An artificial viscosity augmented physics-informed neural network for incompressible flow

Physics-informed neural networks(PINNs)are proved methods that are effective in solving some strongly nonlinear partial differential equations(PDEs),e.g.,Navier-Stokes equations,with a small amount of boundary or interior data.However,the feasibility of applying PINNs to the flow at moderate or high Reynolds numbers has rarely been reported.The present paper proposes an artificial viscosity(AV)-based PINN for solving the forward and inverse flow problems.Specifically,the AV used in PINNs is inspired by the entropy viscosity method developed in conventional computational fluid dynamics(CFD)to stabilize the simulation of flow at high Reynolds numbers.The newly developed PINN is used to solve the forward problem of the two-dimensional steady cavity flow at Re=1000 and the inverse problem derived from two-dimensional film boiling.The results show that the AV augmented PINN can solve both problems with good accuracy and substantially reduce the inference errors in the forward problem.

An adaptive artificial viscosity for the displacement shallow water wave equation

The numerical oscillation problem is a difficulty for the simulation of rapidly varying shallow water surfaces which are often caused by the unsmooth uneven bottom,the moving wet-dry interface,and so on.In this paper,an adaptive artificial viscosity(AAV)is proposed and combined with the displacement shallow water wave equation(DSWWE)to establish an effective model which can accurately predict the evolution of multiple shocks effected by the uneven bottom and the wet-dry interface.The effectiveness of the proposed AAV is first illustrated by using the steady-state solution and the small perturbation analysis.Then,the action mechanism of the AAV on the shallow water waves with the uneven bottom is explained by using the Fourier theory.It is shown that the AVV can suppress the wave with the large wave number,and can also suppress the numerical oscillations for the rapidly varying bottom.Finally,four numerical examples are given,and the numerical results show that the DSWWE combined with the AAV can effectively simulate the shock waves,accurately capture the movements of wet-dry interfaces,and precisely preserve the mass.

Optimization of Artificial Viscosity in Production Codes Based on Gaussian Regression Surrogate Models

To accurately model flows with shock waves using staggered-grid Lagrangian hydrodynamics, the artificial viscosity has to be introduced to convert kinetic energy into internal energy, thereby increasing the entropy across shocks. Determining the appropriate strength of the artificial viscosity is an art and strongly depends on the particular problem and experience of the researcher. The objective of this study is to pose the problem of finding the appropriate strength of the artificial viscosity as an optimization problem and solve this problem using machine learning (ML) tools, specifically using surrogate models based on Gaussian Process regression (GPR) and Bayesian analysis. We describe the optimization method and discuss various practical details of its implementation. The shock-containing problems for which we apply this method all have been implemented in the LANL code FLAG (Burton in Connectivity structures and differencing techniques for staggered-grid free-Lagrange hydrodynamics, Tech. Rep. UCRL-JC-110555, Lawrence Livermore National Laboratory, Livermore, CA, 1992, 1992, in Consistent finite-volume discretization of hydrodynamic conservation laws for unstructured grids, Tech. Rep. CRL-JC-118788, Lawrence Livermore National Laboratory, Livermore, CA, 1992, 1994, Multidimensional discretization of conservation laws for unstructured polyhedral grids, Tech. Rep. UCRL-JC-118306, Lawrence Livermore National Laboratory, Livermore, CA, 1992, 1994, in FLAG, a multi-dimensional, multiple mesh, adaptive free-Lagrange, hydrodynamics code. In: NECDC, 1992). First, we apply ML to find optimal values to isolated shock problems of different strengths. Second, we apply ML to optimize the viscosity for a one-dimensional (1D) propagating detonation problem based on Zel’dovich-von Neumann-Doring (ZND) (Fickett and Davis in Detonation: theory and experiment. Dover books on physics. Dover Publications, Mineola, 2000) detonation theory using a reactive burn model. We compare results for default (currently used values in FLAG) and optimized values of the artificial viscosity for these problems demonstrating the potential for significant improvement in the accuracy of computations.

A DDG Method with a Residual-Based Artificial Viscosity for the Transonic/Supersonic Compressible Flow

In this work,a direct discontinuous Galerkin(DDG)method with artificial viscosity is developed to solve the compressible Navier-Stokes equations for simulating the transonic or supersonic flow,where the DDG approach is used to discretize viscous and heat fluxes.A strong residual-based artificial viscosity(AV)technique is proposed to be applied in the DDG framework to handle shock waves and layer structures appearing in transonic or supersonic flow,which promotes convergence and robustness.Moreover,the AV term is added to classical BR2 methods for comparison.A number of 2-D and 3-D benchmarks such as airfoils,wings,and a full aircraft are presented to assess the performance of the DDG framework with the strong residualbased AV term for solving the two dimensional and three dimensional Navier-Stokes equations.The proposed framework provides an alternative robust and efficient approach for numerically simulating the multi-dimensional compressible Navier-Stokes equations for transonic or supersonic flow.

A Node-Centered Artificial Viscosity Method for Two-Dimensional Lagrangian Hydrodynamics Calculations on a Staggered Grid

This work deals with the simulation of two-dimensional Lagrangian hydrodynamics problems.Our objective is the development of an artificial viscosity that is to be used in conjunction with a staggered placement of variables:thermodynamics variables are centered within cells and position and fluid velocity at vertices.In[J.Comput.Phys.,228(2009),2391-2425],Maire develops a high-order cell-centered scheme for solving the gas dynamics equations.The numerical results show the accuracy and the robustness of the method,and the fact that very few Hourglass-type deformations are present.Our objective is to establish the link between the scheme of Maire and the introduction of artificial viscosity in a Lagrangian code based on a staggered grid.Our idea is to add an extra degree of freedom to the numerical scheme,which is an approximation of the fluid velocity within cells.Doing that,we can locally come down to a cell-centered approximation and define the Riemann problem associated to discrete variable discontinuities in a very natural way.This results in a node-centered artificial viscosity formulation.Numerical experiments show the robustness and the accuracy of the method,which is very easy to implement.

Suitability of artificial viscosity discontinuous Galerkin method for compressible turbulence

A discontinuous Galerkin method based on an artificial viscosity model is investigated in the context of the simulation of compressible turbulence. The effects of artificial viscosity on shock capturing ability, broadband accuracy and under-resolved instability are examined combined with various orders and mesh resolutions. For shock-dominated flows, the superior accuracy of high order methods in terms of discontinuity resolution are well retained compared with lower ones. For under-resolved simulations, the artificial viscosity model is able to enhance stability of the eighth order discontinuous Galerkin method despite of detrimental influence for accuracy. For multi-scale flows, the artificial viscosity model demonstrates biased numerical dissipation towards higher wavenumbers. Capability in terms of boundary layer flows and hybrid meshes is also demonstrated.It is concluded that the fourth order artificial viscosity discontinuous Galerkin method is comparable to typical high order finite difference methods in the literature in terms of accuracy for identical number of degrees of freedom, while the eighth order is significantly better unless the under-resolved instability issue is raised. Furthermore, the artificial viscosity discontinuous Galerkin method is shown to provide appropriate numerical dissipation as compensation for turbulent kinetic energy decaying on moderately coarse meshes, indicating good potentiality for implicit large eddy simulation.

Shock calculation based on second viscosity using local differentialquadrature method

Based on the second viscosity, the local differential quadrature （LDQ） method is applied to solve shock tube problems. It is shown that it is necessary to consider the second viscosity to calculate shocks and to simulate shock tubes based on the viscosity model. The roles of the shear viscous stress and the second viscous stress are checked. The results show that the viscosity model combined with the LDQ method can capture the main characteristics of shocks, and this technique is objective and simple.

Artificial neural network approach for rheological characteristics of coal-water slurry using microwave pre-treatment

Detailed experimental investigations were carried out for microwave pre-treatment of high ash Indian coal at high power level(900 W) in microwave oven. The microwave exposure times were fixed at60 s and 120 s. A rheology characteristic for microwave pre-treatment of coal-water slurry(CWS) was performed in an online Bohlin viscometer. The non-Newtonian character of the slurry follows the rheological model of Ostwald de Waele. The values of n and k vary from 0.31 to 0.64 and 0.19 to 0.81 Pa·sn,respectively. This paper presents an artificial neural network(ANN) model to predict the effects of operational parameters on apparent viscosity of CWS. A 4-2-1 topology with Levenberg-Marquardt training algorithm(trainlm) was selected as the controlled ANN. Mean squared error(MSE) of 0.002 and coefficient of multiple determinations(R^2) of 0.99 were obtained for the outperforming model. The promising values of correlation coefficient further confirm the robustness and satisfactory performance of the proposed ANN model.

Numerical method for simulating rarefaction shocks in the approximation of phase-flip hydrodynamics

A finite-difference algorithm is proposed for numerical modeling of hydrodynamic flows with rarefaction shocks, in which the fluid undergoes a jump-like liquid-gas phase transition. This new type of flow discontinuity, unexplored so far in computational fluid dynamics, arises in the approximation of phase-flip(PF) hydrodynamics, where a highly dynamic fluid is allowed to reach the innermost limit of metastability at the spinodal, upon which an instantaneous relaxation to the full phase equilibrium(EQ) is assumed. A new element in the proposed method is artificial kinetics of the phase transition, represented by an artificial relaxation term in the energy equation for a "hidden"component of the internal energy, temporarily withdrawn from the fluid at the moment of the PF transition. When combined with an appropriate variant of artificial viscosity in the Lagrangian framework, the latter ensures convergence to exact discontinuous solutions, which is demonstrated with several test cases.

Numerical Simulation of Underwater Explosion Loads

Numerical simulation of TNT underwater explosion was carried out with AUTODYN soft-ware. Influences of artificial viscosity and mesh density on simulation results were discussed. Deto-nation waves in explosive and shock wave in water during early time of explosion are high frequency waves. Fine meshes (less than 1 mm) in explosive and water nearby, and small linear viscosity co-efficients and quadratic viscosity coefficients (0.02 and 0.1 respectively, 1/10 of default values) are needed in numerical simulation model. According to these rules, numerical computing pressure profiles can match well with those calculated by Zamyshlyayev empirical formula. Otherwise peak pressure would be smeared off and upstream relative errors would be cumulated downstream to make downstream peak pressure lower.

Solving the Sod Shock Tube Problem Using Localized Differential Quadrature (LDQ) Method

The localized differential quadrature (LDQ) method is a numerical technique with high accuracy for solving most kinds of nonlinear problems in engineering and can overcome the difficulties of other methods (such as difference method) to numerically evaluate the derivatives of the functions.Its high efficiency and accuracy attract many engineers to apply the method to solve most of the numerical problems in engineering.However,difficulties can still be found in some particular problems.In the following study,the LDQ was applied to solve the Sod shock tube problem.This problem is a very particular kind of problem,which challenges many common numerical methods.Three different examples were given for testing the robustness and accuracy of the LDQ.In the first example,in which common initial conditions and solving methods were given,the numerical oscillations could be found dramatically;in the second example,the initial conditions were adjusted appropriately and the numerical oscillations were less dramatic than that in the first example;in the third example,the momentum equation of the Sod shock tube problem was corrected by adding artificial viscosity,causing the numerical oscillations to nearly disappear in the process of calculation.The numerical results presented demonstrate the detailed difficulties encountered in the calculations,which need to be improved in future work.However,in summary,the localized differential quadrature is shown to be a trustworthy method for solving most of the nonlinear problems in engineering.

On the Solution Accuracy Downstream of Shocks When Using Godunov-Type Schemes.I.Sources of Errors in One-Dimensional Problems

The article opens a series of publications devoted to a systematic study of numerical errors behind the shock wave when using high-order Godunov-type schemes,including in combination with the artificial viscosity approach.The proposed paper describes the numerical methods used in the study,and identifies the main factors affecting the accuracy of the solution for the case of one-dimensional gas dynamic problems.The physical interpretation of the identified factors is given and their influence on the grid convergence is analyzed.

Improved Implicit SPH Method for simulating free surface flows of power law fluids

Implicit smoothed particle hydrodynamics method has been proposed to overcome the problem that only very small time steps can be used for high viscosity fluids(such as power law fluids)in order to obtain a stable simulation.However,the pressure field is difficult to simulate correctly with this method because the numerical high-frequency noise on the pressure field cannot be removed.In this study,several improvements,which are the diffusive term in the continuity equation,the artificial viscosity and a simplified physical viscosity term in the momentum equation,are introduced,and a new boundary treatment is also proposed.The linear equations derived from the momentum equation are large-scale,sparse and positive definite but unsymmetrical,therefore,Conjugate Gradient Squared(CGS)is used to solve them.For the purpose of verifying the validity of the proposed method,Poiseuille flows with Newtonian and power law fluids are solved and compared with exact solution and traditional SPH.Drops of different fluid properties impacting a rigid wall are also simulated and compared with VOF solution.All the numerical results obtained by the proposed method agree well with available data.The proposed method shows the higher efficiency than traditional SPH and the less numerical noise on the pressure field and better stability than implicit SPH.

Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme

The aim of the present work is to develop a general formalism to derive staggered discretizations for Lagrangian hydrodynamics on two-dimensional unstructured grids.To this end,we make use of the compatible discretization that has been initially introduced by E.J.Caramana et al.,in J.Comput.Phys.,146(1998).Namely,momentum equation is discretized by means of subcell forces and specific internal energy equation is obtained using total energy conservation.The main contribution of this work lies in the fact that the subcell force is derived invoking Galilean invariance and thermodynamic consistency.That is,we deduce a general form of the sub-cell force so that a cell entropy inequality is satisfied.The subcell force writes as a pressure contribution plus a tensorial viscous contribution which is proportional to the difference between the nodal velocity and the cell-centered velocity.This cell-centered velocity is a supplementary degree of freedom that is solved by means of a cell-centered approximate Riemann solver.To satisfy the second law of thermodynamics,the local subcell tensor involved in the viscous part of the subcell force must be symmetric positive definite.This subcell tensor is the cornerstone of the scheme.One particular expression of this tensor is given.A high-order extension of this discretization is provided.Numerical tests are presented in order to assess the efficiency of this approach.The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of this scheme.

Corrected SPH methods for solving shallow-water equations

The artificial viscosity in the traditional smoothed particle hydrodynamics （SPH） methodology concerns some empirical coefficients, which limits the capability of the SPH methodology. To overcome this disadvantage and further improve the accuracy of shock capturing, this paper introduces two other ways for numerical viscosity, which are the Lax-Friedrichs flux and the two- shock Riemann solver with MUSCL reconstruction to provide stability. Six SPH methods with different kinds of numerical viscosity are tested against the analytical solution for a 1-D dam break with a wet bed. The comparison shows that the Lax-Friedrichs flux with MUSCL reconstruction can capture the shock wave more accurate than other five methods. The Lax-Friedrichs flux and the artificial viscosity with MUSCL reconstruction are finally both applied to a 2-D dam-break test case in a L-shaped channel and the numerical results are compared with experimental data. It is concluded that this corrected SPH method can be used to solve shallow-water equations well.

A Study of 2D Separated Cascade Flow at Large Angles of Attack Using Euler Equations

This paper presents a numerical simulation method developed for separated flow in cascades using the Euler equations and demonstrates the feasibility of this method.MacCormack's two-steps explicit finite difference scheme is used to discretize the equations in conservation form,and the artificial viscosity is added to the dis- cretized inviscid equations by means of the self-adapted filter technique.The initial separation boundary is given according to simple experimental results.The numerical simulation results including subsonic and transonic turbine cascades flow with or without separation show that the fundamental idea of this numerical method is reasonable and simple.The present study indicates that for solving certain engineering problems it is a simple and effective tool for adding some viscosity corrections to inviscid flow model,especially the current when the Navier-Stokes equations have not been solved very effectively for various complicated flows in turbomachinery.

Fourier Collocation and Reduced Basis Methods for Fast Modeling of Compressible Flows

A projection-based reduced order model(ROM)based on the Fourier collocation method is proposed for compressible flows.The incorporation of localized artificial viscosity model and filtering is pursued to enhance the robustness and accuracy of the ROM for shock-dominated flows.Furthermore,for Euler systems,ROMs built on the conservative and the skew-symmetric forms of the governing equation are compared.To ensure efficiency,the discrete empirical interpolation method(DEIM)is employed.An alternative reduction approach,exploring the sparsity of viscosity is also investigated for the viscous terms.A number of one-and two-dimensional benchmark cases are considered to test the performance of the proposed models.Results show that stable computations for shock-dominated cases can be achieved with ROMs built on both the conservative and the skew-symmetric forms without additional stabilization components other than the viscosity model and filtering.Under the same parameters,the skew-symmetric form shows better robustness and accuracy than its conservative counterpart,while the conservative form is superior in terms of efficiency.

Numerical modelling of supercritical flow in circular conduit bends using SPH method

The capability of 1he smoothed-particle hydrodynamics （SPH） method to model supercritical flow in circular pipe bends is considered. The standard SPH method, which makes use of dynamic boundary particles （DBP）, is supplemented with the original algorithm for the treatment of open boundaries. The method is assessed through a comparison with measured free-surface profiles in a pipe bend, and already proposed regression curves for eslimation of the flow-type in a pipe bend. The sensitivity of the model to different parameters is also evaluated. It is shown that an adequate choice of the artificial viscosity coefficient and the initial particle spacing can lead to correct presentation of the flow-type in a bend. Due to easiness of its implementation, the SPH method can he efficiently used in the design of circular conduits with supercritical flow in a bend, such as tunnel spillways, and bottom outlets of dams, or storm sewers.