Discontinuous Galerkin Formulation Research Articles

Very High-Order Accurate Discontinuous Galerkin Method for Curved Boundaries with Polygonal Meshes

Preserving the optimal convergence order of discontinuous Galerkin (DG) discretisations in curved domains is a critical and well-known issue. The proposed approach relies on the reconstruction for off-site data (ROD) method developed originally within the finite volume framework. The main advantages are simplicity, since the PDE solver only considers polygonal domains, and versatility, since any type of boundary condition can be imposed. The developed DG–ROD method consists in splitting the boundary conditions treatment and the leading discrete equations from a classical DG formulation into two independent solvers coupled in a simple and efficient iterative procedure. A numerical benchmark is provided to assess the capability of the method with Dirichlet and Neumann boundary conditions prescribed on curved boundaries, demonstrating that the optimal convergence order is effectively achieved.

A unified discontinuous Galerkin formulation for interfacial multiphysics modeling of thermo-chemically driven fracture

Many engineering and natural materials exhibit coupled thermo-chemo-mechanical phenomena, which can result in embrittlement and fracture. These fractures, in turn, can alter the subsequent thermal, chemical, and mechanical response. We present a theoretical formulation and computational framework for the analysis of thermo-chemically fractured solids, with emphasis on the post-fracture thermal and chemical interfacial behavior. The theoretical model is based on the thermodynamically-consistent formulation of Loeffel and Anand (IJP, 2011). The computational method extends the scalable discontinuous Galerkin/Cohesive Zone Model (DG/CZM) of Radovitzky et al. (CMAME, 2011) to thermo-chemo-mechanics, which facilitates coupled, large-scale simulations of materials and structures containing failed interfaces. In the proposed framework, all balance laws are enforced weakly via the DG formalism, resulting in a unified formulation for multiphysics problems in solids. This naturally enables the incorporation of general interface models, e.g. to account for effects such as the aeolotropic reduction in thermochemical transport due to the presence of fractures, or the acceleration of chemical reactions along crack flanks. The approach is verified against two analytical solutions of boundary value problems drawn from thermo-poro-elasticity and thermally-driven delamination. A scalable, three-dimensional simulation of thermochemically-driven concrete cracking illustrates the complete capabilities of the interfacial multiphysics modeling framework.

High-order accurate transient and free-vibration analysis of plates and shells

The limited availability of analytical solutions and the high cost associated with experimental testing motivate the use of computational tools to assess the dynamic behavior of load-bearing components, especially when a wide design space must be explored, as is often the case with composite structures. In this context, a novel high-order accurate discontinuous Galerkin formulation for transient and free-vibration analysis of multilayered plates and shells is presented and numerical validated. The starting point of the formulation is a generalized structural theory for multilayered shells with arbitrary curvature based on the expansion of the displacement covariant components throughout the shell thickness. The variational statement of three-dimensional elastodynamics allows deriving the strong form of the governing differential equations, which form the basis to obtain the corresponding discontinuous Galerkin weak statements. As the order of the through-the-thickness expansion and the order of the discontinuous Galerkin basis functions are free parameters, the proposed approach allows tuning the order of accuracy of the computed solution throughout both the shell thickness and the shell modeling domain. Numerical results are reported and discussed for several validation test cases in terms of h- and p-convergence analyses, demonstrating the high-order accuracy, robustness, and computational savings of the formulation.

A low-cost, penalty parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations

This paper proposes a low-cost, penalty parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, despite its advantage of symmetry, the symmetric IPDG formulation requires a lot of computational effort to choose an optimal penalty parameter and compute different trace terms. To reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a penalty parameter-free numerical scheme that has reduced computational complexity and conserves the optimal convergence orders. Moreover, we achieve pressure robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples.

A numerical investigation on coupling of conforming and hybridizable interior penalty discontinuous Galerkin methods for fractured groundwater flow problems

The present paper focuses on the numerical modeling of groundwater flows in fractured porous media using the codimensional model description. Therefore, fractures are defined explicitly as a (d−1)-dimensional geometric object immersed in a d-dimensional region and can act arbitrarily as a drain or a barrier. We numerically investigate a novel numerical strategy combining distinctive classes of conforming and nonconforming high-order Galerkin methods, both eligible for static condensation. This procedure is here particularly relevant, leading to a smaller and sparser final system with coupled degrees of freedom solely on the mesh skeleton. Precisely, we combine an inspired hybridizable interior penalty discontinuous Galerkin (HIP) formulation inside the bulk region and a standard continuous Galerkin (CG) approximation on the fracture network. The distinctive discretization of corresponding PDEs and the coupling strategy are rigorously exposed, and the local and global matrix assemblies are detailed. Extensive numerical experiments are then achieved to prove the model’s performances for 2D/3D analytic and realistic benchmarks. Qualitative comparisons are also considered with other discretization methods and commercial software, such as comsol.

Local randomized neural networks with discontinuous Galerkin methods for partial differential equations

Randomized Neural Networks (RNNs) are a variety of neural networks in which the hidden-layer parameters are fixed to randomly assigned values, and the output-layer parameters are obtained by solving a linear system through least squares. This improves the efficiency without degrading the accuracy of the neural network. In this paper, we combine the idea of the Local RNN (LRNN) and the Discontinuous Galerkin (DG) approach for solving partial differential equations. RNNs are used to approximate the solution on the subdomains, and the DG formulation is used to glue them together. Taking the Poisson problem as a model, we propose three numerical schemes and provide convergence analysis. Then we extend the ideas to time-dependent problems. Taking the heat equation as a model, three space–time LRNN with DG formulations are proposed. Finally, we present numerical tests to demonstrate the performance of the methods developed herein. We evaluate the performance of the proposed methods by comparing them with the finite element method and the conventional DG method. The LRNN-DG methods can achieve higher accuracy with the same degrees of freedom, and can solve time-dependent problems more precisely and efficiently. This indicates that this new approach has great potential for solving partial differential equations.

A hybridizable discontinuous Galerkin formulation for the Euler–Maxwell plasma model

This work introduces a hybridizable discontinuous Galerkin formulation for the simulation of ideal plasmas governed by the Euler–Maxwell equations. The approach is based on a monolithic source-based coupling of the fluid and electromagnetic subproblems, along with a fully implicit time integration scheme, and a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. The numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation and validate it against problems of increasing complexity, ranging from single-physics cases to weakly and fully coupled electromagnetic plasma simulations.

A face-centred finite volume approach for coupled transport phenomena and fluid flow

We present a particular derivation of the face-centred finite volume (FCFV) method and study its performance in non-linear, coupled transport problems commonly encountered in geoscientific and geotechnical applications. The FCFV method is derived from the hybridisable discontinuous Galerkin formulation, using a constant degree of approximation for the discretization of the unknowns defined on the mesh faces (edges in two dimensions). The piecewise constant degrees of freedom are determined in a global problem over the mesh skeleton. Then, the solution and its gradient are recovered at the cells centroid in a set of element-by-element independent postprocesses, both exhibiting linear convergence. The formulation of the transient advection-diffusion-reaction equation is presented in detail and the numerical analysis under challenging advective/diffusive regimes is studied. Finally, we use several numerical examples to illustrate the advantages and limitations of the FCFV method to solve problems of geoscientific and geotechnical relevance governed by the non-linear coupling between advection-diffusion-reactive transport and Stokes flow. Our results show that the FCFV method is an attractive and highly competitive alternative to other commonly used methods.

A Discontinuous Galerkin and Semismooth Newton Approach for the Numerical Solution of Bingham Flow with Variable Density

Abstract This paper is devoted to the study of Bingham flow with variable density. We propose a local bi-viscosity regularization of the stress tensor based on a Huber smoothing step. Next, our computational approach is based on a second-order, divergence-conforming discretization of the Huber regularized Bingham constitutive equations, coupled with a discontinuous Galerkin scheme for the mass density. We take advantage of the properties of divergence-conforming and discontinuous Galerkin formulations to effectively incorporate upwind discretizations, thereby ensuring the stability of the formulation. The stability of the continuous problem and the fully discrete scheme are analyzed. Further, a semismooth Newton method is proposed for solving the obtained fully discretized system of equations at each time step. Finally, several numerical examples that illustrate the main features of the problem and the properties of the numerical scheme are presented.

Convex optimization-based structure-preserving filter for multidimensional finite element simulations

In simulation sciences, capturing the real-world problem features as accurately as possible is desirable. Methods popular for scientific simulations such as the finite element method (FEM) and finite volume method (FVM) use piecewise polynomials to approximate various characteristics of a problem, such as the concentration profile and the temperature distribution across the domain. Polynomials are prone to creating artifacts such as Gibbs oscillations while capturing a complex profile. An efficient and accurate approach must be applied to deal with such inconsistencies to obtain accurate simulations. This often entails dealing with negative values for the concentration of chemicals, exceeding a percentage value over 100, and other such problems. We consider these inconsistencies in the context of partial differential equations (PDEs). We propose an innovative filter based on convex optimization to deal with the inconsistencies observed in polynomial-based simulations. In two or three spatial dimensions, additional complexities are involved in solving the problems related to structure preservation. We present the construction and application of a structure-preserving filter with a focus on multidimensional PDEs. Methods used such as the Barycentric interpolation for polynomial evaluation at arbitrary points in the domain and an optimized root-finder to identify points of interest, improve the filter efficiency, usability, and robustness. Lastly, we present numerical experiments in 2D and 3D using discontinuous Galerkin formulation and demonstrate the filter's efficacy to preserve the desired structure. As a real-world application, implementation of the mathematical biology model involving platelet aggregation and blood coagulation has been reviewed and the issues around FEM implementation of the model are resolved by applying the proposed structure-preserving filter.

A discontinuous Galerkin formulation for nonlinear analysis of multilayered shells refined theories

A novel pure penalty discontinuous Galerkin method is proposed for the geometrically nonlinear analysis of multilayered composite plates and shells, modelled via high-order refined theories. The approach allows to build different two-dimensional equivalent single layer structural models, which are obtained by expressing the covariant components of the displacement field through-the-thickness via Taylor’s polynomial expansion of different order. The problem governing equations are deduced starting from the geometrically nonlinear principle of virtual displacements in a total Lagrangian formulation. They are addressed with a pure penalty discontinuous Galerkin method using Legendre polynomials trial functions. The resulting nonlinear algebraic system is solved by a Newton–Raphson arc-length linearization scheme. Numerical tests involving plates and shells are proposed to validate the method, by comparison with literature benchmark problems and finite element solutions, and to assess its features. The obtained results demonstrate the accuracy of the method as well as the effectiveness of high-order elements.

Discontinuous Galerkin Integral Equation Approach for Electromagnetic Modeling of Realistic and Complex Radiating Systems

An electric field integral equation (EFIE) discontinuous Galerkin (DG) formulation for the electromagnetic radiation analysis of realistic antennas is presented. Radiation problems are known to be computationally challenging, involving complex multiscale geometries often made up of a collection of closed and open surfaces. This complexity points to the convenience of separately building and meshing parts with different characteristics, either physical or geometric. This piecewise modelling approach is also desirable in the context of design optimization processes. The proposed formulation fits perfectly into this context by enabling the assembly of independently-meshed surfaces, yet without the need to enforce mesh conformity along the junction contours. The continuity of the normal components of the equivalent current at junctions is enforced using the interior penalty (IP) concept, which is extended to multiple conducting surfaces meeting at a junction.

H-adaptation for high-order discontinuous Galerkin schemes built on local multiwavelet analysis

We develop and analyze error estimators and mesh adaptation strategies within a discontinuous Galerkin formulation. The basic idea of the study is to reduce the computational cost of the simulation by employing mesh adaptation as a better alternative to the use of uniform grids. The novelty of the study resides in the use of multiwavelets and how their remarkable properties may shed new light on driving the adaptation process. This is motivated by the fact that multiwavelets break any input apart into a hierarchy of low resolution data and subsequently finer details. Our error estimator makes use of multiwavelets’ properties while being local to the element, thereby maintaining the parallel efficiency of the solver. Early tests on the one-dimensional viscous Burgers equation have shown convincing results (García Bautista et al. [1]). This work is focused on the laminar backward-facing step configuration to assess the performance of the method in higher dimensions.

Any order spectral volume methods for diffusion equations using the local discontinuous Galerkin formulation

In this paper, we present and study two spectral volume (SV) schemes of arbitrary order for diffusion equations by using the local discontinuous Galerkin formulation to discretize the viscous flux. The basic idea of the scheme is to rewrite the diffusion equation into an equivalent first-order system first, and then use the SV method to solve the system. The SV scheme is designed with control volumes constructed by using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes referred to as LSV and RSV schemes, respectively. The stability analysis for the linear diffusion equations based on alternating fluxes are provided, and optimal error estimates are established for both the exact solution and the auxiliary variable. Furthermore, a rigorous mathematical proof are given to demonstrate that the proposed RSV method is identical to the standard LDG method when applied to constant diffusion problems. Numerical experiments are presented to demonstrate the stability, accuracy and performance of the two SV schemes for both linear and nonlinear diffusion equations.

Dislocation structures formation induced by thermal stress in additive manufacturing: Multiscale crystal plasticity modeling of dislocation transport

The motion of dislocations governs the plastic deformation of crystalline materials, which in turn determines the mechanical properties. The complex thermal history, large temperature gradients and high cooling rates during the process of additive manufacturing (AM) can induce high dislocation density and unique dislocation structures in the material. The origin of these dislocation structures and their stability during mechanical loading are debated. A novel temperature dependent continuum dislocation dynamics (CDD) model is developed, in which four state variables are used for each slip system representing the total dislocation density, edge and screw geometrically necessary dislocation densities and dislocation curvature. The CDD model is fully coupled with a crystal plasticity solver, which captures the plastic deformation induced by the dislocation motion. A hybrid continuous and discontinuous Galerkin formulation is developed to accurately reproduce the dynamics of highly discontinuous dislocation density fields that are typical of dislocation structures. A multiscale modeling approach is used, in which the thermally induced deformation in specific grains of a polycrystal is extracted from larger scale crystal plasticity simulations of the laser powder-bed fusion process, and is then used for single crystal scale dislocation dynamics simulations. Simulation results reveal the dynamics of dislocation structure formation in grains at different positions during laser scanning and cooling stages. The effect of the cyclic thermal stress during multi-layer AM fabrication is also investigated. The simulations provide a new perspective on the specific conditions that should be satisfied during AM process for the formation of stable dislocation structures.

HODG:high-order discontinuous Galerkin methods for solving compressible Euler and Navier-Stokes equations - an open-source component-based development framework

In this work, we present HODG, an open-source component-based development framework based on high order Discontinuous Galerkin (DG) methods for solving compressible Euler and Navier-Stokes equations. This framework is written in pure C++11, and proposes “component” as the basic function unit, which is the key to the Interface-Oriented Programming principle and Aspect-Oriented Programming (AOP) technology. Built on the top-level design of components, HODG is a flexible yet pragmatic development framework that works right out of the box and is easy to use for starters and developers.The current release of HODG supports structured, unstructured, hybrid, and second-order meshes available. It is capable of solving Euler and Navier-Stokes (N-S) equations in two dimensions and three dimensions. In this development framework, spatial accuracy is easy to be extended to higher orders, and DG-P0, DG-P1, DG-P2 (up to third order) are implemented in the current version. Moreover, implicit temporal discretization and explicit discretization are available. Especially for viscous fluxes, the direct Discontinuous Galerkin (DDG) formulation and Bassi and Rebay II (BR2) scheme are implemented. In the case of strong discontinuities or shock waves, artificial viscosity is applied to capture shock waves. Various benchmark numerical examples are provided to demonstrate the full capabilities of HODG. The software is freely available under an MIT license. Program summaryProgram Title: HODG v2.0.0CPC Library link to program files:https://doi.org/10.17632/835732yz8s.1Code Ocean capsule:https://codeocean.com/capsule/4788938Licensing provisions: MITProgramming language: C++11Nature of problem: Compressible Euler and Navier–Stokes equations of fluid dynamics; potential for any advection–diffusion type problem.Solution method: DG-P0, DG-P1, DG-P2 spatial discretization, combining with DDG and BR2 schemes for viscous fluxes, explicit and implicit time schemes, and supporting structured, unstructured, hybrid, and high-order meshes.unusual features: developed based on modern C++11 features; propose the “component” as basic function unit, which follows the practice of Interface-Oriented Programming principle; Special component named “action” is used, and is the key to AOP technology in this framework; Implicitly implements the Dependency Injection (DI) technology; Flexible yet pragmatic, and works right out of the box.

An enriched immersed finite element method for interface problems with nonhomogeneous jump conditions

This article presents the first higher degree immersed finite element (IFE) method with proven optimal convergence for elliptic interface problems with nonhomogeneous jump conditions. It also gives the first analysis for the condition numbers of the resulting systems including the optimal upper bounds with respect to the mesh size and its robustness with respect to small-cut interface elements. In this method, jump conditions are approximated optimally by basic IFE and enrichment IFE which are piecewise pth degree polynomial functions constructed by solving local Cauchy problems on interface elements. The proposed IFE method is based on a discontinuous Galerkin formulation on interface elements and a continuous Galerkin formulation on non-interface elements.

A new symmetric interior penalty discontinuous Galerkin formulation for the Serre–Green–Naghdi equations

AbstractIn this work, we investigate the construction of a new discontinuous Galerkin discrete formulation to approximate the solution of Serre–Green–Naghdi (SGN) equations in the one‐dimensional horizontal framework. Such equations describe the time evolution of shallow water free surface flows in the fully nonlinear and weakly dispersive asymptotic approximation regime. A new non‐conforming discrete formulation belonging to the family of symmetric interior penalty discontinuous Galerkin methods is introduced to accurately approximate the solutions of the second order elliptic operator occurring in the SGN equations. We show that the corresponding discrete bilinear form enjoys some consistency and coercivity properties, thus ensuring that the corresponding discrete problem is well‐posed. The resulting global discrete formulation is then validated through an extended set of benchmarks, including convergence studies and comparisons with data taken from experiments.

Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems

The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. Specifically in the optimal control minimization problem, a tracking-type cost functional is minimized to steer the crack via the phase-field variable into a desired pattern. To achieve such optimal solutions, Neumann type boundary conditions need to be determined. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a challenge is to include regularization terms and the crack irreversibility constraint. The optimal control setting is formulated by means of the Lagrangian approach from which the primal part, adjoint, tangent and adjoint Hessian are derived. Herein the overall Newton algorithm is based on a reduced approach by eliminating the state constraint, namely the displacement and phase-field unknowns, but keeping the control variable as the only unknown. From the low-order discontinuous Galerkin discretization, adjoint time-stepping schemes are finally obtained. Both our formulation and algorithmic developments are substantiated and illustrated with six numerical experiments.

A Unifying Algebraic Framework for Discontinuous Galerkin and Flux Reconstruction Methods Based on the Summation-by-Parts Property

We propose a unifying framework for the matrix-based formulation and analysis of discontinuous Galerkin (DG) and flux reconstruction (FR) methods for conservation laws on general unstructured grids. Within such an algebraic framework, the multidimensional summation-by-parts (SBP) property is used to establish the discrete equivalence of strong and weak formulations, as well as the conservation and energy stability properties of a broad class of DG and FR schemes. Specifically, the analysis enables the extension of the equivalence between the strong and weak forms of the discontinuous Galerkin collocation spectral-element method demonstrated by Kopriva and Gassner (J Sci Comput 44:136–155, 2010) to more general nodal and modal DG formulations, as well as to the Vincent-Castonguay-Jameson-Huynh (VCJH) family of FR methods. Moreover, new algebraic proofs of conservation and energy stability for DG and VCJH schemes with respect to suitable quadrature rules and discrete norms are presented in which the SBP property serves as a unifying mechanism for establishing such results. Numerical experiments are provided for the two-dimensional linear advection and Euler equations, highlighting the design choices afforded for methods within the proposed framework and corroborating the theoretical analysis.