Runge-Kutta Solver Research Articles

A combined neural ODE-Bayesian optimization approach to resolve dynamics and estimate parameters for a modified SIR model with immune memory

We propose a novel hybrid approach that integrates Neural Ordinary Differential Equations (NODEs) with Bayesian optimization to address the dynamics and parameter estimation of a modified time-delay-type Susceptible-Infected-Removed (SIR) model incorporating immune memory. This approach leverages a neural network to produce continuous multi-wave infection profiles by learning from both data and the model. The time-delay component of the SIR model, expressed through a convolutional integral, results in an integro-differential equation. To resolve these dynamics, we extend the NODE framework, employing a Runge-Kutta solver, to handle the challenging convolution integral, enabling us to fit the data and learn the parameters and dynamics of the model. Additionally, through Bayesian optimization, we enhance prediction accuracy while focusing on long-term dynamics. Our model, applied to COVID-19 data from Mexico, South Africa, and South Korea, effectively learns critical time-dependent parameters and provides accurate short- and long-term predictions. This combined methodology allows for early prediction of infection peaks, offering significant lead time for public health responses.

Research on Hydrodynamics of Trans-Media Vehicles Considering Underwater Time-Varying Attitudes

A trans-media vehicle is a new type of equipment that can adapt to two environments, water and air, to maintain optimal hydrodynamic and aerodynamic performance. However, no matter what kind of trans-media vehicle, its dynamics are much more complicated when traversing the interface of the medium, as the parameters are time-varying due to the change in ambient medium and vehicle attitudes. In order to improve the stability and performance of the trans-media vehicle in complex environments, an accurate mathematical model is established to characterize the dynamics of the trans-media vehicle in this process in this study. The time-varying hydrodynamic coefficients with different attitudes or depths are obtained using computational fluid dynamics software. The mathematical model is solved iteratively using a Runge–Kutta solver to calculate the dynamic response. A prototype of trans-media vehicle is fabricated, and motion experiments are performed in the pool. The experimental results confirm the effectiveness of the established model and lay the foundation for further controller design, providing a reference for the dynamic modeling of other similar equipment operating in complex environments. The primary novelty of this study lies in the fact that the established dynamic model considers the complex interaction between the attitude of the trans-media vehicle and the inherent different properties of water and air and utilizes computational fluid dynamics software to accurately obtain time-varying coefficients under different attitudes and depths. This approach not only recognizes the criticality of orientation-dependent hydrodynamic coefficients but also incorporates their temporal variations, which were often overlooked in previous studies.

Onset of instability in Darcy–Forchheimer porous layer with power‐law saturating fluid

AbstractThe paper investigates the effects of the Forchheimer term (form drag) and vertical pressure gradient on the buoyancy‐induced instability of power‐law saturating fluid in a porous plane medium. Two isobaric permeable layers are assumed to sandwich the horizontal porous plane. In the meantime, Dirichlet and Neumann equations are the thermal boundary conditions considered for the lower and upper layers. A base flow developed analytically via the governing equations is just in function of the Péclet number , with no dependence on the characteristic parameter of the power law fluid. A linear stability analysis consists of substituting a base flow with a small perturbation into the governing equations leads to a four‐order eigenvalue problem. An analytical solution is performed for the asymptotic cases of an infinite wavelength. The Runge–Kutta solver is applied together with the shooting technique to evaluate numerical solutions for the general case of nonnegligible wave numbers. Among the findings is the contribution of the Forchheimer term in the variation of the threshold Péclet number whose value can switch the wave numbers from zero to nonzero and increase the stability of the system.

Physics knowledge-based transfer learning between buildings for seismic response prediction

The recent advance in deep learning has attracted considerable interest for employing the state-of-the-art methods to solve engineering problems. However, the applicability of machine learning based models is hindered by the high cost of big data acquisition and task-specific difficulties. This paper presents a framework of physics knowledge-based transfer learning (Phy-KTL) neural networks that integrates the powerful learning capacity of physics-informed neural networks (PINNs) and the flexible transferability of model-based transfer learning technique to enhance structural seismic response prediction in the context of limited labelled datasets. The leverage of physics knowledge (represented by Runge-Kutta solver) allows the neural networks to better capture the structural nonlinear pattern. The use of model-based transfer learning improves the model generality by transferring features extracted from the source building to target buildings. The effectiveness of Phy-KTL in predicting seismic responses between target buildings is numerically validated as compared with Data-driven neural networks, PINNs, and Data-based transfer learning (Data-KTL). A practical application, which uses Phy-KTL to transfer features extracted from the numerical model to the physical building tested on the shaking table, validates that Phy-KTL is robust and effective to improve seismic response prediction of target buildings with limited labelled data.

Efficient Runge–Kutta solver for stochastic crack growth analysis

Stochastic crack growth analysis by simulation may easily require a significant amount of computational effort. The Initial Value ordinary differential equations appearing in crack propagation problems may be solved by the Runge–Kutta (RK) family of solvers. However, crack propagation considering uncertainties imposes additional challenges for RK solvers: possibility of vectorized and/or parallelized computation of sample realizations; and adaptation of time discretization for each sample, to handle changes and discontinuities in the derivatives of crack growth curves, caused by changes in loads and in crack propagation rates. Existing adaptive RK methods, with change in time step length during crack growth computation, struggle with these challenges. In this setting, the present paper proposes a modified adaptive RK method which allows for simultaneous crack growth computations with different time-step discretizations, and aims at efficiently dealing with cases where discontinuities are present in the derivative of the crack growth curve. Application of the proposed method to three crack propagation examples, including one related to weld flaws in pipes, indicates that it is as accurate as other adaptive methods from the literature, while requiring only a fraction of the computational time.

On a sparse and stable solver on graded meshes for solving high-dimensional parabolic pricing PDEs

The purpose of this work is to investigate the multi-dimensional Black-Scholes partial differential equation with variable coefficients numerically. The problem is of practical importance due to option pricing at the presence of multi assets. Since by increasing the dimension, the curse of dimensionality restricts the computations, the proposed solver will be constructed based on sparse arrays. Toward this goal, fourth-order finite difference approximations on graded meshes are introduced and then employed through semi-discretization. Then a sixth-order Runge-Kutta solver is employed for finding the resolution of the derived set of ordinary differential equations. The stability of the proposed scheme is furnished in detail as well. Numerical testings are given to uphold the accuracy and efficacy of the proposed procedure.

Dynamical analysis of nonlinear fractional order Lorenz system with a novel design of intelligent solution predictive radial base networks

In this research work, the convective flow of Lorenz attractor in the fractional domain is modeled with a nonlinear flexible structure of radial basis neural network (RBNN). The physical parameters of the complex fractional-order system are initially computed with fractional order Runge–Kutta solver for different stochastic scenarios of control constraints and designed its parametric model with RBFN under various initial conditions of the chaotic Lorenz system. Multiple chaos of the Lorenz system with the dynamical neural network structure are computed for different fractional order solutions to analyze the sensitivity of chaotic behavior by means of Lyapunov exponents. Phase diagrams of the chaotic pattern show the fractional-order Lorenz system has compact dynamical behavior and has the potential for the design application of a real dynamical system. Embedded dimension and time delay of the fractional Lorenz system are computed using the average mutual information (AMI) technique to evaluate the suitability of the time delay chaotic pattern in the fractional domain, which is affected by the numerical algorithm and fluctuated on minute time step size. The proposed design of RBFN has been confirmed as an exceptionally high-performance tool for soft computing and dynamics analysis of chaotic systems in the fractional order domain.

Lie-group method solutions for a viscous flow in a dilating-squeezing permeable channel with velocity slip

The investigation focuses on the effects of wall velocity slip on the solution of a viscous, laminar, incompressible channel flow subjected to small-scaled contraction and expansion of the weakly permeable walls. The study of such flow systems is often contextual for fluid transport in biological organisms. In the considered flow configuration, the vertically moving porous walls enable the fluid to enter or exit with a constant rate. The tangential slip velocity of the flow at the porous walls is modeled with the Navier slip boundary condition. The flow dynamics inside the channel is governed by the full Navier–Stokes equations. The Lie symmetry analysis and the invariant method are adopted to reduce the number of independent variables in the system of governing equations. Consequently, a single fourth-order ordinary differential equation is obtained, which is solved analytically by the double perturbation method and the variation of iteration method. The solutions are compared for different arrangements. Furthermore, the approximated analytical solutions are likened to the numerical solutions obtained from a fourth-order Runge–Kutta solver embedding the Shooting method to check the accuracy. It is observed that the boundary layers are formed, and the flow rapidly turns near the walls, when suction and wall contraction coexist. Alternatively, if injection and wall expansion are paired, the flow adjacent to the walls is delayed. The existence of wall velocity slip advances the near-wall velocity and cuts down the speed of centerline velocity. It results in a change in the volumetric flow rate and shear rate. The overall pressure is also varied by higher wall velocity slip. The results are explored for different values of the permeation Reynold number and the dimensionless wall dilation rate to capture all possible impacts of the flow parameters. The current analysis rectifies the existing errors in the work of Boutros et al. [“Lie-group method solution for two-dimensional viscous flow between slowly expanding or contracting walls with weak permeability,”Appl. Math. Model. 31(6), 1092–1108 (2007)] with the no-slip boundary condition and discusses the overall influences of slip boundary condition on the Lie symmetry solution of the flow system.

Simulating supercontinua from mixed and cascaded nonlinearities

Nonlinear optical frequency conversion is of fundamental importance in photonics and underpins countless of its applications: Sum- and difference-frequency generation in media with quadratic nonlinearity permits reaching otherwise inaccessible wavelength regimes, and the dramatic effect of supercontinuum generation through cubic nonlinearities has resulted in the synthesis of broadband multi-octave spanning spectra, much beyond what can be directly achieved with laser gain media. Chip-integrated waveguides permit to leverage both quadratic and cubic effects at the same time, creating unprecedented opportunities for multi-octave spanning spectra across the entire transparency window of a nonlinear material. Designing such waveguides often relies on numeric modeling of the underlying nonlinear processes, which, however, becomes exceedingly challenging when multiple and cascading nonlinear processes are involved. Here, to address this challenge, we report on a novel numeric simulation tool for mixed and cascaded nonlinearities that uses anti-aliasing strategies to avoid spurious light resulting from a finite simulation bandwidth. A dedicated fifth-order interaction picture Runge–Kutta solver with adaptive step-size permits efficient numeric simulation, as required for design parameter studies. The simulation results are shown to quantitatively agree with experimental data, and the simulation tool is available as an open-source Python package (pychi).

Modelling and Validation of a Remotely Operated Towed Vehicle and computational cost analysis of Umbilical Cable

This paper deals with modelling of a remotely operated towed vehicle (ROTV) and attached umbilical cable, as a means of exploration and research in the ocean. Fossen's approach is used for modelling the ROTV using XFLR5 for airfoil analysis. The lumped mass-spring approach with fourth-order Runge-Kutta solver is used as umbilical cable model. It is shown that the mean square error exponentially increases when the segment length exceeds 4 meters, and the computation time is linearly increasing with the number of nodes. The results show that the umbilical cable model combined with the ROTV model can reproduce the output of the real data when taking modelling uncertainties into consideration.

Gradient descent algorithm to search for periodic [formula omitted]-body orbits

We describe a gradient-descent based algorithm to compute approximate periodic orbits of an N-body system. Given initial position and velocity vectors, the N-body system is numerically evolved using a Runge–Kutta solver which is end-to-end differentiable, such that loss gradients can be computed. The frame of the trajectories which is most similar to the starting conditions, approximately one orbital period after t=0, is computed, and the L1 norm between the initial conditions and the conditions of this frame is minimized. In this manner, the trajectories are optimized to exhibit periodic behavior. We apply this algorithm to the three-body problem to find quasi-stable orbits, but this algorithm can be applied to other chaotic systems.

Performance analysis of a tuned point absorber using SPH calm water and wave tank simulations

In this paper, two smoothed particle hydrodynamics (SPH) models, namely SPH-W and SPH-C were used to evaluate the motion response of a point absorber wave energy converter (WEC). In the SPH-W model, a long wave flume was constructed and a long simulation was performed to obtain the motion response of the WEC. In the SPH-C model, the SPH method is only used to find the hydrodynamic coefficients of the device by analysing a few seconds of free-decaying motion of WEC in calm water in a much smaller numerical flume. Then, these coefficients were inserted in the equation of motion of a heaving WEC that was solved using a 4th order Runge-Kutta (ODE45) solver in MATLAB. First, the energy conservation property of the WCSPH model was examined through a standing wave benchmark test. Then, the wave-point absorber interaction was simulated. While the simulation time for SPH-C model is much smaller than that of SPH-W, it gave almost similar results for the motion response of WEC. These two models were used to evaluate the effects of the control force and the draft of a cone-cylinder point absorber on its hydrodynamic responses. The results showed that compared to the effect of the supplementary inertia, changes in the draft of the WEC have a small influence on its hydrodynamic responses. The buoy draft has an inverse relationship with both added mass and damping coefficients. However, increasing the supplementary mass increases the added mass and decreases the hydrodynamic damping coefficients.

Design of intelligent computing networks for nonlinear chaotic fractional Rossler system

In a recent development, the legacy of fractional-order multi-dimensional chaotic systems has attracted numerous researchers owing to their valuable applications in cryptology, control, information security, mathematics, and physical sciences. In this study, strength of artificial intelligence (AI) algorithms are explored for stochastic numerical computing for nonlinear fractional Rossler differential system by introducing state of art transformation in the radial basis neural networks (RNFN). Fractional order complex chaotic dynamical system of Rossler model represented by nonlinear coupled fractional differential equations are numerically solved with state of art fourth order fractional Runge-Kutta solver for different scenarios of control parameter and these outcomes are mensurate as reference solutions to model implementing RBFN with the different initial conditions of the chaotic dynamics. The innovative transformations are introduced to enhance the fast convergence of machine learning RBFN algorithm achieved through bimodal chaotic parameters. Performance of designed computing RBFN is authenticated efficaciously by RMSE metric for fractional order chaotic model of Rossler attractor with outcomes matching of the order of 10 to 15 decimal places in term of accuracy as compared to standard results computed from fractional Runge-Kutta solver. It is believed the proposed AI knacks based research work will open new innovative path in fractional order modelling and analysis of natural chaotic dynamic systems.

Efficient momentum conservation constrained PDE-LDDMM with Gauss–Newton–Krylov optimization, Semi-Lagrangian Runge–Kutta solvers, and the band-limited parameterization

This paper proposes three efficient variants of Jacobi EPDiff PDE-LDDMM, where efficiency is achieved through Semi-Lagrangian Runge–Kutta integration and the band-limited parameterization. During Gauss–Newton–Krylov optimization, the method computes the gradient and the Hessian-vector product on the final time sample, and transports these magnitudes towards the initial time using the adjoint Jacobi equations and their incremental counterparts. Then, the optimization is performed on the initial time sample. The proposed methods have effectively achieved a considerable reduction of the computational complexity at a competitive accuracy. These variants constitute a contribution to the efficient computation of diffeomorphisms belonging to geodesics, suitable for statistical shape analysis or the construction of transversal and longitudinal models of shape variability using Principal Geodesic Analysis and Geodesic Regression in spaces of diffeomorphisms.

Higher‐order explicit schemes based on the method of characteristics for hyperbolic equations with crossing straight‐line characteristics

AbstractWe develop method of characteristics schemes based on explicit Runge–Kutta and pseudo‐Runge–Kutta third‐ and fourth‐order solvers along the characteristics. Schemes based on Runge–Kutta solvers are found to be strongly unstable for certain physics‐motivated models. In contrast, schemes based on pseudo‐Runge–Kutta solvers are shown to be only weakly unstable for periodic boundary conditions and essentially stable for the more physically relevant nonreflecting boundary conditions. Our implementation of nonreflecting boundary conditions does not rely on interpolation.

A Fast Time-Step Selection Method for Explicit Solver-Based Simulation of High Frequency Low Loss Circuit and Its Application on EMI Filter

Electromagnetic interference (EMI) modeling and prediction are essential for the design of most power electronics apparatuses. This article aims at finding a fast method to select time step for explicit solver-based simulation of high frequency low loss (HFLL) circuits like EMI filter. The state-space model of HFLL circuit is constructed and its eigenvalues are proved to be very close to the imaginary axis. Both the nondegenerate and degenerate circuit cases are discussed. During the analysis, a circuit lemma is summarized on how to transform degenerate circuit into nondegenerate circuit and the corresponding inversion of its coefficient matrix is derived based on Sherman-Morrison's formula. Then the Laguerre-Samuelson's inequality is employed to find the upper bound of HFLL circuit's eigenvalues. This process only requires two matrix multiplications and traces of the matrix operation results, thus keeping the computational complexity retaining in O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). A typical EMI filter is constructed and its equivalent circuit including the parasitic effects is extracted from ANSYS. This filter is simulated in application between a dc/ac converter and the grid using the fourth-order Runge-Kutta (RK4) solver with a time step selected by the proposed method. Numerical test shows that the spectrum results are very close to those obtained by experiment while being much more efficient than traditional methods, which demonstrates that this time-step selection method could benefit the analysis and time-domain simulation of HFLL circuits.

An enhanced sequential fully implicit scheme for reservoir geomechanics

In this paper, it is proposed an enhanced sequential fully implicit (ESFI) algorithm with a fixed stress split to approximate robustly poro-elastoplastic solutions related to reservoir geomechanics. The constitutive model considers the total strain effect on porosity/permeability variation and associative plasticity. The sequential fully implicit (SFI) algorithm is a popular solution to approximate solutions of a coupled system. Generally, the SFI consists of an outer loop to solve the coupled system, in which there are two inner iterative loops for each equation to implicitly solve the equations. The SFI algorithm occasionally suffers from slow convergence or even convergence failure. In order to improve the convergence (robustness) associated with SFI, a new nonlinear acceleration technique is proposed employing Shanks transformations in vector-valued variables to enhance the outer loop convergence, with a quasi-Newton method considering the modified Thomas method for the internal loops. In this ESFI algorithm, the fluid flow formulation is defined by Darcy’s law including nonlinear permeability based on Petunin model. The rock deformation includes a linear part being analyzed based on Biot’s theory and a nonlinear part being established using Mohr-Coulomb associative plasticity for geomechanics. Temporal derivatives are approximated by an implicit Euler method, and spatial discretizations are adopted using finite element in two different formulations. For the spatial discretization, two weak statements are obtained: the first one uses a continuous Galerkin for poro-elastoplastic and Darcy’s flow; the second one uses a continuous Galerkin for poro-elastoplastic and a mixed finite element for Darcy’s flow. Several numerical simulations are presented to evaluate the efficiency of ESFI algorithm in reducing the number of iterations. Distinct poromechanical problems in 1D, 2D, and 3D are approximated with linear and nonlinear settings. Where appropriate, the results were verified with analytic solutions and approximated solutions with an explicit Runge-Kutta solver for 2D axisymmetric poro-elastoplastic problems.

Numerical Modeling Methods for Large Open Loop Multibody System

A pendulum’s motion was stated to be as a way to illustrate the movement of human body in the studies of multibody system. Therefore, a comparison between the two numerical models in multibody systems were implemented on the articulated pendulums of different sizes. The two numerical models were known as the augmented Lagrangian formulation and fully recursive method. In order to identify the difference performance of the numerical models, various size of articulated pendulums has been tested which are 2, 4, 8, 16, 20 and 40 pendulums. Differential equations developed from both models were solved by using Runge-Kutta 4 and 5. Both models were coded in Matlab and have been optimized in order to ensure only related routine were considered. The performance was evaluated based on the computing time with constant relative and absolute tolerance in Runge-Kutta solver which is 0.01 s. All pendulums were assumed to have the same weight, angle and length. As for the results, the augmented Lagrangian formulation solved the differential equations faster than the fully recursive method when tested up to 20 pendulums. However, fully recursive method started to solve the differential equations faster than the augmented Lagrangian method when it need to deal with a very large system such as 40 pendulums and above. Thus, it can be concluded that the suitable method to solve the small, open loop system such as articulated pendulums is augmented Lagrangian method while for a very large system, the fully recursive method will be more efficient.

Development of a diesel engine’s digital twin for predicting propulsion system dynamics

A digital twin is the essential part of a recent and unavoidable trend in ship operation digitalisation. The digital twin is a virtual replica of real ship or a particular system that coexists with its physical counterpart and maps the dynamic behaviour in real-time. Thus, the digital twin combines physical space real-time data with a set of dynamic models representing the physical counterpart in the cyberspace. The problem of digital twin development is a trade-off between insight into the dynamic process and real-time execution constraint. This paper describes a modelling approach that combines continuous time-domain cycle-mean value engine model with the crank-angle resolved phenomenological combustion model, satisfying the real-time execution constraint. The set of conservation laws, notably energy and mass, supplemented with the phenomenological Wiebe combustion model, is treated in the integral form allowing transformation into a set of nonlinear algebraic equations. The solution of the resulting system exhibits fast speed and accuracy as compared with the traditional approach combining differential equations and Runge-Kutta solver.

Numerical Solution of the Duffin’s Equation based on 4th Order Runge-Kutta Solver, with and without Taylor’s Approximation

Numerical Solution of the Duffin’s Equation based on 4th Order Runge-Kutta Solver, with and without Taylor’s Approximation