Error-driven dynamicalhp-meshes with the Discontinuous Galerkin Method for three-dimensional wave propagation problems

Abstract

An hp-adaptive Discontinuous Galerkin Method for electromagnetic wave propagation phenomena in the time-domain is proposed. The method is highly efficient and allows for the first time the adaptive full-wave simulation of large, time-dependent problems in three-dimensional space. Refinement is performed anisotropically in the approximation order p and the mesh step size h regardless of the resulting level of hanging nodes. For guiding the adaptation process a variant of the concept of reference solutions with largely reduced computational costs is proposed. The computational mesh is adapted such that a given error tolerance is respected throughout the entire time-domain simulation.

An enhanced transient solver with dynamic p‐adaptation and multirate time integration for electromagnetic and multiphysics simulations

“…When the physical variation is smooth, the p ‐refinement is usually superior to the h ‐refinement counterpart in providing a better numerical accuracy for the same efficiency. To achieve a higher computational efficiency while maintaining the same numerical accuracy, the element size and the polynomial order can be adjusted dynamically in real time of a simulation, known as the dynamic h ‐ and p ‐adaptation, respectively, which have been investigated extensively in many areas including EM, elastics, and fluid dynamics . In these efforts, fluid dynamic problems were considered with adaptive Cartesian mesh, elastic wave problems were simulated using a discontinuous Galerkin time‐domain (DGTD) method with p ‐adaptation and local time stepping, and shallow water equations were solved with dynamic p ‐adaptation based on the DGTD method .…”

“…In the numerical simulation of EM problems, adaptive Cartesian meshes (also known as the adaptive mesh refinement) were employed in the finite‐volume time‐domain (FVTD) and the finite‐difference time‐domain (FDTD) methods. A dynamic hp ‐adaptation for EM problems was also developed based on the DGTD method …”

“…To achieve a higher computational efficiency while maintaining the same numerical accuracy, the element size and the polynomial order can be adjusted dynamically in real time of a simulation, known as the dynamic h-and p-adaptation, respectively, which have been investigated extensively in many areas including EM, elastics, and fluid dynamics. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] In these efforts, fluid dynamic problems were considered with adaptive Cartesian mesh, 4 elastic wave problems were simulated using a discontinuous Galerkin time-domain (DGTD) method with p-adaptation and local time stepping, 7 and shallow water equations were solved with dynamic p-adaptation based on the DGTD method. 10 In the numerical simulation of EM problems, adaptive Cartesian meshes (also known as the adaptive mesh refinement) were employed in the finite-volume time-domain (FVTD) 5 and the finite-difference time-domain (FDTD) 12 methods.…”

A p‐adaptive method for electromagnetic wave propagation

“…In parallel to the continuous FEMs (essentially implicit in nature), developments of DG methods, which can be advanced explicitly in time and have lower memory requirements, have been pursued for the numerical solution of the Maxwell equations . The ability of the DG discretization to achieve electromagnetic wave propagation in media with different properties was demonstrated .…”

Efficient Antenna Modeling by DGTD: Leap-frog discontinuous Galerkin timedomain method

“…The schemes aimed to obtain hp-adaptability in the TD [16]- [18], where the use of highorder time integrators must be considered, deserve a special mention.…”