Discontinuous Galerkin Finite Element Research Articles

Hydro-morphodynamic modelling of mangroves imposed by tidal waves using finite element discontinuous Galerkin method

Hydro-morphodynamic modelling of mangroves imposed by tidal waves using finite element discontinuous Galerkin method

Tensor-Product Space-Time Goal-Oriented Error Control and Adaptivity With Partition-of-Unity Dual-Weighted Residuals for Nonstationary Flow Problems

Abstract In this work, the dual-weighted residual method is applied to a space-time formulation of nonstationary Stokes and Navier–Stokes flow. Tensor-product space-time finite elements are being used to discretize the variational formulation with discontinuous Galerkin finite elements in time and inf-sup stable Taylor–Hood finite element pairs in space. To estimate the error in a quantity of interest and drive adaptive refinement in time and space, we demonstrate how the dual-weighted residual method for incompressible flow can be extended to a partition-of-unity based error localization. We substantiate our methodology on 2D benchmark problems from computational fluid mechanics.

Regression and Flame Structure in Cavity Flameholding Solid-Fuel Ramjet Fuel Grains

Introducing cavity flameholders into a solid-fuel ramjet fuel grain demonstrated increased fuel loading with sustained combustion in previously unfavorable geometries. Volumetric fuel loading improvements of up to 26% were demonstrated to sustain combustion. Regression patterns of cavity fuel grains are presented and show that the effect of implementing a cavity flameholder is to change the location of maximum regression and the reattachment point. The addition of a cavity flameholder does not appear to have a significant effect on combustion efficiency. However, it is noteworthy that longer cavities increased the chamber pressure above what was observed for a center-perforated fuel grains as a result of the increased mass addition and higher equivalence ratio associated with the higher regression rate. Large-eddy simulation computations were performed using a fourth-order discontinuous Galerkin finite element solver with a novel flamelet and progress variable formulation. The predictions agree well with the experiments and point to the increased heat transfer for longer cavities as the main flameholder mechanism. The larger heat feedback is supported by the formation of a stronger recirculation region, which leads to increased coherent fluctuations due to the transition between local and global instabilities.

An energy-conservative DG-FEM approach for solid–liquid phase change

We present a discontinuous Galerkin method for melting/solidification problems based on the “linearized enthalpy approach,” which is derived from the conservative form of the energy transport equation and does not depend on the use of a so-called mushy zone. We use the symmetric interior penalty method and the Lax–Friedrichs flux to discretize diffusive and convective terms, respectively. Time is discretized with a second-order implicit backward differentiation formula, and two outer iterations with second-order extrapolation predictors are used for the coupling of the momentum and energy. The numerical method was validated with three different benchmark cases, i.e., the one-dimensional Stefan problem, octadecane melting in a square cavity and gallium melting in a rectangular cavity. The performance of the method was quantified based on the L 2 norm error and the number of iterations needed to convergence the energy equation at each time step. For all three validation cases, a mesh convergence rate of approximately O(h) was obtained, which is below the expected accuracy of the numerical method. Only for the gallium melting case, the use of a higher-order method proved to be beneficial. The results from the present numerical campaign demonstrate the promise of the discontinuous Galerkin finite element method for modeling certain solid–liquid phase change problems where large gradients in the flow field are present or the phase change is highly localized, however, further enhancement of the method is needed to fully benefit from the use of a higher-order numerical method when solving solid–liquid phase change problems.

Hybrid mixed discontinuous Galerkin finite element method for incompressible wormhole propagation problem

Hybrid mixed discontinuous Galerkin finite element method for incompressible wormhole propagation problem

Unified discontinuous Galerkin finite-element framework for transient conjugated radiation-conduction heat transfer.

Research on conjugated radiation-conduction (CRC) heat transfer in participating media is of vital scientific and engineering significance due to its extensive applications. Appropriate and practical numerical methods are essential to forecast the temperature distributions during the CRC heat-transfer processes. Here, we established a unified discontinuous Galerkin finite-element (DGFE) framework for solving transient CRC heat-transfer problems in participating media. To overcome the mismatch between the second-order derivative in the energy balance equation (EBE) and the DGFE solution domain, we rewrite the second-order EBE as two first-order equations and then solve both the radiative transfer equation (RTE) and the EBE in the same solution domain, resulting in the unified framework. Comparisons between the DGFE solutions with published data confirm the accuracy of the present framework for transient CRC heat transfer in one- and two-dimensional media. The proposed framework is further extended to CRC heat transfer in two-dimensional anisotropic scattering media. Results indicate that the present DGFE can precisely capture the temperature distribution at high computational efficiency, paving the way for a benchmark numerical tool for CRC heat-transfer problems.

Error analysis of a novel discontinuous Galerkin method for the two-dimensional Poisson's equation

In this paper, we develop a novel discontinuous Galerkin (DG) finite element method for solving the Poisson's equation uxx+uyy=f(x,y) on Cartesian grids. The proposed method consists of first applying the standard DG method in the x-spatial variable leading to a system of ordinary differential equations (ODEs) in the y-variable. Then, using the method of line, the DG method is directly applied to discretize the resulting system of ODEs. In fact, we propose a fully DG scheme that uses p-th and q-th degree DG methods in the x and y variables, respectively. We show that, under proper choices of numerical fluxes, the method achieves optimal convergence rate in the L2-norm of O(hp+1)+O(kq+1) for the DG solution, where h and k denote, respectively, the mesh step sizes for the x and y variables. Our theoretical results are validated through several numerical experiments.

L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation.

In this paper, a class of discrete Gronwall inequalities is proposed. It is efficiently applied to analyzing the constructed L1/local discontinuous Galerkin (LDG) finite element methods which are used for numerically solving the Caputo-Hadamard time fractional diffusion equation. The derived numerical methods are shown to be -robust using the newly established Gronwall inequalities, that is, it remains valid when . Numerical experiments are given to demonstrate the theoretical statements.

A hybrid mixed finite element method for convection-diffusion-reaction equation with local exponential fitting technique

A new hybrid mixed finite element method is proposed for solving the convection-diffusion-reaction equation with local exponential fitting technique. In this method, the exponential fitting technique in [7] is used to discretize the convection-diffusion-reaction equation by introducing a new variable at element level; then the hybrid mixed discontinuous Galerkin finite element method is used to approximate the discretized problem. The convergence of the proposed method is analyzed, and the corresponding a priori error estimate is derived. The numerical results are presented to confirm our theoretical analysis.

Computationally efficient high-fidelity plasma simulations by coupling multi-species kinetic and multi-fluid models on decomposed domains

A numerical method is developed for coupling a multi-species kinetic plasma model with a 5N-moment multi-fluid plasma model. The simulation domain is decomposed such that the local conditions satisfy the corresponding plasma model's region of validity. The method allows for hybrid simulations by formulating each model as a set of conservation laws and using a continuum numerical method to solve each model's governing equations in the subdomains of the decomposed domain. The models are coupled through fluxes across subdomain interfaces. Two methods are explored for the formulation of the fluxes that can be self-consistently represented by both plasma models. One method allows for flux calculations consistent with the 5N-moment multi-fluid plasma model and assumes thermodynamic equilibrium within each species of the kinetic plasma model. The second method ensures conservation of the distribution function as well as mass, momentum, and energy by formulating the fluxes using a composite underlying distribution function at the subdomain interfaces. The methods are compared in 1D1V simulations of a double rarefaction wave and a plasma sheath using the WARPXM framework, which solves each model using the discontinuous Galerkin finite element method. Both methods for formulating the fluxes perform well as the subdomain interface distribution function approaches a Maxwellian, with the consistent method being more robust to larger deviations. A simulation of the magnetized Kelvin-Helmholtz instability in 2D2V is also performed using the consistent method, which demonstrates the potential of the domain-decomposed hybrid method in facilitating speedup and reduction in required computational resources for high-fidelity plasma simulations, allowing for the investigation of problems that are beyond current capabilities.

A locking-free discontinuous Galerkin method for linear elastic Steklov eigenvalue problem

In this paper, a discontinuous Galerkin finite element method of Nitsche's version for the Steklov eigenvalue problem in linear elasticity is presented. The a priori error estimates are analyzed under a low regularity condition, and the robustness with respect to nearly incompressible materials (locking-free) is proven. Furthermore, some numerical experiments are reported to show the effectiveness and robustness of the proposed method.

Discontinuous Galerkin methods for stochastic Maxwell equations with multiplicative noise

In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results.

A Priori Analysis of a Symmetric Interior Penalty Discontinuous Galerkin Finite Element Method for a Dynamic Linear Viscoelasticity Model

Abstract The stress-strain constitutive law for viscoelastic materials such as soft tissues, metals at high temperature, and polymers can be written as a Volterra integral equation of the second kind with a fading memory kernel. This integral relationship yields current stress for a given strain history and can be used in the momentum balance law to derive a mathematical model for the resulting deformation. We consider such a dynamic linear viscoelastic model problem resulting from using a Dirichlet–Prony series of decaying exponentials to provide the fading memory in the Volterra kernel. We introduce two types of internal variable to replace the Volterra integral with a system of auxiliary ordinary differential equations and then use a spatially discontinuous symmetric interior penalty Galerkin (SIPG) finite element method and – in time – a Crank–Nicolson method to formulate the fully discrete problems: one for each type of internal variable. We present a priori stability and error analyses without using Grönwall’s inequality and with the result that the constants in our estimates grow linearly with time rather than exponentially. In this sense, the schemes are therefore suited to simulating long time viscoelastic response, and this (to our knowledge) is the first time that such high quality estimates have been presented for SIPG finite element approximation of dynamic viscoelasticity problems. We also carry out a number of numerical experiments using the FEniCS environment (https://fenicsproject.org), describe a simulation using “real” material data, and explain how the codes can be obtained and all of the results reproduced.

Solution of Neutron Diffusion Problems by Discontinuous Galerkin Finite Element Method With Consideration of Discontinuity Factors

Abstract Neutronics calculations are the basis of reactor analysis and design. Finite element methods (FEM) have gained increasing attention in solving neutron transport problems for the rigorous mathematical formulation and the flexibility of handling complex geometric domains and boundary conditions. In order to reduce the computational errors caused by the homogenization of cross sections, this paper adopts the discontinuous Galerkin finite element method (DG-FEM) to solve the generalized eigenvalue problem formulated by the neutron diffusion theory and compensates the homogenization error by incorporating discontinuity factors. The results show that the discontinuous Galerkin finite element method can introduce the discontinuity factors with clear mathematical and physical meanings. The computational results of the discontinuous Galerkin finite element method are slightly better than those of the continuous Galerkin finite element method. However, the computation cost of the former is higher than that of the latter. Although good parallel efficiency can be achieved, the discontinuous Galerkin finite element method is not preferable for large-scale problems unless the effect of the discontinuity factors is significant.

Discontinuous Galerkin finite element scheme for solving non-linear lumped kinetic model of non-isothermal reactive liquid chromatography

Discontinuous Galerkin finite element scheme for solving non-linear lumped kinetic model of non-isothermal reactive liquid chromatography

Spherical Harmonics and Discontinuous Galerkin Finite Element Methods for the Three-Dimensional Neutron Transport Equation: Application to Core and Lattice Calculation

The spherical harmonics or PN method is intended to approximate the neutron angular flux by a linear combination of spherical harmonics of degree at most . In this work, the PN method is combined with the discontinuous Galerkin (DG) finite elements method and yield to a full discretization of the multigroup neutron transport equation. The employed method is able to handle all geometries describing the fuel elements without any simplification nor homogenization. Moreover, the use of the matrix assembly-free method avoids building large sparse matrices, which enables producing high-order solutions in a small computational time and less storage usage. The resulting transport solver, called NYMO, has a wide range of applications; it can be used for a core calculation as well as for a precise 281-group lattice calculation accounting for anisotropic scattering. To assess the accuracy of this numerical scheme, it is applied to a three-dimensional (3-D) reactor core and fuel assembly calculations. These calculations point out that the proposed PN -DG method is capable of producing precise solutions, while the developed solver is able to handle complex 3-D core and assembly geometries.

An adaptive coupling modeling between peridynamics and classical continuum mechanics for dynamic crack propagation and crack branching

Current techniques for the objective simulation of dynamic crack propagation are highly complex and remain computationally expensive, making the simulation of dynamic crack propagation challenging. Based on the peridynamic theory, it allows discontinuities in the displacement field and promises the fracture simulation. However, a peridynamic model often involves intractable computational costs. As an alternative, an adaptive coupling modeling between the peridynamics and classical continuum mechanics is developed for the dynamic crack propagation and crack branching in this work, using the Morphing coupling method presented in previous work (Lubineau et al., J Mech Phys Solids 60(6):1088–1102, 2012). The Morphing method is promising for the adaptive introduction of the peridynamic model (i.e., a nonlocal model) during the dynamic crack propagation, while the rest of the structure outside the peridynamic region is described using the local model based on the continuum mechanics. Thus, a hybrid local/nonlocal model system is established through adaptive coupling modeling. Moreover, the discontinuous Galerkin finite elements (DGFEs) are adopted in the pure peridynamic and hybrid regions, while the finite elements are adopted in the rest of the structure. A benchmark dynamic fracture problem containing branching patterns is used to verify the adaptive coupling of dynamic models. The results show that the simulation based on the adaptive coupling of dynamic models captures the main features during the dynamic crack propagation and branching. Furthermore, the convergence studies on time increments and mesh density under uniform grid refinement (m-convergence) and under changing the peridynamic horizon (δ-convergence) are also proposed.

Convergence and Error Estimates of a Mixed Discontinuous Galerkin-Finite Element Method for the Semi-stationary Compressible Stokes System

Convergence and Error Estimates of a Mixed Discontinuous Galerkin-Finite Element Method for the Semi-stationary Compressible Stokes System

The Allen-Cahn equation with a time Caputo-Hadamard derivative: Mathematical and Numerical Analysis

<abstract><p>In this paper, we investigate the local discontinuous Galerkin (LDG) finite element method for the fractional Allen-Cahn equation with Caputo-Hadamard derivative in the time domain. First, the regularity of the solution is analyzed, and the results indicate that the solution of this equation generally possesses initial weak regularity in the time dimension. Due to this property, a logarithmic nonuniform L1 scheme is adopted to approximate the Caputo-Hadamard derivative, while the LDG method is used for spatial discretization. The stability and convergence of this fully discrete scheme are proven using a discrete fractional Gronwall inequality. Numerical examples demonstrate the effectiveness of this method.</p></abstract>

Blast Damage Simulation With the Discontinuous Galerkin Finite Element Method of Bond-Based Peridynamics

Blast Damage Simulation With the Discontinuous Galerkin Finite Element Method of Bond-Based Peridynamics