Approximate finite-element method of stress analysis of non-axisymmetric configurations

A finite-element method of analysis is developed for structural configurations which are derived from axisymmetric geometries but contain definite non-axisymmetric features in the circumferential direction. The purpose of the present analysis is to develop a method which will take into consideration the fact that the stress and strain conditions in these geometires will be related to the corresponding axisymmetric solution. To analyze these structures, the geometry is divided into several segments in the r-θ plane. The axisymmetric displacements are obtained for each segment by solving a related axisymmetric configuration. A perturbation analysis is then performed to match the solutions at certain points between the segments and obtain the perturbation displacements for the total structure. The total displacement is then the axisymmetric displacement plus the perturbation displacement. The stresses and strains are then calculated at any desired point once the total displacements are known. The method is applied to a number of examples to illustrate the accuracy of the method. The results for these examples are presented and discussed. Some of these examples illustrate the difference between the present approach and the well known substructural analysis and it is shown that the present method is much more accurate.     

An elastic-plastic analysis of non-axisymmetric structures

Previously a method of analyzing elastic non-axisymmetric problems was developed. This method involved dividing the structure into segments in the r−θ plane and performing an axisymmetric analysis. The displacements were then forced to match at selected nodes. This yields a set of perturbation displacements for each segment. The total displacement in the structure is the axisymmetric displacement plus the perturbation displacement.

In this study this method is expanded to include elastic-plastic material behavior with strain hardening effects. For the elastic-plastic analysis the von Mises-Hencky yield condition and the Prager, or kinematic, strain hardening rule are used. The method is applied to a number of examples to test its accuracy. Numerical results are presented and discussed.     

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 finite difference method for 3D incompressible flows in cylindrical coordinates

In this work, a finite difference method to solve the incompressible Navier-Stokes equations in cylindrical geometries is presented. It is based upon the use of mimetic discrete first-order operators (divergence, gradient, curl), i.e. operators which satisfy in a discrete sense most of the usual properties of vector analysis in the continuum case. In particular the discrete divergence and gradient operators are negative adjoint with respect to suitable inner products. The axis r = 0 is dealt with within this framework and is therefore no longer considered as a singularity. Results concerning the stability with respect to 3D perturbations of steady axisymmetric flows in cylindrical cavities with one rotating lid, are presented.     

The PCICE-FEM scheme for highly compressible axisymmetric flows

The recently developed PCICE-FEM scheme (Journal of Computational Physics, vol. 198, 659, 2004) is extended to two-dimensional axisymmetric geometries. The main discretization problem for nodal-based axisymmetric formulations lies in deriving a closed form as the radial coordinates approach zero along the axis of symmetry. This problem is addressed by employing the finite element piecewise linear approximations to both the flow variables and (separately) to the nodal values of the radial coordinates. The resulting formulation is an elegant treatment of the axisymmetric coordinate system with out noticeable loss of spatial accuracy and little additional cost in computational effort. An overview of the PCICE algorithm for the axisymmetric governing equations will be followed by a detailed axisymmetric finite element formulation for the PCICE-FEM scheme. The ability of the PCICE-FEM scheme to accurately and efficiently simulate highly compressible axisymmetric flows is demonstrated.     

Transition finite elements with temperature and temperature gradients as primary variables for axisymmetric heat conduction

This paper presents finite element formulation for a special class of elements referred to as “transition finite elements” for axisymmetric heat conduction. The transition elements are necessary in applications requiring the use of both axisymmetric solid elements and axisymmetric shell elements. The elements permit transition from the solid portion of the structure to the shell portion of the structure. A novel feature of the formulation presented here is that nodal temperatures as well as nodal temperature gradients are retained as primary variables. The weak formulation of the Fourier heat conduction equation is constructed in the cylindrical co-ordinate system (r, z). The element geometry is defined in terms of the co-ordinates of the nodes as well as the nodal point normals for the nodes lying on the middle surface of the element. The element temperature field is approximated in terms of element approximation functions, nodal temperatures and the nodal temperature gradients. The properties of the transition elements are then derived using the weak formulation and the element temperature approximation. The formulation presented here permits linear temperature distribution through the element thickness. Convective boundaries as well as distributed heat flux is permitted on all four faces of the element. Furthermore, the element formulation also permits distributed heat flux and orthotropic material behaviour. Numerical examples are presented, first to illustrate the accuracy of the formulation and second to demonstrate its usefulness in practical applications. Numerical results are also compared with the theoretical solutions.     

Mixed finite element formulations of strain-gradient elasticity problems

Theories on intrinsic or material length scales find applications in the modeling of size-dependent phenomena. In elasticity, length scales enter the constitutive equations through the elastic strain energy function, which, in this case, depends not only on the strain tensor but also on gradients of the rotation and strain tensors. In the present paper, the strain-gradient elasticity theories developed by Mindlin and co-workers in the 1960s are treated in detail. In such theories, when the problem is formulated in terms of displacements, the governing partial differential equation is of fourth order. If traditional finite elements are used for the numerical solution of such problems, then C¹ displacement continuity is required. An alternative “mixed” finite element formulation is developed, in which both the displacement and the displacement gradients are used as independent unknowns and their relationship is enforced in an“integral-sense”. A variational formulation is developed which can be used for both linear and non-linear strain-gradient elasticity theories. The resulting finite elements require only C⁰ continuity and are simple to formulate. The proposed technique is applied to a number of problems and comparisons with available exact solutions are made.     

Cluster variation method in the computational materials science

Cluster Variation Method (CVM) has been very successful in the computations of alloy phase diagrams as well as in many problems of the materials science related to the phase transitions. Originally, CVM was developed in the framework of the so-called rigid lattice approximation, but it has recently been extended to include continuous atomic displacements due to thermal lattice vibration and local atomic distortion due to size mismatch of the constituent atoms. In the present study, we focus our attention on the latter continuous displacement treatment of CVM. The continuous displacement (CD) formulation of the CVM is applied to study the phase stability of the binary alloys. The basic idea is to treat an atom which is displaced by r from its reference lattice point as a species designated by r. The effects of continuous atomic displacement on the thermodynamic quantities and phase transitions of binary alloys are investigated in detail. We also discuss the extension of the CD treatment of CVM to the calculations of solid-liquid and gas liquid phases transitions.     

Application of a three-dimensional mixed finite element model to the flexure of sandwich plate

The bending analysis of sandwich plates consisting of very stiff face sheets and a comparatively flexible core material offers challenge due to large variation in the magnitude of stress and strain components in the face and in the core regions of the plate. Similarly, the displacement fields do vary in zigzag manner at the layer interface of stiff face sheet and the soft core, thereby making the transverse strains highly discontinuous at such layer interfaces. All these behavioural aspects indicate that only an individual layerwise model can appropriately analyze sandwich plates. A layerwise (three-dimensional), mixed, 18-node finite element (FE) model developed by Ramtekkar et al. [Mech. Adv. Mater. Struct. 9 (2002) 133] has been employed for the accurate evaluation of transverse stresses in sandwich laminates. The FE model consists of six degrees-of-freedom (three displacement components and three transverse stress components τ_xz, τ_yz, σ_z, where z is the thickness direction) per node which ensures the through thickness continuity of transverse stress and displacement fields. Results obtained by using the FE model have shown excellent agreement with the available elasticity solutions for sandwich plates. Additional results on the variation of transverse strains have also been presented to highlight the magnitude of discontinuity in these quantities due to difference in properties of the face and the core materials of sandwich plates.     

Static,vibration and buckling analysis of axisymmetric circular plates using finite elements

The static, vibration, and buckling analysis of axisymmetric circular plates using the finite element method is discussed. For the static analysis, the stiffness matrix of a typical annular plate element is derived from the given displacement function and the appropriate constitutive relations. By assuming that the static displacement function, which is an exact solution of the circular plate equation ?²?²W = 0, closely represents the vibration and buckling modes, the mass and stability coefficient matrices for an annular element are also constructed. In addition to the annular element, the stiffness, mass, and stability coefficient matrices for a closure element are also included for the analysis of complete circular plates (no center hole). As an extension of the analysis, the exact displacement function for the symmetrical bending of circular plates having polar orthotropy is also given.     

Fast projection methods for minimal design problems in linear system theory

The so-called minimal design problem (or MDP) of linear system theory is to find a proper minimal degree rational matrix solution of the equation H(z)D(z)=N(z), where {N(z),D(z)} are given p×r and m×r polynomial matrices with D(z) of full rank r≦m.We describe some solution algorithms that appear to be more efficient (in terms of number of computations and of potential numerical stability) than those presently known. The algorithms are based on the structure of a polynomial echelon form of the left minimal basis of the so-called generalized Sylvester resultant matrix of {N(z), D(z)}. Orthogonal projection algorithms that exploit the Toeplitz structure of this resultant matrix are used to reduce the number of computations needed for the solution.     

Low-cost calculation of multigroup neutron transport equation for the recalculation of spectrum-averaged cross sections

The procedures of the recalculation of the multigroup equation of neutron transport in the two-dimensional r-z geometry based on the quasi-diffusion method are described. The quasi-diffusion method allows a considerable reduction of the required iterations of the source and increases accuracy of the calculation. The procedure is demonstrated on the calculation results of a two-dimensional model of the active zone of the BN-800 reactor working in the self-controlled neutron-nuclear mode.     

Reliability of the finite element method for calculating free edge stresses in composite laminates

The edge-stress problem for a [±45]_s graphite/epoxy laminate was examined in detail. A review of the literature on this problem showed that the interlaminar normal stress σ_z distributions along the interface between the +45° and -45° plies, obtained by various investigators, disagreed in magnitude and sign. In particular, a finite difference solution and a perturbation solution predicted a tensile σ_z, whereas the finite element methods predicted a compressive stress. Since a stress singularity exists at the intersection of the interface and the free edge, the differences in magnitude of the peak stress were expected, but not the difference in the sign.This paper investigates the reliability of the displacement-formulated finite element method in analyzing the edge-stress problem. Analyses of two well-known elasticity problems, one involving a stress discontinuity and one a singularity, showed that the finite element analysis yields accurate stress distributions everywhere except in two elements closest to the stress discontinuity or singularity. Stress distributions for a [±45]_s laminate showed the same behavior near the singularity as found in the well-known problems with exact solutions. The displacementformulated finite element method, therefore, appears to be a highly accurate technique for calculating interlaminar stresses in composite laminates. The disagreement among the numerical methods was attributed to the unsymmetric stress tensor at the singularity.     

A probabilistic analysis of an error-correcting algorithm for the towers of Hanoi puzzle

Let S be a set of n horizontal and vertical segments on the plane, and let s, t ∈ S. A Manhattan path (of length k) from s to t is an alternating sequence of horizontal and vertical segments, s = r₀, r₁,…,r_k = t, such that r_i intersects r_i+1, 0 ? i < k. An O(n log n) time O(n log n) space algorithm is presented which, given S and t, finds a tree of shortest Manhattan paths from all s ∈ S to t. The algorithm relies on a new data structure which makes it possible to find in O(log n + p) time all p segments currently in S which intersect a given s ∈ S, and which support a deletion of any segment from S in O(log n) time, where we assume that the cost of these operations is accumulated over the whole algorithm. The structure makes use of the recently discovered Gabow and Tarjan's linear time version of the union-find algorithm on consecutive sets. We prove an Ω(n log n) lower bound on the complexity of deciding whether there is a Manhattan path between two given segments, under the linear decision tree model. Finally, some applications of the Manhattan path algorithm are indicated.     

n-tuple complex helical geometry modeling using parametric equations

Complex helical structures are difficult to model in three-dimensional form to conduct finite element analysis. In general, parametric mathematical equations for single and double helical geometries are readily available in existing literature. However, more complex forms such as triple or in general n-tuple helical structures are still not widely studied. In this paper, at first, definitions of single and double helical structures are presented in parametric mathematical forms. Centerlines, curvatures, and torsions of these geometries are found, and these two helical geometries are visualized in three-dimensional structures. Next, one of the untouched helical models, the triple helical geometry, is investigated and a procedure to find the centerline of the triple helical geometry is presented. In addition, the first three-dimensional generated solid model of a triple helix geometry is presented. Finally, the steps used to create triple helix geometry are generalized to find parametric mathematical equations for n-tuple helical geometries.     

Binary collision of drops in simple shear flow at finite Reynolds numbers: Geometry and viscosity ratio effects

The collision of two equal-size drops in an immiscible phase undergoing a shear flow is simulated over a range of viscosity ratios (??) and different geometries. The full Navier-Stokes equations are solved by a finite difference/front tracking method. Based on experimental data, different cases were simulated by changing the offset, size of drops, and viscosity ratio. The distance between drop centres along the velocity gradient direction (z) was measured as a function of time. It was found that ??z increases after collision and reaches a new steady-state value after separation. The values of ??z, during the interaction, increases with increasing initial offset. Our results show that the time of approaching of drops at low initial offset is greater than the other cases, but the maximum deformation is the same for equal drop sizes. The deformation decreases with decreasing the size of drops. As the initial offset increases, the drops rotate more quickly and the available contact time for film drainage decreases. We found that the trajectories of drops in the approaching stage are different owing to the different initial offsets. However, after the drops come into contact, it observed that they follow the same trajectories. As ?? increases, the drops rotate more slowly, and the point at which the drops separate is delayed. The trajectories of drops become more symmetric with the increased ??.     

Signature extension in remote sensing

This paper considers the problem of signature extension in remote sensing. Signature extension is a process of increasing the spatial-temporal range over which a set of training statistics can be used to classify data without significant loss of recognition accuracy.Methods are developed for the selection of segments for obtaining the training data. Selection of the number of segments is treated as the problem of expansion of rectangular matrix with basis matrices. Computational algorithms based on mean minimum square estimation error are developed for the selection of best segments. Furthermore, a combinatorial algorithm for generating all possible r combinations of S in Sc_r steps with a single change at each step is presented.     

Adaptive fairing of digitized point data with discrete curvature

An algorithm for fairing two-dimensional (2D) shape formed by digitised data points is described. The application aims to derive a fair curve from a set of dense and error-filled data points digitised from a complex surface, such that the basic shape information recorded in the original point data is relatively unaffected. The algorithm is an adaptive process in which each cycle consists of several steps. Given a 2D point set, the bad points are identified by analysing the property of their discrete curvatures (D-curvatures) and first-order difference of D-curvatures, in two consecutive fairing stages. The point set is then segmented into single bad point (SBP) segments and multiple bad point (MBP) segments. For each MBP segment, a specially designed energy function is used to identify the bad point to be modified in the current cycle. Each segment is then faired by directly adjusting the geometric position of the worst point. The amount of adjustment in each cycle is kept less than a given shape tolerance. This algorithm is particularly effective in terms of shape preservation when dealing with MBP segments. Case studies are presented that illustrate the efficacy of the developed technique.