Nonlinear Dynamic Problems Research Articles

Piecewise linear value function approximations in nonlinear dynamic scheduling problems with VTOLs

Modern vertical take-off and landing vehicles (VTOLs) could significantly affect the future of mobility. The broad range of their application fields encompasses urban air mobility, transportation and logistics as well as reconnaissance and observation missions in a military context. This article presents an efficient numerical framework for dynamic scheduling of multiple VTOLs. It combines a scheduling problem with optimal trajectory planning. The whole setting is formulated as a mixed-integer bilevel optimization problem. At the upper level, VTOLs are scheduled, and their starting times are computed. The solution of the lower level problem involves the computation of a value function and yields optimal trajectories for every aerial vehicle. In order to solve the bilevel problem, it is recast into a single-level one. The resulting mixed-integer nonlinear program is then piecewise linearized and solved numerically by a mixed-integer linear solver based on the Branch-and-Bound algorithm. Numerical results prove the feasibility of the approach proposed herein.

An Advanced Hybrid Model Based On Stochastic - Eulerian Numerical Approach: Application To Atmospheric Pollution

In this paper, we propose for the first time to the best of our knowledge, extend the application of a stochastic Eulerian numerical approach based on the Extended Kalman Filter (EKFE.N.M.) to address the limitations of the Eulerian air pollution model CHIMERE. This approach integrates a comprehensive set of processes, including advection, turbulence, chemical reactions, emissions, and deposition, to model the dynamics of pollutant mass concentration. The EKF technique is employed to transform nonlinear dynamic problems into a succession of locally linearized ones, which are then used to estimate system states and adjust pollutant concentrations based on measured data. This stochastic approach is tested through two scenarios: one without external forces or control terms, and another that incorporates external factors like temperature, wind speed, and nitrogen dioxide as ozone precursors. A comparison of the obtained results with those from the standard CHIMERE model and studies from the literature demonstrates the accuracy and effectiveness of the proposed method.

Physics‐Informed Active Learning With Simultaneous Weak‐Form Latent Space Dynamics Identification

ABSTRACTThe parametric greedy latent space dynamics identification (gLaSDI) framework has demonstrated promising potential for accurate and efficient modeling of high‐dimensional nonlinear physical systems. However, it remains challenging to handle noisy data. To enhance robustness against noise, we incorporate the weak‐form estimation of nonlinear dynamics (WENDy) into gLaSDI. In the proposed weak‐form gLaSDI (WgLaSDI) framework, an autoencoder and WENDy are trained simultaneously to discover intrinsic nonlinear latent‐space dynamics of high‐dimensional data. Compared with the standard sparse identification of nonlinear dynamics (SINDy) employed in gLaSDI, WENDy enables variance reduction and robust latent space discovery, therefore leading to more accurate and efficient reduced‐order modeling. Furthermore, the greedy physics‐informed active learning in WgLaSDI enables adaptive sampling of optimal training data on the fly for enhanced modeling accuracy. The effectiveness of the proposed framework is demonstrated by modeling various nonlinear dynamical problems, including viscous and inviscid Burgers' equations, time‐dependent radial advection, and the Vlasov equation for plasma physics. With data that contains 5%–10 Gaussian white noise, WgLaSDI outperforms gLaSDI by orders of magnitude, achieving 1%–7 relative errors. Compared with the high‐fidelity models, WgLaSDI achieves 121 to 1779 speed‐up.

A hybrid quantum-classical framework for computational fluid dynamics

Recent advancements in quantum computing provide opportunities to address the challenges of computational resource limitations in computational fluid dynamics (CFD). This work presents a hybrid quantum-classical CFD framework that leverages quantum linear algorithms to practical flow simulations. By transforming nonlinear fluid dynamics problems into linear systems, the framework employs quantum linear algorithms to compute solutions, effectively integrating quantum and classical computing approaches. Furthermore, this framework employs subspace methods to map the original large-scale linear systems to small ones solved by quantum linear algorithms, thereby enabling the solution of large-scale problems using the currently limited quantum resources. We applied the famous Harrow–Hassidim–Lloyd (HHL) algorithm and variational quantum linear solver to simulate complex flows, including unsteady flows around a cylinder, simulation of an aircraft with turbulence model, and combustion flows. The performance and quantum resource consumption of these algorithms were evaluated in practical flow scenarios, demonstrating their effectiveness with an average relative error lower than 0.001%. We conducted quantum resource and convergence analyses to adapt this framework for use with near-term quantum computers. Our framework supports simulations of up to 9.4 × 106 grid cells, marking a significant advancement from previous quantum approaches limited to simple flows. This paper not only offers a way for employing quantum linear algorithms in solving complex flow problems but also provides insights into optimizing quantum algorithms for CFD applications, pushing forward the practical utilization of quantum CFD.

Path Tracking Control for Four-Wheel Independent Steering and Driving Vehicles Based on Improved Deep Reinforcement Learning

We propose a compound control framework to improve the path tracking accuracy of a four-wheel independent steering and driving (4WISD) vehicle in complex environments. The framework consists of a deep reinforcement learning (DRL)-based auxiliary controller and a dual-layer controller. Samples in the 4WISD vehicle control framework have the issues of skewness and sparsity, which makes it difficult for the DRL to converge. We propose a group intelligent experience replay (GER) mechanism that non-dominantly sorts the samples in the experience buffer, which facilitates within-group and between-group collaboration to achieve a balance between exploration and exploitation. To address the generalization problem in the complex nonlinear dynamics of 4WISD vehicles, we propose an actor-critic architecture based on the method of two-stream information bottleneck (TIB). The TIB method is used to remove redundant information and extract high-dimensional features from the samples, thereby reducing generalization errors. To alleviate the overfitting of DRL to known data caused by IB, the reverse information bottleneck (RIB) alters the optimization objective of IB, preserving the discriminative features that are highly correlated with actions and improving the generalization ability of DRL. The proposed method significantly improves the convergence and generalization capabilities of DRL, while effectively enhancing the path tracking accuracy of 4WISD vehicles in high-speed, large-curvature, and complex environments.

A modified neural network method for computing the Lyapunov exponent spectrum in the nonlinear analysis of dynamical systems

This study presents a modified neural network method for computing the Lyapunov exponent spectrum in non-linear dynamical systems. Its mathematical description is an introduction. The proposed modified neural network method allows the addition of bias and constant neurons to the neural network topology and the use of different numbers of activation functions to suit different cases. Various algorithms for computing Lyapunov exponents, such as the Benettin method, the Wolf method, the Rosenstein method, the Kantz method, the Synchronisation method, the Sano-Sawada algorithm and the proposed modification of the neural network method, are used for classical problems in nonlinear dynamics. These problems include the generalised Hénon map, the chaotic attractor of the Baier-Klein map, and the vibrations of mechanical systems such as the flexible Bernoulli-Euler beam and the flexible functionally graded porous closed cylindrical shell under alternating load. The comparative analyses presented in this study are aimed at validating the accuracy and effectiveness of the methods, and at identifying the most relevant approaches for different types of systems and classes of problems. The proposed method is demonstrated to be superior to existing methods based on time-series evaluation in terms of sample size and accuracy. Furthermore, it does not require the initial system equations.

Nonlinear model predictive control for maximum wind energy extraction of semi-submersible floating offshore wind turbine based on simplified dynamics model

The application of floating offshore wind turbines means better exploitation of offshore wind resources. However, the six-degree of freedom motion characteristics of floating platforms bring greater challenges to control system design. Based on the dynamic characteristics of semi-submersible floating offshore wind turbines, this paper establishes a simplified nonlinear dynamic model for control system design. On this basis, a complete framework for nonlinear model predictive control of maximum wind energy extraction for semi-submersible floating offshore wind turbines considering wind and wave disturbances is developed. Based on the previewed wind and wave, a dynamic optimization problem with both state and control constraints is constructed, considering maximum wind energy extraction and torque fluctuation. Then, an improved equilibrium optimizer is proposed to address the nonlinear non-convex dynamic optimization problems, which achieves a better trade-off between exploitation and exploration. Simulation results verify the superiority of the proposed nonlinear model predictive control framework via the improved equilibrium optimizer, and the influences of different control algorithms on the platform motion state, power coefficient, and equivalent wind speed are analyzed.

Uncertainty quantification analysis of bifurcations of the Allen–Cahn equation with random coefficients

In this work we consider the Allen–Cahn equation, a prototypical model problem in nonlinear dynamics that exhibits bifurcations corresponding to variations of a deterministic bifurcation parameter. Going beyond the state-of-the-art, we introduce a random coefficient in the linear reaction part of the equation, thereby accounting for random, spatially-heterogeneous effects. Importantly, we assume a spatially constant, deterministic mean value of the random coefficient. We show that this mean value is in fact a bifurcation parameter in the Allen–Cahn equation with random coefficients. Moreover, we show that the bifurcation points and bifurcation curves become random objects. We consider two distinct modeling situations: (i) for a spatially-homogeneous coefficient we derive analytical expressions for the distribution of the bifurcation points and show that the bifurcation curves are random shifts of a fixed reference curve; (ii) for a spatially-heterogeneous coefficient we employ a generalized polynomial chaos expansion to approximate the statistical properties of the random bifurcation points and bifurcation curves. We present numerical examples in 1D physical space, where we combine the popular software package Continuation Core and Toolboxes (COCO) for numerical continuation and the Sparse Grids Matlab Kit for the polynomial chaos expansion. Our exposition addresses both dynamical systems and uncertainty quantification, highlighting how analytical and numerical tools from both areas can be combined efficiently for the challenging uncertainty quantification analysis of bifurcations in random differential equations.

Design and Simulation Analysis of Bridge Anti-collision Structure based on Nonlinear Numerical Simulation

In order to solve the dynamic nonlinear problem of bridge loads and responses during ship collisions, a design method for bridge anti-collision structures based on nonlinear numerical simulation was proposed. The author describes in detail the entire process of collision force evolution, energy conversion, and plastic deformation of the anti-collision energy dissipator, and conducts a comprehensive simulation of it. The experimental results show that when a ship with a mass of 1000 tons collides forward at speeds of 1, 3, and 5 meters per second, the collision depth is 0.23, 1.46, and 3.95 meters, respectively, less than the maximum allowable collision depth of 4.3 meters, and the collision energy dissipator is still in the protective working state for the bridge pier. When a ship with a mass of 3000 tons collides with the collision avoidance energy dissipator at a speed of 3 or 5 meters per second, the collision depth exceeds the maximum allowable collision depth, and the collision avoidance energy dissipator fails, the ship will directly collide with the wharf. The plastic deformation of the anti-collision energy dissipator provided has important reference value for design.

Adaptive formation control for obstacle avoidance of USVs with asymmetric input saturation

Due to the complex maritime navigation environment, Unmanned Surface Vessels (USVs) are influenced by unknown nonlinear dynamics arising from external disturbances and internal uncertainties. Achieving effective formation control while maintaining obstacle avoidance performance presents significant challenges. This article proposes a Neural Networks (NNs) adaptive formation Artificial Potential Field (APF) obstacle avoidance control method for multiple USVs. By employing online updates of Radial Basis Function (RBF) NNs technology, the unknown nonlinear dynamics are approximated, thus addressing complex nonlinear dynamics problems. In scenarios involving multiple USVs navigating under high wind and wave conditions, collisions with obstacles frequently occur. To tackle this issue, a leader-follower control strategy is designed that effectively addresses risk assessment and obstacle avoidance under such challenging conditions. Additionally, to account for saturation constraints or potential faults in the controller inputs commonly encountered in engineering applications, it implements an asymmetric auxiliary control system. Furthermore, the Lyapunov stability theorem is utilized to ensure the stability of both the formation control and obstacle avoidance algorithms for multiple USVs. Finally, the effectiveness of the proposed algorithm is validated through simulations.

Nonlinear dynamic analyses of sandwich porous FG cylindrical shells with dual-layered FG porous cores

This research examines the nonlinear vibrations and dynamic postbuckling (DPB) analyses of sandwich porous functionally graded (SPFG) cylindrical shells with dual-layered FG porous (FGP) cores exposed to external excitation, using a semi-analytical method. The study investigates two types of FG layers: one with evenly distributed porosities (FG-EPD) and another with unevenly distributed porosities (FG-UEPD). It also studies the FGP cores in four configurations: one featuring symmetric porosity distribution with stiffening in the surface areas (SPD-Stiff), another with softening in the surface areas (SPD-Soft), a third with nonsymmetric porosity distribution (NSPD), and a fourth with uniform porosity distribution (UPD). Therefore, the SPFG cylindrical shells with dual-layered FGP cores with eight different configurations are investigated. Utilizing the classical shell theory (CST) alongside the geometrical nonlinearity in von Kármán–Donnell framework, and Galerkin’s method, this study addresses the nonlinear dynamic problem. An approximate solution for the deflection shape using three terms is selected, and the relationship between frequency and amplitude in nonlinear vibration is clearly defined. The nonlinear dynamic behaviors including vibration and DPB responses are examined using the P-T method, which is named for its use of the piecewise constant argument in conjunction with the Taylor series expansion. The critical dynamic buckling load of SPFG cylindrical shells with dual-layered FGP cores is investigated via the Budiansky–Roth criterion.

Dynamical system prediction from sparse observations using deep neural networks with Voronoi tessellation and physics constraint

Despite the success of various methods in addressing the issue of spatial reconstruction of dynamical systems with sparse observations, spatio-temporal prediction for sparse fields remains a challenge. Existing Kriging-based frameworks for spatio-temporal sparse field prediction fail to meet the accuracy and inference time required for nonlinear dynamic prediction problems. In this paper, we introduce the Dynamical System Prediction from Sparse Observations using Voronoi Tessellation (DSOVT) framework, an innovative methodology based on Voronoi tessellation which combines convolutional encoder–decoder (CED) and long short-term memory (LSTM) and utilizing Convolutional Long Short-Term Memory (ConvLSTM). By integrating Voronoi tessellations with spatio-temporal deep learning models, DSOVT is adept at predicting dynamical systems with unstructured, sparse, and time-varying observations. CED-LSTM maps Voronoi tessellations into a low-dimensional representation for time series prediction, while ConvLSTM directly uses these tessellations in an end-to-end predictive model. Furthermore, we incorporate physics constraints during the training process for dynamical systems with explicit formulas. Compared to purely data-driven models, our physics-based approach enables the model to learn physical laws within explicitly formulated dynamics, thereby enhancing the robustness and accuracy of rolling forecasts. Numerical experiments on real sea surface data and shallow water systems clearly demonstrate our framework’s accuracy and computational efficiency with sparse and time-varying observations.

The α-generalized implicit method associated with Potra–Pták iteration for solving non-linear dynamic problems

Generally, direct time integration procedures are used for solving the equations of motion in transient analysis of structures with large displacements. In this context, we propose an algorithm that combines the α-Generalized implicit integration method with the Potra–Pták two-step iterative scheme. The free Scilab program develops a computer code for the non-linear dynamic analysis of plane frames with large displacements and rotations. The FEM corotational formulation discretizes the structures considering the Euler-Bernoulli beam theory. The null-length connection element described by the axial, translational and rotational stiffnesses simulate the behavior of the beam-column connection. Jacobi’s method and the Scilab’s spec function determine the natural frequencies. The developed program is used for modal and transient dynamic analyses of frame problems available in the literature. The numerical results show that the Potra-Pták scheme obtains approximate solutions with fewer cumulative iterations until convergence and shorter processing time compared to the standard Newton-Raphson scheme. Further-more, the results show that the type of beam-column connection affects the vibratory behavior of the structure as well as the values of its natural frequencies.

Thermomechanical responses of steel frames under aftershock after fires

A novel advanced fiber beam-column element, which incorporates stability functions and the fiber distributed plasticity model to capture geometric and material nonlinearities, respectively, has been successfully developed to predict the nonlinear inelastic thermo-mechanical behaviors of steel frames subjected to aftershock after fires. To address the nonlinear dynamic problems resulting from aftershocks after fires, a sequential nonlinear thermal incremental-iterative solution scheme has been developed, which is based on the Newton-Raphson algorithm and Newmark-β method. In addition, a transcendental probability-based method is employed to express the statistical characteristics of aftershock intensity relative to the mainshock. The proposed method is applied to analyze three framed structures: a steel portal frame, a two-story plane steel frame, and a four-story asymmetric space steel frame. This analysis shows the accuracy, computational performance, and applicability of the proposed method. Remarkably, the results obtained from the code work, using just one or two elements per member, are in close agreement with all the results analyzed by Abaqus. This emphasizes that the proposed method effectively predicts the response of framed structures under aftershock after fires with excellent computational efficiency, significantly surpassing traditional finite element approaches in commercial software like Abaqus. The successful results of the dynamic analysis of the frames under aftershocks following fires underscore the importance of considering aftershocks after a fire incident. It is demonstrated that the occurrence of an aftershock accelerates the failure of the frame compared to the case when the frame is subjected to fire alone. This highlights the critical need to account for aftershocks when assessing the safety and structural integrity of framed structures after a fire event.

Regression based stochastic energy management strategy for a grid connected microgrid using Artificial Electric Field Algorithm

Recent decade has witnessed steep technological advancement in the renewable energy sector due to growing concern of climate change and emission cut target. The lack of availability of fossil fuels and the high price of the same have made the use of Renewable Energy Sources (RES) as a promising approach for generating clean and sustainable energy at local level. This persuades the implementation of Microgrid (MG) supported with RES, to cater various types of load demand. However, planning and optimizing a MG operation have become challenging due to the intermittent nature of RES and uncertain loads. To address this issue, this paper introduces a Stochastic Energy Management Strategy (SEMS) for a grid connected MG comprising Photovoltaic, Wind Turbine, Fuel Cell, Micro Turbine, Battery Energy Storage and electrical as well as heat energy demand. The volatilities of RES power output and energy demand are modeled via Gaussian Process Regression (GPR) and Monte Carlo Simulation (MCS) respectively. The problem is framed as non-linear (NL) dynamic stochastic optimization problem and solved using ‘Artificial Electric Field Algorithm (AEFA)’. Here, minimization of expected value of operational and emission cost are taken as objective functions; while various practicality aspects are modeled as constraints in the problem. Further, the efficacy of the proposed approach is validated around the first central moment of the load. Various case studies are performed to estimate the day ahead optimum operating schedule of the Distributed Energy Resources (DER) and to assess the impact of load uncertainties on DER operation and total MG operation cost. Further, the efficacy of the GPR is verified through comparison with MCS based result.

Algebraically stable SDIRK methods with controllable numerical dissipation for first/second-order time-dependent problems

In this paper, a family of four-stage singly diagonally implicit Runge-Kutta methods are proposed to solve first-/second-order time-dependent problems, exhibiting the following numerical properties: fourth-order accuracy in time, unconditional stability, controllable numerical dissipation, and adaptive time step selection. The BN-stability condition is employed as a constraint to optimize parameters in the Butcher table, having significant benefits, and hence is recommended for nonlinear dynamics problems in contrast to existing methods. Numerical examples involving both first- and second-order linear/nonlinear dynamics problems validate the proposed method, and numerical results reveal that the proposed methods are free from the order reduction phenomenon when applied to nonlinear dynamics problems. The performance of adaptive time-stepping using the embedded scheme is further illustrated by the phase-field modeling problem. Additionally, the advantages and disadvantages of three-stage third-order accurate algebraically stable methods are discussed. The proposed high-order time integration can be readily integrated into high-order spatial discretization methods, such as the high-order spectral element method employed in this paper, to obtain high-order discretization in space and time dimensions.

FreeGSNKE: A Python-based dynamic free-boundary toroidal plasma equilibrium solver

We present a Python-based numerical solver for the two-dimensional dynamic plasma equilibrium problem. We model the time evolution of toroidally symmetric free-boundary tokamak plasma equilibria in the presence of the non-linear magnetohydrodynamic coupling with both currents in the “active” poloidal field coils, with assigned applied voltages, and eddy currents in the tokamak passive structures. FreeGSNKE (FreeGS Newton–Krylov Evolutive) builds and expands on the framework provided by the Python package FreeGS (Free boundary Grad–Shafranov). FreeGS solves the static free-boundary Grad–Shafranov (GS) problem, discretized in space using finite differences, by means of Picard iterations. FreeGSNKE introduces: (i) a solver for the static free-boundary GS problem based on the Newton–Krylov (NK) method, with improved stability and convergence properties; (ii) a solver for the linearized dynamic plasma equilibrium problem; and (iii) a solver for the non-linear dynamic problem, based on the NK method. We propose a novel “staggered” solution strategy for the non-linear problem, in which we make use of a set of equivalent formulations of the non-linear dynamic problem we derive. The alternation of NK solution steps in the currents and in the plasma flux lends this strategy an increased resilience to co-linearity and stagnation problems, resulting in favorable convergence properties. FreeGSNKE can be used for any user-defined tokamak geometry and coil configuration. FreeGSNKE's flexibility and ease of use make it a suitably robust control-oriented simulator of plasma magnetic equilibria. FreeGSNKE is entirely written in Python and easily interfaced with Python libraries, which facilitates machine learning based approaches to plasma control.

Efficient time domain response computation of massive wave power farms

A potential future challenge in the wave energy sector will involve the design and construction of massive wave power farms. That is, collections of several (> 100) wave energy converters (WEC) operating in identical environmental conditions at a distance comparable with typical water wave lengths. In this context, the WECs are likely to be influenced by each another by radiation force effects that are associated with the radiated wave field propagated by WECs operating in the surrounding wave field. These effects are commonly captured by the Cummins’ equation, where the radiation force is expressed as a convolution integral depending on the past values of the WEC response. Due to this mathematical representation, the time domain computation of the wave farm response can become computationally daunting. This article proposes one approach for computing efficiently the wave farm response in the time domain. Specifically, it demonstrates that the values of the radiation force components can be determined at each time step from their previous values by approximating the retardation function matrix elements via the Prony method. A notable advantage of this approach with respect to the ones available in the open literature is that it does not require either the storage of past response values or additional differential equations. Instead, it uses simple algebraic expressions for updating at each time instant the radiation force values. Obviously, this feature can induce significant computational efficiency in analyzing an actual wave farm facility.The reliability and efficiency of the proposed algorithm are assessed vis-à-vis direct time domain comparisons and Monte Carlo data concerning a wave farm composed by an array of U-Oscillating Water Columns. Notably, the proposed methodology can be applied to any linear or nonlinear dynamics problem governed by differential equations involving memory effects.

Large‐scale seismic soil–structure interaction analysis via efficient finite element modeling and multi‐GPU parallel explicit algorithm

AbstractAs urban population increases, integrated underground–aboveground complexes are being constructed at growing paces in major cities. The seismic analysis of such complexes is crucial for the safety and functionality in the threat of potential earthquake disasters. However, fine‐grained numerical modeling and analysis of such large and complex structures are still inefficient due to the consideration of the soil–structure interaction (SSI). To address this challenge, an efficient approach for numerical modeling of large‐scale seismic SSI analysis is presented in this paper to overcome the limitations of existing finite element analysis (FEA) software. Moreover, a multi‐graphic processing unit (GPU) parallel explicit algorithm is implemented for the nonlinear dynamic SSI problems to further increase the computational efficiency. A large underground–aboveground complex project in China is used as an example to demonstrate the capability of the integrated method. The accuracy and reliability of the multi‐GPU parallel explicit finite element algorithm for SSI analysis (GFEA‐SSIA) are verified through a comparative analysis of the linear‐elastic and nonlinear dynamic response of the building calculated by GFEA‐SSIA and common FEA software. Finally, the structural response and structural damage of the underground–aboveground complex are analyzed under multidirectional seismic motions, and the damage distributions of the structures are provided.

A deep learning technique to control the non-linear dynamics of a gravitational-wave interferometer

Abstract In this work we developed a deep learning technique that successfully solves a non-linear dynamic control problem. Instead of directly tackling the control problem, we combined methods in probabilistic neural networks and a Kalman-filter-inspired model to build a non-linear state estimator for the system. We then used the estimated states to implement a trivial controller for the now fully observable system. We applied this technique to a crucial non-linear control problem that arises in the operation of the Laser Interferometer Gravitational-Wave Observatory (LIGO) system, an interferometric gravitational-wave observatory. We demonstrated in simulation that our approach can learn from data to estimate the state of the system, allowing a successful control of the interferometer’s mirror. We also developed a computationally efficient model that can run in real time at high sampling rate on a single modern CPU core, one of the key requirements for the implementation of our solution in the LIGO digital control system. We believe these techniques could be used to help tackle similar non-linear control problems in other applications.