Isoparametric finite difference energy method for plate bending problems

This paper incorporates the concept of isoparametry in finite difference energy method making it more powerful and versatile to tackle complex plate bending problems with curved boundaries. This approach overcomes the drawbacks of the finite difference energy method and its application to practical problems is now feasible. In order to estimate the accuracy and reliability of the present formulation several isotropic plates with a variety of planform are solved and the results are compared with the existing analytical and numerical solutions.     

Free vibration analysis of plates using least-square-based finite difference method

In this paper, we apply the two-dimensional least-square-based finite difference (LSFD) method for solving free vibration problems of isotropic, thin, arbitrarily shaped plates with simply supported and clamped edges. Using the chain rule, we show how the fourth-order derivatives of the plate governing equation can be discretized in two or three steps as well as how the boundary conditions can be implemented directly into the governing equation. By analyzing vibrating plates of various shapes and comparing the solutions obtained against existing results, we clearly demonstrate the effectiveness of LSFD as a mesh-free method for computing vibration frequencies of generally shaped plates accurately.     

Integrated analysis of curved bridge superstructures by variational finite difference method

Static analysis of bridge superstructures which are curved in plan, having radial as well as circumferential beams, and continuous over axially flexible column supports is proposed using a variational based finite difference energy method. The total potential energy of the system composed of different components of the bridge decks is discretized in terms of pivot displacements with the help of finite difference operators with constant order of truncation error. The principle of minimum total potential energy is applied to obtain the force-displacement relationship which was solved for pivot displacements. In order to estimate the accuracy and reliability of the present formulation several problems of curved plates with increasing order of complexity are solved. The results are compared with the existing solutions.     

Trefftz-unsymmetric finite element for bending analysis of orthotropic plates

This work develops a new four-node quadrilateral displacement-based Trefftz-type plate element for bending analysis of orthotropic plates within the framework of the unsymmetric finite element method (FEM). In the present formulation, the modified isoparametric interpolations are employed to formulate the element’s test functions in which the deflection is effectively enriched by the nodal rotation degrees of freedom (DOFs). Meanwhile, the element’s trial functions are determined based on the Trefftz functions that can a prior satisfy the governing equations of orthotropic Mindlin–Reissner plates. Numerical benchmark tests reveal that the new unsymmetric plate element is free of shear locking problem and can produce satisfactory results for both the displacement and stress resultant. In particular, it exhibits quite good tolerances to the gross mesh distortion.

     

A general finite difference method for arbitrary meshes

A two-dimensional finite-difference technique for irregular meshes is formulated for derivatives up to the second order. The domain in the vicinity of a given central point is broken into eight 45 degree pie shaped segments and the closest finite-difference point in each segment to the center point is noted. By utilizing Taylor series expansions about a central point with a unique averaging process for the points in the four diagonal segments, good approximations to all derivatives up to the second order and including the mixed derivatives are obtained. For square meshes the general derivative expressions for arbitrary meshes which were determined reduce to the usual finite difference formulae. In one example problem the Poisson equation is solved for an irregular mesh. In a second example for the first time a problem with a geometric nonlinearity, namely large deflection response of a flat membrane, is solved with an irregular mesh. The solutions compare very favorably with results obtained previously. Some discussion is given on possible approaches for determination of finite difference derivatives higher than the second.     

A finite difference method for a Stefan problem

Many physical problems involve initial boundary value problems for parabolic differential equations in which part of the boundary is not given a priori but is found as part of the solution. These problems have been considered under the name of «Stefan problems». Stefan problems occur in such physical processes as the melting of solids and the crystallizing of liquids. Existence and uniqueness for various one dimensional Stefan problems have been shown by several authors ([1], [4], [5], [8], [9]). Some multidimensional Stefan problems were considered by ([6], [11]). The purpose of this paper is to approximate the solution of a Stefan problem using an implicit finite difference analog for the heat equation and an explicit finite difference analog for the differential equation describing the free boundary. Also, we shall show that the finite difference solution we obtain converges uniformly to the actual solution. Numerous numerical schemes for various Stefan problems have been successfully employed by several authors ([2], [3], [10], [12], [14], [16], [17]). In Chapter I we formulate a continuous time Stefan problem, and in Chapter II we describe a finite difference scheme for approximating results. Chapter IV contains the main result which shows that the finite difference solution converges to the actual solution. Finally, in Chapter V we give a numerical example.     

Preconditioned variational methods on rotated finite difference discretisation

Due to their rapid convergence properties, recent focus on iterative methods in the solution of linear system has seen a flourish on the use of gradient techniques which are primarily based on global minimisation of the residual vectors. In this paper, we conduct an experimental study to investigate the performance of several preconditioned gradient or variational techniques to solve a system arising from the so-called rotated (skewed) finite difference discretisation in the solution of elliptic partial differential equations (PDEs). The preconditioned iterative methods consist of variational accelerators, namely the steepest descent and conjugate gradient methods, applied to a special matrix ‘splitting’ preconditioned system. Several numerical results are presented and discussed.     

A finite difference method for a moving-interface diffusion-reaction problem

A software is described which solves a diffusion-reaction problem with N diffusing elements which can react to form M non-diffusing compounds. The numerical solution of such a problem is complicated by the fact that different systems of partial differential equations hold in regions in which different compounds are present, and by the fact that the interfaces between regions are mobile. This software is applied to a problem involving the diffusion of carbon and chromium in a NiCr alloy, in which multiple chromium carbides may form and dissociate.     

A symmetric variational finite element-boundary integral equation coupling method

This paper is concerned with the development of a mixed variational principle for coupling finite element and boundary integral methods in interface problems, using the generalized Poisson's equation as a prototype situation. One of its primary objectives is to compare the performance of fully variational procedures with methods that use collocation for the treatment of boundary integral equations. A distinctive feature of the new variational principle is that the discretized algebraic equations for the coupled problem are automatically symmetric since they are all derived from a single functional. In addition, the condition that the flux remain continuous across interfaces is satisfied naturally. In discretizing the problem within inhomogeneous or loaded regions, domain finite elements are used to approximate the field variable. On the other hand, only boundary elements are used for regions where the medium is homogeneous and free of external agents. The corresponding integral equations are discretized both by fully variational and by collocation techniques. Results of numerical experiments indicate that the accuracy of the fully variational procedure is significantly greater than that of collocation for the complete interface problem, especially for complex disturbances, at little additional computational cost. This suggests that fully variational procedures may be preferable to collocation, not only in dealing with interface problems, but even for solving integral equations by themselves.     

A collocation finite element-finite difference solution for developing isothermal flow between two parallel plates

This paper presents a finite element-finite difference method for the solution of the boundary layer equations for developing flow between two parallel plates. Due to the parabolic nature of the equations it was possible to discretize the transverse flow direction with one-dimensional Hermite cubic finite elements and the axial flow direction with a backward finite difference approximation. The collocation finite element-finite difference approximation was found to be appropriate for the modeling of the non-linear convection terms in the axial momentum equation. The resulting system of mixed linear and non-linear algebraic equations was solved using the Newton-Raphson method. Several numerical experiments were conducted to study the behavior of the solution with respect to the element size and number, order of finite difference approximation, and the marching step size.     

A Hermitian finite difference method for the solution of parabolic equations

A high accuracy difference method (hermitian method) for the solution of evolution equations of parabolic type is presented. Its most original feature is to use several unknowns (the value of the solution and its spatial derivatives) at every nodal point of the computational grid. It is shown that this method has better computational performance than classical schemes on non-uniform and coarse meshes.     

A finite element method for construction of dynamical theories of layered plates

A procedure is presented for reduction of equations of three-dimensional elasticity to a twodimensional theory for elastodynamic behavior of a transversely heterogeneous plate. The method is based upon the elimination of the thickness coordinate by using, in conjunction with a variational principle, a set of finite element basis functions along the axis normal to the plate; an asymptotic analysis of the resulting semi-discrete equations yields a hierarchy of reduced order models for the plate. Some numerical results illustrate the applicability of the proposed methodology to layered plates.     

A discontinuous finite difference streamline diffusion method for time-dependent hyperbolic problems

In this article, a new finite element method, discontinuous finite difference streamline diffusion method (DFDSD), is constructed and studied for first-order linear hyperbolic problems. This method combines the benefit of the discontinuous Galerkin method and the streamline diffusion finite element method. Two fully discrete DFDSD schemes (Euler DFDSD and Crank–Nicolson (CN) DFDSD) are constructed by making use of the difference discrete method for time variables and the discontinuous streamline diffusion method for space variables. The stability and optimal L² norm error estimates are established for the constructed schemes. This method makes contributions to the discontinuous methods. Finally, a numerical example is provided to show the benefit of high efficiency and simple implementation of the schemes.     

A finite difference method with non-uniform timesteps for fractional diffusion equations

An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the behavior of the solution in order to keep the numerical errors small without the penalty of a huge computational cost. The method is unconditionally stable and convergent. In fact, it is shown that consistency and stability implies convergence for a rather general class of fractional finite difference methods to which the present method belongs. The huge computational advantage of adaptive methods against fixed step methods for fractional diffusion equations is illustrated by solving the problem of the dispersion of a flux of subdiffusive particles stemming from a point source.     

A finite difference discretization method for elliptic problems on composite grids

In this paper we discusss a simple finite difference method for the discretization of elliptic boundary value problems on composite grids. For the model problem of the Poisson equation we prove stability of the discrete operator and bounds for the global discretization error. These bounds clearly show how the discretization error depends on the grid size of the coarse grid, on the grid size of the local fine grid and on the order of the interpolation used on the interface. Furthermore, the constants in these bounds do not depend on the quotient of coarse grid size and fine grid size. We also discuss an efficient solution method for the resulting composite grid algebraic problem.     

A least-squares variational method for full-field reconstruction of elastic deformations in shear-deformable plates and shells

A variational principle is formulated for the inverse problem of full-field reconstruction of three-dimensional plate/shell deformations from experimentally measured surface strains. The formulation is based upon the minimization of a least-squares functional that uses the complete set of strain measures consistent with linear, first-order shear-deformation theory. The formulation, which accommodates for transverse shear-deformation, is applicable for the analysis of thin and moderately thick plate and shell structures. The main benefit of the variational principle is that it is well-suited for C⁰-continuous displacement finite element discretizations, thus enabling the development of robust algorithms for application to complex civil and aeronautical structures. The methodology is especially aimed at the next generation of aerospace vehicles for use in real-time structural health monitoring systems.     

A powerful finite element for plate bending

A powerful finite element formulation for plate bending has been developed using a modified version of the variational method of Trefftz. The notion of a boundary has been generalized to include the interelement boundary. All boundary conditions and the interelement continuity requirements (displacements, slopes, internal forces) have been obtained as natural conditions on the generalized boundary. Coordinate functions have been constructed to satisfy the nonhomogeneous Lagrange equation locally within the elements. Singularities due to isolated loads have been properly taken into account. For practical use a general quadrilateral element has been developed and its accuracy illustrated on several numerical examples. Work is in progress to extend the formulation to anisotropic and moderately thick plates and to vibration analysis.     

A finite difference continuation method for computing energy levels of Bose-Einstein condensates

We study a finite difference continuation (FDC) method for computing energy levels and wave functions of Bose-Einstein condensates (BEC), which is governed by the Gross-Pitaevskii equation (GPE). We choose the chemical potential λ as the continuation parameter so that the proposed algorithm can compute all energy levels of the discrete GPE. The GPE is discretized using the second-order finite difference method (FDM), which is treated as a special case of finite element methods (FEM) using the piecewise bilinear and linear interpolatory functions. Thus the mathematical theory of FEM for elliptic eigenvalue problems (EEP) also holds for the Schrödinger eigenvalue problem (SEP) associated with the GPE. This guarantees the existence of discrete numerical solutions for the ground-state as well as excited-states of the SEP in the variational form. We also study superconvergence of FDM for solution derivatives of parameter-dependent problems (PDP). It is proved that the superconvergence O(ht) in the discrete H¹ norm is achieved, where t=2 and t=1.5 for rectangular and polygonal domains, respectively, and h is the maximal boundary length of difference grids. Moreover, the FDC algorithm can be implemented very efficiently using a simplified two-grid scheme for computing energy levels of the BEC. Numerical results are reported for the ground-state of two-coupled NLS defined in a large square domain, and in particular, for the second-excited state solutions of the 2D BEC in a periodic potential.     

An explicit finite difference method for finite-time observers

In this paper we present a numerical method for estimating the current state of a nonlinear control system. We use finite differences to discretize a modified version of the finite-time observer equations in James. The discretized equations are simple and easily programmed. The convergence and accuracy of the scheme is proved, and the scheme enjoys a number of important properties: availability of rate of convergence estimates, good robustness characteristics, and the ability to handle certain types of discontinuities in the observations. The major disadvantage is that the number of grid points required increases exponentially with the number of state dimensions.