Qiwei Zhan

EB Scheme-Based Hybrid SE-FE DGTD Method for Multiscale EM Simulations

This communication presents an EB scheme subdomain-level discontinuous Galerkin time domain (DGTD) method for multiscale simulations. It is an extension of the previous subdomain-level DGTD research by combining the degree of freedom efficiency of spectral element time domain method and the mesh flexibility of the finite element time domain method. Thus, the multiscale problems can be solved efficiently by separating the geometrically fine and coarse parts and meshing them with hexahedrons and tetrahedrons, respectively, via a nonconformal mesh. The implicit-explicit Runge-Kutta method is applied to the EB scheme DGTD method to obtain an efficient time integration approach.

Wave Equation-Based Implicit Subdomain DGTD Method for Modeling of Electrically Small Problems

A second-order wave equation-based implicit discontinuous Galerkin time-domain (DGTD) method is proposed to efficiently model electrically small problems. The proposed method employs the second-order wave equation for electric field (or magnetic field) as the governing equation of the DG formulation, instead of the first-order Maxwell’s curl equations. A modified version of the Riemann solver (upwind flux) is introduced to evaluate the numerical flux resulting from the weak form of the wave equation. Compared with previous first-order Maxwell’s curl equation-based implicit DGTD methods, which typically solve all electric and magnetic field unknowns for each subdomain, the proposed method only needs to solve for the electric field unknowns plus the surface magnetic field unknowns at subdomain interfaces. This reduces the dimensions of the resultant linear system and thus allows for modeling larger problems. Furthermore, unlike element-based DGTD methods, the proposed method is subdomain-based. The computational region is divided into multiple subdomains based on the domain-decomposition method, and each subdomain may contain multiple elements. Different element types and orders of basis functions can be employed in different subdomains to exploit the geometry property of the model. A nonconformal mesh is allowed between different subdomains to increase meshing flexibility. The Newmark-beta time-integration scheme is used for implicit temporal discretization, and fast direct linear solvers, such as the lower-diagonal-upper decomposition algorithm, are employed to accelerate time integration when all the subdomains are in a sequential order. Numerical results show that the proposed method is more efficient in terms of CPU time, and also saves memory with respect to the previous implicit DGTD method when modeling electrically small problems.

Isotropic Riemann Solver for a Nonconformal Discontinuous Galerkin Pseudospectral Time-Domain Algorithm

We present a discontinuous Galerkin pseudospectral time-domain (DG-PSTD) algorithm to solve elastic-/acoustic-wave propagation problems. The developed DG-PSTD algorithm combines the merits of flexibility from a finite-element method and spectral accuracy and efficiency from a high-order pseudospectral method, while having a flavor closer to a finite-volume method. This numerical approach not only uses structured/unstructured conformal meshes but also handles nonconformal meshes (h-adaptivity) with nonuniform approximation orders (p-adaptivity) in different regions, thus leading to high flexibility and efficiency for heterogeneous multiscale problems. To implement the discontinuous Galerkin algorithm, a concise but more general heterogeneous Riemann solver is provided to effectively and accurately resolve the coupling of multiple subdomains for both elastic–elastic/fluid–fluid and fluid–solid coupling. Finally, numerical results demonstrate the flexibility, high accuracy, and efficiency of our method for elastic-/acoustic-wave simulation.

A Higher Order Hybrid SIE/FEM/SEM Method for the Flexible Electromagnetic Simulation in Layered Medium

A novel hybrid method is developed for the flexible and accurate electromagnetic simulation of penetrable objects in a layered medium (LM). In this method, the original complex simulation domain is first divided into several subdomains, following the spirit of divide-and-conquer. Each subdomain is then meshed and solved independently, where nonconformal mesh is inevitable. The Riemann type transmission condition is utilized at the interfaces of each subdomain to correctly exchange information so that the solutions of all subdomains converge rapidly to the real solution of the original problem. More specifically, in our method, the surface integral equation (SIE) combined with the LM Green’s functions (LMGFs) is adopted for the boundary subdomain, while the finite-element method (FEM) and the spectral element method (SEM) are employed for all the other interior dielectric subdomains. The SIE with LMGFs truncates the simulation domain tightly within the object itself, which drastically decreases the number of unknowns. The interior subdomains are modeled by either FEM or SEM, depending on the geometry and material property of each subdomain. To further enhance the simulation capability, higher order approaches are adopted for all the subdomain solvers in this hybrid method. Several numerical examples are demonstrated, where a high convergence and accuracy of this method is observed. This paper will serve as an efficient and flexible simulation tool for the applications of geophysical exploration.

Efficient Ordinary Differential Equation-Based Discontinuous Galerkin Method for Viscoelastic Wave Modeling

We present an efficient nonconformal-mesh discontinuous Galerkin (DG) method for elastic wave propagation in viscous media. To include the attenuation and dispersion due to the quality factor in time domain, several sets of auxiliary ordinary differential equations (AODEs) are added. Unlike the conventional auxiliary partial differential equation-based algorithm, this new method is highly parallel with its lossless counterpart, thus requiring much less time and storage consumption. Another superior property of the AODE-based DG method is that a novel exact Riemann solver can be derived, which allows heterogeneous viscoelastic coupling, in addition to accurate coupling with purely elastic media and fluid. Furthermore, thanks to the nonconformal-mesh technique, adaptive hp-refinement and flexible memory allocation for the auxiliary variables are achieved. Numerical results demonstrate the efficiency and accuracy of our method.

3-D Domain Decomposition Based Hybrid Finite-Difference Time-Domain/Finite-Element Time-Domain Method With Nonconformal Meshes

A new 3-D domain decomposition based hybrid finite-difference time-domain (FDTD)/finite-element time-domain (FETD) method is introduced to facilitate electromagnetic modeling by exploiting both the computational efficiency of FDTD and the meshing flexibility of FETD. The proposed hybrid method allows the FETD mesh and the FDTD grid to be nonconformal based on domain decomposition technique. It implements the hybridization with a buffer zone, which functions as a transition region between FDTD and FETD. The buffer zone helps the proposed hybrid method obviate the interpolation approach for field coupling of the nonconformal mesh and hence overcome the late-time instability issue. The discontinuous Galerkin method is utilized to couple different regions, thus improving the coupling accuracy compared with that using the Dirichlet boundary condition. Moreover, the hybrid method allows further division of the FETD region into multiple subdomains when the degrees of freedom in this region are large. For temporal discretization, a global leapfrog time integration scheme is implemented to sequentially update the fields in the FDTD, buffer, and FETD regions. The numerical results are shown to demonstrate the meshing flexibility and computational efficiency of the proposed hybrid method inherited from FETD and FDTD methods.

Multiscale Hydraulic Fracture Modeling With Discontinuous Galerkin Frequency-Domain Method and Impedance Transition Boundary Condition

To facilitate the detection of hydraulic fractures by electromagnetic survey, a discontinuous Galerkin frequency-domain (DGFD) method is introduced in this paper to efficiently model the fracture responses under complicated geophysical environments. In the proposed DGFD method, the computational domain can be split into multiple subdomains with nonconformal meshes. The Riemann solver (upwind flux) is introduced to evaluate the numerical flux. The impedance transition boundary condition (ITBC) is employed to facilitate fracture modeling by approximating fractures as surfaces. Numerical results show that the ITBC works well for different fracture conductivities, dipping angles, operation frequencies, as well as different sources. For both small- and large-scale fractures, it also shows good agreement with the references. The responses of fractures increase as their conductivities become larger. Large dipping angles can cause spikes on the responses in a borehole. For a magnetic source, higher operation frequencies can enhance the signal level, while for an electric source, the sensitivity to frequency is small. When no borehole is considered, the responses due to an electric source are in general larger than those due to a magnetic one. However, when a borehole with conductive mud is included, the responses can be reversed for the electric and magnetic sources. For multiple fractures outside a cased borehole, the signal level of an electric source is significantly reduced, while that of a magnetic source remains at a similar level compared with the scenario without a casing. With the proposed technique, multiscale modeling of hydraulic fractures in complicated geophysical environments becomes possible.

Discontinuous Galerkin spectral elemen/finite element time domain (DGSE/FETD) method for anisotropic medium

Time domain methods, such as spectral element time domain (SETD) and finite element time domain (FETD) methods have potentials in solving transient and nonlinear problems. SETD method is suitable for the space with only coarse structures. Because large elements in SETD will not bring large geometric error. However, the high-order hexahedrons in SETD method can achieve good accuracy with low spatial sampling density. While tetrahedrons in finite element time domain (FETD) method can seize the detailed geometry information of the fine structures in the computational region, but the low-order elements in FETD is not high efficient as SETD in DoFs. Thus a hybrid of them via discontinuous Galerkin (DG) method can inherit the advantages of both methods. Riemann Solver is used to deal with the energy communication between adjacent subdomains.