Precise Integration Method Research Articles

Analyses of dynamic characteristics of a fluid-filled thin rectangular porous plate with various boundary conditions

Based on the classical theory of thin plate and Biot theory, a precise model of the transverse vibrations of a thin rectangular porous plate is proposed. The first order differential equations of the porous plate are derived in the frequency domain. By considering the coupling effect between the solid phase and the fluid phase and without any hypothesis for the fluid displacement, the model presented here is rigorous and close to the real materials. Owing to the use of extended homogeneous capacity precision integration method and precise element method, the model can be applied in higher frequency range than pure numerical methods. This model also easily adapts to various boundary conditions. Numerical results are given for two different porous plates under different excitations and boundary conditions.

Enhancement of Precise Integration Method for Dynamic Structural Analysis using Inversion of State Matrix

Enhancement of Precise Integration Method for Dynamic Structural Analysis using Inversion of State Matrix

Homogenization of hexagonal and re-entrant hexagonal structures and wave propagation of the sandwich plates with symplectic analysis

Abstract The aim of this work is to provide closed-form expressions of the effective elastic constants of hexagonal and re-entrant hexagonal structures, which contain the variable dimensional parameters, such as the relative density, aspect ratio, length ratio and the cell wall angle. We also numerically investigate the dynamic properties of the sandwich plates with hexagonal cores. By taking into account the bending, axial and shearing deformations of the unit cell walls, the effective elastic constants are derived. In order to analyze the wave propagation of the sandwich plates, the original governing equations are converted into a set of the first-order governing differential equations in the Hamilton system, by introducing the dual variables and with the help of a variational principle. The precise integration method in conjunction with the extended Wittrick-Williams algorithm is utilized to numerically solve these equations to obtain the frequencies of structures. The effects of relative density, length ratio, cell wall angle and material distribution parameter on the dispersion relations of hexagonal and re-entrant hexagonal structures are investigated. It is found that the stiffness plays a more dominant role on the dispersion relations than that of the mass, and the effects of length ratio and material distribution parameter are more prominent than that of the cell wall angle.

An efficient and accurate method for transient heat conduction in 1D periodic structures

An efficient and accurate method is proposed for solving transient heat conduction in a one-dimensional (1D) periodic structure. Based on the physical features of transient heat conduction, the periodic property of the structure and the physical meaning of the matrix exponential, the sparsity of the matrix exponential corresponding to the 1D periodic structure and the repeatability of the elements in the matrix are proved in detail in this paper. According to the algebraic structure of the matrix exponential and the precise integration method (PIM), an efficient and accurate method is proposed by computing the matrix exponential corresponding to a representative periodic structure (RPS) with a few unit cells instead of computing the matrix exponential corresponding to the entire periodic structure. The proposed method achieves significantly improved computational efficiency in terms of both CPU time and memory. Meanwhile, the method inherits the accuracy and stability of the original PIM. Those merits of the proposed method are verified through two numerical examples.

Thermal performance of stratified fluid-filled geomaterials with compressible constituents around a deep buried decaying heat source

This paper investigates the transient thermal performance of stratified fluid-filled geomaterials with compressible constituents due to a buried decaying heat source. Starting with the governing partial differential equations, the ordinary differential matrix equation is derived by means of Laplace transform and Hankel transform. Using the extended precise integration method, the ordinary differential matrix equation is solved in the transformed domain, and the actual solutions are acquired through the inversion of Laplace and Hankel transforms. Two numerical calculations are given to validate the accuracy and feasibility of the presented approach, and other calculations are performed to study the effects of the heat source’s half-life, compressible constituents, thermal expansion coefficients, and the stratified character on the transient thermal response of medium.

A novel extended precise integration method based on Fourier series expansion for periodic Riccati differential equations

SummaryA new, reliable algorithm for nonnegative, stabilizing solutions for the periodic Riccati differential equation (PRDE) is proposed based on Fourier series expansion and the precise integration method (PIM). Taking full advantages of periodicity, we expand coefficient matrices of the underlying linear time‐varying periodic Hamiltonian system associated with the PRDE in Fourier series, and a novel extended PIM for the transition matrix of linear time‐varying periodic systems is developed by combining the doubling algorithm with the increment‐storage technique. This method needs to compute the matrix exponential and its related integrals only once for all evenly divided subintervals, which greatly improves the computational efficiency. Further, by introducing the Riccati transformation, a fast recursive formula for the PRDE is derived based on the block form of the transition matrix computed by the extended PIM. Finally, two numerical examples are presented to verify the numerical accuracy and efficiency of the proposed algorithm with compared results.

A Highly Accurate Approach for Aeroelastic System with Hysteresis Nonlinearity

We propose an accurate approach, based on the precise integration method, to solve the aeroelastic system of an airfoil with a pitch hysteresis. A major procedure for achieving high precision is to design a predictor-corrector algorithm. This algorithm enables accurate determination of switching points resulting from the hysteresis. Numerical examples show that the results obtained by the presented method are in excellent agreement with exact solutions. In addition, the high accuracy can be maintained as the time step increases in a reasonable range. It is also found that the Runge-Kutta method may sometimes provide quite different and even fallacious results, though the step length is much less than that adopted in the presented method. With such high computational accuracy, the presented method could be applicable in dynamical systems with hysteresis nonlinearities.

基于精细积分法和传递矩阵法的变截面压杆临界力

针对变截面压杆临界力提出了一种基于精细积分法和传递矩阵法的组合法。先将压杆离散为若干段，视每段压杆为等截面，采用材料力学理论、平衡原理及矩阵理论，导出了各段压杆的传递矩阵。然后利用相邻段间变形与内力关系，导出了整个变截面压杆的传递矩阵。最后，利用整个压杆两端约束条件，得到了求解变截面压杆临界力的特征方程。结果表明：本方法是一种易于编程、精度易于控制，也易于使用的方法。 A combination method of critical load of variable cross-section compressive rods was put forward based on precise integration method and transfer matrix method. First of all, the rod was divided into many sections and every section was viewed as uniform cross-section; the transfer matrixes of all the compressive sections were deduced according to theory of material mechanics, equilibrium principle and theory of matrix. Then the transfer matrix of whole variable cross-section compressive rod was obtained by means of deformation and internal force relationships between adjacent sections. Finally, a characteristic equation to solve the critical load of the variable cross-section compressive rod was presented. Results show that the present method is an easy one to program, control precisely and use conveniently.

高精度Fox-Goodwin时间积分法

本文建立了高精度Fox-Goodwin时间积分法。这种方法采用等分步的思想，连乘每分步的Jacobi矩阵建立高精度Jacobi矩阵，然后利用高精度Jacobi矩阵进行递推求解。在构造高精度Jacobi矩阵的过程中，用指数矩阵运算技巧来达到降低计算量，用存贮增量矩阵策略来减小舍入误差的影响。性能分析表明，这种方法与精细积分法相比，具备较高的幅值和相位精度，且格式简单，便于应用。 A Highly Accurate Fox-Goodwin time Integration Method (HAFIM) is presented in this paper. HAFIM creates the highly accurate Jacobi matrix by multi-sub-step notion and uses the highly ac-curate Jacobi matrix for recursive procedure. In the process of constructing the matrix, the exponential matrix operation techniques are employed to reduce the computational efforts and the computer rounding error. The analysis on properties indicates that the proposed method possesses higher amplitude and phase accuracy compared with the Precise time Integration Methods (PIMs). Moreover, the proposed method is very practical given its simplicity.

Dynamic responses of a beam with breathing cracks by precise integration method

The beam structure with breathing cracks subjected to harmonic excitations was modeled by FEM based on Euler-Bernoulli theory, and a piecewise dynamical system was deduced. The precise integration method (PIM) was employed to propose an algorithm for analyzing the dynamic responses of the deduced system. This system was first divided into linear sub-systems, between which there are switching points resulted from the breathing cracks. The inhomogeneous terms due to the external excitations were tackled by introducing auxiliary variables to express the harmonic functions, hence the sub-systems are homogeneous. The PIM was then applied to solve the homogeneous sub-systems one by one. During the procedures, a predictor-corrector algorithm was presented to determine the switching points accurately. The presented method can provide solutions with an accuracy to a magnitude of 10-12 compared with exact solutions obtained by the theories of ordinary differential equations. The PIM results are much more accurate than Newmark ones with the same time step. Moreover, it is found that the PIM can maintain a high level of accuracy even when the time step increases within a relatively wide range.

Thermal shock analysis of 2D cracked solids using the numerical manifold method and precise time integration

The numerical manifold method (NMM), combined with the precise time integration method (PTIM), is proposed for thermal shock fracture analysis. The temperature and displacement discontinuity across crack faces is naturally portrayed attributing to the cover systems in the NMM. The crack tip singularities are characterized through the use of asymptotic bases in the approximations. The discrete equations for transient thermal analysis are firstly solved with the PTIM and then the thermoelastic study is performed. With the interaction integral, the stress intensity factors are computed. Several examples are tested and the nice consistency between the present and existing results is found.

Structure-preserving Analysis on Folding and Unfolding Process of Undercarriage

The main idea of the structure-preserving method is to preserve the intrinsic geometric properties of the continuous system as much as possible in numerical algorithm design. The geometric constraint in the multi-body systems, one of the difficulties in the numerical methods that are proposed for the multi-body systems, can also be regarded as a geometric property of the multi-body systems. Based on this idea, the symplectic precise integration method is applied in this paper to analyze the kinematics problem of folding and unfolding process of nose undercarriage. The Lagrange governing equation is established for the folding and unfolding process of nose undercarriage with the generalized defined displacements firstly. And then, the constrained Hamiltonian canonical form is derived from the Lagrange governing equation based on the Hamiltonian variational principle. Finally, the symplectic precise integration scheme is used to simulate the kinematics process of nose undercarriage during folding and unfolding described by the constrained Hamiltonian canonical formulation. From the numerical results, it can be concluded that the geometric constraint of the undercarriage system can be preserved well during the numerical simulation on the folding and unfolding process of undercarriage using the symplectic precise integration method.

Structural and acoustic response of a finite stiffened submarine hull

After borrowing the idea of precise integration method, a precise integration transfer matrix method (PITMM) is proposed by modifying traditional transfer matrix method. The submarine hull can be modeled as joined conicalcylindrical-spherical shells. By considering the effect of the ring-stiffeners, the field transfer matrixes of shells of revolution are obtained accurately by PITMM. After assembling the field transfer matrixes into an entire matrix, the dynamic model is established to solve the dynamic responses of the joined shell. By describing the sound pressure in fluid by modified wave superposition method (MWSM) and collocating points along the meridian line of the joined shell, finally the structural and acoustic responses of a finite stiffened submarine hull can be predicted by coupled PITMM and MWSM. The effectiveness of the present method has been verified by comparing the structural and acoustic responses of the spherical shell with existing results. Furthermore, the effects of the model truncation, stiffness and thickness on the structural and acoustic responses of the submarine hull are studied.

A survey of the precise integration method

For the initial value and two-point boundary value problems of linear ordinary differential equation, the Precise Integration Method (PIM) gives exact solutions in the computer accuracy sense. In this paper, the basic idea and the further development of PIM are surveyed. For the initial value problems, the basic idea of PIM, the methods for integrating the nonhomogeneous term, the methods for large scale problems and the application of PIM in time-varying and nonlinear system are surveyed. For two-point boundary value problems, the basic idea and the fundamental formula of PIM are given and the application of the PIM for two-point boundary value problems in many fields are surveyed. Finally, the relationship between the PIM for the initial value and two-point boundary value problems is discussed, which gives a new angle for understanding and application of PIM.

Forced Vibration of Surface Foundation on Viscoelastic Isotropic Multi-Layered Stratum

A numerical approach is presented for analyzing the forced vibration of a rigid surface foundation. In the analysis, the foundation is discretized into a number of sub square-elements. The dynamic response within each sub-element is described by the Green’s function, which is obtained by the Fourier–Bessel transform and the precise integration method (PIM). Then, a system of linear algebraic equation in terms of the contact forces within each sub-element is derived, which leads to the desired dynamic impedance functions of the foundation. Numerical results are obtained for the foundation not only with a simple geometry, such as circular one, but also with irregular shapes. Comparisons between the results obtained by the proposed approach and the thin layered method are made, for which good agreement is achieved. Also, parametric studies are performed on the dynamic response of the foundation, considering the effects of the material damping, stratum depth, Poisson’s ratio and the contact condition of the soil–foundation interface. Several conclusions are drawn concerning the significance of each parameter. Further application of the method can be easily extended to the analysis of a foundation on a viscoelastic anisotropic multi-layered stratum because no further complexity is introduced except the constitutive matrix needs to be modified.

Dynamic modeling and vibration characteristics analysis of submerged stiffened combined shells

A semi-analytical method is present to analysis the vibration response of submerged stiffened combined shells. In general, the submarine hull can be modeled as submerged stiffened combined conical-cylindrical-spherical shells. The precise integration method is imported to develop a precise transfer matrix method (PTMM). The dynamic model is established to solve the dynamic responses of the combined shell in vacuo. The fluid load is described by wave superposition method (WSM). Then the structural responses of a submerged stiffened submarine hull can be obtained by coupled PTMM and WSM method. The effectiveness of the present method has been verified by comparing the frequency parameters of the combined shells and the structural responses of the submerged spherical shell with existing results. Furthermore, the effects of the model truncation, stiffness, damping and fluid load on the structural responses of the combined shells are studied.

A study on calculation method for mechanical impedance of air spring

This paper proposes an approximate analytic method of obtaining the mechanical impedance of air spring. The sound pressure distribution in cylindrical air spring is calculated based on the linear air wave theory. The influences of different boundary conditions on the acoustic pressure field distribution in cylindrical air spring are analysed. A 1-order ordinary differential matrix equation for the state vector of revolutionary shells under internal pressure is derived based on the non-moment theory of elastic thin shell. Referring to the transfer matrix method, a kind of expanded homogeneous capacity high precision integration method is introduced to solve the non-homogeneous matrix differential equation. Combined the solved stress field of shell with the calculated sound pressure field in air spring under the displacement harmonic excitation, the approximate analytical expression of the input and transfer mechanical impedance for the air spring can be achieved. The numerical simulation with the Comsol Multiphysics software verifies the correctness of theoretical analysis result.

Transient response analysis of tapered FRP poles with flexible joints by an efficient one-dimensional FE model

This research develops a finite element code for the transient dynamic analysis of tapered fiber reinforced polymer (FRP) poles with hollow circular cross-section and flexible joints used in power transmission lines. The FRP poles are modeled by tapered beam elements and their flexible joints by a rotational spring. To solve the time equations of transient dynamic analysis, precise time integration method is utilized. In order to verify the utilized formulations, a typical jointed FRP pole under step, triangular and sine pulses is analyzed by the developed finite element code and also ANSYS commercial finite element software for comparison. Thereafter, the effect of joint flexibility on its dynamic behavior is investigated. It is observed that by increasing the joint stiffness, the amplitude of the pole tip deflection history decreases, and the time of occurrence of the maximum deflection is earlier.

A novel finite element two-step solution scheme for fully coupled hydro-mechanical processes in poroelastic media

A novel two-step solution scheme (TSSS) is described for fully coupled hydro-mechanical (FCHM) analysis of saturated poroelastic media. The TSSS is based on the pressure formulation in two-step. In step one, the pressure field is obtained directly by solving the sub-problems with a reduced scale of displacement variables. This process is fully decoupled. In step two, the displacement field is calculated by staggered iteration of pressure variables. The finite element method (FEM) is used for discretization of the FCHM differential equations in the space domain. The precise time step integration method is performed for the time derivatives. The stability and convergence of the TSSS are proved using a matrix-based spectral analysis in the time domain. It is demonstrated that the TSSS is unconditionally stable, fully explicit and highly precise. The algorithmic error estimation results indicate that the numerical performance in the time domain can match the computer precision. Theoretically, the algorithmic error is caused only by mesh discretization. The stability and accuracy of the TSSS are verified and calibrated by numerical examples. By comparing with the analytical or reference solutions, it is shown that the TSSS result is highly precise, and it is remarkably better than the standard FEM in terms of precision. In addition, the numerical results are stable and independent of the time step size. The numerical experiments also demonstrate the stability, convergence and precision for the pressure formulation of TSSS. The proposed scheme has great potential in engineering applications for long timescale problems, especially the problems focusing on the evolution of pressure field.

Research on the Vibration Control Algorithms for Large Space Truss Structure Based on Precise Integration Method

In this paper, vibration control algorithms for a large space truss structure are investigated. First, a precise integration method is used to solve the differential equation such that the validity and efficiency of the algorithm are guaranteed. Second, on the basis of the discrete algebraic equation combined with the decentralized control theory, the solving method of the control force based on the linear quadratic (LQ) optimal control theory is improved. Third, a new control method, the eigenvalue control method, is proposed in this paper, and its relationship with the LQ optimal control method is discussed. Finally, these two control methods are applied to the structure to analyze the effectiveness of vibration suppression. The simulation results show that the proposed algorithms can quickly suppress the vibration of a large space truss structure.