Abstract
Abstract Robust design methods have expanded from experimental techniques to include sampling methods, sensitivity analysis and probabilistic optimisation. Such methods typically require many evaluations. We study design and noise variable cross-term second derivatives of a response to quickly identify design variables that reduce response variability. We first compute the response uncertainty and variance decomposition to determine contributing noise variables of an initial design. Then we compute the Hessian second-derivative matrix cross-terms between the variance-contributing noise variables and proposed design change variables. Design variable with large Hessian terms are those that can reduce response variability. We relate the Hessian coefficients to reduction in Sobol indices and response variance change. Next, the first derivative Jacobian terms indicate which design variable can shift the mean to maintain a desired nominal target value. Thereby, design changes can be proposed to reduce variability while maintaining a targeted nominal value. This workflow finds changes that improve robustness with a minimal four runs per design change. We also explore further computation reductions achieved through compounding variables. An example is shown on a Stirling engine where the top four variance-contributing tolerances and design changes identified through 16 Hessian terms generated a design with 20% less variance.
Highlights
robust design method (RDM) is more than a statistical experiment, it involves a multiple step workflow including identifying possible sources of variability, quantifying their relative contribution with noise experiments, generating ideas for design changes that may promote variation reduction and quantifying the ability of design changes to reduce this variability through a further set of experiments
We find that when using this approach, one can estimate in four runs the variation reduction impact that a design change can have due to a causal noise variable
Robust design was introduced by Taguchi as an experimental method to study the effect of different input noise variables on performance variability, and how these can be reduced through design variable selections (Taguchi 1986; Taguchi & Taguchi 2000)
Summary
Parametric robust design has been well researched and developed into what has become the standard experimental robust design method (RDM), making use of design-of-experiments to reduce the performance variability of a design due to multiple causes (Taguchi 1986; Phadke 1989; Taguchi & Taguchi 2000; Thornton 2003; Arvidsson & Gremyr 2008; Wu & Hamada 2011; Montgomery 2017). We explore here using rapidly computed Hessian second derivative terms to rank potential design changes We combine this with computed Jacobian first derivative terms to enable reshifting the mean to remain on target while reducing variation. We outline a workflow and derive the calculations to (i) quantify uncertainty rank contributing noise variables using Sobol indices, (ii) rank design changes using a Hessian derived calculation, (iii) construct variance and mean prediction equations using as few experimental runs as possible, (iv) compute a constrained optimal set of design changes that minimise variance subject to a nominal target constraint and (v) verify the uncertainty variation reduction at the new design configuration.
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