Qingtao Sun

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.

A New 3-D Nonspurious Discontinuous Galerkin Spectral Element Time-Domain (DG-SETD) Method for Maxwell’s Equations

A new discontinuous Galerkin spectral element time-domain (DG-SETD) method for Maxwells equations based on the field variables E and B is proposed to analyze three-dimensional (3-D) transient electromagnetic phenomena. Compared to the previous SETD method based on the field variables E and H (the EH scheme), in which different orders of interpolation polynomials for electric and magnetic field intensities are required, the newly proposed method can eliminate spurious modes using basis functions with the same order interpolation for electric field intensity and magnetic flux density (the EB scheme). Consequently, it can reduce the number of unknowns and computation load. Domain decomposition for the EB scheme SETD method is completed via the DG method. In addition, the EB scheme SETD method is extended to the well-posed time-domain perfectly matched layer (PML) to truncate the computation domain when solving open-region problems. The effectiveness and advantages of the new DG-SETD method are validated by eigenvalue analysis and numerical results.

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.

New Efficient Implicit Time Integration Method for DGTD Applied to Sequential Multidomain and Multiscale Problems

The discontinuous Galerkins (DG) method is an efficient technique for packaging problems. It divides an original computational region into several subdomains, i.e., splits a large linear system into several smaller and balanced matrices. Once the spatial discretization is solved, an optimal time integration method is necessary. For explicit time stepping schemes, the smallest edge length in the entire discretized domain determines the maximal time step interval allowed by the stability criterion, thus they require a large number of time steps for packaging problems. Implicit time stepping schemes are unconditionally stable, thus domains with small structures can use a large time step interval. However, this approach requires inversion of matrices which are generally not positive definite as in explicit shemes for the first-order Maxwells equations and thus becomes costly to solve for large problems. This work presents an algorithm that exploits the sequential way in which the subdomains are usually placed for layered structures in packaging problems. Specifically, a reordering of interface and volume unknowns combined with a block LDU (Lower-Diagonal-Upper) decomposition allows improvements in terms of memory cost and time of execution, with respect to previous DGTD implementations.

Efficient Noniterative Implicit Time-Stepping Scheme Based on E and B Fields for Sequential DG-FETD Systems

The discontinuous Galerkin finite-element time-domain (DG-FETD) method with implicit time integration has an advantage in modeling electrically fine-scale electromagnetic problems. Based on domain decomposition methods, it avoids the direct inversion of a large system matrix as in the conventional FETD method; by employing implicit time integration, it obviates an extremely small time-step interval to maintain stability as in explicit schemes. Based on curl-conforming basis functions for the electric field intensity E field and divergence-conforming basis functions for the magnetic flux density B field, a new noniterative implicit time-stepping scheme is proposed to efficiently solve sequentially ordered systems for electrically fine-scale problems. Compared with the previous EH-based scheme, the new scheme introduces fewer unknowns and, thereby, results in a smaller matrix system. Based on the Crank-Nicholson algorithm for time integration, the matrix system is in a block tridiagonal form. Then, through separating the surface unknowns from the volume unknowns, a block lower-diagonal-upper (LDU) decomposition is implemented, reducing the computational complexity of the original system. The adaptivity of parallel computing in subdomain level during preprocessing further helps shorten the computation time. Numerical results confirm that the proposed LDU scheme presents improved efficiency in terms of memory and CPU time while retaining the same accuracy, compared with the previous implicit block-Thomas method. With respect to the explicit Runge-Kutta method and the standard FDTD, it also shows an advantage in CPU time. The proposed scheme will help improve the performance of DG-FETD in modeling electrically fine-scale problems.

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.

Multiphysics Coupling of Dynamic Fluid Flow and Electromagnetic Fields for Subsurface Sensing

Summary form only given. Electromagnetic (EM) measurement has been extensively applied in subsurface sensing while fluid flow modeling is capable of characterizing subsurface fluid flow behavior. The multiphysics coupling of the EM measurement and dynamic fluid flow analysis has significant potential to improve electromagnetic geophysical exploration with injecting electromagnetic contrast agents.