Artificial Viscosity Research Articles

Numerical Strategy on the Grid Orientation Effect in the Simulation for Two‐Phase Flow in Porous Media by Using the Adaptive Artificial Viscosity Method

ABSTRACTIn the context of numerical simulators for multiphase flow in porous media, there exists a long‐standing issue known as the grid orientation effect (GOE), wherein different numerical solutions can be obtained when considering grids with different orientations under certain unfavorable conditions. The GOE is relevant to the instability near displacement fronts. If numerical oscillations accompanied by sharp fronts are not adequately suppressed, the GOE occurs. To reduce or even eliminate the GOE, we propose augmenting adaptive artificial viscosity in the process of solving the saturation equation. It has been demonstrated that appropriate artificial viscosity can effectively reduce or even eliminate the GOE. The proposed numerical method can be easily applied in practical engineering problems.

An Optimal Control Deep Learning Method to Design Artificial Viscosities for Discontinuous Galerkin Schemes

An Optimal Control Deep Learning Method to Design Artificial Viscosities for Discontinuous Galerkin Schemes

Well-balanced High-order Finite Difference Weighted Essentially Nonoscillatory Schemes for a First-order Z4 Formulation of the Einstein Field Equations

We develop a new class of high-order accurate well-balanced finite difference (FD) weighted essentially nonoscillatory (WENO) methods for numerical general relativity (GR), which can be applied to any first-order reduction of the Einstein field equations, even if nonconservative terms are present. We choose the first-order nonconservative Z4 formulation of the Einstein equations, which has a built-in cleaning procedure that accounts for the Einstein constraints and that has already shown its ability in keeping stationary solutions stable over long timescales. By introducing auxiliary variables, the vacuum Einstein equations in first-order form constitute a partial differential equation system of 54 equations that is naturally nonconservative. We show how to design FD-WENO schemes that can handle nonconservative products. Different variants of FD WENO are discussed, with an eye to their suitability for higher-order accurate formulations for numerical GR. We successfully solve a set of fundamental tests of numerical GR with up to ninth-order spatial accuracy. Due to their intrinsic robustness, flexibility, and ease of implementation, FD-WENO schemes can effectively replace traditional central finite differencing in any first-order formulation of the Einstein field equations, without any artificial viscosity. When used in combination with well-balancing, the new numerical schemes preserve stationary equilibrium solutions of the Einstein equations exactly. This is particularly relevant in view of the numerical study of the quasi-normal modes of oscillations of relevant astrophysical sources. In conclusion, general relativistic high-energy astrophysics could benefit from this new class of numerical schemes and the ecosystem of desirable capabilities built around them.

Comparing Discontinuous Galerkin Shock-Capturing Techniques Applied to Inviscid Three-Dimensional Hypersonic Flows

This paper considers the performance of various shock-capturing schemes when simulating three-dimensional, hypersonic flows using the nodal discontinuous Galerkin (DG) finite element method on unstructured, hexahedral meshes. Simulations use a new code, Cartablanca++, which is verified using the method of manufactured solutions. Three shock-capturing techniques are compared: artificial viscosity (AV), slope limiting, and subcell finite volume limiting. Three test cases are considered, including a shock tube (one-dimensional), a reflecting shock (two-dimensional), and an inclined cylinder with a hemispherical endcap (three-dimensional). The AV formulation was not robust in the sense that it could not maintain pressure positivity after initialization from freestream conditions in the final three-dimensional test case. The slope and subcell limiters performed well in all simulations. Both techniques robustly captured strong shock waves while still benefiting from the use of high-order polynomials. The targeted application of the slope limiter prevented residual convergence to machine precision, while the subcell limiter could achieve residual convergence. The mixed DG/finite volume formulation, inherent to the subcell limiting scheme, appears sensitive to the inviscid flux function. Future work will consider modifications to reduce this sensitivity. Additionally, modifications to the shock detection techniques would improve performance for both the slope and subcell limiters.

A turbulence model: Second-order temporal accuracy

This paper develops the defect-deferred correction method of finite element discretization to solve a turbulence model. The method consists of two phases: (1) implementing an artificial viscosity term as a defect step and (2) applying a correction strategy. This method can not only achieve second-order accuracy in time but also simulate small viscosity problem. Moreover, the stability and convergence of the defect step and the deferred correction step are proven, respectively. Finally, we perform several numerical examples to validate the theoretical analysis and illustrate the effectiveness of the proposed method.

Electron Influence on the Parallel Proton Firehose Instability in 10-moment, Multifluid Simulations

Instabilities driven by pressure anisotropy play a critical role in modulating the energy transfer in space and astrophysical plasmas. For the first time, we simulate the evolution and saturation of the parallel proton firehose instability using a multifluid model without adding artificial viscosity. These simulations are performed using a 10-moment, multifluid model with local and gradient relaxation heat-flux closures in high-β proton–electron plasmas. When these higher-order moments are included and pressure anisotropy is permitted to develop in all species, we find that the electrons have a significant impact on the saturation of the parallel proton firehose instability, modulating the proton pressure anisotropy as the instability saturates. Even for lower β's more relevant to heliospheric plasmas, we observe a pronounced electron energization in simulations using the gradient relaxation closure. Our results indicate that resolving the electron pressure anisotropy is important to correctly describe the behavior of multispecies plasma systems.

Modeling Brittle Failure in Rock Slopes Using Semi‐Lagrangian Nonlocal General Particle Dynamics

ABSTRACTThe nonlocal general particle dynamics (NGPD) has been successfully developed to model crack propagation and large deformation problems. In this paper, the semi‐Lagrangian nonlocal general particle dynamics (SL‐NGPD) is proposed to solve brittle failure in rock slopes. In SL‐NGPD, the interaction between particles due to deformation is calculated in the initial configuration, while the friction contact interaction from discontinuities is calculated in the current configuration. The Van der Waals force model is utilized for friction contact. The bond‐level energy‐based failure criterion is developed to predict tensile/compressive‐shear mix‐mode cracks. The artificial viscosity and damage correction are used to enhance the numerical stability and accuracy when modeling brittle failure. The SL‐NGPD paradigm is numerically implemented through adaptive dynamic relaxation and predictor–corrector schemes for stable numerical solutions. The stability and accuracy of SL‐NGPD are verified by simulating compression tests. Thereafter, the crack coalescence patterns of double‐flaw specimens are investigated to understand the triggering failure mechanism of jointed rock slopes. Finally, the progressive failure process of the rock slope with step‐path joints is simulated to demonstrate its validity and robustness in modeling brittle failure in rockslides. The numerical results illustrate that the proposed SL‐NGPD is promising and performant for analyzing brittle failure problems in geotechnical engineering.

An asymptotic-preserving and exactly mass-conservative semi-implicit scheme for weakly compressible flows based on compatible finite elements

We present a novel asymptotic-preserving semi-implicit finite element method for weakly compressible and incompressible flows based on compatible finite element spaces. The momentum is sought in an H(div)-conforming space, ensuring exact pointwise mass conservation at the discrete level. We use an explicit discontinuous Galerkin-based discretization for the nonlinear convective terms, while treating the pressure and viscous terms implicitly, so that the CFL condition depends only on the fluid velocity. To handle shocks and damp spurious oscillations in the compressible regime, we incorporate an a posteriori limiter that employs artificial viscosity and is based on a discrete maximum principle. By using hybridization, the final algorithm requires solving only symmetric positive definite linear systems. As the Mach number approaches zero and the density remains constant, the method naturally converges to an H(div)-based discretization of the incompressible Navier-Stokes equations in the vorticity-velocity-pressure formulation. Several numerical tests validate the proposed method.

Constraint Realization‐Based Hamel Field Integrator for Geometrically Exact Planar Euler–Bernoulli Beam Dynamics

ABSTRACTIn this article, we first introduce a Hamel field integrator designed for a geometrically exact Euler–Bernoulli beam with infinite‐dimensional holonomic constraints, constructed using a Lagrange multiplier. This method addresses the complexities introduced by constraints, but the additional multiplier introduces a new degree of freedom and hence results in a system with mixed‐type partial differential equations. To address this issue, we further propose a constraint realization method based on perturbation theory for infinite‐dimensional mechanical systems within the framework of Hamel's formalism. This method circumvents the use of additional Lagrange multiplier, significantly reducing the computational complexity of modeling problems. Building on this, we construct a perturbed Hamel field integrator optimized for parallel computing and incorporate artificial viscosity to accelerate constraint convergence. While applicable to three dimensions, our method is demonstrated in a simplified context using planar Euler–Bernoulli beam examples to illustrate the effectiveness of the unified mathematical framework.

Evaluating Fracture Energy Predictions Using Phase-Field and Gradient-Enhanced Damage Models for Elastomers

Abstract Recently, the phase-field method has been increasingly used for brittle fractures in soft materials like polymers, elastomers, and biological tissues. When considering finite deformations to account for the highly deformable nature of soft materials, the convergence of the phase-field method becomes challenging, especially in scenarios of unstable crack growth. To overcome these numerical difficulties, several approaches have been introduced, with artificial viscosity being the most widely utilized. This study investigates the energy release rate due to crack propagation in hyperelastic nearly-incompressible materials and compares the phase-field method and a novel gradient-enhanced damage (GED) approach. First, we simulate unstable loading scenarios using the phase-field method, which leads to convergence problems. To address these issues, we introduce artificial viscosity to stabilize the problem and analyze its impact on the energy release rate utilizing a domain J-integral approach giving quantitative measurements during crack propagation. It is observed that the measured energy released rate during crack propagation does not comply with the imposed critical energy release rate, and shows non-monotonic behavior. In the second part of the paper, we introduce a novel stretch-based GED model as an alternative to the phase-field method for modeling crack evolution in elastomers. It is demonstrated that in this method, the energy release rate can be obtained as an output of the simulation rather than as an input which could be useful in the exploration of rate-dependent responses, as one could directly impose chain-level criteria for damage initiation. We show that while this novel approach provides reasonable results for fracture simulations, it still suffers from some numerical issues that strain-based GED formulations are known to be susceptible to.

A Convergent Semi-Lagrangian Scheme for the Game $$\infty $$-Laplacian

AbstractWe propose a new semi-Lagrangian scheme for the game $$\infty $$ ∞ -Laplacian. We demonstrate the convergence of the scheme to the viscosity solution of the given problem, showing its consistency, monotonicity, and stability. The proof of this result is established following the Barles-Souganidis analysis. This analysis assumes convergence at the boundary in a strong sense and is applied to our proposed scheme, augmented with an artificial viscosity term.

Discovering artificial viscosity models for discontinuous Galerkin approximation of conservation laws using physics-informed machine learning

Finite element-based high-order solvers of conservation laws offer large accuracy but face challenges near discontinuities due to the Gibbs phenomenon. Artificial viscosity is a popular and effective solution to this problem based on physical insight. In this work, we present a physics-informed machine learning algorithm to automate the discovery of artificial viscosity models. We refer to the proposed approach as an “hybrid” approach which stands at the edge between supervised and unsupervised learning. More precisely, the proposed “hybrid” paradigm is not supervised in the classical sense as it does not utilize labeled data in the traditional way but relies on the intrinsic properties of the reference solution. The algorithm is inspired by reinforcement learning and trains a neural network acting cell-by-cell (the viscosity model) by minimizing a loss defined as the difference with respect to a reference solution thanks to automatic differentiation. This enables a dataset-free training procedure. We prove that the algorithm is effective by integrating it into a state-of-the-art Runge-Kutta discontinuous Galerkin solver. We showcase several numerical tests on scalar and vectorial problems, such as Burgers' and Euler's equations in one and two dimensions. Results demonstrate that the proposed approach trains a model that is able to outperform classical viscosity models. Moreover, we show that the learnt artificial viscosity model is able to generalize across different problems and parameters.

DualSPHysics+: An enhanced DualSPHysics with improvements in accuracy, energy conservation and resolution of the continuity equation

This paper presents an enhanced version of the well-known SPH (Smoothed Particle Hydrodynamics) open-source code DualSPHysics for the simulation of free-surface fluid flows, leading to the DualSPHysics+ code. The enhancements are made through incorporation of several schemes with respect to stability, accuracy and energy/volume conservation issues in simulating incompressible free-surface fluid flows within the weakly compressible SPH formalism. The Optimized Particle Shifting (OPS) scheme is implemented to improve the accuracy of particle shifting vectors in the free-surface region. To mitigate energy dissipation and maintain consistency, the artificial viscosity in δ-SPH is substituted with a Riemann stabilization term, leading to the δR-SPH. The Velocity divergence Error Mitigating (VEM) and Volume Conservation Shifting (VCS) schemes are adopted in DualSPHysics+ to mitigate the velocity divergence error and improve the volume conservation, and hence to enhance the resolution of the continuity equation. To further reduce both the instantaneous and accumulated errors in velocity divergence, a Hyperbolic/Parabolic Divergence Cleaning (HPDC) scheme is incorporated in addition to the VEM scheme. The implementations of the introduced schemes on both CPU and GPU-based versions of the DualSPHysics+ code along with details on the compilation, running and computational performance are presented. Validations in terms of accuracy, energy conservation and convergence of DualSPHysics+ are shown via several relevant benchmarks. It is demonstrated that a better velocity divergence error cleaning in both instantaneous and accumulated errors can be achieved by the combination of VEM and HPDC. Meanwhile, the excessive energy dissipation by the artificial viscosity is shown to be suppressed by adopting the Riemann stabilization term. Enhanced resolution of the continuity equation along with improved energy conservation of DualSPHysics+ advance the SPH-based simulation of incompressible free-surface fluid flows.

An operational discontinuous Galerkin shallow water model for coastal flood assessment

Hydrodynamic modeling for coastal flooding risk assessment is a highly relevant topic. Many operational tools available for this purpose use numerical techniques and implementation paradigms that reach their limits when confronted with modern requirements in terms of resolution and performances. In this work, we present a novel operational tool for coastal hazards predictions, currently employed by the BRGM agency (the French Geological Survey) to carry out its flooding hazard exposure studies and coastal risk prevention plans on International and French territories. The model, called UHAINA (wave in the Basque language), is based on an arbitrary high-order discontinuous Galerkin discretization of the nonlinear shallow water equations with SSP Runge–Kutta time stepping on unstructured triangular grids. It is built upon the finite element library AeroSol, which provides a modern C++ software architecture and high scalability, making it suitable for HPC applications. The paper provides a detailed development of the mathematical and numerical framework of the model, focusing on two key-ingredients : (i) a pragmatic P0 treatment of the solution in partially dry cells which garantees efficiently well-balancedness, positivity and mass conservation at any polynomial order; (ii) an artificial viscosity method based on the physical dissipation of the system of equations providing nonlinear stability for non-smooth solutions. A set of numerical validations on accademic benchmarks is performed to highlight the efficiency of these approaches. Finally, UHAINA is applied on a real operational case of study, demonstrating very satisfactory results.

Modeling Cardiovascular Flow with Artificial Viscosity: Analyzing Navier-Stokes Solutions and Simulating Cardiovascular Diseases

In this paper, the numerical solutions of the Navier-Stokes equations (NSE) used for modeling the flow in the cardiovascular system are investigated using the Finite Element Method (FEM). A fully discrete solution scheme of the NSE and its stability and error analysis are presented. Artificial viscosity stabilization is added to the fully discrete scheme to better model the real flow structure and to remove non-physical oscillations. Numerical tests are also presented to demonstrate the effectiveness of the resulting scheme. Simulations analyzing the flow structure in the case of cardiovascular diseases such as atherosclerosis and brain aneurysm are presented in detail along with wall shear stress values.

Supersonic turbulence simulations with GPU-based high-order Discontinuous Galerkin hydrodynamics

ABSTRACT We investigate the numerical performance of a Discontinuous Galerkin (DG) hydrodynamics implementation when applied to the problem of driven, isothermal supersonic turbulence. While the high-order element-based spectral approach of DG is known to efficiently produce accurate results for smooth problems (exponential convergence with expansion order), physical discontinuities in solutions, like shocks, prove challenging and may significantly diminish DG’s applicability to practical astrophysical applications. We consider whether DG is able to retain its accuracy and stability for highly supersonic turbulence, characterized by a network of shocks. We find that our new implementation, which regularizes shocks at subcell resolution with artificial viscosity, still performs well compared to standard second-order schemes for moderately high-Mach number turbulence, provided we also employ an additional projection of the primitive variables on to the polynomial basis to regularize the extrapolated values at cell interfaces. However, the accuracy advantage of DG diminishes significantly in the highly supersonic regime. Nevertheless, in turbulence simulations with a wide dynamic range that start with supersonic Mach numbers and can resolve the sonic point, the low-numerical dissipation of DG schemes still proves advantageous in the subsonic regime. Our results thus support the practical applicability of DG schemes for demanding astrophysical problems that involve strong shocks and turbulence, such as star formation in the interstellar medium. We also discuss the substantial computational cost of DG when going to high order, which needs to be weighted against the resulting accuracy gain. For problems containing shocks, this favours the use of comparatively low DG order.

A modern concept of Lagrangian hydrodynamics

AbstractWe offer a modern interpretation of Lagrangian hydrodynamics as employed in Lagrangian simulations of compressible fluid flow. Our main result is to show that artificial viscosity, traditionally viewed as a numerical artifice to control unphysical oscillations in flows with shocks, actually represents a physical process and is necessary to derive accurate simulations in any compressible flow. We begin by reviewing the origins of two numerical devices, artificial viscosity and finite‐volume methods. We proceed to construct a mathematical (PDE) model that incorporates those numerics and in which a new length scale, the observer, arises representing the discretization. Associated with that length scale, there are new inviscid fluxes that are the artificial viscosity as first formulated by Richtmyer and an artificial heat flux postulated by Noh but typically not included in Lagrangian codes. We discuss the connection of our results to bivelocity hydrodynamics. We conclude with some speculation as to the direction of future developments in multidimensional Lagrangian codes as computers get faster and have larger memories.

Tangential artificial viscosity to alleviate the carbuncle phenomenon, with applications to single-component and multi-material flows

This paper describes a novel approach to alleviate the carbuncle phenomenon which consists in adding to any carbuncle prone Riemann solver an extra viscosity term in tangential momentum flux and its contribution to the energy conservation equation. This term contains one numerical parameter only, a scalar viscosity, which is reduced using a face-based shear detector to preserve shear waves.The idea stems from the investigation of some of the existing Riemann solvers, also presented in the paper. Indeed, when splitting the numerical flux into the face normal and tangential components, we observe that all the carbuncle free Riemann solvers present in the tangential part a numerical viscosity which scales with the sound speed when the normal flow velocity becomes zero. Opposite, in the carbuncle prone solvers this viscosity scales with the normal flow velocity. In particular the carbuncle free HLLCM scheme proposed by Shen et al. can be written by adding to the carbuncle prone HLLC scheme a tangential artificial viscosity term. Then the same can be done for any other Riemann solver, which renders the approach easy to implement in CFD codes for compressible flows.Numerical experiments shows the efficiency of the approach in computing carbuncle free single-component and multi-material flows.

P adé: A Code for Protoplanetary Disk Turbulence Based on Padé Differencing

The Padé code has been developed to treat hydrodynamic turbulence in protoplanetary disks. It solves the compressible equations of motion in cylindrical coordinates. Derivatives are computed using nondiffusive and conservative fourth-order Padé differencing, which has higher resolving power compared to both dissipative shock-capturing schemes used in most astrophysics codes, as well as nondiffusive central finite-difference schemes of the same order. The fourth-order Runge–Kutta method is used for time stepping. A previously reported error-corrected Fargo approach is used to reduce the time step constraint imposed by rapid Keplerian advection. Artificial bulk viscosity is used when shock capturing is required. Tests for correctness and scaling with respect to the number of processors are presented. Finally, efforts to improve efficiency and accuracy are suggested.

Entropy stable discontinuous Galerkin methods for the shallow water equations with subcell positivity preservation

AbstractHigh order schemes are known to be unstable in the presence of shock discontinuities or under‐resolved solution features, and have traditionally required additional filtering, limiting, or artificial viscosity to avoid solution blow up. Entropy stable schemes address this instability by ensuring that physically relevant solutions satisfy a semi‐discrete entropy inequality independently of discretization parameters. However, additional measures must be taken to ensure that solutions satisfy physical constraints such as positivity. In this work, we present a high order entropy stable discontinuous Galerkin (ESDG) method for the nonlinear shallow water equations (SWE) on two‐dimensional (2D) triangular meshes which preserves the positivity of the water heights. The scheme combines a low order positivity preserving method with a high order entropy stable method using convex limiting. This method is entropy stable and well‐balanced for fitted meshes with continuous bathymetry profiles.