Atul Bhaskar

A Knowledge-Based Approach To Response Surface Modelling in Multifidelity Optimization

This paper is concerned with approximations for expensive function evaluation – the expensive functions arising in an engineering design context. The problem of reducing the computational cost of generating sufficient learning samples is addressed. Several approaches of using a priori knowledge to achieve computational economy are presented. In all these, the results of a cheap model are treated as knowledge to be incorporated in the training process. Several approaches are described here: in particular, we focus on neural based systems. This approach is then developed as a new knowledge-based kriging model which is shown to be as accurate as neural based alternatives while being much easier to train. Examples from the domain of structural optimization are given to demonstrate the approach.

A constraint mapping approach to the structural optimization of an expensive model using surrogates (in special issue on surrogate modelling and space mapping for engineering optimization)

The use of response surface methods are well established in the global optimization of expensive functions, the response surface acting as a surrogate to the expensive function objective.In structural design however, the change in objective may vary little between the two models: it is more often the constraints that change with models of varying fidelity. Here approaches are described whereby the coarse model constraints are mapped so that the mapped constraints more faithfully approximate the fine model constraints. The shape optimization of a simple structure demonstrates the approach.

Optimal orthogonal-array-based latin hypercubes

The use of optimal orthogonal array latin hypercube designs is proposed. Orthogonal arrays were proposed for constructing latin hypercube designs by Tang (1993). Such designs generally have better space filling properties than random latin hypercube designs. Even so, these designs do not necessarily fill the space particularly well. As a result, we consider orthogonal-array-based latin hypercube designs that try to achieve optimality in some sense. Optimization is performed by adapting strategies found in Morris & Mitchell (1995) and Ye et al. (2000). The strategies here search only orthogonal-array-based latin hypercube designs and, as a result, optimal designs are found in a more efficient fashion. The designs found are in general agreement with existing optimal designs reported elsewhere.

Wheel-rail dynamics with closely conformal contact Part 1: Dynamic modelling and stability analysis

Abstract Observations on the Vancouver mass transit system suggest that noise, vibration and corrugation of the rail appear to be associated with close conformity between the transverse profiles of the wheel and rail. To investigate this, a dynamic model of the wheel and rail under conditions of close conformity has been developed. Previous work has suggested that motion of the wheel could be neglected, so the model comprises two subsystems: (a) the rail and its supports, and (b) the contact between wheel and rail. A dynamic model of a continuously supported rail is presented, which is consistent with similar models in the literature. Conformal contact has been represented in two ways: (a) as a single highly eccentric elliptical contact, and (b) as a two-point contact. Novel ‘rolling contact mechanics’ have been incorporated in both these models. The complete system is closed: oscillations of the rail give rise to fluctuating contact forces, which in turn excite the rail. A linear stability analysis of the system shows it to be stable under all conditions examined, thus precluding the possibility of self-excited oscillations occurring on a perfectly smooth rail. The model can then be used to investigate the forced response to existing roughness on the railhead, which is the subject of a companion paper (1).

Elastic waves in Timoshenko beams: the ‘lost and found’ of an eigenmode

This paper considers propagating waves in elastic bars in the spirit of asymptotic analysis and shows that the inclusion of shear deformation amounts to singular perturbation in the Euler–Bernoulli (EB) field equation. We show that Timoshenko, in his classic work of 1921, incorrectly treated the problem as one of regular perturbation and missed out one physically meaningful ‘branch’ of the dispersion curve (spectrum), which is mainly shear-wise polarized. Singular perturbation leads to: (i) Timoshenkos solution and (ii) a singular solution ; ε, ω* and k* are the non-dimensional slenderness, frequency and wavenumber, respectively. Asymptotic formulae for dispersion, standing waves and the density of modes are given in terms of ε. The second spectrum—in the light of the debate on its existence, or not—is discussed. A previously proposed Lagrangian is shown to be inadmissible in the context. We point out that Lagrangian densities that lead to the same equation(s) of motion may not be equivalent for field problems: careful consideration to the kinetic boundary conditions is important. A Hamiltonian formulation is presented—the conclusions regarding the validity (or not) of Lagrangian densities are confirmed via the constants of motion.

Turbine blade fir-tree root design optimisation using intelligent CAD and finite element analysis

This paper is concerned with automation and optimisation of the design of a turbine blade fir-tree root by incorporating a knowledge based intelligent computer-aided design system (ICAD®) and finite element analysis. Various optimisation algorithms have been applied in an effort to optimise the shape against a large number of geometric and mechanical constraints drawn from industrial experience in the development of such a structure. Attention is devoted to examining the effects of critical geometric features on the stress distribution at the interface between the blade and disk using a feature-based geometry modelling tool and the optimisation techniques. Various aspects of this problem are presented: (1) geometry representation using ICAD® and transfer of the geometry to a finite element analysis code, (2) application of boundary conditions/loads and retrieval of analysis results, (3) exploration of various optimisation methods and strategies including gradient-based and modern stochastic methods. A product model from Rolls-Royce is used as a base design in the optimisation.

Global Approximation and Optimization Using Adjoint Computational Fluid Dynamics Codes

Approximation methods have found increasing use in the optimization of complex engineering systems. The approximation method provides a surrogate model that, once constructed, can be used in lieu of the original expensive model for the purposes of optimization. These approximations may be defined locally, for example, a low-order polynomial response surface approximation that employs trust region methodology during optimization, or globally, by the use of techniques such as kriging. Adjoint methods for computational fluid dynamics have made it possible to obtain sensitivity information on the model’s response without recourse to finite differencing. This approach then allows for an efficient local optimization strategy where these sensitivities are utilized in gradient-based optimization. The combined use of an adjoint computational fluid dynamics code with approximation methods (incorporating gradients) for global optimization is shown. Several approximation methods are considered. It is shown that an adjoint-based approximation model can provide increased accuracy over traditional nongradientbased approximations at comparable cost, at least for modest numbers of design variables. As a result, these models are found to be more reliable for surrogate assisted optimization.

Forced response statistics of mistuned bladed disks: a stochastic reduced basis approach

This paper presents a stochastic reduced basis approach for predicting the forced response statistics of mistuned bladed-disk assemblies. In this approach, the system response in the frequency domain is represented using a linear combination of complex stochastic basis vectors with undermined coefficients. The terms of the preconditioned stochastic Krylov subspace are used here as basis vectors. Two variants of the stochastic Bubnov–Galerkin scheme are employed for computing the undetermined terms in the reduced basis representation, which arise from how the condition for orthogonality between two random vectors is interpreted. Explicit expressions for the response quantities can then be derived in terms of the random system parameters, which allow for the possibility of efficiently computing the response statistics in the post-processing stage. Numerical studies are presented for mistuned cyclic assemblies of mono-coupled single-mode components. It is demonstrated that the accuracy of the response statistical moments computed using stochastic reduced basis methods can be orders of magnitude better than classical perturbation methods.

Normal modes of carbon nanotubes: similarities and differences with their continuum counterpart

Carbon nanotubes (CNTs) possess a range of unusually interesting and useful physicochemical properties. In this paper, the mechanical properties of single wall CNTs are investigated via free vibration normal modes using molecular mechanics models. The forcefield used is empirical and the usual assumptions of potential energy contributions coming from bondstretching, bond angle bending, and bond twisting for two, three, and four atom interactions respectively, are made. The validity of continuum behaviour is examined by comparing the modal spacing obtained from the molecular mechanics models and that obtained from classical continuum elastodynamics. The breakdown of continuum behaviour is systematically characterised for various combinations of length to diameter ratio as well as for the number of atoms per circumference.

The effective Poisson ratio of random cellular matter having bending dominated architecture

We argue that the effective Poisson ratio of cellular and porous solids is independent of the material of the solid phase, if the mechanism of the cell wall deformation is dominated by beam bending —thus rendering it to be a purely kinematic quantity. Introducing a kinematic simplification and requiring statistical isotropy, we prove a result of remarkable generality that the effective Poisson ratio of irregular planar structures equals 1 for all bending dominated random architectures. We then explore a deeper connection of this behavior with area-preserving deformation of planar closed elastic cells. We show that thin sheets and films made of such microstructured material afford physical realizations of the two-dimensional analogue of incompressible matter. We term such non-stretchable sheet material as well as deformations as isoektasic.