It is important to account for inherent variability in the material properties in the design and analysis of engineering structures. These properties are typically not homogeneous, but vary across the spatial coordinates within a structure, as well as from specimen to specimen. This form of uncertainty is commonly modelled using random fields within the Stochastic Finite Element Method. Simulation within this framework can be complicated by the dependence of a random field's correlation function upon the geometry of the domain over which it is defined. In this paper, a new method is proposed for simulating random fields over a general two-dimension curved surface, represented as a finite element mesh. The covariance function is parametrised using the geodesic distance, evaluated using the solution to the 'discrete geodesic problem,' and a point discretisation approach is subsequently applied in order to sample the random field at the nodes of the model. The major contribution of the present work is the development of a methodology for simulating random fields over curved surfaces of arbitrary geometry, with a focus upon non-intrusive application to industrial finite element models using 'off the shelf' commercial software. In order to demonstrate the potential impact of the the proposed approach, the algorithm is applied in an uncertainty quantification case study concerning vibration and buckling of an industrial composite aircraft wing model.
Reliability-based design is concerned with ensuring that constraints are enforced with acceptable probability under inherent variability in properties. In aircraft design, such a constraint may be that aeroelastic instability does not occur at velocities encountered by the aircraft. This approach can be complicated, as the aeroelastic instability speed is a discontinuous function of material properties, on account of particular modes only becoming unstable for some parameter values. In reliability analysis, it is common to use surrogate models due to the computational expense associated with Monte Carlo Simulation, however, such methods can be inaccurate when emulating discontinuous functions such as the aeroelastic instability speed. In this paper, an alternative approach is proposed in which Gaussian process surrogate models are fitted directly to each of the modal eigenvalues at the design air-speed, and used to emulate a stability margin based upon the most critical eigenvalue. Using this approach, it is shown that the reliability may be estimated for the aeroelastic stability using smooth emulators, thereby overcoming the problems associated with discontinuities. The method is demonstrated for layup optimisation of composite plate wings with uncertain ply angles, in which the probability of aeroelastic instability occurring is minimised for a prescribed air-speed. In uncertainty quantification, a good agreement is found with Monte Carlo Simulation with an order of two magnitudes reduction in model runs. Through reliability-based design, reductions in the probability of failure of up to 99.8% are achieved by increasing the stability margin at the design speed.
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