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Projection-based reduced order models (ROM) based on the weak form and the strong form of the discontinuous Galerkin (DG) method are proposed and compared for shock-dominated problems. The incorporation of dissipation components of DG in a consistent manner, including the upwinding flux and the localized artificial viscosity model, is employed to enhance stability of the ROM. To ensure efficiency, the discrete empirical interpolation method (DEIM) is adopted to enable hyper-reduction, for which the upwinding flux is decomposed into the central part and the dissipation part. The maximum local wave speed in the upwinding dissipation part is compressed and approximated using the DEIM approach, and the same strategy is applied to the artificial viscosity. Energy stability is proved with the strong-form-based ROM prior to hyper-reduction for the one-dimensional scalar cases. Eigenvalue spectrum is analyzed to verify and compare the stability properties of the two proposed ROMs. Several benchmark cases are conducted to test the performance of the proposed models. Results show that stable computations with reasonable acceleration for shock-dominated cases can be achieved with the ROM built on the strong form. (C) 2022 Elsevier Inc. All rights reserved.

In this paper, we present an accurate, stable and robust shock-capturing finite difference method for solving scalar non-linear conservation laws. The spatial discretization uses high-order accurate upwind summation-by-parts finite difference operators combined with weakly imposed boundary conditions via simultaneous-approximation-terms. The method is an extension of the residual-based artificial viscosity methods developed in the finite- and spectral element communities to the finite difference setting. The three main ingredients of the proposed method are: (i) shock detection provided by a residual-based error estimator; ( i i ) first-order viscosity applied in regions with strong discontinuities; ( i i i ) additional dampening of spurious oscillations provided by high-order dissipation from the upwind finite difference operators. The method is shown to be stable for skew-symmetric discretizations of the advective flux. Accuracy and robustness are shown by solving several benchmark problems in 2D for convex and non-convex fluxes.

We present a memory-efficient high-order hybridizable discontinuous Galerkin (HDG) formulation coupled with high-order fully implicit Runge-Kutta schemes for immiscible and incompressible two-phase flow through porous media. To obtain the same high-order accuracy in space and time, we propose using high-order temporal schemes that allow using large time steps. Therefore, we require unconditionally stable temporal schemes for any combination of element size, polynomial degree and time step. Specifically, we use the Radau IIA and Gauss-Legendre schemes, which are unconditionally stable, achieve high-order accuracy with few stages, and do not suffer order reduction in this problem. To reduce the memory footprint of coupling these spatial and temporal high-order schemes, we rewrite the non-linear system. In this way, we achieve a better sparsity pattern of the Jacobian matrix and less coupling between stages. Furthermore, we propose a fix-point iterative method to further reduce the memory consumption. The saturation solution may present sharp fronts. Thus, the high-order approximation may contain spurious oscillations. To reduce them, we introduce artificial viscosity. We detect the elements with high-oscillations using a computationally efficient shock sensor obtained from the saturation solution and the post-processed saturation of HDG. Finally, we present several examples to assess the capabilities of our formulation. This work has been supported by FEDER and the Spanish Government, Ministerio de Economía y Competitividad grant project contract CTM2014-55014-C3-3-R, Ministerio de Ciencia Innovación y Universidades grant project contract PGC2018-097257-B-C33 and the grant BES-2015-072833.

This work reports analysis of complex shock wave diffraction and longtime behavior of shock-vortex dynamics over splitter geometry encountered in both external and internal compressible flows. The simulation resolved the experimental findings of literature, and the insight of the flow topology is being presented with the probability density functions (PDFs) of various contributing terms of enstrophy transport equation and the invariants of the velocity gradient tensor. We use an artificial viscosity (AV)-based explicit discontinuous spectral element method (DSEM)-based compressible flow solver for this purpose. The numerical scheme utilizes entropy generation-based artificial viscosity and thermal conductivity to simulate the conservative form of the governing compressible flow equations. A shock sensor-based switch is used to reduce the addition of AV coefficients in rotation-dominated regions. This work used the resource of Extreme Science and Engineering Discovery Environment (XSEDE) [17] supported by the National Science Foundation of USA, Grant number ACI1053575

This work aims at developing a high-order numerical method for the propagation of acoustic shock waves using the discontinuous Galerkin method. High order methods tend to amplify the formation of spurious oscillations (Gibbs phenomenon) around the discontinuities/shocks, associated to the relative importance of higher-harmonics resulting from nonlinear propagation (in our case). To handle this critical issue, a new shock sensor is introduced for the sub-cell shock capturing. Thereafter, an element-centered smooth artificial viscosity is introduced into the system wherever an acoustic shock wave is sensed. Validation tests in 1D and 2D configurations show that the method is well-suited for the propagation of acoustic shock waves along with other physical effects like geometrical spreading and diffraction. (C) 2018 Elsevier Inc. All rights reserved.

Smoothed particle hydrodynamics (SPH), as a Lagrangian, meshfree method, is supposed to be useful in solving acoustic problems, such as combustion noise, bubble acoustics, etc., and has been gradually used in sound wave computation. However, unphysical oscillations in the sound wave simulation cannot be ignored. In this paper, an artificial viscosity term is added into the standard SPH algorithm used for solving linearized acoustic wave equations. SPH algorithms with or without artificial viscosity are both built to compute sound propagation and interference in the time domain. Then, the effects of the smoothing kernel function, particle spacing and Courant number on the SPH algorithms of sound waves are discussed. After comparing SPH simulation results with theoretical solutions, it is shown that the result of the SPH algorithm with the artificial viscosity term added attains good agreement with the theoretical solution by effectively reducing unphysical oscillations. In addition, suitable computational parameters of SPH algorithms are proposed through analyzing the sound pressure errors for simulating sound waves.

A variationally consistent eddy viscosity discretization is presented in [W.J. Layton, A connection between subgrid scale eddy viscosity and mixed methods, Appl. Math. Comput. 133 (2002) 147–157] for the stationary convection diffusion problem. This discretization is extended to the evolutionary problem in [N. Heitmann, Subgridscale stabilization of time-dependent convection dominated diffusive transport, J. Math. Anal. Appl. 331 (2007) 38–50] with a near optimal error bound. In the following, we couple this discretization with the porous media problem. We present a comprehensive analysis of stability and error for the velocity field derived from the porous media problem. Next, using a backward Euler approximation for the time derivative we follow the inherited error in velocity through the coupling with the convection diffusion problem. The method is shown to be stable and the error near optimal and independent of the diffusion coefficient, ϵ.

This paper considers the effect of artificial viscosity in smoothed particle hydrodynamics (SPH) computations of six different regular waves. The purpose is to improve the modelling of physically real effects and thereby make SPH a more attractive modelling option. The essence of the proposed method is to avoid running the simulation with different values of the empirical coefficient used in artificial viscosity in order to find the optimum value of this parameter for a given problem. Thorough calibration of the SPH's numerical parameters is performed through the comparison between numerical and experimental data. Among the various parameters involved, the smoothing length and the particle resolution are important in shaping the results. The analysis confirms that when the ratio of particle spacing to smoothing length and the particle resolution useful for different computational domains have been defined, the empirical coefficient depends only on the type of wave breaking in term of the Irribarre...

多段階格子法を用いた非粘性ナビエ･ストークス方程式のソルバーを開発した。非圧縮ナビエ･ストークス方程式の定常解を得るため人工圧縮性の手法を用いる。空間離散化は有限容積法により行い、対流項は流束差分分割を用いた3次精度の風上差分により評価した。時間積分はルンゲクッタ法による。Baldwin-Lomaxの代数的乱流モデルを用いた。自由表面流れの問題を解くことができるよう、非線形自由表面条件を解法に組み込んだ。収束加速のため、局所時間刻み、残差平滑化および多段階格子法を用いた。数値解法の概略と船体まわりの流れの計算結果を示した。, A multigrid incompressible Navier-Stokes solver is developed. The solver employs artificial compressibility approach to obtain a steady state solution of the incompressible Navier-Stokes equations. The spatial discretization is based on the finite-volume method. The convection terms are evaluated by the third order upwind scheme with the flux-difference splitting. Time integration is made by the Runge-Kutta method. The algebraic turbulence model by Baldwin and Lomax is used. Nonlinear free surface conditions are implemented in the scheme that makes it possible to solve free surface flow problems. For the convergence acceleration, local time stepping, residual smoothing and a multigrid method are used. Brief description of the numerical procedure is given together with the computational results for ship flows., 資料番号: AA0004174012, レポート番号: NAL SP-27

International audience; The use of finite element and finite difference methods of spatial and temporal discretization for solving structural dynamics problems gives rise to purely numerical errors. Among the many numerical methods used to damp out the spurious oscillations occurring in the high frequency domain, it is proposed here to analyse and compare the well-known Bulk Viscosity method, which modifies the stresses calculations and a method recently presented by Tchamwa and Wielgosz, which is based on a modification of an explicit time integration algorithm. The efficiency of both methods is evaluated in a 2-D axisymmetric compressive test.

In this paper we develop methods based primarily on the work of Kuropatenko and Wilkins to improve the application of artificial viscosity in 3D finite element method (FEM) codes. The primary goal is to obtain better shock predictions for hypervelocity impacts (HVI) and reduce the need for user calibration. We focus on examining factors such as geometric variability with respect to shock direction, dynamic adaptation to changes in compressibility in the shock front, and anisotropic compression in multi-dimensional formulations. We implement the methods in the Velodyne hydro-structural code and investigate the effects on shock propagation using a series of simple flyer impact test cases which cover a range of system responses including strong and weak shocks. Various initial mesh geometries are utilized to examine mesh effects. Energetic materials using the Ignition and Growth Reactive Burn (IGRB) equation of state (EOS) are also examined due to the rapid change in compressibility and energy density which occurs due to reaction. These rapid changes can lead to insufficient damping in artificial viscosity calculations and thus provide an effective test case. We employ the CTH hydrocode to evaluate baseline shock behavior. The regular, ordered mesh of CTH allows for a consistent and precise application of the artificial viscosity. Direct numerical comparisons are used rather than experimental data to eliminate uncertainty due to factors such as material characterizations, EOS models, and mesh resolution. We compare the CTH results against various FEM artificial viscosity implementations to evaluate performance. It is demonstrated that shock response in FEM codes can be significantly improved by using updated artificial viscosity methods.

High-order numerical solvers for conservation laws suffer from Gibbs phenomenon close to discontinuities, leading to spurious oscillations and a detrimental effect on the solution accuracy. A possible strategy to reduce it comprises adding a suitable amount of artificial dissipation. Although several viscosity models have been proposed in the literature, their dependence on problem-dependent parameters often limits their performances. Motivated by the objective to construct a universal artificial viscosity method, we propose a new technique based on neural networks, integrated into a Runge-Kutta Discontinuous Galerkin solver. Numerical results are presented to demonstrate the performance of this network-based technique. We show that it is able both to guarantee optimal accuracy for smooth problems, and to accurately detect discontinuities, where dissipation has to be injected. A comparison with some classical models is carried out, showing that the network-based model is always at par with the best among the traditional optimized models, independently of the selected problem and parameters. (C) 2020 Elsevier Inc. All rights reserved.

While a nonlinear viscosity is used widely to control oscillations when solving conservation laws using high-order elements based methods, such techniques are less straightforward to apply in global spectral methods since a local estimate of the solution regularity is generally required. In this work we demonstrate how to train and use a local artificial neural network to estimate the local solution regularity and demonstrate the efficiency of nonlinear artificial viscosity methods based on this, in the context of Fourier spectral methods. We compare with entropy viscosity techniques and illustrate the promise of the neural network based estimators when solving one- and two-dimensional conservation laws, including the Euler equations. (C) 2021 Elsevier Inc. All rights reserved.