Increasing the discriminatory power of bounding models using problem-specific knowledge when viewing design as a sequential decision process

Abstract

A recent design paradigm seeks to overcome the challenges associated with broadly exploring a design space requiring computationally expensive model evaluations by formally viewing design as a sequential decision process (SDP). With the SDP, a set of computational models of increasing fidelity are used to sequentially evaluate and systematically eliminate inefficient design alternatives from further consideration. Key to the SDP are concept models that are of lower fidelity than the true function and are constructed in such a way that when used to evaluate a given design, they return two-sided limits that bound the precise value of the decision criteria, hence referred to as bounding models. Efficiency in the SDP is achieved by using such low-fidelity, inexpensive models, early in the design process to eliminate inefficient design alternatives from consideration after which a higher fidelity, more computationally expensive model, is executed, but only on those design alternatives that appear promising. In general, low-fidelity models trade off discriminatory power for computational complexity; however, it can be demonstrated that knowledge of the underlying physics and/or mathematics can be used to increase the discriminatory power of the lower fidelity models for a given computational cost. Increasing the discriminatory power of the bounding models directly translates into an increase in the efficiency of the SDP. This paper discusses and demonstrates how knowledge of the underlying physics and/or mathematics, otherwise referred to as “problem-specific knowledge,” such as monotonicity and concavity can be used to increase the discriminatory power of the bounding models in the context of the SDP and for engineering designs characterized by demand and capacity relationships. Furthermore, the concept of constructing the bounding models to systematically defer decisions on a subset of design variables, for example for a subsystem, is demonstrated, while retaining the desirable convergence guarantees to the optimal set. The utility of leveraging knowledge to increase discriminatory power and systematically deferring decisions through bounding models in the context of the SDP is demonstrated through two design problems: (1) the notional design of an engine-propeller combination to minimize takeoff distance for a light civil aircraft, and (2) the design of a building’s seismic force resisting structural-foundation system where the performance is evaluated on the basis of minimizing drift and total system cost.

Appendices

Appendix A: Bounding model

This appendix presents the proof that using concavity produces an upper bound that is less than the upper bound formed using monotonicity. Mathematical expressions are derived using Euclidean geometry and by defining the intersection of two lines using determinants. With respect to the Cartesian coordinates presented in Fig. 17, the upper bound formed by concave bounds is denoted as P₁ and for monotonic bounds as P₂. Hence, the following claim can be made:

$$ P_{1x} < P_{2x} $$

(8)

where P_1x is the x-coordinate of the intersection of two secants for functions known to be concave, and P_2x is the x-coordinate of where two regions intersect for functions known to be monotonic. The inequality relating the two upper bounds can be expanded as follows:

$$ P_{1x} =\frac{\left|\begin{array}{ll} \left|\begin{array}{ll}x_{1}&y_{1}\\x_{2}&y_{2}\\ \end{array}\right|&\left|\begin{array}{ll}x_{1}&{1}\\x_{2}&{1}\\ \end{array}\right|\\ \left|\begin{array}{ll}x_{3}&y_{3}\\x_{4}&y_{4} \end{array}\right|&\left|\begin{array}{ll}x_{3}&{1}\\x_{4}&{1} \end{array}\right| \end{array}\right|} {\left|\begin{array}{ll} \left|\begin{array}{ll}x_{1}&{1}\\x_{2}&{1}\\ \end{array}\right|&\left|\begin{array}{ll}y_{1}&{1}\\y_{2}&{1}\\ \end{array}\right|\\ \left|\begin{array}{ll}x_{3}&{1}\\x_{4}&{1} \end{array}\right|&\left|\begin{array}{ll}y_{3}&{1}\\y_{4}&{1} \end{array}\right| \end{array}\right|}< x_{2} = P_{2x} $$

(9)

Simplifying (9) results in the following expression:

$$ \frac{(x_{3}-x_{4})(x_{1}y_{2}-x_{2}y_{1})-(x_{1}-x_{2})(x_{3}y_{4}-x_{4}y_{3})}{(x_{1}-x_{2}) (y_{3}-y_{4})-(x_{3}-x_{4}) (y_{1}-y_{2})} < x_{2} $$

(10)

Fig. 17

Comparison of an upper bound formed from monotonic bounds to an upper bound formed from concave bounds

Therefore, as long as x₁ < x₂ the above inequality holds true thus confirming that P_1x < P_2x.

In considering the lower bound, similar logic can be applied to additional coordinates and their respective secants due to concavity and monotonicity. However, one difference is that the maximum value of the secant intersections will result in a lower bound that is greater than the lower bound produced by the intersection of the regions. The proof is similar to that of the upper bound for each secant intersection, but is omitted for brevity.

Appendix B: Shallow foundation parameters

The shallow foundations were modeled as Beam-on-Nonlinear-Winkler Foundations (BNWF) (Harden and Hutchinson 2009; Raychowdhury 2008). Dimensions were assigned to the footing using the FootProp command (Raychowdhury 2008). The square footing was assigned a 0.91 m height, and length and width according to the values presented in Table 1. Vertical zero-length QzSimple2 springs were modeled and assigned a bearing capacity based on the Terzaghi equation (1951):

$$ \begin{array}{@{}rcl@{}} q_{ult} &=& cN_{c}F_{cs}F_{cd}F_{ci} + \gamma D_{f}N_{q}F_{q}sF_{qd}F_{qi} \\ && + 0.5\gamma BN_{\gamma}F_{\gamma s}F_{\gamma d}F_{\gamma i} \end{array} $$

(11)

where c is the cohesion factor, γ is the unit weight of soil, D_f is the depth of embedment, and B is the width of footing. The bearing capacity factors (N_c, N_q, N_γ), shape factors (F_cs, F_qs, F_γs), depth factors (F_cd, F_qd, F_γd), and inclination factors (F_ci, F_qi, F_γi) are determined using Meyerhof (1963). Additional footing properties, also assigned through FootProp, are an embedment depth of 0.61 m, a 3118 MPa elastic modulus, and a 0^∘ inclination of the load on the foundation with respect to vertical. A stiffness intensity ratio of 5, end length ratio of 0.3, and a 0.3 vertical spring spacing ratio were assigned to the MeshProp command to generate the foundations mesh properties. Lastly, a footing condition of ‘4’ was assigned using the appropriate notation FootingCondition = 4. The assigned footing condition modifies the QzSimple2 non-linear springs to exhibit non-linear inelastic behavior in the z and 𝜃 degrees-of-freedom while restraining the x degree-of-freedom (Boulanger 2000; Raychowdhury 2008).

Appendix C: Bounding model g_dj

The assumption that all intermediate foundation options are contained in an interval formed from bounds produced from evaluation of maximum and minimum foundation dimensions for a given superstructure-foundation system as described in Section 5.2.2 is supported considering the linearization of a simplified structural-foundation system, as shown in Fig. 18a. First, from the simplified system the total displacement, Δ_Total, is determined from the following relation:

$$ \varDelta_{Total} = \frac{Fh^{3}}{3EI}+\frac{12Fh^{2}}{k_{s}L^{3}} $$

(12)

where F is the force applied at the top of the system, E is the elastic modulus, I is the moment of inertia, k_s is the spring constant, and h is the column height. If all variables in (12) are held constant with the exception of the foundation’s length, L, then it can be seen that by increasing L the total displacement of the system decreases. The effect that the foundation length has on the system can be further extended to the performance of the system using the capacity spectrum method (Applied Technology Council 1996), whereby the acceleration-displacement response spectra (ADRS) (Mahaney et al. 1993) for systems with different foundation lengths are compared in Fig. 18.

Fig. 18

a Simplified structural-foundation system with related b acceleration-displacement response spectrum

If a set of foundation lengths are considered, then the system with the largest foundation will have a greater stiffness than all other systems with smaller foundation dimensions. Therefore, the capacity curve for the system with the largest foundation can be represented by the solid curve labeled $L_{\max \limits }$ shown in Fig. 18b. The capacity curve for the system with the smallest foundation is shown as the dashed curve labeled $L_{\min \limits }$ in Fig. 18b. A system with an intermediate foundation length, L_intm, will be contained within the light grey shaded region between the two capacities related to the maximum and minimum foundation lengths, labeled capacity region in Fig. 18b. Similarly, the inelastic demand for a system with an intermediate foundation will be contained within the light grey shaded region, labeled demand region, formed by the solid and dashed demand curves, which correspond to the system’s demand for the maximum, $L_{\max \limits }$, and minimum, $L_{\min \limits }$, foundation lengths. Hence, the intersection of the capacity region with the demand region, indicated by the dark grey shaded region in Fig. 18b, will contain the performance points of all systems related to the set of foundation lengths considered. Therefore, the spectral displacement values corresponding to the extreme left and right corners of the dark grey shaded region, S_d1 and S_du, bound the performance points’ spectral displacement values. For the purpose of this illustration, the demand and capacity are shown precisely, but can be readily extended to the idea of multi-fidelity bounding models as described in Section 5.2.2. As such, intervals obtained from higher fidelity evaluations will be contained within intervals obtained from the lower fidelity evaluations which can be proven using similar logic presented in Section 5.2.1.