Membrane approximation of the behaviour of the drop shaped tank under symmetrical loading

An algorithm for dealing with membrane equations is applied to the drop shaped tank subjected to symmetrical loading. The results obtained are compared with a finite element analysis of the problem taking bending and shear effects into consideration. An assessment of the usefulness of the membrane method in dealing with this problem is then made.     

Optimization of coupled thermal-structural problems of laminated plates with lamination parameters

The behaviour of a laminated plate with given boundary temperatures and displacement constraints is optimized and the optimization problem is expressed in terms of lamination parameters. Because the thermal conductivity and structural properties of a laminate depend on the lamination parameters of the laminate, the analysis of the plate consists of solving a coupled-field problem. The strain energy, or certain displacements of the laminated plate due to given boundary temperatures and displacement boundary conditions, is optimized with respect to in-plane lamination parameters, and also buckling of the plate is considered. The buckling factors for thermal loading are expressed as a function of four in-plane and four bending lamination parameters, and the smallest factor is maximized with respect to these parameters. In addition to these thermal problems, the natural frequencies of the laminated plate are studied. Since transverse shear deformations are taken into account,the natural frequencies can be expressed as functions of two in-plane and four bending lamination parameters, with respect to which the lowest natural frequency of the plate is maximized. The lay-up for the laminate, corresponding to four optimal in-plane or bending lamination parameters, consists of three layers at most and can be determined using explicit equations. Explicit equations are derived for creating a lay-up having optimal bending lamination parameters. Received May 12, 1999     

Finite elements for 2D problems of pressurized membranes

This paper presents theoretical and numerical developments of finite elements for axisymmetric and cylindrical bending problems of pressurized membranes. The external loading is mainly a normal pressure to the membrane and the developments are made under the assumptions of follower forces, large displacements and finite strains. The numerical computing is carried out in a different way that those used by the conventional finite element approach which consists in solving the non-linear system of equilibrium equations in which appears the stiffness matrix. The total potential energy is here directly minimized, and the numerical solution is obtained by using optimization algorithms. When the derivatives of the total energy with respect to the nodal displacements are calculated accurately, this approach presents a very good numerical stability in spite of the nil bending rigidity of the membrane. Our numerical models show a very good accuracy by comparisons to analytical solutions and experimental results.     

A variational method in design optimization and sensitivity analysis for aerodynamic applications

Variational method is applied to the state equations in order to derive the costate equations and their boundary conditions. Thereafter, the analyses of the eigenvalues of the state and costate equations are performed. It is shown that the eigenvalues of the Jacobean matrices of the state and the transposed Jacobean matrices of the costate equations are analytically and numerically the same. Based on the eigenvalue analysis, the costate equations with their boundary conditions are numerically integrated. Numerical results of the eigenvalues problems of the state and costate equations and of a maximization problem are finally presented.     

Vibration response of geometrically nonlinear elastic beams subjected to pulse and impact loading

The governing equations for the geometrically non-linear deformation of elastic beams subjected to dynamic bending loads are developed and solved for various initial conditions. Of primary interest is the response to pulse loading and simulated impact. Both transient and several cycle solutions are generated for the free vibration response to pulse loading. The results obtained are compared to a first mode analysis approximation.A new model is developed to simulate impact loading by the distribution of additional mass to the elastic system and subjecting it to a velocity pulse. The governing equations are solved using second order finite differences in space and time. The solutions obtained are in reasonable agreement with experimental results previously obtained [1].     

Optimization of the reinforcement of a 3D medium with thin composite plates

The reinforcement with a thin composite plate of a 3D linear elastic medium on its external boundary or inside is considered. A linear analysis of the 3D problem leads to a variational formulation in which the reinforcement is modelled by a Kirchhoff–Love plate. Considering the sum of the compliance and a cost as the design objective, a numerical example of the optimization of this reinforcement is performed taking into account the in-plane membrane rigidity only (i.e. the bending aspects are not treated numerically).     

Creep analysis of cylindrical shell by a finite difference method

The governing equations of the steady state creep of a two-dimensional thin shallow circular cylindrical shell are developed on the basis of Mises' criterion and the power law of creep.Stresses, membrane forces and bending moments expressed in terms of displacements are then linearized by expanding them in the neighbourhood of a certain approximate value of displacement. Substitution of these expressions into the equations of equilibrium reduces the problem into a set of simultaneous linear differential equations with respect to the small perturbation of the displacements. A method that may be interpreted as a modification of the Newton-Raphson method combined with the method of finite differences is used to solve the linear system of equations.Although calculations are made for a cylindrical panel with clamped edges subjected to normal pressure, the method is quite general and other types of shells, boundary conditions and geometries can be treated similarly.     

A New Family of Mixed Methods for the Reissner-Mindlin Plate Model Based on a System of First-Order Equations

The mixed method for the biharmonic problem introduced in (Behrens and Guzmán, SIAM J. Numer. Anal., 2010) is extended to the Reissner-Mindlin plate model. The Reissner-Mindlin problem is written as a system of first order equations and all the resulting variables are approximated. However, the hybrid form of the method allows one to eliminate all the variables and have a final system only involving the Lagrange multipliers that approximate the transverse displacement and rotation at the edges of the triangulation. Mixed finite element spaces for elasticity with weakly imposed symmetry are used to approximate the bending moment matrix. Optimal estimates independent of the plate thickness are proved for the transverse displacement, rotations and bending moments. A post-processing technique is provided for the displacement and rotations variables and we show numerically that they converge faster than the original approximations.     

Application of the extended Kantorovich method to the bending of variable thickness plates

The governing equations of the classical plate theory for a uniform or a unidirectional variable thickness rectangular plate under transverse applied loading are solved by means of the extended Kantorovich method. The plate may be either simply supported or clamped along the edges. The solution procedure is iterative and must be carried out numerically. This necessitates the calculation of the two missing pieces of boundary data along the edges of the plate. The missing boundary data are determined utilizing the method of adjoints of the shooting method. The numerical values of the deflection and bending moments for uniform and variable thickness plates are compared with those from the exact solutions and finite element analysis, respectively.     

Minimum-weight design with displacements constraints in 2-dimensional elasticity

The optimum weight-bounded deflection design of a beam, modelled by means of a set of 2-dimensional elasticity equations is studied. The iterative algorithm based on differentiation with respect to the domain formulae is proposed. The examples of one-sidedly the two-sidedly clamped beams with different loading conditions and solved numerically. The results are compared with those of a straightforward approach.     

一种局部无网格配点法在功能梯度材料板上的应用

针对基于1阶剪切变形理论和Hamilton原理的功能梯度材料(functionally graded material,FGM)板微分控制方程,验证将广义有限差分法(generalized finite difference method,GFDM)用于FGM板弯曲行为数值计算的有效性,利用GFDM对物理域进行离散布点...     

Space–time spectral collocation method for one-dimensional PDE constrained optimisation

This paper solves an optimal control problem governed by a parabolic PDE. Using Lagrangian multipliers, necessary conditions are derived and then space–time spectral collocation method is applied to discretise spatial derivatives and time derivatives. This method solves partial differential equations numerically with errors bounded by an exponentially decaying function which is dependent on the number of modes of analytic solution. Spectral methods, which converge spectrally in both space and time, have gained a significant attention recently. The problem is then reduced to a system consisting of easily solvable algebraic equations. Numerical examples are presented to show that this formulation has exponential rates of convergence in both space and time.     

Numerical investigation based on radial basis function–finite-difference (RBF–FD) method for solving the Stokes–Darcy equations

A meshless method is presented to numerically study an interface problem between a flow in a porous medium governed by Darcy equations and a fluid flow, governed by Stokes equations. In fact, the domain of the problem has two parts, one governed by Stokes equations and the another governed by Darcy law. Governing equations on these two parts are mutually coupled by interface conditions. The approximation solution is based on local radial basis function–finite-difference (RBF–FD) which is carried out within a small influence domain instead of a global one. By this strategy, the final linear system is more sparse and well posed than the global one. Several numerical results are provided to illustrate the good performance of the proposed scheme.

     

Separation-bubble flow solution using Euler/Navier-Stokes zonal approach with downstream compatibility conditions

An Euler/Navier-Stokes zonal scheme is developed to numerically simulate the two-dimensional flow over a blunt leading-edge plate. The computational domain has been divided into inner and outer regions where the Navier-Stokes and Euler equations are used, respectively. On the downstream boundary, compatibility conditions derived from the boundary-layer equations are used. The grid is generated by using conformal mapping and the problem is solved by using a compressible Navier-Stokes code, which has been modified to treat Euler and Navier-Stokes regions. The accuracy of the solution is determined by the reattachment location. Bench-mark solutions have been obtained using the Navier-Stokes equations throughout the optimum computational domain and size. The problem is recalculated with sucessive decrease of the computational domain from the downstream side where the compatibility conditions are used, and with successive decrease of the Navier-Stokes computational region. The results of the zonal scheme are in excellent agreement with those of the benchmark solutions and the experimental data. The CPU time saving is about 15%.     

Disturbance decoupling using constrained Sylvester equations

Presents numerically efficient algorithms for the problem of disturbance decoupling with arbitrary pole placement, using state feedback with or without disturbance measurement. The problem is studied under conditions of generic solvability. The main tool for the development of the proposed algorithm is the notion of constrained Sylvester equations. These equations follow naturally from the already existing geometric conditions for the same problem in the literature     

A new approach to eigenstructure assignment by output feedback

The eigenstructure assignment problem for a linear time invariant multi-input-multi-output system using output feedback is considered. A new approach is developed which identifies the eigenspaces for the desired set of all the closed-loop eigenvalues. For the assignment of this desired set, necessary and sufficient conditions are established. These conditions contain two coupled Sylvester matrix equations, one of which is proven to be a reduced-order square Sylvester matrix equation. This results in an efficient analytical procedure, numerically superior to known techniques, for the determination of the output feedback gain-matrix     

Thermal post-buckling characteristics of clamped skew plates

Thermal post-buckling characteristics of clamped skew plates with restrained edges subjected to planar temperature distributions are studied. The problem is formulated in terms of Von Kármán equations expressed in terms of displacements generalised to include thermal effects. A perturbation method is employed to obtain a linear set of partial differential equations. The solution to this linear set is then obtained by using the Galerkin method. Numerical results are presented for different skew plate configurations for both uniform and two-dimensional temperature distributions. Plots of central deflection, membrane and bending stress distributions are presented and the post-buckling characteristics are discussed in detail.     

An inverse problem for estimation of bending stiffness in Kirchhoff–Love plates

This work deals with the development of a numerical method for solving an inverse problem for bending stiffness estimation in a Kirchhoff–Love plate from overdetermined data. The coefficient is identified using a technique called the Method of Variational Imbedding, where the original inverse problem is replaced by a minimization problem. The Euler–Lagrange equations for minimization comprise higher-order equations for the solution of the displacement and an equation for the bending stiffness. The correctness of the embedded problem is discussed. A difference scheme and a numerical algorithm for solving the parameter identification problem are developed. Numerical results for the obtained values of the bending stiffness as an inverse problem are presented.     

2D shape optimization with static and dynamic constraints

The paper presents an approach that allows us to consider in the shape optimization several static loading conditions together with constraints imposed on eigenfrequencies. The idea of the method is based upon simultaneous solutions of equations and inequalities arising from the Kuhn-Tucker necessary conditions for an optimum problem. The paper is illustrated with four examples in which stress and eigenfrequency are active constraints.     

Finite element dynamic analysis of geometrically exact planar beams

We present a new strain-based finite element formulation for the dynamic analysis of highly flexible elastic planar beams. The formulation employs the geometrically exact Reissner planar beam theory which accounts for finite displacements and rotations, and finite membrane, shear and bending strains. The system of semi-discrete dynamic equations of motion is derived from the modified Hamilton principle in which only the strain variables are interpolated. Such a choice of the interpolated variables is an advantage over approaches, in which the displacements and rotations are interpolated, since the field consistency problem and related locking phenomena do not arise. The performance and accuracy of the formulation are illustrated by several numerical examples.