On the numerical treatment of an ordinary differential equation arising in one-dimensional non-Fickian diffusion problems

In a recent study, Chen and Liu [Comput. Phys. Comm. 150 (2003) 31] considered a one-dimensional, linear non-Fickian diffusion problem with a potential field, which, upon application of the Laplace transform, resulted in a second-order linear ordinary differential equation which was solved by means of a control-volume finite difference method that employs exponential shape functions. It is first shown that this formulation does not properly account for the spatial dependence of the drift forces and results in oscillatory solutions near the left boundary when these forces are large. A piecewise linearized method that provides piecewise analytical solutions, is exact in exact arithmetic for constant coefficients, homogeneous, second-order linear ordinary differential equations and results in three-point finite difference equations is then proposed. Numerical simulations indicate that the piecewise linearized method is free from unphysical oscillations and more accurate than that of Chen and Liu, especially for large drift forces. The method is then applied to non-Fickian diffusion problems with non-constant drift forces in order to determine the effects of the potential field on the concentration distribution.     

Analysis of the deflection of a circular plate with an annular piezoelectric actuator

A new analytical model for predicting the deflection of a circular plate with an annular piezoelectric actuator is presented. The plate and actuator are treated as a mechanically over-constrained system and a structural mechanics approach is applied to establish the relevant equations of geometrical compatibility and static equilibrium, assuming that the interaction forces between the actuator and plate are concentrated at the edges of the actuator annulus. These equations can be solved analytically or numerically to determine the interaction forces. Analytical expressions for plate deflection in terms of the interaction forces are then presented for three sets of plate boundary conditions. The analytical results are shown to be in good agreement with finite element simulations and provide an efficient alternative to finite element analysis for design and optimization studies.     

A numerical method for two phase flows with insoluble surfactants

In many practical multiphase flow problems, i.e. treatment of gas emboli and various microfluidic applications, the effect of interfacial surfactants, or surface reacting agents, on the surface tension between the fluids is important. The surfactant concentration on an interface separating the fluids can be modeled with a time dependent differential equation defined on the moving and deforming interface. The equations for the location of the interface and the surfactant concentration on the interface are coupled with the Navier–Stokes equations. These equations include the singular surface tension forces from the interface on the fluid, which depend on the interfacial surfactant concentration.A new accurate and inexpensive numerical method for simulating the evolution of insoluble surfactants is presented in this paper. It is based on an explicit yet Eulerian discretization of the interface, which for two dimensional flows allows for the use of uniform one dimensional grids to discretize the equation for the interfacial surfactant concentration. A finite difference method is used to solve the Navier–Stokes equations on a regular grid with the forces from the interface spread to this grid using a regularized delta function. The timestepping is based on a Strang splitting approach.Drop deformation in shear flows in two dimensions is considered. Specifically, the effect of surfactant concentration on the deformation of the drops is studied for different sets of flow parameters.     

A finite difference variational method for bending of plates

The method of analysis for bending of plates presented in this paper combines a finite difference scheme for the plate strain components and a variational derivation of the equations of motion or equilibrium. The plate strain components are expressed in terms of discrete nodal displacements with the aid of the two dimensional Taylor expansion. Consequently, the virtual work, or the first variation of the strain energy, in an area element is found as a function of the nodal displacements. The derivation of the element forces or the element stiffness matrices and the assembly of the equations of motion or equilibrium follows closely the steps of the finite element method.     

构造法在机翼气动载荷转换中的应用

针对机翼气动压力分布数据的离散性,构造任意点附近区域3种形式的压力场曲面函数,以有限元单元为积分区域积分得到该单元内的集中气动载荷,并将其按照最小变形能原理分配到有限元的节点之上.结合某型飞机的测压试验数据比较3种构造函数法求得的气动累积内力,结果表明构造连续的压力曲面函数求得的气动累积内力符合实际气动力的分布特性,满足实际工程需求.     

Large eddy simulation of turbulent heat transport in the Strait of Gibraltar

We develop a numerical model for large eddy simulation of turbulent heat transport in the Strait of Gibraltar. The flow equations are the incompressible Navier–Stokes equations including Coriolis forces and density variation through the Boussinesq approximation. The turbulence effects are incorporated in the system by considering the Smagorinsky model. As a numerical solver we propose a finite element semi-Lagrangian method. The solution procedure consists of combining a non-oscillatory semi-Lagrangian scheme for time discretization with the finite element method for space discretization. Numerical results illustrate a buoyancy-driven circulations along the Strait of Gibraltar and the sea-surface temperature is flushed out and move to northeast coast. The Ocean discharge and the temperature difference are shown to control the plume structure.     

Large eddy simulation of turbulent heat transport in the Strait of Gibraltar

We develop a numerical model for large eddy simulation of turbulent heat transport in the Strait of Gibraltar. The flow equations are the incompressible Navier–Stokes equations including Coriolis forces and density variation through the Boussinesq approximation. The turbulence effects are incorporated in the system by considering the Smagorinsky model. As a numerical solver we propose a finite element semi-Lagrangian method. The solution procedure consists of combining a non-oscillatory semi-Lagrangian scheme for time discretization with the finite element method for space discretization. Numerical results illustrate a buoyancy-driven circulations along the Strait of Gibraltar and the sea-surface temperature is flushed out and move to northeast coast. The Ocean discharge and the temperature difference are shown to control the plume structure.     

Adaptive modelling of plates in bending and shear

As an alternative to a direct finite element approximation of the Reissner–Mindlin plate equations we propose an adaptive modeling starting with a finite element solution according to the Germain–Kirchhofff equation and in an adaptive manner adding shear deformations calculated from the Germain–Kirchhoff shear forces. The system matrix need not be changed, so the extra cost, compared to the first solution, can be kept small.     

Pulse control of flexible multibody systems

An active pulse control method is developed to reduce the vibrations of multibody systems resulting from impact loadings. The pulse, which is a function of system generalized coordinates and velocities, is determined analytically using energy and momentum balance equations of the impacting bodies. Elastic components in the multibody system are discretized using the finite element method. The system equations of motions and nonlinear algebraic constraint equations describing mechanical joints between different components are written in the Lagrangian formulation using a finite set of coupled reference position and local elastic generalized coordinates. A set of independent differential equations are identified by the generalized coordinate partitioning of the constraint Jacobian matrix. These equations are written in the state space formulation and integrated forward in time using a direct numerical integration method. Dependent coordinates are then determined using the constraint kinematic relations. Points in time at which impact occurs are monitored by an impact predictor function, which controls the integration algorithms and forces for the solution of the momentum relation, to define the jump discontinuities in the composite velocity vector as well as the system reaction forces. The effectiveness of the active pulse control in reducing the vibration of flexible multibody aircraft during the touchdown impact is investigated and numerical results are presented.     

Numerical solutions of a nonlinear reaction-diffusion system

This paper is concerned with finite difference solutions of a coupled system of nonlinear reaction-diffusion equations. The investigation is devoted to the finite difference system for both the time-dependent problem and its corresponding steady-state problem. The existence and uniqueness of a non-negative finite difference solution and three monotone iterative algorithms for the computation of the solutions are given. It is shown that the time-dependent problem has a unique non-negative solution, whereas the steady-state problem may have multiple non-negative solutions depending on the parameters in the problem. The different non-negative steady-state solutions can be computed from the monotone iterative algorithms by choosing different initial iterations. Also discussed is the asymptotic behaviour of the time-dependent solution in relation to the steady-state solutions. The asymptotic behaviour result gives some conditions ensuring the convergence of the time-dependent solution to a positive or semitrivial non-negative steady-state solution. Numerical results are given to demonstrate the theoretical analysis results.     

Transient analysis of flexible multi-body systems. Part II: Application to aircraft landing

This investigation presents a method for transient analysis of a large-scale multi-body aircraft consisting of interconnected rigid and flexible bodies that undergo large angular rotations. Elastic components of the aircraft are discretized using the finite element method. The system equations of motion and nonlinear algebraic constraint equations describing joints between different components are written in the Lagrangian formulation using a finite set of coupled reference and modal coordinates. The system differential equations of motion and algebraic constraint equations are computer-generated and integrated forward in time using an explicit-implicit direct numerical integration algorithm coupled with a Newton-Raphson type iteration in order to check on constraint violations. Impact and intermittent motion events are accounted for by using a generalized momentum balance that predict jump discontinuities in the generalized velocities as well as jump discontinuities in the system reaction forces. The formulation presented and the computer program developed are used to simulate the impact between the landing gear and the runway. The method is also used to predict the dynamic behavior of the aircraft during the traverse of an abrupt elevation change in the runway.     

Chain Vibration and Dynamic Stress in Three-Dimensional Multibody Tracked Vehicles

A three-dimensional computational finite element procedure for the vibration and dynamic stress analysis of the track link chains of off-road vehicles is presented in this paper. The numerical procedure developed in this investigation integrates classical constrained multibody dynamics methods with finite element capabilities. The nonlinear equations of motion of the three-dimensional tracked vehicle model in which the track link s are considered flexible bodies, are obtained using the floating frame of reference formulation. Three-dimensional contact force models are used to describe the interaction of the track chain links with the vehicle components and the ground. The dynamic equations of motion are first presented in terms of a coupled set of reference and elastic coordinates of the track links. Assuming that the structural flexibility of the track links does not have a significant effect on their overall rigid body motion as well as the vehicle dynamics, a partially linearized set of differential equations of motion of the track links is obtained. The equations associated with the rigid body motion are used to predict the generalized contact, inertia, and constraint forces associated with the deformation degrees of freedom of the track links. These forces are introduced to the track link flexibility equations which are used to calculate the deformations of the links resulting from the vehicle motion. A detailed three-dimensional finite element model of the track link is developed and utilized to predict the natural frequencies and mode shapes. The terms that represent the rigid body inertia, centrifugal and Coriolis forces in the equations of motion associated with the elastic coordinates of the track link are described in detail. A computational procedure for determining the generalized constraint forces associated with the elastic coordinates of the deformable chain links is presented. The finite element model is then used to determine the deformations of the track links resulting from the contact, inertia, and constraint forces. The results of the dynamic stress analysis of the track links are presented and the differences between these results and the results obtained by using the static stress analysis are demonstrated.     

轻负荷磁头气动力分析

本文采用微分离散有限差分法求解滑流条件下的修正雷诺方程.离散后的三对角线性代数方程组用ADI方法求解.本方法适用于任意几何形状、尺寸、姿态角的多轨窄式磁头滑块的气动力计算,并不受大压缩数的限制.计算结果包括即时给出的压强分布立体图.     

A fast immersed interface method for solving Stokes flows on irregular domains

We present a fast immersed interface method for solving the steady Stokes flows involving the rigid boundaries. The immersed rigid boundary is represented by a set of Lagrangian control points. In order to enforce the prescribed velocity at the rigid boundary, singular forces at the rigid boundary are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are approximated using the cubic splines. The strength of singular forces is determined by solving a small system of equations via the GMRES method. The Stokes equations are discretized using finite difference method with the incorporation of jump conditions on a staggered Cartesian grid and solved by the conjugate gradient Uzawa-type method. Numerical results demonstrate the accuracy and ability of the proposed method to simulate Stokes flows on irregular domains.     

Finite element/difference methods in random vibration

Finite element and finite difference analyses of a class of random vibration problem are presented. The random field equations are discretized using the standard finite element/difference techniques. Recursive algorithms for the moments of the response in terms of the moments of the random loading and random initial condition are derived based on the discretized equations. The approach is straightforward and it provides an alternative to the standard normal-mode approach in the analysis of linear random structural systems. However, unlike the standard approach, the finite element/difference approach can be applied to other more complex systems. Two numerical examples are given to illustrate the application. The approximate numerical results are also compared with the exact solutions to check the accuracy of the finite element/difference analyses. It is found that for the problems considered, the finite difference algorithms provide better accuracy and efficiency than the finite element method.     

Constraint forces in holonomic mechanical systems

Constraint forces play important roles in the dynamic formulation of mechanical systems. In robotic and locomotion systems, implementing, maintaining, and deliberately violating contact and support conditions require either on-line sensing of constraint forces or indirect computation of them. In this paper three different methods for calculating constraint forces are compared with respect to computational complexity and accuracy, and computation time. In the first two methods, the nonlinear constraint equations and the system dynamic equations are combined such that the resulting set of equations is uniquely solvable for the constraint forces. In the third method, the mechanical system is embedded in a higher-dimensional state space, chosen in such a way that the constraint equations are linear in the new state of the system. Consequently the computation of the constraint forces simplifies. The complexity of the three methods and the computation time for a particular computer can be inferred from an operation count. The accuracy of the individual method is estimated by providing an upper bound of the relative error and the relative residual of the computed constraint forces.     

Finite element cyclic thermoplasticity analysis by the method of subvolumes

The subject of this paper is the development of an analytical tool capable of economically evaluating the cyclic plasticity which occurs in areas of strain concentration resulting from the combination of both mechanical and thermal stresses. The techniques developed are capable of handling large excursions in temperatures with the associated variations in material properties, including plasticity. The techniques are capable of reproducing real cyclic material behavior including Bauschinger effect, cross-hardening and memory.These analytical techniques have been implemented in a time-sharing finite element computer program. Cyclic plasticity has been introduced into this program using incremental loading and an iterative solution technique. The plasticity theory involved makes use of the von Mises yield criterion and the Prandtl-Reuss flow rule. The major portion of the developmental work in this effort was expended in the establishment of a temperature variable hardening rule and its finite element implementation. The plane stress, constant strain triangle is the finite element used in this work.The incremental plasticity solution is obtained by iteratively revising the right-hand side of the system of finite element equations by the addition of a vector of plastic pseudo forces. The method of subvolumes is used to generate the vector of plastic pseudo forces such that real material cyclic plasticity behavior is mathematically reproduced.The effects of the plastic deformations are introduced into the system of finite element equations by considering them as load terms in much the same way as thermal expansions are usually treated. The nonlinear solution is then attained through solution of a series of elastic problems and by variation of the plastic load terms until the requirements of compatibility, equilibrium and the specified nonlinear stress-strain relations are all met within a given tolerance.     

Finite element cyclic thermoplasticity analysis by the method of subvolumes

The subject of this paper is the development of an analytical tool capable of economically evaluating the cyclic plasticity which occurs in areas of strain concentration resulting from the combination of both mechanical and thermal stresses. The techniques developed are capable of handling large excursions in temperatures with the associated variations in material properties, including plasticity. The techniques are capable of reproducing real cyclic material behavior including Bauschinger effect, cross-hardening and memory.These analytical techniques have been implemented in a time-sharing finite element computer program. Cyclic plasticity has been introduced into this program using incremental loading and an interative technique. The plasticity theory involved makes use of the von Mises yield criterion and the Prandtl-Reuss flow rule. The major portion of the developmental work in this effort was expended in the establishment of a temperature variable hardening rule and its finite element implementation. The plane stress, constant strain triangle is the finite element used in this work.The incremental plasticity solution is obtained by interatively revising and right-hand side of the system of finite element equations by the addition of a vector of plastic pseudo forces. The method of subvolumes is used to generate the vector of plastic pseudo forces such that real material cyclic plasticity behavior is mathematically reproduced.The effects of the plastic deformations are introduced into the system of finite element equations by considering them as load terms in much the same way as thermal expansions are usually treated. The nonlinear solution is then attained through solution of a series of elastic problems and by variation of the plastic load terms until the requirements of compatibility, equilibrium and the specified non-linear stress-strain relations are all met within a given tolerance.     

The FEM optimization of active vibration control in structures by solving the corresponding two-point-boundary-value problem

An algorithm for solving optimal active vibration control problems by the finite element method (FEM) is presented. The optimality equations for the problem are derived from Pontryagin’s principle in the form of a set of the fourth order ordinary differential equations that, together with the initial and final boundary conditions, constitute the boundary value problem in the time domain, which in control is referred to as a two-point-boundary-value problem. These equations decouple in the modal space and can be solved by the FEM technique. An analogy between the optimality equations and the governing equations for a set of certain static beams permits obtaining numerical solutions to the optimal control problem with the help of standard ‘structural’ FEM software. The optimal action of actuators is automatically calculated by applying the independent modal space control concept. The structure’s response to actuation forces is also determined and can independently be verified for spillover effects. As an illustration, the algorithm is used for the analysis of optimal action of actuators to attenuate vibrations of an elastic fin.     

The development of a sliding joint for very flexible multibody dynamics using absolute nodal coordinate formulation

In this paper, a formulation for a spatial sliding joint is derived using absolute nodal coordinates and non-generalized coordinate and it allows a general multibody move along a very flexible cable. The large deformable motion of a spatial cable is presented using absolute nodal coordinate formulation, which is based on the finite element procedures and the general continuum mechanics theory to represent the elastic forces. And the nongeneralized coordinate, which is related to neither the inertia forces nor the external forces, is used to describe an arbitrary position along the centerline of a very flexible cable. Hereby, the non-generalized coordinate represents the arc-length parameter. The constraint equations for the sliding joint are expressed in terms of generalized coordinate and nongeneralized coordinate. In the constraint equations for the sliding joint, one constraint equation can be systematically eliminated. There are two independent Lagrange multipliers in the final system equations of motion associated with the sliding joint. The development of this sliding joint is important to analyze many mechanical systems such as pulley systems and pantograph-catenary systems for high speed-trains.