Linear constraint equations for continuous support conditions in finite element analysis

Discretised structural models such as by finite elements imply discretised support conditions. In some cases such as plates on elastic foundation or slabs on large interacting columns an improved formulation of the continuous support conditions is desirable. This can be achieved by means of linear constraint equations. The numerical treatment of linear constraints is discussed for the method of elimination of variables as well as for the method of Lagrange multipliers. Then specific constraint equations for different accuracy requirements are derived, which can be used to constrain rectangular flat shell elements of arbitrary shape functions. These constraints introduce six generalized displacements according to the rigid body motions of the element and transmit the corresponding generalized reactions on the nodal degrees of freedom in a way consistent with distributed reactions. The effect on the strain energy of a square shell element is shown for the different constraint equations. As an application, the linear constraints are used to represent the continuous interaction of columns with the plate in a flat slab structure. Comparison of the finite element solutions with analytical results shows that the derived constraint equations allow a considerably improved formulation of continuous support conditions.     

基于结式的曲面拼接与三维造型

针对很多几何造型是带有约束条件的曲面拼接问题,在线性连续同伦的基础上提出了利用非线性同伦连续计算拼接曲面以进行三维造型的方法。首先,根据得到的截面(切片)的位置及其曲线方程确定插值点并得到插值多项式;其次,将此插值多项式作为非线性连续同伦映射函数并分别代入主曲面和辅助曲面的多项式方程得到过渡曲面的方程;然后,仅将插值变元作为变元而主、辅助曲面方程的变元作为参数,利用Sylvester结式消去过渡方程中的变元得到关于主曲面的拼接方程即造型曲面。利用该方法能实现带有控制点的曲面造型以及多曲面约束的几何造型,而且它可以确定造型过程中的中间形状及中间形状的位置,从而更加具有实用性。     

Collocation method to solve inequality-constrained optimal control problems of arbitrary order

In this paper, the generalized fractional order of the Chebyshev functions (GFCFs) based on the classical Chebyshev polynomials of the first kind is used to obtain the solution of optimal control problems governed by inequality constraints. For this purpose positive slack functions are added to inequality conditions and then the operational matrix for the fractional derivative in the Caputo sense, reduces the problems to those of solving a system of algebraic equations. It is shown that the solutions converge as the number of approximating terms increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach one. The applicability and validity of the method are shown by numerical results of some examples, moreover a comparison with the existing results shows the preference of this method.

     

MHD flow in a rectangular duct with a perturbed boundary

The unsteady magnetohydrodynamic (MHD) flow of a viscous, incompressible and electrically conducting fluid in a rectangular duct with a perturbed boundary, is investigated. A small boundary perturbation $\epsilon$ is applied on the upper wall of the duct which is encountered in the visualization of the blood flow in constricted arteries. The MHD equations which are coupled in the velocity and the induced magnetic field are solved with no-slip velocity conditions and by taking the side walls as insulated and the Hartmann walls as perfectly conducting. Both the domain boundary element method (DBEM) and the dual reciprocity boundary element method (DRBEM) are used in spatial discretization with a backward finite difference scheme for the time integration. These MHD equations are decoupled first into two transient convection–diffusion equations, and then into two modified Helmholtz equations by using suitable transformations. Then, the DBEM or DRBEM is used to transform these equations into equivalent integral equations by employing the fundamental solution of either steady-state convection–diffusion or modified Helmholtz equations. The DBEM and DRBEM results are presented and compared by equi-velocity and current lines at steady-state for several values of Hartmann number and the boundary perturbation parameter.     

Frobenius-Chebyshev polynomial approximations with a priori error bounds for nonlinear initial value differential problems

In this paper, we present a method for approximating the solution of initial value ordinary differential equations with a priori error bounds. The method is based on a Chebyshev perturbation of the original differential equation together with the Frobenius method for solving the equation. Chebyshev polynomials in two variables are developed. Numerical results are presented.     

Trigonometric approximation of optimal periodic control problems for distributed-parameter systems

The optimal periodic control problem for a system described by first order partial differential equations is approximated by a sequence of discretized optimization problems. Trigonometric polynomials in two variables are used in the latter problems to approximate the state trajectory, the control and functions appearing in differential equations and in the criterion of the basic problem. The state equations and the instantaneous constraints on the state and the control are taken into account by the mixed exterior-interior penalty function. Sufficient conditions are given for the convergence of solutions of discretized problems to the optimal solution of the basic problem. The possibility of applying the method to a class of optimal periodic control problems in chemical engineering is emphasized.     

Direct and indirect boundary element methods for solving the heat conduction problem

The boundary element method is used to solve the stationary heat conduction problem as a Dirichlet, a Neumann or as a mixed boundary value problem. Using singularities which are interpreted physically, a number of Fredholm integral equations of the first or second kind is derived by the indirect method. With the aid of Green's third identity and Kupradze's functional equation further direct integral equations are obtained for the given problem. Finally a numerical method is described for solving the integral equations using Hermitian polynomials for the boundary elements and constant, linear, quadratic or cubic polynomials for the unknown functions.     

车桥耦合振动的摄动分析方法

基于结构摄动理论,推导车桥耦合动力系统的摄动方程,将原有的车桥耦合动力方程变成一组车辆和桥梁之间不相互耦合的方程组,实现车辆与桥梁的之间的解耦．理论上基于摄动分析方法进行严格推导,以简支梁为例,采用1／4车辆模型,验证了方法的可靠性．提出了该方法在大型有限元软件中的应用方法．     

Homotopy perturbation transform method for nonlinear equations using He’s polynomials

In this paper, a combined form of the Laplace transform method with the homotopy perturbation method is proposed to solve nonlinear equations. This method is called the homotopy perturbation transform method (HPTM). The nonlinear terms can be easily handled by the use of He’s polynomials. The proposed scheme finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this algorithm over the decomposition method.     

General formulation for the numerical solution of optimal control problems

A new improved computational method for a class of optimal control problems is presented. The state and the costate (adjoint) variables are approximated using a set of basis functions. A method, similar to a variational virtual work approach with weighing coefficients, is used to transform the canonical equations into a set of algebraic equations. The method allows approximating functions that need not satisfy the initial conditions a priori. A Lagrange multiplier technique is used to enforce the terminal conditions. This enlarges the space from which the approximating functions can be chosen. Orthogonal polynomials are used to obtain a set of simultaneous equations with fewer non-zero entries. Such a sparse system results in substantial computational economy. Two examples, a time-invariant system and a time-varying system with quadratic performance index, are solved using three different sets of orthogonal polynomials and the power series to demonstrate the feasibility and efficiency of this method.     

Topology redesign for performance by large admissible perturbations

A methodology is developed for optimal structural topology design subject to several performance constraints. Eight-node solid elements are used to model the initial structure, which is a uniform solid block satisfying the boundary conditions and subjected to external loading. The Young modulus of each solid element or group of elements is used as redesign variable. A minimum change function is used as an optimality criterion. Performance constraints include static displacements, natural frequencies, forced response amplitudes, and static stresses. These constraints are treated by the large admissible perturbation methodology which makes it possible to achieve the performance objectives incrementally without trial and error or repetitive finite element analyses for changes in the order of 100–300%. Thus, the optimal topology is reached in about four to five iterations, where each iteration includes one finite element analysis and setting of an upper limit for the value of the modulus of elasticity to produce a manufacturable structure. Several numerical applications are presented using three different benchmark structures to demonstrate the methodology and the impact of performance constraints on the generated topology.     

Orthogonal collocation on finite elements for elliptic equations

The method of orthogonal collocation on finite elements (OCFE) combines the features of orthogonal collocation with those of the finite element method. The method is illustrated for a Poisson equation (heat conduction with source term) in a rectangular domain. Two different basis functions are employed: either Hermite or Lagrange polynomials (with first derivative continuity imposed to ensure equivalence to the Hermite basis). Cubic or higher degree polynomials are used. The equations are solved using an LU-decomposition for the Hermite basis and an alternating direction implicit (ADI) method for the Lagrange basis.     

A high order finite-difference solver for investigation of disturbance development in turbulent boundary layers

This paper outlines a velocity–vorticity based numerical simulation method for modelling perturbation development in laminar and turbulent boundary layers at large Reynolds numbers. Particular attention is paid to the application of integral conditions for the vorticity. These provide constraints on the evolution of the vorticity that are fully equivalent to the usual no-slip conditions. The vorticity and velocity perturbation variables are divided into two distinct primary and secondary groups, allowing the number of governing equations and variables to be effectively halved. Compact finite differences are used to obtain a high-order spatial discretization of the equations. Some novel features of the discretization are highlighted: (i) the incorporation of the vorticity integral conditions and (ii) the related use of a co-ordinate transformation along the semi-infinite wall-normal direction. The viability of the numerical solution procedure is illustrated by a selection of test simulation results. We also indicate the intended application of the simulation code to parametric investigations of the effectiveness of spanwise-directed wall oscillations in inhibiting the growth of streaks within turbulent boundary layers.     

An analytical approach to the Fornberg–Whitham type equations by using the variational iteration method

In this paper, He’s variational iteration method (VIM) is applied to solve the Fornberg–Whitham type equations. The VIM provides fast converged approximate solutions to nonlinear equations without linearization, discretization, perturbation, or the Adomian polynomials. This makes the method attractive and reliable for solving the Fornberg–Whitham type equations. Numerical examples related to two initial value problems are presented to show the efficiency of the VIM.     

A method of large finite elements

A modification of a finite element method by considering large elements is presented. For a given discretization of a system, approximating polynomials of an arbitrary degree are constructed for each element. Such an approach allows an automatic generation of matrices for the resulting systems of equations as well as a straightforward evaluation of unknowns with any desired accuracy. It becomes possible moreover to reproduce exactly shapes of curvilinear elements and to make a numerical analysis of unbounded media. Existing algorithms and programs of the finite element method may be adapted to treat problems within the present formulation.     

A new heterosis plate element for geometrically non-linear finite element analysis of laminated plates

In this study, a new nine-node quadrilateral, shear-deformable heterosis element is developed. In order to model this element, Kirchhoff constraints are modified using Reissner-Mindlin theory assumptions. All of the modifications are performed for first-order shear-deformation theory (FSDT). This new heterosis element is developed by modifying 8-node serendipity and twelve-node cubic polynomials. The new heterosis element is used with nine-node Lagrangian elements in finite element analysis of composite plates. A modified element is used in finite element analysis of linear and non-linear analysis considering the advantages of free of ‘shear locking’. Numerical results are presented by comparing Navier's series solution.     

Generation of shape functions for straight beam elements

Straight beam finite elements with greater than two nodes are used for edge stiffening in plane stress analyses and elsewhere. It is often necessary to match the number of nodes on the edge stiffener to the number on a whole plane stress element side. Beam elements employ shape functions which are recognised to be level one Hermitian polynomials. An alternative to the commonly adopted method for determining these shape functions is given in this note, using a formula widely reported in mathematical texts which has hitherto not been applied to this task in the finite element literature. The procedure derives shape functions for beams entirely from the set of Lagrangian interpolating polynomials. Examples are given for the derivation of functions for a three and four-noded beam element.     

Time‐optimal constant speed motion program for multiple cooperating manipulators

This article presents a systematic approach for synthesizing the time‐optimal constant speed motion program for multiple manipulators moving a commonly held object along a specified Cartesian trajectory. In this approach, the motion program is constructed by using piecewise polynomials to blend the acceleration, constant speed, and deceleration periods. The polynomials are interpolated according to the boundary and continuity conditions to obtain a smooth and continuous profile. With this formulation, it is shown that the final form of the motion program can be established in terms of the initial acceleration, the constant operation speed, and the finial deceleration. The optimum values of these terms to allow the given trajectory to be executed in minimum time are determined based on the parametric dynamic equations of the system and the torque constraints of the actuator. This approach is conceptually straightforward and can be applied to various multirobot systems with nonlinear actuator constraints. ©1999 John Wiley & Sons, Inc.