Dynamic Problem Research Articles

Fast Fourier Transform periodic interpolation method for superposition sums in a periodic unit cell

We propose a Fast Fourier Transform based Periodic Interpolation Method (FFT-PIM), a flexible and computationally efficient approach for computing the scalar potential given by a superposition sum in a unit cell of an infinitely periodic array. Under the same umbrella, FFT-PIM allows computing the potential for 1D, 2D, and 3D periodicities for dynamic problems involving the Helmholtz potential and static problems involving Coulomb potential, including problems with and without a periodic phase shift. The computational complexity of the FFT-PIM is of O(Nlog⁡N) for N spatially coinciding sources and observer points. The FFT-PIM uses rapidly converging series representations of the Green's function serving as a kernel in the superposition sum. Based on these representations, the FFT-PIM splits the potential into its near-zone component, which includes a small number of images surrounding the unit cell of interest, and far-zone component, which includes the rest of an infinite number of images. The far-zone component is evaluated by projecting the non-uniform sources onto a sparse uniform grid, performing superposition sums on this sparse grid, and interpolating the potential from the uniform grid to the non-uniform observation points. The near-zone component is evaluated using an FFT-based method, which is adapted to efficiently handle non-uniform source-observer distributions within the periodic unit cell. The FFT-PIM can be used for a broad range of applications, such as periodic problems involving integral equations for wave propagation in electromagnetics and acoustics, micromagnetic solvers, and density functional theory solvers.

Collaborative dynamic scheduling in a self-organizing manufacturing system using multi-agent reinforcement learning

Personalized product demands have made the production mode of many varieties and small batches mainstream. Self-organizing manufacturing systems represented by multi-agent-based manufacturing systems (MAMS) are pursued by manufacturing enterprises to support collaborative dynamic scheduling among various pieces of manufacturing equipment. In contrast to traditional scheduling methods including meta-heuristic algorithms and dispatching rules, reinforcement learning (RL), whether in the form of single-agent RL (SARL) or multi-agent RL (MARL), combines high response and robust performance in dynamic environments. However, the SARL-based methods suffer from the curse of dimensionality in the action space, while current MARL-based methods fail to achieve the optimization of collaborative scheduling decisions among heterogeneous manufacturing equipment. These factors directly hinder the effective resolution of dynamic scheduling in the MAMS. Therefore, we present a novel MARL-based approach that solves the dynamic flexible job-shop scheduling problem in the MAMS (DFJSP-MAMS), with the objective of minimizing the mean tardiness. Here, various items of heterogeneous manufacturing equipment in the MAMS, including the warehouse, machines, buffers, and vehicles, are encapsulated as agents. Moreover, a partially observable Markov game is constructed to specify the collaborative scheduling process among these agents and transfer the DFJSP-MAMS into a MARL task. Furthermore, an action space consisting of several weight variables is designed for each agent to offer the weight selection of dispatching rules. Subsequently, a policy network is built for each agent to determine the weights and scheduling decisions. Finally, the multi-agent deep deterministic policy gradient algorithm is employed to train these agents for the optimization of collaborative scheduling decisions among various pieces of heterogeneous equipment. Experiment results demonstrate that the proposed approach outperforms single dispatching rules, the SARL-based method, and other MARL-based methods under 36 distinct manufacturing environment conditions.

Dynamic robot routing optimization: State–space decomposition for operations research-informed reinforcement learning

There is a growing interest in implementing artificial intelligence for operations research in the industrial environment. While numerous classic operations research solvers ensure optimal solutions, they often struggle with real-time dynamic objectives and environments, such as dynamic routing problems, which require periodic algorithmic recalibration. To deal with dynamic environments, deep reinforcement learning has shown great potential with its capability as a self-learning and optimizing mechanism. However, the real-world applications of reinforcement learning are relatively limited due to lengthy training time and inefficiency in high-dimensional state spaces. In this study, we introduce two methods to enhance reinforcement learning for dynamic routing optimization. The first method involves transferring knowledge from classic operations research solvers to reinforcement learning during training, which accelerates exploration and reduces lengthy training time. The second method uses a state–space decomposer to transform the high-dimensional state space into a low-dimensional latent space, which allows the reinforcement learning agent to learn efficiently in the latent space. Lastly, we demonstrate the applicability of our approach in an industrial application of an automated welding process, where our approach identifies the shortest welding pathway of an industrial robotic arm to weld a set of dynamically changing target nodes, poses and sizes. The suggested method cuts computation time by 25% to 50% compared to classic routing algorithms.

Explicit frictional stick–slip dynamics of elastic contact problem incorporating the LuGre model

Dynamical contact problem with friction is of continuous concern and have been investigated over the years. The effective and accurate description of the development of the friction force is the key to this problem. However, the stick–slip phenomenon, as typical dynamical behaviour, has rarely been considered when developing contact algorithms for frictional contact problems of elastic bodies. The conventionally used Coulomb’s model, with differentiating the static and kinetic coefficient of friction, can lead to inaccurate results due to errors generated when switch between the stick or slip states. In this work, a three-dimensional mortar-based frictional contact algorithm is proposed in an explicit scheme. The proposed algorithm incorporates the LuGre model to account for dynamical frictional behaviours such as stick–slip. The effectiveness and accuracy of the proposed algorithm are validated through several simplified two-dimensional cases. The sticking time ratio and the evolution of the global coefficient of friction with time of a three-dimensional contact problem with a randomly rough contact interface are investigated as an application.

Machine learning in biological physics: From biomolecular prediction to design

Machine learning has been proposed as an alternative to theoretical modeling when dealing with complex problems in biological physics. However, in this perspective, we argue that a more successful approach is a proper combination of these two methodologies. We discuss how ideas coming from physical modeling neuronal processing led to early formulations of computational neural networks, e.g., Hopfield networks. We then show how modern learning approaches like Potts models, Boltzmann machines, and the transformer architecture are related to each other, specifically, through a shared energy representation. We summarize recent efforts to establish these connections and provide examples on how each of these formulations integrating physical modeling and machine learning have been successful in tackling recent problems in biomolecular structure, dynamics, function, evolution, and design. Instances include protein structure prediction; improvement in computational complexity and accuracy of molecular dynamics simulations; better inference of the effects of mutations in proteins leading to improved evolutionary modeling and finally how machine learning is revolutionizing protein engineering and design. Going beyond naturally existing protein sequences, a connection to protein design is discussed where synthetic sequences are able to fold to naturally occurring motifs driven by a model rooted in physical principles. We show that this model is “learnable” and propose its future use in the generation of unique sequences that can fold into a target structure.

Safe reinforcement learning based optimal low-carbon scheduling strategy for multi-energy system

Multi-energy system with distributed energy resources has become the inevitable trend in recent years due to their potential for creating the efficient and sustainable energy infrastructure, with a strong ability on carbon emission reduction. To accommodate the uncertainties of renewable energy generation and energy demand, model-free deep reinforcement learning methods are emerging for energy management in multi-energy system. However, traditional reinforcement learning methods still have operation safety issue of violating the physical constraints of multi-energy system. To address the challenges, a low-carbon scheduling strategy based on safe soft actor-critic algorithm is proposed in this paper. Firstly, an electricity-thermal-carbon joint scheduling framework is constructed, where carbon trading mechanism is incorporated to further motivate carbon emission reductions. Secondly, the energy cost and carbon trading cost are simultaneously integrated in the objective function, and the dynamic optimization problem of multi-energy system is modeled as a constrained Markov decision process by taking into account the diverse uncertainties. Then, a novel safe soft actor-critic method is proposed to achieve the benefits of economic and carbon emissions, where the security networks and Lagrangian relaxation are introduced to deal with operation constraints. The case study validates that the proposed scheduling strategy can reduce the energy cost and carbon trading cost by up to 26.24% and 33.73% within constraints, compared with existing methods.

Optimal trading with regime switching: Numerical and analytic techniques applied to valuing storage in an electricity balancing market

Accurately valuing storage in the electricity market recognizes its role in enhancing grid flexibility, integrating renewable energy, managing peak loads, providing ancillary services and improving market efficiency. In this paper we outline an optimal trading problem for an Energy Storage Device trading on the electricity balancing (or regulating) market. To capture the features of the balancing (or regulating) market price we combine stochastic differential equations with Markov regime switching to create a novel model, and outline how this can be calibrated to real market data available from NordPool. By modelling a battery that can be filled or emptied instantaneously, this simplifying assumption allows us to generate numerical and quasi analytic solutions.We implement a case study to investigate the behaviour of the optimal strategy, how it is affected by price and underlying model parameters. Using numerical (finite-difference) techniques to solve the dynamic programming problem, we can estimate the value of operating an Energy Storage Device in the market given fixed costs to charge or discharge. Finally we use properties of the numerical solution to propose a simple quasi-analytic approximation to the problem. We find that analytic techniques can be used to give a benchmark value for the storage price when price variations during the day are relatively small.

Dynamic modeling and simulation of hard-magnetic soft beams interacting with environment via high-order finite elements of ANCF

Hard-magnetic soft (HMS) beams made of soft polymer matrix embedded with hard-magnetic particles can generate large and fast deformation under magnetic stimulation. Dynamic modeling and simulation of HMS beams interacting with complex environment are challenging in terms of computational accuracy and efficiency. This paper presents a method for high-order modeling and efficient computation of HMS beams. The major contribution of the method is a new three-node HMS beam element of absolute nodal coordinate formulation (ANCF), which applies to two material models of nonlinear and linear elasticities (i.e. neoHookean and St. Venant-Kirchhoff) coupled with magnetic energy. To improve the efficiency of the method, the paper presents how to derive the generalized internal forces and their Jacobians via invariant tensors, and how to determine the generalized external forces to model dynamic loads and interactions including gravity, hydrodynamics in fluids, and frictional contact in pipelines. Afterwards, the paper gives both static and dynamic equations with Rayleigh damping and discusses the numerical algorithms. Finally, the paper makes a comparison of static analysis and the experimental observation to validate the accuracy of the proposed modeling method. The paper also discusses the dynamic simulations, including forced vibration, swimming motion, crawling locomotion, and navigating motion to demonstrate the predictive capability and efficacy of the proposed method for dynamic problems.

TD3-BC-PPO: Twin delayed DDPG-based and behavior cloning-enhanced proximal policy optimization for dynamic optimization affine formation

This article addresses the dynamic optimal design problem of the affine formation for adversarial multi-agent systems in real-time confrontation environments. To maximize the win rate and maximize the battle damage ratio, a novel deep reinforcement learning (DRL) algorithm named TD3-BC-PPO is designed to adaptively control the shape transformation of affine formation, including translation, rotation, and scaling. Specifically, Twin Delayed Deep Deterministic Policy Gradient (TD3) is introduced to enhance the exploratory aspect of the strategy and efficiently leverage the crucial sparse win–loss rewards. Behavior cloning (BC) is used for strategy replication and model migration. Proximal policy optimization (PPO) is utilized for stabilizing policy updates, conducting a secondary optimization of the strategy through dense damage-related rewards. To validate the effectiveness and robustness of the proposed algorithm, extensive simulation experiments are conducted. Comparative analyses are performed against several baselines, encompassing a diverse range of formation confrontation situations. These situations involve various initial formation structures, differing attack angles, unequal combat quantity and expanded confrontation space dimensions. Simulation results show that the proposed TD3-BC-PPO algorithm can engender an impressive surge of at least 5.4% in the win rate and an incremental ascent of at least 0.274 in the battle damage ratio of the affine formation for adversarial multi-agent systems in complex real-time confrontation scenarios.

Assessing effects of climate and technology uncertainties in large natural resource allocation problems

Abstract. The productivity of the world's natural resources is critically dependent on a variety of highly uncertain factors, which obscure individual investors and governments that seek to make long-term, sometimes irreversible, investments in their exploration and utilization. These dynamic considerations are poorly represented in disaggregated resource models, as incorporating uncertainty into large-dimensional problems presents a challenging computational task. In this paper, we apply the SCEQ algorithm (Cai and Judd, 2023) to solve a large-scale dynamic stochastic global land resource use problem with stochastic crop yields due to adverse climate impacts and limits on further technological progress. For the same model parameters and bounded shocks, the range of land conversion is considerably smaller for the dynamic stochastic model than for deterministic scenario analysis.

Dynamic capacity allotment planning for airlines via travel agencies

In this paper, we study the dynamic capacity allocation problem of an airline using allotment contracts among travel agencies. The capacity allocation problem is formulated as a dynamic programming model where the initial fixed allotment is determined on the basis of a contract between agencies and the airline while the variable allotment is distributed over the planning horizon. We develop approximation methods based on linear programming and Lagrangian relaxation to solve the underlying dynamic programming model. The computational experiments are designed to illustrate the performance of the proposed approaches using real data. The numerical results show that the capacity allotment policy in terms of fixed and variable allotments is significantly affected by price and demand patterns. Even though fixed allotments can provide a cushion against the loss due to potential empty seats, they may not be in the best interest of the airline. Moreover, the booking rules and the airline’s capacity allocation strategy have an impact on the airline’s revenue and the agency’s profit.

Intelligent diving guidance with terminal angle and velocity constraints via deep reinforcement learning

An adaptive guidance strategy that integrates optimal guidance and deep reinforcement learning to address a highly dynamic terminal guidance problem that entails meeting terminal position, angle, and velocity constraints. The proposed strategy leverages optimal guidance commands to accomplish position and angle control while introducing a deep reinforcement learning-based bias for the velocity constraint. In the training process, a dual-velocity state space is constructed to enhance the adaptability of the strategy to different guidance tasks, while training is optimized using the prediction-correction and expert knowledge to improve the training efficiency and optimality of the strategy. Simulations demonstrate that the proposed guidance strategy can achieve simultaneous control of terminal position, angle and velocity, and adapt to different guidance tasks.

SCAN Algorithm to Optimize Dynamic Instructional Information Network for Predicting Student Behavior

With the integration of work, study and life on campus taking shape gradually in major universities across China, the concept of education from classroom to life is more widely accepted and the smart education model has become a trend in education informatization. However, most of the existing studies are not applicable to student behaviour data in the campus environment, and the temporal as well as cyclical characteristics in the data are not accessible in the student behaviour prediction problem. In this study, a hypergraph-based dynamic campus behaviour information network is designed to address the needs of the student behaviour prediction problem, and a student campus behaviour prediction algorithm is proposed in the dynamic campus behaviour information network-based student behaviour prediction problem. The effectiveness and rationality of the algorithm is verified through experiments with real campus data sets. The experimental results demonstrated that the periodic nature of the data acquired by the cycle gated cyclic unit module needs to be built on top of the snapshot gated cyclic unit module to help the algorithm achieve better results. The area under the curve of the algorithm proposed in the study achieves more advantageous results on both the 21-day and 35-day datasets. The cycle gated cyclic unit module of the student campus behaviour prediction algorithm proposed in this study can more effectively extract the cyclic features present in the dynamic campus behaviour information network and better accomplish the prediction of student campus behaviour.

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.

Novel analytical approach for vertical dynamic response of floating pile groups based on a fictitious soil pile model

A novel analytical model for the vertical dynamic response of floating pile groups is established based on a fictitious group soil pile model (FGSP). Analytical solutions of the pile-to-pile interaction factor and dynamic impedance for the FGSP are obtained by combining the superposition and matrix transform methods. The obtained solutions are then validated by comparison with model test results and published analytical solutions. A numerical analysis is further performed to examine the effects of pile and soil parameters on the dynamic response of the FGSP. Results indicate that the effects of pile end soil on the pile-to-pile interaction factor and dynamic impedance of the FGSP are significant. Consequently, the established novel analytical model has better applicability to the dynamic problems of floating pile groups due to its consideration of the effects of pile end soil.

Complex dynamics in prey-predator systems with cross-coupling: Exploring nonlinear interactions and population oscillations

This study investigates the problem of ecosystem dynamics in fragmented landscapes, specifically focusing on a two-patch environment with interacting prey and predators. The research examines the impact of cross-predation on these interactions. Using bifurcation analysis, we explored the structural arrangement of attractors and identified complex dynamics such as symmetric, asymmetric, and asynchronous attractors induced by varying cross-coupling levels. Notably, our study describes a novel mechanism for the formation of anti-phase synchrony in the patches. Unlike typical occurrences of a cycle following Hopf bifurcation, our model reveals that the anti-phase cycle stabilizes via Neimark-Sacker (NS) bifurcation of a two-period unstable cycle branch emanating from the synchronous cycle branch. Our findings also demonstrate that cross-feeding can lead to significant ecosystem asymmetry and branching, culminating in the dominance of a single cross-feeding chain. These results challenge traditional models and highlight the presence of multistability and the potential for ecosystem evolution towards distinct subsystem branches due to cross-predation. The study’s insights offer valuable contributions to population and evolutionary biology, enhancing our understanding of the intricate dynamics within fragmented ecosystems.

Damage identification technique by model enrichment for structural elastodynamic problems

Structural Health Monitoring (SHM) techniques are key to monitor the health state of engineering structures, where damage type, location and severity are to be estimated by applying sophisticated techniques to signals measured by sensors. However, very localized damage detection algorithms applied to dynamics problems when dealing with rigid structures at low-frequency range remains still a big challenge. The last due to the low influence of very localized damage on the overall response of the structure (Saint-Venant principle). In this context, in the present work, we propose a methodology for locally correcting the models from collected data for elastodynamics problems at low-frequency range which is able to predict very localized damage. The proposed technique consists in enriching the structural model in a sparse way and solving the identification problem in the frequency domain, where the influence of damage over a large frequency band is exploited to improve the prediction of the damage location. The advantages and potential of the proposed technique are illustrated for the damage detection in a plate problem, demonstrating the advantages of the method in detecting very localized damage. The proposed technique is limited to a methodological description, and further developments should be considered to approach its applicability in an industrial scenario.

Matrix‐based implementation and GPU acceleration of hybrid FEM and peridynamic model for hydro‐mechanical coupled problems

AbstractThe hybrid finite element‐peridynamic (FEM‐PD) models have been evidenced for their exceptional ability to address hydro‐mechanical coupled problems involving cracks. Nevertheless, the non‐local characteristics of the PD equations and the required inversion operations when solving fluid equations result in prohibitively high computational costs. In this paper, a fast explicit solution scheme for FEM‐PD models based on matrix operation is introduced, where the graphics processing units (GPUs) are used to accelerate the computation. An in‐house software is developed in MATLAB in both CPU and GPU versions. We first solve a problem related to pore pressure distribution in a single crack, demonstrating the accuracy of the proposed method by a comparison of FEM‐PD solutions with those of PD‐only models and analytical solutions. Subsequently, several examples are solved, including a one‐dimensional dynamic consolidation problem and the fluid‐driven hydraulic fracture propagation problems in both 2D and 3D cases, to comprehensively validate the effectiveness of the proposed methods in simulating deformation and fracture in saturated porous media under the influence of hydro‐mechanical coupling. In the presented numerical results, some typical strong dynamic phenomena, such as stepwise crack advancement, crack branching, and pressure oscillations, are observed. In addition, we compare the wall times of all the cases executed on both the GPU and CPU, highlighting the substantial acceleration performance of the GPU, particularly when tackling problems with a significant computational workload.

Dynamics of ternary nanofluid through radiated sensor surface: Numerical investigation

Application: The impact of flow, heat transfer, and magneto hydrodynamics on sensor surfaces between two parallel compressing plates with porous walls has been examined in this study. This study focuses on understanding unsteady compressed flow in two dimensions, utilizing Aluminum oxide, copper oxide, and titanium dioxide with base fluid polymers as the base fluid. Nanofluids, known as nanometer suspensions in traditional nanoscale fluid transfer, are explored for their potential application in improving lubricative and cooling properties. Purpose and methodology: This study aims to investigate the behavior of a tri-hybrid nanofluid (Aluminum oxide, copper oxide, and titanium dioxide with base fluid polymers) in terms of flow dynamics, heat transfer, and magneto hydrodynamics. Energy and momentum equations, considering magneto hydrodynamic forms and heat transfer, are analyzed. The study employs numerical methods, including similarity transforms and a shooting approach, to solve the governing equations. Core findings: Several parameters, including permeable parameter, magnetic parameter, squeeze flow index parameter, volume fraction by nanoparticles, and radiation parameter, are investigated for their effects on temperature profile and velocity profile. The study illustrates these effects graphically and discusses the influence of these parameters on different components of velocity and temperature fields. Additionally, the impact of the radiation parameter ([Formula: see text] on temperature fields is examined for both positive. Future work: Future research may focus on further optimizing the tri-hybrid nanofluid composition for specific applications, exploring additional parameters that may affect flow behavior, heat transfer, and entropy generation. Additionally, experimental validation of the numerical findings and the development of more advanced numerical techniques for solving complex fluid dynamics problems could be the areas of interest for future work.

Moduli Spaces of Quadratic Maps: Arithmetic and Geometry

Abstract We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$th Gleason polynomial $G_{n}\in{\mathbb{Q}}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $\operatorname{Per}_{n}(0)$ is the $1$-dimensional moduli space of quadratic rational maps on ${\mathbb{P}}^{1}$ with an $n$-periodic critical point. We show that if $G_{n}$ is irreducible over ${\mathbb{Q}}$, then $\operatorname{Per}_{n}(0)$ is irreducible over ${\mathbb{C}}$. To do this, we exhibit a ${\mathbb{Q}}$-rational smooth point on a projective completion of $\operatorname{Per}_{n}(0)$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $n$, $\operatorname{Per}_{n}(0)$ itself has no ${\mathbb{Q}}$-rational points.