Non-physical Oscillations Research Articles

Improve neural representations with general exponential activation function for high-speed flows

Characterizing flow fields with neural networks has witnessed a considerable surge in recent years. However, the efficacy of these techniques is typically constrained when applied to high-speed compressible flows, due to the susceptibility of nonphysical oscillations near shock waves. In this work, we focus on a crucial fundamental component of neural networks, the activation functions, to improve the physics-informed neural representations of high-speed compressible flows. We present a novel activation function, namely, the generalized exponential activation function, which has been specifically designed based on the intrinsic characteristics of high-speed compressible flows. Subsequently, the performance of the proposed method is subjected to a comprehensive analysis, encompassing training stability, initialization strategy, and the influence of ancillary components. Finally, a series of representative experiments were conducted to validate the efficacy of the proposed method, including the contact-discontinuity problem, the Sod shock-tube problem, and the converging–diverging nozzle flow problem.

SPH simulations of non-isothermal viscoplastic free-surface flows incorporating Herschel-Bulkley-Papanastasiou model

In this paper, an improved smoothed particle hydrodynamics (SPH) method is employed to accurately simulate non-isothermal viscoplastic free surface flows, wherein the viscoplastic behavior of the fluid is precisely captured through the incorporation of the Herschel-Bulkley-Papanastasiou constitutive model. To suppress the non-physical oscillation arising from the weakly compressible hypothesis within the pressure field, the density dissipation term is incorporated into the mass conservation equation. To address the tensile instability arising from the uneven distribution of particles, the particle shifting technique is incorporated as a solution. To enhance the precision and ensure numerical stability of the gradient operator, a kernel gradient correction algorithm is implemented. The improved SPH method is employed for numerically simulating the non-isothermal viscoplastic mixed convection, dam-break flow and droplet impacting the solid wall. The effectiveness of the improved SPH method in tackling the complexities of non-isothermal viscoplastic fluid is validated through a rigorous comparison of its outcomes with those derived from alternative numerical methodologies. The assessment of the numerical convergence of the improved SPH method is undertaken through the utilization of varying initial particle spacings. The numerical outcomes demonstrate that the improved SPH method adeptly and precisely delineates the heat transfer mechanisms, intricate rheological properties, as well as the dynamic variation characteristics of the free surface in non-isothermal viscoplastic free surface flows.

Non-inertial computational framework for long-distance shock-driven object dynamics

In the realm of dynamic separation problems, the motion of a body triggered by shock interactions is a common phenomenon. This is particularly important in terms of the safe separation of two-stage-to-orbit vehicles, where the motion must remain stable despite long-distance disturbances from shock waves. The flow field in these cases is complex, marked by interactions between hypersonic shock waves and a moving boundary. This leads to significant unsteady effects due to the body's translation and rotation over extended distances. Existing simulation techniques fall short in rapidly and accurately predicting the aerodynamic force and thermal properties for these problems, largely due to the overwhelming computational demands that result from oversize computational domains and the necessity of grid deformation. This paper presents a novel non-deforming grid method to address these challenges. The central concept is to anchor the reference frame to the moving object itself and to approach the problem from a non-inertial frame perspective. This accounts for the motion of the object solely via the inertial source term, circumventing the complexities of mesh manipulation typically required to link flow and motion equations. The moving shock boundary is designed to be closely compatible with selected shock-captured schemes, which reduces non-physical oscillations compared to the traditional method of direct assembly with theoretical shock relations. Other boundary conditions and the solution process are also refined to specifically target the unsteady, shock-dominated flow. These modifications significantly alleviate the computational burden. The effectiveness of the proposed method is demonstrated through several test cases. To showcase the method's practical application, a scenario is simulated wherein an ellipse is dislodged from a wedge by an incident shock wave, covering a long distance. These tests confirm the method's feasibility in aerospace engineering problems.

A high-fidelity finite volume scheme for ideal magnetohydrodynamics equations using boundary variation diminishing algorithm

A high-fidelity finite volume scheme based on the BVD (boundary variation diminishing) principle is proposed in this study to solve ideal magnetohydrodynamics (MHD) equations. A hybrid spatial reconstruction profile, based on a quadratic polynomial and a steepness-adjustable hyperbolic tangent function, is adopted to reproduce accurate solutions of complex magnetohydrodynamics flows. The BVD principle is used to find an optimal combination of these two types of functions by comparing variations between reconstructed interface values. Non-physical oscillations around discontinuities are removed by switching the quadratic polynomial to the step-shaped function. Additionally, an eight-wave method is applied in this study to control divergence errors of the magnetic field. The widely used numerical tests in one- and two-dimensional cases were checked in this study. The proposed scheme can achieve the accuracy of a third-order linear scheme in convergence tests for both advection and MHD equations and capture strong MHD shock waves without spurious oscillations. In comparison with a third-order WENO (weighted essentially non-oscillatory) scheme, the proposed one gains more accurate solutions for not only strong discontinuities but also smooth structures across scales. To our knowledge, this is the first attempt to build a high-fidelity model for ideal MHD equations by the BVD algorithm. Numerical results demonstrate that the BVD algorithm has promising potential to build practical models for various MHD flows.

Upwind block-centered multistep differences for semiconductor device problem with heat and magnetic influences

In this paper, a mathematical model of semiconductor device with two important factors is discussed. Heat and magnetic influences are considered together. Its numerical approximation is important in information science and other technological innovation fields. The major physical unknowns (the potential, electron concentration, hole concentration, and the heat) are defined by four nonlinear PDEs: an elliptic equation, two convection-diffusion equations and a heat conductor equation. A conservative block-centered method is used to approximate the elliptic equation. The derivatives to time t, ∂∂t(⋅), are discretized by multistep differences for improving the computational accuracy. The diffusions and convection terms are treated by block-centered differences and upwind approximations, respectively. This composite numerical method is suitable for this nonlinear system. Firstly, numerical dispersion and nonphysical oscillations are avoided. Secondly, the unknowns and their adjoint vectors are obtained simultaneously. Thirdly, the law of conservation is preserved. An error estimates is derived. Finally, simple examples are given for showing the efficiency and accuracy.

An Improved PINN Algorithm for Shallow Water Equations Driven by Deep Learning

Solving shallow water equations is crucial in science and engineering for understanding and predicting natural phenomena. To address the limitations of Physics-Informed Neural Network (PINN) in solving shallow water equations, we propose an improved PINN algorithm integrated with a deep learning framework. This algorithm introduces a regularization term as a penalty in the loss function, based on the PINN and Long Short-Term Memory (LSTM) models, and incorporates an attention mechanism to solve the original equation across the entire domain. Simulation experiments were conducted on one-dimensional and two-dimensional shallow water equations. The results indicate that, compared to the classical PINN algorithm, the improved algorithm shows significant advantages in handling discontinuities, such as sparse waves, in one-dimensional problems. It accurately captures sparse waves and avoids smoothing effects. In two-dimensional problems, the improved algorithm demonstrates good symmetry and effectively reduces non-physical oscillations. It also shows significant advantages in capturing details and handling complex phenomena, offering higher reliability and accuracy. The improved PINNs algorithm, which combines neural networks with physical mechanisms, can provide robust solutions and effectively avoid some of the shortcomings of classical PINNs methods. It also possesses high resolution and strong generalization capabilities, enabling accurate predictions at any given moment.

Robust 3D multi‐material hydrodynamics using discontinuous Galerkin methods

AbstractA high‐order discontinuous Galerkin (DG) method is presented for nonequilibrium multi‐material () flow with sharp interfaces. Material interfaces are reconstructed using the algebraic THINC approach, resulting in a sharp interface resolution. The system assumes stiff velocity relaxation and pressure nonequilibrium. The presented DG method uses Dubiner's orthogonal basis functions on tetrahedral elements. This results in a unique combination of sharp multimaterial interfaces and high‐order accurate solutions in smooth single‐material regions. A novel shock indicator based on the interface conservation condition is introduced to mark regions with discontinuities. Slope limiting techniques are applied only in these regions so that nonphysical oscillations are eliminated while maintaining high‐order accuracy in smooth regions. A local projection is applied on the limited solution to ensure discrete closure law preservation. The effectiveness of this novel limiting strategy is demonstrated for complex three‐dimensional multi‐material problems, where robustness of the method is critical. The presented numerical problems demonstrate that more accurate and efficient multi‐material solutions can be obtained by the DG method, as compared to second‐order finite volume methods.

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.

A robust hybridizable discontinuous Galerkin scheme with harmonic averaging technique for steady state of real-world semiconductor devices

Solving real-world nonlinear semiconductor device problems modeled by the drift-diffusion equations coupled with the Poisson equation (also known as the Poisson-Nernst-Planck equations) necessitates an accurate and efficient numerical scheme which can avoid non-physical oscillations even for problems with extremely sharp doping profiles. In this paper, we propose a flexible and high-order hybridizable discontinuous Galerkin (HDG) scheme with harmonic averaging (HA) technique to tackle these challenges. The proposed HDG-HA scheme combines the robustness of finite volume Scharfetter-Gummel (FVSG) method with the high-order accuracy and hp-flexibility offered by the locally conservative HDG scheme. The coupled Poisson equation and two drift-diffusion equations are simultaneously solved by the Newton method. Indicators based on the gradient of net doping and solution variables are proposed to switch between cells with HA technique and high-order conventional HDG cells, utilizing the flexibility of HDG scheme. Numerical results suggest that the proposed scheme does not exhibit oscillations or convergence issues, even when applied to heavily doped and sharp PN-junctions. Devices with circular junctions and realistic doping profiles are simulated in two dimensions, qualifying this scheme for practical simulation of real-world semiconductor devices.

Denoising graph neural network based on zero-shot learning for Gibbs phenomenon in high-order DG applications

With the availability of high-performance computing technology and the development of advanced numerical simulation methods, Computational Fluid Dynamics (CFD) is becoming more and more practical and efficient in engineering. As one of the high-precision representative algorithms, the high-order Discontinuous Galerkin Method (DGM) has not only attracted widespread attention from scholars in the CFD research community, but also received strong development. However, when DGM is extended to high-speed aerodynamic flow field calculations, non-physical numerical Gibbs oscillations near shock waves often significantly affect the numerical accuracy and even cause calculation failure. Data driven approaches based on machine learning techniques can be used to learn the characteristics of Gibbs noise, which motivates us to use it in high-speed DG applications. To achieve this goal, labeled data need to be generated in order to train the machine learning models. This paper proposes a new method for denoising modeling of Gibbs phenomenon using a machine learning technique, the zero-shot learning strategy, to eliminate acquiring large amounts of CFD data. The model adopts a graph convolutional network combined with graph attention mechanism to learn the denoising paradigm from synthetic Gibbs noise data and generalize to DGM numerical simulation data. Numerical simulation results show that the Gibbs denoising model proposed in this paper can suppress the numerical oscillation near shock waves in the high-order DGM. Our work automates the extension of DGM to high-speed aerodynamic flow field calculations with higher generalization and lower cost.

An improved high-order low dissipation weighted compact nonlinear scheme for compressible flow simulations

Recently, Zhang developed a low-dissipation high-order scheme that employs Fu's targeted essentially non-oscillatory approach. this method incorporates an essentially non-oscillatory-like stencil selection strategy within the framework of the weighted compact nonlinear scheme proposed by Deng. Although this new scheme, termed targeted compact nonlinear scheme (TCNS), can recover ideal weights in smooth regions, it reduces accuracy at the discontinuities by discarding less smooth sub-stencils. Acker et al. showed in a recent study that increasing the weight of less smooth sub-stencils on a relatively coarse grid can further improve the wave resolution. Luo further improved Acker's scheme to fully exploit its potential in spectral characteristics. By following this idea, the less smooth stencil weights were incorporated to overcome problems of classical TCNS and to further improve the scheme's ability to resolve high-frequency waves. Moreover, by introducing an adaptive factor, we further optimized the spectral properties to improve the TCNS. A series of standard cases were used to measure the resolution of the new method for multiscale structures and the ability to suppress nonphysical oscillations at shock waves. The new scheme was applied to three cases involving complex configurations and multiscale flows, demonstrating that its low dissipation characteristics offer advantages in solving real-life problems.

A SUPS formulation for simulating unsteady natural/mixed heat convection phenomena in square cavities under intense magnetic forces

This paper presents stabilized finite element simulations of unsteady natural and mixed heat convection phenomena occurring in square enclosures. It is assumed that the enclosures are subject to intense uniform magnetic fields applied perpendicular to the vertical walls. It is well known that the classical Galerkin finite element method (GFEM) is prone to yielding nonphysical oscillations when simulating convection-dominated flows. Therefore, in order to handle such flows occurring at high Hartmann (0≤Ha≤100\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0 \\le {\\rm Ha} \\le 100$$\\end{document}) and Rayleigh (103≤Ra≤108\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10^{3} \\le {\\rm Ra} \\le 10^{8}$$\\end{document}) numbers, the stabilization of the GFEM is accomplished with the streamline-upwind/Petrov–Galerkin and pressure-stabilizing/Petrov–Galerkin formulations. Temporal discretizations are performed with the backward Euler formula. At each time step, the nonlinear systems are linearized with the Newton–Raphson (N–R) method, and the linearized systems are solved directly using the sparse lower–upper decomposition. The incompressible flow solvers are developed in-house, run in parallel within the FEniCS ecosystem, and tested on a comprehensive set of numerical experiments with various thermal boundary conditions. Numerical simulations and comparisons reveal that the proposed formulation performs quite well at high Hartmann and Rayleigh numbers without exhibiting any significant localized or globally spread numerical instabilities. Moreover, it achieves this by using uniform meshes and only linear elements, making the formulation much more computationally efficient. Mesh convergence analyses are also performed to demonstrate the reliability of the numerical results obtained.

A shock capturing artificial viscosity scheme in consistent with the compact high-order finite volume methods

This paper presents a shock capturing artificial viscosity scheme for the compact high-order finite volume methods in terms of the variational reconstructions on unstructured grids. The key for the design of the present artificial viscosity is the smoothness indicator, which is based on the concept of interfacial jump integration, measuring the discontinuities of the reconstruction polynomial and its spatial derivatives across a cell interface. Since the variational reconstruction is carried out by minimizing the functional in terms of the interfacial jump integration, the present smoothness indicator gives a discretization-consistent measurement of the smoothness of the flow fields that is sufficiently large in the region near discontinuities, and is in the same order of magnitude as the spatial truncation error of the finite volume scheme in smooth regions. These properties ensure that the newly developed artificial viscosity scheme has the problem-independent capability to suppress non-physical oscillations near discontinuities and preserve the theoretical order of accuracy for smooth flow. The shock capturing capability of the proposed artificial viscosity scheme has been demonstrated by a number of numerical examples confirming its essentially non-oscillatory and high-resolution properties. Additionally, the proposed artificial viscosity scheme exhibits higher computational efficiency than the approach based on a traditional limiter.

New limiter regions for multidimensional flows

Accurate transport algorithms are crucial for computational fluid dynamics and more accurate and efficient schemes are always in development. One dimensional limiting is commonly employed to suppress nonphysical oscillations. However, the application of such limiters can reduce accuracy. It is important to identify the weakest set of sufficient conditions required on the limiter as to allow the development of successful numerical algorithms.The main goal of this paper is to identify new less restrictive sufficient conditions for flux form in-compressible advection to remain monotonic. We identify additional necessary conditions for incompressible flux form advection to be monotonic, demonstrating that the Spekreijse limiter region is not sufficient for incompressible flux form advection to remain monotonic. Then a convex combination argument is used to derive new sufficient conditions that are less restrictive than the Sweby region for a discrete maximum principle. This allows the introduction of two new more general limiter regions suitable for flux form incompressible advection.

A locking-free numerical method for the quasi-static linear thermo-poroelasticity model

In this paper, we propose a locking-free numerical method to solve the quasi-static linear thermo-poroelasticity model, which exhibits two types of locking phenomena: Poisson locking and nonphysical oscillations. By analyzing the regularity of solution of the original model, we find that the Poisson locking is caused by divu≈0 as λ→+∞. Moreover, when discretizing the diffusion equations using backward Euler method for time, one can see that divu1≈0 for some special parameters. If we directly use the continuous Galerkin mixed finite element method (FEM) to solve the original model, the pressure and temperature fields exhibit numerical oscillations at early times. To overcome these two locking phenomena, we introduce a new variable ξ=αp+βT−λdivu to reformulate the original problem into a new one. This new problem includes a built-in mechanism to maintain stability for the continuous Galerkin mixed FEM. We further prove the existence and uniqueness of the weak solution by using the standard Galerkin method in conjunction with regularity estimates. We also design a fully discrete time-stepping scheme that employs mixed FEM with P2−P1−P1−P1 element pairs for the space variables and the backward Euler method for the time variable. Optimal convergence is demonstrated in both space and time. Finally, numerical examples are provided to demonstrate the optimal convergence rates of the variables and the robustness of the proposed method with respect to ν, and to verify the absence of the locking phenomenon.

A discontinuous Galerkin method for a coupled Stokes–Biot problem

The interaction between fluid and poroelastic structure is a complex problem that couples the Stokes equations with the Biot system. In this work, we rewrite the poroelastic equations using three fields (displacement, fluid pressure, and total pressure) to account for locking phenomenon in fluid–structure interaction and the oscillation of numerical solutions. Then, in order to solve the coupled system, we use the weighted discontinuous Galerkin finite element method for spatial discretization and propose a semi-discrete scheme. In addition, we propose a fully discrete scheme using the backward Euler’s method for temporal discretization. In the framework of the Galerkin approximation, the stability and errors of semi- and fully discrete schemes are analyzed using the theory of differential algebraic equations and weak compactness arguments. Through numerical experiments, we validate the theoretical rate of convergence, explain the behavior of the method in modeling the interaction between surface and groundwater hydrological systems, investigate the robustness of the method with respect to physical parameters, and simulate channel filtering. In particular, the numerical results are free of locking features, reducing non-physical numerical oscillations.

The application of a non-hydrostatic RANS model for simulating irregular wave breaking on a barred and sloping beach

In the current study, a non-hydrostatic two-dimensional vertical (2DV) numerical model with a shock-capturing technique is developed to simulate irregular wave breaking on a barred beach. The complete form of the 2DV Reynolds-averaged Navier-Stokes (RANS) equations is discretized using the finite volume approach. In the spatial discretization phase, a flux limiter function is employed for the velocity advection terms to prevent non-physical oscillations in regions with steep gradients. A novel method has been introduced during the discretization stage, leading to a significant reduction in computational expenses. For temporal discretization, the Leapfrog method, known for its second-order accuracy in time, is employed to address wave-damping issues. Model validation involves a comparison between simulation results and experimental data pertaining to irregular wave breaking, affirming the satisfactory performance of the model. To evaluate the accuracy of the model, the Root Mean Square Error (RMSE) was calculated. The assessment showed that the model is capable of estimating the significant wave height reported in the experiments used with an RMSE between 0.002 and 0.089, and the mean wave periods with an RMSE between 0.061 and 0.252.

A multislope MUSCL method for vectorial reconstructions

Variables interpolation is one of the key concepts of MUSCL schemes. Originally developed for one-dimensional frameworks, many improvements have been made over the last decades to extend these methods to general unstructured multi-dimensional meshes. It results that scalar interpolation in a finite volume cell-centered framework is already quite well understood. It is known that linear reconstructions have to be limited in order to prevent non-physical oscillations of the solutions while ensuring a spatial second-order accuracy for smooth solutions. This is done thanks to a limiting function which allows the reconstruction to satisfy a monotonicity property, which then ensure the stability of the scheme. Nevertheless, some difficulties arise when we try to extend this process to vectorial variables. Generally, vectorial reconstructions are done componentwise, but this process reveals to be frame-dependent and leads to a loss of precision due to false detection of extrema. In this paper, we present a new method dealing with vectorial reconstructions in a multislope MUSCL context.

Vector form of intrinsic finite element method for incompressible fluids

Vector form of intrinsic finite element (VFIFE) is a numerical method widely used in solid mechanics. However, it's hard to extend the VFIFE method to fluid mechanics since the traditional VFIFE method fails to reflect the analytical equilibrium of multiple variables in the continuum. Therefore, under the framework of analytical mechanics, this paper proposes Lagrange's equation of the second kind in fluid mechanics with the extremum condition of Lagrange power functional. And a vectorized motion equation of incompressible viscous fluids is deduced from Lagrange's equation. By using several efficient algorithms in the finite difference method (FDM) and the finite element method (FEM), the NS equation is decomposed into four governing equations of vector form for fluid mechanics. In addition, with the application of the classic Smagorinsky sub-grid scale model in large eddy simulation (LES), this paper puts forward turbulence modelling with VFIFE procedure, and a corresponding MATLAB program is developed. Two typical examples are given to demonstrate the applicability and efficiency of the proposed large eddy simulation with VFIFE method. The proposed algorithm can effectively eliminate the non-physical oscillation of the pressure, and obtain much accurate results with a small number of grids.

An enhanced compressible two-phase flow model with detailed chemistry under the adaptive mesh refinement frame

In the present study, an enhanced compressible two-phase flow model is advanced, considering the effect of chemical reactions within a detailed mechanism. In this model, two immiscible fluids (liquid and gaseous mixture) are accurately separated with the resolved interface. Unlike the classical five-equation two-phase flow model, the thermal properties of gases are no longer assumed to be constant but rather vary as functions of temperature. A modified mechanical relaxation procedure is proposed and employed at the gas-liquid interface to prevent the occurrence of nonphysical pressure oscillation. In the gaseous mixture, numerous gas components are included and resolved by their mass fraction among the gaseous mixture. In this model, the heat release effect is simulated by a detailed chemistry. Furthermore, the numerical results of several benchmark problems in one dimension and two dimensions demonstrate the efficacy of the proposed compressible multiphase flow model, such as the air shock tube, the gaseous detonation tube, the shock-droplet interaction, and especially the detonation-droplet interaction that has received little focused interest and investigations. Moreover, a self-developed adaptive mesh refinement strategy is performed for a high efficiency of numerical solving.