Discontinuous Finite Element Method Research Articles

Space-Time Finite Element Method for Transient and Unconfined Seepage Flow Analysis

This paper aims to develop a moving-mesh type Finite Element Method for the computation of the transient unconfined seepage flow through the porous medium. The proposed method is based on the time discontinuous Galerkin Space-Time Finite Element Method (ST/FEM). It solves the seepage problem in the saturated region. The primary unknown in ST/FEM is piezometric pressure. Fluid velocities are derived from the pressure using Darcy's law. Further, an iterative algorithm has been proposed in this paper to implement the proposed method. In each iteration step, the computation domain is updated according to the flow velocity on the phreatic boundary. Subsequently, internal nodes are moved using the mesh moving technique to accommodate the newly updated computation domain. The mesh moving technique, which is discussed in this paper, is based on an elasticity problem. ST/FEM is employed to analyze several unconfined seepage flow problems, and results of steady state solutions are compared with those available in the literature to demonstrate the efficacy of the proposed scheme. • Moving mesh type finite element method is developed for transient and unconfined seepage flow analysis. • The proposed scheme requires 2 to 3 iterations in a time step to achieve convergence. • Elasticity equation based automatic mesh moving technique is employed to move the internal nodes. • Initial computation domain significantly affects the deformation characteristics of mesh. • Proposed iterative scheme can be employed with any type of mesh moving technique.

An asymptotic preserving method for the linear transport equation on general meshes

While many numerical methods for the linear transport equation are available in the literature in 1D or on Cartesian meshes, fewer works are dedicated to the resolution of this model on unstructured meshes. In the context of radiative hydrodynamics, we need a method capable to handle a wide range of radiation regimes going from free-streaming to diffusion and to be coupled with a Lagrangian hydrodynamics solver. In this paper we design a method based on the micro-macro paradigm and to the Discrete Ordinates (SN) angular discretization, which fulfills these requirements. It allows to choose the limit transport scheme and the limit diffusion scheme. It is compared on challenging test problems to a Discontinuous Finite Element (DFE) method.

Spatial convergence study of Lax-Friedrichs WENO fast sweeping method on the SN transport equation with nonsmoothness

Spatial convergence study of the Lax-Friedrichs fast sweeping method based on the classical third order WENO scheme (WENO3) without a priori smoothness indicators is performed on the variants of Larsen’s benchmark problem with nonsmooth solutions. For comparison, third order upwind (UPWD3), weighted diamond difference, step method, bilinear discontinuous finite element method are also included. In optically thick regimes of the cases with continuous angular fluxes, WENO3 detects steep gradients of angular fluxes and handles them as discontinuities to suppress the unphysical oscillation, which improves the accuracy significantly compared with UPWD3. In the cases with discontinuous angular fluxes, WENO3 achieves non-oscillatory solutions with limited numerical diffusion. Nonlinear Gauss-Seidel iteration is necessary to obtain convergent solution for the nonlinearity of WENO3, which leads to low computational efficiency. The convergence behavior of nonlinear Gauss-Seidel iteration is affected by nonsmoothness of the exact solution, mesh number and the relaxation parameter.

Higher Order Composite DG approximations of Gross–Pitaevskii ground state: Benchmark results and experiments

Discontinuous Galerkin composite finite element methods (DGCFEM) are designed to tackle approximation problems on complicated domains. Partial differential equations posed on complicated domain are common when there are mesoscopic or local phenomena which need to be modelled at the same time as macroscopic phenomena. In this paper, an optical lattice will be used to illustrate the performance of the approximation algorithm for the ground state computation of a Gross–Pitaevskii equation, which is an eigenvalue problem with eigenvector nonlinearity. We will adapt the convergence results of Marcati and Maday 2018 to this particular class of discontinuous approximation spaces and benchmark the performance of the classic symmetric interior penalty hp-adaptive algorithm against the performance of the hp-DGCFEM.

A discontinuous finite element method based on B-spline wavelet on the interval for solving first-order neutron transport equation with discrete ordinate (SN) angular discretisation

In this paper, a discontinuous Galerkin finite element method (DGFEM) based on B-spline wavelet on the interval (BSWI) is developed and applied to solve the first-order neutron transport equation with discrete ordinate (SN) angular discretization. Then, using the scaling function of BSWI function as the basis function instead of the usual Lagrange basis function, the neutron transport equation based on BSWI discontinuous finite element is derived. Owing to the high precision characteristic of BSWI function, the method proposed in this paper can obtain fast convergence and statistical numerical accuracy under the condition that few spatial elements are used, and comparing with the results of the traditional discontinuous finite element method. Finally, some numerical examples are utilized to verify the accuracy and efficiency of the proposed method, and the results show that the method has high efficiency, reliable and stability, and the calculation accuracy is higher than that of the conventional discontinuous finite element method.

Coupled Electromagnetic and Hydrodynamic Modeling for Semiconductors Using DGTD

A hydrodynamic model (HDM) solver based on discontinuous Galerkin time domain finite element method (DGTD-FEM) has been developed in order to simulate the transient charge transport in semiconductors. The bipolar transport equations have been numerically solved together with the Poisson equation to realize ballistic charge transport in semiconductor devices. Furthermore, the developed solver is coupled with an FEM-based full-wave Maxwell solver in order to model the behavior of semiconductors under external electromagnetic illumination. This multiphysics coupled solver is capable of simulating transient behavior of photoactive semiconductor devices that are illuminated by externally modulated light at high frequencies. With the time domain HDM solver, the inertia effects and ballistic transport are also accounted in the accurate transient simulations of high-frequency photoactive devices.

A stabilizer free weak Galerkin finite element method on polytopal mesh: Part II

A stabilizer free weak Galerkin (WG) finite element method on polytopal mesh has been introduced in Part I of this paper (Ye and Zhang (2020)). Removing stabilizers from discontinuous finite element methods simplifies formulations and reduces programming complexity. The purpose of this paper is to introduce a new WG method without stabilizers on polytopal mesh that has convergence rates one order higher than optimal convergence rates. This method is the first WG method that achieves superconvergence on polytopal mesh. Numerical examples in 2D and 3D are presented verifying the theorem.

Discontinuous finite element method for efficient three-dimensional elastic wave simulation

AbstractThe existing discontinuous Galerkin (DG) finite element method (FEM) for the numerical simulation of elastic wave propagation is primarily implemented in two dimensions. Here, a discontinuous FEM (DFEM) for efficient three-dimensional (3D) elastic wave simulation is presented. First, the velocity–stress equations of 3D elastic waves in isotropic media are transformed into first-order coefficient-changed partial differential equations. A DG discretisation method for wave field values on a unit boundary is then defined using the local Lax–Friedrichs flux format. The equations are first transformed into equivalent integral equations, and subsequently into a spatial semi-discrete ordinary differential equation system using a hierarchical orthogonal basis function. The DFEM is extended to an arbitrary high-order accuracy in the time domain using the exponential integrator technique and the explicit optimal strong-stability-preserving Runge–Kutta method. Finally, an efficient method for selecting the calculation area of the geometry of the current shot record is realised. For the computation, a multi-node parallelism with improved resource utilisation and parallelisation efficiency is implemented. The numerical results show that the proposed method can improve both the accuracy of the simulation and the efficiency of the calculation compared with existing methods.

Discontinuous finite element method applied to transient pure and coupled radiative heat transfer

The discontinuous finite element method (DFEM) is applied to transient pure and coupled radiative heat transfer problems in participating media. The DFEM discretization is presented for the time-dependent governing equations that take the full time-dependent term into account. Transient pure radiative transfer problems in a slab medium illuminated by a collimated Gaussian pulse and in a vacuum gapped medium subjected to diffuse irradiation is solved via the DFEM. The DFEM is further extended to transient radiation-conduction problems in two-dimensional media. The DFEM solutions of time-dependent radiation-intensity-based quantities are verified to be highly accurate and they are presented in the form of tabulated numbers with four decimal digits.

USING ARTIFICIAL NEURAL NETWORKS TO ACCELERATE TRANSPORT SOLVES

Discontinuous Finite Element Methods (DFEM) have been widely used for solving SN radiation transport problems in participative and non-participative media. In this method, small matrix-vector systems are assembled and solved for each cell, angle, energy group, and time step while sweeping through the computational mesh. In practice, these systems are generally solved directly using Gaussian elimination, as computational acceleration for solving this small systems are often inadequate. Nonetheless, the computational cost of assembling and solving these local systems, repeated for each cell in the phase-space, can amount to a large fraction of the total computing time. In this paper, a Machine Learning algorithm is designed to accelerate the solution of local systems. This one is based on Artificial Neural Networks (ANNs). Its key idea is training an ANN with a large set of solutions to random one-cell transport problems and, then, replacing the assembling and solution of the local systems by the feedforward evaluation of the trained ANN. It is observed that the optimized ANNs are able to reduce the compute times by a factor of ~ 3:6 per source iteration, while introducing mean absolute errors between 0:5 – 2% in transport solutions.

The Influence of Various Structure Surface Boundary Conditions on Pressure Characteristics of Underwater Explosion

The shock wave of the underwater explosion can cause severe damage to the ship structure. The propagation characteristics of shock waves near the structure surface are complex, involving lots of complex phenomena such as reflection, transmission, diffraction, and cavitation. However, different structure surface boundaries have a significant effect on the propagation characteristics of pressure. This paper focuses on investigating the behavior of shock wave propagation and cavitation from underwater explosions near various structure surfaces. A coupled Runge–Kutta discontinuous Galerkin (RKDG) and finite element method (FEM) is utilized to solve the problem of the complex waves of fluids and structure dynamic response, considering the fluid compressibility. The level set (LS) method and the ghost fluid (GF) method are combined to capture the moving interface and deal with the stability of the coupling between the shock wave and structure surface. Besides, a cut-off cavitation model is introduced to the RKDG method. The validation of the numerical calculation model is discussed by comparing it with the known solution to verify the numerical solutions. Then, crucial kinds of structure surface boundary conditions include shallow-water single layer elasticity plate, double-layer crevasse elasticity plate, single layer curved elasticity plate, and double-layer curved elasticity plates are analyzed and discussed. The results and analysis can provide references for underwater explosion pressure characteristics, the impacting response of different boundary structures, and designing structures.

A Three-Dimensional Discontinuous Galerkin Time-Domain Finite Element Method for Electromagnetic Modeling of Wireless Power Transfer Coils

Inductive coupling based wireless power transfer (WPT) is a popular short-range power delivery mechanism for many industrial, biomedical, and home electronic appliances applications. A numerical methodology is needed for the analysis of the electromagnetic propagation, radiation, scattering and coupling of highly efficient WPT systems. This study is based on the discontinuous Galerkin time-domain (DGTD) finite element method. A brief survey of the DGTD method is given, which is a finite element method using discontinuous piecewise polynomials as basis functions and numerical fluxes based on up-winding for stability. DGTD method is characterized by the fact that no continuity is enforced between the elements, then it is easy to parallelize and results in fast solvers. Even though the finite element method is used by a few researchers to study WPT problems, we found no study using the DGTD method to study WPT problems, which is surprising given that this discretization technique seems particularly well suited for these problems. A design of two coils at the frequency of 3 MHz is introduced, and the effects of the distance and misalignment between two coils on the mutual coupling are studied. The numerical results are validated by experimental and analytical results.

A Broad Class of Conservative Numerical Methods for Dispersive Wave Equations

We develop a general framework for designing conservative numerical methods based on summation by parts operators and split forms in space, combined with relaxation Runge-Kutta methods in time. We apply this framework to create new classes of fully-discrete conservative methods for several nonlinear dispersive wave equations: Benjamin-Bona-Mahony (BBM), Fornberg-Whitham, Camassa-Holm, Degasperis-Procesi, Holm-Hone, and the BBM-BBM system. These full discretizations conserve all linear invariants and one nonlinear invariant for each system. The spatial semidiscretizations include finite difference, spectral collocation, and both discontinuous and continuous finite element methods. The time discretization is essentially explicit, using relaxation Runge-Kutta methods. We implement some specific schemes from among the derived classes, and demonstrate their favorable properties through numerical tests.

波动方程的高阶间断有限元方法

波动方程的高阶间断有限元方法

A Comparative Analysis of Neutron Transport Calculations Based on Variational Formulation and Finite Element Approaches

The application of continuous and discontinuous approaches of the finite element method (FEM) to the neutron transport equation (NTE) has been investigated. A comparative algorithm for analyzing the capability of various types of numerical solutions to the NTE based on variational formulation and discontinuous finite element method (DFEM) has been developed. The developed module is coupled to the program discontinuous finite element method for neutron (DISFENT). Each variational principle (VP) is applied to an example with drastic changes in the distribution of neutron flux density, and the obtained results of the continuous and discontinuous finite element (DFE) have been compared. The comparison between the level of accuracy of each approach using new module of DISFENT program has been performed based on the fine mesh solutions of the multi-PN (MPN) approximation. The obtained results of conjoint principles (CPs) have been demonstrated to be very accurate in comparison to other VPs. The reduction in the number of required meshes for solving the problem is considered as the main advantage of this principle. Finally, the spatial additivity to the context of the spherical harmonics has been implemented to the CP, to avoid from computational error accumulation.

A new discontinuous Galerkin mixed finite element method for compressible miscible displacement problem

A new combined discontinuous Galerkin method is proposed for compressible miscible displacement problem in porous media. Here, a splitting positive definite mixed finite element method is used for the pressure and Darcy velocity, while an interior penalty discontinuous Galerkin (IPDG) method is used for the transport equation. The stability and convergence of this algorithm are considered, and the optimal a priori error estimate in l∞(L2) for velocity, pressure and concentration are given. Finally we provide some numerical results to confirm our theoretical analysis, and simulate compressible fluid flows through homogeneous and isotropic porous media.

Quadratic axial expansion function with sub-plane acceleration scheme for the high-fidelity transport code PROTEUS-MOC

PROTEUS-MOC is a pin-resolved high-fidelity transport code that employs the method of characteristics to discretize the radial variables and the discontinuous finite element method to discretize the axial variable. In this study, we extended the hybrid transport solution method to axial quadratic trial functions from the current linear functions. Compared with the linear approximation, the quadratic approximation reduces the memory requirement by 50% by allowing coarser axial meshes, but the coarser axial meshes deteriorate the performance of the CMFD and pCMFD acceleration schemes. To remedy the deteriorated performance of the acceleration schemes, a sub-plane acceleration method was implemented. The sub-plane acceleration scheme can improve the computational performance of the quadratic approximation in conjunction with the GMRES method. It was also observed that for a fixed radial mesh configuration, the flat leakage source approximation error increases with axial mesh refinement while the truncation error of the finite functional expansion decreases.

A combined hybrid mixed element method for incompressible miscible displacement problem with local discontinuous Galerkin procedure

AbstractIn this article, we propose a combined hybrid discontinuous mixed finite element method for miscible displacement problem with local discontinuous Galerkin method. Here, to obtain more accurate approximation and deal with the discontinuous case, we use the hybrid mixed element method to approximate the pressure and velocity, and use the local discontinuous Galerkin finite element method for the concentration. Compared with other combined methods, this method can improve the efficiency of computation, deal with the discontinuous problem well and keep local mass balance. We study the convergence of this method and give the corresponding optimal error estimates in L∞(L2) for velocity and concentration and the super convergence in L∞(H1) for pressure. Finally, we also present some numerical examples to confirm our theoretical analysis.

Discontinuous finite volume element method for Darcy flows in fractured porous media

This paper presents a numerical simulation of the single phase Darcy flow model in two-dimensional fractured porous media. Under some physically consistent coupling conditions, the model can be described as a reduced problem by coupling the bulk problem in porous matrix and the fracture problem in fractures. Flows are governed by the primal form of the Darcy’s equations for both the bulk and fractures. The coupled discontinuous finite volume element methods and conforming finite element method are adopted to solve the bulk problem and fracture problem, respectively. We theoretically analyze the well-posedness of the discrete problem, and derive optimal error estimates in standard L2 error and broken H1 error. Numerical experiments include not only the fractures with high permeability as the prior flow conduit, but also the fractures with low permeability as the flow barrier, which demonstrate the accuracy, flexibility and robustness of our discrete formulation for complicated networks of fractures in porous media domain.

Calculated radiance errors induced by neglecting the polarization of the irradiation beam exposed to an atmosphere

The radiance calculation via the solution of the scalar approximation form of the rigorous vector radiative transfer equation (VRTE) leads to non-negligible errors in atmospheric radiative transfer problems. In this work, for the first time, a quantitative investigation of the calculated radiance errors induced by neglecting the polarization of incident irradiation beams exposed to a plane-parallel atmosphere is presented. The discontinuous finite element method, which is highly accurate in the spatial and angular spaces, was employed to solve both the VRTE and its scalar approximation. The calculated values of the radiance obtained via the scalar model are compared with the exact solutions calculated via the vector model and their relative errors are analyzed. The results show that neglecting the linearly polarized component of the incident irradiation leads to relative errors up to 94.00%. These values are much larger than the errors induced by neglecting the circularly polarized component.