A novel 2.5D spectral approach for studying thin-walled waveguides with fluid-acoustic interaction

This paper presents a novel formulation of two spectral elements to study guided waves in coupled problems involving thin-walled structures and fluid-acoustic enclosures. The aim of the proposed work is the development of a new efficient computational method to study problems where geometry and properties are invariant in one direction, commonly found in the analysis of guided waves. This assumption allows using a two-and-a-half dimensional (2.5D) spectral formulation in the wavenumber-frequency domain. The novelty of the proposed work is the formulation of spectral plate and fluid elements with an arbitrary order in 2.5D. A plate element based on a Reissner-Mindlin/Kirchhoff-Love mixed formulation is proposed to represent the thin-walled structure. This element uses approximation functions to overcome the difficulties to formulate elements with an arbitrary order from functions. The proposed element uses a substitute transverse shear strain field to avoid shear locking effects. Three benchmark problems are studied to check the convergence and the computational effort for different strategies. Accurate results are found with an appropriate combination of element size and order of the approximation functions allowing at least six nodes per wavelength. The effectiveness of the proposed elements is demonstrated studying the wave propagation in a water duct with a flexible side and an acoustic cavity coupled to a Helmholtz resonator.Ministerio de Economía y Competitividad BIA2013-43085-P y BIA2016-75042-C2-1-RCentro Informático Científico de Andalucía (CICA

A local projection stabilized method for fictitious domains

In this work a local projection stabilization method is proposed to solve a fictitious domain problem. The method adds a suitable fluctuation term to the formulation thus rendering the natural space for the Lagrange multiplier stable. Stability and convergence are proved and these results are illustrated by a numerical experiment.Comment: Submitted Preprin

Stabilised finite element methods for the Oseen problem on anisotropic quadrilateral meshes

In this work we present and analyse new inf-sup stable, and stabilised, finite element methods for the Oseen equation in anisotropic quadrilateral meshes. The meshes are formed of closed parallelograms, and the analysis is restricted to two space dimensions. Starting with the lowest order QIn this work we present and analyse new inf-sup stable, and stabilised, finite element methods for the Oseen equation in anisotropic quadrilateral meshes. The meshes are formed of closed parallelograms, and the analysis is restricted to two space dimensions. Starting with the lowest order Q2 1 × P0 pair, we first identify the pressure components that make this finite element pair to be non-inf-sup stable, especially with respect to the aspect ratio. We then propose a way to penalise them, both strongly, by directly removing them from the space, and weakly, by adding a stabilisation term based on jumps of the pressure across selected edges. Concerning the velocity stabilisation, we propose an enhanced grad-div term. Stability and optimal a priori error estimates are given, and the results are confirmed numerically. Q21 × P0 pair, we first identify the pressure components that make this finite element pair to be non-inf-sup stable, especially with respect to the aspect ratio. We then propose a way to penalise them, both strongly, by directly removing them from the space, and weakly, by adding a stabilisation term based on jumps of the pressure across selected edges. Concerning the velocity stabilisation, we propose an enhanced grad-div term. Stability and optimal a priori error estimates are given, and the results are confirmed numerically

Numerical wave propagation for the triangular P1DGP1_{DG}-P2P2 finite element pair

Inertia-gravity mode and Rossby mode dispersion properties are examined for discretisations of the linearized rotating shallow-water equations using the $P1_{DG}$-$P2$ finite element pair on arbitrary triangulations in planar geometry. A discrete Helmholtz decomposition of the functions in the velocity space based on potentials taken from the pressure space is used to provide a complete description of the numerical wave propagation for the discretised equations. In the $f$-plane case, this decomposition is used to obtain decoupled equations for the geostrophic modes, the inertia-gravity modes, and the inertial oscillations. As has been noticed previously, the geostrophic modes are steady. The Helmholtz decomposition is used to show that the resulting inertia-gravity wave equation is third-order accurate in space. In general the \pdgp finite element pair is second-order accurate, so this leads to very accurate wave propagation. It is further shown that the only spurious modes supported by this discretisation are spurious inertial oscillations which have frequency $f$, and which do not propagate. The Helmholtz decomposition also allows a simple derivation of the quasi-geostrophic limit of the discretised $P1_{DG}$-$P2$ equations in the $\beta$-plane case, resulting in a Rossby wave equation which is also third-order accurate.Comment: Revised version prior to final journal submissio

Algorithmic Algebraic Geometry and Flux Vacua

We develop a new and efficient method to systematically analyse four dimensional effective supergravities which descend from flux compactifications. The issue of finding vacua of such systems, both supersymmetric and non-supersymmetric, is mapped into a problem in computational algebraic geometry. Using recent developments in computer algebra, the problem can then be rapidly dealt with in a completely algorithmic fashion. Two main results are (1) a procedure for calculating constraints which the flux parameters must satisfy in these models if any given type of vacuum is to exist; (2) a stepwise process for finding all of the isolated vacua of such systems and their physical properties. We illustrate our discussion with several concrete examples, some of which have eluded conventional methods so far.Comment: 41 pages, 4 figure

Multiphase modelling of vascular tumour growth in two spatial dimensions

In this paper we present a continuum mathematical model of vascular tumour growth which is based on a multiphase framework in which the tissue is decomposed into four distinct phases and the principles of conservation of mass and momentum are applied to the normal/healthy cells, tumour cells, blood vessels and extracellular material. The inclusion of a diffusible nutrient, supplied by the blood vessels, allows the vasculature to have a nonlocal influence on the other phases. Two-dimensional computational simulations are carried out on unstructured, triangular meshes to allow a natural treatment of irregular geometries, and the tumour boundary is captured as a diffuse interface on this mesh, thereby obviating the need to explicitly track the (potentially highly irregular and ill-defined) tumour boundary. A hybrid finite volume/finite element algorithm is used to discretise the continuum model: the application of a conservative, upwind, finite volume scheme to the hyperbolic mass balance equations and a finite element scheme with a stable element pair to the generalised Stokes equations derived from momentum balance, leads to a robust algorithm which does not use any form of artificial stabilisation. The use of a matrix-free Newton iteration with a finite element scheme for the nutrient reaction-diffusion equations allows full nonlinearity in the source terms of the mathematical model. Numerical simulations reveal that this four-phase model reproduces the characteristic pattern of tumour growth in which a necrotic core forms behind an expanding rim of well-vascularised proliferating tumour cells. The simulations consistently predict linear tumour growth rates. The dependence of both the speed with which the tumour grows and the irregularity of the invading tumour front on the model parameters are investigated