Discontinuous Method Research Articles

A body-of-revolution discontinuous Galerkin time domain method based on curved mesh for Maxwell equations

ABSTRACT Numerical simulation of body-of-revolution (BoR) or axisymmetric structures for scattering and radiation problems has a broad spectrum of technology applications in microwave and optical engineering. In this work, we propose a high-order BoR discontinuous Galerkin time-domain method (BoR-DGTD) for solving Maxwell equations in the cylindrical coordinate system. The presented approach is based on or (m is the mode number and φ is the azimuth) expansions for different field components and thus yields a real-valued final system of equations. It saves half of the unknowns as compared to the existing method in which all the field components are assumed to have the same φ variation, i.e. . In contrast to the common practice of deriving variational formulation for the BoR Maxwell equations on two-dimensional grids in the meridian plane, we build it on three-dimensional coaxial toroidal subdomains. As a result, the singularity difficulty induced by the cylindrical coordinates is eliminated naturally. In addition, for BoR with curved material interfaces in the angular cross-section, a solution using curved grids in conjunction with the low-storage weight-adjusted approximation scheme is provided, which significantly enhances the simulation quality. Particularly, to achieve a set of optimized interpolation shape functions for the isoparametric transformation, an essential assistant in calculating the coefficient matrices, a heuristic procedure is developed to properly position the geometric control points on the curved grids. Some numerical experiments are conducted to demonstrate the accuracy and performance of our proposed BoR-DGTD method.

Integrating the effect of abutments in estimating the average vertical stress of elastic hard rock pillars by combining numerical modelling and artificial neural networks

Estimating average vertical pillar stresses is a critical step in designing room-and-pillar mines. Several analytical methods can be used to estimate the vertical stresses acting on the pillars. However, the present analytical methods fail to adequately account for the influence of abutments on the distribution of vertical stresses, especially when applied to narrow panel widths and pillar layouts comprising evenly spaced barriers. In this study, a multi-layer perceptron neural network (MLPNN) was applied to predict the vertical loads of regular pillars more accurately. Hundreds of room-and-pillar mine layouts were modeled using a displacement discontinuity method (DDM), and a database of 2355 sampled pillar cases was compiled. The MLPNN was trained based on this database, and its prediction capabilities were further validated using simulations by a finite difference code (i.e., FLAC3D). The model predictions and the FLAC3D simulations reasonably agreed with a regression coefficient of 0.99. The model was also adapted for mine cases with evenly spaced barrier pillars, and its application to a real case study mine has shown to provide accurate pillar stress estimations; hence, this model is suitable for practical use at mines. Even though the MLPNN model cannot be applied universally to all mine situations, it seems as a significant improvement over existing analytical techniques in terms of accounting for the influence of abutments on pillar stresses.

A Discontinuous Petrov-Galerkin Method for One-Dimensional Hyperbolic Conservation Law Equations Based on HWENO Limiter

The discontinuous Petrov-Galerkin method (DPG) is a new type of numerical solution to the hyperbolic conservation law equations, which distinguishes itself from the DG method by its high accuracy based on compact stencils. However, in order to overcome the unphysical oscillations of the higher order linear schemes near the large gradient solution, the DPG method often needs to incorporate limiter functions to obtain a high-resolution image of the numerical solution. This paper attempts to combine HWENO as limiter function with DPG to solve the discontinuous initial value problems for the hyperbolic conservation law equations. The single-step high-accuracy SSP Runge-Kutta method is used for time discretization, and a new HWENO-based process is used as the limiter of the RKDPG methods, which requires only one reconstruction to complete the update of the higher-order moments without calculating the linear weight coefficients. . Since the accuracy does not meet the design requirements, the HWENO limiter above is partially improved for the identification of the problem cells in the HWENO limiter above, with the original numerical solution used at the smooth solution. This paper only gives the calculation results of P1 ~P3, and the limiter is also applicable to the DPG method for higher elements. Numerical examples show that the HWENO limiter can effectively suppress non-physical oscillations in the problem cells and keep the original accuracy at the non-problem cells. The high accuracy and compactness characteristics of the DPG method are maintained. The numerical calculation solutions are efficient and accurate.

Time parallelism and Newton-adaptivity of the two-derivative deferred correction discontinuous Galerkin method

In this work, we consider a high-order discretization of compressible viscous flows allowing parallelization both in space and time.The discontinuous Galerkin spectral element method, which is well-suited for massively parallel simulations, is used for spatial discretization. The main novelty in this work is the additional demonstration of time-parallel capabilities within an implicit two-derivative timestepping procedure to further increase the parallel speedup. Temporal parallelism is made possible by a predictor-corrector-type time discretization that allows to split the associated workload onto multiple processors.We identify a homogeneous load balance with respect to the linear (GMRES) iterations on each processor as a key for parallel efficiency. To homogenize the load and to enable practical simulations, an adaptive strategy for Newton’s method is introduced. It is shown that the time-parallel method provides a parallel efficiency of approx. 60−70% on 4−7 computational partitions. Moreover, the capabilities of the novel method for the simulation of large-scale problems are illustrated with a mixed temporal and spatial parallelization on more than 1000 processors.

High-order methods for hypersonic flows with strong shocks and real chemistry

We compare high-order methods including spectral difference (SD), flux reconstruction (FR), the entropy-stable discontinuous Galerkin spectral element method (ES-DGSEM), modal discontinuous Galerkin methods, and WENO to select the best candidate to simulate strong shock waves characteristic of hypersonic flows. We consider several benchmarks, including the Leblanc and modified shock-density wave interaction problems that require robust stabilization and positivity-preserving properties for a successful flow realization. We also perform simulations of the three-species Sod problem with simplified chemistry with the chemical reaction source terms introduced in the Euler equations. The ES-DGSEM scheme exhibits the highest stability, negligible numerical oscillations, and requires the least computational effort in resolving reactive flow regimes with strong shock waves. Therefore, we extend the ES-DGSEM to hypersonic Euler equations by deriving a new set of two-point entropy conservative fluxes for a five-species gas model. In this paper, hypersonic Euler equations refer to the multi-species Euler equations for which the internal energy and thermodynamic properties are computed using the Rigid-Rotor Harmonic-Oscillator model. Stabilization for capturing strong shock waves occurs by blending high-order entropy conservative fluxes with low-order finite volume fluxes constructed using the HLLC Riemann solver. The hypersonic Euler solver is verified using the non-equilibrium chemistry Sod problem. To this end, we adopt the Mutation++ library to compute the reaction source terms, thermodynamic properties, and transport coefficients. We also investigate the effect of real chemistry versus ideal chemistry, and the results demonstrate that the ideal chemistry assumption fails at high temperatures, hence real chemistry must be employed for accurate predictions. Finally, we consider a viscous hypersonic flow problem to verify the transport coefficients and reaction source terms determined by the Mutation++ library.

Dynamic stability assessment of consequent rock slopes considering multiple transmissions and reflections of stress waves

In this work, Based on the conservation of momentum at the wave fronts and the displacement discontinuity method, a theoretical approach for calculating the dynamic safety factor of consequent rock slopes is proposed. The multiple transmissions and reflections of stress waves on the joints and free surfaces are taken into account. A comparison of UDEC simulation and theoretical calculation was carried out to verify the feasibility of the proposed method. Moreover, parametric studies were conducted to investigate the effects of free surfaces, joint stiffness, amplitude of incident waves, and frequency of incident waves. The results show that the proposed method can predict the time history curve of the safety factor of a slope readily and accurately. Waves reflected from both the slope and top surfaces are found to have a significant effect on the stability of the slope – considering them or not can cause a difference in the safety factors up to 32.5%. The greater the joint stiffness is, the greater the safety factor fluctuates with time. However, when the joint stiffness is sufficiently high, the effect of joint stiffness on the safety factor can be neglected. The approach proposed in this work provides a theoretical tool for the dynamic assessment and design of consequent rock slopes. Highlights An analytical method is proposed for calculating the dynamic safety factor of rock slopes. Multiple transmissions and reflections of stress waves are considered. The proposed method has high calculation accuracy and can be applied in practice.

Advanced Numerical Methods for Graphene Simulation with Equivalent Boundary Conditions: A Review

Since the discovery of graphene, due to its excellent optical, thermal, mechanical and electrical properties, it has a broad application prospect in energy, materials, biomedicine, electromagnetism and other fields. A great quantity of researches on the physical mechanism of graphene has been applied to engineering in electromagnetism and optics. To study the properties of graphene, different kinds of numerical methods such as the mixed finite element method (Mixed FEM), the mixed spectral element method (Mixed SEM), Method of Auxiliary Sources (MAS), discontinuous Galerkin time-domain method (DGTD) and interior penalty discontinuous Galerkin time domain (IPDG) have been developed for simulating the electromagnetic field effects of graphene and equivalent boundary conditions such as impedance transmission boundary condition (ITBC), surface current boundary condition (SCBC), impedance matrix boundary condition (IMBC) and surface impedance boundary condition (SIBC) have been employed to replace graphene in the computational domain. In this work, the numerical methods with equivalent boundary conditions are reviewed, and some examples are provided to illustrate their applicability.

Numerical modeling of low-frequency distributed acoustic sensing signals for mixed-mode reactivation

In recent years, the development of distributed acoustic sensing (DAS) technology has enabled direct monitoring of subsurface strain during hydraulic fracturing (HF) operations. Most low-frequency (<1 Hz) DAS (LFDAS) signals exhibit strain or strain-rate patterns that are characteristic of propagating tensile hydraulic fractures. However, mixed-mode fault reactivation, consisting of shear slip (mode II) with tensile opening (mode I), can occur in cases where propagating hydraulic fractures intersect preexisting natural fractures or faults. In this study, we show an anomalous LFDAS signal that was observed during an HF operation, with characteristics that differ from typical signals from tensile hydraulic fractures. Anomalous characteristics include the onset of an asymmetric compression-extension doublet after pumping was terminated. The location of the signals, coupled with the image log and seismic data, suggests that mixed-mode reactivation occurred on a preexisting fault. We use a simplified numerical model based on the displacement discontinuity method (DDM) to simulate the anomalous LFDAS signal. Results find that the first-order characteristics of the anomalous signal can be approximated as an initial tensile fault opening (mode I) followed by a shear slip (mode II) on a fault. Therefore, we demostrate the approach of using DDM to investigate mixed-mode fault reactivation during HF operations.

Water Erosion and Extension of Ground Fissures in Weihe Basin Based on DEM-CFD Coupled Modeling

The Weihe Basin is one of the regions with the highest concentrations of ground fissure development and the most serious disasters in China. Hidden ground fissures are formed in the shallow soil layer due to preexisting fissures generated by tectonism, hidden ground fissures suddenly exposed to the surface after heavy rainfall. Because the details of the water erosion process cannot be replicated by geological survey methods, the erosion and extension mechanism has not fully developed and the discontinuous medium numerical simulation method is well suited for simulating large deformations and facilitates a microscopic perspective in elucidating the underlying causal mechanisms. This paper deploys the discrete element method (DEM)–computational fluid dynamics (CFD) fluid–solid coupled method modeling the growth process of hidden fissures containing different soil types (sand and clay) under heavy rainfall, revealing a mechanism for the development of hidden fissures into surface fissures. The findings include: (1) the emergence process of hidden fissures into surface fissures under heavy rainfall can be summarized into four stages: subsurface erosion into hidden holes, traction into arches, collapse into sinkholes, and horizontal extension; (2) the emergence process of clay is slower than that of sand due to the constraint of cohesion; (3) the shape of the bottom seepage point affects the fissure emergence process, which is an important factor in the macroscopic performance of the exposed surface fissures. The intuitive and reproducible DEM-CFD coupled modeling used in this paper possesses important reference value for the study and prevention of water erosion ground fissures.

A comparison of three error indicators for adaptive high-order large eddy simulation

We use a discontinuous high-order method to evaluate three error indicators, specifically designed for mesh adaptation in large eddy simulation (LES). The indicators are based on specific physical and numerical arguments. The first indicator is based on the unsteady residual. The second indicator is based on a local smoothness indicator. The third indicator is based on an estimate for small scale turbulent kinetic energy. We test the three error indicators at driving autonomous mesh refinement on implicit LES, evaluating the error indicators on two widely used benchmark simulations of the T106 LP turbine cascade. We compare the adaptively meshed simulations against direct numerical simulation (DNS) results to judge their effectiveness at reducing solution error in LES.

Study on the Spatiotemporal Evolution Characteristics and Influencing Factors on Green Building Development of City Clusters in the Yangtze River Delta Region in China

This paper explores the spatiotemporal evolution characteristics and spatial correlation patterns of green building development differences in 41 cities in the Yangtze River Delta region from 2012 to 2020 by means of the gravity center analysis model and spatial autocorrelation analysis. In addition, it further clarifies the impact factors of the spatial differentiation pattern of green building development in combination with GeoDetector based on four dimensional factors of population and economy, market environment, policy, and other factors. The results showed that: (1) According to the analysis of the number of green buildings in each city from 2012 to 2020 and the natural discontinuity method, the development pattern of green buildings in the Yangtze River Delta region city clusters shows an imbalance, being highly concentrated in the eastern coastal areas with Suzhou (1) and Shanghai as the core. The overall trajectory of the center of gravity shows the development from southeast to northwest. (2) The global Moran’s I of green buildings in the Yangtze River Delta region city clusters is greater than 0, and all passed the significance test (Z > 1.96, p < 0.01), indicating that the green buildings in the Yangtze River Delta region city clusters show typical spatial aggregation characteristics. By using the local LISA index, it is found that in the H-H spatial autocorrelation distribution pattern with Suzhou (1) and Shanghai as the core, the core city has a strong attraction ability and relatively low radiation ability. (3) Based on the explanatory power mean, the main driving factors of the spatial differentiation pattern of green building development in the Yangtze River Delta region city clusters are education level (0.6656), technical level (0.6269), and the gross domestic product (0.6091). The factor interaction shows a two-factor enhancement and nonlinear enhancement effect, and there is neither a weakening nor an independent relationship.

Incorporation of anisotropic scattering into Fourier moment expanded axial DGFEM SN based 2D/1D coupling method

Fourier moment expanded axial DGFEM (Discontinuous Galerkin Finite Element Method) SN based 2D/1D coupling method is a promising variant for the pin-by-pin neutron transport calculation, due to its excellent computational efficiency and stability. In this study, the incorporation of anisotropic scattering based on spherical harmonic expansion is developed for the axial DGFEM SN based 2D/1D transport approach, to further capture the anisotropic scattering accurately for the heterogeneous fuel rod or strong absorber. Several well-known benchmarks are used to evaluate its computational performance. Numerical results shows that the implementation of anisotropic scattering can significantly improve the accuracy of 2D/1D coupling method. Moreover, compared with the traditional 1D SN based 2D/1D method, the newly developed method could achieve high computational efficiency and convergence rate, as well as a relatively low storage requirement.

Local discontinuous Galerkin method for the Riesz space distributed-order Sobolev equation

We present and analyze a local discontinuous Galerkin finite element method to solve the Riesz space distributed-order Sobolev equation. To approximate the distributed-order Riesz space derivative, the Gauss quadrature as the high computational accuracy method is proposed. A multi-term fractional equation is then constructed from the considered equation by approximating the Riesz space derivative. Moreover, stability analysis is provided for a semi-discrete scheme. We provide numerical results to justify the theoretical analysis by solving the ordinary differential equation. We obtain this after implementing the local discontinuous Galerkin scheme on the Riesz space distributed-order Sobolev equation with the Crank–Nicolson scheme as a time marching method and the Laplace transform technique.

Entropy Stable DGSEM Schemes of Gauss Points Based on Subcell Limiting.

The discontinuous Galerkin spectral element method (DGSEM) is a compact and high-order method applicable to complex meshes. However, the aliasing errors in simulating under-resolved vortex flows and non-physical oscillations in simulating shock waves may lead to instability of the DGSEM. In this paper, an entropy-stable DGSEM (ESDGSEM) based on subcell limiting is proposed to improve the non-linear stability of the method. First, we discuss the stability and resolution of the entropy-stable DGSEM based on different solution points. Second, a provably entropy-stable DGSEM based on subcell limiting is established on Legendre-Gauss (LG) solution points. Numerical experiments demonstrate that the ESDGSEM-LG scheme is superior in non-linear stability and resolution, and ESDGSEM-LG with subcell limiting is robust in shock-capturing.

Effect of aggregate sphericity on the mechanical properties of ERCC based on highly detailed simulation

Electrically conductive roller-compacted concrete (ERCC) is a novel functional construction material with self-heating ability, which can be applied to solve problems of low efficiency of crushed snow removal and deicing on expressway and airport runway, and difficult temperature control of dam structures. Natural aggregates with high sphericity and crushed aggregates with low sphericity are the sources of ERCC construction. However, the impact of aggregate sphericity on ERCC is currently unclear. This study performed the macro and micro experiments of the core samples from a large-scale ERCC specimen with rolling construction technology. Based on the experimental results, the mechanical properties of ERCC with different aggregate sphericities are analyzed with the highly detailed simulation method by coupling continuous medium method and discontinuous medium method. The results show that increased sphericity of the aggregate can lead to a decrease in the number of aggregates generated, and the simplified spherical model adopted in the traditional mesoscale simulation method will underestimate the strength of concrete. Furthermore, When the sphericity of aggregate is 0.645, the compressive strength and splitting tensile strength of ERCC are increased by 25.34% and 35.04% compared with that of spherical aggregate respectively. In engineering, the crushed aggregate is suggested for rolling construction of ERCC. The results are in accordance with the statistical laws of the field statistical data in engineering practice.

Impedance Sensitivity Analysis Based on Discontinuous Isogeometric Boundary Element Method in Automotive Acoustics

Abstract Acoustic sensitivity analysis is an essential technique to determine the direction of structural-acoustic optimization by evaluating the gradient of the objective functions with respect to the design variables. However, acoustic sensitivity analysis with respect to acoustic impedance, which is an important parameter representing the interior absorbent material in automotive acoustics, is lacking in the study. Moreover, acoustic sensitivity analysis implemented with conventional numerical methods is time and effort-consuming in automotive acoustics, due to the large-scale mesh generation. In this work, the impedance sensitivity analysis for automotive acoustics based on the discontinuous isogeometric boundary element method is presented. The regularized boundary integral equation with impedance boundary conditions is established, then the sensitivity is derived by differentiating the boundary integral equation. The efficiency of the proposed method is improved by employing the parallel technique and generalized minimal residual solver. A long duct example with an analytical solution validates the accuracy of the proposed method, and an automotive passenger compartment subjecting to impedance boundary conditions illustrates that the computing time of the proposed method is one order of magnitude less than the conventional method. This work presents an easily implementable and efficient tool to investigate acoustic sensitivity with respect to impedance, showing great potential in the application of automotive acoustics.

Optimal error estimate of discontinuous Galerkin methods for advection-diffusion-reaction problems with low regularity

In this paper, we present a class of discontinuous Galerkin finite element methods for advection-diffusion-reaction problems and establish a priori error estimates when the solution is only in H1+s(Ω) with s∈(0,1/2].

Finite-Time Discontinuous Control of Nonholonomic Chained-Form Systems

A finite-time discontinuous control method is proposed for controlling nonholonomic chained-form systems. Through a general discontinuous coordinate transformation, the nonholonomic system is transformed into two decoupled subsystems that are independently stabilized with control laws. Sufficient conditions for the existence of decoupled subsystems and finite-time convergence of system states are given. Furthermore, the finite-time discontinuous control laws are proposed to ensure that the system states reach the equilibrium in a finite time. Simulation results are presented to demonstrate the effectiveness and advantages of the proposed approach.

Numerical simulation of proppant migration in horizontal wells with multi-fracture fracturing

For horizontal wells fracturing in unconventional reservoirs, the fracture morphology and proppant migration are severely affected by the interfering effects among multiple fractures propagation. Current numerical simulation for proppant migration did not reveal the interrelation among flow distribution, fracture length and sandbank shape during multiple fractures propagation. In this study, a proppant migration model is proposed by combining the displacement discontinuity method (DDM) and the Eulerian-Lagrangian method (E-L M), which implement dynamic multiple fractures propagation simulation as well as the changing state of flow distribution for each fracture. The accuracy of the numerical model is verified by laboratory experiments. The sensitivity parameters of proppant settlement are analyzed, and two dimensionless parameters, height ratio γ and area ratio η are proposed for proppant migration evaluation. The results show that with a lower fracturing fluid flow rate, fluid viscosity, the height ratio γ is lower, and the area ratio η is higher, which also results in the proppant being more likely to accumulate at the fracture entrance. As the proppant particle size increases from 100 μm to 500 μm, the dimensionless height ratio γ and the dimensionless area ratio η decrease significantly. Increase in fracture spacing during horizontal well multi-cluster fracturing results in a limited improvement for proppant settlement. Formation elastic modulus has an appreciable impact on proppant migration in multi-fracture propagation, high elastic modulus formation leads to an ununiform proppant placement over long distances.

Displacement Discontinuity Method Taking into Account the Curvature of the Crack

Displacement Discontinuity Method Taking into Account the Curvature of the Crack