International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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Please note that all times are shown in the time zone of the conference. The current conference time is: 9th Dec 2022, 01:02:38am CET
Z9b: wave propagation problems
4:10pm - 4:30pm
Trefftz method for electromagnetic wave simulation in three dimensions
1ONERA; 2Projet Makutu (INRIA,CNRS,E2S-UPPA)
The simulation of time-harmonic electromagnetic waves requires a matrix inversion whose cost, espe-
cially in three-dimensional cases, increases quickly with the size of the computational domain. This
is a tangible issue regarding memory consumption when the size of the domain comes of a few do-
zen wavelengths in each direction. The so-called pollution effect tends to accentuate this problem.
This phenomena forces to increase the number of discretization points per wavelength to ensure the
accuracy of the numerical solution when the size of the domain increases.
A conventional idea consists in reducing computing costs by performing domain decomposition. This
method requires the resolution of smaller auxiliary problems on each subdomain. These subdomains
are then coupled thanks to Robin fluxes which ensure the convergence of the method. These methods
are efficient but not flexible enough to be integrated easily in industrial codes.
Recently, lots of authors have been studying Trefftz methods. Such methods offer a good flexi-
bility of the mesh regarding both forms and sizes of cells. Therefore, they can deal with complex
geometrical constraints of industrial environment. Trefftz methods consist in using a discontinuous
Galerkin method whose basis functions are defined as local solutions of the studied equation. They
can be given either analytically by a sum of plane waves or numerically by an auxiliary solver. These
basis functions are specific to the considered physical problem and thus reduce numerical dispersion
phenomena. Trefftz methods are also particularly adapted to domain decomposition methods. It is
then possible to come up with an iterative Trefftz solver.
In this presentation, we will show different Trefftz methods for solving time-harmonic Maxwell problem
in three dimensions. We will deal with the case of an auxiliary analytical solver using plane waves, and
with the case of an auxiliary numerical solver using high order Nédélec finite elements. A comparison
between different formulations will be given. We will also pay particular attention to the accuracy
of the method and to the memory necessary for the resolution. Finally, reasons why these numerical
methods are adapted to modern architectures will be brought to the fore.
4:30pm - 4:50pm
Multi-scale time domain Full Waveform Inversion using evolutive hp-adaptivity
1Inria, France; 2Total R&D, France
Geophysical exploration makes extensive use of numerical simulations to determine the physical characteristics of the subsurface. The parameters of interest (e.g., velocity, density) are obtained by solving an optimization problem under constraints. The Full Waveform Inversion (FWI) relies on minimizing a cost function that quantifies the distance between experimental and numerical data. Time-harmonic FWI has demonstrated its ability in retrieving physical parameters of complex media [Faucher, Ph.D. thesis, 2017]. However, its effectiveness depends on low-frequency information that is often missing in real data. Besides, the latter is collected in the time domain, which implies additional processing to get time-harmonic observations. Finally, time-harmonic solvers still use a lot of memory, which can even be overloaded in 3D cases.
In this work, we propose a FWI governed by time-domain acoustic full-wave equations. Both the direct and the adjoint wave-fields are computed in the time domain. Concerning the space discretization, we are using a solver based on the Discontinuous Galerkin method that allows for a natural h and p adaptivity [Wilcox, Journal of Computational Physics, 2010] and good scalability properties. However, in the course of the FWI workflow, the physical model keeps evolving, and to aid convergence towards the global minimum of the cost function, we employ a multi-scale reconstruction by increasing step by step the frequency of the simulated phenomena [Bunks et al. Geophysics, 1995, vol. 60 n 5, pp. 1457 -- 1473]. Automation of the meshing process in the FWI loop is then crucial to deal with the variations of the physical parameters and the increase of the frequency component of the waves simulated. This way, we guarantee accurate direct and adjoint simulations, which are intrinsic to the FWI loop, with an adapted computational burden for each step of the inversion. For that purpose, we propose to adapt the mesh adaptation tools (mmgtools.org) regarding the unknowns of the inverse problem instead of direct problem ones as it is classically employed. This new feature enables to have an evolutive h and p adaptivity through the different simulations required in the inversion loop.
During this talk, we will present the industrial workflow of the multi-scale time-domain FWI implemented in the Total Research and Development platform, and its extension to handle mesh adaptation. The mesh evolution impacts on the parametrization of the physical parameters. This is the reason why the discretization must be chosen by taking into account the parametrization of the inverse problem while getting efficient and accurate direct and adjoint simulations. In our presentation, we will then discuss in detail the choice of the discretization either in terms of parametrization which can be constant or varying linearly inside each element thanks to the Weight Adjusted Discontinuous Galerkin (WADG) techniques [Chan et al, SIAM journal on Scientific Computing, 2017, vol. 39, n 6, pp. 2935 -- 2961] but also in terms of numerical schemes more particularly in the choice of nodal or modal polynomial bases [Chan et al, SIAM journal on Scientific Computing, 2017, vol. 39, n 2, pp. 628 -- 654]. These methods allow the use of high order polynomial approximations while preserving a sufficient parametrization and affordable computational cost. The contribution of this work relies on the design of an evolutive hp-adaptivity in an inversion workflow based on mesh adaptation tools and Discontinuous Galerkin methods properties.
4:50pm - 5:10pm
Hybridizable Discontinuous Galerkin Methods for modeling Seismoelectric effects
1Inria Bordeaux Sud-Ouest; 2E2S-UPPA
Near-surface exploration using wave propagation poses significant challenges and has been investigated for many years. Classically, for computational reasons, wave propagating in the subsurface are modelled as solutions to the elastic or acoustic equations. However, to improve the accuracy of the simulation, it is now necessary to consider more complex models such as conducting poroelastic media. The poroelastic materials are composed of an elastic solid frame and pores filled with fluid. Wave propagation in poroelastic materials is described by Biot’s model [Biot, 1956, Journal of the Acoustical Society of America], which employs a much larger number of physical parameters compared to acoustics or elasticity, and thus provides a more accurate representation of geophysical layers by taking into consideration physical phenomena such as porosity or attenuation. When the fluid inside the pores is polarized, the propagation of a seismic wave generates a mesurable electric current, the so-called coseismic wave. This is part of the seismoelectric effects and it can be modeled using Pride's equations [Pride, 1994, Physical Review], which couple Maxwell’s equations and Biot's equations.
In natural geophysical media, the coupling between the seismic and electromagnetic phenomenon is small, but it has been observed in the field and in laboratory experiments. The study of the seismokinetic conversions is very interesting because it can bring more information on the material in consideration, that we could not have access to by covering separately the electromagnetic or seismic wave propagation. The coseismic wave travels at the same speed as the seismic wave and carries the same information on the media. However, when these waves impinge an interface between two porous media, they generate a converted electromagnetic wave that propagates at a much faster speed. This wave can highlight interfaces that would not be detected using the seismic fields only.
As we have mentioned above, the characterization of conducting poroelastic media is complex and involves many physical parameters. Some of these parameters depend non-linearly on the frequencies. In addition, the seismic and electromagnetic velocities are significantly different, which causes difficulties for time domain simulations. Hence, we have chosen to solve the equations in the frequency domain and to use a Fourier transform to generate the seismograms in time domain. The counterpart is that we must invert one linear system for each frequency, which contributes to increase the computational burden because of the complexity of the equations and of the high number of unknowns.
In second-order formulation, the equations are described by the solid velocity, the fluid velocity, the electric field, hence nine scalar unknowns by degrees of freedom in three dimensions. For instance, a cell of order 3 contains 20 degrees of freedom. This means that we have 180 unknowns per element. A typical geophysical mesh is composed of several millions of cells, so that a classical Discontinuous Galerkin (DG) method would lead to a linear system of one billion of unknowns. For this reason, we have chosen to consider the Hybridizable Discontinuous Galerkin (HDG) method, which possesses all the advantages of DG method (hp-adaptivity, robustness, easily parallelizable, etc.) without a drastic increase in the number of degrees of freedom. A HDG method is a DG method whose interior degrees of freedom can be removed to end up with a global system involving a discrete hybrid unknown defined on the skeleton of the mesh. The hybridization is implemented thanks to transmission conditions set on the interfaces between elements. HDG methods have been first introduced for Helmholtz equation [Cockburn, 2009, SIAM Journal on Numerical Analysis], then have been developed for acoustics and elastodynamics [Nguyen et al, 2011, Journal of Computational Physics], electromagnetism [Li et al, 2013, COMPEL] and poroelasticity [Fu, 2019, Computers & Mathematics with Applications, Hungria, 2019 PhD Thesis, Barucq et al, 2021, IJNME]. In this talk, we will present the development and implementation of a HDG method for solving Pride’s equations. We will validate the code in two dimensions in circular geometry thanks to analytical solutions that we have developed. Using these analytical solutions, we will show that the method has an optimal order of convergence in the L2 norm. In addition, for the treatment of infinite domains, we will propose a new absorbing boundary condition. Finally, we will present comparisons of our numerical results with laboratory experiments.
5:10pm - 5:30pm
Solving 3D time-harmonic multiple scattering problems using high order difference potentials
1North Carolina State University, United States of America; 2Tel Aviv University, Israel
We consider scattering of a given monochromatic wave about two spheres at a distance from one another. The solution is governed by the three-dimensional Helmholtz equation in the region exte-rior to the spheres. Boundary conditions on the spheres can be different. The computational domain is terminated by a spherical artificial outer boundary that encloses both spheres. At this boundary, we set local high order artificial boundary conditions (ABCs) of Bayliss-Gunzburger-Turkel type (BGT). The BGT conditions are imposed directly and do not require any auxiliary variables to be introduced at the artificial outer boundary.
The Helmholtz equation is discretized with sixth order accuracy on a Cartesian grid using a com-pact finite difference scheme. The non-conforming spherical boundaries of the two scatterers, as well as the non-conforming spherical outer boundary, are handled by the method of difference po-tentials (MDP), which guarantees no deterioration of accuracy.
In MDP, one computes the discrete counterparts of Calderon’s boundary projection operators us-ing a system of basis functions chosen at the boundary. In the specific setting we are considering, the overall boundary is composed of three spheres. The corresponding projection operators apply to a broad variety of formulations and need to be computed only once for a given geometry (size and distance between the spheres). After that, the scattering solution for any impinging wave and any type/combination of the boundary conditions on the surface of the scatterers can be obtained at a very low cost. Moreover, taking spherical harmonics as basis functions (eigenfunctions of the Beltrami operator on the sphere) allows us to eliminate the auxiliary variables in our implementa-tion of the high order BGT artificial boundary conditions.
The proposed methodology has been tested numerically. The simulations corroborate its design level of performance.
Work supported by the US Army Research Office (ARO) under grant number W911NF-16-1-0115 and by the US--Israel Binational Science Foundation (BSF) under grant number 2014048.
5:30pm - 5:50pm
A Discontinuous Galerkin Method for Wave Equations on Curved Geometries with Dirac Delta Source Terms
University of Massachusetts Dartmouth, United States of America
Linear wave equations with source terms containing Dirac delta functions and their derivatives occur in many areas of physics. For example, in extreme mass ratio binary (EMRB) systems solving the full Einstein equation is currently impossible, and a tractable approach is to linearize the equation around a non-flat geometry. The resulting wave equation is defined on a curved geometry and sourced by Dirac delta functions of potentially many derivatives. We develop a spectrally-convergent Discontinuous Galerkin method to numerically solve such wave equations. As proof of concept, we apply our method to an ordinary wave equation sourced by arbitrary nth-order derivatives of a Dirac delta function and a scalar wave equation defined on a Kerr geometry.
5:50pm - 6:10pm
Sparse spectral methods for the helically reduced Einstein equations with gauge-preserving boundary conditions
University of New Mexico, United States of America
Numerical relativity (NR) computationally solves the Einstein equations, often for configurations of two massive objects (binaries). We consider a nonstandard problem in NR, construction of binary configurations which solve the helically reduced Einstein equations, as formulated by Beetle, Bromley, and Price (BBP). Although ultimately unphysical, helically symmetric solutions (or approximations) are of mathematical interest, in particular since they involve balance of incoming and outgoing radiation. We have earlier described a multidomain spectral-element approach for solving the BBP equations via relaxation, despite these equations featuring a mixed-type operator L. To rapidly solve the linear systems which arise in our iterative scheme, we have developed sparse modal methods based on the application of spectral integration matrices. We also employ fast inversion of our subdomain approximations of L, either through modal-based preconditioning or fast direct schemes. After an overview, we will describe a new aspect of our work: incorporation of the "gauge-preserving" boundary conditions (BCs) based on those developed by Kreiss, Reula, Sarbach, and Winicour. Adaptation of these BCs to helical symmetry and their implementation through spectral-tau methods is a challenge. We report on some details and the extent to which the new BCs yield numerical solutions which obey the harmonic gauge condition assumed in the BBP formulation.
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