Horizon Sizes Research Articles

Plate buckling analysis in peridynamic framework

AbstractPeridynamics (PD) is a generalized continuum theory that is developed to account for long range internal force/moment interactions. PD equations of motion are of the integro‐differential type, and only several closed‐form analytical solutions are available for elementary structures. In this paper, governing equations for stability of Kirchhoff type (shear rigid) plates in the PD framework are derived. For several types of boundary conditions, closed form analytical solutions for the buckling loads are presented. The results are compared with the solutions according to the classical plate theory. A very good agreement between the non‐local and the classical theories is observed for the case of the small horizon sizes which shows the capability of the developed approach.

Modeling cosmological perturbations of thermal inflation

We consider a simple system consisting of matter, radiation and vacuum components to model the impact of thermal inflation on the evolution of primordial perturbations. The vacuum energy magnifies the primordial modes entering the horizon before its domination, making them potentially observable, and the resulting transfer function reflects the phase changes and energy contents. To determine the transfer function, we follow the curvature perturbation from well outside the horizon during radiation domination to well outside the horizon during vacuum domination and evaluate it on a constant radiation density hypersurface, as is appropriate for the case of thermal inflation. The shape of the transfer function is determined by the ratio of vacuum energy to radiation at matter-radiation equality, which we denote by , and has two characteristic scales, and , corresponding to the horizon sizes at matter radiation equality and the beginning of the inflation, respectively. If , the Universe experiences radiation, matter and vacuum domination eras and the transfer function is flat for , oscillates with amplitude for and oscillates with amplitude 1 for . For , the matter domination era disappears, and the transfer function reduces to being flat for and oscillating with amplitude 1 for .

A dual-horizon peridynamic model for Reissner–Mindlin plates with arbitrary horizon sizes and shapes

A dual-horizon peridynamic (DHPD) model with arbitrary horizon sizes and shapes is proposed for predicting structural responses of thin-walled structures subjected to out-of-plane loadings. The proposed DHPD method can diminish the influence arising from ghost forces in irregular particle distributions and lessen the computational consumption with optimal refinement discretizations. Meanwhile, the variable PD parameters of each discrete particle that derive based on its own current horizon domain are adopted to mitigate the PD surface effect. Several static and dynamic structural responses with different geometric and boundary conditions are compared between the reference and DHPD approaches. It is concluded that the proposed DHPD method can effectively reproduce structural behaviors in fracture mechanics problems with less computational effort.

Closed-Form Analytical Solutions for the Deflection of Elastic Beams in a Peridynamic Framework

Peridynamics is a continuum theory that operates with non-local deformation measures as well as long-range internal force/moment interactions. The resulting equations are of the integral type, in contrast to the classical theory, which deals with differential equations. The aim of this paper is to analyze peridynamic governing equations for elastic beams. To this end, the strain energy density is formulated as a function of the non-local curvature. By applying the Lagrange principle, the peridynamic equations of motion are derived. Examples of non-local boundary conditions, including simple support, clamped edge, roller clamped edge, and free edge, are presented by introducing the interaction domain. Novel closed-form analytical solutions to integral equations are presented for beams with various boundary conditions, including clamped—simply supported, clamped–clamped, simply supported–roller-clamped, and clamped–roller-clamped beams. Furthermore, different types of loadings, including uniformly distributed load, concentrated force, and concentrated moment, are considered. The results are validated by comparing the derived solutions against solutions to the classical Bernoulli–Euler beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes, which shows the capability of the derived equations of motion and proposed boundary conditions.

Peridynamic analysis of curved elastic beams

Peridynamics is an extended continuum theory which operates with non-local deformation measures as well as long range internal force/moment interactions. The aim of this paper is to present and to analyze peridynamic governing equations for elastic curved beams. To this end the strain energy density is formulated as a function of two non-local deformation measures including the axial deformation and curvature. Applying the Lagrangian formalism, peridynamic equations of motion are derived. To analyze curved beams of finite length peridynamic boundary conditions are required. Examples of boundary conditions including simple support and clamped edge are presented by introducing a fictitious domain. Solutions to PD equations of motion are presented for various problems including static analysis of simply supported and clamped arc beams as well as modal analysis of a slender ring and a simply supported arc beam. To validate peridynamic formulations, solutions for displacements, natural frequencies and vibration modes are compared with those by the classical beam theory. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes which shows the capability of the derived equations of motion and proposed boundary conditions.

Double horizon peridynamics

In this study, a new double horizon peridynamics formulation was introduced by utilising two horizons instead of one as in traditional peridynamics. The new approach allows utilisation of large horizon sizes by controlling the size of the inner horizon size. To demonstrate the capability of the current approach, four different numerical cases were examined by considering static and dynamic conditions, different boundary and initial conditions, and different outer and inner horizon size values. For both static and dynamic cases, it was observed that as the inner horizon decreases, peridynamic solutions converge to classical continuum mechanics solutions even by using a larger horizon size value. Therefore, the proposed approach can serve as an alternative approach to improve computational efficiency of peridynamic simulations by obtaining accurate results with larger horizon sizes.

Predictive Energy Management Strategy for Hybrid Electric Air-Ground Vehicle Considering Battery Thermal Dynamics

Hybrid electric air-ground vehicles (HEAGVs), which can run on the land and fly in the air, are considered a promising future transportation. The operation of HEAGVs, accompanied by high energy consumption, could lead to increasing battery temperature, which may affect the lifespan of the battery. To make the battery last longer and improve energy efficiency, an effective energy management strategy (EMS) is necessary for the operation of HEAGVs. In this regard, this paper proposes a predictive EMS based on model predictive control (MPC). Firstly, speed information is obtained by intelligent network technology to achieve a prediction of power demand, and then the state of charge (SOC) reference trajectory is planned. Secondly, a Pontryagin’s minimum principle-based model predictive control (PMP-MPC) framework is proposed, including battery thermal dynamics. Under the framework, fuel efficiency is improved by reducing the temperature of the battery. Finally, the proposed method is compared to PMP, dynamic programming (DP), and rule-based (RB) methods. The effect of different preview horizon sizes on fuel economy and battery temperature is analyzed. Verification results under two driving cycles indicate that compared with the rule-based method, the proposed method improves fuel economy by 5.14% and 5.2% and decreases the temperature by 5.9% and 4.9%, respectively. The results demonstrate that the proposed PMP-MPC method can effectively improve fuel economy and reduce temperature.

Unfolding AIS Transmission Behavior for Vessel Movement Modeling on Noisy Data Leveraging Machine Learning

The oceans are a source of an impressive mixture of complex data that could be used to uncover relationships yet to be discovered. Such data comes from the oceans and their surface, such as Automatic Identification System (AIS) messages used for tracking vessels’ trajectories. AIS messages are transmitted over radio or satellite at ideally periodic time intervals but vary irregularly over time. As such, this paper aims to model the AIS message transmission behavior through neural networks for forecasting upcoming AIS messages’ content from multiple vessels, particularly in a simultaneous approach despite messages’ temporal irregularities as outliers. We present a set of experiments comprising multiple algorithms for forecasting tasks with horizon sizes of varying lengths. Deep learning models (e.g., neural networks) revealed themselves to adequately preserve vessels’ spatial awareness regardless of temporal irregularity. We show how convolutional layers, feed-forward networks, and recurrent neural networks can improve such tasks by working together. Experimenting with short, medium, and large-sized sequences of messages, our model achieved <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$36/37/38\%$ </tex-math></inline-formula> of the Relative Percentage Difference– the lower, the better, whereas we observed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$92/45/96\%$ </tex-math></inline-formula> on the Elman’s RNN, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$51/52/40\%$ </tex-math></inline-formula> on the GRU, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$129/98/61\%$ </tex-math></inline-formula> on the LSTM. These results support our model as a driver for improving the prediction of vessel routes when analyzing multiple vessels of diverging types simultaneously under temporally noise data.

An extended ordinary state-based peridynamics for non-spherical horizons

This work presents an extended ordinary state-based peridynamics (XOSBPD) model for the non-spherical horizons. Based on the OSBPD, we derive the XOSBPD by introducing the Lagrange multipliers to guarantee the non-local dilatation and non-local strain energy density (SED) are equal to local dilatation and local SED, respectively. In this formulation, the XOSBPD removes the limitation of spherical horizons and is suitable for arbitrary horizon shapes. In addition, the presented XOSBPD does not need volume and surface correction and allows non-uniform discretization implementation with various horizon sizes. Four classic examples demonstrate the accuracy and capability for complex dynamical fracture analysis. The proposed method provides an efficient tool and in-depth insight into the failure mechanism of structure components and solid materials.

Some closed form series solutions to peridynamic plate equations

Peridynamics is a generalized continuum theory which is developed to account for long range internal force/moment interactions. Peridynamic equations of motion are of integro-differential type and only several closed-form analytical solutions are available for elementary structures. The aim of this paper is to derive analytical solutions to peridynamic Kirchhoff type (shear rigid) plates equations for both static and dynamic cases. Applying trigonometric series with respect to in-plane coordinates, solutions for the deflection function of a rectangular simply supported plate are derived. The coefficients in the series are presented in a closed analytical form such that the equations of motion are exactly satisfied. For the dynamic case the solution is derived applying the variable separation with respect to the time and the in-plane coordinates. Several numerical cases are presented to illustrate the derived solutions. Furthermore, results of peridynamic plate equations are compared against results obtained from the classical plate theory. A very good agreement between these two theories is obtained for the case of the small horizon sizes which shows the capability of the presented approach.

Beam buckling analysis in peridynamic framework

Peridynamics is a non-local continuum theory which accounts for long-range internal force/moment interactions. Peridynamic equations of motion are integro-differential equations, and only few analytical solutions to these equations are available. The aim of this paper is to formulate governing equations for buckling of beams and to derive analytical solutions for critical buckling loads based on the nonlinear peridynamic beam theory. For three types of boundary conditions, explicit expressions for the buckling loads are presented. The results are compared with the classical results for buckling loads. A very good agreement between the non-local and the classical theories is observed for the case of the small horizon sizes which shows the capability of the current approach. The results show that with an increase of the horizon size the critical buckling load slightly decreases for the fixed overall stiffness of the beam.

Some analytical solutions to peridynamic beam equations

AbstractPeridynamics (PD) has been introduced to account for long range internal force/moment interactions and to extend the classical continuum mechanics (CCM). PD equations of motion are derived in the form of integro‐differential equations and only few analytical solutions to these equations are presented in the literature. The aim of this paper is to present analytical solutions to PD beam equations for both static and dynamic loading conditions. Applying trigonometric series, general solutions for the deflection function are derived. For several examples in the static case including simply supported beam and cantilever beam, the coefficients in the series are presented in a closed analytical form. For the dynamic case, the solution is derived for a simply supported beam applying the variable separation with respect to the time and the axial coordinate. Several numerical cases are presented to illustrate the derived solutions. Furthermore, PD results are compared against results obtained from the classical beam theory (CBT). A very good agreement between these two different approaches is observed for the case of the small horizon sizes (HSs), which shows the capability of the current approach.

A General Numerical Method to Model Anisotropy in Discretized Bond-Based Peridynamics

This work presents a novel and general method of determining bond micromoduli for anisotropic linear elastic bond-based peridynamics. The problem of finding a discrete distribution of bond micromoduli that reproduces an anisotropic peridynamic stiffness tensor is cast as a least-squares problem. The proposed numerical method is able to find a distribution of bond micromoduli that is able to exactly reproduce a desired anisotropic stiffness tensor provided conditions of Cauchy’s relations are met. Examples of all eight possible elastic material symmetries, from triclinic to isotropic are given and discussed in depth. Parametric studies are conducted to demonstrate that the numerical method is robust enough to handle a variety of horizon sizes, neighborhood shapes, influence functions and lattice rotation effects. Examples of a 2D single edge cracked plate are used to demonstrate the use of the proposed method for practical problems.

A Method of Developing Quantile Convolutional Neural Networks for Electric Vehicle Battery Temperature Prediction Trained on Cross-Domain Data

The energy consumption caused by battery thermal management of electric vehicles can be reduced using predictive control. A predictive controller needs a prediction model of the battery temperature, for example for different battery cooling and heating thresholds. In the proposed method, cross-domain data from simulation, vehicle fleet and weather stations were analyzed and processed as training data for a Convolutional Neural Network (CNN). The CNN took data from previous road segments and predictions for following road segments as input and predicted the change in battery temperature as quantile sequences over a prediction horizon. Properties of the collected cross-domain data sets were analyzed and considered during preprocessing, before 150 models were trained, of which the best performing model was further analyzed. Point-forecast metrics and quantile-related metrics were used for model comparison and evaluation. For example, the median prediction achieved a mean absolute error (MAE) of 0.27 &#x2218;C and the true values were below the median prediction in 47% of the test data. Possible improvements of the method such as increasing data size, using more complex architectures as well as optimizing the horizon sizes were discussed. In conclusion, the method was able to well predict battery temperatures for different battery cooling thresholds.

Predictive energy management for plug-in hybrid electric vehicles considering electric motor thermal dynamics

Energy management is essential for improving the fuel economy of plug-in hybrid electric vehicles (PHEVs). Some existing efforts have focused on optimizing fuel consumption and battery degradation, but without adequately considering the onboard electric motor's (EM) thermal dynamics. To address this research gap, this paper proposes a predictive energy management strategy considering EM thermal control. Specifically, we make three main contributions that distinguish our study from the existing studies. First, we design four velocity predictors based on the artificial neural network (ANN) and examine their prediction accuracy and computational efficiency. Second, we present a Pontryagin's Minimum Principle-based model predictive control (PMP-MPC) framework that includes EM thermal dynamics. The framework minimizes the operating costs while ensuring that the EM temperature is less than the limit value. Finally, we analyze and compare the effects of different reference temperature thresholds and preview horizon sizes on the fuel economy and EM temperature. The results demonstrate that the proposed PMP-MPC approach can effectively control the EM temperature rise and realize online applications with high computational efficiency.

Numerical Modeling on Crack Propagation Based on a Multi-Grid Bond-Based Dual-Horizon Peridynamics

Peridynamics (PD) is a novel nonlocal theory of continuum mechanics capable of describing crack formation and propagation without defining any fracture rules in advance. In this study, a multi-grid bond-based dual-horizon peridynamics (DH-PD) model is presented, which includes varying horizon sizes and can avoid spurious wave reflections. This model incorporates the volume correction, surface correction, and a technique of nonuniformity discretization to improve calculation accuracy and efficiency. Two benchmark problems are simulated to verify the reliability of the proposed model with the effect of the volume correction and surface correction on the computational accuracy confirmed. Two numerical examples, the fracture of an L-shaped concrete specimen and the mixed damage of a double-edged notched specimen, are simulated and analyzed. The simulation results are compared against experimental data, the numerical solution of a traditional PD model, and the output from a finite element model. The comparisons verify the calculation accuracy of the corrected DH-PD model and its advantages over some other models like the traditional PD model.

A vector form conjugated-shear bond-based peridynamic model for crack initiation and propagation in linear elastic solids

A vector form conjugated-shear bond-based peridynamic model is proposed to overcome the limitation of the fixed Poisson’s ratio based on the assumption of central force between material points in bond-based peridynamic theory. In the virtue of this proposed approach, the angle variation of a pair of conjugated-shear bonds is derived as a vector in terms of the displacements at three points located at the conjugated-shear bonds instead of the direct use of the angle, which improves the computational efficiency in both two and three dimensional conditions. Emphasis is placed on deriving the relationship between micro moduli in the proposed model and macro constants by equating peridynamic strain energy to the macroscopic strain energy. It is found that the micro moduli in this proposed approach can reduce to classical bond-based peridynamic parameters when the rotation effect of conjugated-shear bonds is not considered. A comparative study of the proposed approach and the FEM or analytical solutions on a square isotropic plate and a three-dimensional rectangular block under uniaxial tension shows that this proposed approach can well investigate the behavior of elastic solids under both two and three dimensional conditions. We test the numerical convergence behavior of the proposed model to classical solutions in the varying horizon/grid spacing ratios or horizon sizes. Moreover, the improvement of computational efficiency of the proposed model is verified by comparison of the CPU time computed from the scalar form conjugated bond-based peridynamic model. The capability of the proposed model in simulating dynamic brittle fracture problems is demonstrated by comparing the results obtained by the proposed model with the previous numerical results in the case of Kalthoff-Winkler experiment.

Crack nucleation in brittle and quasi-brittle materials: A peridynamic analysis

Peridynamic (PD) models of bodies without pre-cracks, based on a single fracture parameter (associated with the critical fracture energy), produce different strengths when different horizon sizes are used to simulate crack nucleation under quasi-static conditions. To maintain the same strength and fracture energy under different horizon sizes, extra parameters have to be introduced in the failure model. Bilinear and trilinear bond force-strain relationships have been proposed in the literature for crack propagation in quasi-brittle materials. In this paper we study crack nucleation in a plate with a hole under quasi-static loading using bilinear and trilinear PD models. We provide analytical formulas to calibrate the models to measurable material properties. We show convergence for both strength and fracture toughness. The bilinear PD constitutive model works well for both brittle (e.g. ceramics) and quasi-brittle (e.g. concrete) systems, while the trilinear version is more suited for quasi-brittle fracture behavior. We also find that for quasi-brittle fracture, a model that accounts, stochastically, for the presence of small-scale pores/defects performs better than a homogenized model. A wedge-splitting test in concrete and crack nucleation in a quasi-isotropic composite plate with a circular hole are used to demonstrate the model’s performance. In contrast with other models, the current formulation does not depend on the sample geometry.

Thermal diffusion analysis by using dual horizon peridynamics

In this study, Dual Horizon Peridynamics formulation is presented for thermal diffusion analysis. Lagrangian formalism is utilized to derive the governing equations. The proposed formulation allows utilization of variable discretization and horizon sizes inside the solution domain which can result in significant benefits in terms of computational time. To demonstrate the capability of the Dual Horizon Peridynamics formulation, three different example problems are considered including a square plate with temperature and no flux boundary conditions, a square plate under thermal shock loading, and a square plate with an insulated crack. For all problems that are considered good agreement is obtained between peridynamics (PD) predictions and finite element method (FEM) results.

An in-depth investigation of critical stretch based failure criterion in ordinary state-based peridynamics

This study presents an in-depth investigation of the critical stretch based failure criterion in ordinary state-based peridynamics for both static and dynamic conditions. Seven different cases are investigated to determine the effect of the failure parameter on peridynamic forces between material points and dilatation. Based on crack opening displacement (COD) results from both peridynamics and finite element analysis, it was found that one of the seven cases provides the best agreement between the two approaches. This particular case is further investigated by considering the influence of the discretisation and the horizon sizes on COD and crack propagation speeds. Moreover, PD predictions of COD for PMMA material is analysed with the theory of dynamic fracture mechanics and compared with the fracture experiments. It is shown that the peridynamic model can correctly model, simulate and predict the behaviour of the crack under different loading conditions. Furthermore, the presented PD models capture accurate fracture phenomena, specifically the crack path, branching angles and crack propagation speeds, which are in good agreement with experimental results.