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International Journal of Approximate Reasoning ••• (••••) •••–•••
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International Journal of Approximate Reasoning
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A decoupled design approach for complex systems under lack-of-knowledge uncertainty
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Technische Universität München, Arcisstrasse 21, 80333 Munich, Germany b Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom
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Marco Daub a , Fabian Duddeck a,b
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Article history: Received 7 April 2019 Received in revised form 4 September 2019 Accepted 13 January 2020 Available online xxxx Keywords: Epistemic uncertainty Systems engineering Early-phase Crashworthiness
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a b s t r a c t This paper introduces a special approach for the design of complex systems in early development stages accounting for lack-of-knowledge uncertainty. As complex systems have to be broken down into their components by a decoupling methodology, the presented work regards so-called box-shaped solution spaces as subsets of the total set of permissible designs not violating any design constraints. Hereby, the design variables are decoupled and their design intervals are maximized. Then, each aspect related to the corresponding interval can be studied independently in a subsequent development step by different stakeholders (design groups or designers). Especially in early design phases, the consideration of uncertainty is crucial; this is not so much related to aleatoric uncertainty as probability functions are often unavailable. More important and more difficult to handle is epistemic uncertainty, i.e. lack-of-knowledge uncertainty. Here, uncertainties which occur later in the development and the facts that the current design stage does not include smaller design features and that the available models represent only coarsely the later designs are important. This paper complements prior work by providing a complete methodology for relevant uncertainties. This includes uncertainties in controllable design variables as well as in uncontrollable parameters all captured by interval arithmetic. Furthermore, it extends existing worst-case approaches by best-case approaches. The user can now base design decisions on (a) a deterministic solution space without consideration of lack-of-knowledge uncertainties, (b) the same with consideration of uncertainties in uncontrollable parameters only or controllable variables only, or (c) the most complete approach where uncertainties in both controllable variables and uncontrollable parameters are considered. The corresponding scenarios are exemplified via examples from automotive engineering (design for crashworthiness). © 2020 Elsevier Inc. All rights reserved.
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1. Introduction
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A challenge in complex systems design is to break down large system problems into smaller problems which can then be solved independently. Such an approach regarding the decomposition of the design process, was introduced in [21]. Here, an optimal box-shaped component solution space is computed which provides intervals for the design variables.
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E-mail addresses:
[email protected] (M. Daub),
[email protected] (F. Duddeck). https://doi.org/10.1016/j.ijar.2020.01.006 0888-613X/© 2020 Elsevier Inc. All rights reserved.
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M. Daub, F. Duddeck / International Journal of Approximate Reasoning ••• (••••) •••–•••
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These intervals enable decoupled design decisions. If all design variable are selected within their corresponding intervals, the resulting system design fulfills all requirements on the complex system. Investigated applications of the method of box-shaped solution spaces include problems for crashworthiness, e.g., [7], and driving dynamics, e.g., [5], each yielding an efficient design process for the corresponding complex system. Further decoupled design approaches are for example presented in [6,13,20]. Moreover, uncertainty plays a major role in complex systems design and must be taken into account. Particular in early-phase development, lack-of-knowledge uncertainty is present where only little or no information on uncertainties is available. Using box-shaped solution spaces for decoupled design, uncertainties in design variables, which occur because of a low degree of detailedness of the design in early phase or variations in manufacturing, can be treated by targeting the most interior design, compare [18]. In addition, an approach to optimize the intervals of box-shaped solutions spaces depending on the available knowledge on these uncertainties is presented in [4]. It enhances the application of box-shaped solution spaces under uncertainty. Besides uncertainties in design variables, also referred to as controllable variables, uncertainties in uncontrollable parameters are of special interest in complex systems design [17]. These uncertainties arise for example from uncertainties in parameters where designers have limited control like variations in environmental or operation conditions. Ideas to include uncertainties in uncontrollable parameters into the framework of box-shaped solution spaces are presented in [3]. The goal of this paper is to provide a completed methodology for decoupled design using box-shaped solution spaces under lack-of-knowledge uncertainties. It addresses both uncertainties in uncontrollable parameters and controllable variables modeled as intervals. Here, it is distinguished between worst-case and best-case solution spaces, where worst-case solution spaces must account for all realizations of the uncertainties and best-case solution spaces must only guarantee the existence of one realization. The prior work in [3,4] can be assigned to worst-case solution spaces. In this paper, the two approaches are combined and complemented by best-case solution spaces which are more difficult to handle in general. The paper is organized as follows: Section 2 introduces basic definitions of complex systems design, presents the method of box-shaped solutions spaces for decoupled design, and gives an uncertainty quantification for complex systems. Section 3 proposes box-shaped worst- and best-case solution spaces, which enable decoupled design decisions under lack-of-knowledge uncertainties in uncontrollable parameters and controllable variables. The method is illustrated via simple examples. Section 4 applies the proposed method to a simple and a realistic problem in crash design demonstrating its effectiveness in engineering practice.
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In complex systems design, a design is sought for a given system model. This design is expressed as a d-dimensional vector x ∈ Rd . The entries of x are the design variables xi , i = 1, . . . , d. They are selected by one or multiple designers. Thus, these design variables are also called controllable variables. Besides controllable variables, there are uncontrollable parameters, see [17]. Those are collected in a q-dimensional vector p ∈ Rq where their values cannot be controlled by designers. Typical examples for uncontrollable parameters are systemspecific or environmental parameters. If design variables become fixed at during development, they can be also considered as uncontrollable parameters afterwards. The behavior of the system depends on both controllable variables and uncontrollable parameters. In particular, there are m constraints, which must be fulfilled in complex systems design that depend on x and p. Typically, the constraints consist of design space constraints and system performance requirements, [21]. Design space constraints usually stem from technical design limitations and system performance requirements from regulated or desired bounds on system performance functions. Together, the constraints can be expressed mathematically as g (x, p ) ≤ 0 where g are the constraint functions with g (x, p ) ∈ Rm . Hence, for given p ∈ Rq , the set of all permissible designs, i.e. the designs that fulfill the constraints, is
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c = {x ∈ Rd | g (x, p ) ≤ 0}.
(1)
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This set is referred to as complete solution space, cf. [21]. Other names for c are e.g., feasible solution set [16] or permissible design space [9]. Due to design space constraints, the complete solution space is bounded.
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2.2. Decoupled design
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2.1. Definitions
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2. Complex systems design
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Furthermore, complex systems design is concerned with reducing complexity in the design process. By decomposing the decision for a design x into d decoupled decisions for the design variables xi , i = 1, . . . , d, up to d designers (this includes persons or design groups) can be involved in the design process. In case of only one designer, x must be selected from c in order to be permissible. This paper limits its considerations to the case of d designers, where the designers should be able to select the corresponding design variable from intervals [xli , xui ] with xui ≥ xli , i = 1, . . . , d. The approach presented below was introduced in [21].
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Fig. 1. Complete solution space c , optimal box-shaped solution space [xl , xu ], and corresponding intervals [xl1 , xu1 ] and [xl2 , xu2 ].
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Note that for d designers more general one-dimensional sets could be taken into account as well. For designers addressing multiple design variables, yielding less than d designers, corresponding design variables may remain coupled. Thus, the designers select their associated design variables from multi-dimensional sets. Approaches to tackle this problem are proposed, e.g., in [2,5]. The methodology introduced in this paper can be transferred to these approaches in general. In order to guarantee that a permissible design x ∈ c is obtained for selected xi ∈ [xli , xui ], i = 1, . . . , d, a necessary and sufficient condition is that
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(2)
where [xl , xu ] = [xl1 , xu1 ] × · · · × [xdl , xdu ]. A set [xl , xu ] which fulfills condition (2) is called a box-shaped solution space. The
corresponding intervals [xli , xui ], i = 1, . . . , d, enable decoupled design decisions among the designers, which can be made independently. In general, there are infinite options to choose box-shaped solution spaces. Among these, an optimal [xl , xu ] which provides the most flexibility in selecting designs is preferred, i.e. wide ranges for design decisions done later in the development process. This can be quantified by the volume of [xl , xu ] where
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holds, see [15]. The larger this volume, the greater the possibilities for the designers to select xi ∈ can be expressed as a mathematical optimization problem
maximize
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x ≤x , ∀x ∈ [xl , xu ] : x ∈ c .
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A solution of problem (4) forms an optimal box-shaped solution space [xl , xu ]. In literature, various approaches to solve problem (4) can be found, cf. [10,11,16,18,21]. Depending on the constraints functions, the proposed algorithms differ. In Subsection 3.3, an approach to solve problem (4) is presented for constraint functions which are linear in x for fixed p ∈ Rq . In Fig. 1, an example of an optimal box-shaped solution space is shown. Note that single intervals of [xl , xu ] might become small compared to others when solving problem (4). If no uncertainties in the corresponding design variables are present, this is negligible in favor of overall design flexibility. The presence of uncertainties in design variables is considered in [4] and is addressed in Subsection 3.2 of this paper. Moreover, other objective functions for problem (4) are also conceivable like maximizing the weighted minimal interval width of [xl , xu ] like proposed in [8]. A major advantage of using box-shaped solution spaces for complex systems design is that selected design variables can be adapted a-posteriori. This arises from the independence in decoupled design decisions. However, as uncertainty in complex systems design is usually present, a realized design might be non-permissible. The consequences for a decoupled design are discussed subsequently.
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2.3. Uncertainty quantification
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There are various ways to classify or quantify uncertainty, [1]. One classification approach is to distinguish between uncertainties from a system point of view. Here, uncertainties can be found for example in design variables, uncontrollable parameters, and the constraints. Uncertainties in design variables occur because of a low degree of detailedness of the design or if they cannot be realized accurately. This is for example due to lack of knowledge on geometrical model details in early development stages or imprecise or inaccurate manufacturing in later development stages. Examples for uncertainties in uncontrollable parameters are variations in environmental or operating conditions. Uncertainties in the constraints may result from the approximations of real physical objects in order to use models, also referred to as model uncertainty. In addition, also new constraints can occur during the development process and old ones might be adapted or suspended. Further classification schemes distinguish for example between aleatoric or epistemic uncertainty, [19]. In this work, the early-phase of complex systems design is considered where usually no or only limited information about uncertainties is available. This is mainly due to imprecise or incomplete knowledge, i.e. epistemic or lack-of-knowledge uncertainty. Therefore, a proper mathematical quantification is not possible in most cases. However, if it can be ensured that the realized values are at least bounded, uncertainties can be modeled as intervals. For uncontrollable parameters, known nominal values pˇ l ∈ R, l = 1, . . . , q, and known interval widths are assumed. It holds
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pl ∈ [ pˇ l − γl , pˇ l + γl ],
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(6)
where δi ∈ R≥0 , i = 1, . . . , d, are fixed. Note that this simplification for the interval widths can be critically examined in general, however it can be also used to represent, e.g., an average or lower or upper bounds. Approaches to handle unknown δi , i = 1, . . . , d are proposed for example in [4,12]. Using box-shaped solution spaces for a decoupled design of complex systems, some of the above mentioned uncertainties can be treated without a separate uncertainty consideration. If a constraint depends only on one single design variable xi , i ∈ {1, . . . , d}, uncertainties due to changes in this constraint can be treated for example by updating the corresponding interval [xli , xui ]. Hence, only the corresponding design variable xi must be revised a-posteriori and not the whole design of the complex system. Furthermore, large uncertainties in controllable variables can be treated by selecting target design variables in the center of the intervals [xli , xui ], i = 1, . . . , d. More general, box-shaped solution spaces under uncertainties in controllable variables are investigated in [4]. In contrast to this, uncertainties in uncontrollable parameters are more difficult to handle as the intervals for the controllable variables [xli , xui ], i = 1, . . . , d, do not carry information about the uncontrollable parameters, i.e., p. Hence, the main focus of this paper is put on uncertainties in uncontrollable parameters for the computation of box-shaped solution spaces. This complements the achievements of the earlier publications [3,4,21].
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c,wc = {x ∈ Rd | ∀ p ∈ [ pˇ − γ , pˇ + γ ] : g (x, p ) ≤ 0}
(7)
and is called worst-case complete solution space. Regardless which values p ∈ [ pˇ − γ , pˇ + γ ] are assumed, any design in c,wc can be realized exactly due to the assumed absence of uncertainties in controllable variables. Now, decoupled design decisions in a worst-case scenario can be enabled similar to the approach in Subsection 2.2 where the mathematical optimization problem
maximize
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For interval-type uncertainty, it must be established how constraints can be interpreted. The standard approach is to consider hard constraints what corresponds to a worst-case scenario. If, uncertainties in controllable variables are neglected, the constraints are fulfilled for a design x ∈ Rd if they hold for all p ∈ [ pˇ − γ , pˇ + γ ] where pˇ = ( pˇ 1 , . . . , pˇ q ) and γ = (γ1 , . . . , γq ). The set of all designs that fulfill g (x, p ) ≤ 0 for all p ∈ [ pˇ − γ , pˇ + γ ] is
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Fig. 2. Complete worst- and best-case solution spaces c,wc and c,bc , corresponding optimal box-shaped solution spaces, and their intervals.
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must be solved. A solution of problem (8) forms an optimal box-shaped worst-case solution space [xlwc , xuwc ], cf. [3]. For constraints which are linear in x and monotonic in p, problem (8) can be reduced to a linear optimization problem, see Subsection 3.3. In contrast to conservative hard constraints, a best-case scenario is considered in this paper where at least one p ∈ [ pˇ − γ , pˇ + γ ] exists for a design x ∈ Rd such that the constraints are fulfilled. The set of all designs that fulfill g (x, p ) ≤ 0 for at least one p ∈ [ pˇ − γ , pˇ + γ ] is
(9)
[xlbc , xubc ],
and is called best-case complete solution space. In order to compute an optimal box-shaped best-case solution space problem (8) can be solved where c,wc is replaced with c,bc and any other index wc with bc accordingly. However, by using problem (8) to compute both optimal box-shaped worst- and best-case solution spaces, it might be the case that [xlwc , xuwc ] [xlbc , xubc ] holds although c,wc ⊆ c,bc is always guaranteed. This stems from the objective to maximize flexibility in selecting designs, which is done independently for the worst- and best-case. An option to overcome this issue is to introduce the condition [xlwc , xuwc ] ⊆ [xlbc , xubc ], i.e.,
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(10)
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[xlwc , xuwc ]
In this case however, it is not ensured that is non-empty, even if c,wc is non-empty. In Fig. 2, an example of optimal box-shaped worst- and best-case solution spaces are shown. The exemplary intervals of the box-shaped best- and worst-case solution spaces from Fig. 2 can now be used for a decoupled design under uncertainties with respect to uncontrollable parameters. It is visualized where design variables that are permissible for all p ∈ [ pˇ − γ , pˇ + γ ], i.e., xi ∈ [xlwc,i , xuwc,i ], i = 1, 2, and that are permissible for at least one
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selected. Starting from
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comparing c,wc and c,bc .
], i.e., xi ∈ [xlwc,i , xuwc,i ], i = 1, 2, can be x1 for [xlbc , xubc ]. This becomes clear when
[xlwc , xuwc ],
more flexibility is added in
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3.2. Additional uncertainties in controllable variables
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c,bc = {x ∈ Rd | ∃ p ∈ [ pˇ − γ , pˇ + γ ] : g (x, p ) ≤ 0}
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(11)
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Then, decoupled design decisions for target designs can be enabled similar like before. The corresponding mathematical optimization problem reads
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ˇ c,wc and ˇ c,bc of target designs and corresponding optimal box-shaped solution spaces and their Fig. 3. Complete worst- and best-case solution spaces intervals.
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maximize xˇ lwc ,ˇxuwc
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A solution of problem (13) forms an optimal box-shaped worst-case solution space of target designs. Note that flexibility in selecting target designs is maximized in problem (13). In Subsection 3.3, it is described how this problem can be solved for constraints, which are linear in x and monotonic in p. When [ˇxlwc , xˇ uwc ] is available, [xlwc , xuwc ] can be obtained by
[xlwc , xuwc ] = [ˇxlwc − δ, xˇ uwc + δ].
= {ˇx ∈ R | ∃x ∈ [ˇx − δ, xˇ + δ] ∃ p ∈ [ pˇ − γ , pˇ + γ ] : g (x, p ) ≤ 0}. d
(15) (16)
ˇ c,wc with ˇ c,bc in problem (13) and any other index wc accordingly, an optimal box-shaped best-case solution By replacing space of target designs [ˇxlbc , xˇ ubc ] can be computed. When [ˇxlbc , xˇ ubc ] is available, [xlbc , xubc ] can be obtained by
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[xlbc , xubc ] = [ˇxlbc + δ, xˇ ubc − δ].
(17)
Again, it might be the case that [xlwc , xuwc ] [xlbc , xubc ] holds. This problem can be overcome by introducing condition (10)
into the computation of [ˇxlbc , xˇ ubc ]. In Fig. 3, an example of optimal box-shaped worst- and best-case solution spaces of target designs is shown. Again, the intervals of the solution spaces from Fig. 3 can now be used for a decoupled design under uncertainties. Additional to Fig. 2, there is information about the worst- and best-case solution spaces of target designs. In the example chosen, the intervals for [xlwc , xuwc ] in Fig. 3 are larger in x1 -direction and smaller in x2 -direction compared to Fig. 2. This is due to larger uncertainty in x1 than x2 , expressed by δ1 and δ2 . For [xlbc , xubc ], this effect is reversed where δ1 > δ2 provides smaller intervals in x1 -direction and larger intervals in x2 -direction. However, due to the relatively flat upper right constraint in Fig. 3, this effect is balanced and the intervals for [xlbc , xubc ] in Fig. 2 and Fig. 3 are almost the same.
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3.3. Simplifications for monotonic constraints
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Similarly in a best-case, the set of all target designs can be determined where for any target design xˇ at least one x ∈ c,bc exists with x ∈ [ˇx − δ, xˇ + δ]. This set is called best-case complete solution space of target designs and is denoted by ˇ c,bc , the constraints g (x, p ) ≤ 0 are fulfilled for at least one p ∈ [ pˇ − γ , pˇ + γ ] and one x ∈ [ˇx − δ, xˇ + δ]. ˇ c,bc . For every xˇ ∈ It holds
ˇ c,bc = {ˇx ∈ Rd | ∃x ∈ [ˇx − δ, xˇ + δ] : x ∈ c,bc }
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For arbitrary non-linear constraints, the problem of computing box-shaped solution spaces can be tackled by a stochastic algorithm, cf. [10]. Starting from an initial candidate box-shaped solution space, its volume is enlarged based on a Monte
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Carlo sampling in its neighborhood. This is done iteratively until an optimal solution is found. However, the algorithm does not guarantee to converge to a local optimum and is computationally expensive. This aggravates if additionally uncertainties are incorporated which can be done for example by also using sampling strategies to represent the uncertainties modeled as intervals. In this subsection, a theorem is presented that helps in solving the optimization problems of this paper for constraints which are linear in x and monotonic in p. As shown in Section 4, there are relevant cases in engineering design. These are characterized by constraint functions g j (x, p ) of the form
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v bc,l =
(22)
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Theorem 1. Let the constraint functions g be linear in x and monotonic in pl , cf. Equations (18)-(20), where pˇ ∈ R and γ ∈ R≥0 are provided. Then, it is
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(a) : ∀x ∈ ⇔
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(23)
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for
aTj ( pˇ
xlwc , xuwc , xlbc , xubc
+ V bc γ
) W lj xlbc
∈ R with d
xlwc
≤
41 42
(b) : ∀x ∈ [xlbc , xubc ] : x ∈ c,bc
43
43
+ aTj ( pˇ
xuwc
and
+ V bc γ
xlbc
≤
xubc .
) W uj xubc
− b j ( pˇ + V bc γ ) ≤ 0, j = 1, . . . , m
The matrices
W lj ,
W uj ,
(24)
V wc , and V bc are defined by Equations (21) and (22).
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48
Proof. It holds
49 50
50
∀x ∈ [xlwc , xuwc ] : x ∈ c,wc
51 52
51
⇔ ∀x ∈ [xlwc , xuwc ] ∀ p ∈ [ pˇ − γ , pˇ + γ ] : aTj ( p ) x − b j ( p ) ≤ 0, j = 1, . . . , m
53 54
⇔
55 56
⇔
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max
x∈[xlwc ,xuwc ]
max
x∈[xlwc ,xuwc ]
max
p ∈[ pˇ −γ , pˇ +γ ]
aTj ( p ) x − b j ( p )
≤ 0, j = 1 , . . . , m
aTj ( pˇ + V wc γ ) x − b j ( pˇ + V wc γ ) ≤ 0, j = 1, . . . , m
⇔ aTj ( pˇ + V wc γ ) W lj xlwc + aTj ( pˇ + V wc γ ) W uj xuwc − b j ( pˇ + V wc γ ) ≤ 0, j = 1, . . . , m,
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8
32
−1 if g mon. increasing in pl , 1 if g mon. decreasing in pl . q
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6
26
0 if a j ,i ≤ 0, 1 if a j ,i > 0,
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5
22
∈ [ pˇ − γ , pˇ + γ ].
j = 1, . . . , m. Moreover, q × q-diagonal auxiliary matrices V wc and V bc are defined where the lth entries on the diagonals are given by
32
4
20
(20)
In order to simplify problem (8), d × d-diagonal auxiliary matrices W lj and W uj are defined where the ith entries on the diagonals are given by
3
19
holds for all x ∈ c,bc with l ∈ {1, . . . , q} where p and p
only differ in the lth entry and p , p
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2
18
pl ≤ pl
⇒ g (x, p ) ≥ g (x, p
)
21
1
16
(19)
and monotonically decreasing in pl if
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(18)
pl ≤ pl
⇒ g (x, p ) ≤ g (x, p
)
17 19
= aTj ( p ) x − b j ( p ),
j = 1, . . . , m where a j ( p ) is a d-dimensional vector with a j ( p ) = (a j ,1 ( p ), . . . , a j ,d ( p )), and b j ( p ) is a scalar. For fixed p ∈ R, the constraints g j (x, p ) ≤ 0 build a system of linear inequalities, i.e., A ( p )x ≤ b( p ). The rows of A ( p ) consist of a j ( p ) and the entries of b( p ) are the b j ( p ) respectively, j = 1, . . . , m. Furthermore, the constraint functions are said to be monotonically increasing in pl if
16 18
7
and
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∀x ∈ [xlbc , xubc ] : x ∈ c,bc
1 2 3 4 5 6 7 8 9 10 11
⇔ ∀x ∈ ⇔ ⇔
[xlbc , xubc ]
max
∃ p ∈ [ pˇ − γ , pˇ + γ ]
min
x∈[xlbc ,xubc ] p ∈[ pˇ −γ , pˇ +γ ]
max
x∈[xlbc ,xubc ]
1
: aTj ( p ) x − b j ( p )
2
≤ 0, j = 1 , . . . , m
3
aTj ( p ) x − b j ( p ) ≤ 0, j = 1, . . . , m
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aTj ( pˇ + V bc γ ) x − b j ( pˇ + V bc γ ) ≤ 0, j = 1, . . . , m
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⇔ aTj ( pˇ + V bc γ ) W lj xlbc + aTj ( pˇ + V bc γ ) W uj xubc − b j ( pˇ + V bc γ ) ≤ 0, j = 1, . . . , m, using the monotonic properties of the constraint functions g.
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2
11 12
12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
By substituting the constraints of problem (8) according to Theorem 1, a linear optimization problem is obtained, which can be solved numerically by standard linear optimization techniques. Note that problem (4) is a special case of problem (8) with p = pˇ and γ = 0. Hence, the results of Theorem 1 can be also used to solve problem (4). In order to compute box-shaped worst-case solution spaces of target designs, the equivalence
∀ˇx ∈
[ˇxlwc , xˇ uwc ]
ˇ c,wc : xˇ ∈
(25)
for provided δ ∈ Rd≥0 can be obtained similar to the proof of Theorem 1. For box-shaped best-case solution spaces of target designs however, a corresponding result usually does not hold because constraint functions which are linear in x are not necessarily monotonic in xi according to above definitions, i = 1, . . . , d. They are only component-wise monotonic in x, i.e. g j (x , p ) ≤ g j (x
, p ) or g j (x , p ) ≥ g j (x
, p ) for x ≤ x
where g j (x , p ) ≤ g j
(x
, p ) and g j (x , p ) ≥ g j
(x
, p ) for j = j
is possible. By neglecting this property and using a similar equivalence for the best-case scenario, an approximate set is obtained which contains the best-case solution space of target designs. Note that the results would be similar for box-shaped best-case solution spaces if g is only component-wise monotonic. This is the case in the following application. Nevertheless, such simplifications for the best-case scenario can be used as approximations which provide simple computation results. Furthermore, a box-shaped best-case solution with only permissible designs might be obtained as it is only a subset of the complete best-case solution space. Though, the designs of the computed box-shaped best-case solution space must be tested if they fulfill the constraints g (x, p ) ≤ 0 for at least one p ∈ [ pˇ − γ , pˇ + γ ] in doing so. Note that in contrast to this, component-wise monotonic behavior of g is not crucial for the computation of worst-case solution spaces.
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22 23 24 25 26 27 28 29 30 31 32 33 34 35
38
4.1. Simple problem
39 40
A vehicle that has to perform in a crash can be considered as a complex systems design problem. Here, the focus is put on a front crash scenario where the vehicle is driven against a rigid wall at full overlap with an initial velocity v 0 , like done in crash tests. Significant constraints that must be fulfilled concern with the energy absorption, the maximum acceleration, and the order of deformation of the components. First, a simple crash design problem, introduced in [21], is taken into account, where the vehicle’s front structure is modeled as two components and its mass m is lumped behind, see Fig. 4. It is assumed that the deformation behaviors of the components in horizontal direction are responsible for the crash performance, i.e., the fulfillment of the constraints. These deformation behaviors can be described by the force-deformation characteristics of the components which are regarded as the vehicle’s design variables. For reasons of simplicity constant force-deformation characteristics are considered for the components. In the crash case, the first component deforms between s0 and s1 by applying the force F 1 and the second component deforms between s1 and s2 by applying the force F 2 with respect to a local vehicle coordinate system. Here, s¯ 1 = s1 − s0 and s¯ 2 = s2 − s1 are the components’ possible deformation lengths in horizontal direction which are assumed to be uncontrollable and not dependent on F 1 and F 2 . Subsequently, the mentioned constraints for front crash are formulated with respect to F 1 and F 2 , cf. [21]. Together, they yield the complete solution space for the simple crash design problem.
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• Energy absorption: The complete impact energy must be absorbed, i.e.,
58 59
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37
57 58
17
36
4. Application to crash design
40 41
16
20
38 39
15
19
36 37
14
18
⇔ aTj ( pˇ + V wc γ ) W lj xˇ lwc − aTj ( pˇ + V wc γ ) W lj δ + aTj ( pˇ + V wc γ ) W uj xˇ uwc + aTj ( pˇ + V wc γ ) W uj δ − b j ( pˇ + V wc γ ) ≤ 0, j = 1, . . . , m
13
−¯s1 F 1 − s¯ 2 F 2 ≤ −
1 2
mv 20 .
(26)
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Table 1 Nominal values of the uncontrollable parameters for the simple crash design problem.
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quantity
s0
s1
s2
m
v0
19
value
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ac mm ms
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• Maximum acceleration: The acceleration of the vehicle must be upper bounded by a critical acceleration ac , i.e., F1 m
≤ ac ,
F2 m
≤ ac .
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(27)
• Order of deformation: The deformation of the components must be ordered beginning at the front of the vehicle, i.e., F 1 − F 2 ≤ 0.
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By providing intervals for the single design variables F 1 and F 2 , i.e., computing a box-shaped solution space [ F l , F u ] = [ F 1l , F 1u ] × [ F 2l , F 2u ], decoupled design decisions are enabled for the components. This is done under lack-of-knowledge uncertainty. The nominal values of the uncontrollable parameters are stated in Table 1. In the following, three different examples of uncertainties are considered.
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(a) No uncertainties: There are neither uncertainties in the uncontrollable parameters nor uncertainties in the forcedeformation characteristics of the components. (b) Uncertainties in uncontrollable parameters only: There are only uncertainties in uncontrollable parameters. It is assumed mm that γm = 50 kg, γ v 0 = 0.1 mm , and γac = 0.01 ms 2 hold. ms (c) Uncertainties in both uncontrollable parameters and controllable variables: In addition to the uncertainties given in (b), there are uncertainties in the force-deformation characteristics of the components. It is assumed that δ F 1 = δ F 2 = 25 kN holds.
35 36 37 38 39 40 41
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25 27
(28)
34 35
24
As the constraints are component-wise monotonic in the uncontrollable parameters and linear in the design variables, results from Subsection 3.3 can be used for computing an optimal box-shaped solution space [ F l , F u ] for (i), optimal boxu l l u shaped worst-case and best-case solution spaces [ F wc , F wc ] and [ F bc , F bc ] for (ii), and optimal box-shaped worst-case and
u l l u best-case solution spaces of target designs [ Fˇ wc , Fˇ wc ] and [ Fˇ bc , Fˇ bc ] for (iii). In Fig. 5, optimal box-shaped solution spaces and their corresponding intervals are shown. By using box-shaped worst- and best-case solution spaces for decoupled design decisions in complex systems design, l u it can be visualized where designs which are permissible for all uncertainty scenarios, i.e., within [ F wc , F wc ], and designs u l which are permissible for at least one uncertainty scenario, i.e., within [ F bc , F bc ], are located. This information is missing if uncertainties in uncontrollable parameters are present and only the nominal values are used for modeling the uncertainties, which then corresponds to the case in example (a). In case of both uncertainties in uncontrollable parameters and controlu l l u lable variables, the intervals of [ Fˇ wc , Fˇ wc ] and [ Fˇ bc , Fˇ bc ] provide additional information on where to select target designs; dashed lines in Fig. 5 (c). The intervals of the worst-case box-shaped solution space of example (b) compared to the corresponding intervals of example (c) are larger for F 1 and smaller for F 2 . This is due to the objective of enlarging flexibility in selecting target designs if uncertainties in the controllable variables are present, compare [4]. These results transfer to the best-case solution spaces u u l l l u as they must fulfill condition [ F wc , F wc ] ⊆ [ F bc , F bc ] which fixes F bc ,1 and F bc,2 here. Note that in general the opposite effect is observed for best-case solution spaces, as discussed in Subsection 3.2. Furthermore, the influence of componentwise-monotonic constraints in the uncontrollable parameters can be spotted in the upper left part of c,bc in examples (b) and (c). There is an extra constraint that ensures the existence of a mass m ∈ [1450 kg, 1550 kg] for which both constraints (26) and (27) can be fulfilled. Moreover, there are further additional constraints
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Fig. 5. Optimal box-shaped solution spaces within their complete solution spaces (left) and corresponding intervals (right) for the simple crash design problem.
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ˇ c,bc . In the computation of [ F l , F u ] and [ Fˇ l , Fˇ u ] following Subsection 3.3, they are all neglected. However, as the for bc bc bc bc designs of the computed best-case solution spaces for examples (b) and (c) do not violate these constraints, compare Fig. 5 (b) and (c), the neglection of the additional constraints is reasonable here. 4.2. Realistic problem
53 54 55 56 57 58 59 60 61
46 47 48 50 51
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Next, a realistic crash design problem is considered. Here, the model of the vehicle’s front structure consists of components that represent the real structural components which are relevant in a front crash scenario, see Fig. 6. In contrast to the simple problem, the components only deform partially here, illustrated by the lightly colored parts. The darkly colored parts illustrate their non-deformable lengths. In addition, a discrete mass distribution is assumed for the vehicle that approximates its real mass distribution. The plastic parts of the force-deformation characteristics of the components are regarded as the vehicle’s design. Note that in general force-deformation characteristics cannot be designed. They are responses of lower level design variables, like material or geometrical properties. Therefore, an appropriate calculation of the lower level design variables must follow in a second step which is not covered by this paper. The framework for these types of realistic crash design models was introduced in [7] and also considered in, e.g., [4,14].
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Fig. 6. Vehicle front structure with six crash relevant components. There are deformable parts of every component (light) and non-deformable parts (dark).
28 29 30 31 32 33 34 35 36
In order to formulate relevant constraints with respect to the force-deformation characteristics, the non-deformable lengths are removed from the structure. Hence, coordinates in geometry space which deform simultaneously are aligned vertically in deformation space, see Fig. 6. Like for the simple problem, the deformation lengths are considered as uncontrollable parameters here. The kth component of the vehicle deforms between sk0 and skend where sk0 , skend ∈ {s0 , s1 , s2 , s4 }, k = 1, . . . , 6. Thus, it holds F k (s) ≥ 0 if s ∈ [sk0 , skend ) and F k (s) = 0 else for the force-deformation characteristic of the kth component. The mass that
is located between the kth and (k + 1)th component in the same load path is denoted by mk , where mk ∈ {0, m1 , m2 }, k = 1, . . . , 5. Furthermore, F (s) denotes the sum of the deformation forces over all load paths and m∗ (s) the active mass of the vehicle at a position s ∈ [s0 , s4 ).
39
• Energy absorption: The associated constraint can be calculated using the integral of the sum of deformation forces over
44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
− s0
F (s) m∗ (s )
1
≤ − v 20 .
61
31 32 33 34 35 36 38
2
41
(29)
44
• Maximum acceleration: The acceleration is calculated by the sum of deformation forces over all load paths divided by the active mass. The corresponding constraint is
F (s) m∗ (s )
42 43 45 46 47
≤ ac
(30)
F k (s) −
48 49
for all s ∈ [s0 , s4 ). • Order of deformation: The corresponding constraints compare the deformation force of the kth component at any position s ∈ [sk0 , skend ) to the deformation force of the (k + 1)th component at sk0+1 by paying respect to the influence of the further deformation forces at position s. It is
mk m∗ (s )
F (s) ≤ F k+1 (sk0+1 )
50 51 52 53 54 55
(31)
for all s ∈ [sk0 , skend ) where the kth and (k + 1)th component must share the same load path, k = 1, . . . , 5.
59 60
30
40
s4
41 43
29
39
all load paths divided by the active mass from s0 to s4 . It reads
40 42
28
37
37 38
26 27
27
In order to provide decoupled design decisions for the components by applying the framework proposed in this work, the force-deformation characteristics are parametrized. This is done by dividing the deformation space between the start
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Table 2 Nominal values of the uncontrollable parameters for the realistic crash design problem.
2
quantity
s0
s1
s2
s3
s4
m
m3
m3
v0
value
0 mm
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15.6
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mm ms2
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Fig. 7. Intervals of the box-shaped solution spaces for the realistic crash design problem. The inner white regions represent the intervals of the worst-case solution space and together with the gray region, the intervals of the best-case solution space are represented. Moreover, the intervals of the worst-case solution space of target designs are bounded by the inner dashed lines and the intervals of the best-case solution space of target designs by the inner dashed lines.
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28 29 30 31
31 32
27
and end of deformation of the vehicle into 16 equidistant sections and modeling the force-deformation characteristics of the components as constant in each section. Thus, these 35 constant force-levels are the degrees of freedom of the force-deformation characteristics and form design variables F i , i = 1, . . . , 35, which must assume positive values. They are labeled numerically along the load paths from left to right, starting at the top load path. Defining s0 = 0 mm, it holds s4 s4 s4 s4 F k (s) = F i , s ∈ [(i + 3) 16 , (i + 4) 16 ) for the upper load path, s ∈ [(i − 13) 16 , (i − 12) 16 ) for the middle load path, and s4 s4 s ∈ [(i − 20) 16 , (i − 19) 16 ) for the lower load path. For piece-wise constant force-deformation characteristics, the constraints (29)-(31) can be formulated using these design variables, i.e., F i , i = 1, . . . , 35. Then, one inequality for constraint (29), 16 inequalities for constraint (30), and 15 inequalities for constraint (31) are yielded as they must hold for multiple positions s. This sums up to 32 inequalities in total. For the resulting system of inequalities, the underlying functions are both component-wise monotonic in the uncontrollable parameters and linear in the controllable variables. Note that a different parametrization of the force-deformation characteristics is also possible. Considering piece-wise constant force-deformation characteristics has the advantage that the intervals [ F il , F iu ] of a box-shaped solution space also determine lower and upper bounds for the real force-deformation characteristics of the vehicle at a corresponding position s that guarantee fulfilling the constraints (29)-(31), compare [7]. Conversely, force-deformation characteristics can be section-wise approximated by its mean, yielding piece-wise constant characteristics. In the following, the case where there are both uncertainties in uncontrollable parameters and controllable variables is considered, which illustrates the overall approach of this paper for complex systems design under lack-of-knowledge uncertainty. The nominal values of the uncontrollable parameters are stated in Table 2. It is assumed that γm1 = γm2 = 10 kg, mm γm3 = 50 kg, γ v 0 = 0.1 mm , and γac = 0.01 ms 2 hold for the uncertainties in the uncontrollable parameters, and δi = 15 kN for ms the first and forth, δi = 45 kN for the second, fifth, and sixth, and δi = 10 kN for the third component regarding uncertainties in controllable variables. Like for the simple example, the results from Subsection 3.3 can be used for computing the worst-case and best-case u l l u solution spaces of target designs [ Fˇ wc , Fˇ wc ] and [ Fˇ bc , Fˇ bc ] In Fig. 7, the corresponding intervals are shown. Again, it must be tested if for all designs within the box-shaped best-case solution space at least one p ∈ [ pˇ − γ , pˇ + γ ] exists such that the constraints g (x, p ) ≤ 0 are fulfilled. As this is the case, the approach taken in this paper is also reasonable for this crash design problem. In general, the interpretation of the intervals of the box-shaped solution spaces for the realistic crash design problem matches the discussions from above. The incorporation of uncertainties in uncontrollable parameters provides two different design regions, i.e., worst-case and best-case solution spaces, and the incorporation of uncertainties in controllable variables
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provides regions for the corresponding target designs by ensuring a minimal width for the intervals of the worst-case box-shaped solution space for F i , e.g., i = 6, 17, 22, 29. Here, it could be critically remarked that the lower bounds for the fifth and sixth component are non-viable in a realistic scenario. This could be avoided by additionally bounding the target design variables. 5. Conclusions
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In this paper, a new completed methodology for a decoupled design approach in complex systems design that allows considering the full set of lack-of-knowledge uncertainties is proposed. Prior work on worst-case box-shaped solution spaces [3,4] is complemented by best-case solution spaces. The former is on the conservative side and offers design support via a set of permissible designs, which allows decoupled development for complexity management accounting for the full intervals of the uncontrollable parameters describing the lack-of-knowledge uncertainties. The newly proposed complementary best-case scenario enlarges the intervals and therefore the design flexibility by assuming that it is sufficient that at least a single parameter value exists for all uncontrollable lack-of-knowledge parameters. The worst-case scenario may be used for cases where a feasible design is sought for whatever value of the lack-of-knowledge parameters and the best-case scenario is appropriate for cases where the current lack-of-knowledge parameters can be freely manipulated later in the development process, e.g., via more design details as provided by local structural reinforcements. The complete set of methods is illustrated via a simple and a realistic crash design problem. Besides, it can be transferred directly to a wider range of engineering applications addressing early phase development of complex systems. Declaration of competing interest
24
We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome. Acknowledgements
29
This work was supported by the SPP 1886 “Polymorphic uncertainty modeling for the numerical design of structures” of the German Research Foundation, DFG. References
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