The second-generation Shifted Boundary Method and its numerical analysis

Recently, the Shifted Boundary Method (SBM) was proposed within the class of unfitted (or immersed, or embedded) finite element methods. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we present enhanced variational SBM formulations for the Poisson and Stokes problems with improved flexibility and robustness. These simplified variational forms allow to relax some of the assumptions required by the mathematical proofs of stability and convergence of earlier implementations. First, we show that these new SBM implementations can be proved asymptotically stable and convergent even without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive. Second, we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at Dirichlet boundaries, therefore avoiding the calibration of an additional stabilization parameter. Finally, we prove enhanced L2-estimates without the cumbersome assumption – of earlier proofs – that the surrogate domain is convex. Instead we rely on a conventional assumption that the boundary of the true domain is smooth, which can also be replaced by requiring convexity of the true domain. The aforementioned improvements open the way to a more general and efficient implementation of the Shifted Boundary Method, particularly in complex three-dimensional geometries. We complement these theoretical developments with numerical experiments in two and three dimensions

Exploring the Ambiguity of Operation Sophia Between Military and Search and Rescue Activities

Over the past decade, for the purpose of managing the phenomenon of migration by sea, a wide number of different measures have been adopted by the European Union and its Member States. Notwithstanding the persistent need and the legal obligation to save people's lives at sea, Europe remains stocked on the protection of the security of its internal and external borders and goes ahead with the launch of Eunavfor Med--Operation Sophia, the first naval mission aimed to disrupt the business model of migrant smuggling and human trafficking in the Mediterranean. The following chapter examines the factual and legal background behind the establishment of this military mission and focuses on two sensitive and interrelated aspects: the use of enforcement powers against alleged smugglers and traffickers on the one hand and the rescue of irregular migrants at sea on the other hand. While various challenges prevent the activation of the crucial military phase of Operation Sophia, the operational and legal framework applicable to incidental search and rescue interventions carried out by its naval forces appears rather unclear and problematic under different perspectives of international law, especially if the Operation will continue into Libyan territorial waters in cooperation with its unstable authorities

Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations

A new SUPG-stabilized formulation for Lagrangian Hydrodynamics of materials satisfying the Mie-Gruneisen equation of state was presented in the first paper of the series [7]. This article investigates in more detail the design of SUPG stabilization, focusing on its multiscale and physical interpretations. Connections with Kuropatenko's [5] analysis of shock-capturing operators in the limit of weak shocks are shown. Galilean invariance requirements for the SUPG operator are explored and corroborated by numerical evidence. This work is intended to elucidate the profound physical significance of the SUPG operator as a subgrid interaction model. Acknowledgments This research was partially funded by the DOE NNSA Advanced Scientific Computing Program and the Computer Science Research Institute at Sandia National Laboratories. The author would like to thank Tom Hughes, Mark Christon, and John Shadid for providing helpful comments and suggestions on a preliminary draft of the paper. Contents

A geometrically-conservative, synchronized, flux-corrected remap for arbitrary Lagrangian–Eulerian computations with nodal finite elements

This article describes a conservative synchronized remap algorithm applicable to arbitrary Lagrangian–Eulerian computations with nodal finite elements. In the proposed approach, ideas derived from flux-corrected transport (FCT) methods are extended to conservative remap. Unique to the proposed method is the direct incorporation of the geometric conservation law (GCL) in the resulting numerical scheme. It is shown here that the geometric conservation law allows the method to inherit the positivity preserving and local extrema diminishing (LED) properties typical of FCT schemes. The proposed framework is extended to the systems of equations that typically arise in meteorological and compressible flow computations. The proposed algorithm remaps the vector fields associated with these problems by means of a synchronized strategy. The present paper also complements and extends the work of the second author on nodal-based methods for shock hydrodynamics, delivering a fully integrated suite of Lagrangian/remap algorithms for computations of compressible materials under extreme load conditions. Extensive testing in one, two, and three dimensions shows that the method is robust and accurate under typical computational scenarios

The shifted boundary method for solid mechanics

We propose a new embedded/immersed framework for computational solid mechanics, aimed at vastly speeding up the cycle of design and analysis in complex geometry. In many problems of interest, our approach bypasses the complexities associated with the generation of CAD representations and subsequent body-fitted meshing, since it only requires relatively simple representations of the surface geometries to be simulated, such as collections of disconnected triangles in three dimensions, widely used in computer graphics. Our approach avoids the complex treatment of cut elements, by resorting to an approximate boundary representation and a special (shifted) treatment of the boundary conditions to maintain optimal accuracy. Natural applications of the proposed approach are problems in biomechanics and geomechanics, in which the geometry to be simulated is obtained from imaging techniques. Similarly, our computational framework can easily treat geometries that are the result of topology optimization methods and are realized with additive manufacturing technologies. We present a full analysis of stability and convergence of the method, and we complement it with an extensive set of computational experiments in two and three dimensions, for progressively more complex geometries