1 Introduction

There are many approaches to the design of technical systems like vehicles and airplanes. One of them is designing the system as a whole by using classical optimization where a system performance function is minimized while fulfilling quantitative system requirements, see, e.g., (Avriel et al. 1973). However, if uncertainties are present, it is possible that the optimal design deteriorates, shows poor performance or requirement violation. This is especially the case if the design is selected in an early phase of development where epistemic uncertainties exist. Uncertainties are associated, for example, with controllable design variables and uncontrollable parameters, cf. Parkinson et al. (1993). Reasons for uncertainties in controllable variables are, for instance, variations from target design values due to manufacturing or more detailed design modifications in later development stages. Variations in parts of the system where the system designer has limited control, like changing operation conditions or modifications in the system not in the responsibility of the designer, can be considered as uncertainties in uncontrollable parameters. These circumstances may result in a time-consuming and cost-intensive design process.

In Beyer and Sendhoff (2007), a survey of various approaches, related to robust design, is given on how uncertainties can be treated in order to obtain an optimal system design. What these approaches have in common is that they are based on specific mathematical quantifications of uncertainty which are often non-available in the early phase of systems design. An approach that is not based on a mathematical quantification of uncertainty is presented in Hendrix et al. (1996). Here, a feasible design with maximum distance to the region that violates the system requirements is sought in order to tolerate maximum uncertainties in controllable variables. If the Chebyshev distance is used, a maximum box that is entirely contained in the permissible region is spanned around the robust design. In Zimmermann and von Hoessle (2013), a box inside the permissible region that is maximized with respect to its volume is called an optimal box-shaped solution space. Along with being able to encompass uncertainties, box-shaped solution spaces have the property to fully decouple system requirements as they consist of permissible intervals for each design variable. This allows a flexible and independent selection of design variables. Hence, this approach is capable of improving the design process regarding costs and time, cf. Zimmermann et al. (2017).

However, a major drawback of using box-shaped solution spaces is that many designs which do not violate the constraints cannot be found within box-shaped solution spaces. This limits the options for systems design. To reduce this loss, optimal component solution spaces are proposed. They are defined as the set of permissible design variable values associated with components. Each component specifies a group of design variables that may interact with each other. Like box-shaped solution spaces, component solution spaces are based on an optimal system requirement decomposition. However, the requirements are not fully decoupled for component solution spaces; they are only decoupled between components. This increases the available solution space and therefore the freedom for design work. Moreover, larger uncertainties can be treated.

The paper is organized as follows: In Section 2, basics used for systems design are defined and the method of optimal component solution spaces is introduced. A differentiation is made between box-shaped and arbitrarily shaped component solution spaces. Furthermore, numerical algorithms are proposed to obtain optimal component solution spaces. A simple crash design problem is described in Section 3 and optimal component solution spaces are computed and compared. In Section 4, the same is done for a realistic crash design problem. In Section 5, a conclusion is given.

2 Component solution spaces

2.1 Definitions

Assume a system designer’s perspective in the early development phase. Suppose that a system model exists; however, design variables have not yet assumed a fixed value. The system is composed of n components where each design variable belongs to one of the components. Thus, the k th component is described by dk-independent design variables \({x^{k}_{i}}\in \mathbb {R}\), i = 1,…,dk, which assigns dk degrees of freedom to the component. Its design variables are collected in a vector \(x^{k}\in \mathbb {R}^{d^{k}}\) and named component design of the k th component. Together, the component designs constitute the system design(x1,…,xn). In Fig. 1, such a system is visualized.

Fig. 1
figure 1

A system composed of n components each specified by a design variable vector xk, k = 1,…,n

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For given design variables, the system responses, denoted by \(z\in \mathbb {R}^{m}\), are determined uniquely by the system model. This relationship is expressed by the system performance function f where design variables serve as inputs and system responses as outputs, typically z = f(x1,…,xn).

In systems design, several constraints must be met. In general, constraints on design variables and on system responses are distinguished:

  • The set of system designs with \(x_{\text {ds},i}^{\mathrm {l},k} \leq {x_{i}^{k}} \leq x_{\text {ds},i}^{\mathrm {u},k}\) for i = 1,…,dk, k = 1,…,n is called design space Ωds of the system. Similarly, the design space for the k th component design is named \({\Omega }^{k}_{\text {ds}}\).

  • The system responses must fulfill performance requirements which can be expressed mathematically as fj(x1,…,xn) ≤ fc,j for j = 1,…,m. Here, only upper bounds are considered as thresholds because any lower bound can be transformed into an upper bound by multiplication with − 1.

A system design that satisfies both constraints is defined as permissible; otherwise, it is said to be non-permissible. The set of all permissible system designs is named complete system solution spaceΩc, i.e.,

$$ {\Omega}_{\mathrm{c}} = \{(x^{1}, \ldots, x^{n}) \in {\Omega}_{\text{ds}}:~ f(x^{1},\ldots,x^{n})\leq f_{\mathrm{c}} \}. $$
(1)

In the literature, Ωc is also called feasible solution set (Milanese et al. 1996) or permissible design space (Graf et al. 2018).

2.2 Problem statement

Systems design is concerned with enabling independent component development to reduce overall design complexity. This can be realized by decoupling the performance requirements for the system into requirements for the components yielding component solution spaces Ωk, k = 1,…,n. This means that instead of having one system designer who targets designs within the complete system solution space Ωc, there are n component designers who target component designs within component solution spaces Ωk, k = 1,…,n.

In order to guarantee that the component designers can work independently, it must be ensured that \((x^{1},\ldots , x^{n})\in {\Omega }_{\mathrm {c}}\) holds for xk ∈Ωk, k = 1,…,n. Therefore, it is necessary and sufficient that a system solution space formed by the Cartesian product of the single Ωk is a subset of Ωc, i.e.,

$$ {\Omega}^{1} \times {\cdots} \times {\Omega}^{n}\subseteq {\Omega}_{\mathrm{c}}. $$
(2)

The overall idea is visualized in Fig. 2.

Fig. 2
figure 2

Decomposition of the complete system solution space into component solution spaces

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Among all possibilities to choose component solution spaces, optimal component solution spaces that provide most the flexibility for component designers are preferred. Here, flexibility is quantified by an objective function, which measures the infinite amount of permissible system designs covered in Ω1 ×⋯ ×Ωn. This can be represented by the volume of Ω1 ×⋯ ×Ωn. The resulting mathematical optimization problem reads as follows:

$$ \begin{array}{ll} \underset{{\Omega}^{1}, \ldots, {\Omega}^{n}}{\text{maximize}}&\text{vol}({\Omega}^{1} \times {\cdots} \times {\Omega}^{n})\\ \mathrm{subject~to} & {\Omega}^{1} \times {\cdots} \times {\Omega}^{n} \subseteq{\Omega}_{\mathrm{c}}. \end{array} $$
(3)

In Fig. 3, an example of two optimal component solution spaces Ω1 and Ω2 is shown.

Fig. 3
figure 3

Example of optimal component solution spaces Ω1 and Ω2 for a system composed of two components with one degree of freedom each

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The particular challenge of solving problem (3) is that the optimization variables are sets instead of values in real coordinate space. An approach to tackle this problem is to simplify the problem such that a so-called design centering problem, see Harwood and Barton (2017), is obtained where the optimization variables are in real coordinate space.

2.3 Box-shaped component solution spaces

In this approach, component solution spaces are considered which are box-shaped, i.e., high-dimensional intervals. Box-shaped Ωk can be uniquely defined by the vectors \(x^{\mathrm {l},k}, x^{\mathrm {u},k}\in \mathbb {R}^{d^{k}}\) via

$$ {\Omega}^{k} = [x^{\mathrm{l},k},x^{\mathrm{u},k}], $$
(4)

where [xl,k, xu,k] is a dk-dimensional interval for k = 1,…,n, providing a simple and real-valued description for the component solution spaces. Thus, corresponding shape constraints must be included in problem (3), and the new optimization problem reads as follows:

$$ \begin{array}{ll} \underset{x^{\mathrm{l},k}, x^{\mathrm{u},k}}{\text{maximize}}&\prod\limits_{k=1}^{n} \prod\limits_{i=1}^{d^{k}} (x^{\mathrm{u},k}_{i}- x^{\mathrm{l},k}_{i})\\ \mathrm{subject~to} & [x^{\mathrm{l},1}, x^{\mathrm{u},1}] \times {\cdots} \times [x^{\mathrm{l},n}, x^{\mathrm{u},n}] \subset{\Omega}_{\mathrm{c}} \end{array} $$
(5)

where the optimization variables are in real coordinate space. Problem (5) is identical to the optimization problem for computing box-shaped solution spaces proposed in Zimmermann and von Hoessle (2013). Hence, the Cartesian product of box-shaped component solution spaces can always be considered as a box-shaped solution space in terms of Zimmermann and von Hoessle (2013) and the other way around. In the literature, the calculation of solutions to problems similar to problem (5) has been studied extensively and is used for systems design. Further examples can be found in Bemporad et al. (2004), Daub (2017), Fender (2013), Götz et al. (2012), Graf et al. (2018), Graff et al. (2016), Harwood and Barton (2017), and Milanese et al. (1996). The associated algorithms mainly differ depending on the type of the system performance function f, e.g., linear, non-linear, and black-box. Particularly, if the functions fj, j = 1,…,m are linear, i.e.,

$$ f(x^{1},\ldots,x^{n})=\sum\limits_{k=1}^{n} A^{k} x^{k} $$
(6)

with \(A^{k}\in \mathbb {R}^{m\times d^{k}}\), k = 1,…,n, problem (5) can be simplified, see Harwood and Barton (2017). Defining b = fc, it yields:

$$ \begin{array}{ll} \underset{x^{\mathrm{l},k}, x^{\mathrm{u},k}}{\text{maximize}}&\prod\limits_{k=1}^{n} \prod\limits_{i=1}^{d^{k}} (x^{\mathrm{u},k}_{i}- x^{\mathrm{l},k}_{i})\\ \mathrm{subject~to} & x^{\mathrm{l},k}\leq x^{\mathrm{u},k},~ -x^{\mathrm{l},k}\leq - x_{\text{ds}}^{\mathrm{l},k}, ~ x^{\mathrm{u},k}\leq x_{\text{ds}}^{\mathrm{u},k},~k=1,\ldots,n,\\ &\sum\limits_{k=1}^{n} (a_{j}^{k,\mathrm{T}} W_{j}^{\mathrm{l},k} x^{\mathrm{l},k}+a_{j}^{k,\mathrm{T}} W_{j}^{\mathrm{u},k} x^{\mathrm{u},k})\leq b_{j},~j=1,\ldots,m, \end{array} $$
(7)

where \(a_{j}^{k,\mathrm {T}}\) is the j th row of Ak and \(W_{j}^{\mathrm {l},k}\), \(W_{j}^{\mathrm {u},k}\) are diagonal dk × dk matrices where their i th entries on the diagonals are given by the following:

$$ w^{\mathrm{l},k}_{j,i}= \left\{\begin{array}{rl} 1&\text{for}~a_{j,i}^{k}\leq 0,\\ 0&\text{for}~a_{j,i}^{k}> 0 \end{array}\right., ~~~w^{\mathrm{u},k}_{j,i}= \left\{\begin{array}{rl} 0&\text{for}~a_{j,i}^{k}\leq 0,\\ 1&\text{for}~a_{j,i}^{k}> 0 \end{array}\right. $$
(8)

for i = 1,…,dk, k = 1,…,n, j = 1,…,m. Problem (7) can be solved numerically by using standard gradient-based optimization algorithms.

2.4 Arbitrarily shaped component solution spaces

In contrast to box-shaped component solution spaces, there are no shape constraints in this approach. Thus, arbitrary shapes can be assumed. If Ωk, k = 1,…,n, are represented by

$$ {\Omega}^{k} = \{x^{k}\in {\Omega}^{k}_{\text{ds}}:~ f^{k}(x^{k})\leq f^{k}_{\mathrm{c}} \}, $$
(9)

the goal is to find optimal functions fk and corresponding thresholds \(f^{k}_{\mathrm {c}}\) for k = 1,…,n to obtain an optimal solution of problem (3). The functions fk, k = 1,…,n can be interpreted as component performance functions that contribute to the total system performance. If the optimal functions fk are known a-priori, only the values of \(f^{k}_{\mathrm {c}}\) must be sought in order to obtain optimal component solution spaces. This, for example, is the case when the system performance function can be written in the form as follows:

$$ f(x^{1},\ldots,x^{n}) = \sum\limits_{k=1}^{n} f^{k}(x^{k}), $$
(10)

cf. Daub (2017). To express the dependency of Ωk on \(f^{k}_{\mathrm {c}}\) in here, the notation \({\Omega }^{k}(f^{k}_{\mathrm {c}})\) is used. Then, problem (3) reduces to the optimization problem as follows:

$$ \begin{array}{ll} \underset{f^{1}_{\mathrm{c}},\ldots, f^{n}_{\mathrm{c}}}{\text{maximize}}&\text{vol}({\Omega}^{1}(f^{1}_{\mathrm{c}})\times\cdots\times {\Omega}^{n}(f^{n}_{\mathrm{c}}))\\ \mathrm{subject~to} & \sum\limits_{k=1}^{n} f^{k}_{\mathrm{c}}\leq f_{\mathrm{c}}. \end{array} $$
(11)

Again, this is an optimization problem where the optimization variables are in real coordinate space. The optimization constraints guarantee that condition (2) is fulfilled. As the objective function increases if any entry of \(f^{k}_{\mathrm {c}}\), k = 1,…,n, is increased, a maximum of problem (11) is always obtained for \({\sum }_{k=1}^{n} f^{k}_{\mathrm {c}}= f_{\mathrm {c}}\). Note that linear system performance functions fj are incorporated in this approach as they can be written in the form of (10), compare equation (6). For more general system performance functions, a reduction of problem (3) is more difficult.

2.5 Numerical implementation

A challenge for solving problem (11) is the computation of the volume of \({\Omega }^{1}(f^{1}_{\mathrm {c}})\times \cdots \times {\Omega }^{n}(f^{n}_{\mathrm {c}})\). If the single dk-dimensional volumes of \({\Omega }^{k}({f_{\mathrm {c}}^{k}})\), k = 1,…,n are known, the total volume is obtained by multiplication. The single volumes can be approximated or calculated analytically:

  • Volume approximation: Monte Carlo integration can be used for an efficient approximation of the volume of \({\Omega }^{k}({f_{\mathrm {c}}^{k}})\), see (Evans and Swartz 2000). Therefore, Nt independent, uniformly distributed sample points are generated in \({\Omega }_{\text {ds}}^{k}\), where fk is evaluated at each sample point. If \(f^{k}(x^{k})\leq f^{k}_{\mathrm {c}}\) holds, the sample point xk is considered as permissible. Now, the volume of \({\Omega }^{k}({f_{\mathrm {c}}^{k}})\) can be calculated by dividing the number of permissible sample points, denoted by Np, by Nt, and multiplying with the volume of \({\Omega }_{\text {ds}}^{k}\), i.e.,

    $$ \text{vol}({\Omega}^{k}({f_{\mathrm{c}}^{k}})) = \frac{N_{\mathrm{p}}}{N_{\mathrm{t}}}\prod\limits_{i=1}^{d^{k}} (x^{\mathrm{u},k}_{\text{ds},i}-x^{\mathrm{l},k}_{\text{ds},i}). $$
    (12)

  • Exact volume computation: For linear performance functions, Ωk(bk), i.e.,

    $$ {\Omega}^{k}(b^{k}) = \{x^{k}\in {\Omega}^{k}_{\text{ds}}:~ A^{k} x^{k}\leq b^{k} \} $$
    (13)

    with \(b^{k}=f^{k}_{\mathrm {c}}\) is a polytope. Its volume can be computed exactly by using the method proposed in Lasserre (1983). Here, the volume of a polytope which is described by Ax ≤ b, \(A\in \mathbb {R}^{m\times d}\), \(b\in \mathbb {R}^{m}\), is computed via a recursion scheme. Let V (d,A,b) be the volume of this polytope, then:

    $$ V(d,A,b)=\frac{1}{d}\sum\limits_{j=1}^{m}\frac{b_{j}}{|a_{j,i(j)}|}V_{j}(d-1,\tilde{A}_{i(j)},\tilde{b}), $$
    (14)

    where \(V_{j}(d-1,\tilde {A}_{i(j)},\tilde {b})\) is the volume of the j th face, obtained by eliminating xi(j) in Ax ≤ b. For this purpose, \(a^{\mathrm {T}}_{i(j)}x=b_{i(j)}\) is used where \(a^{\mathrm {T}}_{i(j)}\) is the i th row of A which yields \(\tilde {A}_{i(j)}x\leq \tilde {b}\), \(\tilde {A}_{i(j)}\in \mathbb {R}^{(m-1)\times (d-1)}\), \(\tilde {b}\in \mathbb {R}^{m-1}\). Other methods to calculate the volume of a polytope require the calculating of its corner points, see Büeler et al. (2000) for an overview. For arbitrary functions fk, analytical formulae to calculate the volume of \({\Omega }^{k}({f_{\mathrm {c}}^{k}})\), k = 1,…,n are either not available or more complex.

In general, the objective function of problem (11) is not differentiable, see Erschen (2018) for an example. Therefore, non-gradient-based optimization algorithms such as direct search methods are preferred.

Suitable initial values \((f^{1}_{\mathrm {c},0},\ldots , f^{n}_{\mathrm {c},0})\) for problem (11) can be obtained by solving the minimax problem as follows:

$$ \begin{array}{ll} \underset{x^{1},\ldots, x^{n}}{\text{minimize}} & \max\{ f_{j}(x^{1},\ldots,x^{n})-f_{\mathrm{c},j}:~j=1,\ldots,m\} \\ \mathrm{subject~to}& x^{\mathrm{l},k}_{\text{ds}}\leq x^{k}\leq x^{\mathrm{u},k}_{\text{ds}},~k=1,\ldots,n. \end{array} $$
(15)

and setting \(f^{k}_{\mathrm {c},0}=f^{k}(x^{k})+\epsilon ^{k}\), \(\epsilon ^{k}\in \mathbb {R}^{m}_{+}\), k = 1,…,n with \(\sum \limits _{k=1}^{n} (f^{k}(x^{k})+\epsilon ^{k}) \leq f_{\mathrm {c}}\) for an optimal solution of problem (15). Note that problem (15) can be also used for choosing initial values for problem (5). Moreover, if a solution of problem (5) is already available, i.e., box-shaped component solution spaces, initial values for problem (11) can be also obtained by setting \(f^{k}_{\mathrm {c},j,0}=\max \limits \{{f_{j}^{k}}(x^{k}):~x^{\mathrm {l},k}\leq x^{k}\leq x^{\mathrm {u},k}\}\), j = 1,…,m, k = 1,…,n.

In the following two sections, optimal box-shaped and optimal arbitrarily shaped component are computed and compared for crash design problems. In doing so, the numerical approaches presented in this paper are used.

3 Application to a simple crash design problem

3.1 System specifications

The problem of designing a vehicle front structure consisting of two sections modeled as components is considered, compare (Zimmermann and von Hoessle 2013). The problem is visualized in Fig. 4. The vehicle mass m is lumped at the end of the two front end components. Together with the deformation length of the components \(\bar {s}^{1}\) and \(\bar {s}^{2}\), they are assumed to be uncontrollable parameters. They are assumed without deeper knowledge for the early design stage and define the frame of the first decisions with m = 1500kg, \(\bar {s}^{1}=\bar {s}^{2}=300 \text {mm}\). Further detailed properties of the components still need to be chosen such that the overall vehicle meets crash requirements. These are formulated with respect to quantities measured in crash tests. For example, in the US-NCAP crash load case, a vehicle is driven against a rigid barrier with a velocity of \(v_{0}=15.6 \mathrm {\frac {mm}{ms}}\). Here, there are requirements on the system performances concerning the absorbed energy, the maximum acceleration of the vehicle, and the progressive order of deformation, see Fender (2013). All of these responses can be computed from force–deformation characteristics which are functional properties of the two components and represent the force levels required for plastic deformation. Design variables are used to define the force–deformation characteristics F1(s) for \(s\in [0 \text {mm},\bar {s}^{1})\) and F2(s) for \(s\in [0 \text {mm},\bar {s}^{2})\). Note that in this simple example, the mass m.

Fig. 4
figure 4

Vehicle front structure for the simple crash design problem with two components and force-deformation characteristics F1(s), \(s\in [0 \text {mm},\bar {s}^{1})\) and F2(s), \(s\in [0 \text {mm},\bar {s}^{2})\)

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Due to technical limitations in this crash design problem, it is not possible to build force–deformation characteristics Fk(s) that go below 0kN and above 500kN for \(s\in [0 \text {mm},\bar {s}^{k})\), k = 1,2. The requirements on the responses can be formulated mathematically as follows:

  • Energy absorption: The impact energy \(\frac {1}{2}m{v_{0}^{2}}\) must be completely absorbed, meaning

    $$ -\int\limits_{0 \text{mm}}^{\bar{s}^{1}}F^{1}(s) \mathrm{d}s -\int\limits_{0 \text{mm}}^{\bar{s}^{2}}F^{2}(s) \mathrm{d}s \leq - \frac{1}{2}{{mv}_{0}^{2}}. $$
    (16)

  • Maximum acceleration: The deceleration must be smaller than a critical threshold value ac, i.e.,

    $$ F^{1}(s) \leq {ma}_{\mathrm{c}} $$
    (17)

    for all \(s\in [0 \text {mm},\bar {s}^{1})\) and

    $$ F^{2}(s) \leq {ma}_{\mathrm{c}} $$
    (18)

    for all \(s\in [0 \text {mm},\bar {s}^{2})\). Here, \(a_{\mathrm {c}}=0.3 \mathrm {\frac {mm}{ms^{2}}}\) is used.

  • Progressive order of deformation: The ordered deformation of the vehicle must start at the front, meaning

    $$ F^{1}(s) \leq F^{2}(0 \text{mm}) $$
    (19)

    for all \(s\in [0 \text {mm},\bar {s}^{1})\).

For computing optimal component solution spaces with the approaches presented in this work, force–deformation characteristics with a finite number of design variables are considered. In the following, the crash design problem is investigated under a various number of design variables. Both approaches from the previous chapter are used for this purpose and are compared.

3.2 Components with one degree of freedom

First, constant force–deformation characteristics are considered for the simple crash design problem. Each of them has one degree of freedom which means \(F^{k}(s)={F^{k}_{1}}\) for \(s\in [0 \text {mm},\bar {s}^{k})\). Here, \({F^{k}_{1}}\), k = 1,2, are the design variables. In Fig. 7, examples of such characteristics are shown.

The design space of the system for constant force–deformation characteristics is Ωds = [0kN,500kN]2, and the system performance functions become linear of the form f(x1, x2) = A1x1 + A2x2. Hence, the performance requirements represent a system of linear inequalities, i.e.,

$$ \small \underbrace{\left( \begin{array}{c} -\bar{s}^{1}\\ 1\\ 0\\ 1 \end{array}\right)}_{=A^{1}} \left( \begin{array}{c} {F^{1}_{1}} \end{array}\right) + \underbrace{\left( \begin{array}{c} -\bar{s}^{2}\\ 0\\ 1\\ -1 \end{array}\right)}_{=A^{2}} \left( \begin{array}{c} {F^{2}_{1}} \end{array}\right) \leq \underbrace{\left( \begin{array}{c} - \frac{1}{2}{mv_{0}^{2}}\\ma_{\mathrm{c}}\\ma_{\mathrm{c}}\\0 \end{array}\right)}_{=b} $$
(20)

where the first row belongs to inequality (16), the second row to (17), the third row to (18), and the last row to (19).

Now, optimal component solution spaces Ωk can be computed for \({F^{k}_{1}}\), k = 1,2. Box-shaped component solution spaces \({\Omega }_{\text {bs}}^{k}\) are accomplished by solving problem (7) and arbitrarily shaped component solution spaces \({\Omega }_{\text {as}}^{k}\) by using the exact method from Section 2.5. In Fig. 5, the corresponding component solution spaces are visualized.

Fig. 5
figure 5

Decomposition of the complete system solution space Ωc (thick black) for the simple crash design problem with dk = 1 into \({\Omega }_{\text {bs}}^{1}\), \({\Omega }_{\text {bs}}^{2}\) (thin black) and \({\Omega }_{\text {as}}^{1}\), \({\Omega }_{\text {as}}^{2}\) (gray)

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The component solution spaces for both approaches coincide, i.e., are identical intervals. This is due to the consistence of the optimization problems (5) and (11) for convex system performance functions and dk = 1, compare (Daub 2017).

3.3 Components with two degrees of freedom

Second, the simple crash design problem is considered where each force–deformation characteristic is modeled as a linear function. In detail, it is as follows:

$$ F^{k}(s)= {F^{k}_{1}} \underbrace{\frac{\bar{s}^{k}-s}{\bar{s}^{k}}}_{={\phi^{k}_{1}}(s)} + {F^{k}_{2}}\underbrace{\frac{s}{\bar{s}^{k}}}_{={\phi^{k}_{2}}(s)} $$
(21)

for \(s\in [0 \text {mm},\bar {s}^{k})\), k = 1,2. This means that every force–deformation characteristic has two degrees of freedom. The functions \({\phi ^{k}_{1}}\) and \({\phi ^{k}_{2}}\) are similar to linear basis functions in finite element theory. Examples of such characteristics are shown in Fig. 7.

Here, the design space of the system is Ωds = [0kN,500kN]4 and the system performance functions are linear again. Thus, the performance requirements can be written as a system of linear inequalities, i.e.,

$$ \small \underbrace{\left( \begin{array}{cc} -\frac{\bar{s}^{1}}{2}&-\frac{\bar{s}^{1}}{2}\\ 1&0\\ 0&1\\ 0&0\\ 0&0\\ 1&0\\ 0&1 \end{array}\right)}_{=A^{1}} \left( \begin{array}{c} {F^{1}_{1}}\\ {F^{1}_{2}} \end{array}\right) + \underbrace{\left( \begin{array}{cc} -\frac{\bar{s}^{2}}{2}&-\frac{\bar{s}^{2}}{2}\\ 0&0\\ 0&0\\ 1&0\\ 0&1\\ -1&0\\ -1&0 \end{array}\right)}_{=A^{2}} \left( \begin{array}{c} {F^{2}_{1}}\\ {F^{2}_{2}} \end{array}\right) \leq \underbrace{\left( \begin{array}{c} - \frac{1}{2}{mv_{0}^{2}}\\ma_{\mathrm{c}}\\ma_{\mathrm{c}}\\ma_{\mathrm{c}}\\ma_{\mathrm{c}}\\0\\0 \end{array}\right)}_{=b} $$
(22)

where the first row belongs to inequality (16), the second and third row to (17), the following two rows to (18), and the last two rows to (19). Note that other force–deformation characteristics with two degrees of freedom may be also reasonable, for example, piece-wise constant functions \(F^{k}(s)={F^{k}_{1}}\), \(s\in [0,\frac {\bar {s}^{k}}{2})\) and \(F^{k}(s)={F^{k}_{2}}\), \(s\in [\frac {\bar {s}^{k}}{2},\bar {s}^{k})\), k = 1,2.

Now, optimal component solution spaces \({\Omega }_{\text {as}}^{k}\) and \({\Omega }_{\text {bs}}^{k}\) are computed for \(({F^{k}_{1}},{F^{k}_{2}})\), k = 1,2, like in the previous case and visualized in Fig. 6.

Fig. 6
figure 6

Decomposition of the complete system solution space Ωc for the simple crash design problem with dk = 2 into \({\Omega }_{\text {bs}}^{1}\), \({\Omega }_{\text {bs}}^{2}\) (black) and \({\Omega }_{\text {as}}^{1}\), \({\Omega }_{\text {as}}^{2}\) (gray)

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The volume of \({\Omega }_{\text {as}}^{1}\times {\Omega }_{\text {as}}^{2}\), denoted by Vas, is approximately 2.40 times bigger than the volume of \({\Omega }_{\text {bs}}^{1}\times {\Omega }_{\text {bs}}^{2}\), denoted by Vbs. This increase in volume makes system design more flexible and more uncertainty can be tolerated.

3.4 Components with arbitrary degrees of freedom

This procedure can be extended to force–deformation characteristics with arbitrary, but finitely many degrees of freedom. Here, piece-wise linear and continuous characteristics are as follows:

$$ F^{k}(s)= \sum\limits_{i=1}^{d^{k}} {F^{k}_{i}}\phi_{i}(s) $$
(23)

for \(s\in [0 \text {mm},\bar {s}^{k})\) and dk > 1 are considered. The functions \({\phi ^{k}_{i}}\), i = 1,…,dk are similar to linear basis functions from finite element theory. They are piece-wise linear, continuous, and \(\phi _{i}((j-1) \frac {\bar {s}^{k}}{d^{k}-1})=0\) for i≠j, \(\phi _{i}((j-1) \frac {\bar {s}^{k}}{d^{k}-1})=1\) for i = j, i,j = 1,…,dk. In Fig. 7, examples of such characteristics are shown. As in the previous cases, the system performance functions are linear and the performance requirements given by the inequalities (16)–(19) can be formulated as a system of linear inequalities. The optimal component solution spaces \({\Omega }_{\text {as}}^{k}\) and \({\Omega }_{\text {bs}}^{k}\), k = 1,2, can then be computed as done previously. However, the visualization of these sets as geometric shapes is difficult for dk > 3.

Fig. 7
figure 7

Component solution spaces Ωk and their corresponding region of possibly permissible characteristics both bounded by solid gray lines for different dk. Inside, there are permissible (white dots) or non-permissible (gray dots) force–deformation characteristics. Outside, they are always non-permissible (black dots)

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To overcome this problem, regions of possibly permissible characteristics can be used which are related to parallel coordinate plots. A force–deformation characteristic given by (23) is said to be permissible if Fk ∈Ωk holds where \(F^{k}=({F_{1}^{k}},\ldots , F_{d^{k}}^{k})\). The region of possibly permissible characteristics is then defined as the region between the two bounding characteristics defined by \(F^{\mathrm {l},k}_{\text {out},i}=\min \limits \{{F_{i}^{k}}:~F^{k}\in {\Omega }^{k}\}\) and \(F^{\mathrm {u},k}_{\text {out},i}=\max \limits \{{F_{i}^{k}}:~F^{k}\in {\Omega }^{k}\}\) for i = 1,…,dk. The dk-dimensional interval [F outl,k,F outu,k] forms the minimal outer box of the component solution space Ωk. However, any force–deformation inside this region is not guaranteed to be permissible. In order to get an idea about the quantity of permissible force–deformation characteristics within the region of possibly permissible ones, the average edge lengths of Ωk and [F outl,k,F outu,k] can be calculated and compared. For its ratio rk, it holds as follows:

$$ r^{k}=\sqrt[d^{k}]{\frac{\text{vol}({\Omega}^{k})}{{\prod}_{i=1}^{d^{k}} (F^{\mathrm{u},k}_{\text{out},i}-F^{\mathrm{l},k}_{\text{out},i})}} $$
(24)

In Fig. 7, Ωk is visualized for different degrees of freedom together with the region of possibly permissible characteristics.

3.5 Box-shaped vs. arbitrarily-shaped component solution spaces

An advantage of using box-shaped component solution spaces is that the regions of possibly permissible characteristics do not contain non-permissible force–deformation characteristics as \({\Omega }^{k}_{\text {bs}}=[F^{\mathrm {l},k}_{\text {out}},F^{\mathrm {u},k}_{\text {out}}]\), and therefore rk = 1 always holds. However, the volume Vbs and therefore the regions of possibly permissible characteristics, become rapidly smaller with growing dk compared to the volume of Vas. This difference can also be assessed in terms of average edge length of \({\Omega }_{\text {bs}}^{1}\times {\Omega }_{\text {bs}}^{2}\), \(l_{\text {bs}}=\sqrt [2d^{k}]{V_{\text {bs}}}\), and of \({\Omega }_{\text {as}}^{1}\times {\Omega }_{\text {as}}^{2}\), \(l_{\text {as}}=\sqrt [2d^{k}]{V_{\text {as}}}\), see Fig. 8.

Fig. 8
figure 8

Total volume (left) and average edge length (right) of box-shaped (bs) and arbitrarily shaped (as) component solution spaces

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This means, the greater dk, the more permissible force–deformation characteristics can be found within the regions of possibly permissible characteristics for arbitrarily shaped compared to box-shaped component solution spaces. However, with increasing dk, calculating arbitrarily shaped component solution spaces with exact volume computation requires much more computing time than the calculation of box-shaped solution spaces, compare Fig. 9.

Fig. 9
figure 9

CPU time for Intel(R) Xeon(R) CPU E5-1660 v4 @ 3.20 GHz to calculate box-shaped (bs) and arbitrarily shaped (as) component solution spaces with analytically calculated volume

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4 Application to a realistic crash design problem

4.1 System specifications

A realistic crash design problem is considered where the frontal structure of the vehicle is composed of 8 crash-relevant components, i.e., n = 8, which are distributed to three different load paths, denoted by nlp = 3. In contrast to Section 3, these components are not fully deformable in the event of a crash. They are characterized by a deformation length in horizontal direction, denoted by \(\bar {s}^{k}\), k = 1,…,8, before they behave as if they were rigid. From knowledge about the crash behavior of similar components, their deformation lengths are given in Table 1. The sum of the deformation length and the non-deformable length yield the real length of the component. Furthermore, the mass of the vehicle can be distributed to 5 points, denoted by mi, i = 1,…,5. The values of these discrete mass points are also given in Table 2. They add up to the total mass of the vehicle. The masses mi, i = 1,…,4, are each part of one load path, the mass m5 acts on all load paths as it represents the rear end of the vehicle. All in all, a model in geometry space is obtained which is visualized together with the underlying vehicle structure in Fig. 10a.

Table 1 Values of deformation lengths and bounds of force-deformation characteristics for the realistic crash problem
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Table 2 Values of masses, initial velocity, and critical acceleration for the realistic crash problem.
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Fig. 10
figure 10

Vehicle front structure for the realistic crash design problem with eight components. The gray parts indicate deformable structure and the black parts non-deformable structure

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From geometry space, the parts of the structure that deform simultaneously during a vehicle crash can be deduced. For that reason, the non-deformable lengths of the components are subtracted from the real component length. Hence, components which are shortened to their deformation lengths \(\bar {s}^{k}\), k = 1,…,8 are considered. They are stapled on the right. Now, the parts of the vehicle that deform simultaneously are aligned vertically. This yields a model in deformation space (see Fig. 10b).

For more information about this modeling approach see, e.g., Fender (2013). Note that any interaction between the different load paths due to structural connection is assumed to be negligible here. This is, for example, the case for the connection between the first and forth component and the connection between the second and sixth component (see Fig. 10a). An extension of this approach for non-negligible interactions is proposed in Lange et al. (2019).

Like for the simple crash design problem from Section 3, requirements from crash tests are considered. These are the energy absorption, the maximal acceleration, and the order of deformation of the components. Again, they are formulated with respect to the force–deformation characteristics of the components. Due to technical limitation, the force–deformation characteristics of the k th component are bounded by a lower limit \(F^{\mathrm {l},k}_{\text {ds}}\) and an upper limit \(F^{\mathrm {u},k}_{\text {ds}}\). The corresponding values are given in Table 1. Compared to Section 3 where local positions of deformation are used as input for the force–deformation characteristics of the components, positions of deformation are globally defined here. The deformation of the vehicle starts at s0 = 0mm and ends at send = 750mm. The deformation of the k th starts at \({s^{k}_{0}}\in [s_{0},s_{\text {end}})\) and ends at \(s^{k}_{\text {end}}\in (s_{0},s_{\text {end}}]\) with \({s^{k}_{0}}<s^{k}_{\text {end}}\), compare Fig. 4. Moreover, the mass of the vehicle that is active at position s ∈ [s0, send) is denoted by m∗(s), which is the mass in the interval (s,send]. Every component can be identified by the deformation position s ∈ [s0, send) and the load path j ∈{1,…,nlp} which is denoted by k(j,s). Thus, the requirements can be mathematically formulated as follows:

  • Energy absorption:

    $$ -\int\limits_{s_{0}}^{s_{\text{end}}}\frac{1}{m^{*}(s)}\sum\limits_{j=1}^{n_{\text{lp}}}F^{k(j,s)}(s) \mathrm{d}\text{s} \leq - \frac{1}{2}{v_{0}^{2}} $$
    (25)

    where v0 is given in Table 2.

  • Maximum acceleration:

    $$ \frac{1}{m^{*}(s)}\sum\limits_{j=1}^{n_{\text{lp}}}F^{k(j,s)}(s) \leq a_{\mathrm{c}} $$
    (26)

    for all s ∈ [s0, send) where ac is given in Table 2.

  • Progressive order of deformation:

    $$ F^{k}(s)-\frac{m^{k,*}(s)}{m^{*}(s)}\sum\limits_{j=1}^{n_{\text{lp}}}F^{k(j,s)}(s) \leq F^{k+1}(s^{k+1}_{0}) $$
    (27)

    for all \(s{\in [s^{k}_{0}},s^{k}_{\text {end}})\), k = 1,…,(n − 1) where the k th and (k + 1)th component share the same load path and mk,∗(s) is the mass in the interval \((s,s^{k}_{\text {end}}]\) in the load path of the k th component.

The requirements (16)–(19) from Section 3 are special cases of the requirements (25)–(27) where there is only one load path and the active mass m∗ does not change.

The values given in Tables 1 and 2 are uncontrollable parameters. The controllable variables, which can be targeted by component designers, are the degrees of freedom of the force–deformation characteristics of the components. Again, piece-wise linear force–deformation characteristics with dk degrees of freedom are considered which are given by (23) for k = 1,…,8. According to the deformation length of the components, the degrees of freedom of the components, i.e., the number of their design variables are chosen as d1 = d3 = d4 = 3, d5 = 4, d2 = 5, d6 = d8 = 7, and d7 = 8.

Therefore, the design space has 40 dimensions and the requirements on the system performance reduce to a system of linear inequalities. Now the optimal component solution spaces Ωk, k = 1,…,8, for the design variables of the force–deformation characteristics can be computed like before. The results are visualized by using the lower and upper bounds of box-shaped component solution spaces and the regions of possibly permissible characteristics for arbitrarily shaped component solution spaces (see Fig. 11).

Fig. 11
figure 11

Regions of possibly permissible characteristics from optimal component solution spaces for optimal arbitrarily shaped component solution spaces (gray) and lower and upper bounds of the optimal box-shaped component solution spaces (black)

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Here, the findings for the simple crash design problem of Section 3 are validated. Again, the volume of arbitrarily shaped component solution spaces is significantly greater than the volume of box-shaped ones, as is their average edge length. Details are stated in Table 3. However, the design variables of any force–deformation characteristic within the region of possibly permissible characteristics need to be tested if they are permissible, meaning to be in Ωk, k = 1,…,8. In the case of box-shaped component solution spaces, this is not necessary. Furthermore, the computation time for arbitrarily shaped component solution spaces is much greater than the computation time for box-shaped component solution spaces.

Table 3 Comparison of volume V, average edge length l, and CPU time (Intel(R) Xeon(R) CPU E5-1660 v4 @ 3.20 GHz) for different approaches to compute component solution spaces
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To determine the regions of possibly permissible characteristics in Fig. 11, the exact algorithm for the computation of the volume of Ωk, k = 1,…,8, from Section 3 is used. In Table 3, this is extended by also using the Monto Carlo approach with 103, 104, 105, and 106 local sample points per component to compute the volume of each Ωk, k = 1,…,8. Their computation time is shown together with the volume of Ω1 ×⋯ ×Ω8 and its average edge length. The table is complemented by the results for the initial values for computing arbitrarily shaped component solution spaces from box-shaped solution spaces.

Table 3 demonstrates that arbitrarily shaped component solution spaces computed by using the exact algorithm for volume computation comprise the largest volume V. Although the same optimization problem is solved by using the Monte Carlo method for volume computation, the volume V is smaller. The less sample points to compute the volume of Ωk, k = 1,…,8, the less computational effort is required to solve problem (11). However, the optimal volume found is also less. If 106 sample points are used, the volume gets close to the volume which is obtained by using the exact approach. If 103 sample points are used, the volume is still close to the volume for the initial values.

As the initial values for the computation of arbitrarily shaped component solution spaces are deducted directly from box-shaped component solution spaces, their computation time is the same. Furthermore, the volume V for the initial values is approximately 104 times greater than the volume for box-shaped solution spaces. In contrast to the other approaches which solve problem (11), the computational effort to obtain component solution spaces for the initial values is very small. Overall, it is always a trade-off between the volume V and the CPU time to decide which approach to choose when computing arbitrarily shaped component solution spaces. In case the volume for box-shaped solution spaces is sufficient, such a decision is not required.

5 Conclusion

In this paper, a new approach for optimal decomposition of system requirements into component requirements is presented, yielding component solution spaces. Hence, a more appropriate and more flexible decoupling-based development of components is enabled. This decoupling is necessary to break down complexity and to allow for independent design of components. Previous work relied on strong decoupling of all design variables, which often leads to an unnecessary loss of solution space associated with the loss of feasible designs. Hence, this work aims at a better balance between decoupling and flexibility. For this, a new hybrid approach is proposed where engineering knowledge is combined with mathematical procedures such that the decoupling can follow the engineering definition of components, which are motivated by the organization of the different stakeholders contributing to the overall system development.

In contrast to existing box-shaped solution space approaches by Zimmermann and von Hoessle (2013) and Fender et al. (2016), the proposed decomposition scheme for computing component solution spaces does not decouple all considered design variables. By not decoupling design variables of one component, larger solution spaces with potentially more feasible designs, and thus higher design flexibility, are achieved. The originally box-shaped solution spaces for the fine-grain decoupling are now replaced by the Cartesian product of arbitrarily shaped sets as a subset of the complete solution space following the engineering definitions of meaningful components.

The effectiveness of the new approach is demonstrated by small examples and the full potential is illustrated by a complex structural design problem for crashworthiness. The proposed method can be transferred to other problems in systems engineering where a clear decomposition into components is meaningful.