Multidisciplinary Reliability Design Optimization Modeling Based on SysML

Abstract

Model-Based Systems Engineering (MBSE) supports the system-level design of complex products effectively. Currently, system design and optimization for complex products are two distinct processes that must be executed using different software or platforms, involving intricate data conversion processes. Applying multidisciplinary optimization to validate system optimization often necessitates remodeling the optimization objects, which is time-consuming, labor-intensive, and highly error-prone. A critical activity in systems engineering is identifying the optimal design solution for the entire system. Multidisciplinary Design Optimization (MDO) and reliability analysis are essential methods for achieving this. This paper proposes a SysML-based multidisciplinary reliability design optimization modeling method. First, by analyzing the definitions and mathematical models of multidisciplinary reliability design optimization, the SysML extension mechanism is employed to represent the optimization model based on SysML. Next, model transformation techniques are used to convert the SysML optimization model generated in the first stage into an XML description model readable by optimization solvers. Finally, the proposed method’s effectiveness is validated through an engineering case study of an in-vehicle environmental control integration system. The results demonstrate that this method fully utilizes SysML to express MDO problems, enhancing the efficiency of design optimization for complex systems. Engineers and system designers working on complex, multidisciplinary projects can particularly benefit from these advancements, as they simplify the integration of design and optimization processes, facilitating more reliable and efficient product development.

1. Introduction

With the continuous advancement of technology and growing economic demands, the complexity of engineering systems is on the rise. Designing and developing complex engineering systems involve multiple disciplines, each having varying degrees of coupling. Multidisciplinary Design Optimization (MDO) aims to fully account for the interactions and coupling effects between disciplines, employing suitable optimization methods, strategies, and distributed computing technologies to organize and manage the entire design process. By balancing the various disciplines during the design phase and deeply analyzing the coupling relationships and synergistic effects, MDO achieves the system’s optimal overall solution, enhancing system performance, reducing design costs, and shortening development cycles. This approach is effective for solving complex engineering system optimization problems [1]. Conversely, document-based systems engineering can no longer meet current R&D needs when dealing with the complexities of interdisciplinary coupling. Thus, adopting Model-Based Systems Engineering (MBSE) has become an inevitable trend. MBSE involves a model-centric approach adopted by different disciplines (including mechanical, electrical, and software fields) to integrate various models into the systems engineering process, enabling model-driven product system development and full lifecycle integration. This model-centric approach is poised to replace traditional document-centric methods [2]. MBSE supports systems engineering activities across the lifecycle through formalized modeling, characterized by managing system complexity, ensuring information consistency, and maintaining global traceability. Presently, the aerospace, automotive, and shipbuilding industries are actively exploring and applying MBSE technology in product development [3,4,5].

The MBSE approach enables designers to focus on developing system functional and behavioral models while automatically extracting multidisciplinary optimization models from the established system models [6]. This improves optimization verification efficiency and reduces the programming burden on designers. MBSE supports the integration of various models, allowing detailed design processes across different fields to be linked with the abstract system level. Therefore, in the system design process for complex products, optimization information is incorporated into the design information to build formal optimization models, achieving automatic system optimization [7]. Additionally, MDO can be employed at all stages of product design. Establishing MDO models based on MBSE facilitates the extraction of optimization models and allows for the reuse of established MDO models in iterative design processes. System models can define disciplinary interfaces, data flows, strategy process information, and abstract design parameter information.

Therefore, this paper analyzes the definitions and mathematical models of MDO and reliability analysis [8]. It implements the representation of optimization models based on the SysML language through the SysML extension mechanism, including the construction of optimization variables, optimization constraints, optimization goal metamodels, and the formal expression of MDO problems [9]. Additionally, the paper analyzes the representation mechanism of SysML parametric diagrams and uses model transformation to convert the SysML optimization models generated in the first stage into XML description models readable by optimization solvers. Furthermore, the multidisciplinary optimization modeling of an in-vehicle environmental control integration system demonstrates the effectiveness of the proposed method.

The rest of the paper is organized as follows: Section 2 presents the related research on Reliability-Based Multidisciplinary Design Optimization (RBMDO) and the integration of system design and optimization. Section 3 discusses the SysML-based RBMDO modeling method. Section 4 describes the XML-based approach to optimization model extraction. Section 5 validates the proposed method through a case study. Section 6 summarizes the findings and suggests future research directions.

2. Related Works

2.1. RBMDO

In 1982, Sobieski first proposed the concept of Multidisciplinary Design Optimization (MDO), laying the foundation for its development [10]. Subsequently, Reliability-Based Multidisciplinary Design Optimization (RBMDO), aimed at improving the design quality of complex engineering systems by considering the impact of uncertainties, was introduced [8]. RBMDO comprehensively takes into account various uncertainty factors in a system, reasonably assessing their impact on the output functions of the multidisciplinary system to ensure that the MDO design results meet specified reliability requirements.

MDO has developed over nearly 30 years, yet there is no universally accepted definition of MDO internationally. Nonetheless, its characteristics can be summarized as follows: it involves a thorough exploration and utilization of interactions within the system; it requires the analysis of interactions between disciplines or subsystems and the use of these interactions for integrated system optimization [11]. Essentially, MDO is a comprehensive method for addressing complex engineering system design challenges. By exploring and leveraging the synergistic mechanisms of interactions within the engineering system and considering the interplay between disciplines, MDO aims to optimize the design of complex engineering systems from a systemic perspective, with goals of improving product performance, reducing costs, and shortening design cycles. MDO leverages the interactions between disciplines to achieve an overall optimal solution for the system.

The traditional Deterministic Multidisciplinary Design Optimization (DMDO) mathematical model can be described as Equation (1) [8]:

$$\begin{gathered} \underset{\left(DV={\mathit{d}}_s,\mathit{d}\right)}{min} f\left({\mathit{d}}_s,\mathit{d},\mathit{y}\right) \\ s.t.g^{\left(i\right)}\left({\mathit{d}}_s,{\mathit{d}}_i,{\mathit{y}}_{\cdot i}\right)\le 0 \\ {h}^{\left(i\right)}\left({\mathit{d}}_s,{\mathit{d}}_i,{\mathit{y}}_{\cdot i}\right)=0 \\ {\mathit{d}}_s^L\le {\mathit{d}}_s\le {\mathit{d}}_s^U, {\mathit{d}}_i^L\le {\mathit{d}}_i\le {\mathit{d}}_i^U \\ i=1,2,\cdots ,nd \end{gathered}$$

(1)

The deterministic design variables in Equation (1) are ${\mathit{d}}_s$ and $\mathit{d}$. ${\mathit{d}}_s$ is the shared deterministic design variable, $\mathit{d}=\left({\mathit{d}}_i, i=1~nd\right)$ is the local design variable consisting of all disciplines, and ${\mathit{d}}_i$ is the deterministic local design variable of the discipline $i$. The term “shared design variables” refers to design variables that are common to all disciplines, while “local design variables” refers to design variables of a particular discipline. ${\mathit{d}}_s^U$ and ${\mathit{d}}_s^L$ are the upper and lower bounds of ${\mathit{d}}_s$, and ${\mathit{d}}_i^U$ and ${\mathit{d}}_i^L$ are the upper and lower bounds of ${\mathit{d}}_i$. $f\left(\mathbf{•}\right)$ are the optimization objectives of DMDO, $g^{\left(i\right)}$ and $h^{\left(i\right)}$ are inequality constraints and equation constraints in the discipline $i,i=1~nd$, and $nd$ is the total number of all disciplines included in the complex engineering system. $y=\left\{y_{1•},y_{2•},\dots ,y_{nd•}\right\}$ is the coupled variable that discipline $i$ outputs to other disciplines and is the set of all coupled state variables of the MDO optimization problem. $y_{i•}=\left\{y_{ij},j=1~nd,j\ne i\right\}$, and $y_{i•}$ is the coupled state variable of the discipline $i$ output to all other disciplines. Similarly, $y_{•i}$ is the coupled state variable of all other disciplines input to the discipline $i$.

Uncertainties commonly exist in all stages of engineering design, and as the complexity of products increases, uncertainties (including types and quantities) also increase. Uncertainty in engineering design has the following characteristics: first, there are many sources of uncertainty (many channels causing uncertainty generation), such as various uncertainties caused by dimensional errors, material properties, assembly errors, stress loads, etc., of products; second, there are various forms of uncertainty in the design process, such as aleatory uncertainty obeying statistical distribution, cognitive uncertainty with only partial fuzzy information, and time-varying uncertainty that considers time factors, etc.

Unlike traditional deterministic design optimization, in RBMDO, continuous design variables and design parameters are considered to have stochastic, cognitive uncertainty or time-varying uncertainty. The design constraints consist of deterministic constraints and probabilistic reliability constraints. Under the condition that the reliability of the design constraint is greater than a given value, RBMDO is generally constructed as a problem of minimizing the total product cost or weight, etc. Reliability design optimization ensures that the reliability of the designed engineering system or product meets the given reliability requirement, thus reducing the probability of system or product failure to an acceptable level. A typical multidisciplinary reliability design optimization model with RBMDO containing stochastic uncertainty variables is shown in Equation (2) as an example [8]:

$$\begin{gathered} min f\left({\mathit{d}}_s,{\mathit{d}}_i,{\mathit{x}}_s,{\mathit{x}}_i,\mathit{p}\right) \\ s.t. \mathit{Pr}\left(G^{\left(i\right)}\left({\mathit{d}}_s,{\mathit{d}}_i,{\mathit{x}}_s,{\mathit{x}}_i,\mathit{p},{\mathit{y}}_i\right)\ge 0\right)\ge R_i \\ {g}^{\left(i\right)}\left({\mathit{d}}_s,{\mathit{d}}_i,{\mathit{y}}_{\cdot i}\right)\ge 0 \\ {\mathit{d}}_s^L\le {\mathit{d}}_s\le {\mathit{d}}_s^U,{\mathit{d}}_i^L\le {\mathit{d}}_i\le {\mathit{d}}_i^U \\ {\mathit{x}}_s^L\le {\mathit{x}}_s\le {\mathit{x}}_s^U,{\mathit{x}}_i^L\le {\mathit{x}}_i\le {\mathit{x}}_i^U \\ i=1,2,3 \end{gathered}$$

(2)

where ${\mathit{x}}_s$ is the set of the randomized input design variables shared by all disciplines, and ${\mathit{x}}_s^U$ and ${\mathit{x}}_s^L$ are the upper and lower bounds of ${\mathit{x}}_s$, respectively. ${\mathit{x}}_i$ are the local randomized design variables of the discipline $i$, and ${\mathit{x}}_i^U$ and ${\mathit{x}}_i^L$ are the upper and lower bounds of ${\mathit{x}}_i$, respectively. $p$ is the set of the design parameters, and $R_i$ is the reliability requirement of the discipline $i$. $g^{\left(i\right)}\left(\mathbf{•}\right)$ is the deterministic design constraint of the discipline $i$. $\mathit{Pr}\left(G^{\left(i\right)}\left(\mathbf{•}\right)\ge 0\right)$ is the probabilistic reliability non-failure model of the discipline $i$, and $\mathit{Pr}\left(G^{\left(i\right)}\left(\mathbf{•}\right)\ge 0\right)\ge R_i$ is the probabilistic reliability design constraint of the discipline $i$.

The reliability design optimization model with cognitive uncertainty and time-varying uncertainty is similar to Equation (2) and is not repeated here. From Equation (2), it can be seen that a typical multidisciplinary reliability design optimization problem includes optimization variables, optimization objectives, and optimization constraints, where optimization constraints are divided into two types: reliability constraints and deterministic constraints, and the difference between the two is whether the constraints contain uncertainty variables or not.

2.2. Integration of System Design and System Optimization

In model-based system design, system modeling is the primary way to present the outcomes of system design. The International Council on Systems Engineering (INCOSE) first defined “Model-Based Systems Engineering” (MBSE) in its “Vision 2020” document [12]: MBSE is a formalized approach to modeling that supports system requirements, design, analysis, verification, and validation activities from the conceptual design phase and continues throughout the entire development process and subsequent lifecycle stages. This definition highlights the application of modeling methods, where modeling involves creating models using specific languages and tools. While systems engineering was not the first to adopt modeling methods, MBSE’s unique approach involves constructing system architecture models using modeling languages during the conceptual design phase, which is a significant innovation. The three pillars of MBSE are system modeling languages, system modeling methodologies, and system modeling software. SysML has emerged as the standard modeling language for MBSE. SysML [13] is characterized by its object-oriented, graphical, and platform-independent nature, making it suitable for describing, analyzing, designing, and validating complex systems that include both software and hardware aspects.

Model-based approaches are tools for system design, but achieving system optimization is the ultimate goal of design. As illustrated in Figure 1, the development of complex products involves an iterative process of continually optimizing design solutions. Advanced system design methods and optimization techniques have always been key drivers in the innovative development of complex products. Efficient integration and application of system design and optimization are crucial for enhancing the design efficiency of complex products [14]. Research on the integration of system design and optimization can be discussed from two perspectives: application platforms and theoretical studies.

When using MBSE for system modeling, there may be many design solutions that satisfy system-level requirements that have conflicting objectives at the parameter level, and the existing MBSE does not allow for functional verification and performance optimization of the design solutions. Therefore, MBSE needs to be linked with multidisciplinary reliability design optimization to determine the best design solution [15]. Integrating system design with system optimization has been a hot research topic for scholars in various countries [9,14].

In recent years, several scholars at home and abroad have developed the SysML language to provide an integrated depiction of design and optimization. NASA’s Jet Propulsion Laboratory (JPL) researched an MBSE-based spacecraft-mission operating system architecture to improve the ability to make trade-offs and optimization evaluations of the overall system [16,17]. The metamodel of SysML was enhanced by JPL in the design of tiny satellites to facilitate the trade-off analysis of solutions [18]. Yusheng Liu et al. [19] proposed a pattern-based approach for integrating system design and system optimization by extending the SysML metamodel to define a system optimization model and using pattern information containing basic information, problem descriptions, solutions, impacts, and other attributes to find a solution to the optimization problem through semantic matching. Leserf et al. [20] explored the constraint satisfaction multicriteria optimization problem (CSMOP) resulting from the SysML model. Several configurations of model variability including continuous and discrete variables have been proposed.

Table 1. Main contributions and limitations of references.

Originator	Key Contributions	Limitations
JPL [16,17,18]	Expanded SysML metamodel for trade-off analysis in tiny satellite design and mission architecture optimization	Primarily focused on trade-off analysis, with limited application to comprehensive MDO scenarios
Yusheng Liu [19]	Developed a pattern-based approach to integrate system design and optimization using an extended SysML metamodel	Limited scalability and effectiveness in highly complex systems with undefined patterns
Leserf [20]	Explored CSMOP within SysML, proposing configurations for model variability to solve optimization problems	Focused mainly on CSMOPs, with less applicability to broader MDO challenges

showcases the key contributions and limitations of the referenced works.

Integration of numerical computation programs with system conceptual design modeling tools [21] is another method to integrate system design with system optimization. For example, integration of SysML modeling tools with Multidisciplinary Analysis and Optimization (MDAO) analytical optimization and decision tools [22], integration of SysML with trade-off analysis tools [23], integration of SysML with ModelCenter v13.5 software [24], integration of SysML with MATLAB R2018a tools [25], and integration of SysML with Constraint Satisfaction Problem (CSP) solvers [26] have been performed.

3. Representation of Optimization Problems Using SysML

To integrate MDO models with system design models, it is essential to develop a domain metamodel that is appropriate for expressing MDO-related issues. A metamodel essentially defines a modeling language, and when this language is designed for a specific domain issue like multidisciplinary optimization, it is known as a Domain-Specific Language (DSL). There are three main strategies for constructing a metamodel for a specific DSL: (1) elaborating on an existing DSL metamodel within the same domain to translate more general concepts into detailed ones; (2) extending a generic, domain-neutral modeling language (such as SysML) to incorporate MDO-related concepts; and (3) designing a completely new metamodel for a language from scratch.

Based on the integration needs of system design and system optimization, the second approach can be chosen—extending the existing general-purpose modeling language SysML to define a metamodel for complex product Multidisciplinary Design Optimization. To extend SysML, two methods can be employed [27]: (1) defining new metaclasses within the SysML metamodel to create new modeling elements with specific attributes, constraints, and usages; or (2) extending existing SysML elements using Stereotypes [28] and Tagged Values. The first method allows for precise representation of the new modeling elements and their semantics but may not be supported by existing modeling tools. In contrast, the second method—using Stereotypes to extend the semantics of SysML model elements and Tagged Values to assign attributes not included in the SysML metamodel—offers greater flexibility and generality. Consequently, this paper adopts the latter method to extend SysML to support the expression of elements included in optimization models.

The process of defining a multidisciplinary optimization metamodel based on the SysML extended version (Stereotype) mainly includes the following steps: (1) extracting MDO domain knowledge and identifying the concepts, relationships, and constraints involved according to MDO definitions and application scenarios; (2) defining the Abstract Syntax of the above concepts using a formal approach; (3) defining the Concrete Syntax, which defines the corresponding SysML representation and notation for all the above-summarized model elements based on the Abstract Syntax, and this process can be realized by extending similar model elements in SysML; (4) describing the model elements and usage methods in the extended MDO metamodel to form the corresponding specification.

The underlying structure of the Multidisciplinary Design Optimization problem is shown in Figure 2, which is generally divided into a system level and a subsystem level, although of course there may be more layers for complex problems. The optimization model at each level contains parameters, objectives, variables, constraints, and other information. The system level also includes the number of disciplines and disciplinary coupling information. The interdisciplinary coupling information stored at the system level will transmit the coupling information to each discipline when building the multidisciplinary optimization mathematical model.

3.1. Definition of Extension Types for Optimization Types

Multidisciplinary optimization information includes two types, an optimization object model and an optimization process model. The optimization object model consists mainly of the basic elements needed to construct an optimization problem, including optimization objectives, constraint functions, design variables, etc. The optimization process model is mainly applied to specific solvers, such as MATLAB, ISIGHT, etc. It is worth noting that these commercial solvers are very mature for optimization solving, while the main functions of SysML modeling software mainly focus on modeling the requirements, structure, function, and behavior information of the system. Therefore, we do not need to consider the process model of optimization too much. Here, we mainly introduce the SysML extension of the optimization object model. The optimization object model is mainly used to construct the basic elements of an optimization problem. As mentioned above, the optimization object information mainly includes parameters, objectives, variables, and constraints, where the parameter element in SysML has the same meaning as the element in optimization, so only the optimization objectives, optimization variables, and optimization constraints are extended here for the metamodel.

3.1.1. Optimization Variable Metamodel

SysML lacks direct elements to represent variables. In multidisciplinary systems, a design variable refers to quantities to be optimized in the design process, which are independent of each other and unaffected by other variables. Design variables can be categorized based on their scope of influence into system design variables and local design variables. System design variables, also known as shared deterministic variables, have an effect across the entire system, whereas local design variables only impact a specific discipline. Local design variables are sometimes referred to as disciplinary variables or subspace design variables. Design variables can also be classified based on uncertainty into four types: deterministic design variables, aleatory uncertainty (AU) variables, epistemic uncertainty (EU) variables, and time-variant uncertainty variables. Additionally, variables can be classified as either discrete variables (DVs) or continuous variables (CVs) based on their value continuity. Since system design variables and local design variables can be defined and explored within optimization solvers, there is no need to define them separately in SysML system design models. Therefore, this paper focuses primarily on defining design variables based on their uncertainty. Another special type of variable in multidisciplinary systems is the coupled state variable (CSV). This refers to state variables from other disciplines that are used as inputs for the current discipline being analyzed. Since state variables can be represented by parameters, there is no need to extend the metamodel for them.

Based on the above analysis, it is necessary to enhance the “parameter” type in SysML with additional attributes to define variable types suitable for multidisciplinary system optimization. This can be achieved by adding Tagged Values to specify the type of optimization variable a module attribute belongs to, as illustrated in Figure 3.

To facilitate the definition of design variables in the SysML design model, as depicted in Figure 4, the “Variable” class extends the SysML metaclass “Block”. Given that coupled state variables typically have associated equations and can be expressed through a continuous variable and a constraint block, no additional extensions are needed for them. Hence, the optimization variable metamodel “Variable” is instantiated into four types: continuous variables, discrete variables, aleatory variables, and epistemic variables.

As shown in Figure 4, for the continuous variable configuration shapes in the SysML design model, the minimum and maximum values must be specified for their conformations to limit the range of the solver exploration space.

For aleatory variables, at present, the use of probability theory to quantify aleatory uncertainty has been widely recognized in academia and engineering circles and has achieved great success. For aleatory uncertainties subject to other distributions, the most appropriate probability distribution (such as log-normal distribution, Weibull distribution, Gumbel distribution, etc.) should be selected for characterization, and the Rosenblatt method should be used to convert it to standard normal space. The conversion relationship of different probability distributions is shown in

Table 2. Probability distributions and their x-u spatial transformation relationships.

Distribution	Parameters	Convert Relationships
Normal distribution	$\mu =mean value, \sigma =standard deviation$	$X=\mu +\sigma U$
Log-normal distribution	${\overline{\sigma }}^2=\ln \left[1+{\left({\displaystyle \frac{\sigma }{\mu }}\right)}^2\right]$ $\overline{\mu }=\ln \left(\mu \right)-0.5{\overline{\sigma }}^2$	$X=e\left(\overline{\mu }+\overline{\sigma }U\right)$
Weibull distribution	$k>0, \mu =\lambda \mathsf{\Gamma }\left(1+{\displaystyle \frac{1}{k}}\right)$ ${\sigma }^2={\lambda }^2\left[\Gamma \left(1+2/k\right)-{\Gamma }^2\left(1+1/k\right)\right]$	$X=\lambda {\left[-\ln \left(\mathsf{\Phi }{\left(-U\right)}^{*}\right)\right]}^{{\displaystyle \frac{1}{k}}}$
Gumbel distribution	$\mu =\nu +{\displaystyle \frac{0.577}{\alpha }}, \sigma ={\displaystyle \frac{\pi }{\sqrt{6}\alpha }}$	$X=\nu -{\displaystyle \frac{1}{\alpha }}\ln \left[-\ln \left(\mathsf{\Phi }\left(U\right)\right)\right]$
Uniform distribution	$\mu ={\displaystyle \frac{a+b}{2}}, \sigma ={\displaystyle \frac{b-a}{\sqrt{12}}}$	$X=a+\left(b-a\right)\mathsf{\Phi }\left(U\right)$

* $\mathsf{\Phi }\left(U\right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{{\displaystyle \int }}_{-\infty }^U{e}^{-{\displaystyle \frac{u^2}{2}}}du$.

.

For the characterization of aleatory variables, it is necessary to specify their means and variances, as well as their distribution types. As shown in Table 2, this paper defines the distribution type of aleatory variables as integer types from 1 to 5. When extracting the optimization model, its distribution type is determined by judgment. When the distribution type is 1 to 5, its distribution is defined as a normal distribution, a logarithmic distribution, a Weibull distribution, a Gumbel distribution, or a uniform distribution.

Regarding cognitive variables, they can be classified into fuzzy uncertainty and interval uncertainty according to their quantification types, etc. For the convenience of processing, interval uncertainty is used uniformly in this paper for characterization. Because interval uncertainty requires fewer quantification parameters, only the range of its values needs to be determined, while other types of cognitive uncertainty can remove some information to become cognitive uncertainty. Therefore, in the SysML optimization definition, it is only necessary to define the value boundaries of cognitive variables.

In the SysML design model, discrete variables often correspond to the selection of components, while the selection of components has two processes: the selection of a specific type of component and the selection of a specific number of components. The use of multiplicity and instances can be used here to characterize that, for a discrete variable, there is a class and a set of instances, INS = $\left[{\mathrm{ins}}_1,{\mathrm{ins}}_2,\dots ,{\mathrm{ins}}_m\right]$, associated with it for characterizing the choice of the value taken by the variable. There are three situations:

(1) The model element has only one instance decision. An instance, ${\mathrm{ins}}_1$, is identified by defining a bounded integer variable $x\in \left[1\dots m\right]$;

(2) The existence of a multiplicity of model element decisions. Identify the multiplicity of this model element by defining a bounded integer variable $y\in \left[1\dots n\right]$;

(3) Model elements have both instance decisions and multiplicity decisions. For this combination problem, create a Boolean value $x_{ij}$, where

$$\begin{array}{c}{x}_{ij}=\left\{\begin{array}{l}1 {\mathrm{ins}}_j \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{s} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}\mathrm{y} i\\ 0 \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{s} \end{array}\right.\\ i\in \left[1 \dots n\right], j\in \left[1 \dots m\right]\end{array}$$

(3)

3.1.2. Optimization Constraint Metamodel

Constraints that must be satisfied during the system design process are categorized into equality constraints and inequality constraints. The extended definition of optimization constraints is illustrated in Figure 5. As shown, optimization constraints inherit from the SysML constraint block and include six attributes: name, equation, owner, variable, parameter, and disciplinary. The Equation attribute is linked to the definition of equations in SysML, allowing a constraint to be represented by a formula composed of multiple equations. The Owner attribute supports the hierarchical relationships between structures; for example, we establish a relationship owner_ownedConstraint between the class model and itself, indicating that a constraint can own other constraints, thus defining global system constraints and local disciplinary constraints. A constraint can contain multiple variables and parameters, and a variable or parameter can be assigned to multiple constraints, allowing the definition of disciplinary-shared variables, or interdisciplinary coupling information.

The constraints are divided into equation constraints and inequality constraints, where the equation constraints need to determine the value of the state function in addition to the above definition, and the inequality constraints need to set the upper and lower boundaries of the state function. The state function is derived from the equation in the attribute. The extended relationship between the equation constraint and the inequality constraint is shown in Figure 6.

3.1.3. Optimization Objective Metamodel

The extension of optimization objectives is similar to that of optimization constraints. In some cases, optimization objectives and constraints can be interchangeable. For example, the mass of a module can serve as either a system optimization constraint or an optimization objective. As illustrated in Figure 7, optimization objectives also inherit from the SysML constraint block and include seven attributes: name, equation, owner, variable, parameter, disciplinary, and direction. The direction attribute defines the value orientation of the objective function, indicating whether the objective is to be maximized or minimized. We define its data type as Boolean, where 0 represents minimization and 1 represents maximization.

3.2. Formal Expression of MDO Problems Using Extended SysML

As previously mentioned, SysML can be extended according to the fundamental elements of a multidisciplinary optimization information model to express optimization objectives, optimization constraints, and optimization variables. However, it is also necessary to define a metamodel for multidisciplinary optimization problems, providing a method to represent the overall semantic information of the optimization problem organization through the extended framework.

Among the nine types of diagrams in SysML, parametric diagrams capture the internal structure of constraint blocks based on the parameters and the connections between them. A Block Definition Diagram (BDD) defines the constraint blocks and their relationships. Both contain the most features of the optimization information representation, and thus they can be used as the basis for defining the metamodel of the optimization problem using the parameter diagram and the module definition diagram. The MDO problem represents the system model situation that needs to be optimized in order to maximize or minimize a set of objective functions. As shown in Figure 8, the defined metamodel of the optimization problem is inherited from the parameter diagram. In this figure, the top-level “MDO Problem” block extends the original SysML construct type “ConstraintBlock”. The “MDO Problem” block has a reference to “SystemBlock” and “SubsystemBlock”, which can be used to represent and analyze the hierarchical relationship of the system. In addition, the “MDO Problem” contains elements such as optimization objectives, optimization constraints, and optimization variables.

Since the MDO problem diagram is inherited from the parameter diagram, it is possible to bind each variable or parameter in the optimization constraint or optimization objective expression to a value present in the SysML model based on the inherited value binding property, and these constraints can be applied to a specific module. The value attributes that are bound can belong to the module itself or to component attributes or reference attributes of the module. In this way, it is possible to create an arbitrarily complex mathematical model and bind it to an arbitrarily complex institutional model component. In addition, a parameter is defined for each objective function to bind the output to the “Opt Solution Block”, which allows the optimized results to be subsequently returned to the system design model.

4. System Optimization Model Extraction Method

SysML is object-oriented and supports a standardized approach for exchanging data between tools (OMG XML metadata exchange standard) [29], so XML is used here as an intermediate format for SysML optimization information extraction and integration with the optimization solver (MATLAB is used in this paper.) The data for the MDO problem can all be organized in a tree structure, using XML for storage and transfer. The translation mechanism is shown in Figure 9.

4.1. SysML Parametric Diagram Expression Mechanism

The metamodel of the system optimization problem is inherited from the parameter graph. When extracting the optimization model from the system design model, it is necessary to parse the SysML parameter graph, so it is necessary to analyze the mechanism of the parameter graph expression in SysML, which is related to the completeness and validity of the mathematical properties of the output target model.

Table 3. SysML parameter diagram model elements and their meanings.

SysML Parametric Graph Model Elements	Description
Constraint	Common mechanism for expressing system constraints, applied to model elements in the form of mathematical expressions, containing equations or inequalities represented in text
Constraint module	Encapsulation of constraint expressions, which can be applied in different contexts to facilitate the reuse of constraint information, usually including constraints and constraint parameters
Constraint properties	One of the properties of the system structure module, defined by the constraints module, used to bind the value attribute parameters of the structure
Constraint parameters	One of the properties of the constraint module, expressing explicitly the parameters in the constraint expression
Value properties	One of the properties of the system structure module, which can be bound to the constraint parameters in the constraint module
Binding connector	Represents the equivalence between the elements at the two ends of the connector, either as constraint value attributes or as constraint arguments

summarizes the model elements of the parameter graph in SysML.

The principle of equation or inequality constraint extraction from a parameter map is illustrated with the example of a constraint module for a spring component that follows Hooke’s law, F = −KX, where F is the elastic force, K is the coefficient of elasticity, and X is the amount of displacement for spring extension or compression. This is only the equation extraction for a single component, and when the component is inside a subsystem associated with other components through an interface, the parameters in this component constraint may be influenced by the value properties of other components, thus increasing the complexity of the equation transformation. Therefore, this paper adopts the following principles to extract the optimization model from the constraint module of SysML: (1) the equation constraints and inequality constraints in the extended constraint module correspond to the equation constraints and inequality constraints in optimization, and the constraint expressions and the values in their attributes and the upper and lower boundaries constitute the complete optimization equation constraints and inequality constraints; (2) to facilitate the understanding and extraction of the optimization model, each extended constraint module includes only one constraint expression and all its related variables and constraint parameters; (3) the constraint attributes expressed by each extended constraint module correspond to a parameter map used for equation extraction; (4) the binding connector is fully utilized to bind the constraint parameters to the value attributes of the module, so that the equation variables can be named using the value attribute names when the constraint expressions are converted into mathematical equation codes.

4.2. XML-Based Optimization Information Extraction

In order to implement the extraction of the SysML optimization model, specific rules for extracting each type of element can be defined. Taking the extraction rule Var_Rule for optimization variables as an example, the pseudo-code of the implementation algorithm is shown in Algorithm 1. In Algorithm 1, $ov$ denotes optimization variables, $cv$ denotes continuous variables, $av$ denotes aleatory variables, and $ev$ denotes cognitive variables. The algorithm can extract the information of variables contained in the SysML model by traversing each block of the SysML model.

Algorithm 1. Pseudo-code of the optimization variable extraction rule algorithm

01: Input: BS = {Blocks of SysML model} 02: boolean ErrorFound ← False 03: optimization problem P ← Ø 04: OptimizationVariable cv ← Ø 05: For b in BS 06: cv ← b.GetVariable () 07: If(cv ≠ Ø) 08: Switch(ov.GetType()) 09: case cv: 10: P.AddNumVar(cv.enum) 11: P.AddValue(cv. min,max) 12: case av: 13: P.AddNumVar(av.enum) 14: P.AddValueVar(av. mean,variance,distribution) 15: case ev: 16: P.AddNumVar(ev.enum) 17: P.AddValueVar(ev. . min,max) 18: default: 19: ErrorFound ← True 20: Else 21: AT ← b.GetRealAttrList() 22: For at in 23: cv ← at.GetVariability() 24: If (ov.GetType() = cvv) 25: Switch(at.GetType()) 26: case real: 27: P.AddRealVar(cv.min,cv.max) 28: case integer: 29: P.AddIntVar(cv.min,cv.max) 30: Endif 31: Endfor 32: Endif 33: Endfor 34: Return P,ErrorFound

The process of optimization can be summarized and abstracted from the actual model extraction activities to obtain a set of modeling elements, and the activity diagram of optimized model extraction can be constructed visually, as shown in Figure 10, which can support designers to carve out the mapping relationships between SysML models and XML in model transformation by defining extraction rules.

5. Evaluation and Discussion

5.1. SysML Optimization Metamodel Extension Based on CSM

Stereotype defines how to extend existing metaclasses. This section extends the optimization metamodel defined above using the Cameo System Modeler 19.0 (CSM) software. By adding extension elements to the Profile Diagram and based on the theory described in Section 3, the Multidisciplinary Design Optimization metamodel was constructed, as shown in Figure 11. It includes the optimization variable metamodel, the optimization constraint metamodel, and the optimization objective metamodel.

The extended optimization element metamodel can be utilized in future optimization modeling by publishing it. For example, the metamodel for an aleatory design variable includes attributes such as name, mean, variance, and distribution type. In practice, the metamodel for an aleatory design variable can be directly dragged in, and its attribute information can be set through the specification. The specific extension relationships of the optimization metamodel are shown in Figure 12.

In Figure 12, the MDO metamodel encompasses design variables, discipline classification, optimization constraint, and optimization objective. The design variables are further divided into continuous variables, aleatory variables, and epistemic variables. The constraints are categorized into equalities and inequalities, with both optimization constraints and objectives classified under the constraint block stereotype.

5.2. MDO Modeling Application for In-Vehicle Environmental Control Integration System

This section demonstrates the effectiveness of the approach proposed in this paper by using an example of multidisciplinary optimization modeling of an integrated system for in-vehicle environmental control of a vehicle. The integrated in-vehicle environmental control system is an important component of a vehicle. Its main components include the air conditioning subsystem and the integrated air supply subsystem, among others. The air conditioning subsystem primarily handles temperature and humidity control, and its structural composition is shown in Figure 13.

The integrated air supply subsystem provides functions including cabin air supply, internal and external air circulation conversion, overpressure control, dust filtration, flexible air delivery, and biological deactivation. Its structural composition is depicted in Figure 14.

With the overall quality of the two analyzed systems as the goal, a Multidisciplinary Design Optimization model formulation for the integrated in-vehicle environmental control system was constructed as follows.

(1) Air conditioning subsystem.

Design variables include expansion valve parameters ($x_{PR}$), condenser heat exchange ($x_{HE}$), condensing fan air volume ($x_{AirV}$), evaporator volume ($x_V$), compressor cooling capacity ($x_{cc}$), and the objective function. $M_1=m_1+m_2+m_3+m_4+m_5$ in the equation $M_1$ denotes the overall mass of the air conditioning subsystem, and $m_1, m_2, m_3, m_4, \mathrm{and} m_5$ denote the masses of the expansion valve, condenser, condenser fan, evaporator, and compressor, respectively. The empirical formula for the mass of each component and the design variables can be summarized as follows:

$$\begin{gathered} m_1=\alpha \cdot {x^2}_{PR} \\ {m}_2=k\cdot {\displaystyle \frac{\sqrt{x_{HE}}}{2}} \\ {m}_3=243.1\cdot x_{AirV}\cdot \eta \\ {m}_4=\rho {\displaystyle \frac{x_V}{64}} \\ {m}_5=\epsilon \cdot x_{cc} \end{gathered}$$

(4)

where $\alpha ,k,\eta ,\rho , \mathrm{and} \epsilon$ are the design parameters, taking the values 5.5, 1.4, 62, 3.8, and 7. Taking the compressor displacement, $g_1$, to be greater than $260{ \mathrm{cm}}^2/\mathrm{r}$ and the air supply volume of the air conditioning, $g_2$, to be greater than $450{ \mathrm{cm}}^3/\min$ as the constraints, the disciplinary optimization expression of the air conditioning subsystem can be expressed as follows:

$$\begin{array}{c}Find:x_{PR},x_{HE},x_{AirV},x_V,x_{cc}\\ \mathit{min} M_1=m_1+m_2+m_3+m_4+m_5\\ \mathrm{s}.\mathrm{t}. \hspace{0.33em}{g}_1>260,\hspace{1em}{g}_2>450\end{array}$$

(5)

(2) Integrated air supply subsystem

The design variables of the integrated air supply subsystem include ventilation fan air pressure ($x_{WP}$), the air volume of the internal circulator ($x_{Vicm}$), integrated duct diameter ($x_d$), and dust collector volume ($x_{Vdc}$); the objective function selects the mass of the integrated air supply subsystem, whose empirical expression can be expressed as $M_2=0.45\mathit{exp}(x_{wp}-12)+x_{wp}{x}_{Vicm}^2+1.67x_d^2+14.94x_{vdc}$; the constrained air supply volume, $g_3$, is greater than $121{ \mathrm{cm}}^3/\min$, and the pressure relief velocity, $g_4$, is greater than $15 \mathrm{pa}/\min$. The optimization model of the integrated air supply subsystem can be expressed as follows:

$$\begin{array}{c}Find:x_{WP},x_{Vicm},x_d,x_{Vdc}\\ \mathit{min} M_2=0.45\mathit{exp} (x_{wp}-12)+x_{wp}{x}_{Vicm}^2+1.67x_d^2+14.94x_{vdc}\\ \mathrm{s}.\mathrm{t}. \hspace{0.33em}{g}_3>121,g_4>15\end{array}$$

(6)

A Multidisciplinary Design Optimization model for the integrated in-vehicle environmental control system was constructed based on the above information and the completed system design model. As shown in Figure 15, the system optimization model for the air conditioning subsystem includes the subsystem objective function, the optimization constraints, and the attribute value constraint blocks within the optimization objectives (optimization variables are not included here for clarity of display).

The optimization model of the air supply subsystem, as shown in Figure 16, includes the subsystem optimization objectives and two inequality constraints along with their design variables. The attributes contained within the inequality constraints include the constraint expression, the associated discipline, and upper and lower boundaries, among other details.

As mentioned in Section 4.1, each extended optimization constraint block and optimization objective block is associated with a parametric diagram for equation extraction. As illustrated in Figure 17, using the air supply subsystem’s optimization objectives as an example, binding connectors can be effectively used to bind the constraint parameters to the value properties of the modules.

Once the optimization models for the two subsystems are built, the defined optimization problem diagram can be utilized to capture the internal structure of the entire optimization problem based on the connections between variables. Figure 18 presents the defined optimization problem for the in-vehicle environment control integrated system, detailing the relationships between system-level optimization objectives, subsystem objectives, design variables, optimization constraints, and optimization objectives. For example, the overall optimization goal is composed of two subsystem objectives, with the subsystem objective $M_2$ associated with the variables $WP$, $Vicm$, $d$, and $Vdc$. Additionally, the compressor displacement constraint is related to the variables $HE$ and $WP$.

In order to parse the optimization information model constructed by SysML modeling tools such as MagicDraw, it is necessary to convert the file to an intermediate XML format. The XML file generated from the constructed SysML optimization model is shown in Figure 19. This XML file contains the complete optimization information, including information on optimization variables, optimization objectives, and optimization constraints. The optimization information in this XML format can be extracted to an optimization solver such as MATLAB for optimization solving.

5.3. Discussion

Table 4. Optimization results of MDO problem in-vehicle environmental control integrated system.

Discipline	Variable	Symbol	Variable Type	Range	Initial Value	CO	SysML-CEA	SysML-RBMDO
Air conditioning	Expansion valve parameters	$x_{PR}$	Discrete	{2,3,4}	2	3	2	2
	Condenser heat exchange	$x_{HE}$	Continuous	[5,10]	8	7.53	6.86	6.97
	Condensing fan air volume	$x_{AirV}$	Continuous	[50,80]	80	70.32	65.22	63.81
	Evaporator volume	$x_V$	Continuous	[60–100]	80	65.75	62.88	61.47
	Compressor cooling capacity	$x_{cc}$	Aleatory	$\mu =1.25$ $\sigma =2$	1.5	1.42	1.23	1.18
Integrated air supply	Ventilation fan air pressure	$x_{WP}$	Continuous	[120,500]	300	153.46	138.25	142.35
	Air volume of the internal circulator	$x_{Vicm}$	Continuous	[25,60]	50	38.10	35.24	34.17
	Integrated duct diameter	$x_d$	Continuous	[50,100]	100	67.24	65.78	63.58
	Dust collector volume	$x_{Vdc}$	Continuous	[30,55]	50	40.54	40.20	38.94
Mass	Total mass	M	Continuous	/	/	81.43	78.41	72.63

presents the optimization results for the traditional CO algorithm, the SysML-CEA algorithm, and the SysML-RBMDO approach applied to the in-vehicle environmental control integration system.

From Table 4, it is evident that the SysML-RBMDO method significantly outperforms both the CO and SysML-CEA algorithms in optimizing total mass. The SysML-RBMDO approach reduces the total mass by 8.80 kg compared to the CO algorithm, representing an improvement of approximately 10.8%. Compared to the SysML-CEA algorithm, the reduction is 5.78 kg, or approximately 7.4%.

This section discusses the implications of extending SysML for multidisciplinary optimization (MDO) and reflects on the effectiveness of the proposed modeling approach based on evaluation results and case studies. The extension of SysML to include optimization variables, constraints, and objectives represents a significant advancement in handling complex Multidisciplinary Design Optimization (MDO) problems [30]. Building upon previous research, which primarily focused on basic SysML applications, our work introduces comprehensive metamodels for optimization variables, constraints, and objectives. This extension provides a clearer and more structured framework, improving both the clarity and manageability of optimization issues. By enabling the precise representation of various optimization scenarios, our approach enhances the integration of system design with optimization processes, facilitating a more holistic and effective analysis across multiple disciplines.

Traditional optimization modeling methods often struggle with the integration and management of complex problems. Our research addresses these challenges by offering a more consistent and integrated SysML-based approach. Previous methods typically lacked the ability to effectively capture constraint expressions and interactions between variables [15]. In contrast, our use of SysML’s parametric diagrams simplifies the definition and analysis of optimization problems. Additionally, the implementation of an XML-based information extraction method in our approach boosts model flexibility and scalability, streamlining data processing and enabling efficient integration with other tools. This advancement reduces the complexity of model design and enhances the overall manageability of the optimization process, distinguishing our work from earlier approaches.

This study’s application of the SysML-based optimization model to the in-vehicle environmental control integration system validates the practical effectiveness of our proposed method. While previous research laid the groundwork for SysML applications, our work extends these foundations by defining detailed system-level optimization objectives and subsystem-specific goals. This approach demonstrates how effectively our method manages and optimizes various aspects of real-world engineering systems. The case study not only highlights the potential of the SysML-based approach in addressing practical engineering challenges but also suggests its applicability to other domains. By bridging theoretical models with practical applications, our research provides valuable insights that advance the field and offer practical solutions for complex engineering systems.

6. Conclusions

This paper presents a novel approach to multidisciplinary reliability design optimization (MDO) based on SysML, with a focus on extending SysML to handle optimization problems more effectively. The key contributions and findings of this study can be summarized as follows: (1) Extension of SysML for Optimization. The proposed extensions to SysML, including metamodels for optimization variables, constraints, and objectives, significantly enhance the language’s capability to represent complex optimization problems. These extensions provide a comprehensive framework for defining and analyzing various optimization scenarios, improving the integration of system design and optimization processes. (2) Formal Expression and Extraction Methods. The formal expression of MDO problems using extended SysML offers a structured and systematic approach to capturing the internal structure of optimization problems. The use of SysML parametric diagrams and XML-based extraction methods demonstrates a practical approach to managing and processing optimization data, facilitating better model organization and data integration. (3) Practical Application and Validation. The application of the SysML-based optimization model to the in-vehicle environmental control integration system validates the effectiveness of the proposed method. The case study confirms that the extended SysML framework can successfully manage system-level and subsystem-specific optimization objectives, illustrating its potential for practical use in complex engineering systems.

Despite the significant advantages of the proposed approach, there are limitations to consider. The complexity of extending SysML and the potential need for specialized tools to support extended metamodels may pose implementation challenges. Additionally, the effectiveness of the method in different application contexts and its performance with respect to larger-scale problems need further investigation. Future work should focus on refining SysML extensions, developing automated tools for model generation, and validating the approach with a wider range of case studies to ensure its reliability and effectiveness in broader applications. While the proposed approach offers significant benefits, several areas for further research and development are identified to enhance its effectiveness and applicability: (1) Refinement of SysML Extensions. Enhancing the flexibility and applicability of SysML extensions to accommodate a broader spectrum of optimization problems is crucial. This includes addressing dynamic constraints, non-linear relationships, and time-varying objectives. Investigating ways to make these extensions more adaptable to different engineering contexts will improve their utility and effectiveness. (2) Development of Automated Tools. There is a need for developing automated tools that can streamline the process of model generation, optimization setup, and real-time integration. These tools should facilitate easier creation and management of SysML-based optimization models, reducing manual effort and improving efficiency. Automation in model generation and solver integration can significantly enhance usability and reduce errors. (3) Expansion of Validation Efforts. To ensure the robustness and generalizability of the approach, further validation is necessary. This includes applying the SysML-based optimization model to a wider range of case studies, including larger-scale and more complex engineering problems. Such validation efforts will provide insights into the approach’s performance across different domains and its scalability. (4) Integration with Other Tools and Standards. Exploring the integration of SysML-based optimization models with other tools and data exchange standards is essential for improving interoperability. Investigating how the SysML framework can interact with various software tools, such as optimization solvers and simulation platforms, will enhance its practical applicability and streamline workflows. (5) Addressing Computational Challenges. As optimization problems grow in complexity, addressing computational challenges such as efficiency and scalability becomes critical. Future research should focus on developing techniques and algorithms that can handle large-scale optimization problems effectively, ensuring that the proposed approach remains viable for complex and high-dimensional systems. In conclusion, the SysML-based approach to MDO represents a significant advancement in managing complex optimization problems. By addressing these future research directions, we aim to further refine and enhance the approach, ensuring its continued relevance and applicability in diverse engineering fields.