Precise Integration Method Research Articles

The performance analysis of energy piles in cross-anisotropic soils

Energy piles possess two important properties – structural load-bearing and ground heat exchangers, which is a cost-effective building energy-saving solution. Besides, natural soils generally exhibit anisotropic properties which impact the overall performance of energy piles. Therefore, the performance of energy piles in layered cross-anisotropic soils is investigated in this paper. Firstly, based on the basic equations of elasticity and heat transfer theory, the extended precise integration method (EPIM) is applied to acquire the thermo-elastic fundamental solution of layered cross-anisotropic soils, which will be utilized as the kernel function of the boundary element method (BEM). Then, the stiffness matrix equation of the energy pile considering the thermal strain and mechanical loads is derived with the aid of the finite element method (FEM). Afterwards, according to soil-pile deformation coordination conditions, the coupled BEM-FEM formulation to represent the soil-pile interaction is developed. Compared with in-situ tests and numerical simulations, the accuracy of the proposed method is verified. Finally, parametric analyses are conducted on the performance of energy piles subjected to the mechanical and temperature load simultaneously, and influences of pile-soil stiffness ratio, length-diameter ratio and soil cross-anisotropy on energy pile responses are discussed.

Parameter Identification Method for Nonsmooth Aeroelastic System

In this paper, a parameter identification method merged with the precise integration method and the predictor–corrector strategy is proposed to identify the parameters of a nonsmooth aeroelastic system quickly and accurately. Considering the aeroelastic system subject to unsteady aerodynamic force, the parameters that need to be identified include the structural and nonsmooth parameters. The work of this paper mainly includes three parts. First, based on the precise integration method and combined with the predictor–corrector strategy, a new technology that can accurately locate the switching point is proposed to obtain the response of the aeroelastic system with high precision. Then, the problem for identification of parameters of the nonsmooth aeroelastic system is modeled as a nonlinear least-squares problem between the measured data and the calculated response. A new strategy that can obtain the nonsmooth parameters sensitivity matrix accurately is proposed, and then the parameters are identified through the enhanced response sensitivity approach. Finally, the nonsmooth aeroelastic system with freeplay and hysteresis nonlinearity are taken as the numerical examples to demonstrate the feasibility and robustness of the present method.

Dynamic analysis on flexible hub-beam with step-variable cross-section

Decreasing the vibration amplitude of the manipulator in the Space Manipulator Systems (SMS) with a given mass limitation is an effective way to reduce the control costs for the SMS. In this paper, the dynamic response of the SMS modeled by a flexible hub-beam with a step-variable cross-section excited by an external torque is investigated employing a new complex structure-preserving method. The dynamic model of the flexible hub-beam with a step-variable cross-section is deduced firstly. Then, a new structure-preserving method connecting the symplectic precise integration method and the generalized multi-symplectic method is proposed, in which, the time-consuming iteration is avoided and the simulation time is saved. In the numerical simulations, several simple cases for the step-variable cross-section are considered and the stable vibration amplitudes of the flexible beam at the free end are reported. The main contribution of this work is proposing a structure-preserving method for the solving the coupling dynamic problems of the hub-beam system to provide some suggestions on the structure design for the SMS directly.

Modelling the thermal–hydro-mechanical behaviour of unsaturated soils with a high degree of saturation using an extended precise integration method

Unsaturated soils with a high degree of saturation (HDS) are commonly encountered in marine and lacustrine sediments. In these soils, the gas phase generally exists in the state of discrete bubbles, which is sensitive to stress and temperature changes and can dramatically change the soil's engineering properties. This paper explores the thermal-induced behaviour of HDS soils using an efficient extended precise integration method (XPIM). Biot poroelasticity theory, extended to include thermal effects and compressibility of gas–water mixture, is employed to analyze the soil behaviour under non-isothermal conditions. Based on the Laplace–Fourier transform and Taylor series expansion, such problems can be solved by XPIM. The robustness of XPIM was confirmed by comparing the present results with analytical solutions and test data. Extensive parametric studies are undertaken to examine both the effects of soil grain thermal expansion and anisotropic permeability on soil behaviour and temperature effects on the degree of saturation ( Sr). The thermal-induced variations in Sr are more pronounced with lower initial values (e.g., 90% compared to 99%). These quantitative results demonstrate the benefits of the proposed method, which proves to be extremely efficient and several orders more precise than conventional numerical approaches, with its precision limited only by the computational effort used.

A Generalized Method for Dispersion Analysis of Guided Waves in Multilayered Anisotropic Magneto-Electro-Elastic Structures

Based on the symplectic structure of the Hamiltonian matrix, the precise integration method (PIM), and the Wittrick–Williams (W-W) algorithm, a generalized method for computing the dispersion curves of guided waves in multilayered anisotropic magneto-electro-elastic (MEE) structures for different types of mechanical, electrical, and magnetical boundaries is developed. A strictly theoretical analysis shows that the W-W algorithm cannot be applied directly to the MEE structure. This is because a block of the Hamiltonian matrix is not positive definite for MEE structures so that the eigenvalue count of the sublayer is not zero when the divided sublayer is sufficiently thin. To overcome this difficulty, based on the symplectic structure of the Hamiltonian matrix, a symplectic transformation is introduced to ensure that the W-W algorithm can be applied conveniently to solve wave propagation problems in multilayered anisotropic MEE structures. The application of the PIM based on the mixed energy matrix to solve the wave equation can ensure the stability and efficiency of the method, and all eigenfrequencies are found without the possibility of any being missed using the W-W algorithm. This research provides the necessary insight to apply the W-W algorithm in wave propagation and vibration problems of MEE structures.

Elastodynamic analyses of transversely isotropic unsaturated subgrade–pavement system under moving loads

AbstractIn this paper, elastodynamic responses of the layered transversely isotropic unsaturated subgrade–pavement system subjected to the moving load are investigated. The unsaturated subgrade soil is described via a triphasic Biot‐type model introducing the effect of matric suction, immiscible flows of water and air, and transverse isotropy. The rigid pavement is modeled by the Kirchhoff plate theory, while the flexible pavement is modeled by the stratification medium theory. First, the governing equations of the unsaturated subgrade soil are obtained after algebraical manipulations. The precise integration method (PIM) is further combined with the double Fourier transform to solve the governing equations. The subgrade soil solution is utilized as the flexible coefficient to describe the plate–soil interaction, and the solution of the pavement is further acquired via the double inverse Fourier transform. Four verification examples regarding literatures and COMSOL are presented to verify the correctness of the theory. Finally, numerical examples are conducted to investigate the effect of various parameters on the dynamic behavior of the unsaturated subgrade–pavement system.

A new two-dimensional transient forecast model of wellbore temperature based on precise time step integration method

Dramatic temperature changes in the wellbore can lead to the severe deformation and breakage of the wellbore string, packer seal malfunction, and other operational accidents. A new two-dimensional (2D) transient forecast model of the wellbore temperature is established to provide accurate temperature parameters for the precision mechanical analysis of the wellbore string and the safety evaluation of well integrity. Furthermore, a precise time step integration scheme of the forecast model using the precise time step integration method of a one-point subdomain is proposed. A solution method for the 2D transient forecast model of the wellbore temperature is proposed by combining the precise time step integration method. Finally, a case study focused on a high-temperature deep gas well with a transient wellbore temperature is presented by comparing the calculation results of the different models and solution methods. The sensitivity analysis of the wellbore temperature is performed as well. The results indicate that the forecast model in this paper has higher accuracy than the existing two-dimensional transient model. By contrast analysis of the finite difference method and the solution method that combines the precise time step integration method, the solution method has the advantages of better convergence and faster calculation speed. The major sensitive parameters that affect the stability time of wellbore temperature are ranked as follows: gas production rate, tubing size, specific heat, temperature gradient, gas relative density, and thermal conductivity. The work presented can provide a theoretical foundation and technological basis for the transient forecast of the wellbore temperature, which is critical for well completion design and wellbore safety evaluation.

Dynamics of plates resting on layered transversely isotropic poroelastic media under moving loads

This paper investigates the dynamic interaction between a plate and multi-layered transversely isotropic (TI) hysteretically damped poroelastic media under a moving rectangular load. Following Kirchhoff's thin plate theory, the plate deflection is derived utilizing the Fourier series expansion and the integral transform technique. The governing equation for the layered TI hysteretically damped poroelastic media is formulated through the standard hysteretic damping model and Biot's theory. The solution for the plate-soil dynamic interaction in the wavenumber domain considering the plate-soil interface condition is obtained by the extended precise integration method (PIM) for the layered TI saturated porous media. This solution can be further converted into the spatial domain by the inverse transform technique. We have verified the efficiency and accuracy of the proposed method. Several numerical examples are presented to investigate the influences of the plate thickness, plate modulus ratio, surface layer, speed of the moving load, and hysteretic damping of the solid skeleton on the dynamic behaviors of the plate on layered TI poroelastic media.

Dynamic behaviour of layered transversely isotropic poroelastic media subjected to rectangular harmonic loads

AbstractThis paper aims to investigate the dynamic behaviour of layered transversely isotropic (TI) poroelastic media subjected to rectangular harmonic loads. First, governing equations of a medium layer derived from Biot's dynamic theory of porous media and constitutive equations of TI materials are converted into the ordinary differential domain by means of integral transform techniques. Next, by introducing boundary and load conditions, the extended precise integration method (PIM) is applied to obtain the solution for a multi‐layered system. Then, after an inverse integral treatment, the numerical results in the physical domain are verified with published work, and some examples are designed to unravel the effect of transverse isotropy, load frequencies and stratification on the dynamic behaviour of media.

Dynamic analysis of eccentrically loaded rigid strip foundation on layered transversely isotropic poroelastic media

This paper presents an analytical method for analyzing vertical and rocking motions of a rigid strip foundation on layered transversely isotropic poroelastic media. The extended precise integration method is employed to obtain the plain strain dynamic responses of layered poroelastic media due to the reaction force from the rigid strip. Then, the contact stresses for dynamic soil-structure interactions (SSI) are modelled by the Bessel series expansions, and a set of linear equations can be developed from the dual integral equations of mixed boundary value problems. The explicit expressions for the contact stresses are further obtained by solving these linear equations. Finally, the presented method is verified, and numerical examples are provided to demonstrate the effects of stratifications, transverse isotropy and permeability of media on the dynamic responses of eccentrically loaded rigid strip foundation.

An isogeometric boundary element method for transient heat transfer problems in inhomogeneous materials and the non-iterative inversion of loads

In this paper, the transient thermal analysis theory of isogeometric dual reciprocity boundary element method (IG-DRBEM) of the multi-patch inhomogeneous complex model is established and the non-iterative inversion method of load identification is proposed based on the IG-DRBEM. The establishment of the multi-patch analysis theory framework extended the ability of IG-DRBEM to analyze complex geometry to some extent. The multi-patch analysis method adopted herein is simple and operable, and still retains the advantages of seamless connection between CAD generated geometry and response analysis of isogeometric boundary element method (IGBEM), which is very beneficial to expand IG-DRBEM to practical engineering application fields in the future. The inversion method based on IG-DRBEM makes full use of the good geometrical and field representation ability of NURBS, and further expands the application scope of IG-DRBEM. Furthermore, the introduction of the precision integration method for both forward and inverse problems improves the accuracy and stability of calculation to a certain extent. The numerical results show that the proposed method has good stability and accuracy even when solving complex geometric problems such as a fighter model. Moreover, the implementation of basis function expansion and regularization scheme further improves the accuracy and noise resistance of boundary condition inversion.

Axis-symmetric analysis of layered transversely isotropic saturated elastic soils containing a monopile under time-harmonic vibration

The axis-symmetric dynamic soil-pile interaction analysis is of fundamental significance in geotechnical, seismic and transportation engineering. In addition, soils tend to show an obvious stratification after the long-term deposition. Physical and mechanical properties of soils are generally transversely isotropic due to the orientation of soil particles in deposition. Therefore, the axis-symmetric analysis of layered transversely isotropic saturated elastic soils containing a monopile under time-harmonic vibration is carried out in this study. Firstly, the extended precise integration method (EPIM) is employed to obtain the dynamic fundamental solution for layered transversely isotropic saturated soils, which would be utilized as the kernel function of the boundary element method (BEM). Moreover, by modeling the pile as a one-dimensional bar, the motion equation of the monopile under steady-state vibration is acquired based on the finite element method (FEM). Finally, the coupled BEM-FEM formulation representing the soil-pile interaction is developed and the axis-symmetric solution for layered soils containing a monopile is acquired. Compared with existing results, the accuracy of the proposed method is verified. Numerical examples are conducted to analyze the influence of frequency, transverse isotropy, the stratification of soils and the length-radius ratio of piles on soil-pile interaction behaviors.

Numerical simulation of dynamic fractures in 2D FGMs using the numerical manifold method

In this study, the numerical manifold method is used with an explicit time integration scheme to implement dynamic fracture analysis of 2D functionally graded materials. The framework of the numerical manifold method is used to discretize the dynamic equilibrium equation in the spatial domain. Instead of the commonly used Newmark method, the precise time step integration method is used for temporal discretization, and the detailed implementations of the proposed technique are described in detail in this paper. The dynamic stress intensity factors that can describe the inertia effect are extracted by the domain form of the interaction integral with nonequilibrium formulation. For verification purposes, four numerical examples with increasing complexity are investigated, in which different material gradients, crack configurations and loading forms are considered. Simulation results show that the proposed method achieves high reliability and precision for dynamic problems of 2D cracked functionally graded materials.

Three‐dimensional dynamic response of transversely isotropic saturated media subjected to transient loads

AbstractThis paper investigates three‐dimensional (3D) transient responses of stratified transversely isotropic (TI) saturated media. Firstly, we transform the partial differential equations in Cartesian coordinates into ordinary differential equations (ODEs) for the convenience of solving. Then, we consider the transient load forms and apply the extended precise integration method (EPIM) to study the response of stratified TI saturated media. This paper involves the transverse isotropy of solid constituent, the permeable anisotropy of pore fluid, and various load forms including concentrated loadings and uniform rectangular loadings. The solution in the physical domain can be obtained through the numerical inversion of Laplace‐Fourier transform. Several examples are set up to verify the proposed theory and to investigate the influences of the transverse isotropy of solid constituent, the permeable anisotropy of pore fluid and the stratification of media.

Dynamic response analysis of isolated curved beam bridge under multi-dimensional non-stationary random excitation considering the torsion component

The coupling effect of bending and torsion caused by plane irregularity of the isolated curved beam bridge makes its seismic response more complex than that of the isolated straight beam bridge. In order to study the influence of multi-dimensional non-stationary random excitation considering torsion component on seismic response of tqiongw@mls.sinanet.com he isolated curved beam bridge. The classical Bouc-Wen model was used to simulate the hysteretic characteristics of the curved beam bridge, and the nonlinear dynamic equation considering the eccentricity of superstructure was established. The pseudo excitation was transformed into harmonic external load by Euler formula, and combining with the precise integration method, the precise integration forms of the special solution under different modulation functions were derived. By using these integral forms to solve the response at each time, the time-varying variance of displacement of the isolated curved beam bridge under multi-dimensional non-stationary random excitation was obtained, and the law of time-varying variance of displacement, including the upper and lower structures of the isolated curved beam bridge, was analyzed with different deck width, curvature radius and yield ratio of bearing. The results show that the multi-dimensional non-stationary random excitation with torsion component has significant influence on the seismic response of the isolated curved beam bridge. Under the rare non-stationary random excitation, the dynamic response of the isolated curved beam bridge presents strong non-stationarity, and the time-lag phenomenon is apparent. For the isolated curved beam bridge with large deck width and small yield ratio of bearing, the influence of non-stationary random excitation on the structure is particularly obvious. The influence of non-stationary random excitation considering torsion component on dynamic response of the isolated curved beam bridge with large deck width, small curvature radius and large yield ratio of bearing is more significant.

Research on Shear Lag Warping Displacement Modes of Frame-Tube Structures Based on the Hamiltonian Mechanics

Based on the equivalent continuity method, the accuracy of the shear lag warping displacement functions for frame-tube structures was studied under the Hamiltonian mechanics. Different types of functions were selected to describe the shear lag warping displacement of the flange plate, and the shear deformation and longitudinal warping of the equivalent plate were considered. The total potential energy of the structure and the corresponding Lagrangian function under different displacement modes were obtained. Not with the traditional variational methods, the problem was studied under the Hamiltonian mechanics system. The Hamiltonian canonical equation for the frame-tube structure was derived and solved with the precise integration method, then the column axial force was calculated and the accuracy was analyzed. The verification results of the calculation examples show that, this method is simple and feasible to analyze the shear lag effects of the frame-tube structures. The choice of different warping displacement functions has little effect on the lateral displacement calculation results, but has great influence on the axial force solution, and the quadratic parabola can best reflect the actual warping displacement distribution of the flange. Comparison of the stress distributions in the equivalent flange under different types of loads indicate that, with the increase of the position of the external load resultant force, the negative shear-lag effect on the top gradually weakens to disappear.

Vertical Dynamic Response of a Pile Embedded in Layered Transversely Isotropic Unsaturated Soils

This paper investigates the dynamic response of a pile embedded in layered transversely isotropic unsaturated soils subjected to a vertical external excitation. The quintessential Biot biphasic model for saturated media is replaced with a triphasic model for unsaturated media, which simulates more realistic terrestrial subsurface soil characteristics. With the aid of the boundary element method (BEM), the soil around the pile is divided into a series of elements and the Green’s function is obtained by the extended precise integration method (EPIM), while the pile is modeled as a 1D bar based on the finite element method (FEM). Thereafter, the soil–pile interaction formulation is acquired and further solved by the coupling of the BEM-FEM. The correctness of the proposed theory is verified by a comparison example with the existing results. A series of parametric studies are further provided to discuss the influences of degree of saturation, pore size distribution, and air-entry pressure on the dynamic behavior of the pile.

Analysis of Sound Propagation Across Acoustic Ducts With Discontinuities Using a High Precision Finite Element Method

A high-precision numerical method is presented to model acoustic wave propagation along a duct with discontinuities. The discontinuous duct can be divided into homogeneous and inhomogeneous substructures. The inhomogeneous substructures are purely discretized by the conventional finite element method, while only cross-sectional areas need to be discretized for homogeneous substructures. The Legendre transformation is then used to transform the semi-discretized problem from the Lagrangian system into the Hamiltonian system. A Riccati equation-based high precision integration method is taken to perform the integral along the longitudinal direction, i.e., the homogeneous direction, to generate stiffness matrices of substructures. The final system stiffness matrix is obtained by assembling stiffness matrices of homogeneous substructures with stiffness matrices of inhomogeneous substructures. Numerical examples are provided to validate the proposed method, and these examples have shown high efficiency and accuracy compared with the conventional finite element method.

An optimal control method for time-delay feedback control of 1/4 vehicle active suspension under random excitation

Time-delay feedback control can effectively broaden the damping frequency band and improve the damping efficiency. However, the existing time-delay feedback control strategy has no obvious effect on multi-frequency random excitation vibration reduction control. That is, when the frequency of external excitation is more complicated, there is no better way to obtain the best time-delay feedback control parameters. To overcome this issue, this paper is the first work of proposing an optimal calculation method that introduces stochastic excitation into the process of solving the delay feedback control parameters. It is a time-delay control parameter with a better damping effect for random excitation. In this paper, a 2 DOF one-quarter vehicle suspension model with time-delay is studied. First, the stability interval of time-delay feedback control parameters is solved by using the Lyapunov stability theory. Second, the optimal control parameters of the time-delay feedback control under random excitation are solved by particle swarm optimization (PSO). Finally, the simulation models of a one-quarter vehicle suspension simulation model are established. Random excitation and harmonic excitation are used as inputs. The response of the vehicle body under the frequency domain damping control method and the proposed control method is compared and simulated. To make the control precision higher and the solution speed faster, this paper simulates the model by using the precise integration method of transient history. The simulation results show that the acceleration of the vehicle body in the proposed control method is 13.05% less than the passive vibration absorber under random excitation. Compared with the time-delay feedback control optimized by frequency response function, the damping effect is 12.99%. The results show that the vibration displacement, vibration velocity, and vibration acceleration of the vehicle body are better than the frequency domain function optimization method, whether it is harmonic excitation or random excitation. The ride comfort of the vehicle is improved obviously. It provides a valuable tool for time-delay vibration reduction control under random excitation.

Research on the mechanical model of cord-reinforced air spring with winding formation

Abstract In this article, the parametric model for the stiffness characteristic and burst pressure of cord-reinforced air spring with winding formation is developed. Based on the non-geostrophic winding model and the assumption of cord cross-stability, the cord winding trajectory model of the capsule is established. Then, the anisotropic and nonlinear mechanics model of the capsule with complex cord winding trajectory variation characteristics is constructed by the classical thin-shell theory. The capsule state vector is solved by the extended homogeneous capacity precision integration method. Due to the complex coupling relationship between the capsule state vector and the internal air pressure, the stiffness characteristic is solved by the iterative integration method. The burst pressure of the air spring is solved by the Tsai–Hill strength theory. Eventually, the accuracy and reliability of the proposed method are verified by the experimental results. The effects of the material properties, winding parameters, and geometric structure parameters on stiffness characteristics and burst pressure are discussed. The results of this article provide an important theoretical basis for the performance design of cord-reinforced air springs with winding formation.