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

Transient Propagation and Scattering of Quasi-Rayleigh Waves in Plates: Quantitative comparison between Pulsed TV-Holography Measurements and FC(Gram) elastodynamic simulations

We study the scattering of transient, high-frequency, narrow-band quasi-Rayleigh elastic waves by through-thickness holes in aluminum plates, in the framework of ultrasonic nondestructive testing (NDT) based on full-field optical detection. Sequences of the instantaneous two-dimensional (2-D) out-of-plane displacement scattering maps are measured with a self-developed PTVH system. The corresponding simulated sequences are obtained by means of an FC(Gram) elastodynamic solver introduced recently, which implements a full three-dimensional (3D) vector formulation of the direct linear-elasticity scattering problem. A detailed quantitative comparison between these experimental and numerical sequences, which is presented here for the first time, shows very good agreement both in the amplitude and the phase of the acoustic field in the forward, lateral and backscattering areas. It is thus suggested that the combination of the PTVH system and the FC(Gram) elastodynamic solver provides an effective ultrasonic inspection tool for plate-like structures, with a significant potential for ultrasonic NDT applications.Comment: 46 pages, 16 figures, corresponding author Jos\'e Carlos L\'opez-V\'azquez, [email protected]. Changes: 1st, 4th, 5th paragraphs (intro), 3rd, 4th paragraphs (sec. 4); [59-60] cited only in appendixes; old ref. [52] removed; misprints corrected in the uncertainty of c_L (subsec. 3.1), citation to fig. 10 (sec. 4), size of images (caption fig.15); reference to Lam\'e constants removed in subsec. 3.

The VOLNA code for the numerical modelling of tsunami waves: generation, propagation and inundation

A novel tool for tsunami wave modelling is presented. This tool has the potential of being used for operational purposes: indeed, the numerical code \VOLNA is able to handle the complete life-cycle of a tsunami (generation, propagation and run-up along the coast). The algorithm works on unstructured triangular meshes and thus can be run in arbitrary complex domains. This paper contains the detailed description of the finite volume scheme implemented in the code. The numerical treatment of the wet/dry transition is explained. This point is crucial for accurate run-up/run-down computations. Most existing tsunami codes use semi-empirical techniques at this stage, which are not always sufficient for tsunami hazard mitigation. Indeed the decision to evacuate inhabitants is based on inundation maps which are produced with this type of numerical tools. We present several realistic test cases that partially validate our algorithm. Comparisons with analytical solutions and experimental data are performed. Finally the main conclusions are outlined and the perspectives for future research presented.Comment: 47 pages, 27 figures. Other author's papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh

A wildland fire model with data assimilation

A wildfire model is formulated based on balance equations for energy and fuel, where the fuel loss due to combustion corresponds to the fuel reaction rate. The resulting coupled partial differential equations have coefficients that can be approximated from prior measurements of wildfires. An ensemble Kalman filter technique with regularization is then used to assimilate temperatures measured at selected points into running wildfire simulations. The assimilation technique is able to modify the simulations to track the measurements correctly even if the simulations were started with an erroneous ignition location that is quite far away from the correct one.Comment: 35 pages, 12 figures; minor revision January 2008. Original version available from http://www-math.cudenver.edu/ccm/report

Cavitation Induction by Projectile Impacting on a Water Jet

The present paper focuses on the simulation of the high-velocity impact of a projectile impacting on a water-jet, causing the onset, development and collapse of cavitation. The simulation of the fluid motion is carried out using an explicit, compressible, density-based solver developed by the authors using the OpenFOAM library. It employs a barotropic two-phase flow model that simulates the phase-change due to cavitation and considers the co-existence of non-condensable and immiscible air. The projectile is considered to be rigid while its motion through the computational domain is modelled through a direct-forcing Immersed Boundary Method. Model validation is performed against the experiments of Field et al. [Field, J., Camus, J. J., Tinguely, M., Obreschkow, D., Farhat, M., 2012. Cavitation in impacted drops and jets and the effect on erosion damage thresholds. Wear 290–291, 154–160. doi:10.1016/j.wear.2012.03.006. URL http://www.sciencedirect.com/science/article/pii/S0043164812000968 ], who visualised cavity formation and shock propagation in liquid impacts at high velocities. Simulations unveil the shock structures and capture the high-speed jetting forming at the impact location, in addition to the subsequent cavitation induction and vapour formation due to refraction waves. Moreover, model predictions provide quantitative information and a better insight on the flow physics that has not been identified from the reported experimental data, such as shock-wave propagation, vapour formation quantity and induced pressures. Furthermore, evidence of the Richtmyer-Meshkov instability developing on the liquid-air interface are predicted when sufficient dense grid resolution is utilised

A trajectory mechanics approach for the study of wave propagation in an anisotropic elastic medium

We derive equations describing the path and traveltime of a coherent elastic wave propagating in an anisotropic medium, generalizing expressions from conventional high-frequency asymptotic ray theory. The methodology is valid across a broad range of frequencies and allows for subwavelength variations in the material properties of the medium. The primary difference from current ray methods is the retention of a term that is neglected in the derivation of the eikonal equation. The additional term contains spatial derivatives of the properties of the medium and of the amplitude field, and its presence couples the equations governing the evolution of the amplitude and phase along the trajectory. The magnitude of this term provides a measure of the validity of expressions based upon high-frequency asymptotic methods, such as the eikonal equation, when modelling wave propagation dominated by a band of frequencies. In calculations involving a layer with gradational boundaries, we find that asymptotic estimates do deviate from those of our frequency-dependent approach when the width of the layer boundaries become sufficiently narrow. For example, for a layer with boundaries that vary over tens of meters, the term neglected by a high-frequency asymptotic approximation is significant for frequencies around 10 Hz. The visible differences in the paths of the rays that traverse the layer substantiate this conclusion. For a velocity model derived from an observed well log, the majority of the trajectories calculated using the extended approach, accounting for the frequency-dependence of the rays, are noticeably different from those produced by the eikonal equation. A suite of paths from a source to a specified receiver, calculated for a range of frequencies between 10 and 100 Hz, define a region of sensitivity to velocity variations and may be used for an augmented form of tomographic imaging