Multi-fidelity optimization via surrogate modelling

This paper demonstrates the application of correlated Gaussian process based approximations to optimization where multiple levels of analysis are available, using an extension to the geostatistical method of co-kriging. An exchange algorithm is used to choose which points of the search space to sample within each level of analysis. The derivation of the co-kriging equations is presented in an intuitive manner, along with a new variance estimator to account for varying degrees of computational ‘noise’ in the multiple levels of analysis. A multi-fidelity wing optimization is used to demonstrate the methodology

Engineering design applications of surrogate-assisted optimization techniques

The construction of models aimed at learning the behaviour of a system whose responses to inputs are expensive to measure is a branch of statistical science that has been around for a very long time. Geostatistics has pioneered a drive over the last half century towards a better understanding of the accuracy of such ‘surrogate’ models of the expensive function. Of particular interest to us here are some of the even more recent advances related to exploiting such formulations in an optimization context. While the classic goal of the modelling process has been to achieve a uniform prediction accuracy across the domain, an economical optimization process may aim to bias the distribution of the learning budget towards promising basins of attraction. This can only happen, of course, at the expense of the global exploration of the space and thus finding the best balance may be viewed as an optimization problem in itself. We examine here a selection of the state of-the-art solutions to this type of balancing exercise through the prism of several simple, illustrative problems, followed by two ‘real world’ applications: the design of a regional airliner wing and the multi-objective search for a low environmental impact hous

Optimization with missing data

Engineering optimization relies routinely on deterministic computer based design evaluations, typically comprising geometry creation, mesh generation and numerical simulation. Simple optimization routines tend to stall and require user intervention if a failure occurs at any of these stages. This motivated us to develop an optimization strategy based on surrogate modelling, which penalizes the likely failure regions of the design space without prior knowledge of their locations. A Gaussian process based design improvement expectation measure guides the search towards the feasible global optimum

Response surface model evolution

Methods are presented for reducing time and effort when performing aerodynamic optimisation using response surface models. Significant time savings are made possible by monitoring the convergence of computational fluid dynamics simulations and omitting regions of poor designs. In so doing, optimal regions of the design space can be highlighted and surface refinement commenced early in the convergence of the design point set. A strategy employing surface updates with new data at points of maximum expected improvement is shown to perform more efficiently than reducing the design space to the region of the optimum. The response surface evolution methods are demonstrated through an example two parameter optimisation of a flap track fairing on a commercial airliner wing

Wireless sensor network for determining boat motions and hydroelastic responses

A flexible segmented ship model has been tested in a series of head sea wave conditions including regular, irregular and a rogue wave. The model has a typical frigate form and is constructed using a flexible central spine against which four segments are mounted. A three node wireless sensor system, which currently acquires data at up to 30Hz was also attached to the model. These wireless development sensors were supplied as part of the ESPRIT programme and incorporate 9 degrees of freedom using rotational and translational accelerometers and a 3 component magnetometer. The data acquired was compared to those obtained using the traditional pitch and heave potentiometers and a standard heave accelerometer. Overall, the system, although still requiring further development to improve data acquisition rate, performed well with good correlation observed between the various measurement components. The practical advantages are the low mass and low power requirements of such a wireless sensor networ

Global optimization of deceptive functions with sparse sampling

This paper introduces a new method of calculating the expected improvement infill criterion, which does not rely on accurate model parameter estimation. The parameter estimation is embedded within the search of the infill criterion, wherein parameter changes are assessed using likelihood ratio tests. Unlike the traditional expected improvement, a new formulation we present cannot be 'fooled' by unlucky sampling or deceptive functions. The new method is introduced both mathematically and illustratively using a one-variable test function. It is then shown to outperform traditional expected improvement when optimizing the geometry of a passive vibration isolating truss

Novel passive vibration isolators

We present the design of a novel anti-vibration mounting. The mounting takes the form of a triangular truss which is ‘folded’ such that each section extends in the opposite direction to the previous section. The geometry of this complex, compact structure is optimised to exploit the reflections that occur in vibrational waves travelling through the structure to provide significant levels of vibration isolation. In this study we have achieved a 15dB reduction in vibration energy compared to a baseline regular structure

Review of efficient surrogate infill sampling criteria with constraint handling

This paper discusses the benefits of different infill sampling criteria used in surrogate-model-based constrained global optimization. Here surrogate models are used to approximate both the objective and constraint functions with the assumption that these are computationally expensive to compute. The construction of these surrogates (also known as meta models or response surface models) involves the selection of a limited number of designs, evaluated using the original expensive functions. Conventionally this involves two stages. First the surrogate is built using an initial sampling plan; the second stage uses infill sampling criteria to select further designs that offer model improvement. This paper provides a comparison of three different infill criteria previously used in constrained global optimization problems. Particular attention is paid to the need to balance the needs of wide ranging exploration and focussed exploitation during global optimization if good results are to be achieve

Applications of algorithmic differentiation within surrogate model generation

The construction of a surrogate model for the purposes of design optimisation often involves some form of sub-optimisation of the surrogate's controlling parameters. The construction of a kriging model, for example, can require a series of O(n^3) factorisations of the correlation matrix when performing the likelihood maximisation. Due to the smooth nature of the likelihood function, gradient information can be used to accelerate the likelihood optimisation when employed within a gradient enhanced global optimisation strategy. To this end a series of adjoints of the likelihood function of a variety of kriging based surrogate models are presented.An adjoint of the likelihood function derived via algorithmic differentiation is presented for traditional kriging. Recent extensions of this formulation to the likelihood functions for co-kriging and gradient enhanced kriging are also presented. Gradient enhanced kriging may be of particular interest to those wishing to employ derivative information from computational simulations, which itself may be a result of an algorithmic differentiation, within a design optimisation

Generative topographic mapping for dimension reduction in engineering design

Multi-variate design optimization is plagued by the problem of a design space which increases exponentially with number of variables. The computational burden caused by this 'curse of dimensionality' can be avoided by reducing the dimension of the problem. This work describes a dimension reduction method called generative topographic mapping. Unlike the earlier practice of removing irrelevant design variables for dimension reduction, this method transforms the high dimensional data space to a low dimensional one. Hence there is no risk of missing out on information relating to any variables during the dimension redution. The method is demonstrated using the two dimensional Branin function and applied to a problem in wing design