Precise Integration Method Research Articles

Axisymmetric consolidation behavior of multilayered unsaturated soils with transversely isotropic permeability

Time-dependent consolidation behavior of unsaturated soils is a vital problem in the geotechnical engineering. With the aid of the Fredlund consolidation theory, this work further assumes the total stress of soils skeleton freely change, and extends the Fredlund consolidation theory to a Biot-type theory, establishing the fully-coupled equation model of multilayered unsaturated poroelastic media with transversely isotropic permeability. To convert the partial differential governing equation into ordinary differential equations, the integration transform technology is applied. Subsequently, the precise integration method is used to acquire the time-dependent consolidation solution of multilayered unsaturated media with transversely isotropic permeability in the transformed domain, which is further solved in the actual domain by the inverse Hankel transform. A verification examples is provided to compare the present results with the existing work in the literature, showing a great coincidence and proving the feasibility of the present solution. Finally, numerous numerical examples are presented to investigate the evolution of excess pore pressure and settlement under quasi-static loads, revealing the consolidation behavior of unsaturated soils. The results demonstrates that the ramping time, stratification, permeability, depth and m1w have a significant effect on the consolidation behavior.

One-step sparse-matrix method for full wave equation depth migration

By completely solving the second-order full-acoustic wave equation, two-way/full wave equation migration generates images up to arbitrary angles in inhomogeneous media. Full wave equation depth migration (FWDM) necessitates less wavefield storage during imaging, generates clearer imaging results, and has more computational flexibility compared with conventional reverse time migration. Conventionally, FWDM uses Runge-Kutta or two explicit one-way operators to extrapolate the data, which involves either a multirecursive process or complex square root computations of the Helmholtz operator, causing implementation complexities and inconveniences. To overcome these challenges, a novel algorithm called the one-step sparse-matrix method for FWDM (OSM-FWDM) is developed and executed. This method involves a straightforward calculation of the full wave equation propagator through a Taylor expansion and using the so-called precise integration method. The computational efficiency can be significantly improved by using sparse-matrix multiplication in the implementation of wavefield extrapolation. Building upon that, the method serves as a basic solution complemented by FWDM, offering straightforward, efficient, and easy coding characteristics. Our OSM-FWDM method is simpler and faster compared with conventional FWDM, all while maintaining imaging accuracy. In addition, a new solution that uses an extended velocity model to remove source-related artifacts in FWDM is developed and illustrated. These improvements are demonstrated through a series of numerical examples and a real data application.

Dynamic analysis of fractional poroviscoelastic reinforced subgrade under moving loading

This paper conducts the dynamic analysis of fractional poroviscoelastic reinforced subgrade under moving loading. Based on the Biot theory and transversely isotropic (TI) parameter expression of the geogrid reinforced subgrade, the governing equations of the poroelastic reinforced subgrade are established in the wavenumber domain by the double Fourier transform. Considering the viscosity of the soil skeleton and the flow-dependent viscosity between the soil skeleton and pore water, the governing equations are extended to the fractional poroviscoelastic medium by introducing the Zener viscoelastic model, fractional calculus theory and the dynamic elastic-viscoelastic correspondence principle. Combining boundary conditions and interlayer continuity conditions, the extended precise integration method (PIM) and double Fourier integral transform are employed to obtain the solution of fractional poroviscoelastic reinforced subgrade in the spatial domain. After the numerical validation, a sensitivity analysis of the relaxation time, permeability, reinforcement ratio and the load velocity are conducted.

Dynamic responses of a plate on layered transversely isotropic poroelastic subgrade under moving thermal and mechanical loads

In this paper, the dynamic responses of a plate on layered transversely isotropic (TI) poroelastic subgrade under moving thermo-mechanical loads are investigated. Based on the thermodynamics, poroelastic mechanics and the Kirchhoff's thin plate theory, the governing equations of the plate and layered TI saturated subgrade are derived in the wavenumber domain. Then, by introducing the flexibility coefficient of the layered TI poroelastic subgrade and the plate-subgrade interface condition, the solution of the plate-subgrade dynamic interaction is obtained through the extended precise integration method. After the presented method is validated, parametric studies are carried out to confirm the influences of the thermal load, plate-subgrade modulus ratio, plate thickness, and the speed of moving load. Parametric analysis shows that the effect of the thermal load cannot be ignored when considering the plate-subgrade dynamic interaction. In addition, the plate-subgrade modulus ratio, plate thickness, and the speed of moving load have a significant effect on the plate-subgrade dynamic interaction.

Precise integration boundary element method for solving nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity

The precise integration boundary element method (PIBEM) is used to solve nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity in this article. At first, a boundary-domain integral equation can be obtained by using the fundamental solution for steady linear heat conduction problems. Secondly, the domain integrals are converted into boundary integrals by using the radial integration method to get a pure boundary integral equation, which can retain the advantage of dimension reduction of the boundary element method. Then, after discretization, a first-order nonlinear differential equation system in time is obtained, and the precise integration method with predictor-corrector technique is used to solve the system of nonlinear equations. In the end, several numerical examples are presented to verify the accuracy and stability of the presented method. The results obtained by the presented method are less affected by the time step size and satisfactory results can be obtained even for a large time step size.

Time-frequency analysis of nonlinear structures under nonstationary excitation by directly solving structural dynamic equation through absolute displacement

The time-frequency evolution of seismic ground motion energy under strong earthquakes, which can be described by the evolutionary power spectral density (EPSD) matrix, will amplify the most unfavorable responses of members in nonlinear structural systems. However, the existing general finite element software fails to reduce the dimension of the EPSD matrix and then decouple time from frequency in frequency domain analysis. This failure greatly limits the simulation and calculation of multi-dimensional nonlinear structures with multiple degrees of freedom by the stochastic vibration theory. In this study, the pseudo excitation method (PEM), which can direct solve structural dynamic equations using absolute displacement, is used to derive the self-spectral and cross-spectral densities of seismic ground motions in one-dimensional and multi-dimensional cases. In MATLAB, the EPSD matrix is transformed into a four-dimensional uniform modulation excitation matrix independent of the time variable. Nonlinear iteration is conducted and the precise integration method (PIM) is employed through a transient analysis module. Furthermore, a simple one-dimensional example is presented to verify the correctness of the proposed algorithm herein. Finally, as an example, we select important parameters of the coherence model at different supports of a half-through three-span concrete-filled steel tubular (CFST) tied-arch bridge in view of real engineering conditions, and calculate the time-varying power spectrum and variance of the responses of arch ribs, arch feet and the bridge deck system. The results show that the theoretical derivation and the algorithm constructed in this paper can simulate the time-delay effect of the nonlinear structural system. This proves the feasibility of general finite element software in solving and analyzing random responses to multi-dimensional and multi-support non-stationary excitation of nonlinear structures.

New unconditionally stable and high-accuracy finite element procedures for high-frequency solid-fluid coupled dynamic systems

To achieve high-precision computation of the dynamic response of saturated poroelastic media under high-frequency loads, this paper develops new, unconditionally stable and high-accuracy finite element procedures based on the u-w and u-U solid-fluid coupled dynamic systems. In the procedures the solution domain is spatially discretized using the standard finite element method, while a precise time step integration method is introduced for temporal domain computations. A matrix-based spectral analysis is conducted to validate the stability of the proposed procedures. Subsequently, an error estimation and convergence analysis of the procedures are performed. Four classical examples involving different load types with varying time dependences are used to validate the numerical performance of the proposed procedures. Through rigorous comparison with analytical/reference solutions, the proposed procedures have been convincingly demonstrated to accurately capture the relative motion between the solid skeleton and the pore fluid under high-frequency loads. Furthermore, the procedures have remarkable advantages in both accuracy and efficiency compared to the commonly employed Newmark schemes.

Fractional derivative modeling for dynamic response of layered viscoelastic media under moving time-harmonic loads

The dynamic response of layered viscoelastic media under moving time-harmonic loads is an important issue in transportation geotechnics. However, the existing research often has limitations in terms of the constant creep rate or time-independent relaxation modulus. To fill this gap, this study introduces a fractional Zener model to simulate the viscoelastic properties of media. First, the governing equations of the elastic medium are converted into ordinary differential equations using an integral transform. Subsequently, by introducing the fractional viscoelastic model and extended precise integration method, a solution to the dynamic response of layered viscoelastic media under moving time-harmonic loads is obtained. After comparing the results with those in existing literature, the reliability and accuracy of the proposed solution were verified. Finally, a parametric analysis was performed to further explore the influence of load velocity and fractional order on the dynamic response of the viscoelastic media.

Performance analysis of multilayered transversely isotropic saturated media under temperature and horizontally circular loads

AbstractThis study constructs a multilayered transversely isotropic saturated model under thermal and horizontally circular loads, and further investigates the model's thermo‐hydro‐mechanical coupling response. Firstly, the ordinary differential matrix equations of thermoelastic saturated media in the integral transformed domain are derived. Secondly, the solution for multilayered thermoelastic saturated media is developed using the extended precise integration method (EPIM), along with the boundary conditions at both ends of the foundation and the continuity conditions between adjacent layers. After that, the solution in the physical domain is further attained with the use of Laplace‐Hankel integral transform inversion. Finally, the accuracy of the proposed theory is confirmed through numerical examples, and the influences of anisotropic parameters, the soil's stratification and porosity on the thermo‐hydro‐mechanical coupling response of the media are studied.

Behavior analysis of energy piles in layered transversely isotropic saturated soils

This paper conducts theoretical research on energy piles in layered saturated transversely isotropic soils under different load forms. Firstly, the Timoshenko beam coupled axial force rod elements is used to simulate the pile elements, and the finite element equilibrium equation for energy single piles is derived accordingly. Then, the extended precise integration method (EPIM) is applied to obtain the fundamental solution of soil, which is regarded as the kernel function of the boundary element method to construct the boundary integral equation. Based on the theory of the coupled finite element method - boundary element method (FEM-BEM), and combined with the equilibrium conditions of force and displacement coordination at the pile-soil interface, the boundary integral equation of soil is coupled with the finite element equilibrium equation of the pile to establish the interaction equation. The calculation results of this method are compared with theoretical solutions and in-site field tests to verify the correctness of this theory. Finally, numerical examples are designed to explore the influence of the parameters of energy piles and soils on the behavior of energy piles in layered saturated transversely isotropic soils under vertical and horizontal loadings.

Dynamic analysis on an asymmetric spatial dumbbell-type model

The structural symmetric breaking of the spatial structure will enhance the coupling dynamic behaviors and bring new challenges for the dynamic analysis on the spatial structure inevitably. The main contribution of this paper is proposing a structure-preserving iteration method to investigate the effects of the dynamic symmetry-breaking factors on the dynamic behaviors of the rigid-flexible coupling systems. Firstly, the spatial flexible damping beam assembled with two particles on both ends is simplified as an asymmetric spatial dumbbell-type model. For this model, the coupling dynamic equations are presented based on the Hamiltonian variational principle. Then, connecting the symplectic precise integration method for the ordinary differential equations mainly controlling the plane motion of the model and the generalized multi-symplectic scheme for the partial differential equation mainly controlling the transverse vibration of the beam, a structure-preserving numerical iteration method is proposed, which provides a new way to analysis the dynamic behaviors of the coupling problems with the symmetric breaking. Finally, the effects of the absolute mass and the mass ratio of the additional particles on the orbit radius, the orbit true anomaly velocity, the attitude angle velocity of the model and the transverse vibration of the flexible beam are reproduced by using the structure-preserving numerical iteration method in the numerical simulations. Particularly, the effects of the additional particles on the attitude stability of the model and the vibration dissipation of the flexible damping beam are investigated, which gives some guidance for the attitude adjustment strategy and the vibration control method for the spatial flexible structure with the structural symmetric breaking.

The consolidation behavior of layered fractional viscoelastic soils considering groundwater

AbstractThis paper investigates the consolidation behavior of multi‐layered viscoelastic soils considering groundwater. First, the fractional Merchant viscoelastic model is introduced to describe the behavior of multi‐layered viscoelastic soils considering groundwater. Later, the governing equations are extended to a viscoelastic medium by virtue of the elastic‐viscoelastic corresponding principle in the Laplace–Hankel domain. According to the extended precise integration method, the soil layer is divided into a series of layer units. Then the relationship between general stress vector and general displacement vector on the top and bottom planes is established. Every two adjacent layer units are combined into one layer in each computational iteration. The solutions in the Laplace–Hankel domain are obtained by considering the boundary conditions, and numerical inversion is performed to obtain the solutions in the physical domain. The practicability of the present method is assessed by comparing the numerical results with those in the existing literature and done by ABAQUS. Finally, the effects of groundwater table, properties of the soils above groundwater table, load depth, viscoelastic parameters, and soil stratification are investigated.

Nonstationary time-frequency characteristics of vehicle suspension systems based on evolutionary power spectral density

Improving the ride comfort for vehicles that travel at varying speeds is a nonstationary problem and has been an increasingly important topic for suspension system research. Accurate identification of frequencies associated with the resonant responses is needed for effective design of active suspension control strategies. However, traditional time-frequency (TF) methods have resolution limitations in either the time domain or the frequency domain. This paper proposes a new methodology to employ the evolutionary power spectral density (EPSD) to describe the responses of vehicles in the time-frequency domain. The key to calculate the responses in nonstationary cases is to derive the nonuniform modulation function which is obtained by the pseudo excitation method. To illustrate the applicability of the proposed method, an example of a seven-degree-of-freedom vehicle model is investigated in which the nonstationary excitations acting on the wheels are computed by the nonuniform modulation function and a precise integration method. Results show that resolutions obtained by the EPSD are more precise than the traditional TF methods with less than 0.2 Hz of bandwidth in both the lower frequency range at the first natural frequency of 1 Hz and in the higher frequency around 10 Hz. Moreover, the changing of the resonant frequency with varying speeds provides the insights, from the perspective of human body’s sensitivity to frequencies of vibration, to improve the ride comfort for vehicles under nonstationary driving conditions.

Dynamic stiffness matrix perturbation theory for time-varying structural analysis

Matrix perturbation theory (MPT) is widely used in sensitivity analysis and reanalysis of structural eigenvalue problems when small changes are imposed on parameters. However, the conventional MPT based on the finite element method is only applicable to the cases with small parameter variations, and it is difficult to avoid cumulative errors caused by multi-step perturbations. A key issue for the application of MPT is how to evaluate the error of perturbation solution and correct the results during the process of multi-step perturbation solution. By invoking the dynamic stiffness method (DSM), Wittrick-Williams algorithm, and the precise integration method (PIM), in this paper, we propose an improved method with controllable accuracy, self-checking and elimination of accumulated errors, referred to as the dynamic stiffness matrix perturbation theory (DSMPT). In present method, the rationality of the results can be estimated preliminarily according to dynamic stiffness equation, which is not present in the original MPT, can greatly enhance the accuracy of the perturbation results and evaluate the errors. Case studies prove that the DSMPT combined with PIM provides a convenient way to realize accuracy self-inspection and rectification of the perturbation results, and the maximum relative error of eigenvalue and response can be controlled to be smaller than 0.2% and 0.025%, respectively. Besides, DSMPT can also be applied to the analysis of coupled and nonlinear dynamic system with time-varying modal characteristics. Finally, a thermally-induced vibration analysis of tensegrity modules during space deployment is taken as an example to further illustrate the advancements of the DSMPT in solving time-varying dynamic problems. Results show that the DSMPT effectuate an almost sixfold reduction in computational time.

A new time integration method based on state formulations for dynamic analysis of nonviscously damped systems

The mode superposition method (MSM), which is used for dynamic response calculations of nonviscously damped systems, requires state–space eigensolutions, which incur a high computational cost. This study proposed an alternative symmetric formulation-based time integration method for analyzing structural dynamic systems. An anelastic displacement field (ADF) model was introduced to capture damping mechanisms in nonviscous systems. The differential equation describing the ADF model was transformed into an augmented coupled system via the introduction of dissipative variables. Two symmetric state formulations based on coupled augmented systems were derived for nonviscous systems. Consequently, alternative asymmetric state formulations were derived for nonviscous systems involving the ADF model. Thereafter, a new method was developed based on these state–space formulations by assuming linear approximations. A numerical implementation of the new method for dynamic systems was demonstrated. Three benchmark examples of discrete and continuous systems were considered to evaluate the asymmetric formulation and the computational efficiency, accuracy, stability, and time-step-size sensitivity of the new method. Further, the response of the new method was validated using MSM and precise time integration (PTI) methods. An asymmetric state formulation was also validated against MSM and direct frequency response methods. The results revealed that the new method accurately matched the MSM and PTI methods at various time steps. The results also showed that the new method was more efficient than the MSM and PTI methods, rendering it suitable for solving large-scale structural dynamics problems.

Shannon‐Cosine wavelet precision integration method for image enhancement based on fractional partial differential equations

AbstractFractional order partial differential equations have a wide range of applications in image processing. The solution of partial differential equations is generally obtained using the finite difference method, which still requires improvement in terms of efficiency and effectiveness. In this work, a multi‐scale interpolative wavelet operator is constructed by means of Shannon‐Cosine wavelets with interpolation, smoothness, compact support and other excellent properties. We use the wavelet operator instead of the finite difference operator to solve fractional‐order partial differential equations. The proposed method can limit the artefacts and remove the noises appeared in the processed images effectively. In addition, for the loss of texture in the denoised image, we enhance the image by employing fractional order differential equations to improve the quality of the image. Finally, the biological sections images are taken as the examples to illustrate the effectiveness of the proposed method.

Dynamic response of fractional poroviscoelastic layered media subjected to moving loads

This study investigates the dynamic response of fractional poroviscoelastic layered media subjected to a moving harmonic load. The material viscosity includes two parts: the flow-dependent viscosity between the skeleton and pore water and the skeleton viscosity. Firstly, the governing equations of poroelastic media are derived based on Biot's theory and then transformed into ordinary differential equations by using the double Fourier integral transform. By incorporating fractional calculus theory and the dynamic correspondence principle, the applicability of these equations is extended to fractional poroviscoelastic media. The extended precise integration method (PIM) is employed to obtain solutions. Subsequently, the fractional Zener model and the presented method are validated. Finally, parametric studies are conducted to analyze the effects of fractional order, moving velocity, load frequency, and stratification on the dynamic behaviors of the media.

MLS-based numerical manifold method based on IPIM for 3D transient heat conduction of FGMs

In this study, the moving least squares based numerical manifold method (MLS-NMM) is presented to solve three-dimensional (3D) transient heat conduction problems of functionally graded materials (FGMs), where the increment dimensional precise integration method (IPIM) is applied to integrate the ordinary differential equations derived from the semi-discrete form. In the 3D MLS-NMM, the influence domains of the MLS-nodes are taken as the mathematical patches which are used to construct the mathematical cover (MC), and the shape functions of the MLS-nodes are employed as the weight functions subordinate to the MC. MLS-NMM is utilized for spatial discretization of the weak form of the problem. The IPIM overcomes the shortcomings of traditional backward difference scheme dependent on the time step and greatly improves the computing efficiency. Meanwhile, a mass lumping technique is proposed for the 3D MLS-NMM. Furthermore, a number of numerical experiments on transient thermal conduction are conducted, suggesting that the 3D MLS-NMM based on IPIM and the proposed mass lumping has excellent accuracy, absolutely stable and high computing efficiency.

Mesh reduction methods for thermoelasticity of laminated composite structures: Study on the B-spline based State Space Finite Element Method and Physics-Informed Neural Networks

Calculating the mechanical characteristics such as interlaminar stresses of composite laminated structures is a major challenge. To avoid the disadvantages of the traditional finite element method (FEM) of discretizing the whole structure too cumbersome, we implemented a method which only need to discretize on horizontal planes. The state space differential equation is derived from the Hellinger–Reissner variational principle, and both quadratic and cubic B-splines are used to interpolate the in-plane physical quantities, while the precise integration method is carried out along thickness to obtain accurate numerical results efficiently. The method is hence treated as the B-spline based State Space FEM (B-SSFE). To further reduce degree of freedoms, we implemented another method based on physics-informed neural networks (PINNs), which is completely meshless. Deep Learning is combined with differential equations carrying physical information, and neural networks are trained by constraining the loss function. We calculated different boundary conditions for a 3D composite laminated plate under thermal loading. Comparison is conduct among B-SSFE, PINNs and FEM. Numerical results demonstrate that in terms of computational accuracy, B-SSFE can be more accurate than FEM dealing with complex boundary conditions, and PINNs can reflects the stress fields well.

Consolidation behaviors of gas-bearing sediments with modulus varying along depth under horizontal loads

Gas-bearing sediments are widely distributed in marine and lacustrine areas. The engineering properties of gas-bearing sediments, containing enclosed bubbles, are significantly different with that of saturated and unsaturated sediments. This paper investigates the consolidation behaviors of gas-bearing sediments with modulus varying along depth subjected to horizontal load using an extended precise integration method (XPIM). The compressibility of enclosed bubble is introduced to a new derived seepage equation and a varying modulus along depth is considered in governing equations. With the aid of integral transformation and Taylor expansion, such problems are solved by XPIM, which is proved to be markedly efficient and precise for boundary value problems of porous media than traditional numerical methods. Detailed comparisons against analytical solutions are performed to confirm the accurateness of XPIM. Extensive parametric investigations are conducted to examine the influence of saturation degree, varying modulus along depth and type of external load on the consolidation behaviors of gas-bearing sediments. The present work is conveniently utilized to evaluate the behavior of gas-bearing sediments, and subsequently the response of foundations built in gas-bearing soil areas.