Shear flexible element based on coupled displacement field for large deflection analysis of laminated plates

A shear deformable theory accounting for the transverse-shear (in the sense of Reissner-Mindlin’s thick plate theory) and large deflections (in the sense of von Karman theory) is employed in the construction of variational statement. A four-node, lock-free, shear-flexible rectangular plate element based on the coupled displacement field is developed in this paper to carry out the large deflection analysis. The displacement field of the element is derived by making use of the linearized equations of static equilibrium. A bi-cubic polynomial distribution is assumed for the transverse displacement ‘w’. The field distribution for the in-plane displacements (u,v) and plate normal rotations (θ_x, θ_y) and twist (θ_xy) is derived using equilibrium of composite strips parallel to the plate edges. The displacement fields so derived are coupled through material couplings. The transverse shear strain fields of the proposed element do not contain inconsistent terms, so that the element predicts even shear-rigid bending accurately.The element is validated for a series of numerical problems and results for deflections and stresses are presented for rectangular composite plates with various boundary conditions, loading and lay-ups. The influence of the sign of the loading on the deflection of unsymmetrically laminated plates, in the large deflection regime is also investigated.     

An accurate algorithm for computing the eigenvalues of a polygonal membrane

For a polygonal domain Ω, we consider the eigenvalue problem Δu + λu = 0 in Ω, u = 0 on the boundary of Ω. Ω is decomposed into subdomains Ω₁, Ω₂,...; on each $\text{Ω}{}_{\mathrm{i}}$, u is approximated by a linear combination of functions which satisfy the equation Δu + Δu = 0 and continuity conditions are imposed at the boundaries of the subdomains. We propose a non-conventional method based on the use of a Rayleigh quotient. We present numerical examples and a proof of the exponential convergence of the algorithm.     

Exact solution of mixed problems for variable coefficient one-dimensional diffusion equation

This paper deals with diffusion problems modeled by the equation a(t)u_xx = u_t, x > 0, t > 0, u(x, 0) = c(x) together with the boundary condition u(0, t) = b(t) or u_x(0, t) = b(t). By using Fourier transforms, existence conditions and exact solutions of the above mixed problems are given.     

Asymptotic behavior and blow up of solutions for semilinear parabolic equations at critical energy level

In this paper we study the initial boundary value problem of semilinear parabolic equations with semilinear term f(u). By using the family of potential wells method we prove that if f(u) satisfies some conditions, J(u₀) ≤ d and I(u₀) > 0, then the solution decays to zero exponentially as t → ∞. On the other hand, if J(u₀) ≤ d, I(u₀) < 0, then the solution blows up in finite time.     

Fixed sign solutions of second-order difference equations with Neumann boundary conditions

In this paper, we study the existence of solution of the nonlinear second-order difference problem with Neumann boundary conditions. Assuming the existence of a pair of ordered lower and upper solutions γ and β, we obtain optimal existence results for the case γ ≤ β and even for γ ≥ β.To this end, we do an exhaustive study of the values of the real parameters α and μ, for which the associated Green's function to the linear operator L[α,μ] u_κ ≡ u_κ+2 − 2 α u_κ+1 + μ u_κ with Neumann boundary conditions has fixed sign.     

Maximum subset intersection

Consider the following problem. Given u sets of sets A₁,…,Au with elements over a universe E={e₁,…,en}, the goal is to select exactly one set from each of A₁,…,Au in order to maximize the size of the intersection of the sets. In this paper we present a gap-preserving reduction from Max-Clique which enables us to show that our problem cannot be approximated within an n1−? multiplicative factor, for any ?>0, unless P=NP (Zuckerman, 2006 [12]).     

The finite volume element method with quadratic basis functions

The paper presents a box scheme with quadratic basis functions for the discretisation of elliptic boundary value problems. The resulting discretisation matrix is non-symmetrical (and also not an M-matrix). The stability analysis is based on an elementwise estimation of the scalar product <A _h u _h ,u _h >. Sufficient conditions placed on the triangles of the triangulation lead to discrete ellipticity. Proof of anO(h ²) error estimate is given for these conditions.     

The formula of the solution for some classes of initial boundary value problems for the hyperbolic equation with two independent variables

A new representation is proved of the solutions of initial boundary value problems for the equation of the form u _xx (x, t) + r(x)u _x (x, t) ? q(x)u(x, t) = u _tt (x, t) + μ(x)u _t (x, t) in the section (under boundary conditions of the 1st, 2nd, or 3rd type in any combination). This representation has the form of the Riemann integral dependent on the x and t over the given section.     

Sufficient conditions for the existence and asymptotic behaviour of solution to a quasi-linear elliptic problem

In this article, we combine the existing regularity theory, perturbation method and the lower and upper solutions method to study the existence and asymptotic behaviour of positive solution to a boundary value problem for the p-Laplacian operator. More exactly, we study the existence and asymptotic behaviour of the positive solution to a quasi-linear elliptic problem of the form?Δ _p u=λ a(x)g(u) in D′(Ω), u>0 in Ω, lim _{x→? Ω} u(x)=0. Under some conditions on a and g, we show that there is a non-negative number Λ₀ such that for all λ∈(0, Λ₀], the problem has a solution u _λ in the sense of distribution, which is bounded from above by some positive numbers μ(λ). Such estimates and the asymptotic behaviour are important in computer programs when we know an algorithm for determining the solution.     

Upper and lower solutions for a superlinear singular boundary value problem

The singular boundary value problem u″ = −g(t, u), for t ϵ (0, 1), u(0) = u(1) = 0, is studied in this paper. The singularity may appear at u = 0, t = 0, and t = 1, the function g may be superlinear at u = ∞ and change sign. The existence of solutions is obtained via an upper and lower solution method.     

Consistent initial conditions for lattice Boltzmann simulations

We propose a simple and effective iterative procedure to generate consistent initial conditions for the lattice Boltzmann equation (LBE) for incompressible flows with a given initial velocity field u₀. Using the Chapman-Enskog analysis we show that not only the proposed procedure effectively solves the Poisson equation for the pressure field p₀ corresponding to u₀, it also generates at the same time the initial values for the nonequilibrium distribution functions {f_α} in a consistent manner. This procedure is validated for the decaying Taylor–Green vortex flow in two dimensions and is shown to be particularly effective when using the generalized LBE with multiple relaxation times.     

Bang-bang strategies using interrupted observations for steering a random motion of a point

Consider the random motion in the plane of a point M, whose velocity u = (u₁, u₂) is perturbed by an R²-valued Gaussian white noise. The point is equipped with an observation mechanism which provides interrupted noisy observations of a given target. The interruptions are characterized by a jump Markov process taking on the values 0 or 1, and using these observations the point M wishes to steer itself into the target. Sufficient conditions on optimal velocity laws (strategies) u are derived, and suboptimal strategies computed. The numerical results obtained show that the suboptimal strategies obey the separation principle of optimal stochastic control theory.     

A study of nonlinear dispersive equations with solitary-wave solutions having compact support

With the use of Adomian decomposition method, the prototypical, genuinely nonlinear K(m,n) equation, u_t+(u^m)_x+(uⁿ)_xxx=0, which exhibits compactons — solitons with finite wavelength — is solved exactly. Two numerical illustrations, K(2,2) and K(3,3), are investigated to illustrate the pertinent features of the proposed scheme. The technique is presented in a general way so that it can be used in nonlinear dispersive equations.     

Spurious solutions for discrete superlinear boundary value problems

We consider finite dimensional nonlinear eigenvalue problems of the typeAu=λFu whereA is a matrix and(Fu) _i =f(u _i ),i=1, ...,m. These may be thought of as discretizations of a corresponding boundary value problem. We show that positive, spurious solution branches of the discrete equations (which have been observed in some cases in [1, 7]) typically arise iff increases sufficiently strong and ifA ^?1 has at least two positive columns of a certain type. We treat in more detail the casesf(u)=e ^u andf(u)=u ^α where also discrete bifurcation diagrams are given.     

Sorting a sequence of strong kings in a tournament

A king in a tournament is a player who beats any other player directly or indirectly. According to the existence of a king in every tournament, Wu and Sheng [Inform. Process. Lett. 79 (2001) 297-299] recently presented an algorithm for finding a sorted sequence of kings in a tournament of size n, i.e., a sequence of players u₁,u₂,…,u_n such that u_i→u_i+1 (u_i beats u_i+1) and u_i is a king in the sub-tournament induced by {u_i,u_i+1,…,u_n} for each i=1,2,…,n−1. With each pair u,v of players in a tournament, let b(u,v) denote the number of third players used for u to beat v indirectly. Then, a king u is called a strong king if the following condition is fulfilled: if v→u then b(u,v)>b(v,u). In the sequel, we will show that the algorithm proposed by Wu and Sheng indeed generates a sorted sequence of strong kings, which is more restricted than the previous one.     

Parametric quintic spline solution of third-order boundary value problems

In this paper, we develop parametric quintic spline function to approximate the solution of third-order boundary value problems of the form u″′=f(x, u), a≤x≤b, subject to the boundary conditions u(a)=k ₁, u′(a)=k ₂ and u(b)=k ₃. The class of methods are second-, fourth- and sixth-order accurate. End equations of the splines are derived and truncation error is given. Three numerical examples are presented to illustrate the practical use of our methods as well as their accuracies when compared with some existing spline function methods. It is shown that the new methods give approximations, which are better than those produced by other methods.     

Maximum principle and uniform convergence for the finite element method

The solution of the Dirichlet boundary value problem over a polyhedral domain Ω ? Rⁿ, n ≥ 2, associated with a second-order elliptic operator, is approximated by the simplest finite element method, where the trial functions are piecewise linear. When the discrete problem satisfies a maximum principle, it is shown that the approximate solution u_h converges uniformly to the exact solution u if u ? W_1,p (Ω), with p > n, and that $\text{∥u?u}{}_{\mathrm{h}}\text{∥}{}_{\mathrm{L}}\text{∞(Ω) = O(h)}$ if u ? W²,p(Ω), with 2p > n. In the case of the model problem ?Δu+au = f in Ω, u = u_o on δΩ, with a ? 0, a simple geometrical condition is given which insures the validity of the maximum principle for the discrete problem.     

A coarse preconditioner for multi-domain boundary element equations

This paper is concerned with the application of a coarse preconditioner, the generalised minimal residual (GMRES) method and a generalised successive over-relaxation (GSOR) method to linear systems of equations that are derived from boundary integral equations. Attention is restricted to systems of the form ∑^N_j=1H_ijx_j=c_i, i=1,2,…,N, where H_ij are matrices, x_j and c_i are column vectors. The integer N denotes the number of domains and these systems are solved by adapting techniques initially devised for solving single-domain problems. These techniques include parameter matrix accelerated GMRES and GSOR in combination with a multiplicative Schwarz method for non-overlapping domains. The multiplicative Schwarz method is a generalised form of the block Gauss–Seidel method and is called the generalised multi-domain iterative procedure. A new form of coarse grid preconditioning is applied to limit the convergence dependence on block numbers. The coarse preconditioner is obtained from a crude representation of the global system of equations. Attention is restricted to thermal problems with domains connected through resistive thermal barriers. The effect of lowering and increasing the thermal resistance between domains is investigated. The coarse preconditioner requires a more accurate representation on interfaces with lower thermal resistance. Computation times are determined for the iterative procedures and for elimination techniques indicating the relative benefits for problems of this nature.     

Remarks on the periodic boundary value problems for first-order differential equations

This paper is concerned with the existence of solutions and the monotone method of first-order periodic boundary value problems when the lower and upper solutions α and β violate the boundary conditions α(0) ≤ α(2π) and β(0) ≥ β(2π). Using the topological degree theory, two existence theorems are established under weaker conditions than the one-side Lipschitz conditions. An example is given, which illustrates that PBVP may not have solutions between α and β without further restrictions to f(t, u). The monotone method is also discussed with some new results.     

A Fourth Order Accurate Finite Difference Scheme for the Elastic Wave Equation in Second Order Formulation

We present a fourth order accurate finite difference method for the elastic wave equation in second order formulation, where the fourth order accuracy holds in both space and time. The key ingredient of the method is a boundary modified fourth order accurate discretization of the second derivative with variable coefficient, (??(x)u _x ) _x . This discretization satisfies a summation by parts identity that guarantees stability of the scheme. The boundary conditions are enforced through ghost points, thereby avoiding projections or penalty terms, which often are used with previous summation by parts operators. The temporal discretization is obtained by an explicit modified equation method. Numerical examples with free surface boundary conditions show that the scheme is stable for CFL-numbers up to 1.3, and demonstrate a significant improvement in efficiency over the second order accurate method. The new discretization of (??(x)u _x ) _x has general applicability, and will enable stable fourth order accurate approximations of other partial differential equations as well as the elastic wave equation.