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Scottish Computational
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Nevertheless, there are many modelling situations where a Lagrangian framework is useful, for example in nonlinear problems having moving boundaries or evolving singularities. Moreover, there are good mathematical reasons why moving meshes should be employed, for example to maintain the scale invariance of the original problem.
It is only relatively recently that researchers have made a concerted effort to exploit moving mesh methods. An early method was the Moving Finite Element method of Miller, and this has been followed by MMPDEs (Moving Mesh Partial Differential Equations) based on equidistribution and its generalisations.
In this talk I shall discuss the application of a Lagrangian moving mesh finite element method to moving boundary problems. The basic idea is to partition a conserved global quantity and maintain it in time (as in Lagrangian fluid dynamics). The method inherits the scale invariance of the original problem and can be generalised to include artificial densities (acting as monitor functions) and vorticities. The mesh velocities are generated via a distributed form of the Reynolds Transport Theorem.
The advantages and limitations of the approach will be described.
In this talk, I will discuss meshfree, kernel based methods. After a short introduction along the lines of optimal recovery, I will concentrate on results concerning convergence orders and stability. After that I will address efficient numerical algorithms. Finally, I will present some examples, including one from fluid-structure interaction, which will demonstrate why these methods are currently becoming Airbus's preferred solution in Aeroelasticity.