Hybridizable Discontinuous Galerkin Research Articles

An Adaptive and Quasi-periodic HDG Method for Maxwell’s Equations in Heterogeneous Media

With the aim to continue developing a hybridizable discontinuous Galerkin (HDG) method for problems arisen from photovoltaic cells modeling, in this manuscript we consider the time harmonic Maxwell’s equations in an inhomogeneous bounded bi-periodic domain with quasi-periodic conditions on part of the boundary. We propose an HDG scheme where quasi-periodic boundary conditions are imposed on the numerical trace space. Under regularity assumptions and a proper choice of the stabilization parameter, we prove that the approximations of the electric and magnetic fields converge, in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{L}^2$$\\end{document}-norm, to the exact solution with order hk+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^{k+1}$$\\end{document} and hk+1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h^{k+1/2}$$\\end{document}, resp., where h is the meshsize and k the polynomial degree of the discrete spaces. Although, numerical evidence suggests optimal order of convergence for both variables. An a posteriori error estimator for an energy norm is also proposed. We show that it is reliable and locally efficient under certain conditions. Numerical examples are provided to illustrate the performance of the quasi-periodic HDG method and the adaptive scheme based on the proposed error indicator.

Adjoint-based aerodynamic shape optimization with hybridized discontinuous Galerkin methods

We present a discrete adjoint approach to aerodynamic shape optimization (ASO) based on a hybridized discontinuous Galerkin (HDG) discretization. Our implementation is designed to tie in as seamlessly as possible into a solver architecture written for general balance laws, thus adding design capability to a tool with a wide range of applicability. Design variables are introduced on designated surfaces using the knots of a 2D spline-based geometry representation, while gradients are computed from the adjoint solution using a difference approximation of residual perturbations. A suitable optimization algorithm, such as an in-house steepest descent or the Preconditioned Sequential Quadratic Programming (PSQP) approach from the pyOpt framework, is then employed to find an improved geometry. We present verification of the implementation, including drag or heat flux minimization in compressible flows, as well as inverse design.

Numerical analysis of a hybridized discontinuous Galerkin method for the Cahn–Hilliard problem

Abstract The mixed form of the Cahn–Hilliard equations is discretized by the hybridized discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg–Landau chemical energy density and for convex domains. The paper also contains discrete functional tools, namely discrete Agmon and Gagliardo–Nirenberg inequalities, which are proved to be valid in the hybridizable discontinuous Galerkin spaces.

Homogeneous multigrid for HDG applied to the Stokes equation

Abstract We propose a multigrid method to solve the linear system of equations arising from a hybrid discontinuous Galerkin (in particular, a single face hybridizable, a hybrid Raviart–Thomas, or a hybrid Brezzi–Douglas–Marini) discretization of a Stokes problem. Our analysis is centered around the augmented Lagrangian approach and we prove uniform convergence in this setting. Beyond this, we establish relations, which resemble those in Cockburn & Gopalakrishnan (2008, Error analysis of variable degree mixed methods for elliptic problems via hybridization. Math. Comput., 74, 1653–1677) for elliptic problems, between the approximates that are obtained by the single-face hybridizable, hybrid Raviart–Thomas and hybrid Brezzi–Douglas–Marini methods. Numerical experiments underline our analytical findings.

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.

Optoelectronic optimization of graded-bandgap thin-film AlGaAs solar cells. Part II: optimal antireflection front-surface texturing.

In Part I [Appl. Opt.59, 1018 (2020)APOPAI0003-693510.1364/AO.381246], we used a coupled optoelectronic model to optimize a thin-film AlGaAs solar cell with a graded-bandgap photon-absorbing layer and a periodically corrugated Ag backreflector combined with localized ohmic Pd-Ge-Au backcontacts, because both strategies help to improve the performance of AlGaAs solar cells. However, the results in Part I were affected by a normalization error, which came to light when we replaced the hybridizable discontinuous Galerkin scheme for electrical computation by the faster finite-difference scheme. Therefore, we re-optimized the solar cells containing an n-AlGaAs photon-absorbing layer with either a (i) homogeneous, (ii) linearly graded, or (iii) nonlinearly graded bandgap. Another way to improve the power conversion efficiency is by using a surface antireflection texturing on the wavelength scale, so we also optimized four different types of 1D periodic surface texturing: (i) rectangular, (ii) convex hemi-elliptical, (iii) triangular, and (iv) concave hemi-elliptical. Our new results show that the optimal nonlinear bandgap grading enhances the efficiency by as much as 3.31% when the n-AlGaAs layer is 400nm thick and 1.14% when that layer is 2000nm thick. A hundredfold concentration of sunlight can enhance the efficiency by a factor of 11.6%. Periodic texturing of the front surface on the scale of 0.5-2 free-space wavelengths provides a small relative enhancement in efficiency over the AlGaAs solar cells with a planar front surface; however, the enhancement is lower when the n-AlGaAs layer is thicker.

An [formula omitted] error analysis of a hybrid discontinuous mixed Galerkin method for linear viscoelasticity

We present a hybrid discontinuous Galerkin method for the velocity/stress formulation of Zener’s model in dynamic viscoelasticity. Our approach utilizes a spatial discretization that enforces strongly the symmetry of the stress tensor, and that allows for efficient handling of heterogeneous materials comprising both purely elastic and viscoelastic components. We provide an hp error analysis of the semidiscrete scheme, which yields quasi-optimal error estimates for the stress tensor and sub-optimal error estimates for the velocity field in the L2-norm. Next, we apply the Crank–Nicolson rule as a time-stepping scheme and analyze its primary convergence properties. Finally, we present the results of numerical experiments to validate our approach and confirm the theoretical rates of convergence.

An adaptive viscosity regularization approach for the numerical solution of conservation laws: Application to finite element methods

We introduce an adaptive viscosity regularization approach for the numerical solution of systems of nonlinear conservation laws with shock waves. The approach seeks to solve a sequence of regularized problems consisting of the system of conservation laws and an additional Helmholtz equation for the artificial viscosity. We propose a homotopy continuation of the regularization parameters to minimize the amount of artificial viscosity subject to positivity-preserving and smoothness constraints on the numerical solution. The regularization methodology is combined with a mesh adaptation strategy that identifies the shock location and generates shock-aligned meshes, which allows to further reduce the amount of artificial dissipation and capture shocks with increased accuracy. We use the hybridizable discontinuous Galerkin method to numerically solve the regularized system of conservation laws and the continuous Galerkin method to solve the Helmholtz equation for the artificial viscosity. We show that the approach can produce approximate solutions that converge to the exact solution of the Burgers' equation. Finally, we demonstrate the performance of the method on inviscid transonic, supersonic, hypersonic flows in two dimensions. The approach is found to be accurate, robust and efficient, and yields very sharp yet smooth solutions in a few homotopy iterations.

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 hybridizable discontinuous Galerkin method with characteristic variables for Helmholtz problems

A new hybridizable discontinuous Galerkin method, named the CHDG method, is proposed for solving time-harmonic scalar wave propagation problems. This method relies on a standard discontinuous Galerkin scheme with upwind numerical fluxes and high-order polynomial bases. Auxiliary unknowns corresponding to characteristic variables are defined at the interface between the elements, and the physical fields are eliminated to obtain a reduced system. The reduced system can be written as a fixed-point problem that can be solved with stationary iterative schemes. Numerical results with 2D benchmarks are presented to study the performance of the approach. Compared to the standard HDG approach, the properties of the reduced system are improved with CHDG, which is more suited for iterative solution procedures. The condition number of the reduced system is smaller with CHDG than with the standard HDG method. Iterative solution procedures with CGNR or GMRES required smaller numbers of iterations with CHDG.

Hybridizable discontinuous Galerkin methods for second-order elliptic problems: overview, a new result and open problems

Hybridizable discontinuous Galerkin methods for second-order elliptic problems: overview, a new result and open problems

Hybridizable discontinuous Galerkin methods for the coupled Stokes–Biot problem

We present and analyze a hybridizable discontinuous Galerkin (HDG) finite element method for the coupled Stokes–Biot problem. Of particular interest is that the discrete velocities and displacement are H(div)-conforming and satisfy the compressibility equations pointwise on the elements. Furthermore, in the incompressible limit, the discretization is strongly conservative. We prove well-posedness of the discretization and, after combining the HDG method with backward Euler time stepping, present a priori error estimates that demonstrate that the method is free of volumetric locking. Numerical examples further demonstrate optimal rates of convergence in the L2-norm for all unknowns and that the discretization is locking-free.

A Stable and Convergent Hybridized Discontinuous Galerkin Method for Time-Fractional Telegraph Equations

In this paper, we extend the application of the hybridized discontinuous Galerkin (HDG) method to solve time-fractional telegraph equations. In fact, we use an HDG method for space discretization and L1 and L2 finite difference schemes using non-uniform meshes for time discretization. Thanks to a special kind of discrete Gronwall inequality, we prove that the HDG method is unconditionally stable and it is convergent with the optimal spatial order of convergence. Two numerical experiments are tested to confirm the theoretical results.

Hybridizable discontinuous Galerkin reduced order model for the variable coefficient advection equation

Hybridizable discontinuous Galerkin reduced order model for the variable coefficient advection equation

Deep reinforcement learning for adaptive mesh refinement

Finite element discretizations of problems in computational physics often rely on adaptive mesh refinement (AMR) to preferentially resolve regions containing important features during simulation. However, these spatial refinement strategies are often heuristic and rely on domain-specific knowledge or trial-and-error. We treat the process of adaptive mesh refinement as a local, sequential decision-making problem under incomplete information, formulating AMR as a partially observable Markov decision process. Using a deep reinforcement learning (RL) approach, we train policy networks for AMR strategy directly from numerical simulation. The training process does not require an exact solution or a high-fidelity ground truth to the partial differential equation (PDE) at hand, nor does it require a pre-computed training dataset. The local nature of our deep RL (DRL) allows the policy network to be trained inexpensively on much smaller problems than those on which they are deployed. The new DRL-AMR method is not specific to any particular PDE, problem dimension, or numerical discretization. The RL policy networks, trained on simple examples, can generalize to more complex problems, and can flexibly incorporate diverse problem physics. To that end, we apply the method to a range of PDEs, using a variety of high-order discontinuous Galerkin and hybridizable discontinuous Galerkin finite element discretizations. We show that the resultant DRL policies are competitive with common AMR heuristics and strike a favorable balance between accuracy and cost such that they often lead to a higher accuracy per problem degree of freedom, and are effective across a wide class of PDEs and problems.

Hybridizable discontinuous Galerkin method with mixed-order spaces for non-linear diffusion equations with internal jumps

We formulate a hybridizable discontinuous Galerkin method for parabolic equations with non-linear tensor-valued coefficients and jump conditions (Henry’s law). The analysis of the proposed scheme indicates the optimal convergence order for mildly non-linear problems. The same order is also obtained in our numerical studies for simplified settings. A series of numerical experiments investigate the effect of choosing different order approximation spaces for various unknowns.

Numerical modeling of anisotropic ferroelectric materials with hybridizable discontinuous Galerkin methods

We investigate a gradient flow structure of the Ginzburg–Landau–Devonshire (GLD) model for anisotropic ferroelectric materials by reconstructing its energy form. We show that the modified energy form admits at least one minimizer. Under some regularity assumptions for the electric charge distribution and the initial polarization field, we prove that the gradient flow structure has a unique solution. To simulate the GLD model numerically, we propose an energy‐stable semi‐implicit time‐stepping scheme and a hybridizable discontinuous Galerkin method for space discretization. Numerical tests are provided to verify the stability and convergence of the proposed numerical scheme as well as some properties of ferroelectric materials.

A matrix‐free macro‐element variant of the hybridized discontinuous Galerkin method

SummaryWe investigate a macro‐element variant of the hybridized discontinuous Galerkin (HDG) method, using patches of standard simplicial elements that can have non‐matching interfaces. Coupled via the HDG technique, our method enables local refinement by uniform simplicial subdivision of each macro‐element. By enforcing one spatial discretization for all macro‐elements, we arrive at local problems per macro‐element that are embarrassingly parallel, yet well balanced. Therefore, our macro‐element variant scales efficiently to n‐node clusters and can be tailored to available hardware by adjusting the local problem size to the capacity of a single node, while still using moderate polynomial orders such as quadratics or cubics. Increasing the local problem size means simultaneously decreasing, in relative terms, the global problem size, hence effectively limiting the proliferation of degrees of freedom. The global problem is solved via a matrix‐free iterative technique that also heavily relies on macro‐element local operations. We investigate and discuss the advantages and limitations of the macro‐element HDG method via an advection‐diffusion model problem.

Adaptive Model Reduction of High-Order Solutions of Compressible Flows via Optimal Transport

The solution of conservation laws with parametrised shock waves presents challenges for both high-order numerical methods and model reduction techniques. We introduce an r-adaptivity scheme based on optimal transport and apply it to develop reduced order models for compressible flows. The optimal transport theory allows us to compute high-order r-adaptive meshes from a starting reference mesh by solving the Monge–Ampère equation. A high-order discretization of the conservation laws enables high-order solutions to be computed on the resulting r-adaptive meshes. Furthermore, the Monge–Ampère solutions contain mappings that are used to reduce the spatial locality of the resulting solutions and make them more amenable to model reduction. We use a non-intrusive model reduction method to construct reduced order models of both the mesh and the solution. The procedure is demonstrated on three supersonic and hypersonic test cases, with the hybridisable discontinuous Galerkin method being used as the full order model.

A priori error analysis of new semidiscrete, Hamiltonian HDG methods for the time-dependent Maxwell’s equations

We present the first a priori error analysis of a class of space-discretizations by Hybridizable Discontinuous Galerkin (HDG) methods for the time-dependent Maxwell’s equations introduced in Sánchez et al. [Comput. Methods Appl. Mech. Eng. 396 (2022) 114969]. The distinctive feature of these discretizations is that they display a discrete version of the Hamiltonian structure of the original Maxwell’s equations. This is why they are called ``Hamiltonian’’ HDG methods. Because of this, when combined with symplectic time-marching methods, the resulting methods display an energy that does not drift in time. We provide a single analysis for several of these methods by exploiting the fact that they only differ by the choice of the approximation spaces and the stabilization functions. We also introduce a new way of discretizing the static Maxwell’s equations in order to define the initial condition in a manner consistent with our technique of analysis. Finally, we present numerical tests to validate our theoretical orders of convergence and to explore the convergence properties of the method in situations not covered by our analysis.