The finite element method is a useful tool to predict the behaviour of a structure under static and dynamic loads. Reliable finite element analyses can reduce the need for prototype testing and thus reduce the design validation cost and time.
In many real life situations however, a deterministic analysis is not sufficient to assess the quality of a design. In a design stage, some physical properties of the model may not be determined yet. But even in a design ready for production, design tolerances and production inaccuracies introduce variability and uncertainty. In these cases, a non-deterministic analysis procedure is required, either using a probabilistic or a possibilistic approach. In the former case, Monte Carlo simulation is best known. In the latter case, interval arithmetic and global optimisation can be used. Unless precautions are taken, the conservatism of interval arithmetic approaches and the computational cost of global optimisation approaches are prohibitively high for practical applications.
The authors developed a hybrid (global optimisation and interval arithmetic) interval finite element procedure [
] to predict the bounds on frequency response functions (FRFs) of problems with interval inputs. In a first step, the bounds on the modal parameters are determined using a global optimisation approach. In a second step, the bounds on the FRF are calculated using an interval arithmetic approach. This hybrid approach reduces the conservatism compared to a full interval arithmetic approach and reduces the computational cost compared to a full global optimisation approach.
Still, the optimisation of the modal parameters is by far computationally the most expensive step of the hybrid algorithm. Therefore, highly efficient optimisation algorithms are necessary to perform analyses on industrial sized applications. Response surface based optimisation algorithms take advantage of the fact that - using a standard modal FE solver — the computational cost to calculate all modal parameters for all modes of interest is almost equal to the computational cost to calculate only one modal parameter for only one mode.
This paper discusses the application of a response surface optimisation algorithm in the context of interval and fuzzy finite element analysis and compares them to classical optimisation algorithms (line search and trust region) on a reference model.