Discrete flow mapping: transport of phase space

Wave Motion, 2014

Journal of Computational Physics, 2012

Dynamical energy analysis was recently introduced as a new method for determining the distribution of mechanical and acoustic wave energy in complex built up structures. The technique interpolates between standard statistical energy analysis and full ray tracing, containing both of these methods as limiting cases. As such the applicability of the method is wide ranging and additionally includes the numerical modelling of problems in optics and more generally of linear wave problems in electromagnetics. In this work we consider a new approach to the method with enhanced versatility, enabling three-dimensional problems to be handled in a straightforward manner. The main challenge is the high dimensionality of the problem: we determine the wave energy density both as a function of the spatial coordinate and momentum (or direction) space. The momentum variables are expressed in separable (polar) coordinates facilitating the use of products of univariate basis expansions. However this is not the case for the spatial argument and so we propose to make use of automated mesh generating routines to both localise the approximation, allowing quadrature costs to be kept moderate, and give versatility in the code for different geometric configurations.

Mathematics of Computation, 2009

We propose fully discrete schemes to approximate the harmonic map heat flow and wave maps into spheres. The finite-element based schemes preserve a unit length constraint at the nodes by means of approximate discrete Lagrange multipliers, satisfy a discrete energy law, and iterates are shown to converge to weak solutions of the continuous problem. Comparative computational studies are included to motivate finite-time blow-up behavior in both cases.

Bulletin of the Seismological Society of America, 2007

We present a new numerical method for elastic wave modeling in 3D isotropic and anisotropic media, which is called the 3D optimal nearly analytic discrete method (ONADM) in this article. This work is an extension of the 2D ONADM that models acoustic and elastic waves propagating in 2D media. The formulation is derived by using a multivariable truncated Taylor series expansion and high-order interpolation approximations. Our 3D ONADM enables wave propagation to be simulated in three dimensions through isotropic and anisotropic models. Promising numerical results show that the error of the ONADM for the 3D case is less than those of the conventional finite-difference (FD) method and the so-called Lax-Wendroff correction (LWC) schemes, measured quantitatively by the rootmean-square deviation from analytical solution. The seismic wave fields in the 3D isotropic and anisotropic media are simulated and compared with those obtained by using the fourth-order LWC method and exact solutions for the acoustic wave case. We show that, compared with the conventional FD method and the LWC schemes, the ONADM for the 3D case can reduce significantly the storage space and computational costs. Numerical experiments illustrate that the ONADM provides a useful tool for the 3D large-scale isotropic and anisotropic problems and it can suppress effectively numerical dispersions caused by discretizing the 3D wave equations when too-coarse grids are used, which is the same as the 2D ONADM. Numerical modeling also implies that simultaneously using both the wave displacement and its gradients to approximate the high-order derivatives is important for both decreasing the numerical dispersion caused by the discretization of wave equations and compensating the important wave-field information included in wave-displacement gradients.

Engineering Analysis with Boundary Elements, 2002

This paper formulates a computational procedure based on the Trefftz method for the solution of nonlinear transport phenomena. This new unified approach is particularly important when solving coupled, nonlinear, inhomogenous, anisotropic, multiphase, and multifield heat and mass transfer problems. Physical system represents the general transport equation, standing for a broad spectra of mass, energy, momentum, and species transfer problems. This equation is cast into non-linear Poisson form and expanded with respect to the transport variable. Fully implicit time-discretization is used. The particular solution method is applied as a general solution framework. The solution of the inhomogenous part is based on the radial basis function global approximation, and the solution of the homogenous part is based on the Trefftz method Laplace equation fundamental solution global approximation. The discrete approximate method results in a global point-collocation based grid and hence eliminates the need for polygonization of the computational boundary and domain. q

GEOPHYSICS, 2015

Reverse time migration (RTM) has become an extremely important imaging method due to its ability of handling complicated velocity models. In recent years, there has been an increasing need to perform RTM with anisotropic velocity models such as transversely isotropic media with a vertical symmetry axis (VTI). Finite difference (FD) is one of the most widely used numerical methods in RTM. However, it often suffers from serious numerical dispersion, resulting in significant loss of imaging resolution. Over the years, a nearly analytic discrete operator has been developed to approximate the high-order spatial partial differential operators, which is effective in suppressing numerical dispersion. Using this operator, we developed a nearly analytic symplectic (NAS) method to numerically solve acoustic VTI wave equations, which preserves the symplectic structure. We have investigated its properties including the stability condition, numerical dispersion relation, and computational efficiency. Also, we applied the NAS method to solve wave equations in RTM in acoustic VTI media. This new method not only reduced numerical dispersion and improved computational efficiency, compared to the conventional numerical methods such as the Lax-Wendroff correction (LWC) method, but it also was symplectic. Our forward-modeling results showed that the computational efficiency of NAS is about four times and its overall memory requirement is about 75% of those of LWC. The RTM results of the 2D prestack Hess acoustic VTI model showed that RTM by using NAS obtained better image quality than that by using LWC when coarse grids were used, which demonstrated that the NAS method can be promising in large-scale anisotropic RTM.

2017

This work deals with a novel three-dimensional finite-volume non-hydrostatic shock-capturing model for the simulation of wave transformation processes and wave-structure interaction. The model is based on an integral formulation of the Navier-Stokes equations solved on a coordinate system in which the vertical coordinate is varying in time. A finite-volume shock-capturing numerical technique based on high order WENO reconstructions is adopted in order to discretize the fluid motion equations.

Proceedings of the ICA congress, 2019

Dynamical Energy Analysis (DEA) is a mesh-based high frequency method modelling structure borne sound for complex built-up structures. This has proven to enhance vibro-acoustic simulations considerably by making it possible to work directly on existing finite element meshes circumventing time-consuming and costly remodeling strategies. In addition , DEA provides detailed spatial information about the vibrational energy distribution within a complex structure in the mid-to-high frequency range. DEA was successfully applied and validated to a structure borne sound calculation of an assembled agricultural tractor. Modelling solid structures is still a challenge for DEA, however, as it is based on 2D wave transmission calculations. We propose a novel method to generate DEA elements based on measurement data in order to model solid parts of a complex structure. Advanced Transfer Path Analysis (ATPA) is employed to extract energy transmission characteristics of a structure. First, Frequency Response Functions are measured between interface points on a structure. Then a direct transfer function between interface points is calculated by using ATPA. Finally, DEA elements connecting interface points and representing energy transmission characteristics of the structure are created based on the ATPA result. Applications of the method will be presented.

GPU Computing Gems Jade Edition, 2012

Waves are all around us-be it in the form of sound, electromagnetic radiation, water waves, or earthquakes. Their study is an important basic tool across engineering and science disciplines. Every wave solver serving the computational study of waves meets a trade-off of two figures of meritits computational speed and its accuracy. Discontinuous Galerkin (DG) methods fall on the high-accuracy end of this spectrum. Fortuitously, their computational structure is so ideally suited to GPUs that they also achieve very high computational speeds. In other words, the use of DG methods on GPUs significantly lowers the cost of obtaining accurate solutions. This article aims to give the reader an easy on-ramp to the use of this technology, based on a sample implementation which demonstrates a highly accurate, GPU-capable, real-time visualizing finite element solver in about 1500 lines of code.

2008

This review paper presents a unified treatment of the numerical time-domain modeling of acoustic, electromagnetic, and elastodynamic waves and of the combination thereof: this means coupled wave fields, like piezoelectric (elasto-electric) and electromagnetic-ultrasonic wave fields. Since the famous paper by Yee [1966], the time-domain solution of Maxwell’s equations has become very popular, and this initiated the Finite-Difference Time-Domain (FDTD) Method [Taflove and Hagness, 2000; fdtd.org, 2001]. Ten years later, Madariaga [1976] independently developed a similar method in elastodynamics. Both methods start from the governing equations in differential form, using standard second-order finite-difference stencils in space and time. In electromagnetics, Weiland [1977] introduced a different approach, which starts from the full set of Maxwell’s equations in integral form. Today, this method is commonly called the Finite Integration Technique (FIT). In the 1990s, based on Weiland’s ...