogate model configurations analyzed in Case 2 (HNL).

Figure 11. MAPE values of the different surrogate model configurations analyzed in Case 2 (HNL).

Figure 12. NRMSE values of the different surrogate model configurations analyzed in Case 2 (HNL).

Figure 12. NRMSE values of the different surrogate model configurations analyzed in Case 2 (HNL).

Figure 13. RMAE values of the different surrogate model configurations analyzed in Case 2 (HNL).

Figure 13. RMAE values of the different surrogate model configurations analyzed in Case 2 (HNL).

The models that fulfil the established NRMSE and RMAE metric criteria for Case 2 (HNL) are the ANN_10, URBaM, UKRI_Const, PPS-GA, and RBF_MQ_1/N models. According to the MAPE metric criterion, the ANN_10 (20.2%) and URBaM (21.4%) models present the smallest error percentage values and are close to the 20% error limit. Thus, both models could be considered representative for modelling Case 2.

4.2.3. Case 3 (MNL): Slab Dovetail Dimensioning in a Downstream Slab Valve Family

The MAPE, NRMSE, and RMAE values of Case 3 (MNL) are shown in Figure 14, Figure 15, and Figure 16, respectively. The highlights of the histograms are set out as follows:

• The models that present an MAPE below 10% are the RBF_MQ_1/N (3.4%) and SVMR (7%) models. However, several models are close to that value: the RF (10.3%), URBaM (10.6%), POL2 (11.4%), MARS (12%), and UKRI_Const (12.1%) models.
• The models that present an NRMSE below 0.5 are the RBF_MQ_1/N (0.1), URBaM (0.2), POL2 (0.2), UKRI_Const (0.2), SVMR (0.2), MARS (0.2), RF (0.2), ANN_3 (0.4), PPS-GA (0.5), and ANN_5 (0.5) models.
• The RMAE of the RBF_MQ_1/N (0.3) and URBaM (0.5) models are below 1.

The RBF_MQ_1/N model fulfils the established three metric criteria completely for Case 3 (MNL). The URBaM model fulfils the RMAE and NRMSE criteria, and it only surpasses in 0.6% the MAPE criterion of 10%. Therefore, both of them could be considered representative for modelling Case 3.

Figure 14. MAPE values of the different surrogate model configurations analyzed in Case 3 (MNL).

Figure 14. MAPE values of the different surrogate model configurations analyzed in Case 3 (MNL).

Figure 15. NRMSE values of the different surrogate model configurations analyzed in Case 3 (MNL).

Figure 15. NRMSE values of the different surrogate model configurations analyzed in Case 3 (MNL).

Figure 16. RMAE values of the different surrogate model configurations analyzed in Case 3 (MNL).

Figure 16. RMAE values of the different surrogate model configurations analyzed in Case 3 (MNL).

4.2.4. Case 4 (MNL): Stem–Ball Joint Dimensioning for a High-Pressure Ball Valve Family

The MAPE, NRMSE, and RMAE values of Case 4 (MNL) are shown in Figure 17, Figure 18, and Figure 19, respectively. The highlights of the histograms are set out as follows:

• All the analyzed models have an MAPE value higher than 10%. The models that present an error value within the 10–20% domain are the POL2 (11.3%), URBaM (12.9%), UKRI_Const (13.9%), UKRI_Var (14%), ANN_10 (14.2%), RBF_MQ_1/N (14.6%), RBF_MQ_1 (14.7%), PPS-GA (14.8%), RBF_IMQ_1 (14.9%), SVMR (15%), MARS (17.5%), ANN_5 (19.2%), and ANN_3 (19.7%) models.
• The NRMSE of all the analyzed models is below 0.5.
• The models that present an RMAE below 1 are the URBaM (0.5), POL2 (0.6), SVMR (0.7), PPS-GA (0.7), RBF_MQ_1 (0.7), RBF_MQ_1/N (0.7), RBF_IMQ_1 (0.7), UKRI_Var (0.8), UKRI_Const (0.8), RBF_IMQ_1/N (0.8), MARS (0.8), and ANN_3 (1) models.

Figure 17. MAPE values of the different surrogate model configurations analyzed in Case 4 (MNL).

Figure 17. MAPE values of the different surrogate model configurations analyzed in Case 4 (MNL).

Figure 18. NRMSE values of the different surrogate model configurations analyzed in Case 4 (MNL).

Figure 18. NRMSE values of the different surrogate model configurations analyzed in Case 4 (MNL).

Figure 19. RMAE values of the different surrogate model configurations analyzed in Case 4 (MNL).

Figure 19. RMAE values of the different surrogate model configurations analyzed in Case 4 (MNL).

All the evaluated models, excluding ANN5, ANN10, RBF_IMQ_1/N, and RF, have an MAPE in the 10–20% range. These models, the ones which have an MAPE in the 10–20% range, also have NRMSE and RMAE values below 0.5 and 1, respectively. Therefore, they meet the acceptance criteria and can be considered appropriate to represent Case 4. However, it is worth highlighting that the models that present the lowest MAPE within the 10–20% domain are the POL2 (11.3%) and URBaM (12.9%) models. These two models also present the lowest NRMSE and RMAE values. Therefore, both models are considered the most appropriate ones to represent Case 4 (MNL).

4.2.5. Case 5 (LNL): Flange Dimensioning in a Bonnet to Body Joint for a High-Pressure Ball Valve Family

The MAPE, NRMSE, and RMAE values of Case 5 (LNL) are shown in Figure 20, Figure 21, and Figure 22, respectively. The highlights of the histograms are set out as follows:

• The models that present an MAPE below 10% are the RBF_MQ_1/N (2.8%), UKRI_Const (2.9%), PPS-GA (3%), UKRI_Var (3.2%), SVMR (3.9%), POL2 (6.2%), MARS (6.2%), and URBaM (7.3%) models. However, several models are close to that value: RBF_IMQ_1/N (11.2%), RF (12%), and ANN_5 (12.8%).
• The models that present an NRMSE below 0.5 are the PPS-GA (0.1), UKRI_Var (0.2), UKRI_Const (0.2), SVMR (0.2), RBF_MQ_1/N (0.2), MARS (0.2), URBaM (0.3), POL2 (0.3), RBF_IMQ_1/N (0.4), ANN_10 (0.5), and RF (0.5) models.
• The models that report an RMAE below 1 are the PPS-GA (0.4), UKRI_Var (0.4), UKRI_Const (0.5), SVMR (0.6), RBF_MQ_1/N (0.7), MARS (0.7), and URBaM (0.9) models.

The models that fulfil the established three metric criteria for Case 5 (LNL) are the URBaM, PPS-GA, UKRI_Var, UKRI_Const, SVMR, RBF_MQ_1/N, and MARS models. Hence, all of them are representative for modelling Case 5.

Figure 20. MAPE values of the different surrogate model configurations analyzed in Case 5 (LNL).

Figure 20. MAPE values of the different surrogate model configurations analyzed in Case 5 (LNL).

Figure 21. NRMSE values of the different surrogate model configurations analyzed in Case 5 (LNL).

Figure 21. NRMSE values of the different surrogate model configurations analyzed in Case 5 (LNL).

Figure 22. RMAE values of the different surrogate model configurations analyzed in Case 5 (LNL).

Figure 22. RMAE values of the different surrogate model configurations analyzed in Case 5 (LNL).

4.2.6. Case 6 (LNL): Gate–Seat Dimensioning for a High-Pressure Slab Valve Family

The MAPE, NRMSE, and RMAE values of Case 6 (LNL) are shown in Figure 23, Figure 24, and Figure 25, respectively. The highlights of the histograms are set out as follows:

• The models that present an MAPE below 10% are the UKRI_Var (1.1%), UKRI_Const (1.1%), POL2 (1.3%), URBaM (1.5%), PPS-GA (1.7%), ANN_3 (1.7%), te ANN_5 (1.7%), MARS (1.8%), ANN_10 (2.9%), SVMR (3%), RF (3.6%), and RBF_IMQ_1/N (4%) models.
• The models that present an NRMSE below 0.5 are the UKRI_Var (0.1), UKRI_Const (0.1), POL2 (0.1), URBaM (0.2), PPS-GA (0.2), ANN_3 (0.2), ANN_5 (0.2), MARS (0.2), ANN_10 (0.4), SVMR (0.4), and RF (0.5) models. The NRMSE of the RBF_IMQ_1/N (0.6) model is also close to 0.5.
• The models that present an RMAE below 1 are the POL2 (0.3), UKRI_Var (0.4), UKRI_Const (0.4), ANN_3 (0.4), URBaM (0.5), PPS-GA (0.5), ANN_5 (0.5), MARS (0.5), ANN_10 (0.7), SVMR (0.8), and RF (1) models. The RMAE of the RBF_IMQ_1/N (1.2) model is also close to 1.

Figure 23. MAPE values of the different surrogate model configurations analyzed in Case 6 (LNL).

Figure 23. MAPE values of the different surrogate model configurations analyzed in Case 6 (LNL).

Figure 24. NRMSE values of the different surrogate model configurations analyzed in Case 6 (LNL).

Figure 24. NRMSE values of the different surrogate model configurations analyzed in Case 6 (LNL).

Figure 25. RMAE values of the different surrogate model configurations analyzed in Case 6 (LNL).

Figure 25. RMAE values of the different surrogate model configurations analyzed in Case 6 (LNL).

The models that fulfil the established three metric criteria for Case 6 (LNL) are the UKRI_Var, UKRI_Const, POL2, URBaM, PPS-GA, ANN_3, ANN_5, MARS, ANN_10, SVMR, and RF models. Thus, all of them are representative for modelling Case 6. The RBF_IMQ_1/N model slightly surpasses the NRMSE and RMAE criteria and the MAPE value is far below 10%. Therefore, this model could be considered also representative for modelling Case 6.

4.2.7. Comparison Overview

Figure 26 summarizes the representativeness of the evaluated surrogate modelling techniques in accordance with the defined MAPE, NRMSE, and RMAE limits. The models that fulfil all criteria for a case study are marked with a black dot. In addition, the models that are considered still representative, although with one of the defined criteria slightly surpassed, are marked with a white dot. It can be observed that no model completely fulfils the established criteria for Case 2, which is a high non-linear case (HNL). However, the predictions of the URBaM model and the ANN_10 models slightly surpassed the established limits and thus can still be considered representative. Figure 26 shows that most of the modelling techniques analyzed are representative for Cases 5 and 6 (low non-linear cases), but when the non-linearity of the case increases, the number of representative models drastically decreases. However, it is worth highlighting that the URBaM modelling technique is the only technique that provided representative predictions for all the six case studies, independent of the level of non-linearity.

With the aim of analyzing the error stability according to different non-linearity levels, the MAPE (Figure 27), NRMSE (Figure 28), and RMAE (Figure 29) of the most representative surrogate models were studied. Therefore, the PPS-GA, UKRI_Const, and RBF_MQ_1/N models, which were representative in four out of six cases, were selected for comparing against the URBaM method. These models presented average MAPE values of 23.9%, 19.9%, and 13.9% with a standard deviation of 32.9%, 28%, and 11.1%, respectively; average NRMSE values of 0.26, 0.22, and 0.75 with a standard deviation of 0.12, 0.07, and 1.23, respectively; and average RMAE values of 0.88, 0.85, and 1.77 with a standard deviation of 0.61, 0.39, and 2.48, respectively. It can be observed that the URBaM model not only presents the lowest average values of MAPE, NRMSE, and RMAE metrics with values of 10.5%, 0.22, and 0.66 but also presents the highest stability from case to case with standard deviation values of 6%, 0.07, and 0.2, respectively.

In addition, the charts show that the optimum model choice results to be different from case to case. However, the obtained accuracy metric values of the URBaM model are always close to those of the optimum model. Thus, the average of the deviation of MAPE, NRMSE, and RMAE of the URBaM model with respect to the optimum model from case to case are 3.1%, 0.1, and 0.23, with standard deviation values of 2.3%, 0.06, and 0.17, respectively.

5. Conclusions

In this paper, a new surrogate modelling technique to assess the definition of reliable product family scaling rules, named URBaM, is presented. The model was evaluated and compared with 14 configurations of eight widely studied surrogate models in six real engineering problems with different non-linearity levels: two low (LNL), two medium (MNL), and two high (HNL):

• The URBaM model adapts with average MAPE, NRMSE, and RMAE errors of 10.5%, 0.22, and 0.66, respectively, with a case-to-case standard deviation of 6%, 0.07, and 0.2, demonstrating good capability in adapting to different non-linearity levels with a single configuration.
• The URBaM model was not the optimum choice in all cases but resulted in being the unique model capable of accurately representing the six analyzed cases of different non-linearity levels. The model was always close to the optimum model with an average deviations MAPE, NRMSE, and RMAE of 3.1%, 0.1, and 0.23, respectively.

Consequently, the URBaM technique was validated to assist the design process of scalable mechanical product families for problems with different non-linearities contributing in two different ways:

• Allows for efficiently determining reliable scaling rules for mechanical product families. The cumbersome and time-consuming optimum surrogate model type and configuration election process is not required with the URBaM model in the process of determining product family scaling rules, where non-linearity level is unknown a priori.
• The result accuracy level shown by the URBaM model allows for reducing the design–analysis iterations close to zero for a new family member. Consequently, the design delivery time and the cost of the product are minimized, which is crucial in product delivery strategies where product dimensions are stablished by the customer, as in Engineer To Order (ETO) strategies.

The time required in the two scenarios—(i) not defining a scaling rule, resulting in nearly zero design-analysis iterations, and (ii) performing model configuration and comparison tasks—can extend to days or even weeks, depending on the complexity of the design case. While quantifying the time savings in such frameworks is challenging, significant reductions are ensured, as surrogate model training takes only minutes and model execution requires only few seconds.

The URBaM modelling technique is currently limited to the use of full factorial DOEs. However, as reported by F. A. Viana et al. [64], full factorial designs are not recommended when more than five design variables are involved. Therefore, in this work, the authors recommend restricting the number of design variables to fewer than five when applying the URBaM method. In this context, it is important to emphasize the role of sensitivity analysis in eliminating unnecessary variables in surrogate modelling, as highlighted by J. Zhai and F. Boukouvala [65]. Future research should focus on overcoming the dependency on full factorial DOEs, thereby enabling the broader application of the URBaM technique to frameworks involving more than five variables.