Discontinuous Finite Element Method Research Articles

Introduction to the Distributional Finite Difference Method for 3D Seismic Wave Propagation and Comparison With the Spectral Element Method

AbstractWe have extended the distributional finite difference method (DFDM) to simulate the seismic‐wave propagation in 3D regional earth models. DFDM shares similarities to the discontinuous finite element method on a global scale and to the finite difference method locally. Instead of using linear staggered finite‐difference operators, we employ DFDM operators based on B‐splines and a definition of derivatives in the sense of distributions, to obtain accurate spatial derivatives. The weighted residuals method used in DFDM's locally weak formalism of spatial derivatives accurately and naturally accounts for the free surface, curvilinear meshing, and solid‐fluid coupling, for which it only requires setting the shear modulus and the continuity condition to zero. The computational efficiency of DFDM is comparable to the spectral element method (SEM) due to the more accurate mass matrix and a new band‐diagonal mass factorization. Numerical examples show that the superior accuracy of the band‐diagonal mass and stiffness matrices in DFDM enables fewer points per wavelength, approaching the spectral limit, and compensates for the increased computational burden due to four Lebedev staggered grids. Specifically, DFDM needs 2.5 points per wavelength, compared to the five points per wavelength required in SEM for 0.5% waveform error in a homogeneous model. Notably, while maintaining the same accuracy in the solid domain, DFDM demonstrates superior accuracy in the fluid domain compared to SEM. To validate its accuracy and flexibility, we present various 3D benchmarks involving homogeneous and heterogeneous regional elastic models and solid‐fluid coupling in both Cartesian and spherical settings.

On robust boundary treatments for wall‐modeled LES with high‐order discontinuous finite element methods

SummaryTo robustly and accurately simulate wall‐bounded turbulent flows at high Reynolds numbers, we propose suitable boundary treatments for wall‐modeled large‐eddy simulation (WMLES) coupled with a high‐order flux reconstruction (FR) method. First, we show the need to impose an auxiliary boundary condition on auxiliary variables (solution gradients) that are commonly introduced in high‐order discontinuous finite element methods (DFEMs). Auxiliary boundary conditions are introduced in WMLES, where the grid resolution is too coarse to resolve the inner layer of a turbulent boundary layer. Another boundary treatment to further enhance stability with under‐resolved grids, is the use of a modal filter only in the wall‐normal direction of wall‐adjacent cells to remove the oscillations. A grid convergence study of turbulent channel flow with a high Reynolds number () shows that the present WMLES framework accurately predicts velocity profiles, Reynolds shear stress, and skin friction coefficients at the grid resolutions recommended in the literature. It was confirmed that a small amount of filtering is sufficient to stabilize computation, with negligible influence on prediction accuracy. In addition, non‐equilibrium periodic hill flow with a curved wall, including flow separation, reattachment, and acceleration at a high Reynolds number (), is reported. Considering stability and the prediction accuracy, we recommend a loose auxiliary wall boundary conditions with a less steep velocity gradient for WMLES using high‐order DFEMs.

TARS: A parallel tetrahedral discontinuous finite element code for the solution of the discrete ordinates neutron transport equation

In this study, we introduce TARS, a neutron transport code based on the discrete-ordinates discontinuous finite element (SN-DFEM) method, which tackles existing challenges in the field by providing accurate and efficient solutions to the neutron transport equation. The use of DFEM enables higher-order spatial discretization, improving numerical stability, while parallelizing transport sweeping reduces computational time and resource requirements. TARS' performance was evaluated using the Takeda benchmark problems. The results showed that TARS generates highly accurate solutions, with eigenvalue errors no more than 90 pcm compared to MC solutions. The relative differences of region-averaged fluxes were below 1% in most regions, except those with severe flux gradients. Parallel calculation efficiency analysis revealed a PCE of over 70% even at 1024 CPU cores. However, hybrid MPI and OpenMP parallel programming did not outperform pure MPI programming, suggesting that the latter might be more suitable for TARS.

A treatment of particle–electrolyte sharp interface fracture in solid-state batteries with multi-field discontinuities

In this work, we present a computational framework for coupled electro-chemo-(nonlinear) mechanics at the particle scale for solid-state batteries. The framework accounts for interfacial fracture between the active particles and solid electrolyte due to intercalation stresses. We extend discontinuous finite element methods for a sharp interface treatment of discontinuities in concentrations, fluxes, electric fields and in displacements, the latter arising from active particle–solid electrolyte interface fracture. We model the degradation in the charge transfer process that results from the loss of contact due to fracture at the electrolyte–active particle interfaces. Additionally, we account for the stress-dependent kinetics that can influence the charge transfer reactions and solid state diffusion. The discontinuous finite element approach does not require a conformal mesh. This offers the flexibility to construct arbitrary particle shapes and geometries that are based on design, or are obtained from microscopy images. The finite element mesh, however, can remain Cartesian, and independent of the particle goemetries. We demonstrate this computational framework on micro-structures that are representative of solid-state batteries with single and multiple anode and cathode particles.

An energy stable discontinuous Galerkin time-domain finite element method in optics and photonics

In this paper, a time-domain discontinuous Galerkin (TDdG) finite element method for the full system of Maxwell’s equations in optics and photonics is investigated, including a complete proof of a semi-discrete error estimate. The new capabilities of methods of this type are to efficiently model linear and nonlinear effects, for example of Kerr nonlinearities. Energy stable discretizations both at the semi-discrete and the fully discrete levels are presented. In particular, the proposed semi-discrete scheme is optimally convergent in the spatial variable on Cartesian meshes with Qk-type elements, and the fully discrete scheme is conditionally stable with respect to a specially defined nonlinear electromagnetic energy.

An adaptive discontinuous finite volume element method for the Allen-Cahn equation

An adaptive discontinuous finite volume element method for the Allen-Cahn equation

Modeling shear thinning polymer flooding using a dynamic viscosity model

Two distinct effects that polymers exhibit are shear thinning and viscoelasticity. The shear thinning effect is important as the polymers used in chemical enhanced oil recovery usually have this property. We propose a novel approach to incorporate shear thinning effect through effective dynamic viscosity of the shear thinning polysolution. The procedure of viscosity calculation of the polysolution, although based on a very basic power law model, is based on empirical coefficients that depend on a spatio-temporally evolving variable, namely, concentration of polymer. Since viscosity calculation is performed in space and time, the results obey correct physics and are more accurate than what exists in the literature. This method has been integrated with an existing method for a Newtonian physics based model of porous media flows. The solver uses a hybrid numerical method developed by Daripa and Dutta [“DFEM-MMOC based EOR code in MATLAB” (2020); P. Daripa and S. Dutta, “Modeling and simulation of surfactant–polymer flooding using a new hybrid method,” J. Comput. Phys. 335, 249–282 (2017); and P. Daripa and S. Dutta, “On the convergence analysis of a hybrid numerical method for multicomponent transport in porous media,” Appl. Numer. Math. 146, 199–220 (2019)]. The above method solves a system of coupled elliptic and transport equations modeling Darcy's law based polymer flooding process using a discontinuous finite element method and a modified method of characteristics. Simulations show (i) competing effects of shear thinning and mobility ratio; (ii) injection conditions, such as injection rate and injected polymer concentration, influence the choice of polymers to optimize cumulative oil recovery; (iii) permeability affects the choice of polymer; (iv) dynamically evolving traveling viscosity waves; and (v) shallow mixing regions of small scale viscous fingers in homogeneous porous media. The overall goal of this study is to develop an effective yet easy approach to make design choices of polymers in any given flooding condition, which has been shown here.

Totally volume integral of fluxes for discontinuous Galerkin method-two dimensional Euler equations

In this paper, the scheme of constructing high-order accurate totally volume discontinuous finite element method for the numerical solution of the 1D Euler equations is extended to 2D Euler equations on Cartesian meshes. In the present work, the boundary integral fluxes are transformed into volume integral by applying divergence theorem to the boundary integral of the Riemann fluxes. Therefore, the totally volume discontinuous finite element is independent on the boundary integral fluxes at the element boundaries as opposed to the classical discontinuous Galerkin method. The accuracy is obtained by applying high-order polynomial approximations within elements using the tensor product of Lagrange polynomial. For temporal integration, strong stability preserving Runge-Kutta method SSPRK (3, 3) is applied. The scheme is stabilized by using Streamline Upwind Petrov Galerkin (SUPG) stabilization technique. For the spatial discretization, the polynomial of order 1and 2 are used, the shape function is constructed for the master (computational) element after applying the coordinate transformation for the physical domain, the transformation for the governing equations is performed to get it in the function of computational Cartesian, then the governing equations are put in conservative form. The numerical results of applying totally volume integral discontinuous Galerkin method for two-dimensional Euler equations presented in this paper show that the scheme is very accurate, fast, and effective even with shock appearance.

A novel time‐domain numerical methodology for the electromagnetic analysis of an H‐plane tee power divider

A time-domain numerical methodology for the transient analysis of multisection waveguide power dividers is proposed. Waveguide power dividers are essential components of microwave systems for high-power applications. A numerical strategy based on the 3D discontinuous Galerkin time-domain finite element method is used to estimate the electromagnetic propagation inside a power divider with three ports. The proposed approach leads to simulations that are fast, robust, and accurate, this being the result of a parallel local discretisation of the full-wave time-domain Maxwell's equations on unstructured tetrahedral meshes. The stability of the overall numerical scheme is obtained using an upwind numerical flux, an implicit second-order accurate time integration scheme, and Whitney's edge elements. Good agreements are obtained when the computed results are compared with other HFSS simulation results, and with measurement results found in the literature.

Coupled Electromagnetic and Hydrodynamic Semiconductor Modeling for Terahertz Generation

Terahertz (THz) electrical signal generation from a photoactive semiconductor device illuminated by an optical pulse is modeled and simulated. Hydrodynamic equations in time domain are numerically solved for both electrons and holes using the discontinuous Galerkin time-domain finite element method (DGTD-FEM) for the high-frequency charge transport that occurs in the semiconductor device. The obtained frequency spectra of photocurrent in various semiconductor materials under illumination of an ultrashort light pulse are presented. The inertia effects and ballistic transport of carriers play an important role to determine the frequency response of these materials, and the developed hydrodynamic model (HDM) solver delivers a transport analysis by predicting higher and wideband frequency capabilities for GaAs and Ge detectors over Si-photodetectors.

Discontinuous finite element method for transient conductive heat transfer in layered media with thermal contact resistance

Research on accurately solving the transient conductive heat transfer problem in layered media has both scientific and engineering significance. Numerical algorithms up to date usually suffer from additional dubious treatments when dealing with the temperature discontinuity induced by the thermal contact resistance (TCR) between the inner interface. In this paper, we extend the discontinuous finite element method (DFEM) to solve the problem of transient conductive heat transfer in a layered slab with TCR interfaces. The correctness of the DFEM for handling the TCR-induced temperature discontinuity is verified at first. Then we utilize the DFEM to solve the transient conductive heat transfer problem in a three-layered medium subjected to Dirichlet, Neumann, and Robin boundary conditions, respectively. The time-dependent temperature along the slab and the temperature discontinuity across the TCR interfaces are captured and analyzed. Results indicate the present DFEM can naturally capture the TCR interface-induced temperature discontinuities accurately. The time-dependent temperature and temperature discontinuities across the TCR interfaces presented in this work can serve as benchmarks for later research on conductive heat transfer in layered media.

From computed tomography to finite element space: A unified bone material mapping strategy

BackgroundThe spatially varying mechanical properties in finite element models of bone are most often derived from bone density data obtained via quantitative computed tomography. The key step is to accurately and efficiently map the density given in voxels to the finite element mesh. MethodsThe density projection is first formulated in least-squares terms and then discretized using a continuous and discontinuous variant of the finite element method. Both discretization variants are compared with the nodal and element approaches known from the literature. FindingsIn terms of accuracy in the L2 norm, energy distance and efficiency, the discontinuous zero-order variant appears to be the most advantageous. The proposed variant sufficiently preserves the spectrum of density at the edges, while keeping computational cost low. InterpretationThe continuous finite element method is analogous to the nodal formulation in the literature, while the discontinuous finite element method is analogous to the element formulation. The two variants differ in terms of implementation, computational cost and ability to preserve the density spectrum. These differences cannot be described and measured by known indirect methods from the literature.

Ultimate Bearing Capacity Analysis of Manned Submersible Based on the Genetic Algorithm Discontinuous and Galerkin Finite Element Method

The pressure hull of deep manned submersible is the most basic component to ensure its intended function. It is necessary to study the influence of initial geometric defects on the bearing capacity of pressure hull of manned submersible with different depths. According to the idea of discontinuous Galerkin finite element method, the theoretical model is constructed and the corresponding algorithm is designed, and the genetic algorithm is combined with discontinuous Galerin finite element method to establish the inverse method to obtain the ultimate bearing capacity of manned submersible. First, the discontinuous Galerkin finite element model is constructed, the inversion model is also established through combing the discontinuous Galerkin finite element method and genetic algorithm, and then the corresponding solution algorithm is designed. Moreover, then, the ultimate bearing analysis of manned submersible for different deep is carried out based on the inversion model combing discontinuous Galerkin finite element method and genetic algorithm. The effect of defect parameters on ultimate bearing capacity of manned submersible is obtained.

Mitigation of post-shock oscillations induced by artificial viscosity in discontinuous finite element methods

Current shock-capturing techniques for high-order discontinuous finite element methods based on modal energy shock sensors coupled to an artificial viscosity operator provide efficient and relatively simple solutions to tackle high-speed flows with discontinuities. Yet, due to inherent inhomogeneities in modal detection of shocks within elements, post-shock spurious oscillations can occur. These can be particularly severe in the case of slow-moving shocks. In the present paper, the origin of such oscillations is identified in the way the artificial viscosity is distributed across shocks by the modal shock detector. A novel approach is hence proposed to mitigate post-shock oscillations by introducing, via exponential averaging, a time delay in the artificial viscosity injection. Numerical one- and two-dimensional tests will show that a calibration of the time delay based on the shock propagation velocity can suppress, almost completely, post-shock oscillations, even in the most severe cases. Inspired by machine learning techniques, an efficient way to evaluate the propagation velocities of different shocks is also proposed, leading to a method to suppress post-shock oscillations, induced by the artificial viscosity, which is extremely simple, and relatively inexpensive.

Modeling landslide generated waves using the discontinuous finite element method

AbstractA new two‐layer model for impulsive wave generation by deformable granular landslides is developed based upon a discontinuous Galerkin finite element discretization. Landslide motion is modeled using a depth‐averaged formulation for a shallow subaerial debris flow, which considers the bed curvature represented by the local slope angle variable and accounts for inter‐granular stresses using Coulomb friction. Wave generation and propagation are simulated with the three‐dimensional non‐hydrostatic coastal ocean model Thetis to accurately capture key features such as wave dispersion. Two different techniques are used in treating wetting and drying (WD) processes during the landslide displacement and wave generation, respectively. For the lower‐layer landslide motion across the dry bed a classical thin‐layer explicit WD method is implemented, while for the resulting free‐surface waves interacted with the moving landslide an implicit WD scheme is utilized to naturally circumvent the artificial pressure gradient problem which may appear in the dynamic interaction between the landslide and water if using the thin‐layer method. The two‐layer model is validated using a suite of test cases, with the resulting good agreement demonstrating its capability in describing both the complex behaviors of granular landslides from initiation to deposition, and the consequent wave generation and propagation.

High Order Discontinuous Cut Finite Element Methods for Linear Hyperbolic Conservation Laws with an Interface

We develop a family of cut finite element methods of different orders based on the discontinuous Galerkin framework, for hyperbolic conservation laws with stationary interfaces in both one and two space dimensions, and for moving interfaces in one space dimension. Interface conditions are imposed weakly and so that both conservation and stability are ensured. A CutFEM with discontinuous elements in space is developed and coupled to standard explicit time stepping schemes for linear advection problems and the acoustic wave problem with stationary interfaces. In the case of moving interfaces, we propose a space-time CutFEM based on discontinuous elements both in space and time for linear advection problems. We show that the proposed CutFEM are conservative and energy stable. For the stationary interface case an a priori error estimate is proven. Numerical computations in both one and two space dimensions support the analysis, and in addition demonstrate that the proposed methods have the expected accuracy.

Adaptive configurational force‐based propagation for brittle and fatigue crack analysis

AbstractThis article presents a ‐adaptive crack propagation method for highly accurate 2D crack propagation paths which requires no a priori knowledge of the tip solution. The propagation method is designed to be simple to implement, only ‐adaptivity is required, with the propagation step size independent of the initial mesh allowing users to obtain high fidelity crack path predictions for domains containing multiple cracks propagating at different rates. The proposed method also includes a crack path derefinement scheme, where elements away from the crack tip are derefined whilst elements close to the crack tip are small so capture the fidelity of the crack path. The result is that the propagation of cracks over an increasingly larger distances has negligible increased computational effort and effect on the propagation path prediction. The linear elastic problem is solved using the hp discontinuous Galerkin symmetric interior penalty finite element method, which is post‐processed to obtain the configurational force at each tip to a user defined accuracy. Several numerical examples are used to demonstrate the accuracy, efficiency, and capability of the method. Due to the method's high accuracy crack path solutions of benchmark problems that are prolifically used in the literature are challenged.

A LIMITER OF DISCONTINUOUS FINITE ELEMENT METHOD FOR THERMAL RADIATION HEAT TRANSFER

Due to the irregular shape of the high-temperature equipment, the thermal radiation processes are susceptible to discontinuous and shielding effects. Achieving the high accuracy of numerical solution for such processes is still a significant challenge. In this paper, the discontinuous finite element method is developed to discretize the angle and space for the numerical solution of the radiative transfer equation. A limiter, which is based on the Barth-Jespersen limiter borrowing the hierarchical limiting strategy, is imposed for the angular and spatial domains of radiative intensity to suppress the nonphysical oscillations of numerical solutions. Several benchmark problems involving discontinuous boundary conditions or irregular geometrical systems with the participating medium are solved. Besides, by comparing numerical results with exact solutions of radiative intensity in azimuthal angle, it is verified that the limiter can effectively suppress the numerical oscillation caused by the shielding effect. Finally, the three-dimensional angular distributions of radiative intensity on space nodes are depicted. In a word, the limiter, which is specially designed for the discontinuous finite element method, is a useful technical means to solve radiative heat transfer with discontinuous boundary conditions or irregular geometrical systems.

ORDER TWO SUPERCONVERGENCE OF THE CDG METHOD FOR THE STOKES EQUATIONS ON TRIANGLE/TETRAHEDRON

<abstract> A new conforming discontinuous Galerkin (CDG) finite element method is introduced for solving the Stokes equations. The CDG method gets its name by combining good features of both conforming finite element method and discontinuous finite element method. It has the flexibility of using discontinuous approximation and simplicity in formulation of the conforming finite element method. This new CDG method is not only stabilizer free but also has two order higher convergence rate than the optimal order. This CDG method uses discontinuous $ P_k $ element for velocity and continuous $ P_{k+1} $ element for pressure. Order two superconvergence is derived for velocity in an energy norm and the $ L^2 $ norm. The superconvergent $ P_k $ solution is lifted elementwise to a $ P_{k+2} $ velocity which converges at the optimal order. The numerical experiments confirm the theories. </abstract>

A Full-Wave Discontinuous Galerkin Time-Domain Finite Element Method for Electromagnetic Field Mode Analysis

A Full-Wave Discontinuous Galerkin Time-Domain Finite Element Method for Electromagnetic Field Mode Analysis