Nonphysical Solutions Research Articles

Proximal Gauss-Newton Method for Box-constrained Parameter Identification of a Nonlinear Railway Suspension System

The identification of railway vehicle components’ characteristics from measured data is a challenging task with compelling applications in health monitoring, fault detection, and system prognosis. Usually, though, such systems are highly nonlinear, and naive identification techniques may lead to unstable methods and inaccurate results. In this paper, we show that these issues can be easily tackled with the recently introduced proximal Gauss–Newton method, which we employ to identify the parameters of a railway nonlinear suspension system. In the proposed model, the parameters are subject to safety bounds in form of box constraints, which allows preventing nonphysical solutions. The suspension system we consider is highly nonlinear due to the presence of an airspring in the secondary suspension, which we introduce in a simplified Berg model. Numerical examples, featuring data corrupted by various noise levels, demonstrate the accuracy and efficiency of our proposed method. Comparisons with state-of-the-art approaches are also provided.

Finite Element Method on locally refined composite meshes for Dirichlet fractional Laplacian

It is known that the solution of the Dirichlet fractional Laplacian in a bounded domain exhibits singular behavior near the boundary. Consequently, numerical discretizations on quasi-uniform meshes lead to low accuracy and nonphysical solutions. We adopt a finite element discretization on locally refined composite meshes, which consist in a combination of graded meshes near the singularity and uniform meshes where the solution is smooth. We also provide a reference strategy on parameter selection of locally refined composite meshes. Numerical tests confirm that finite element method on locally refined composite meshes has higher accuracy than uniform meshes, but the computational cost is less than that of graded meshes. Our method is applied to discrete the fractional-in-space Allen–Cahn equation and the fractional Burgers equation with Dirichlet fractional Laplacian, some new observations are discovered from our numerical results.

Hermitian Dynamic Mode Decomposition - Numerical Analysis and Software Solution

The Dynamic Mode Decomposition (DMD) is a versatile and increasingly popular method for data driven analysis of dynamical systems that arise in a variety of applications in, e.g., computational fluid dynamics, robotics or machine learning. In the framework of numerical linear algebra, it is a data driven Rayleigh-Ritz procedure applied to a DMD matrix that is derived from the supplied data. In some applications, the physics of the underlying problem implies hermiticity of the DMD matrix, so the general DMD procedure is not computationally optimal. Furthermore, it does not guarantee important structural properties of the Hermitian eigenvalue problem and may return non-physical solutions. This paper proposes a software solution to the Hermitian (including the real symmetric) DMD matrices, accompanied with a numerical analysis that contains several fine and instructive numerical details. The eigenpairs are computed together with their residuals, and perturbation theory provides error bounds for the eigenvalues and eigenvectors. The software is developed and tested using the LAPACK package.

Positivity-Preserving Discontinuous Galerkin Methods on Triangular Meshes for Macroscopic Pedestrian Flow Models

The macroscopic models for solving the pedestrian flow problem can be generally classified into two categories as follows: first-order continuum models and high-order continuum models. In first-order continuum models, the density satisfies the mass conservation law, the speed is defined by a flow-density relationship, and the desired directional motion of pedestrians is determined by an Eikonal-type equation. In contrast, in high-order models, the velocity is governed by the momentum conservation law. In this study, we summarize existing first-order and high-order models and rewrite them in the form of unified scalar or system hyperbolic conservation laws. Next, we apply high-order discontinuous Galerkin methods with a positivity-preserving limiter on unstructured triangular meshes to solve the conservation law and a second-order fast-sweeping scheme to solve the Eikonal equations. Our method can efficiently model real-life complex computational regions and avoid nonphysical solutions and simulation blow-ups. Finally, numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed solution algorithm. The numerical results validate the reliability of the proposed numerical method and highlight the advantages of triangular meshes.

An Evolution Model for Polygonal Tessellations as Models for Crack Networks and Other Natural Patterns

We introduce and study a general framework for modeling the evolution of crack networks. The evolution steps are triggered by exponential clocks corresponding to local micro-events, and thus reflect the state of the pattern. In an appropriate simultaneous limit of pattern domain tending to infinity and time step tending to zero, a continuous time model, specifically a system of ODE is derived that describes the dynamics of averaged quantities. In comparison with the previous, discrete time model, studied recently by two of the present three authors, this approach has several advantages. In particular, the emergence of non-physical solutions characteristic to the discrete time model is ruled out in the relevant nonlinear version of the new model. We also comment on the possibilities of studying further types of pattern formation phenomena based on the introduced general framework.

Radial basis function-differential quadrature-based physics-informed neural network for steady incompressible flows

In this work, a radial basis function differential quadrature-based physics-informed neural network (RBFDQ-PINN) is proposed to simulate steady incompressible flows. The conventional physics-informed neural network (PINN) makes use of the physical equation as a constraint to ensure that the solution satisfies the physical law and the automatic differentiation (AD) method to calculate derivatives at collocation points. Although the AD-PINN is expedient in evaluating derivatives at arbitrary points, it is time-consuming with higher-order derivatives and may lead to nonphysical solutions with sparse samples. Alternatively, the finite difference (FD) method can facilitate the calculation of derivatives, but the FD-PINN will increase the computational cost when handling random point distributions, especially with higher-order discretization schemes. To address these issues, the radial basis function differential quadrature (RBFDQ) method is incorporated into the PINN to replace the AD method for the calculation of derivatives. The RBFDQ method equips with high efficiency in the calculation of high-order derivatives as compared with the AD method and great flexibility in the distribution of mesh points as compared with the FD method. As a result, the proposed RBFDQ-PINN is not only more efficient and accurate but also applicable to irregular geometries. To demonstrate its effectiveness, the RBFDQ-PINN is tested in sample problems such as the lid-driven cavity flow, the channel flow over a backward-facing step, and the flow around a circular cylinder. Numerical results reveal that the RBFDQ-PINN achieves satisfactory accuracy without any labeled collocation points, whereas the AD-PINN struggles to solve some cases, especially for high Reynolds number flows.

An efficient multiple harmonic balance method for computing quasi-periodic responses of nonlinear systems

Quasi-periodic responses composed of multiple base frequencies widely exist in science and engineering problems. The multiple harmonic balance (MHB) method is one of the most commonly used approaches for such problems. However, it is limited by low-order estimations due to complex symbolic operations in practical uses. Many variants have been developed to improve the MHB method, among which the time domain MHB-like methods are regarded as crucial improvements because of their high efficiency and simple derivation. But there is still one main drawback remaining to be addressed. The time domain MHB-like methods negatively suffer from non-physical solutions, which have been shown to be caused by aliasing (mixtures of the high-order into the low-order harmonics). Inspired by the collocation-based harmonic balancing framework recently established by our group, we herein propose a reconstruction multiple harmonic balance (RMHB) method to reconstruct the conventional MHB method using discrete time domain collocations. Our study shows that the relation between the MHB and time domain MHB-like methods is determined by an aliasing matrix, which is non-zero when aliasing occurs. On this basis, a conditional equivalence is established to form the RMHB method. Three numerical examples demonstrate that this new method is more robust and efficient than the state-of-the-art methods.

On the accuracy of power flow and load margin calculation caused by incorrect logical PV/PQ switching: Analytics and improved methods

Power Flow (PF) and Continuation Power Flow (CPF) are key tools for power system static calculation and analysis. This paper shows that the widely used logical PV/PQ switching method can lead to calculating non-physical power flow solutions and incorrect load margin calculations. From a computational viewpoint, several small and large numerical examples are presented to show the non-physical power flow solution and inaccurate load margin calculation made by the outer loop PF method. The existence of a non-physical power flow solution caused by logical PV/PQ switching and its impact on incorrect load margin calculation will be demonstrated. To address these issues, two smooth functions to approximate the generator Q-V curve and an active-set scheme will be presented. The Type-I smooth model is suitable for calculating physically feasible power flow solution, in addition it will be shown to contribute to faster convergence than the logical PV/PQ switching methods. The type-II smooth model is suitable for continuation power flow calculation for accurate load margin and speed improvement. A smooth-model-based active-set-assisted power flow method is then developed and its effectiveness in converging to the feasible (physical) power flow solution and the correct calculation of load margins will be shown. Numerical studies on several test systems were conducted to demonstrate the effectiveness of the proposed method in computing physical power flow solutions and achieving faster speed. For example, the proposed method for 70,000-bus PF computation is 1.7 times faster than that of the inner loop-based method and for the 25,000-bus CPF computation is about 3 times faster than that of the outer loop-based method.

An efficient bound-preserving and energy stable algorithm for compressible gas flow in porous media

Due to the great significance of the natural gas and shale gas, it is becoming increasingly important to simulate compressible gas flow in porous media. The model obeys an energy dissipation law as well as molar density must be positive and bounded in terms of the equation of state. For the purpose of eliminating nonphysical solutions as well as improving the stability in practical simulation, preservation of these properties is essential for a promising numerical method, but it is actually challenging due to the strong nonlinearity and complexity of the model. In this paper, we propose an efficient linearized numerical scheme that inherits the energy dissipation law as well as preserves the boundedness of molar density. Specifically, to treat the logarithmic type Helmholtz free energy density determined by the Peng-Robinson equation of state, we propose a novel adaptive stabilization approach involving the second derivatives of the convex energy terms. At each time step, the stabilization parameter is adaptively updated by a simple and explicit formula to ensure the energy dissipation law. The stabilized and linearized chemical potential allows to formulate the local mass conservation equation as an equivalent convection-diffusion form, and from this, an adaptive time step strategy is proposed to preserve the positivity and boundedness of molar density. The calculation of the time step size is fully explicit and easy to implement. Additionally, the fully discrete scheme is constructed using the conservative cell-centered finite difference method with the upwind strategy, and thus, it enjoys the local mass conservation. Numerical results are also presented to demonstrate the excellent performance of the proposed scheme.

A model of isotope transport in the unsaturated zone, case study

This work deals with a groundwater flow and solute transport model in the near-surface (predominantly unsaturated) zone. The model is implemented so that it allows simulations of contamination transport from a source located in a geological environment of a rock massif. The groundwater flow model is based on Richards’ equation. Evaporation is computed using the Hamon model. The transport model is able to simulate advection, diffusion, sorption and radioactive decay. Besides the basic model concept, the article also discusses potential cases that could lead to non-physical solutions. On three selected examples, which include, for example, rapidly changing boundary conditions, the article shows the solvability of such cases with the proposed model without unwanted effects, such as negative concentrations or oscillations of solution, that do not correspond to inputs.

An Optical Engine Used as a Physical Model for Studies of the Combustion Process Applying a Two-Color Pyrometry Technique

This work describes an experimental installation for the investigation of the combustion and injection processes. This installation is based on a two-stroke direct injection diesel engine with a total displacement of 3 L and a cylinder head equipped with three quartz windows. The windows are optical accesses that allow studying the process of injection, the atomization and evaporation of the fuel jet in an inert atmosphere (nitrogen), and the combustion process in a reactive atmosphere (ambient air). Additionally, the application of a two-color pyrometry technique to measure soot formation in this facility is presented. A methodological study is carried out regarding the influence of the dynamic range of the detectors and the wavelengths used. Maps of KL2C, flame temperature, and error probability are presented. The use of cameras with high dynamic range provides better results since the system seems to be less sensitive to measurement noise, and fewer points are obtained with a non-physical solution. Moreover, an appropriate combination of interference filters can improve the reliability of the solution. The greater the difference between the wavelengths of both interference filters, the fewer points with a non-physical solution, which improves the reliability of results.

Reconsideration of Nonphysical Solution in Jiles–Atherton Model for Real-Time Hysteresis Estimation

The Jiles–Atherton (JA) model is widely used in real-time hysteresis estimation for its better physical understanding and less experimental efforts. However, the inherent defects of JA model may lead to some nonphysical solutions, including negative differential permeability, non-closure of minor loops, and asymmetry of centered minor loops. To deal with such problems, a limiter for differential permeability and a scaling factor for loop closure are generally adopted. However, the existing methods may cause loops’ edges overlapping and cannot solve the asymmetry of centered minor loops. Therefore, a new equation for irreversible magnetization is proposed to solve the negative differential permeability and the edges overlapping. Also, to improve the fitting performance, the JA coefficients are adjusted for different fields and different loops’ turning points. In addition, a scaling factor for centered minor loops is developed to deal with the asymmetry. The factor only needs prior knowledge of minor loops’ starting points. Besides, a modified stack model is proposed, which can extract enough information for coefficients lookup and scaling factors calculation and can realize real-time hysteresis estimation for arbitrary field. Finally, the estimation results are verified by experiments on a toroidal transformer.

A spacetime formulation for unsteady aerodynamics with geometry and topology changes

AbstractA spacetime formulation is presented to solve unsteady aerodynamic problems involving large deformation or topological change such as store separation, slat and flap deployment or spoiler deflection. This technique avoids complex CFD meshing methods, such as Chimera, by the use of a finite-volume approach both in space and time, and permits a locally varying real timestep. The use of a central-difference scheme in the time direction can yield non-physical transient solutions as a consequence of information travelling backwards in time. Therefore, an upwind formulation is provided and validated against one-dimensional and two-dimensional test cases. A hybrid formulation (central in space, upwind in time) is also given and unsteady cases are computed for a spoiler and spoiler/flap deployment, with all three formulations compared, demonstrating that the use of an upwind time stencil yields more representative physical solutions and improves the rate of convergence.

Calibration of the Transitional Turbulence Model

In the present paper, the turbulent transition model from Langtry and Menter (“Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes,” AIAA Journal, Vol. 47, No. 12, 2009, pp. 2894–2906) is implemented in the structured finite volume Reynolds-averaged Navier–Stokes code FANSC and coupled with the shear-stress transport turbulence model. The model is modified and recalibrated to better match experimental results. The local turbulent intensity is replaced with a more consistent turbulent fluctuation representation to avoid nonphysical solutions at the stagnation point that would contaminate the downstream boundary-layer predictions. A calibration is devised by customizing several constants involved in the equation for the intermittency and the destruction term for the turbulent kinetic energy . An optimization algorithm is thus employed to facilitate the calibration on well-known two-dimensional test cases, such as the flat plates of the T3 series and of Schubauer and Klebanoff (“Contributions on the Mechanics of Boundary-Layer Transition,” NACA TN 3489, 1955), as well as the NACA 0012 airfoil. The calibrated model is then validated on the NLF-0416 and S809 airfoil cases that have not been involved in the optimization process.

Numerical solutions of a gradient-elastic Kirchhoff plate model on convex and concave geometries using isogeometric analysis

Abstract In this work, we study numerical solutions of a gradient-elastic Kirchhoff plate model on convex and concave geometries. For a convex plate, we first show the well-posedness of the model. Then, we split the sixth-order partial differential equation (PDE) into a system of three second-order PDEs. The solution of the resulting system coincides with that of the original PDE. This is verified with convergence studies performed by solving the sixth-order PDE directly (direct method) using isogeometric analysis (IGA) and the system of second-order PDEs (split method) using both IGA and C0 finite elements. Next, we study a concave pie-shaped plate, which has one re-entrant point. The well-posedness of the model on the concave domain is proved. Numerical solutions obtained using the split method differ significantly from that of the direct method. The split method may even lead to nonphysical solutions. We conclude that for gradient-elastic Kirchhoff plates with concave corners, it is necessary to use the direct method with IGA.

Data assimilation of volcanic aerosol observations using FALL3D+PDAF

Abstract. Modelling atmospheric dispersal of volcanic ash and aerosols is becoming increasingly valuable for assessing the potential impacts of explosive volcanic eruptions on buildings, air quality, and aviation. Management of volcanic risk and reduction of aviation impacts can strongly benefit from quantitative forecasting of volcanic ash. However, an accurate prediction of volcanic aerosol concentrations using numerical modelling relies on proper estimations of multiple model parameters which are prone to errors. Uncertainties in key parameters such as eruption column height and physical properties of particles or meteorological fields represent a major source of error affecting the forecast quality. The availability of near-real-time geostationary satellite observations with high spatial and temporal resolutions provides the opportunity to improve forecasts in an operational context by incorporating observations into numerical models. Specifically, ensemble-based filters aim at converting a prior ensemble of system states into an analysis ensemble by assimilating a set of noisy observations. Previous studies dealing with volcanic ash transport have demonstrated that a significant improvement of forecast skill can be achieved by this approach. In this work, we present a new implementation of an ensemble-based data assimilation (DA) method coupling the FALL3D dispersal model and the Parallel Data Assimilation Framework (PDAF). The FALL3D+PDAF system runs in parallel, supports online-coupled DA, and can be efficiently integrated into operational workflows by exploiting high-performance computing (HPC) resources. Two numerical experiments are considered: (i) a twin experiment using an incomplete dataset of synthetic observations of volcanic ash and (ii) an experiment based on the 2019 Raikoke eruption using real observations of SO2 mass loading. An ensemble-based Kalman filtering technique based on the local ensemble transform Kalman filter (LETKF) is used to assimilate satellite-retrieved data of column mass loading. We show that this procedure may lead to nonphysical solutions and, consequently, conclude that LETKF is not the best approach for the assimilation of volcanic aerosols. However, we find that a truncated state constructed from the LETKF solution approaches the real solution after a few assimilation cycles, yielding a dramatic improvement of forecast quality when compared to simulations without assimilation.

On Overcoming the Transverse Boundary Error of the SU/PG Scheme for Moving Conductor Problems

Conductor moving in magnetic field is quite common in electrical equipment. The numerical simulation of such problem is vital in their design and analysis of electrical equipment. The Galerkin finite element method (GFEM) is a commonly employed simulation tool, nonetheless, due to its inherent numerical instability at higher velocities, the GFEM requires upwinding techniques to handle moving conductor problems. The Streamline Upwinding/Petrov-Galerkin (SU/PG) scheme is a widely acknowledged upwinding technique, despite its error-peaking at the transverse boundary. This error at the transverse-boundary, is found to be leading to non-physical solutions. Several remedies have been suggested in the allied fluid dynamics literature, which employs non-linear, iterative techniques. The present work attempts to address this issue, by retaining the computational efficiency of the GFEM. By suitable analysis, it is shown that the source of the problem can be attributed to the Coulomb's gauge. Therefore, to solve the problem, the Coulomb's gauge is taken out from the formulation and the associated weak form is derived. The effectiveness of this technique is demonstrated with pertinent numerical results.

A New Numerical Technique for Index‐3 DAEs Arising from Constrained Multibody Mechanical Systems

Index‐3 differential‐algebraic equations (DAEs) are mathematical models for the dynamics of constrained multibody mechanical systems that arise in many applications. These DAEs are known to pose a challenge to numerical methods. The purpose of this paper is to propose a new approach to efficiently solve these equations. This approach relies on an effective combination of the power series method (PSM) with the Adomian polynomials. Here, the PSM is directly applied to these DAEs without using the usual index reduction techniques, which are costly and often lead to nonphysical solutions. We expand the nonlinear terms in a series form using the Adomian polynomials to overcome the limitation of the PSM in collecting the coefficients of the power series solution. This technique has led to a simple and efficient algorithm. The domain of convergence of the power series solution is expanded by developing a multistage PSM (MSPSM). To demonstrate the efficiency of the MSPSM and show its applicability, an index‐3 nonlinear DAE problem describing a two‐link planar robot arm is solved. The numerical results show that the MSPSM is a powerful tool for solving the index‐3 DAEs arising from constrained multibody mechanical systems.

Advancing Artificial Neural Network Parameterization for Atmospheric Turbulence Using a Variational Multiscale Model

AbstractData‐driven parameterizations offer considerable potential for improving the fidelity of General Circulation Models. However, ensuring that these remain consistent with the governing equations while still producing stable simulations remains a challenge. In this paper, we propose a combined Variational‐Multiscale (VMS) Artificial Neural Network (ANN) discretization which makes no a priori assumptions on the model form, and is only restricted in its accuracy by the precision of the ANN. Using a simplified problem, we demonstrate that good predictions of the required closure terms can be obtained with relatively compact ANN architectures. We then turn our attention to the stability of the VMS‐ANN discretization in the context of a single implicit time step. It is demonstrated that the ANN parameterization introduces nonphysical solutions to the governing equations that can significantly affect or prevent convergence. We show that enriching the training data with nonphysical states from intra‐time step iterations is an effective remedy. This indicates that the lack of representative ANN‐induced errors in our original, exact training inputs underpin the observed instabilities. In turn, this suggests that data set enrichment might aid in resolving instabilities that develop over several time steps.

Analysis of Frank-Kamenetskii thermal explosion system of non-class a geometry based on Boltzmann method

Exothermic reaction systems of non-class A geometries are very common, with an endless rectangular rod typical. As a strong nonlinear source word is included in the governing equation, which is sensitive to the frank-kamenetskii parameter, there is no analytical solution. Many methods were previously suggested. However, with them are often non-physical solutions obtained. In this paper, the lattice Boltzmann process provides us with complete physical and precise solutions. We also analysed the sensitivity of the strong nonlinear source term and suggested advice for similar numerical calculations and experiments with thermal explosion.