Interaction balance optimization in multidisciplinary design optimization problems

Abstract

As a bi-level optimization method, collaborative optimization can solve multidisciplinary design optimization problems in practical engineering effectively. However, if there are high-dimensional couplings in a multidisciplinary design optimization problem, a large number of compatibility constraints will be required in collaborative optimization. In this situation, collaborative optimization will not be suitable to be utilized because of low computational efficiency or divergence issue. To solve this problem, an efficient interaction balance optimization method is proposed in this article. In interaction balance optimization method, the simple coordination strategy of interaction balance principle and the distributed optimization strategy of collaborative optimization can be integrated effectively. Lagrange multipliers are used instead of compatibility constraints to maintain the consistency between any two coupled disciplines. Two examples are given to show the effectiveness of the proposed method.

Keywords

collaborative optimization multidisciplinary design optimization compatibility constraint interaction balanced optimization Lagrange multiplier

Introduction

With the development of engineering systems, concurrent engineering (CE) has received increasing attention. Recently, many CE technologies have been developed to solve integrated, simultaneous design problems of engineering systems (Chhabra and Emami, 2014; Deshmukh et al., 2013; Huang et al., 2011; Hwang et al., 2014; Meng et al., 2014; Ostrosi et al., 2014; Zhang et al., 2011, 2013). As a specific implementation of CE technologies, multidisciplinary design optimization (MDO) is a powerful method to design complex and coupled engineering systems (Huang et al., 2011; Hwang et al., 2014; Meng et al., 2014; Sobieszczanski-Sobieski and Haftka, 1997; Zhang et al., 2011, 2013). In MDO, a cost-type objective function is minimized while multidisciplinary constraints are satisfied. Furthermore, a coordination strategy is used to make interaction variables (including shared design variables and linking variables) satisfying the consistency requirements between any two coupled disciplines (Jeremy and Panos, 2005). Based on the coordination strategies, MDO methods can be categorized into two types: single-level methods and multilevel methods (Balling and Wilkinson, 1997; Chen et al., 2002; Hulme and Bloebaum, 2000; Martins and Lambe, 2013; Tosserams et al., 2009). Single-level methods generally have a single optimizer and use a non-hierarchical structure directly. Multidisciplinary feasible (MDF; Cramer, 1992; Lambe and Martins, 2012), individual discipline feasible (IDF; Cramer et al., 1994; Dener and Hicken, 2014; Kodiyalam and Sobieszczanski-Sobieski, 2001; Yu et al., 2014), and all-at-once (AAO; Haftka, 1994; Roshanian et al., 2014) are single-level methods. Multilevel methods modify the relationship of a non-hierarchical structure into a hierarchical structure. Generally, they have system and subsystem levels. Each level has an optimizer. Therefore, each discipline has a separate analyzer and can determine design solutions. Concurrent subspace optimization (CSSO; Liang et al., 2012; Renaud and Gabriele, 1994; Wujek et al., 1996; Zhang and Tang, 2015), bi-level integrated system synthesis (BLISS; Barthelemy and Sobieszczanski-Sobieski, 1983; Sobieszczanski-Sobieski et al., 2003; Yi et al., 2008; Zhao and Cui, 2011; Zhou et al., 2014), and collaborative optimization (CO; Alexandrov and Lewis, 2002; Braun, 1996; Braun and Kroo, 1995; Li et al., 2014; Tappeta and Renaud, 1997; Zhao et al., 2012) are multilevel methods.

Although many methods have been proposed, the developments of MDO are still necessary. On one hand, the development of engineering systems may make the existing MDO methods difficult to deal with all new challenges efficiently. On the other hand, because of the limited time and cost, designers have to select an appropriate method among various MDO methods to solve their problems. More new MDO methods should be developed to provide more options for designers. For example, the CO method is suitable to solve a MDO problem for the distributed engineering system with low couplings. However, if the distributed engineering system is large scale, there will be a large number of interaction variables in the corresponding MDO model. In this situation, CO method will be low efficient or even cannot converge.

This work is aimed to address the low-efficient issue of CO. We will focus on the situation where the architecture of engineering system is distributed and CO is feasible to use. The question is how can we enhance the efficiency of CO to solve MDO problems for large-scale engineering systems? To answer the question, we integrate the coordination strategy of interaction balance principle (IBP) and the collaborative mechanism of CO, and propose a new MDO method, which is named as interaction balance optimization (IBO).

This article is organized as follows. In section “The formulation of MDO,” the general MDO formulation is given. In section “The CO method,” the CO method is briefly reviewed. In section “IBO,” the basic idea of IBO is discussed in detail. Then the solution procedure of IBO is proposed. In section “Examples,” a numerical example and an aircraft conceptual design problem are used to show the effectiveness of the proposed method. Some conclusions are presented in section “Conclusion.”

The formulation of MDO

In this study, we consider the case that the system objective is a linear sum of all discipline objectives. Then a MDO problem can be formulated as

\begin{matrix} min_{X_{s}, X_{i}, Y_{ji}, Y_{ij}} f = \sum_{i = 1}^{n} f_{i} (X_{s}, X_{i}, Y_{ji}, Y_{ij}) \\ s . t . g_{i} (X_{s}, X_{i}, Y_{ji}, Y_{ij}) > 0 \\ h_{i} (X_{s}, X_{i}, Y_{ji}, Y_{ij}) = 0 \\ X_{i}^{L} \leq X_{i} \leq X_{i}^{U}, X_{s}^{L} \leq X_{s} \leq X_{s}^{U} \\ Y_{ij}^{L} \leq Y_{ij} \leq Y_{ij}^{U}, Y_{ji}^{L} \leq Y_{ji} \leq Y_{ji}^{U}, \\ i, j = 1, 2, \dots, n (i \neq j) \end{matrix}

(1)

where $f (•)$ is the system objective; $f_{i} (•)$ is the ith discipline objective; $g_{i} (•)$ is the vector of inequality constraints in the ith discipline, $g_{i} (•) > 0$ denotes the feasible region; $h_{i} (•)$ is the vector of equality constraints in the ith discipline; $X_{s}$ is the vector of shared design variables; $X_{i}$ is the vector of local design variables in the ith discipline; $Y$ is the vector of linking variables, $Y_{ij}$ denotes the inputs of the jth discipline and the outputs of the ith discipline, $Y_{ji}$ denotes the inputs of the ith discipline and the outputs of the jth discipline, the relationship of $Y_{ij}$ and $Y_{ji}$ can be denoted as $Y_{ij} = Y_{ij} (X_{s}, X_{i}, Y_{ji})$ ; superscripts $L$ and $U$ denote the lower and upper bounds, respectively; $n$ is the total number of disciplines. The interactions between the coupled disciplines in a MDO problem is shown in Figure 1.

Figure 1.

Interactions between coupled disciplines.

The CO method

In MDO, the performance of an engineering system is driven not only by the performance of individual disciplines but also by their interactions. Considering these interactions in an optimization problem generally requires a special coordination strategy. In CO, a complex system is decomposed into multiple disciplines. Then, the original MDO problem in equation (1) is modified into a system optimization problem as

\begin{matrix} min_{X'_{s}, X'_{i}, Y'_{ji}, Y'_{ij}} f = \sum_{i = 1}^{n} f_{i} (X_{s}^{'}, X_{i}^{'}, Y_{ji}^{'}, Y_{ij}^{'}) \\ s . t . J_{i} = {(X_{i}^{'} - X_{i})}^{2} + {(X_{s}^{'} - X_{s})}^{2} \\ + {(Y_{ji}^{'} - Y_{ji})}^{2} + {(Y_{ij}^{'} - Y_{ij})}^{2} \leq ε, i, j = 1, 2, \dots, n (i \neq j) \end{matrix}

(2)

and some corresponding discipline optimization problems as

\begin{matrix} min_{X_{s}, X_{i}, Y_{ji}} & J_{i} = {(X_{i}^{'} - X_{i})}^{2} + {(X_{s}^{'} - X_{s})}^{2} \\ + {(Y_{ji}^{'} - Y_{ji})}^{2} + {(Y_{ij}^{'} - Y_{ij})}^{2} \\ s . t . & g_{i} (X_{s}, X_{i}, Y_{ji}, Y_{ij}) > 0 \\ h_{i} (X_{s}, X_{i}, Y_{ji}, Y_{ij}) = 0 \\ X_{i}^{L} \leq X_{i} \leq X_{i}^{U}, X_{s}^{L} \leq X_{s} \leq X_{s}^{U} \\ Y_{ij}^{L} \leq Y_{ij} \leq Y_{ij}^{U}, Y_{ji}^{L} \leq Y_{ji} \leq Y_{ji}^{U}, i, j = 1, 2, \dots, n (i \neq j) \end{matrix}

(3)

where $J_{i}$ is the compatibility constraint in the ith discipline and $ε$ is a very small positive number.

The system optimization problem in equation (2) manages the overall optimization process at system level. It minimizes the system objective $f$ with the results $(X_{s}, X_{i}, Y_{ji}, Y_{ij})$ from subsystem level and determines the target values $(X_{s}^{'}, X_{i}^{'}, Y_{ji}^{'}, Y_{ij}^{'})$ for discipline optimization problems. Each discipline optimization problem finds the values of $(X_{s}, X_{i}, Y_{ji}, Y_{ij})$ which can satisfy their own constraints while trying to match the target values from system level as closely as possible. The optimization strategy and the coordination strategy of CO are shown in Figures 2 and 3, respectively.

Figure 2.

Optimization strategy of CO.

Figure 3.

Coordination strategy of CO (Kroo, 2004).

The optimization strategy of CO fits the architecture of distributed engineering systems (Perez et al., 2004). Thus, it can be applied in practical easily. However, if there are a large number of interaction variables in a MDO problem, the CO method will be low efficient or even cannot converge (Kim, 2001). It is because that the high dimensionality of interaction variables makes it difficult or even impossible to satisfy compatibility constraints.

IBO

To solve the above problem in CO, IBP, a coordination method for large-scale systems, is developed in this study. Similar to CO, IBP decomposes a large-scale system into multiple subsystems and has system and subsystem levels (Amgai and Abdelwahed, 2014; Findeisen et al., 1980; Mehrotra and Abdelwahed, 2014; Sadati and Marvast, 2006). At subsystem level, each subsystem problem is solved concurrently while the Lagrange multipliers $λ$ are considered as coordination parameters. The system coordinator manipulates $λ$ at system level to manage the overall coordination process. The coordination strategy of IBP is shown in Figure 4.

Figure 4.

Coordination strategy of IBP.

We combine the coordination strategy of IBP and the distributed optimization strategy of CO and propose the IBO method to solve MDO problems. In IBO, we treat sharing design variable $X_{s}$ as linking variables which are denoted as $X_{s, ij}$ and $X_{s, ji}$ , respectively. The coupled relationship between $X_{s, ij}$ and $X_{s, ji}$ can be considered as $X_{s, ij} = X_{s, ji}$ . Moreover, we use $I$ to denote the interaction variables in a MDO problem, $I_{ij} = {X_{s, ij}, Y_{ij}}$ .

Then, based on the linear relationship $f = \sum_{i = 1}^{n} f_{i}$ , the MDO problem in equation (1) can be decomposed into multiple coupled discipline optimization problems as in equation (4)

\begin{matrix} min_{X_{i}, I_{ji}} & f_{i} = f_{i} (X_{i}, I_{ji}, I_{ij} (X_{i}, I_{ji})) \\ s . t . & g_{i} (X_{i}, I_{ji}, I_{ij} (X_{i}, I_{ji})) > 0 \\ h_{i} (X_{i}, I_{ji}, I_{ij} (X_{i}, I_{ji})) = 0 \\ X_{i}^{L} \leq X_{i} \leq X_{i}^{U}, I_{ji}^{L} \leq I_{ji} \leq I_{ji}^{U} \\ I_{ij}^{L} \leq I_{ij} (X_{i}, I_{ji}) \leq I_{ij}^{U}, i, j = 1, 2, \dots, n (i \neq j) \end{matrix}

(4)

In equation (4), the value of $I_{ij}$ can be obtained by the ith discipline analysis $I_{ij} = I_{ij} (X_{i}, I_{ji})$ . To control the discipline optimization process at subsystem level, we introduce $λ$ to discipline optimization problems. Moreover, the discipline Lagrange function $L_{i}$ which is corresponding to the original discipline objective $f_{i} (•)$ is constructed as

L_{i} (X_{i}, I_{ji}, I_{ij} (X_{i}, I_{ji})) = f_{i} (X_{i}, I_{ji}, I_{ij} (X_{i}, I_{ji})) + λ_{i}^{T} \times I_{ji} - \sum_{j = 1, j \neq i}^{n} λ_{j}^{T} \times I_{ij} (X_{i}, I_{ji})

(5)

Then, the discipline optimization problems in equation (4) are changed to

\begin{matrix} min_{X_{i}, I_{ji}} & L_{i} = L_{i} (X_{i}, I_{ji}, I_{ij} (X_{i}, I_{ji})) \\ s . t . & g_{i} (X_{i}, I_{ji}, I_{ij} (X_{i}, I_{ji})) > 0 \\ h_{i} (X_{i}, I_{ji}, I_{ij} (X_{i}, I_{ji})) = 0 \\ X_{i}^{L} \leq X_{i} \leq X_{i}^{U}, I_{ji}^{L} \leq I_{ji} \leq I_{ji}^{U} \\ I_{ij}^{L} \leq I_{ij} (X_{i}, I_{ji}) \leq I_{ij}^{U}, i, j = 1, 2, \dots, n (i \neq j) \end{matrix}

(6)

It should be noted that the isolated discipline optimization problems in equation (6) can be solved concurrently at subsystem level and $λ$ are considered as design parameters. At system level, we construct the system Lagrange function $L$ as a linear sum of the discipline Lagrange functions

\begin{matrix} L (X_{i}, I_{ji}, I_{ij}) & = \sum_{\underset{(i \neq j)}{i, j = 1}}^{n} L_{i} (X_{i}, I_{ji}, I_{ij}) = \sum_{i = 1}^{n} {f_{i} (X_{i}, I_{ji}, I_{ij}) + λ_{j}^{T} \times (I_{ij} - I_{ij} (X_{i}, I_{ji}))} \end{matrix}

(7)

In equation (6), each discipline optimization problem is independent. Thus

\sum_{i = 1}^{n} min L_{i} = min \sum_{i = 1}^{n} L_{i} = min L

(8)

Moreover, we denote the Lagrange dual function of the system Lagrange function in equation (7) as

ϕ (λ) = L (X_{i} (λ), I_{ji} (λ), I_{ij} (λ))

(9)

Then, the equivalence relationship in equation (10) can be obtained based on the Lagrange duality theorem if there is a saddle point in the system Lagrange function

min_{X_{i}, I_{ji}, I_{ij}} L (X_{i}, I_{ji}, I_{ij}) = \max_{λ} ϕ (λ)

(10)

Substituting equation (8) into equation (10), we obtain $\sum_{i = 1}^{n} min_{X_{i}, I_{ji}} L_{i} (X_{i}, I_{ji}) = \max_{λ} ϕ (λ)$ . We assume that equation (9) is derivable. Then $ϕ' (λ)$ will be equal to zero if the maximum of $ϕ (λ)$ is obtained, $ϕ' (λ) = I_{ij} (λ) - I_{ij} (X_{i} (λ), I_{ji} (λ)) = 0$ . Moreover, the system Lagrange function will be equal to the system objective function

\begin{array}{l} L (X_{i}, I_{j i}, I_{i j}) = \sum_{i = 1}^{n} {f_{i} (X_{i}, I_{j i}, I_{i j}) + λ_{j}^{T} \times (I_{i j} - I_{i j} (X_{i}, I_{j i}))} \\ = \sum_{i = 1}^{n} {f_{i} (X_{i}, I_{j i}, I_{i j})} = f (X_{i}, I_{j i}, I_{i j}) \end{array}

(11)

Thus, we can update $λ$ according to the solutions in equation (6) at system level and manage the discipline optimization problems at subsystem level. The consistency requirements between coupled disciplines will be satisfied if the optimum of system objective is obtained in IBO. The distributed optimization strategy of IBO is illustrated in Figure 5.

Figure 5.

Optimization strategy of IBO.

The detailed solution procedure of IBO is given as follows:

Step 1. Set the initial values for design variables $X_{i}^{(k - 1)}$ , $I_{ji}^{(k - 1)}$ , and $I_{ij}^{(k - 1)}$ ; the initial values for Lagrange multipliers $λ_{i}^{(k - 1)}$ ; and the cycle number $k = 1$ . To keep the original gradient information in the revised optimization problems, $λ_{i}^{(k - 1)}$ can be equal to zero at the beginning of optimization process.

Step 2. Solve the system optimization problem $\max_{λ} ϕ (λ)$ in equation (10) at system level. The design variables $X_{i}^{(k - 1)}$ , $I_{ji}^{(k - 1)}$ , and $I_{ij}^{(k - 1)}$ are taken as design parameters during the system optimization process. After the optimization convergences, the solutions $λ_{i}^{(k)}$ are sent to the discipline optimization problems at subsystem level.

Step 3. Solve the discipline optimization problems in equation (6) at subsystem level. Using the distributed optimization strategy of IBO, discipline optimizations can be performed concurrently. $λ_{i}^{(k)}$ are used as design parameters during the discipline optimization process. After the optimization convergences, the solutions $X^{(k)}$ and $I^{(k)}$ can be obtained.

Step 4. Check the convergence. Calculate $G = ‖ L (X_{i}^{(k)}, I_{ji}^{(k)}, I_{ij}^{(k)}) - L (X_{i}^{(k - 1)}, I_{ji}^{(k - 1)}, I_{ij}^{(k - 1)}) ‖$ , $i, j = 1, 2, \dots, n$ and $i \neq j$ . If $G \leq ε$ , $g > 0$ , $h = 0$ , and the value of system objective is stable, go to Step 5; otherwise, set $k = k + 1$ and go to Step 2.

Step 5. Stop the overall optimization process. Output the solutions $X^{(k)}$ and $I^{(k)}$ .

The flowchart of IBO is illustrated in Figure 6.

Figure 6.

Flowchart of IBO.

Examples

In this section, the effectiveness of the proposed method is illustrated by a numerical example and an aircraft conceptual design problem. We also use other multilevel MDO methods, CO, CSSO, and BLISS, as comparisons to solve these MDO problems. All optimization processes are conducted under the platform of ${iSIGHT}^{TM}$ .

A numerical example

A numerical example is provided to show the proposed method in detail. The formulation of this example is given in equation (12)

\begin{matrix} \min & f = {(y_{21} - 2)}^{2} + x_{1}^{2} + x_{2}^{2} + {(y_{12} - 1)}^{2} + x_{3}^{2} \\ s . t . & y_{12} = x_{1} - x_{2} + 2 y_{21}, y_{21} = x_{3} - y_{12} \\ - 1 \leq x_{1} \leq 1, - 1 \leq x_{2} \leq 1, 0 \leq x_{3} \leq 3 \\ 0 \leq y_{12} \leq 1, 0 \leq y_{21} \leq 2 \end{matrix}

(12)

where $f$ is the system objective; $x_{1}$ , $x_{2}$ , $x_{3}$ , $y_{12}$ , and $y_{21}$ are the design variables.

We modify the original optimization problem into a MDO problem including two coupled disciplines. The discipline optimization problems are given in equations (13) and (14), and the coupled relationship is shown in Figure 7.

Figure 7.

Coupled relationship of two disciplines.

The first discipline optimization problem is

\begin{matrix} \min_{x_{1}, x_{2}, y_{21}} & f_{1} = {(y_{21} - 2)}^{2} + x_{1}^{2} + x_{2}^{2} \\ s . t . & - 1 \leq x_{1} \leq 1, - 1 \leq x_{2} \leq 1 \\ 0 \leq y_{12} \leq 1, 0 \leq y_{21} \leq 2 \end{matrix}

(13)

where $y_{12} = x_{1} - x_{2} + 2 y_{21}$ .

The second discipline optimization problem is

\begin{matrix} \min_{x_{3}, y_{12}} & f_{2} = {(y_{12} - 1)}^{2} + x_{3}^{2} \\ s . t . & 0 \leq x_{3} \leq 3, 0 \leq y_{12} \leq 1, 0 \leq y_{21} \leq 2 \end{matrix}

(14)

where $y_{21} = x_{3} - y_{12}$ .

Using the distributed optimization strategy of IBO, there are a system optimization problem and two modified discipline optimization problems as follows.

The system optimization problem at system level is

\begin{matrix} max_{λ_{1}, λ_{2}} L (λ_{1}, λ_{2}) = {(y_{21} - 2)}^{2} + x_{1}^{2} + x_{2}^{2} + {(y_{12} - 1)}^{2} + x_{3}^{2} \\ + λ_{1} \times (y_{21} - (x_{3} - y_{12})) + λ_{2} \times \\ (y_{12} - (x_{1} - x_{2} + 2 \times y_{21})) \end{matrix}

(15)

The first modified discipline optimization problem at subsystem level is

\begin{matrix} \min_{x_{1}, x_{2}, y_{21}} L_{1} (x_{1}, x_{2}, y_{21}) = {(y_{21} - 2)}^{2} + x_{1}^{2} + x_{2}^{2} + λ_{1} \\ \times y_{21} - λ_{2} \times (x_{1} - x_{2} + 2 \times y_{21}) \\ s . t . - 1 \leq x_{1} \leq 1, - 1 \leq x_{2} \leq 1, 0 \leq x_{1} - x_{2} \\ + 2 \times y_{21} \leq 1, 0 \leq y_{21} \leq 2 \end{matrix}

(16)

The second modified discipline optimization problem at subsystem level is

\begin{matrix} \min_{x_{3}, y_{12}} L_{2} (x_{3}, y_{12}) = {(y_{12} - 1)}^{2} + x_{3}^{2} + λ_{2} \times y_{12} \\ - λ_{1} \times (x_{3} - y_{12}) \\ s . t . 0 \leq x_{3} \leq 3, 0 \leq y_{12} \leq 1, 0 \leq x_{3} - y_{12} \leq 2 \end{matrix}

(17)

We select two initial points to show the effectiveness of the proposed method. The first point $(x_{1}, x_{2}, x_{3}, y_{12}, y_{21}) = (0, 0, 0, 0, 0)$ is in the feasible region and the second point $(x_{1}, x_{2}, x_{3}, y_{12}, y_{21}) = (- 1, 1, - 1, 1, - 1)$ is in the non-feasible region. The initial Lagrange multipliers are $λ = (λ_{1}, λ_{2}) = (0, 0)$ . The solutions obtained by different methods are shown in Table 1. We can see that all methods can obtain reasonable solutions. Compared with CSSO and BLISS, CO and IBO enjoy the higher computational efficiency. This is because that both CSSO and BLISS use the system analysis to maintain the consistency between two coupled disciplines. The system analysis is effective, however less efficiency. On the other hand, instead of system analysis, compatibility constraints and Lagrange multipliers are used in CO and IBO, respectively, which can enhance the efficiency.

Table 1.

Solutions of the numerical example.

	The first point				The second point
	IBO	CO	CSSO	BLISS	IBO	CO	CSSO	BLISS
$x_{1}$	−0.2999	−0.3001	−0.2999	−0.2993	−0.3000	−0.3000	−0.3000	−0.2993
$x_{2}$	0.2995	0.2999	0.2995	0.2990	0.2995	0.2998	0.2995	0.2990
$x_{3}$	0.8989	0.8998	0.8989	0.9005	0.8988	0.8998	0.8988	0.9005
$y_{12}$	0.3996	0.3999	0.3995	0.3995	0.3995	0.3999	0.3994	0.4009
$y_{21}$	0.4995	0.4999	0.4994	0.4989	0.4995	0.4999	0.4994	0.4996
$f$	3.6005	3.6000	3.6000	3.5977	3.6005	3.6000	3.6000	3.6000
$Time (s)$	66	91	151	127	74	106	138	130

IBO: interaction balance optimization; CO: collaborative optimization; CSSO: concurrent subspace optimization; BLISS: bi-level integrated system synthesis.

Aircraft conceptual design problem

The second example is an aircraft conceptual design problem which is developed by the MDO research group at the University of Notre Dame (Agarwal et al., 2004). The optimization problem involves aerodynamic discipline, weight discipline, and performance discipline. The tightly coupled relationships among these disciplines are shown in Figure 8.

Figure 8.

Aircraft conceptual design problem.

There are five shared design variables $x_{1} ~ x_{4}, x_{7}$ , six linking variables $y_{1} ~ y_{6}$ , and two discipline design variables $x_{5}, x_{6}$ . The descriptions of the variables are given in Table 2.

Table 2.

List of variables in the aircraft conceptual design problem.

	Description (unit)		Description (unit)
$x_{1}$	Aspect ratio of the wing	$y_{1}$	Total aircraft wetted area $({ft}^{2})$
$x_{2}$	Wing area $({ft}^{2})$	$y_{2}$	Maximum lift to drag ratio
$x_{3}$	Fuselage length $(ft)$	$y_{3}$	Stall speed $(ft / s)$
$x_{4}$	Fuselage diameter $(ft)$	$y_{4}$	Aircraft range $(miles)$
$x_{5}$	Density of air at cruise altitude $({slug / ft}^{3})$	$y_{5}$	Gross take-off weight $(lbs)$
$x_{6}$	Cruise speed $(ft / s)$	$y_{6}$	Empty weight $(lbs)$
$x_{7}$	Fuel weight $(lbs)$	–	–

The optimal objective in this example is to determine the least gross take-off weight within the bounded design space subject to two performance constraints. The constraints are the range and stall speed of the aircraft. The optimization model of this example can be formulated as follows (Agarwal et al., 2004)

\begin{matrix} \min_{x_{1} ~ x_{7}} & f = Weight = y_{5} \\ s . t . & g_{1} = 1 - \frac{y_{3}}{V_{req}^{stall}} \geq 0 \\ g_{2} = 1 - \frac{{Range}_{req}}{y_{4}} \geq 0 \end{matrix}

(18)

where $V_{req}^{stall} = 70 ft / s$ and ${Range}_{req} = 560 miles$ . The solutions of this example from IBO, CO, CSSO, and BLISS are shown in Table 3. We can see that the similar results can be obtained using different MDO methods. IBO converges using 556 s, while CO, CSSO, and BLISS require 784, 920, and 874 s, respectively. The investigation shows that IBO can enjoy higher efficiency. CSSO and BLISS need longer computational time because of the system analysis. The extra computational burden is brought into MDO problems by system analysis, leading to low efficiency. Furthermore, the high dimensionality of interaction variables increases the non-linearity of compatibility constraints, which makes it difficult for CO to converge. Thus, the efficiency of CO is lower than IBO in this example.

Table 3.

Solutions of the aircraft conceptual design problem.

	IBO	CO	CSSO	BLISS		IBO	CO	CSSO	BLISS
$x_{1}$	5	5	5	5	$y_{1}$	711.48	710.68	709.14	711.11
$x_{2}$	176.62	177.14	175.93	176.28	$y_{2}$	10.997	11.021	10.975	10.819
$x_{3}$	20	20	20	20	$y_{3}$	70.002	70.014	69.994	70.000
$x_{4}$	4	4	4	4	$y_{4}$	561.76	560.91	560.47	562.01
$x_{5}$	0.0017	0.0015	0.0018	0.0014	$y_{5}$	1744.2	1738.6	1746.1	1744.8
$x_{6}$	200	200	200	200	$y_{6}$	1207.2	1201.6	1210.4	1207.8
$x_{7}$	142.88	141.79	143.02	140.57	$Time (s)$	556	784	920	874

IBO: interaction balance optimization; CO: collaborative optimization; CSSO: concurrent subspace optimization; BLISS: bi-level integrated system synthesis.

Conclusion

In this article, we propose an effective IBO method to solve MDO problems for large-scale distributed engineering systems. The main difference of IBO from CSSO and BLISS is that IBO eliminates system analysis. Thus, IBO can enjoy higher efficiency. The main difference of IBO from CO is that IBO uses Lagrange multipliers instead of compatibility constraints to maintain the consistency between coupled disciplines. Thus, IBO is more effective than CO to solve a MDO problem with high-dimensional couplings. Furthermore, the simple coordination strategy of IBP and the distributed optimization strategy of CO are integrated effectively in IBO. Thus, IBO fits the architecture of distributed engineering systems. Using the distributed optimization strategy of IBO and taking the advantage of advanced computational analysis tools, designers can simultaneously improve the design and reduce the time and cost of the design cycle. However, the assumption that the system objective is a linear sum of all discipline objectives may limit the application of IBO. In practical engineering, the relationship between system objective and discipline objectives are non-linear generally. To solve this problem, we will try to use powerful multi-objective optimization methods to enhance IBO in future works.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported partially by the National Natural Science Foundation of China under contract No. U1330130, the National High Technology Research and Development Program of China (863 Program) under contract No. 2007AA04Z403, the China Postdoctoral Science Foundation under contract No. 2015M580780 and the Fundamental Research Funds for the Central Universities under contract No. ZYGX2015 KYQD045.

Author biographies

Debiao Meng is currently working at University of Electronic Science and Technology of China (UESTC) as an Assistant Professor. He received his Ph.D. in Mechatronic Engineering from UESTC in 2014. His main research interests includes in the area of reliability based optimization, multidisciplinary design and optimization.

Xiaoling Zhang is currently working at UESTC as an Associate Professor. She received her Ph.D. in Mechatronic Engineering from UESTC in 2011. Her main research interests includes in the area of reliability based optimization.

Yuan-Jian Yang is currently a Ph.D. student at UESTC. His main research interests includes in the area of reliability based optimization.

Huanwei Xu is currently working at UESTC as an Associate Professor. He received his Ph.D. in Mechanical Engineering from Dalian University of Technology, China in 2009. His main research interests includes in the area of reliability based optimization.

Hong-Zhong Huang is currently working at UESTC as a Professor. He received his Ph.D. in Reliability Engineering from Shanghai Jiaotong University, China in 1999. His main research interests includes in the area of system reliability analysis, warranty, maintenance planning and optimization, and computational intelligence in product design.

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