Two Guaranteed Equilibrated Error Estimators for Harmonic Formulations in Eddy Current Problems

International audienceIn this paper, two guaranteed equilibrated error estimators are proposed and compared for the 3D harmonic magnetodynamic problem of Maxwell's system. This system is recasted in the classical A − ϕ potential formulation or, equivalently , in the T − Ω potential formulation, and it is solved by the Finite Element method. The first equilibrated estimator presented is built starting from these two complementary problems, the other one is built starting from the A − ϕ numerical solution uniquely by a flux reconstruction technique. The equivalence between errors and estimators is established. Afterwards, an analytical benchmark test illustrates the obtained theoretical results and a physical benchmark test shows the efficiency of these two estimators

Adaptive Algorithms

Overwhelming empirical evidence in computational science and engineering proved that self-adaptive mesh-generation is a must-do in real-life problem computational partial differential equations. The mathematical understanding of corresponding algorithms concerns the overlap of two traditional mathematical disciplines, numerical analysis and approximation theory, with computational sciences. The half workshop was devoted to the mathematics of optimal convergence rates and instance optimality of the Dörfler marking or the maximum strategy in various versions of space discretisations and time-evolution problems with all kind of applications in the efficient numerical treatment of partial differential equations

Schnelle Löser für partielle Differentialgleichungen

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SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Error estimation and adaptivity for finite element structural dynamics models under parameter uncertainty

The optimisation of discretisation and stochastic errors under a single criterion is not a simple task. The nature of the errors derived from both phenomena is totally different and so are the measures needed to assess them. Nonetheless, they are related and if either of the errors dominates a problem, any obtained solution is suboptimal. Error estimation research is focused on optimising and bounding the discretisation error only. On the other hand, stochastic research treats error estimation as a black box that ensures enough accuracy to avoid interference with the stochastic process and/or the surrogate of the numerical model, with the only exception of stochastic finite element method. This dissertation presents an adaptive approach to optimise locally the relation between the aforementioned numerical approximations in any stochastic framework. The main novel contribution of this thesis is the development of an algorithm that ensures that all errors are of the same scale after an adaptive process. The numerical problem posed is a structure vibrating steadily under parametric uncertainty, although any partial differential equation could have been selected modifying the refinement strategy. Steady dynamic problems were chosen because they tend to need less intuitive concentration of refinements, the lack of time dependency allows non-conforming meshes and yet, natural frequencies highly influence the solution. The definitions of all measures of error are linked to the relative discretisation error, and are therefore controlled by the algorithm under this single criterion. Another novelty is a new family of residual error estimators based on the Saint- Venant principle rather than on limiting the support of the test function. This new approach allows to unlink the definition of the patch sub-domain from the split of the residual. In addition to the resulting freedom of patch choice, it is proven than the new approach provides enhanced stability to some element centered patch estimators proposed in the past. iii Finally, two minor new contributions are a discrete way to obtain an indicator of refinement for quantities of interest not involving gradients (simpler than the choices already present in literature), and the testing of analogy between error estimators and preconditioners

Mechanisms of natural and forced variability in the southern ocean

The Southern Ocean is an important regulator of global climate, and accurately predicting its future evolution under climate change constitutes a critical scientific challenge. Mesoscale eddies are key to the dynamics of the Southern Ocean, but the mechanisms and time scales of their natural and forced variability are not completely understood. Motivated by the dynamical analogy between the Antarctic Circumpolar Current and the tropospheric jet stream, the natural variability of eddymean flow interaction is studied by adapting a two-dimensional model of storm track variability to the oceanic case. It is found that eddies and the mean flow interact according to a predator-prey oscillatory relationship in both an idealised, eddy-resolving, channel configuration and the SOSE state estimate product of the Southern Ocean. The oscillatory nature of the dynamics reflects in the structure of the phase space diagrams, where quasi-periodic cycles with typical timescales of a few weeks can be observed. The simplified mathematical model qualitatively captures the statistical properties of the interaction well. The time scales of forced adjustment are investigated by means of an ensemble of wind step-change experiments run with the idealised channel configuration. It is found that the temperature response is driven largely, but not exclusively, by changes in the ocean’s circulation, with enhanced mixing also playing an important role. Circulation changes have a rich spatial structure, and vertical/meridional displacements of the residual overturning circulation cells have a large impact on the temperature response even though the channel is strongly eddy-compensated. The time scales of the response vary across the domain, and are set by the spin-up of baroclinic eddies. The results presented in this Thesis bring the fundamental mechanisms of eddy variability into clearer focus, and inform the interpretation of more realistic numerical simulations of the Southern Ocean