Discontinuous Finite Element Research Articles

Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation

Abstract In [L. Bourhrara, A new numerical method for solving the Boltzmann transport equation using the PN method and the discontinuous finite elements on unstructured and curved meshes, J. Comput. Phys. 397 2019, Article ID 108801], a numerical scheme based on a combined spherical harmonics and discontinuous Galerkin finite element method for the resolution of the Boltzmann transport equation is proposed. One of its features is that a streamline weight is added to the test function to obtain the variational formulation. In the present paper, restricting our attention to the advective part of the Boltzmann equation, we prove the convergence and provide error estimates of this numerical scheme. To this end, the original variational formulation is restated in a broken functional space. The use of broken functional spaces enables to build a conforming approximation, that is the finite element space is a subspace of the broken functional space. The setting of a conforming approximation simplifies the numerical analysis, in particular the error estimates, for which a Céa’s type lemma and standard interpolation estimates are sufficient for our analysis. For our numerical scheme, based on ℙ k {\mathbb{P}^{k}} discontinuous Galerkin finite elements (in space) on a mesh of size h and a spherical harmonics approximation of order N (in the angular variable), the convergence rate is of order 𝒪 ⁢ ( N - t + h k ) {\mathcal{O}(N^{-t}+h^{k})} for a smooth solution which admits partial derivatives of order k + 1 {k+1} and t with respect to the spatial and angular variables, respectively. For k = 0 {k=0} (piecewise constant finite elements) we also obtain a convergence result of order 𝒪 ⁢ ( N - t + h 1 2 ) {\mathcal{O}(N^{-t}+h^{\frac{1}{2}})} . Numerical experiments in one, two and three dimensions are provided, showing a better convergence behavior for the L 2 {L^{2}} -norm, typically of one more order, 𝒪 ⁢ ( N - t + h k + 1 ) {\mathcal{O}(N^{-t}+h^{k+1})} .

A new method for the prediction of vibration responses of thin plates coupled with discontinuous connections

Due to the limitation caused by the assumption of ideal continuous line junctions between plates or the high computation cost, the existing methods cannot provide reliable prediction solutions of vibration responses of coupled plates with real connections, such as bolts and corner irons. To this end, a discontinuous energy finite element analysis (dEFEA) method with high accuracy is proposed to predict the vibration responses of discontinuously connected plates, which can revert the real connection. This method developes the coupling loss factors (CLFs) on the steady-state energy flow technique to characterize vibration transmission between plates with different connections. In addition, a discontinuous energy finite element equation of coupled plates is established to obtain the vibration responses of those. The efficacy of the dEFEA method is demonstrated by experiments and the method can improve the prediction accuracy of vibration, which is verified by the comparison between discontinuous and ideal continuous connections. To sum up, vibration transmission of connected structures varies greatly depending on the way of connection and cannot be simplified to a line connection.

A type of multigrid method for semilinear elliptic problems based on symmetric interior penalty discontinuous Galerkin method

AbstractThis article introduces a new kind of multigrid approach for semilinear elliptic problems, which is based on the symmetric interior penalty discontinuous Galerkin (SIPDG) method. We first give an optimal error estimate of the SIPDG method for the problem. Then, we design a type of multigrid method, which is called the multilevel correction method, and derive a priori error estimates. The primary idea of this method is to take the solution of the semilinear problem and utilize it to establish a sequence of solutions for associated linear boundary value problem on discontinuous finite element spaces and a newly defined low dimensional augmented subspace. Lastly, numerical experiments are offered to confirm the suggested method's precision and effectiveness.

A hybrid Monte Carlo, discontinuous Galerkin method for linear kinetic transport equations

We present a hybrid method for time-dependent particle transport problems that combines Monte Carlo (MC) estimation with deterministic solutions based on discrete ordinates. For spatial discretizations, the MC algorithm computes a piecewise constant solution and the discrete ordinates use bilinear discontinuous finite elements. From the hybridization of the problem, the resulting problem solved by Monte Carlo is scattering free, resulting in a simple, efficient solution procedure. Between time steps, we use a projection approach to “relabel” collided particles as uncollided particles. From a series of standard 2-D Cartesian test problems we observe that our hybrid method has improved accuracy and reduction in computational complexity of approximately an order of magnitude relative to standard discrete ordinates solutions.

Goal-based angular adaptivity with contributon theory for the discrete ordinates transport equation

A goal-based angular adaptivity has been described for solving the discrete ordinates transport equation. This method uses the linear discontinuous finite element quadrature sets, allowing for local angular refinement across space-energy dimensions. Anisotropy quantified factor derived by contributon theory is introduced in this adaptivity to locate spatial regions that require more angular unknowns in every energy group for a given goal region. The goal-based error metric is considered to trigger local refinement in angle, which requires the solutions of the contributon transport equation to determine the importance to a user-defined functional. Several problems that include both one-group and multi-group transport calculations are displayed to demonstrate the effectiveness of this method. Results indicate that our goal-based adaptivity can significantly reduce computational expenses in terms of angular unknowns and computational times for a similar accuracy, as compared to uniform quadrature sets. It is expected that this adaptivity has the potential to enhance the efficiency of an even wider range of transport problems.

Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method

This article presents a simple efficient and asynchronously correcting a posteriori error approximation for discontinuous finite element solutions of the second-order hyperbolic partial differential problems on triangular meshes. This study considersthe basis functions for error spaces corresponding to some finite element spaces. The discretization error of each triangle is estimated by solving the local error problem. It also shows global super convergence for discontinuous solution on triangular lattice. In this article, the triangular elements are classify into three types: (i) elements with one inflow and two outflow edges are of type I, (ii) elements with two inflows and one outflow edges are of type II and (iii) elements with one inflow edge, one outflow edge, and one edge parallel to the characteristics are of type III. The article investigated higher-dimension discontinuous Galerkin methods for hyperbolic problems on triangular meshes and also studied the effect of finite element spaces on the superconvergence properties of DG solutions on three types of triangular elements and it showed that the DG solution is <i>O(h<sup>p+2</sup>)</i> superconvergent at Legendre points on the outflow edge on triangles having one outflow edge using three polynomial spaces. A posteriori error estimates are tested on a number of linear and nonlinear problems to show their efficiency and accuracy under lattice refinement for smooth and discontinuous solutions.

Port-Hamiltonian discontinuous Galerkin finite element methods

Abstract A port-Hamiltonian (pH) system formulation is a geometrical notion used to formulate conservation laws for various physical systems. The distributed parameter port-Hamiltonian formulation models infinite dimensional Hamiltonian dynamical systems that have a nonzero energy flow through the boundaries. In this paper, we propose a novel framework for discontinuous Galerkin (DG) discretizations of pH-systems. Linking DG methods with pH-systems gives rise to compatible structure preserving semidiscrete finite element discretizations along with flexibility in terms of geometry and function spaces of the variables involved. Moreover, the port-Hamiltonian formulation makes boundary ports explicit, which makes the choice of structure and power preserving numerical fluxes easier. We state the Discontinuous Finite Element Stokes–Dirac structure with a power preserving coupling between elements, which provides the mathematical framework for a large class of pH discontinuous Galerkin discretizations. We also provide an a priori error analysis for the port-Hamiltonian discontinuous Galerkin Finite Element Method (pH-DGFEM). The port-Hamiltonian discontinuous Galerkin finite element method is demonstrated for the scalar wave equation showing optimal rates of convergence.

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.

Homogenization of Nondivergence-Form Elliptic Equations with Discontinuous Coefficients and Finite Element Approximation of the Homogenized Problem

Homogenization of Nondivergence-Form Elliptic Equations with Discontinuous Coefficients and Finite Element Approximation of the Homogenized Problem

On the necessity of the inf-sup condition for a mixed finite element formulation

Abstract We study a nonstandard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in $\textbf{H}(\operatorname{\textrm{div}};\varOmega )$ for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup stable, but we can show existence and uniqueness of the solution, as well as optimal error estimates for the gradient variable when suitable regularity assumptions are made. Several additional remarks complete the paper, shedding some light on the sources of instability for mixed formulations.

A multithreaded parallel upwind sweep algorithm for the SN transport equations discretized with discontinuous finite elements

A multithreaded parallel upwind sweep algorithm for the SN transport equations discretized with discontinuous finite elements

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.

Non-overlapping domain decomposition algorithms with only primal velocity unknowns for the discontinuous viscosity Stokes problem

In this paper, non-overlapping domain decomposition algorithms for the Stokes problem with a discontinuous viscosity coefficient are proposed and analyzed. A pair of inf–sup stable finite element spaces with discontinuous pressure finite element functions is employed to obtain a discrete Stokes system. By introducing a non-overlapping subdomain partition for the given finite element spaces, the discrete Stokes system is reformulated into an algebraic system on dual unknowns, where the continuity on the velocity unknowns at the subdomain vertices is enforced strongly and the continuity on the remaining interface velocity unknowns is enforced weakly using the dual unknowns. The resulting algebraic system on the dual unknowns is then solved by an iterative method. The coarse problem in the resulting algebraic system is only related to the strongly coupled primal velocity unknowns and it is thus obtained as a symmetric positive definite matrix equation. To accelerate the iteration convergence, practical versions of lumped and Dirichlet preconditioners are proposed and their condition numbers are also analyzed for the resulting dual algebraic system. Numerical results are also included to confirm our condition number estimates.

A continuous hp-mesh model for discontinuous Petrov-Galerkin finite element schemes with optimal test functions

A continuous hp-mesh model for discontinuous Petrov-Galerkin finite element schemes with optimal test functions

Efficient hyperbolic–parabolic models on multi‐dimensional unbounded domains using an extended DG approach

AbstractWe introduce an extended discontinuous Galerkin discretization of hyperbolic–parabolic problems on multidimensional semi‐infinite domains. Building on previous work on the one‐dimensional case, we split the strip‐shaped computational domain into a bounded region, discretized by means of discontinuous finite elements using Legendre basis functions, and an unbounded subdomain, where scaled Laguerre functions are used as a basis. Numerical fluxes at the interface allow for a seamless coupling of the two regions. The resulting coupling strategy is shown to produce accurate numerical solutions in tests on both linear and nonlinear scalar and vectorial model problems. In addition, an efficient absorbing layer can be simulated in the semi‐infinite part of the domain in order to damp outgoing signals with negligible spurious reflections at the interface. By tuning the scaling parameter of the Laguerre basis functions, the extended DG scheme simulates transient dynamics over large spatial scales with a substantial reduction in computational cost at a given accuracy level compared to standard single‐domain discontinuous finite element techniques.

Non-intrusive model order reduction for parametric radiation transport simulations

We present a parametric, non-intrusive reduced-order model (ROM) for multigroup radiation transport, addressing the high computational cost and resource requirements associated with transport simulations in scenarios involving multiple queries. To achieve this, we employ Proper Orthogonal Decomposition (POD) to learn a groupwise reduced basis representation of the observed data (obtained from full-order simulations). We utilize Gaussian Process Regression and Multivariate Adaptive Regression Splines to learn the parametric, low-dimensional latent code (the reduced coordinates in the subspace spanned by the observed data). The approach is non-invasive, making it applicable to any software without the need for accessing the source code. We provide numerical results for various parametric problems, including a 3D multigroup transport case. Comparing the ROM-based surrogate to the full-order radiation transport model, solved using discrete ordinates in angle and discontinuous finite elements in space, we achieve a scalar flux solution that is 20,000 times faster with less than 1% relative error in a 3D transport scenario. Moreover, by utilizing the ROM, we demonstrate how a simple uncertainty quantification study can be completed within minutes, which would otherwise require significantly larger computational resources.

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