Manohar P. Kamat

Sensitivity Analysis and Optimal Design of Nonlinear Beams and Plates

ABSTRACT A uniform formulation of sensitivity analysis for beams and plates is presented in terms of generalized stresses and strains. Both physical and geometric nonlinearities can be treated within this formulation. Next, optimal design problems for stress and deflection constraints are formulated and the relevant optimality conditions are derived using the concept of a linear adjoint structure. Finally, several numerical solutions of optimal design problems of beams are presented.

Role of Maggi's Equations in Computational Methods for Constrained Multibody Systems

This paper presents a unified theoretical basis for a class of methods that generate the governing equations of constrained dynamical systems by eliminating the constraints. By using Maggis equations in conjunction with a common projective theory from numerical analysis, it is shown that members of the class are precisely characterized by the basis they choose for the null-space of the variational form of the constraints. For each method considered, the specific basis chosen for the null-space of the variational constraints is derived, as well as a dual basis for the orthogonal complement. The latter basis is of particular interest since it is shown that its knowledge theoretically enables one to generalize certain methods of the class to calculate constraint forces and torques. Practical approaches based on orthogonal transformations to effect this strategy are also outlined. In addition, since the theory presented herein stresses a common, fundamental structure to the various methods, it is especially useful as a means of comparing and evaluating individual numerical algorithms. The theory presented makes clear the relationship between certain numerical instabilities that have been noted in some methods that eliminate a priori constraint contributions to the governing equations by selecting an independent subset of unknowns. It is also briefly indicated how this formalism can be extended, in principle, to the wider class of nonlinear nonholonomic constraints.

OPTIMIZATION OF SPACE TRUSSES AGAINST INSTABILITY USING DESIGN SENSITIVITY DERIVATIVES

This paper addresses the problem of the maximization of the critical load of shallow space trusses of given configuration and volume. Through implicit differentiation of the nonlinear equilibrium equations and the stability criterion, the sensitivity derivatives of the critical load parameter with respect to the design variables are developed. Optimum designs are generated using a projected Lagrangian technique known as the Variable Metric method for constrained optimization (VMCON).

Concurrent multiprocessing for calculating nullspace and range spacebases for multibody simulation

It is shown that highly efficient parallel variants of the QR decomposition and modified Gram-Schmidt procedure can be derived by using a graph theoretic description of system connectivity. The methods developed herein regularly order the directed graph of the mechanical system, such that the transpose of the constraint matrix has block upper triangular form, and subsequently assign independent processors tasks based upon the zero fill structure of the constraint matrix

A quasi-Newton versus a homotopy method for nonlinear structural analysis☆

Abstract The authors assess the relative merits and demerits of the minimization techniques using globally convergent quasi-Newton algorithms on the one hand and the homotopy algorithms on the other hand for the solution of problems of nonlinear structural analysis. Like the homotopy algorithms the globally convergent quasi-Newton algorithms are equally well suited for the solution of the nonlinear equations of structural analysis directly without having to pose the problem as an equivalent minimization problem. It is only in the close neighborhood of the limit and bifurcation points, however, that quasi-Newton algorithms experience difficulties. Homotopy algorithms on the other hand are robust for practically all types of nonlinear problems but are computationally not as cost-effective since they provide an extremely accurate prediction of the response by calculating it at a large number of points. Globally convergent quasi-Newton algorithms can perform well with very approximate Hessians, while homotopy algorithms require extremely accurate Hessians. Finally, while quasi-Newton algorithms can be very easily structured to exploit sparsity and symmetry, homotopy algorithms are not presently so structured and would require special modifications for exploitation of such features without sacrificing robustness and global convergence.

Optimization of an Asymmetric Two-Bar Truss Against Instability∗

ABSTRACT This note considers a two-bar truss with bars whose lengths are not necessarily equal. A vertical load is applied at the internal joint. At the critical load, one (or both) of the bars buckles or the truss exhibits snap-through instability. For given.total volume of the truss, the areas of the bars that maximize the critical load are determined.

Recent Developments in Quasi-Newton Methods for Structural Analysis and Synthesis

Unlike the Newton-Raphson method, quasi-Newton methods by virture of the updates and step length control procedures are globally convergent and hence better suited for the solution of nonlinear problems of structural analysis and synthesis. Extension of quasi-Newton algorithms to large scale problems has led to the development of sparse update algorithms and to economical strategies for evaluating sparse Hessians. Ill-conditioning problems have led to the development of self-scaled variable metric and conjugate gradient algorithms, as well as the use of the singular perturbation theory. This paper emphasizes the effectiveness of such quasi-Newton algorithms for nonlinear structural analysis and synthesis.

Location of stable and unstable equilibrium configurations using a model trust region quasi-Newton method and tunnelling

Abstract A hybrid method for locating multipole equilibrium configurations has been proposed recently. The hybrid method combined the efficiency of a quasi-Newton method capable of locating stable and unstable equilibrium solutions with a robust homotopy method capable of tracking equilibrium paths with turning points and exploiting sparsity of the Jacobian matrix at the same time. A quasi-Newton method in conjunction with a deflation technique is proposed here as an alternative to the hybrid method. The proposed method not only exploits sparsity and symmetry, but also represents an improvement in efficiency.

OPTIMIZATION OF HIGHLY FLEXIBLE BEAMS

The authors propose three different measures of structure stiffness in the nonlinear range and proceed to optimize these measures. Optimization is performed using the simple optimally criterion approach in one case and mathematical programming in the other two cases. It is found that although the three measures can lead to identical designs in the linear case the same is definitely not true in the nonlinear case. For the purposes of illustration a highly flexible centrally loaded clamped beam is chosen.

Energy minimization versus pseudo force technique for nonlinear structural analysis

The effectiveness of using minimization techniques for the solution of nonlinear structural analysis problems is discussed and demonstrated by comparison with the conventional pseudo force technique. The comparison involves nonlinear problems with a relatively few degrees of freedom. A survey of the state-of-the-art of algorithms for unconstrained minimization reveals that extension of the technique to large scale nonlinear systems is possible.