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: 30th Nov 2022, 09:09:23pm CET
Z9a: wave propagation problems
4:00pm - 4:20pm
Complex-Scaled Infinite Elements for Resonance Problems in Open Systems
1ENSTA Paris; 2TU Wien
Complex scaling is a popular method to treat scattering and resonance problems in open domains. Thereby the unbounded domain is decomposed into a bounded interior and an unbounded exterior part. Subsequently the technique of complex scaling is applied to the exterior domain to obtain exponentially decreasing solutions. Finally, when using so-called perfectly matched layers (PMLs), the complex-scaled exterior is usually truncated and discretized using finite elements or finite differences.
Contrary to the use of PMLs we suggest the use of complex-scaled infinite elements which are closely related to Hardy space infinite elements. For discretizing the exterior complex-scaled problems we use a tensor product method based on the decomposition of the exterior domain into a surface- and a generalized radial coordinate. The surface component is discretized using standard finite element basis functions while for the discretization of the radial component we use certain spectral ansatz functions. Due to this ansatz we evade introducing the error due to the truncation of the exterior domain and obtain super-algebraic approximation properties. Moreover, we are able to treat a class of inhomogeneous exterior domains which is not straightforward when using Hardy space infinite elements.
4:20pm - 4:40pm
Asymptotic Green's function methods for wave propagations
Iowa State University, United States of America
We will present asymptotic methods for simulating high frequency wave propagation. The Huygens' principle or the Feymann's path integral is used as the time propagator for the wavefunction, where the Green's functions are approximated by asymptotic approximations. Upon obtaining analytic approximations for the phase and amplitude of the Green's functions, the resulting integral can be evaluated by fast Fourier transform after appropriate lowrank approximations. The perfectly matched layer method is incorporated to restrict the computation onto a bounded domain of interest. Numerical example will be presented to demonstrate the asymptotic Green's function methods.
4:40pm - 5:00pm
A matrix-free high-order spectral implementation of the acoustic wave equation with a perfectly matched layer.
1Universidade de São Paulo, Brazil; 2Imperial College London
This work studies the application of a matrix-free method, using sum-factorization, to a high-order spectral element implementation of the acoustic wave problem with a perfectly matched layer. Matrix-free methods have shown to be effective for increasing computational performance in high-order problems that require parallel scalability. Using spectral elements we discuss the computational cost and memory storage at different polynomial degrees and in different architectures. In this talk we will also compare the matrix-free high-order spectral element implementation in quadrilaterals with a matrix-free mass-lumped higher-order simplicial finite element wave propagation.
The acoustic wave equation is solved using a bi-dimensional and tri-dimensional mesh and a perfectly matched layer is applied in order to reduce reflections at the boundary. The added Perfectly-Matching Layer increases the degrees of freedom of the problem by adding a vector-valued auxiliary variable, in the two-dimensional case, or a vector-valued and scalar auxiliary variables, in the tri-dimensional case, therefore increasing the computational cost. The problem was solved using a open source coding package called spyro which utilizes the Firedrake framework and a discretization relevant to the forward part of Full Waveform Inversion in active source seismic imaging.
This research was carried out in association with the ongoing R\&D project registered as ANP 20714-2, “Software technologies for modeling and inversion, with applications in seismic imaging” (University of São Paulo / Shell Brasil / ANP).
5:00pm - 5:20pm
Divergence error based adaptivity in discontinuous Galerkin solution of time-domain Maxwell’s equations
1Dept. of Aerospace Engineering, Indian Institute of Technology Bombay; 2Dept. of Mechanical Engineering, Indian Institute of Technology Bombay
A variety of physical phenomena are modelled as systems of partial differential equations that admit divergence-free solutions. In some of these, like the incompressible Euler and Navier-Stokes equations, this condition of the solution being divergence-free is enforced explicitly. In certain other systems like the time-domain Maxwell’s equations, the usual practice is to incorporate the solenoidal condition within the evolution equations, combined with the requirement that the initial conditions be solenoidal. It relies on the reasoning that if field variables are initially divergence-free, they remain so when evolved in time using the first order div-curl equations. Jiang et. al.  have challenged the arguments that this system is overdetermined and satisfying the solenoidal condition initially, ensures that it is satisfied at all times. Setting aside issues of formulation, not all numerical schemes satisfy the solenoidal condition.
The finite difference time domain (FDTD) method proposed by Yee , satisfies the divergence-free condition by design. It uses a grid where different components of the field variables are computed at staggered spatial and temporal points. Advancements in higher order Godunov schemes for problems in computational electromagnetism (CEM) gave rise to the finite volume time domain (FVTD)  and discontinuous Galerkin time domain (DGTD) methods . These schemes do not account for the divergence constraint. In literature, there are various approaches to meet the constraint imposed by Gauss’ law. A divergence cleaning step is often added that solves a Poisson equation for a correction potential. Assous  used a constrained variational formulation of Maxwell’s equations and applied a penalization technique. In , Munz et. al. reformulated the constrained Maxwell’s equations and introduced a coupling term into Gauss’ law, rendering a perfectly hyperbolic system of equations. This made for a natural extension of the explicit methods for Maxwell’s equations to a purely hyperbolic system. In DGTD, with standard piecewise polynomial spaces used and no dedicated measures for constraint preservation taken, it is observed that global divergence errors are kth order small when using polynomial bases of degree k to represent the solution .
Divergence errors accruing in conservative higher-order formulations do not significantly impact the overall accuracy of the solution  and are often disregarded in practice. In , Cioni et. al used a mixed finite volume/finite element method to show that divergence error, despite being linked to the accuracy of the solver and the underlying discretization, does not hamper the formal accuracy of the solution.
In this paper, we propose another point of view, that of constructively using
this error in divergence to improve spatial accuracy, rather than of either ignoring or eliminating the naturally occurring divergence error in
conservative, non FDTD frameworks. We establish that relative divergence and relative truncation errors are related and propose that divergence error can be used as an effective truncation error indicator. Since solving the evolution equations does not decrease divergence error in computations, it constantly tracks truncation error. The proposed divergence based error indicator may be utilized to drive adaptive methods that assign spatial operators of varying accuracy in the computational domain, with the motive of achieving desired levels of accuracy using fewer degrees of freedom.
We begin with the transverse magnetic (TM) mode of the time-domain Maxwell’s equations to formulate the formal relation between the relative truncation error in the residual appearing in the semi-discrete system and the propagation of the associated relative divergence errors. Here, relative refers to the difference between quantities computed using different discretizations . We extend this definition to incorporate it in a p-adaptive DGTD framework. Different levels of discretizations are obtained by operators formed using polynomial bases of varying degrees. The formulated relation is applied on plane wave solutions and the resultant simplified expressions obtained, are verified by solving canonical problems using DGTD. The correlation between discretization error in solutions and their divergence is also shown with numerical examples.
 Bo Nan Jiang, Jie Wu, and L. A. Povinelli. “The origin of spurious solutions in computational electromagnetics”. In: Journal of Computational Physics 125.1 (1996), pp. 104–123. issn: 00219991. doi: 10.1006/jcph.1996.0082.
 Kane Yee. “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media”. In: IEEE Transactions on
Antennas and Propagation 14.3 (1966), pp. 302–307. issn: 0018-926X. doi:
10.1109/TAP.1966.1138693. url: http : / / ieeexplore.ieee.org/document/1138693/.
 Vijaya Shankar, Alireza H Mohammadian, and William F Hall. “A Time-
Domain, Finite-Volume Treatment for the Maxwell Equations”. In: Electromagnetics 10.1-2 (1990), pp. 127–145. doi: 10.1080/02726349008908232. url: https://doi.org/10.1080/02726349008908232.
 Jan S. Hesthaven and Tim Warburton. Nodal Discontinuous Galerkin Methods. Vol. 54. Texts in Applied Mathematics. New York, NY: Springer New York, 2008. isbn: 978-0-387-72065-4. doi: 10.1007/978-0-387-72067-8.
 F. Assous et al. “On a Finite-Element Method for Solving the Three-Dimensional Maxwell Equations”. In: Journal of Computational Physics
109.2 (1993), pp. 222–237. issn: 00219991. doi: 10.1006/jcph.1993.1214.
 C.-D. Munz et al. “Divergence Correction Techniques for Maxwell Solvers
Based on a Hyperbolic Model”. In: Journal of Computational Physics 161.2
(2000), pp. 484–511. issn: 00219991. doi: 10.1006/jcph.2000.6507. url:
 Bernardo Cockburn, Fengyan Li, and Chi Wang Shu. “Locally divergence-free discontinuous Galerkin methods for the Maxwell equations”. In: Journal of Computational Physics 194.2 (2004), pp. 588–610. issn: 00219991. doi: 10.1016/j.jcp.2003.09.007.
 J.P. Cioni, Loula Fezoui, and Herve Steve. “A Parallel Time-Domain Maxwell Solver Using Upwind Schemes and Triangular Meshes”. In: IMPACT of Computing in Science and Engineering 5.3 (1993), pp. 215–247. issn: 08998248. doi: 10.1006/icse.1993.1010. url: https://linkinghub.elsevier.com/retrieve/pii/S0899824883710104.
 Avijit Chatterjee. “A Multilevel Numerical Approach with Application in
Time-Domain Electromagnetics”. In: Communications in Computational Physics 17.3 (2015), pp. 703–720. issn: 1815-2406. doi: 10.4208/cicp.181113.271114a. url: https://www.cambridge.org/core/product/
5:20pm - 5:40pm
High-Order Finite Element Simulations of Optical Fiber Amplifiers
1The University of Texas at Austin, United States of America; 2Air Force Research Laboratory, United States of America
Fiber laser amplifiers are of interest in communication technology, medical applications, and military defense capabilities. Silica fiber amplifiers can achieve high-power operation with great efficiency. At high optical intensities, multi-mode amplifiers suffer from undesired thermal coupling effects which pose a major obstacle in power-scaling of such devices. In this talk, both modeling and computational advancements to a unique three-dimensional finite element (FE) model for the simulation of laser amplification in a fiber amplifier are presented. This model is based on the time-harmonic Maxwell equations for two weakly coupled electromagnetic fields, and it incorporates thermal effects via coupling with the heat equation. The high-frequency nature of the wave propagation problem requires the use of high-order discretizations to effectively counter numerical pollution. The discontinuous Petrov-Galerkin (DPG) FE method provides a stable discretization with a built-in error indicator. For simulating a significant fiber length, a scalable parallel implementation is critical. The model is parallelized with an MPI/OpenMP implementation, based on a parallel nested dissection solver suited for the DPG linear system. Weak scalability for the model is shown with up to 24,576 cores on manycore compute architectures, enabling the solution of the nonlinear 3D fiber model with more than 8,000 wavelengths.
5:40pm - 6:00pm
An Energy Conservative High Order Method for Liouville’s Equation of Geometrical Optics
1Eindhoven University of Technology, Netherlands, The; 2Signify
Traditionally, in geometrical optics the illuminance or luminous intensity at a target is computed using Monte Carlo ray tracing. This approach exhibits slow convergence, and in general does not obey energy conservation when using forward ray tracing. An alternative approach is based on Liouville’s equation which states the conservation of energy and describes the advection of basic luminance in phase space. The basic luminance is roughly the luminous flux per unit area, per unit solid angle, and phase space describes the set of positions and direction coordinates. From the basic luminance integrated quantities, such as the illuminance or luminous intensity, can be computed. Ray tracing can directly compute these integrated quantities, whereas for Liouville’s equation one has to compute the basic luminance on phase space first and subsequently integrate to obtain these quantities. Despite the increased dimensionality compared to ray tracing, Liouville’s equation can be solved using a high order numerical method and, therefore, potentially yield faster time to error.
At an optical interface the laws of optics, such as specular reflection and Snell’s law, have to be applied. These laws describe a discontinuous change in the direction coordinate of a ray, i.e., a jump in phase space. Hence, the basic luminance is discontinuous across an optical interface. A suitable method to solve Liouville’s equation is the discontinuous Galerkin spectral element method (DGSEM). At an optical interface non-local boundary conditions arise as a consequence of the laws of optics. Moreover, these non-local boundary conditions must also satisfy energy conservation constraints. Due to these non-local boundary conditions the elements close to the optical interface are connected in a non-trivial manner. A method was developed to handle optical interfaces for two-dimensional optics ensuring energy conservation up to machine precision. In an example numerical experiments show that the high convergence rate of the DGSEM is preserved. Moreover, the DGSEM is compared to ray tracing. The results show that the DGSEM significantly outperforms ray tracing for time to error, yielding a much faster convergence to accurate solutions.
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