Discontinuous Galerkin Approach Research Articles

Computational Study of Shocked V-Shaped N2/SF6 Interface across Varying Mach Numbers

The Mach number effect on the Richtmyer–Meshkov instability (RMI) evolution of the shocked V-shaped N2/SF6 interface is numerically studied in this research. Four distinct Mach numbers are taken into consideration for this purpose: Ms=1.12,1.22,1.42, and 1.62. A two-dimensional space of compressible two-component Euler equations is simulated using a high-order modal discontinuous Galerkin approach to computational simulations. The numerical results show good consistency when compared to the available experimental data. The computational results show that the RMI evolution in the shocked V-shaped N2/SF6 interface is critically dependent on the Mach number. The flow field, interface deformation, intricate wave patterns, inward jet development, and vorticity generation are all strongly impacted by the shock Mach number. As the Mach number increases, the V-shaped interface deforms differently, and the distance between the Mach stem and the triple points varies depending on the Mach number. Compared to lower Mach numbers, higher ones produce larger rolled-up vortex chains. A thorough analysis of the Mach number effect identifies the factors that propel the creation of vorticity during the interaction phase. Moreover, kinetic energy and enstrophy both dramatically rise with increasing Mach number. Lastly, a detailed analysis is carried out to determine how the Mach number affects the temporal variations in the V-shaped interface’s features.

Numerical Study of Shock Wave Interaction with V-Shaped Heavy/Light Interface

This paper investigates numerically the shock wave interaction with a V-shaped heavy/light interface. For numerical simulations, we choose six distinct vertex angles (θ=40∘,60∘,90∘,120∘,150∘, and 170∘), five distinct shock wave strengths (Ms=1.12,1.22,1.30,1.60, and 2.0), and three different Atwood numbers (At=−0.32,−0.77, and −0.87). A two-dimensional space of compressible two-component Euler equations are solved using a third-order modal discontinuous Galerkin approach for the simulations. The present findings demonstrate that the vertex angle has a crucial influence on the shock wave interaction with the V-shaped heavy/light interface. The vertex angle significantly affects the flow field, interface deformation, wave patterns, spike generation, and vorticity production. As the vertex angle decreases, the vorticity production becomes more dominant. A thorough analysis of the vertex angle effect identifies the factors that propel the creation of vorticity during the interaction phase. Notably, smaller vertex angles lead to stronger vorticity generation due to a steeper density gradient, while larger angles result in weaker, more dispersed vorticity and a less complex interaction. Moreover, kinetic energy and enstrophy both dramatically rise with decreasing vortex angles. A detailed analysis is also carried out to analyze the vertex angle effects on the temporal variations of interface features. Finally, the impacts of different Mach and Atwood numbers on the V-shaped interface are briefly presented.

Time-domain simulation of sound propagation through anisotropic porous media for room acoustic applications

Wave-based room acoustic simulations require accurate modeling of sound propagation in the presence of extended (bulk) reacting porous absorbers. The sound absorption in porous media is typically addressed by means of equivalent fluid models that provide frequency-dependent effective material parameters, such as density and bulk modulus. In the time domain, these frequency dependent properties result in the computation of convolution integrals, which is effectively handled via an auxiliary differential equations (ADE) method. In this work, we apply the time-explicit discontinuous Galerkin approach to modelling of sound propagation in anisotropic porous media (with an anisotropic equivalent fluid model) together with the ADE method. We demonstrate the importance of using the extended approach for porous media, where the sound absorption depends not only on the frequency and incident angle, but also on the travelling wave direction.

H(div)-conforming and discontinuous Galerkin approach for Herschel–Bulkley flow with density-dependent viscosity and yield stress

This paper presents a comprehensive study on Herschel–Bulkley flow, where the flow parameters are dependent on the density. The Herschel–Bulkley model is a generalized power-law model used to simulate viscoplastic fluids defined by a plasticity threshold. We consider the case where the plasticity threshold and the viscosity depend on the shear rate and fluid density. To analyze this model, we use a Huber regularization of the stress and propose an H(div)-conforming and discontinuous Galerkin (DG) numerical approximation for the coupled equations governing the flow. We discuss the stability and existence of discrete solutions and propose a semismooth Newton linearization for the numerical solution of the discretized system. Our numerical scheme is validated through several experiments that explore the behavior of Herschel–Bulkley flow under different conditions. The results demonstrate the robustness of our numerical method.

Efficiency analysis of discontinuous Galerkin approaches for the application onto quantum Liouville-type equations

The simulation of nanodevices is computationally inefficient with current algorithms. The discontinuous Galerkin approach has been demonstrated in the field of computational fluid dynamics to deliver high order accuracy and efficiency due to its reliance on matrix–vector multiplications. Previously, the discontinuous Galerkin approach was successfully used in conjunction with the finite volume technique to solve the Liouville–von Neumann equation in center-mass coordinates and thus simulate nanodevices. To exploit its full potential regarding high-performance computing, this work aims to substitute the aforementioned finite volume technique with the discontinuous Galerkin method. To arrive at the said formalism, a finite element method is implemented as an intermediate step.

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.

Amplitude-preserving P-/S-wavefield separation with the discontinuous Galerkin method on unstructured meshes

The decoupling of P and S waves is an essential prerequisite for elastic reverse time migration, which effectively reduces crosstalk artifacts, but most wavefield separation algorithms are implemented on uniform rectangular grids. We develop an amplitude- and phase-preserving P- and S-wavefield separation approach on unstructured meshes, which can effectively decompose the original elastic wavefield into P and S wavefields. The isotropic case is considered. With the aid of viscoelastic theory, we choose to attenuate P or S waves and preserve the other wave mode, so as to achieve wavefield decomposition. Viscoelastic wave equations are first reformulated as decoupling wave equations with a selective strong attenuation. We then use the discontinuous Galerkin (DG) method to simulate decoupling P or S wavefield propagation on triangular and tetrahedral meshes. We adopt a quadrature-free DG approach and the arbitrary mesh is mapped into the reference mesh for numerical calculation, where no additional volume and surface integrations are involved. The amplitude and phase information of this vector decomposition agrees with that of the original elastic data. Four numerical examples are used to demonstrate the superior performance of this vector decomposition algorithm. The isotropic example indicates the applicability and correctness of our scheme and the second example displays superiority in handling strong velocity contrasts. The third example exhibits mesh flexibility in dealing with complex structures, such as caves, faults, and undulating surfaces. The last example indicates the effectiveness of our developed algorithm extended to a 3D case.

A discontinuous Galerkin approach for atmospheric flows with implicit condensation

We present a discontinuous Galerkin method for moist atmospheric dynamics, with and without warm rain. By considering a combined density for water vapour and cloud water, we avoid the need to model and compute a source term for condensation. We recover the vapour and cloud densities by solving a pointwise non-linear problem each time step. Consequently, we enforce the requirement for the water vapour not to be supersaturated implicitly. Together with an explicit time-stepping scheme, the method is highly parallelisable and can utilise high-performance computing hardware. Furthermore, the discretisation works on structured and unstructured meshes in two and three spatial dimensions. We illustrate the performance of our approach using several test cases in two and three spatial dimensions. In the case of a smooth, exact solution, we illustrate the optimal higher-order convergence rates of the method.

Computational aeroelastic analysis of wings based on the structural discontinuous Galerkin and aerodynamic vortex lattice methods

An original computational framework for the aeroelastic analysis of wings featuring general transverse section is developed. The framework is based on the coupling between a novel discontinuous Galerkin structural model and an aerodynamic vortex lattice method, which is implemented in both the planar and non-planar version. The structural model, which constitutes the novelty of the present work, allows generalized kinematics and is thus able to capture higher-order structural deformation modes. With respect to other more used structural representations, the discontinuous Galerkin approach is based on the use of discontinuous basis functions and suitably-defined boundary terms to enforce the inter-element continuity and boundary conditions. Such features naturally enable high-order accuracy, ease of parallelization and, specifically for this work, straightforward coupling with the vortex lattice method. The framework is validated through benchmark tests, providing favourable matching with reference literature data.

A high‐order discontinuous Galerkin approach for physics‐based thermospheric modeling

AbstractThe accurate prediction of aerodynamic drag on satellites orbiting in the upper atmosphere is critical to the operational success of modern space technologies, such as satellite‐based communication or navigation systems, which have become increasingly popular in the last few years due to the deployment of constellations of satellites in low‐Earth orbit. As a result, physics‐based models of the ionosphere and thermosphere have emerged as a necessary tool for the prediction of atmospheric outputs under highly variable space weather conditions. This paper proposes a high‐fidelity approach for physics‐based space weather modeling based on the solution of the Navier–Stokes equations using a high‐order discontinuous Galerkin method, combined with a matrix‐free strategy suitable for high‐performance computing on GPU architectures. The approach consists of a thermospheric model that describes a chemically frozen neutral atmosphere in nonhydrostatic equilibrium driven by the external excitation of the Sun. A novel set of variables is considered to treat the low densities present in the upper atmosphere and to accommodate the wide range of scales present in the problem. At the same time, and unlike most existing approaches, radial and angular directions are treated in a nonsegregated approach. The study presents a set of numerical examples that demonstrate the accuracy of the approximation and validate the current approach against observational data along a satellite orbit, including estimates of established empirical and physics‐based models of the ionosphere‐thermosphere system. Finally, a one‐dimensional radial derivation of the physics‐based model is presented and utilized for conducting a parametric study of the main thermal quantities under various solar conditions.

On the Spatiotemporal Pattern Formation in Nonlinear Coupled Reaction–Diffusion Systems

Nonlinear coupled reaction–diffusion (NCRD) systems have played a crucial role in the emergence of spatiotemporal patterns across various scientific and engineering domains. The NCRD systems considered in this study encompass various models, such as linear, Gray–Scott, Brusselator, isothermal chemical, and Schnakenberg, with the aim of capturing the spatiotemporal patterns they generate. These models cover a diverse range of intricate spatiotemporal patterns found in nature, including spots, spot replication, stripes, hexagons, and more. A mixed-type modal discontinuous Galerkin approach is employed for solving one- and two-dimensional NCRD systems. This approach introduces a mathematical formulation to handle the occurrence of second-order derivatives in diffusion terms. For spatial discretization, hierarchical modal basis functions premised on orthogonal scaled Legendre polynomials are used. Moreover, a novel reaction term treatment is proposed for the NCRD systems, demonstrating an intrinsic feature of the new DG scheme and preventing erroneous solutions due to extremely nonlinear reaction terms. The proposed approach reduces the NCRD systems into a framework of ordinary differential equations in time, which are addressed by an explicit third-order TVD Runge–Kutta algorithm. The spatiotemporal patterns generated with the present approach are comparable to those found in the literature. This approach can readily be expanded to handle large multi-dimensional problems that appear as model equations in developed biological and chemical applications.

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.

Acoustics in Flameholding Solid-Fuel-Ramjet Fuel Grains

Experiments are conducted to determine how acoustic perturbations affect the performance and flameholding of solid-fuel ramjets with nonstandard combustion chambers. The focus is on the effect of wall cavities carved in the fuel grain using additive manufacturing. An improved understanding of how the wall geometry contributes to the establishment of acoustic modes is sought. A novel combustion mechanism was developed using a counterflow burner to study the combustion and regression of solid model fuel polymethyl methacrylate. The diffusion flame between the fuel and oxidizer was studied numerically using a solid-fuel decomposition and melt layer model to simulate convection and pyrolysis of the material. This model was validated using new experimental data as well as previously published works. The foam layer parameters are critical to the success of the validation, showing that the increased residence time of the gas in the bubbles facilitates the fuel breakdown. Fourth-order computational simulations of ramjet combustion without regressing fuel walls using a novel discontinuous Galerkin approach are performed with a fully conjugate solution for the thermal wave in the solid. Turbulent transport strongly affects the heat feedback to the walls, and low-frequency vortical modes (e.g., with a vortical wavemaker) associated with a recirculation region at the injector upstream wall are linked to an increase in chamber pressure and fuel mass flux.

A IETI-DP method for discontinuous Galerkin discretizations in isogeometric analysis with inexact local solvers

We construct solvers for an isogeometric multi-patch discretization, where the patches are coupled via a discontinuous Galerkin approach, which allows for the consideration of discretizations that do not match on the interfaces. We solve the resulting linear system using a Dual-Primal IsogEometric Tearing and Interconnecting (IETI-DP) method. We are interested in solving the arising patch-local problems using iterative solvers since this allows for the reduction of the memory footprint. We solve the patch-local problems approximately using the Fast Diagonalization method, which is known to be robust in the grid size and the spline degree. To obtain the tensor structure needed for the application of the Fast Diagonalization method, we introduce an orthogonal splitting of the local function spaces. We present a convergence theory for two-dimensional problems that confirms that the condition number of the preconditioned system only grows poly-logarithmically with the grid size. The numerical experiments confirm this finding. Moreover, they show that the convergence of the overall solver only mildly depends on the spline degree. We observe a mild reduction of the computational times and a significant reduction of the memory requirements in comparison to standard IETI-DP solvers using sparse direct solvers for the local subproblems. Furthermore, the experiments indicate good scaling behavior on distributed memory machines. Additionally, we present an extension of the solver to three-dimensional problems and provide numerical experiments assessing good performance also in that setting.

Investigation of aspect ratio effects on flow characteristics and vorticity generation in shock-induced rectangular bubble

In shock-induced Richtmyer–Meshkov instability, the polygonal bubbles, including the rectangular interface can offer favorable circumstances for shock refraction research. Also, the aspect ratio is crucial in characterizing the physics behind this instability. Therefore, the aspect ratio effects on the flow characteristics and vorticity generation in the shock-induced rectangular light gas bubbles are investigated numerically in this study. This investigation is performed for a wide range of aspect ratios varying from 0.25 to 3.0. These effects are highlighted in the interface morphology, wave paradigms, vorticity generation, enstrophy evolution, and a qualitative examination of interface deformation. For simulating two-component gas flows, a high-order explicit modal discontinuous Galerkin approach is employed to solve a two-dimensional system of compressible inviscid Euler equations. Existing experimental results for shock-induced square and cylindrical light bubbles are used to validate the numerical model and solver. The results indicate that the effects of aspect ratio are critical in characterizing the interface morphology during the interaction between the shock wave and rectangular bubble. The interface morphology evolves faster as the aspect ratio increases, and the process of transverse jet formation is observed to be significantly different. The aspect ratio effects result in a substantial change in the flow field with bubble deformation, vortex development, vorticity generation, and transverse jet formation. The effects of aspect ratio are further investigated in detail through the mechanism of vorticity generation and enstrophy evolution. Finally, these effects on shock trajectories and interface characteristics over time are thoroughly explored.

Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation

<p style='text-indent:20px;'>We consider one-dimensional distributed optimal control problems with the state equation being given by the viscous Burgers equation. We discretize using a space-time discontinuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.</p>

Hybrid Discontinuous Galerkin Approach for the Solution of Quantum Liouville-Type Equations

Hybrid Discontinuous Galerkin Approach for the Solution of Quantum Liouville-Type Equations

Hybridizable discontinuous Galerkin methods for space-time fractional advection-dispersion equations

This work focuses on the numerical solution of the initial and boundary value problems for space-time fractional advection-diffusion equations. The well-posedness of the weak solutions is shown by Lax-Milgram lemma. Two fully discrete methods are established. The main idea is based on a hybridizable discontinuous Galerkin approach in spatial direction and two finite difference schemes in temporal direction: L1 formula, the weighted and shifted Grünwald-Letnikov formula. The stability and convergence analyses of the proposed methods are derived in detail. Several numerical experiments are provided to illustrate the theoretical results.

Finite element method for the quasiclassical theory of superconductivity

The Eilenberger-Larkin-Ovchinnikov-Eliashberg quasiclassical theory of superconductivity is a powerful method enabling studies of a wide range of equilibrium and non-equilibrium phenomena in conventional and unconventional superconductors. We introduce here a finite element method, based on a discontinuous Galerkin approach, to self-consistently solve the underlying transport equations for general device geometries, arbitrary mean free path and symmetry of the superconducting order parameter. We present exemplary results on i) the influence of scalar impurity scattering on phase crystals in $d$-wave superconducting grains at low temperatures and ii) the current flow and focusing in $d$-wave superconducting weak links, modeling recent experimental realizations of grooved high-temperature superconducting Dayem bridges. The high adaptability of this finite element method for quasiclassical theory paves the way for future investigations of superconducting devices and new physical phenomena in unconventional superconductors.

Elasto-acoustic modeling and simulation for the seismic response of structures: The case of the Tahtalı dam in the 2020 [formula omitted]zmir earthquake

As a mean to assess the risk dam structures are exposed to during earthquakes, we employ an abstract mathematical, three dimensional, elasto-acoustic coupled wave-propagation model taking into account (i) the dam structure itself, embedded into (ii) its surrounding topography, (iii) different material soil layers, (iv) the seismic source as well as (v) the reservoir lake filled with water treated as an acoustic medium. As a case study for extensive numerical simulations we consider the magnitude 7 seismic event of the 30th of October 2020 taking place in the Icarian Sea (Greece) and the Tahtalı dam around 30km from there (Turkey). A challenging task is to resolve the multiple length scales that are present due to the huge differences in size between the dam building structure and the area of interest, considered for the propagation of the earthquake. Interfaces between structures and highly non-conforming meshes on different scales are resolved by means of a discontinuous Galerkin approach. The seismic source is modeled using inversion data about the real fault plane. Ultimately, we perform a real data driven, multi-scale, full source-to-site, physics based simulation based on the discontinuous Galerkin spectral element method, which allows to precisely validate the ground motion experienced along the Tahtalı dam, comparing the synthetic seismograms against actually observed ones. A comparison with a more classical computational method, using a plane wave with data from a deconvolved seismogram reading as an input, is discussed.