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The purpose of this paper is to design a simple and accurate a-posteriori Lagrangian-based error estimator is developed for the class of backward differentiation formula (BDF) algorithms with variable time step size, and the adaptive time-stepping in BDF algorithms is demonstrated for efficient time-dependent simulations in fluid flow and heat transfer.

The Lagrange interpolation polynomial is used to predict the time derivative, and then the accurate primary result is obtained by the Gauss integral, which is applied to evaluate the local error. Not only the generalized formula of the proposed error estimator is presented but also the specific expression for the widely applied BDF1/2/3 is illustrated. Two essential executable MATLAB functions to implement the proposed error estimator are appended for practical applications. Then, the adaptive time-stepping is demonstrated based on the newly proposed error estimator for BDF algorithms.

The validation tests show that the newly proposed error estimator is accurate such that the effectivity index is always close to unity for both linear and nonlinear problems, and it avoids under/overestimation of the exact local error. The applications for fluid dynamics and coupled fluid flow and heat transfer problems depict the advantage of adaptive time-stepping based on the proposed error estimator for time-dependent simulations.

In contrast to existing error estimators for BDF algorithms, the present work is more accurate for the local error estimation, and it can be readily extended to practical applications in engineering with a few changes to existing codes, contributing to efficient time-dependent simulations in fluid flow and heat transfer.

This paper aims to describe a modelling and solving methodology of a (static converter–electric motor–control) system for its sizing by optimization, considering the dynamic thermal heating of the machine.

The electrical drive sizing model is composed of two simulators (electrical and thermal) that are co-simulated with a master−slave relationship for the time step management. The computation is stopped according to simulation criteria.

This paper details a methodology to represent and size an electrical drive using a multiphysics and multidynamics approach. The thermal simulator is the master and calls the electrical system simulator at a fixed exchange time step. The two simulators use a dedicated dynamic time solver with adaptive time step and event management. The simulation automatically stops on pre-established criteria, avoiding useless simulations.

This paper aims to present a generic methodology for the sizing by optimization of electrical drives with a multiphysics approach, so the precision and computation time highly depend on the modelling method of each components. A genetic multiobjective optimization algorithm is used.

The methodology can be applied to size electrical drives operating in a thermally limited zone. The power electronics converter and electrical machine can be easily adapted by modifying their sub-model, without impacting the global model and simulation principle.

The approach enables to compute a maximum operating duration before reaching thermal limits and to use it as an optimization constraint. These system considerations allow to over constrain the electrical machine, enabling to size a smaller machine while guaranteeing the same output performances.

The purpose of this study is to introduce a new numerical scheme with no stability condition and high-order accuracy for the solution of two-dimensional coupled groundwater flow and transport simulation problems with regular and irregular geometries and compare the results with widely acceptable programs such as Modular Three-Dimensional Finite-Difference Ground-Water Flow Model (MODFLOW) and Modular Three-Dimensional Multispecies Transport Model (MT3DMS).

The newly proposed numerical scheme is based on the method of lines (MOL) approach and uses high-order approximations both in space and time. Quintic B-spline (QBS) functions are used in space to transform partial differential equations, representing the relevant physical phenomena in the system of ordinary differential equations. Then this system is solved with the DOPRI5 algorithm that requires no stability condition. The obtained results are compared with the results of the MODFLOW and MT3DMS programs to verify the accuracy of the proposed scheme.

The results indicate that the proposed numerical scheme can successfully simulate the two-dimensional coupled groundwater flow and transport problems with complex geometry and parameter structures. All the results are in good agreement with the reference solutions.

To the best of the authors' knowledge, the QBS-DOPRI5 method is used for the first time for solving two-dimensional coupled groundwater flow and transport problems with complex geometries and can be extended to high-dimensional problems. In the future, considering the success of the proposed numerical scheme, it can be used successfully for the identification of groundwater contaminant source characteristics.

The purpose of this study is to develop a new numerical algorithm to simulate the phase-field model.

First, the derivative of the temporal direction is discretized by a second-order linearized finite difference scheme where it conserves the energy stability of the mathematical model. Then, the isogeometric collocation (IGC) method is used to approximate the derivative of spacial direction. The IGC procedure can be applied on irregular physical domains. The IGC method is constructed based upon the nonuniform rational B-splines (NURBS). Each curve and surface can be approximated by the NURBS. Also, a map will be defined to project the physical domain to a simple computational domain. In this procedure, the partial derivatives will be transformed to the new domain by the Jacobian and Hessian matrices. According to the mentioned procedure, the first- and second-order differential matrices are built. Furthermore, the pseudo-spectral algorithm is used to derive the first- and second-order nodal differential matrices. In the end, the Greville Abscissae points are used to the collocation method.

In the numerical experiments, the efficiency and accuracy of the proposed method are assessed through two examples, demonstrating its performance on both rectangular and nonrectangular domains.

This research work introduces the IGC method as a simulation technique for the phase-field crystal model.

In this study, the authors propose a novel digital twinning approach specifically designed for controlling transient thermal systems. The purpose of this study is to harness the combined power of deep learning (DL) and physics-based methods (PBM) to create an active virtual replica of the physical system.

To achieve this goal, we introduce a deep neural network (DNN) as the digital twin and a Finite Element (FE) model as the physical system. This integrated approach is used to address the challenges of controlling an unsteady heat transfer problem with an integrated feedback loop.

The results of our study demonstrate the effectiveness of the proposed digital twinning approach in regulating the maximum temperature within the system under varying and unsteady heat flux conditions. The DNN, trained on stationary data, plays a crucial role in determining the heat transfer coefficients necessary to maintain temperatures below a defined threshold value, such as the material’s melting point. The system is successfully controlled in 1D, 2D and 3D case studies. However, careful evaluations should be conducted if such a training approach, based on steady-state data, is applied to completely different transient heat transfer problems.

The present work represents one of the first examples of a comprehensive digital twinning approach to transient thermal systems, driven by data. One of the noteworthy features of this approach is its robustness. Adopting a training based on dimensionless data, the approach can seamlessly accommodate changes in thermal capacity and thermal conductivity without the need for retraining.

The main aim of the current paper is to find a numerical plan for hydraulic fracturing problem with application in extracting natural gases and oil.

First, time discretization is accomplished via Crank-Nicolson and semi-implicit techniques. At the second step, a high-order finite element method using quadratic triangular elements is proposed to derive the spatial discretization. The efficiency and time consuming of both obtained schemes will be investigated. In addition to the popular uniform mesh refinement strategy, an adaptive mesh refinement strategy will be employed to reduce computational costs.

Numerical results show a good agreement between the two schemes as well as the efficiency of the employed techniques to capture acceptable patterns of the model. In central single-crack mode, the experimental results demonstrate that maximal values of displacements in x- and y- directions are 0.1 and 0.08, respectively. They occur around both ends of the line and sides directly next to the line where pressure takes impact. Moreover, the pressure of injected fluid almost gained its initial value, i.e. 3,000 inside and close to the notch. Further, the results for non-central single-crack mode and bifurcated crack mode are depicted. In central single-crack mode and square computational area with a uniform mesh, computational times corresponding to the numerical schemes based on the high order finite element method for spatial discretization and Crank-Nicolson as well as semi-implicit techniques for temporal discretizations are 207.19s and 97.47s, respectively, with 2,048 elements, final time T = 0.2 and time step size τ = 0.01. Also, the simulations effectively illustrate a further decrease in computational time when the method is equipped with an adaptive mesh refinement strategy. The computational cost is reduced to 4.23s when the governed model is solved with the numerical scheme based on the adaptive high order finite element method and semi-implicit technique for spatial and temporal discretizations, respectively. Similarly, in other samples, the reduction of computational cost has been shown.

This is the first time that the high-order finite element method is employed to solve the model investigated in the current paper.

Robotic arms play a crucial role in various industrial operations, such as sorting, assembly, handling and spraying. However, traditional robotic arm control algorithms often struggle to adapt when faced with the challenge of dynamic obstacles. This paper aims to propose a dynamic obstacle avoidance method based on reinforcement learning to address real-time processing of dynamic obstacles.

This paper introduces an innovative method that introduces a feature extraction network that integrates gating mechanisms on the basis of traditional reinforcement learning algorithms. Additionally, an adaptive dynamic reward mechanism is designed to optimize the obstacle avoidance strategy.

Validation through the CoppeliaSim simulation environment and on-site testing has demonstrated the method's capability to effectively evade randomly moving obstacles, with a significant improvement in the convergence speed compared to traditional algorithms.

The proposed dynamic obstacle avoidance method based on Reinforcement Learning not only accomplishes the task of dynamic obstacle avoidance efficiently but also offers a distinct advantage in terms of convergence speed. This approach provides a novel solution to the obstacle avoidance methods for robotic arms.

The purpose of this study is to advance the computational modeling of liquid splashing dynamics, while balancing simulation accuracy and computational efficiency, a duality often compromised in high-fidelity fluid dynamics simulations.

This study introduces the fluid efficient graph neural network simulator (FEGNS), an innovative framework that integrates an adaptive filtering layer and aggregator fusion strategy within a graph neural network architecture. FEGNS is designed to directly learn from extensive liquid splash data sets, capturing the intricate dynamics and intrinsically complex interactions.

FEGNS achieves a remarkable 30.3% improvement in simulation accuracy over traditional methods, coupled with a 51.6% enhancement in computational speed. It exhibits robust generalization capabilities across diverse materials, enabling realistic simulations of droplet effects. Comparative analyses and empirical validations demonstrate FEGNS’s superior performance against existing benchmark models.

The originality of FEGNS lies in its adaptive filtering layer, which independently adjusts filtering weights per node, and a novel aggregator fusion strategy that enriches the network’s expressive power by combining multiple aggregation functions. To facilitate further research and practical deployment, the FEGNS model has been made accessible on GitHub (https://github.com/nanjinyao/FEGNS/tree/main).

Thermal fractures initiated under cooling at the surfaces of a 2-D or 3-D structure propagate, arrest and coalesce, leading to its structural failure and material-property changes, while the same processes can happen in the rock mass between parallel hydraulic fractures filled with cold fluid, leading to enhanced fracture connectivity and permeability.

This study used a 2-D plane strain fracture model for mixed-mode thermal fractures from two parallel cooling surfaces. Fracture propagation was governed by the theory of linear elastic fracture mechanics, while the displacement and temperature fields were discretized using the adaptive finite element method. This model was validated using two numerical benchmarks with strong fracture curvature and then used to simulate the propagation and coalescence of thermal fractures in a long rock mass.

Modeling results show two regimes: (1) thermal fractures from a cooling surface propagate and arrest by following the theoretical solutions of half-plane fractures before the unfractured portion decreases to 20% rock-mass width and (2) some pairs of fractures from the opposite cooling surfaces tend to eventually coalesce. The fracture coalescence time is in a power law with rock-mass width.

These findings are relevant to both subsurface engineering and material engineering: structure failure is a key concern in the latter, while fracture coalescence can enhance the connectivity of thermal and hydraulic fractures and thus reservoir permeability in the former.

Voltage source inverter-fed permanent magnet synchronous motors (VSI-PMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion). The dynamic model of VSI-PMSMs is multivariable and exhibits complicated nonlinear dynamics. The inverters’ currents, which are generated through a pulsewidth modulation process, are used to control the stator currents of the PMSM, which in turn control the rotational speed of this electric machine. So far, several nonlinear control schemes for VSI-PMSMs have been developed, having as primary objectives the precise tracking of setpoints by the system’s state variables and robustness to parametric changes or external perturbations. However, little has been done for the solution of the associated nonlinear optimal control problem. The purpose of this study/paper is to provide a novel nonlinear optimal control method for VSI-fed three-phase PMSMs.

The present article proposes a nonlinear optimal control approach for VSI-PMSMs. The nonlinear dynamic model of VSI-PMSMs undergoes approximate linearization around a temporary operating point, which is recomputed at each iteration of the control method. This temporary operating point is defined by the present value of the voltage source inverter-fed PMSM state vector and by the last sampled value of the motor’s control input vector. The linearization relies on Taylor series expansion and the calculation of the system’s Jacobian matrices. For the approximately linearized model of the voltage source inverter-fed PMSM, an H-infinity feedback controller is designed. For the computation of the controller’s feedback gains, an algebraic Riccati equation is iteratively solved at each time-step of the control method. The global asymptotic stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control for this system, the H-infinity Kalman filter is proposed as a state observer. The proposed control method achieves fast and accurate tracking of the reference setpoints of the VSI-fed PMSM under moderate variations of the control inputs.

The proposed H-infinity controller provides the solution to the optimal control problem for the VSI-PMSM system under model uncertainty and external perturbations. Actually, this controller represents a min–max differential game taking place between the control inputs, which try to minimize a cost function that contains a quadratic term of the state vector’s tracking error, the model uncertainty, and exogenous disturbance terms, which try to maximize this cost function. To select the feedback gains of the stabilizing feedback controller, an algebraic Riccati equation is repetitively solved at each time-step of the control algorithm. To analyze the stability properties of the control scheme, the Lyapunov method is used. It is proven that the VSI-PMSM loop has the H-infinity tracking performance property, which signifies robustness against model uncertainty and disturbances. Moreover, under moderate conditions, the global asymptotic stability properties of this control scheme are proven. The proposed control method achieves fast tracking of reference setpoints by the VSI-PMSM state variables, while keeping also moderate the variations of the control inputs. The latter property indicates that energy consumption by the VSI-PMSM control loop can be minimized.

The proposed nonlinear optimal control method for the VSI-PMSM system exhibits several advantages: Comparing to global linearization-based control methods, such as Lie algebra-based control or differential flatness theory-based control, the nonlinear optimal control scheme avoids complicated state variable transformations (diffeomorphisms). Besides, its control inputs are applied directly to the initial nonlinear model of the VSI-PMSM system, and thus inverse transformations and the related singularity problems are also avoided. Compared with backstepping control, the nonlinear optimal control scheme does not require the state-space description of the controlled system to be found in the triangular (backstepping integral) form. Compared with sliding-mode control, there is no need to define in an often intuitive manner the sliding surfaces of the controlled system. Finally, compared with local model-based control, the article’s nonlinear optimal control method avoids linearization around multiple operating points and does not need the solution of multiple Riccati equations or LMIs. As a result of this, the nonlinear optimal control method requires less computational effort.

Voltage source inverter-fed permanent magnet synchronous motors (VSI-PMSMs) are widely used in industrial actuation and mechatronic systems in water pumping stations, as well as in the traction of transportation systems (such as electric vehicles and electric trains or ships with electric propulsion), The solution of the associated nonlinear control problem enables reliable and precise functioning of VSI-fd PMSMs. This in turn has a positive impact in all related industrial applications and in tasks of electric traction and propulsion where VSI-fed PMSMs are used. It is particularly important for electric transportation systems and for the wide use of electric vehicles as expected by green policies which aim at deploying electromotion and at achieving the Net Zero objective.

Unlike past approaches, in the new nonlinear optimal control method, linearization is performed around a temporary operating point, which is defined by the present value of the system’s state vector and by the last sampled value of the control input vector and not at points that belong to the desirable trajectory (setpoints). Besides, the Riccati equation, which is used for computing the feedback gains of the controller, is new, as is the global stability proof for this control method. Comparing with nonlinear model predictive control, which is a popular approach for treating the optimal control problem in industry, the new nonlinear optimal (H-infinity) control scheme is of proven global stability, and the convergence of its iterative search for the optimum does not depend on initial conditions and trials with multiple sets of controller parameters. It is also noteworthy that the nonlinear optimal control method is applicable to a wider class of dynamical systems than approaches based on the solution of state-dependent Riccati equations (SDRE). The SDRE approaches can be applied only to dynamical systems that can be transformed to the linear parameter varying form. Besides, the nonlinear optimal control method performs better than nonlinear optimal control schemes which use approximation of the solution of the Hamilton–Jacobi–Bellman equation by Galerkin series expansions.