Discontinuous Galerkin Finite Element Method Research Articles

High-order discontinuous Galerkin method with immersed boundary treatment for compressible flows on parallel adaptive Cartesian grids

Adaptive mesh refinement (AMR) technology and high-order methods are important means to improve the quality of simulation results and have been hotspots in the computational fluid dynamics community. In this paper, high-order discontinuous Galerkin (DG) and direct DG (DDG) finite element methods are developed based on a parallel adaptive Cartesian grid to simulate compressible flow. On the one hand, a high-order multi-resolution weighted essentially nonoscillatory limiter is proposed for DG and DDG methods. This limiter can enhance the stability of DG/DDG methods for compressible flows dominated by shock waves. It is also compact, making it suitable for the implementation of AMR with frequent refinement/coarsening. On the other hand, a coupling method of DG and immersed boundary method is proposed to simulate flow around objects. Due to the compactness of DG, the physical quantities of image points can be directly obtained through the DG/DDG polynomial of the corresponding cells. It avoids the wide interpolation stencil of traditional IBM and makes it more suitable for the parallel adaptive Cartesian grid framework in this paper. Finally, the performance of the proposed method is verified through typical two- and three-dimensional cases. The results indicate that the method proposed in this paper has low numerical dissipation in smooth areas and can effectively handle compressible flow dominated by discontinuities. Moreover, for transonic flow over a sphere, the error of results between the proposed method and direct numerical simulation is within 1%, fully validating the accuracy of the method presented in this paper.

Comparing Discontinuous Galerkin Shock-Capturing Techniques Applied to Inviscid Three-Dimensional Hypersonic Flows

This paper considers the performance of various shock-capturing schemes when simulating three-dimensional, hypersonic flows using the nodal discontinuous Galerkin (DG) finite element method on unstructured, hexahedral meshes. Simulations use a new code, Cartablanca++, which is verified using the method of manufactured solutions. Three shock-capturing techniques are compared: artificial viscosity (AV), slope limiting, and subcell finite volume limiting. Three test cases are considered, including a shock tube (one-dimensional), a reflecting shock (two-dimensional), and an inclined cylinder with a hemispherical endcap (three-dimensional). The AV formulation was not robust in the sense that it could not maintain pressure positivity after initialization from freestream conditions in the final three-dimensional test case. The slope and subcell limiters performed well in all simulations. Both techniques robustly captured strong shock waves while still benefiting from the use of high-order polynomials. The targeted application of the slope limiter prevented residual convergence to machine precision, while the subcell limiter could achieve residual convergence. The mixed DG/finite volume formulation, inherent to the subcell limiting scheme, appears sensitive to the inviscid flux function. Future work will consider modifications to reduce this sensitivity. Additionally, modifications to the shock detection techniques would improve performance for both the slope and subcell limiters.

Modelling of courtyard acoustics using the nodal discontinuous Galerkin finite element method to validate a newly developed PML

Evaluating the acoustics of courtyards has gained increasing attention from a sustainable urban design perspective, which requires numerical predictions tools for estimation of noise pollution and assessment of the influence of acoustic conditions. Accurate modeling of the acoustics of a courtyard is scarce because of the variety of factors to include: meteorological effects, building material properties, and geometry of the inner façades. We consider the nodal Discontinuous Galerkin finite element method framework in a 3D static and uniform medium and use it to model the sound propagation in a courtyard. A new and efficient Perfectly Matched Layer formulation is implemented at the open boundary of the courtyard model to absorb outgoing acoustic waves and this way limiting the size of the computational domain. The numerical results are validated against experimental measurements carried out in a scaled model of a courtyard with an elementary design to implement an accurate corresponding numerical model. The experiments were conducted in an anechoic chamber where the meteorological effects are controlled, and the courtyard facades are modelled as flat surfaces with known material properties.

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})} .

Discontinuous Galerkin finite element method for the continuum radiative transfer problem inside axis-symmetric circumstellar envelopes

Context. The study of the continuum radiative transfer problem inside circumstellar envelopes is both a theoretical and numerical challenge, especially in the frequency-dependent and multi-dimensional case. While approximate methods are easier to handle numerically, they often fail to accurately describe the radiation field inside complex geometries. For these cases, it is necessary to directly solve the radiative transfer equation numerically. Aims. We investigate the accuracy of the discontinuous Galerkin finite element method (DGFEM hereafter) applied to the frequency-dependent two-dimensional radiative transfer problem, and coupled with the radiative equilibrium equation. We next used this method in the context of axis-symmetric circumstellar envelopes. Methods. The DGFEM is a variant of finite element methods. It employs discontinuous elements and flux integrals along their boundaries, ensuring local flux conservation. However, as opposed to the classical finite element methods, the solution is discontinuous across element edges. We implemented this approach in a code and tested its accuracy by comparing our results with the benchmarks from the literature. Results. For all the tested cases, the temperatures profiles agree within one percent. Additionally, the emerging spectral energy distributions (SEDs) and images, obtained by ray-tracing techniques from the DGFEM emissivity, agree on average within 5% and 10%, respectively. Conclusions. We show that the DGFEM can accurately describe the continuum radiative transfer problem inside axis-symmetric circumstellar envelopes. Consecutively the emerging SEDs and images are also well reproduced. The DGFEM provides an alternative method (other than Monte-Carlo methods for instance) for solving the radiative transfer equation, and it could be used in cases that are more difficult to handle with the other methods.

A stable decoupled perfectly matched layer for the 3D wave equation using the nodal discontinuous Galerkin method

In outdoor acoustics, the calculations of sound propagating in air can be computationally heavy if the domain is chosen large enough for the sound waves to decay. The computational cost is lowered by strategically truncating the computational domain with an efficient boundary treatment. One commonly used boundary treatment is the perfectly matched layer (PML), which dampens outgoing waves without polluting the computed solution in the inner domain. The purpose of this study is to propose and assess a new perfectly matched layer formulation for the 3D acoustic wave equation, using the nodal discontinuous Galerkin finite element method. The formulation is based on an efficient PML formulation that can be decoupled to further increase the computational efficiency and guarantee stability without sacrificing accuracy. This decoupled PML formulation is demonstrated to be long-time stable, and an optimization procedure for the damping functions is proposed to enhance the performance of the formulation.

Finite element modelling of linear rolling contact problems

The present work is devoted to the finite element modelling of linear hyperbolic rolling contact problems. The main equations encountered in rolling contact mechanics are reviewed in the first part of the paper, with particular emphasis on applications from automotive and vehicle engineering. In contrast to the common hyperbolic systems found in the literature, such equations include integral and boundary terms, as well as time-varying transport velocities, that require special treatment. In this context, existence and uniqueness properties are discussed within the theoretical framework offered by the semigroup theory. The second part of the paper is then dedicated to recovering approximated solutions to the considered problems, by combining discontinuous Galerkin finite element methods (DGMs) with explicit Runge–Kutta (RK) schemes of the first and second order for time discretisation. Under opportune assumptions on the smoothness of the sought solutions, and owing to certain generalised Courant–Friedrichs–Lewy (CFL) conditions, quasi-optimal error bounds are derived for the complete discrete schemes. The proposed algorithms are then tested on simple scalar equations in one space dimension. Numerical experiments seem to suggest the theoretical error estimates to be sharp.

A discontinuous Galerkin method for the Brinkman–Darcy-transport problem

In this paper, a weighted discontinuous Galerkin finite element method and an upwind format are presented to solve the coupled Brinkman–Darcy flow and transport model. The flow in a highly permeable region of the model is governed by the Brinkman equations, and the percolation in the porous media is controlled by the Darcy equations, which are coupled by three interface conditions. The permeability coefficients in this model are strongly nonhomogeneous, anisotropic, and discontinuous; the transport equation is a convection-dominated problem; and the velocity field in the porous media domain does not satisfy the divergence-free condition. A weighted discontinuous Galerkin finite element method is used to solve the complex permeability coefficient problem and an upwind scheme is used to solve the convection dominated problem. The interface conditions can be naturally incorporated into the discrete formulation without introducing additional variables. Optimal error estimates are obtained for the semi-discretization and the full discretization with the backward Euler scheme in suitable energy norm. A series of numerical experiments are provided to illustrate the proposed method, including testing the convergence and accuracy of various types of meshes; simulating fluid flow in complex porous media such as obstacles, layered media, and curved interfaces; and studying the coupled flow behavior of surface-subsurface flows with contaminant transport.

On Error Estimates of a discontinuous Galerkin Method of the Boussinesq System of Equations

Abstract In this paper, we propose and analyze a discontinuous Galerkin finite element method for solving the transient Boussinesq incompressible heat conducting fluid flow equations. This method utilizes an upwind approach to handle the nonlinear convective terms effectively. We discuss new a priori bounds for the semidiscrete discontinuous Galerkin approximations. Furthermore, we establish optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in L 2 \mathbf{L}^{2} and energy norms, the temperature approximation in L 2 L^{2} and energy norms and pressure approximation in L 2 L^{2} -norm for t > 0 t>0 . Additionally, under the smallness assumption on the data, we prove uniform in time error estimates. We also consider a backward Euler scheme for full discretization and derive fully discrete error estimates. Finally, we provide numerical examples to support the theoretical conclusions.

Efficient High-Order Space-Angle-Energy Polytopic Discontinuous Galerkin Finite Element Methods for Linear Boltzmann Transport

We introduce an hp-version discontinuous Galerkin finite element method (DGFEM) for the linear Boltzmann transport problem. A key feature of this new method is that, while offering arbitrary order convergence rates, it may be implemented in an almost identical form to standard multigroup discrete ordinates methods, meaning that solutions can be computed efficiently with high accuracy and in parallel within existing software. This method provides a unified discretisation of the space, angle, and energy domains of the underlying integro-differential equation and naturally incorporates both local mesh and local polynomial degree variation within each of these computational domains. Moreover, general polytopic elements can be handled by the method, enabling efficient discretisations of problems posed on complicated spatial geometries. We study the stability and hp-version a priori error analysis of the proposed method, by deriving suitable hp-approximation estimates together with a novel inf-sup bound. Numerical experiments highlighting the performance of the method for both polyenergetic and monoenergetic problems are presented.

A discontinuous Galerkin method for a coupled Stokes–Biot problem

The interaction between fluid and poroelastic structure is a complex problem that couples the Stokes equations with the Biot system. In this work, we rewrite the poroelastic equations using three fields (displacement, fluid pressure, and total pressure) to account for locking phenomenon in fluid–structure interaction and the oscillation of numerical solutions. Then, in order to solve the coupled system, we use the weighted discontinuous Galerkin finite element method for spatial discretization and propose a semi-discrete scheme. In addition, we propose a fully discrete scheme using the backward Euler’s method for temporal discretization. In the framework of the Galerkin approximation, the stability and errors of semi- and fully discrete schemes are analyzed using the theory of differential algebraic equations and weak compactness arguments. Through numerical experiments, we validate the theoretical rate of convergence, explain the behavior of the method in modeling the interaction between surface and groundwater hydrological systems, investigate the robustness of the method with respect to physical parameters, and simulate channel filtering. In particular, the numerical results are free of locking features, reducing non-physical numerical oscillations.

Analysis of two discontinuous Galerkin finite element methods for the total pressure formulation of linear poroelasticity model

In this paper, we develop two discontinuous Galerkin (DG) finite element methods to solve the linear poroelasticity in the total pressure formulation, where displacement, fluid pressure, and total pressure are unknowns. The fully-discrete standard DG and conforming DG methods are presented based on the discontinuous approximations in space and the implicit Euler discretization in time. Compared to the standard DG method with penalty terms, the conforming DG method removes all stabilizers and maintains conforming finite element formulation by utilizing weak operators defined over discontinuous functions. The two methods provide locally conservative solutions and achieve locking-free properties in poroelasticity. We also derive the well-posedness and optimal a priori error estimates, which show that our methods satisfy parameter-robustness with respect to the infinitely large Lamé constant and the null-constrained specific storage coefficient. Several numerical experiments are performed to verify these theoretical results, even in heterogeneous porous media.

Capturing Local Temperature Evolution During Additive Manufacturing Through Fourier Neural Operators

Abstract High-fidelity, data-driven models that can quickly simulate thermal behavior during additive manufacturing (AM) are crucial for improving the performance of AM technologies in multiple areas, such as part design, process planning, monitoring, and control. However, complexities of part geometries make it challenging for current models to maintain high accuracy across a wide range of geometries. In addition, many models report a low mean-square error (MSE) across the entire domain of a part. However, in each time-step, most areas of the domain do not experience significant changes in temperature, except for the regions near recent depositions. Therefore, the MSE-based fidelity measurement of the models may be overestimated. This article presents a data-driven model that uses the Fourier neural operator to capture the local temperature evolution during the AM process. Besides MSE, the model is also evaluated using the R2 metric, which places great weight on the regions where the temperature changes significantly than MSE. The model was trained and tested on numerical simulations based on the discontinuous Galerkin finite element method for the direct energy deposition AM process. The results shows that the model maintains 0.983−0.999 R2 over geometries not included in the training data, which is higher than convolutional neural networks and graph convolutional neural networks we implemented, the two widely used architectures in data-driven predictive modeling.

Operator-splitting finite element method for solving the radiative transfer equation

AbstractAn operator-splitting finite element scheme for the time-dependent radiative transfer equation is presented in this paper. The streamline upwind Petrov-Galerkin finite element method and discontinuous Galerkin finite element method are used for the spatial-angular discretization of the radiative transfer equation, whereas the backward Euler scheme is used for temporal discretization. Error analysis of the proposed numerical scheme for the fully discrete radiative transfer equation is presented. The stability and convergence estimates for the fully discrete problem are derived. Moreover, an operator-splitting algorithm for the numerical simulation of high-dimensional equations is also presented. The validity of the derived estimates and implementation is illustrated with suitable numerical experiments.

Discontinuous Galerkin finite element method for solving non-linear model of gradient elution chromatography

Discontinuous Galerkin finite element method for solving non-linear model of gradient elution chromatography

Discontinuous Galerkin finite element method for dynamic viscoelasticity models of power‐law type

AbstractLinear viscoelasticity can be characterized by a stress relaxation function. We consider a power‐law type stress relaxation to yield a fractional order viscoelasticity model. The governing equation is a Volterra integral problem of the second kind with a weakly singular kernel. We employ spatially discontinuous Galerkin methods, symmetric interior penalty Galerkin method (SIPG) for spatial discretization, and the implicit finite difference schemes in time, Crank–Nicolson method. Further, in order to manage the weak singularity in the Volterra kernel, we use a linear interpolation technique. We present a priori stability and error analyses without relying on Grönwall's inequality, and so provide high quality bounds that do not increase exponentially in time. This indicates that our numerical scheme is well‐suited for long‐time simulations. Despite the limited regularity in time, we establish suboptimal fractional order accuracy in time as well as optimal convergence of SIPG. We carry out numerical experiments with varying regularity of exact solutions to validate our error estimates. Finally, we present numerical simulations based on real material data.

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.

Quadratic Discontinuous Galerkin Finite Element Methods for the Unilateral Contact Problem

Abstract In this article, we employ discontinuous Galerkin methods for the finite element approximation of the frictionless unilateral contact problem using quadratic finite elements over simplicial triangulation. We first develop a posteriori error estimates in the energy norm wherein, the reliability and efficiency of the proposed a posteriori error estimator is addressed. The suitable construction of the discrete Lagrange multiplier 𝝀 𝒉 {\boldsymbol{\lambda_{h}}} and some intermediate operators play a key role in developing a posteriori error analysis. Further, we establish an optimal a priori error estimates under the appropriate regularity assumption on the exact solution 𝒖 {\boldsymbol{u}} . Numerical results presented on uniform and adaptive meshes illustrate and confirm the theoretical findings.

Discontinuous Galerkin method for the coupled dual-porosity-Brinkman model

In this paper we use the weighted discontinuous Galerkin finite element method to solve the dual-porosity-Brinkman coupling model. The dual-porosity model describes fluid flow in a dual-porosity medium composed of matrix and microfractures, while the Brinkman equation describes fluid flow in hydraulic fractures or conduits containing “debris filling” or “proppant filling”. The two models are coupled by the four physically effective interface conditions, which are constructed based on the fundamental properties of the dual-porosity model and the Brinkman-Darcy model. We provide a weighted discontinuous discrete scheme for the model and derive the optimal error estimates. Through numerical experiments, we verify the properties of the model and the advantages of the numerical method, such as the optimal convergence rate of the numerical solution and the simulation of the flow characteristics around hydraulic fractures and conduits.

The effect of peristalsis on dispersion in Casson fluid flow

The present study examines the peristaltic flow of a non-Newtonian (Casson) fluid and solute transport through a flexible tube. Using the long-wavelength approximation, an analytical solution for the Casson fluid velocity is obtained in the axial and radial directions. Then, a numerical solution to the solute concentration is computed using the standard Galerkin finite element method (FEM) for the case of solutal convection-diffusion and the Discontinuous Galerkin FEM for the pure convection case. Graphical results are presented for the axial and local solute concentrations, and the influence of peristaltic wave amplitude, volumetric flow rate, modified Péclet number, time, yield stress, and pertinent initial condition parameters were investigated.