Edward L. Wilson

Incompatible Displacement Models

Publisher Summary This chapter introduces incompatible displacement modes at the element level in order to improve element accuracy. One of the main causes of inaccuracies in lower-order finite elements is their inability to represent certain simple stress gradients. The same basic method of introducing incompatible displacement modes in order to improve the bending properties can be used in three dimensions. The first eight are the standard compatible interpolation functions. The last three are incompatible and are associated with linear shear and normal strains. The nine incompatible modes are eliminated at the element stiffness level by static condensation. As the three-dimensional element degenerates to the same approximation as in the two-dimensional element, the same improvement in accuracy is obtained. This element has been found to be extremely effective in the analysis of massive three-dimensional structures subjected to bending. One element in the thickness direction of arch dams or thick pipe joints has been found to be adequate.

Application of the finite element method to heat conduction analysis

Abstract A variational principle is applied to the transient heat conduction analysis of complex solids of arbitrary shape with temperature and heat flux boundary conditions. The finite element discretization technique is used to reduce the continuous spatial solution into a finite number of time-dependent unknowns. The continuum is divided into subregions (elements) in which the temperature field variable is approximated by Rayleigh-Ritz polynomial expansions in terms of the values of the temperatures at prescribed boundary points of the subregions. These temperatures at the node points act as generalized coordinates of the system and, being common to adjacent subregions, enable appropriate continuity requirements to be satisfied over the entire continuum. The variational principle yields Euler equations of the Lagrangian form which result in the development of an equivalent set of first order, ordinary differential equations in terms of the nodal temperatures. A unique method of numerical solution of these equations is introduced which is stable and requires a minimum of computer effort. Elements of various shapes and their associated temperature fields are discussed for one, two and three-dimensional bodies. The method is developed in detail for two-dimensional bodies which are idealized by systems of triangular elements. The development of a digital computer program is discussed and several examples are given to illustrate the validity and practicality of the method.

A parallel active column equation solver

Abstract General purpose parallel Fortran subroutines are presented for the solution of sparse and dense symmetric systems of linear equations. They constitute an efficient tool that can be easily implemented on already existing sequential out-of-core direct solvers. They are designed to run on the whole class of multiprocessors with local and/or shared memory. Numerical experiments are carried out on both Intel Personal Super Computer (local memory) and Encore Multimax (shared memory). The impact of hardware architecture on software implementation and performance is discussed. Important issues for the efficient use of parallel computers are addressed.

An eigensolution strategy for large systems

Abstract A solution strategy is presented for the evaluation of frequencies and mode shapes for very large structural systems. The subspace iteration method is modified to calculate the eigenpairs in groups near different shift points. If the bandwidth is large and only a few vectors are required, the cost of factorization of the stiffness matrix will dominate and the standard subspace iteration method is effective. However, for the case where the bandwidth is of comparable size to the number of eigenvectors required, there is significant advantage to shifting and evaluating the eigenpairs in groups. It is demonstrated that a good approximation for the optimum number of iteration vectors is given by the square root of the bandwidth. Therefore, if 80 eigenpairs are required of a system with a bandwidth of 400, 20 iteration vectors would be used and the eigenpairs would be found in approximate groups of 10 near 8 shift points. In addition, for the solution of a special class of dynamic response problems, it has been shown that a direct superposition of Ritz vectors yields a more accurate solution than a superposition of the exact eigenvectors. Since this approach eliminates the need to solve for the exact eigenpairs the numerical algorithm, which automatically generates the series of orthogonal Ritz vectors, is presented as an alternative to the solution for the exert eigenpairs. The method does produce an approximate set of frequencies which are excited by a specified load pattern.

Use of incompatible displacement modes for the calculation of element stiffnesses or stresses

Abstract The addition of incompatible displacement modes to lower-order displacement-based elements is re-evaluated. Recent research has indicated that a simple numerical correction can be applied to the shape functions in order that the constant-strain patch test is passed. In this paper a new method of stress recovery is presented in which incompatible modes are introduced. A least-square approximation is used to calculate element stresses which are in microscopic equilibrium and tend to be in global equilibrium with the applied nodal loads on the finite element assemblage. Also, a consistent and robust method for the evaluation of thermal stresses is presented. The numerical methods presented are general and can be applied to all displacement-based finite elements. The basic formulation and examples which are presented in this paper are in three-dimensional elasticity using the eight-node isoparametric elements. Accuracy of the element is illustrated and it is demonstrated that both displacements and stresses are almost identical to those produced by Pians hybrid stress elements.

Direct solution of large systems of linear equations

Abstract A very efficient computer subroutine for the direct solution of large numbers of simultaneous linear equations is presented. Basically the program uses Gauss elimination on positive-definite symmetrical systems. The specific features are that systems of very large size and bandwidth can be solved and that all operations on zero elements are eliminated. Also, the program is very simple and can be incorporated into existing programs with a minimum of effort. The amount of backup storage available on the computer used will govern the maximum size of the system which can be solved. A FORTRAN IV listing of the subroutine is given.

Dynamic analysis of large structural systems with local nonlinearities

Abstract In many types of structures that exhibit significant nonlinearity during dynamic response the nonlinear stiffness property is confined to a few predetermined localities. This physical characteristic may be exploited by making use of substructure concepts in the dynamic response analysis. The elastic components may be represented by a small matrix of stiffness coefficients coupling them to the nonlinear elements, and only the properties of the few nonlinear elements need be modified during the response analysis. Both direct and indirect techniques for taking advantage of localized nonlinearity are reviewed here. In the direct methods the equations of motion are integrated by step-by-step procedures. Various methods of reducing the effective number of degrees of freedom are discussed. In the indirect methods the response to simple loadings is determined, and the response to the prescribed loading is obtained by superposition. A technique for applying superposition to the linear portion of the structure while treating the nonlinear region by a step-by-step procedure is described.

Simple numerical algorithms for the mode superposition analysis of linear structural systems with non-proportional damping

Abstract For linear structural systems it is possible to use the undamped eigenvectors or load-dependent Ritz vectors to produce a set of modal response equations. When arbitrary viscous damping exists the modal equations are coupled with the modal damping matrix. A robust and efficient numerical algorithm is presented, which solves the coupled modal equations by iteration. It is shown that the numerical integration algorithm always converges. The method produces an exact solution for proportional damping and for loading that varies linearly within an arbitrary time interval. In addition, the algorithm has been modified to incorporate automatically the mode acceleration method and periodic loading. Two numerical examples are presented to illustrate the practical application of the algorithm. A FORTRAN listing of a subroutine is given to facilitate easy implementation of the method in existing computer programs for dynamic response analysis.

Solution of finite element systems on concurrent processing computers

A new computer program architecture for the solution of finite element systems using concurrent processing is presented. The basic approach involves the automatic creation of substructures. A host provides control over a set of processors, each of which is assigned initially to one substructure, then dynamically reassigned to the common interface for the solution of the complete system of substructures. Algorithm details are presented fo each phase of the analysis.Results of analysis of large plate bending problems on a hypercube multicomputer are reported. For a system with 2,000 equations, an efficiency of 80 percent of the maximum theoretical value was obtained using 16 processors.

Evaluation of the use of resonant frequencies to characterize physical properties of human long bones

Human tibiae were subjected to steady state vibration over a frequency range to evaluate the use of clinical measurements of resonant frequency to characterize osteoporosis. Displacement and force transmitted to the bone were monitored and used to obtain measures of dynamic mass, stiffness, damping and resonant frequency. Initial results indicate that resonant frequency F is a less sensitive indicator of change in state than either generalized mass M or generalized stiffness K due to the functional relationship of these three parameters.