An Efficient Implementation of Perfectly Matched Layers within a High-order Discontinuous Galerkin Time Domain Method
Permanent link to this recordhttp://hdl.handle.net/10754/662547
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AbstractThe perfectly matched layer (PML) is one easy-to-implement and also accurate technique used for domain truncation in wave propagation and scattering simulations carried out using differential-equation based Maxwell equation solvers. The most popular solvers include finite difference methods, finite element methods, and more recently-developed pseudo-spectral time domain method and discontinuous Galerkin time domain method (DGTD). To achieve high-accuracy in the solution (i.e., to reduce the reflection back into the computation domain as much as possible), the PML ideally requires a high conductivity value and/or has to be thick. But, in practice, one cannot increase the thickness too much which would result in higher computational requirements and cannot use a constant high value of conductivity which would increase the numerical reflection at the interface between the computation domain and the PML. Therefore a smoothly increasing conductivity profile is often used for increased accuracy. The DGTD implements the smooth conductivity profile in two different ways. The first method considers the spatial variation of the conductivity inside each element, thus gives a high-order accurate approximation of the material property. However, this results in a different mass matrix for each element and significantly increases the memory requirement since every elemental mass matrix (or its inverse) has to be stored individually. For example, for the stretched-coordinate PML , the memory cost of the elemental mass matrix is 3 × 5 × N p × N p, where N p is the number of interpolating nodes in each element, 5 comes from the coefficients in front of the field and the auxiliary variable, and 3 comes from the field components in Cartesian coordinate system. The second method assumes that the conductivity in each element is constant. This significantly reduces the memory cost (no need for storing individual mass matrices) and simplifies the implementation of the DGTD. However, it introduces material discontinuity between neighboring elements, which eventually leads to large reflections that destroy the high-order convergence of the error in the solution. Thus, to reach a similar level of accuracy as in the first method, the conductivity jump between neighboring elements has to be reduced and therefore more PML layers (and elements) are required. In this work, we present an efficient implementation of the stretched-coordinate PML within the nodal DGTD method. This implementation considers the local variation of the conductivity in each element (as in the first method above) and approximate the local mass matrix with a weight-adjusted approximation (WAA) . By using WAA, only (additional) storage of 15 × N q is required for each element, where N q is the number of quadrature points close to N p. The PML performance is same as the one implemented by the first method. The reflection decreases exponentially with the increasing the interpolation polynomial order, i.e., showing high-order convergence.
SponsorsKing Abdullah University of Science and Technology (KAUST) Office of Sponsored Research (OSR) under Award No 2016-CRG5-2953.
Conference/Event namePhotonIcs & Electromagnetics Research Symposium 2019
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