Uncertainty Propagation Through Large Nonlinear Models

Abstract

Uncertainty analysis in computer models has seen a rise in interest in recent years as
a result of the increased complexity of (and dependence on) computer models in the
design process. A major problem however, is that the computational cost of propagating
uncertainty through large nonlinear models can be prohibitive using conventional methods
(such as Monte Carlo methods). A powerful solution to this problem is to use an emulator,
which is a mathematical representation of the model built from a small set of model runs
at speci�ed points in input space. Such emulators are massively cheaper to run and
can be used to mimic the "true" model, with the result that uncertainty analysis and
sensitivity analysis can be performed for a greatly reduced computational cost. The work
here investigates the use of an emulator known as a Gaussian process (GP), which is
an advanced probabilistic form of regression, hitherto relatively unknown in engineering.
The GP is used to perform uncertainty and sensitivity analysis on nonlinear �nite element
models of a human heart valve and a novel airship design. Aside from results specifi�c to
these models, it is evident that a limitation of the GP is that non-smooth model responses
cannot be accurately represented. Consequently, an extension to the GP is investigated,
which uses a classi�cation and regression tree to partition the input space, such that
non-smooth responses, including bifurcations, can be modelled at boundaries. This new
emulator is applied to a simple nonlinear problem, then a bifurcating �nite element model.
The method is found to be successful, as well as actually reducing computational cost,
although it is noted that bifurcations that are not axis-aligned cannot realistically be dealt
with.