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A precise integration algorithm is used in combination with the extended Wittrick–Williams algorithm to solve the differential equations governing stationary random wave propagation in layered anisotropic material. The natural frequencies and corresponding modes of vibration are calculated to arbitrary precision. Four examples are given, two of which highlight a phenomenon corresponding to beats.

The linear buckling problems of plates and shells were analysed using a recently developped quadrilateral, 16-degrees of freedom flat shell element (called DKQ16). The geometrical stiffness matrix was established. Comparison of the numerical results for several typical problems shows that the DKQ16 element has a very good precision for the linear buckling problems of plates and shells.

The numerical integration for the ordinary differential equations is extremely important for applications. So far, almost all the numerical integration methods apply the finite difference approximation even for a time-invariant system.The precise integration method (PIM) solves the time step integration for the time-invariant system first. For such a problem, precise integration gives a highly precise numerical result, which approaches the full computer precision. After the integration of time-invariant system is solved, various approximate methods can be applied to other problems such as time-variant or nonlinear time integration.The numerical integration of two-point boundary value problems (TPBVP) is also very important in applications. Such as wave propagation, optimal control, structural mechanics, electro-magnetic wave guide problems, etc. The PIM can also be applied to solve the TPBVP. The TPBVP induced initial value problems such as the Lyapunov differential equation, the Riccati differential equation and the Kalman–Bucy filter equation, etc. can also be solved along the same way.In this paper, the essence of precise integration will be explained.

For the constrained nonlinear optimal control problem, by taking the first term of Taylor series, the dynamic equation is linearized. Thus by introducing into the dual variable (Langrange multiplier vector), the dynamic equation can be transformed into Hamilton system from Lagrange system on the basis of the original variable. Under the whole state, the problem discussed can be described from a new view, and the equation can be precisely solved by the time precise integration method established in linear dynamic system. A numerical example shows the effectiveness of the method.

The solution of 3 D elastic-plastic frictional contact problems belongs to the unspecified boundary problems where the interaction between two kinds of nonlinearities should occur. Considering the difficulties for the solution of 3 D frictional contact problems, the key part is the determination of the tangential slip states at the contact points, and a great amount of computing work is needed for a high accuracy result. A new method based on a combination of programming and iteration methods, which are respectively known as two main kinds of methods for contact analysis, was put forward to deal with 3 D elastic-plastic contact problems. Numerical results demonstrate the efficiency of the algorithm illustrated here.

The variational approach is further applied to the measurement feedback H∞ control problems. Based on the induced norm description of the system Γ, the variational functionals Jc and Jp of state feedback H∞ control for the future time interval (t, tf], and of H∞ filter for the past time interval [0, t), respectively, are combined together to generate the variational functional of the measurement feedback for the whole time interval [0, tf]. The connection condition at the present time t is that the estimated state vector\(\bar x\)(t) must be continuously extended to be the initial condition of the future state vector estimation. Another connection condition for the dual vector λ(t) can be naturally derived from the variational principle. The equations thus derived show that the third condition for the optimal parameter γ α -2 is again a bound of the smallest Rayleigh quotient. Therefore, the precise integration method developed formerly to determine the optimal parameter γ α -2 of H∞ control and of H∞ filter respectively can be further applied to the determination of the optimal parameter.

By using the precise integration method, the numerical solution of linear quadratic Gaussian (LQG) optimal control problem was discussed. Based on the separation principle, the LQG control problem decomposes, or separates, into an optimal state-feedback control problem and an optimal state estimation problem. That is the off-line solution of two sets of Riccati differential equations and the on-line integration solution of the state vector from a set of time-variant differential equations. The present algorithms are not only appropriate to solve the two-point boundary-value problem and the corresponding Riccati differential equation, but also can be used to solve the estimated state from the time-variant differential equations. The high precision of precise integration is of advantage for the control and estimation. Numerical examples demonstrate the high precision and effectiveness of the algorithm.

Pseudo-division algorithm for matrix multivariable polynomial are given, thereby with the view of differential algebra, the sufficient and necessary conditions for transforming a class of partial differential equations into infinite dimensional Hamiltonianian system and its concrete form are obtained. Then by combining this method with Wu's method, a new method of constructing general solution of a class of mechanical equations is got, which several examples show very effective.

This paper is concerned with the formulation and numerical realization of elastic—plastic frictional contact problems. Comparing with two-dimensional elastic-plastic contact problems, three-dimensional elastic-plastic contact problems with friction are more difficult to deal with, because of the unknown slip direction of the tangential force and exhausting computing time when the double nonlinear problems are considered. In order to avoid these difficulties, a combined PQP (Parametric Quadratic Programming) and iteration method is constructed in this paper. The stiffness matrix of a 3-D contact element is introduced by means of a penalty function expression of the contact pressure and frictional force. The contact condition and the flow rule are expressed by the same form as in a non-associated plastic flow problem, and the penalty factors can be eliminated by using a definite numerical technique. The iteration algorithm, whose functions are to give a precise discretization of space Coulomb's friction law, is used along with the PQP method and alleviates the difficulty of unknown slip direction and cuts down computing costs. Also, the additional complementary conditions are discussed here, so that the non-physical ray solutions caused by original mathematical model can be eliminated. Numerical examples are given to demonstrate the validity of the present method.

Non-uniform spatial grids are commonly used to resolve locally fast-changing physical phenomena in space. If a traditional explicit time integration scheme is used to advance solutions in the temporal dimension, the step size is restricted by the stability criterion, which is in turn dictated by the smallest grid spacing in the spatial dimensions. It turns out that the excessively small time step enforced by the smallest spatial grid spacing in a domain is usually unnecessary for most of the grid points with larger grid spacing. A new adaptive time integration method is introduced in this paper to improve the computational efficiency. The basic idea is to use different time step sizes at different spatial grid points. The stability criterion is still satisfied at all grid points by using different time step sizes. In this way, the time step size adjusts automatically based on the local spatial grid spacing. Complexity analysis and implementation details are also discussed in the paper. The non-linear Burger's equation is used in the numerical experiment. Both the complexity analysis and numerical computations demostrate significant improvement of computational efficiency.

A precise integration algorithm has recently been proposed by Zhong (1994) for dynamic stiffness matrix computations, but he did not give a corresponding eigenvalue count method. The Wittrick-Williams algorithm gives an eigenvalue count method for pure displacement formulations, but the precise integration method uses a mixed variable formulation. Therefore the Wittrick-Williams method is extended in this paper to give the eigenvalue count needed by the precise integration method and by other methods involving mixed variable formulations. A simple Timoshenko beam example is included.

In this paper, a new Banach space ZH is defined, and it is proved that there is completeness of eigenfunction systems (symplectic orthogonal system) of a class of Hamiltonian system in ZH space. We have also proved the following results: ZH space can be continuously imbedded toL 2 [0, 1]×L 2 [0, 1], butZH ≠ L 2 [0,1] × L 1 [0, 1].

The traditional semi-inverse solution method of the Saint Venant problem, which is described in the Euclidian space under the Lagrange system formulation, is updated to be solved in the symplectic space under the conservative Hamiltonian system. It is proved in the present paper that all the Saint Venant solutions can be obtained directly via the zero eigenvalue solutions and all their Jordan normal form of the corresponding Hamiltonian operator matrix.

Comparing with two-dimensional contact problems, three-dimensional frictional contact problems are more difficult to deal with, because of the unknown slip direction of the tangential force and enormous computing time. In order to overcome these difficulties, a combined PQP (Parametric Quadratic Programming) and iteration method is derived in this paper. The iteration algorithm, which alleviates the difficulty of unknown slip direction, is used along with the PQP method to cut down computing costs. Numerical example is given to demonstrate the validity of the present algorithm.

The governing equations of plane elasticity in sectorial domain are derived to be in Hamiltoinan form via variable substitutes and variational principles. The method of separation of variables and eigenfunction expansion method are derive1 to solve the finite element analytically for the sectorial domain elasticity problem, so that such kind of analytical element can be installed into FEM program systems. It demonstrates the potential of the Hamiltonian system theory and symplectic mathematics.

The regular equations on the constraint variables are established for LQ and nonlinear control problems in this paper, then the extreme-value principles of the constraint variables are discussed for the equality and unequality constraint cases respectively. At last, the given example verifies the conclusion of this paper. The work in this paper will lay the foundation for the further study about the constrained LQ and nonlinear control systems.

The wave propagation constants of periodic structures are computed using the exact dynamic stiffness matrix of a typical substructure. The approach used is to show that wave propagation and the natural vibration eigenproblem are similar to such an extent that methods used to find the natural frequencies of a structure can be applied to find its wave propagation constants. The Wittrick-Williams algorithm has been incorporated into a finite element program, JIGFEX, in conjunction with exact dynamic member stiffnesses, to ensure that no phase propagation eigenvalues are missed during computation. The accuracy of the present approach is then demonstrated by comparing the results that it gives to analytically determined wave propagation curves for a Timoshenko beam on periodic simple supports. Finally, phase propagation curves are given for a complex Timoshenko beam structure of a type that would be very difficult to analyze analytically.

The methodology presented in Part 1 is employed to deal with flexure and free vibration of anisotropic plates.

In the theory of mechanics and/or mathematical physics problems in a prismatic domain, the method of separation of variables ususally leads to the Sturm–Liouville-type eigenproblems of self-adjoint operators, and then the eigenfunction expansion method can be used in equation solving. However, a number of important application problems cannot lead to self-adjoint operator for the transverse coordinate. From the minimum potential energy variational principle, by selection of the state and its dual variables, the generalized variational principle is deduced. Then, based on the analogy between the theory of structural mechanics and optimal control, the present article leads the problem to the Hamiltonian system. The finite-dimensional theory for the Hamiltonian system is extended to the corresponding theory of the Hamiltonian operator matrix and adjoint symplectic spaces. The adjoint symplectic orthonormality relation is proved for the whole state eigenfunction vectors, and then the expansion of an arbitrary whole state function vector by the eigenfunction vectors is established. Thus the range of classical method of separation of variables is considerably extended. The eigenproblem derived from a plate bending problem in a strip domain is used for illustration. © 1993 John Wiley & Sons, Inc.

Fundamental theory presented in Part (I)[8] is used to analyze anisotropic plane stress problems. First we construct the generalized variational principle to enter Hamiltonian system and get Hamiltonian differential operator matrix; then we solve eigen problem; finally, we present the process of obtaining analytical solutions and semi-analytical solutions for anisotropic plane stress problems on rectangular area.

For the first time, Hamiltonian system used in dynamics is introduced to formulate statics and Hamiltonian equation is derived corresponding to the original governing equation, which enables separation of variables to work and eigen function to be obtained for the boundary problem. Consequently, analytical and semi-analytical solutions can be got. The method is especially suitable to solve rectangular plane problem and spatial prism in elastic mechanics.

The consolidation analysis of interaction between structure and saturated soil foundation is discussed. With the use of substructure technique, the structure is condensed onto the interface of the soil, and then the consolidation governing equations to describe the interaction between soil and structure are derived. The solution with non-iterative algorithm is proposed in this paper. The pressure Master-Slave relation method is used to deal with the non-permeability conditions on soil boundaries. A numerical example is illustrated. Based on this paper, the interactive consolidation analysis between large structure and soil has been more practical.

The theory of optimal control and the semianalytical method of elliptic partial differential equation (PDE) in a prismatic domain are mutually simulated issues. The simulation of discrete-time linear quadratic (LQ) control with the substructural chain problem in static structural analysis is given first. From the minimum potential energy variational principle of substructural chain, the generalized variational principle with two kinds of variables and the dual equations are derived. The simulation relation is then recognized by comparing the variational principle and dual equations of the LQ control theory. The simulation between elliptic PDE in the prismatic domain and continuous-time LQ control is established in the same way, and the interval energy is naturally introduced, as in the case of substructural chain. The assembling and condensation equations can help one to derive the differential equations of the submatrices of potential energy and mixed energy. The well known Riccati equation is one of them. The interval assembling and condensation algorithm can be used to solve the Riccati equation. Some numerical examples are given to illustrate the method.

The theory of optimal control and the theory of a substructural chain in static structural analysis are mutually simulated issues. From the minimum potential energy variational principle of the substructural chain, the generalized variational principle with two kinds of variables is derived first. By comparing that generalized variational principle with the variational principle in LQ control theory, the simulation relation is established. Based on that relation, the potential energy and mixed energy formulation of the algebraic Riccati equations are derived, then iterative algorithms are proposed which give the upper and lower bounds to the solution matrix. By using the solutions of the positive and negative co-ordinate algebraic Riccati equations, the canonical transformation matrices for the eigenproblems of the substructural chain and LQ control are constructed respectively, which reduce the eigenproblem to half-size. The properties of the solutions are analysed, which establishes the basis for expansion solutions.

This paper presents the parametric variational principle for Perzyna model which is one of the main constitutive relations of viscoplasticity.The principle,by which the potential energy function is minimized under a constrained condition transformed by the constitutive relations of viscoplasticity, is free from the bound of Drucker's postulate of plastic flow and consequently suitable for solving the nonassociated plastic flow problems. Furthermore, the paper has proven the presented principle and discussed the creep problem.

In this paper, a new method for dealing with substructure chain is presented. Here, a substructure chain means a structure consisting of a number of identical substructures connected in series. This method is parallel to the expansion method in the structural analysis according to the transverse section. The method of shift invariance and that of initial parameter and transfer matrix, their mutual-relation and combined use, as well as the analysis of the eigenproblem on the transverse section, and the expansion method based on the eigenvectors are described in this paper.

The degeneration of the eigenvalue equation of the discrete-time linear quadratic control problem to the continuous-time one when Δt→0 is given first. When the continuous-time n-dimensional eigenvalue equation, which has all the eigenvalues located in the left half plane, has beeh reduced from the original 2n-dimensional one, the present paper proposes that several of the eigenvalues nearest to the imaginary axis be obtained by the matrix transformation Ae=eA. All the eigenvalues of Ae are in the unit circle, with the eigenvectors unchanged and the original eigenvalues can be obtained by a logarithm operation. And several of the eigenvalues of Ae nearest to the unit circle can be calculated by the dual subspace iteration method.

A new mode superposition substructure method with dynamic superelements is presented for forced vibration analysis of multilevel substructure systems. The algorithms derived in this paper are further development on the studies of dynamic superelement method for free vibration analysis by the authors. Thus a compatible method for both free vibration and forced vibration analyses of substructure systems with multilevel superelements has been implemented. In this method, one can take full advantage of a simple static substructure method and eliminate the errors resulting from static condensations. It has no restriction on substructuring. The formulations are frequency independent and not involved with nonlinear eigen problems. Therefore, the method is adapted to arbitrary multilevel substructure systems for free vibration and forced vibration analyses with both simplicity and accuracy.