Spectral/hp element methods: recent developments, applications, and perspectives

The spectral/hp element method combines the geometric flexibility of the classical h-type finite element technique with the desirable numerical properties of spectral methods, employing high-degree piecewise polynomial basis functions on coarse finite element-type meshes. The spatial approximation is based upon orthogonal polynomials, such as Legendre or Chebychev polynomials, modified to accommodate C0-continuous expansions. Computationally and theoretically, by increasing the polynomial order p, high-precision solutions and fast convergence can be obtained and, in particular, under certain regularity assumptions an exponential reduction in approximation error between numerical and exact solutions can be achieved. This method has now been applied in many simulation studies of both fundamental and practical engineering flows. This paper briefly describes the formulation of the spectral/hp element method and provides an overview of its application to computational fluid dynamics. In particular, it focuses on the use the spectral/hp element method in transitional flows and ocean engineering. Finally, some of the major challenges to be overcome in order to use the spectral/hp element method in more complex science and engineering applications are discussed

Cellular Automata Models of Road Traffic

In this paper, we give an elaborate and understandable review of traffic cellular automata (TCA) models, which are a class of computationally efficient microscopic traffic flow models. TCA models arise from the physics discipline of statistical mechanics, having the goal of reproducing the correct macroscopic behaviour based on a minimal description of microscopic interactions. After giving an overview of cellular automata (CA) models, their background and physical setup, we introduce the mathematical notations, show how to perform measurements on a TCA model's lattice of cells, as well as how to convert these quantities into real-world units and vice versa. The majority of this paper then relays an extensive account of the behavioural aspects of several TCA models encountered in literature. Already, several reviews of TCA models exist, but none of them consider all the models exclusively from the behavioural point of view. In this respect, our overview fills this void, as it focusses on the behaviour of the TCA models, by means of time-space and phase-space diagrams, and histograms showing the distributions of vehicles' speeds, space, and time gaps. In the report, we subsequently give a concise overview of TCA models that are employed in a multi-lane setting, and some of the TCA models used to describe city traffic as a two-dimensional grid of cells, or as a road network with explicitly modelled intersections. The final part of the paper illustrates some of the more common analytical approximations to single-cell TCA models.Comment: Accepted for publication in "Physics Reports". A version of this paper with high-quality images can be found at: http://phdsven.dyns.cx (go to "Papers written"

An edge-based unstructured mesh discretisation in geospherical framework

An arbitrary finite-volume approach is developed for discretising partial differential equations governing fluid flows on the sphere. Unconventionally for unstructured-mesh global models, the governing equations are cast in the anholonomic geospherical framework established in computational meteorology. The resulting discretisation retains proven properties of the geospherical formulation, while it offers the flexibility of unstructured meshes in enabling irregular spatial resolution. The latter allows for a global enhancement of the spatial resolution away from the polar regions as well as for a local mesh refinement. A class of non-oscillatory forward-in-time edge-based solvers is developed and applied to numerical examples of three-dimensional hydrostatic flows, including shallow-water benchmarks, on a rotating sphere

Subgrid scale modelling of transport processes.

Consideration of stabilisation techniques is essential in the development of physical models if they are to faithfully represent processes over a wide range of scales. Careful application of these techniques can significantly increase flexibility of models, allowing the computational meshes used to discretise the underlying partial differential equations to become highly nonuniform and anisotropic, for example. This exibility enables a model to capture a wider range of phenomena and thus reduce the number of parameterisations required, bringing a physically more realistic solution. The next generation of fluid flow and radiation transport models employ unstructured meshes and anisotropic adaptive methods to gain a greater degree of flexibility. However these can introduce erroneous artefacts into the solution when, for example, a process becomes unresolvable due to an adaptive mesh change or advection into a coarser region of mesh in the domain. The suppression of these effects, caused by spatial and temporal variations in mesh size, is one of the key roles stabilisation can play. This thesis introduces new explicit and implicit stabilisation methods that have been developed for application in fluid and radiation transport modelling. With a focus on a consistent residual-free approach, two new frameworks for the development of implicit methods are presented. The first generates a family of higher-order Petrov-Galerkin methods, and the example developed is compared to standard schemes such as streamline upwind Petrov-Galerkin and Galerkin least squares in accurate modelling of tracer transport. The dissipation generated by this method forms the basis for a new explicit fourth-order subfilter scale eddy viscosity model for large eddy simulation. Dissipation focused more sharply on unresolved scales is shown to give improved results over standard turbulence models. The second, the inner element method, is derived from subgrid scale modelling concepts and, like the variational multiscale method and bubble enrichment techniques, explicitly aims to capture the important under-resolved fine scale information. It brings key advantages to the solution of the Navier-Stokes equations including the use of usually unstable velocity-pressure element pairs, a fully consistent mass matrix without the increase in degrees of freedom associated with discontinuous Galerkin methods and also avoids pressure filtering. All of which act to increase the flexibility and accuracy of a model. Supporting results are presented from an application of the methods to a wide range of problems, from simple one-dimensional examples to tracer and momentum transport in simulations such as the idealised Stommel gyre, the lid-driven cavity, lock-exchange, gravity current and backward-facing step. Significant accuracy improvements are demonstrated in challenging radiation transport benchmarks, such as advection across void regions, the scattering Maynard problem and demanding source-absorption cases. Evolution of a free surface is also investigated in the sloshing tank, transport of an equatorial Rossby soliton, wave propagation on an aquaplanet and tidal simulation of the Mediterranean Sea and global ocean. In combination with adaptive methods, stabilising techniques are key to the development of next generation models. In particular these ideas are critical in achieving the aim of extending models, such as the Imperial College Ocean Model, to the global scale

Output-Feedback Control for a Class of Stochastic High-Order Feedforward Nonlinear Systems with Delay

The problem of global output-feedback stabilization for a class of stochastic high-order time-delay feedforward nonlinear systems with different power orders is investigated. By combining the adding one power integrator technique with the homogeneous domination approach, an output-feedback controller design is proposed, which ensures the global asymptotical stability in probability of the closed-loop system

Meshless methods for shear-deformable beams and plates based on mixed weak forms

Thin structural theories such as the shear-deformable Timoshenko beam and Reissner-Mindlin plate theories have seen wide use throughout engineering practice to simulate the response of structures with planar dimensions far larger than their thickness dimension. Meshless methods have been applied to construct numerical methods to solve the shear deformable theories. Similarly to the finite element method, meshless methods must be carefully designed to overcome the well-known shear-locking problem. Many successful treatments of shear-locking in the finite element literature are constructed through the application of a mixed weak form. In the mixed weak form the shear stresses are treated as an independent variational quantity in addition to the usual displacement variables. We introduce a novel hybrid meshless-finite element formulation for the Timoshenko beam problem that converges to the stable first-order/zero-order finite element method in the local limit when using maximum entropy meshless basis functions. The resulting formulation is free from the effects shear-locking. We then consider the Reissner-Mindlin plate problem. The shear stresses can be identified as a vector field belonging to the Sobelov space with square integrable rotation, suggesting the use of rotated Raviart-Thomas-Nedelec elements of lowest-order for discretising the shear stress field. This novel formulation is again free from the effects of shear-locking. Finally we consider the construction of a generalised displacement method where the shear stresses are eliminated prior to the solution of the final linear system of equations. We implement an existing technique in the literature for the Stokes problem called the nodal volume averaging technique. To ensure stability we split the shear energy between a part calculated using the displacement variables and the mixed variables resulting in a stabilised weak form. The method then satisfies the stability conditions resulting in a formulation that is free from the effects of shear-locking.Open Acces

The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications

International audienceHybrid High-Order (HHO) methods are new generation numerical methods for models based on Partial Differential Equations with features that set them apart from traditional ones. These include: the support of polytopal meshes including non star-shaped elements and hanging nodes; the possibility to have arbitrary approximation orders in any space dimension; an enhanced compliance with the physics; a reduced computational cost thanks to compact stencil and static condensation. This monograph provides an introduction to the design and analysis of HHO methods for diffusive problems on general meshes, along with a panel of applications to advanced models in computational mechanics. The first part of the monograph lays the foundation of the method considering linear scalar second-order models, including scalar diffusion, possibly heterogeneous and anisotropic, and diffusion-advection-reaction. The second part addresses applications to more complex models from the engineering sciences: non-linear Leray-Lions problems, elasticity and incompressible fluid flows

Adaptive low and high-order hybridized methods for unsteady incompressible flow simulations

Tesi en modalitat de cotutela: Universitat Politècnica de Catalunya i Università degli Studi di PaviaSimulations of incompressible flows are performed on a daily basis to solve problems of practical and industrial interest in several fields of engineering, including automotive, aeronautical, mechanical and biomedical applications. Although finite volume (FV) methods are still the preferred choice by the industry due to their efficiency and robustness, sensitivity to mesh quality and limited accuracy represent two main bottlenecks of these approaches. This is especially critical in the context of transient phenomena, in which FV methods show excessive numerical diffusion. In this context, there has been a growing interest towards high-order discretisation strategies in last decades. In this PhD thesis, a high-order adaptive hybidisable discontinuous Galerkin (HDG) method is proposed for the approximation of steady and unsteady laminar incompressible Navier-Stokes equations. Voigt notation for symmetric second-order tensors is exploited to devise an HDG method for the Cauchy formulation of the momentum equation with optimal convergence properties, even when low-order polynomial degrees of approximation are considered. In addition, a postprocessing strategy accounting for rigid translational and rotational modes is proposed to construct an element-by-element superconvergent velocity field. The discrepancy between the computed and postprocessed velocities is utilised to define a local error indicator to drive degree adaptivity procedures and accurately capture localised features of the flow. The resulting HDG solver is thus extended to the case of transient problems via high-order time integration schemes, namely the explicit singly diagonal implicit Runge-Kutta (ESDIRK) schemes. In this context, the embedded explicit step is exploited to define an inexpensive estimate of the temporal error to devise an efficient timestep control strategy. Finally, in order to efficiently solve the global problem arising from the HDG discretisation, a preconditioned iterative solver is proposed. This is critical in the context of high-order approximations in three-dimensional domains leading to large-scale problems, especially in transient simulations. A block diagonal preconditioner coupled with an inexpensive approximation of the Schur complement of the matrix is proposed to reduce the computational cost of the overall HDG solver. Extensive numerical validation of two and three-dimensional steady and unsteady benchmark tests of viscous laminar incompressible flows is performed to validate the proposed methodology.Simulaciones de flujo incompresible se emplean a diario para resolver problemas de interés práctico e industrial en varios campos de la ingeniería, p.ej. en aplicaciones automovilísticas, aeronáuticas, mecánicas y biomédicas. Aunque los métodos de volúmenes finitos (FV) siguen siendo la opción preferida por la industria debido a su eficiencia y robustez, la sensibilidad a la calidad de la malla y la baja precisión representan dos limitaciones importantes para estas técnicas. Estas limitaciones son todavía más críticas en el contexto de simulaciones de fenómenos transitorios, donde los FV están penalizados por su excesiva difusión numérica. En este contexto, las estrategias de discretización de alto orden han ganado una popularidad creciente en las últimas décadas para problemas transitorios dónde se necesitan soluciones precisas. Esta tesis propone un método de Galerkin discontinuo híbrido (HDG), de alto orden y adaptativo para la aproximación de las ecuaciones de Navier-Stokes incomprensible laminar, en el caso estacionario y transitorio en el entorno de aplicaciones ingenieriles. Para ello, la notación de Voigt para tensores simétricos de segundo orden (habituales en mecánica de los medios continuos) permite introducir un método HDG para la formulación de Cauchy de la ecuación de momento. La novedad de este resultado reside en la convergencia óptima alcanzada por el método, incluso para aproximaciones de orden polinómico bajo. Además, se desarrolla una estrategia de post-proceso local para construir elemento a elemento un campo de velocidad súper-convergente, tomando en cuenta los modos rígidos de traslación y rotación. La discrepancia entre el campo de velocidad calculado y el súper-convergente, obtenido a través del post-proceso, permite definir un indicador del error local. De esta forma, se desarrolla una estrategia para realizar adecuar elemento a elemento el grado de la aproximación polinómica y así mejorar la precisión adaptándose a las características localizadas del flujo. Seguidamente, se extiende el método HDG propuesto al tratamiento de problemas dependientes del tiempo. Más concretamente, se consideran los esquemas de integración temporal de alto orden explicit singly diagonal implicit Runge-Kutta (ESDIRK). En este contexto, se utiliza el paso explícito embedded para calcular una estimación computacionalmente eficiente del error temporal y definir una estrategia de adaptividad del paso de tiempo. Finalmente, se desarrolla un precondicionador adaptado a la estrategia HDG que acelera la convergencia del método iterativo empleado y, de esta forma, obtener resoluciones eficaces del problema global surgido de la discretización HDG. Es importante resaltar la importancia de una herramienta de resolución eficiente para problemas de gran escala en el contexto de aproximaciones de alto orden y en dominios tridimensionales. Estas herramientas se hacen aún más criticas en simulaciones transitorias. Más concretamente, se proponen un precondicionador diagonal por bloques y una aproximación eficiente del complemento Schur de la matriz para reducir el coste computacional del método HDG. Para validar la metodología propuesta, se realizan varias simulaciones numéricas de flujo incompresible laminar viscoso, para problemas estacionarios y transitorios, en dos y tres dimensiones.Postprint (published version

Computational modelling of iron-ore mineralisation with stratigraphic permeability anisotropy

This study develops a computational framework to model fluid transport in sedimentary basins, targeting iron ore deposit formation. It offers a simplified flow model, accounting for geological features and permeability anisotropy as driving factors. A new finite element method lessens computational effort, facilitating robust predictions and cost-effective exploration. This methodology, applicable to other mineral commodities, enhances understanding of genetic models, supporting the search for new mineral deposits amid the global energy transition