Uri Kirsch

Optimal topologies of truss structures

Abstract The topology of truss structures is optimized by assuming zero lower bounds on cross-sections. It is shown that the optimal topology might correspond to a singular solution even for simple structures. Assuming the force method analysis formulation, the problem can be stated in a linear programming (LP) form under certain assumptions. It is then possible to derive analytically some conditions related to optimal topologies. In addition, the difficulty of singular optimal solutions is eliminated. The effect of compatibility conditions on optimal topologies is studied. It is shown that for particular geometries or loading conditions, where some element forces change from tension to compression or vice versa, the optimal topology might represent an unstable structure. Analytical conditions are derived to obtain geometries of multiple optimal topologies. Part of the latter solutions usually represent statically determinate structures. It is shown that a transition in the set of active constraints at the optimum occurs at these particular geometries. The phenomena presented in this study might lead to a better understanding of some properties associated with optimum structural topologies, and to improved design procedures.

Efficient sensitivity analysis for structural optimization

Abstract Efficient sensitivity analysis, based on high quality approximations of displacements, is presented. The improved approximations are achieved by combining the efficiency of local approximations (series expansion) and the quality of global approximations (the reduced basis method). Three approximate sensitivity analysis methods are developed and compared: the direct method, the adjoint-variable method and the finite-difference method. All the approximations are based on the results of a single exact analysis of the initial design, and they can be used with a general finite element system. The methods are easy to implement and suitable for different types of design variables and structures. It is shown that high quality approximations of derivatives can be achieved with a relatively small computational effort for very large changes in the design variables.

Efficient Finite Difference Design Sensitivities

The problem considered in this study is to evaluate efficiently displacement derivatives using global finite differences. Given the displacements for an initial design, the displacements for various modified designs are evaluated by the recently developed combined approximations method. Calculations of finite difference sensitivity coefficients are demonstrated for static problems and eigenproblems. The presented solution procedure is easy to implement, efficient, and can be used to calculate derivatives for various designs where the exact displacements are not known. Some numerical examples show that the accuracy of the results is similar to the accuracy obtained by finite difference calculations based on exact analysis.

OPTIMAL TOPOLOGIES OF FLEXURAL SYSTEMS

Abstract The main objects of this paper are: a)To introduce some relationships between optimal topologies and the geometric parameters of flexural systems.b) To investigate the effect of compatibility conditions on the optimal topology. Assuming the force method analysis, a linear programming formulation can be obtained under certain circumstances. In such cases where the active constraints can be determined a priori, a direct solution in the space of redundant forces might be possible. It is shown that the optimal design might correspond to a singular point in the design space. Neglecting compatibility conditions, multiple optimal topologies might be obtained for certain geometries. In such cases some of the resulting solutions usually represent statically determinate structures, therefore compatibility conditions do not affect the optimum Numerical examples illustrate these phenomena and how the optimal topology and its corresponding load path change with the geometric parameters.

Optimum design of prestressed beams

Abstract A method is developed for the optimum design of prestressed indeterminate beams with uniform cross section. Optimum values of prestressing force, tendon configuration and cross sectional dimensions are determined subject to constraints on design variables and stresses. A transformation of variables is employed to reduce the optimization to a solution of a linear programming problem. The method is suitable for practical design of structures with no restriction on the number and types of loading conditions, tendon configuration or number of prestressing stages. Applications of the proposed method are illustrated by examples.

Synthesis of structural geometry using approximation concepts

Abstract Some approximation concepts for efficient synthesis of structural geometry are presented. Using the force method of analysis and neglecting temporarily the implicit compatibility conditions, an approximate explicit problem (AEP) is introduced. Solving the AEP, a lower bound of the optimum is efficiently obtained. To evaluate the true optimum of the implicit problem, the compatibility conditions are considered for the final geometry of the AEP. Choosing the geometric variables as the independent ones, multilevel solution procedures are proposed. To improve the solution efficiency, the number of independent variables is reduced by geometric variable linking. Also, the number of trial geometries is reduced by introducing a coarse grid in the independent variables space. Several approximation concepts are proposed for efficient solution of the explicit fixed geometry problem. Linear programming models and approximate treatment of the displacement constraints are presented. The proposed solution procedures do not involve multiple implicit analyses of the structure. Numerical examples show that in a variety of structures, where the optimal geometry is not appreciably affected by the compatibility conditions, a single exact analysis is sufficient to evaluate the final optimum. The efficiency of the solution process and the quality of the approximations used are demonstrated.

Approximate structural reanalysis based on series expansion

Abstract In most optimal design procedures the analysis of the structure must be repeated many times. This operation, which involves much computational effort, is one of the main difficulties in applying optimization methods to large systems. This study deals with approximate reanalysis methods based on series expansion. Both design variables and inverse variables formulations are presented. It is shown that a Taylor series expansion of the nodal displacements or the redundant forces is equivalent to a series obtained from a simple iteration procedure. The series coefficients can readily be computed, providing efficient and high-degree polynomial approximations. To further improve the quality of the approximations, a modified nonpolynomial series is proposed. To reduce the amount of calculations, the possibility of reanalysis along a given line in the variables space is demonstrated. All the proposed procedures require a single exact analysis to obtain an explicit behaviour model along a line. The relationship between the various methods is discussed and numerical examples demonstrate applications. The results obtained are encouraging and indicate that the proposed methods provide efficient and high quality approximations for the structural behavior. This may lead to a wider use of optimization methods in the design of large structural systems.

An improved reanalysis method for grillage-type structures

Abstract An efficient reanalysis method for grillage structures is presented. The method is suitable for structural modifications in general, and topological changes in particular. Combining the computed terms of the binomial series expansion, used as basis vectors, and coefficients of a reduced basis expression, an effective solution procedure is achieved. The method uses the stiffness analysis formulation and it can be integrated into general finite element programs. The calculations are based on results of a single exact analysis, and each reanalysis involves the solution of a small system of equations. Thus, the computational effort is significantly reduced. High quality approximations have been achieved for large changes in the structure, including elimination of elements. Applications for design modifications and damage analysis are illustrated.

Implementation of combined approximations in structural optimization

Abstract Some approximation concepts, intended to reduce the computational effort during optimization of structural systems, are presented. High quality approximations of the response functions are introduced and used to evaluate both the constraint values and constraint derivatives. The various approximations are then integrated into an effective procedure for structural optimization. The solution is carried out by selecting a sequence of direction vectors in the design space. For each selected direction, optimization is carried out in a corresponding two-dimensional design plane, with only a single independent variable. As a result, the number of directions needed to reach the optimum, and the overall computational effort involved in the solution process are significantly reduced. Assuming second-order approximations in some typical examples, it has been found that only two to three exact analyses are needed to achieve the optimum. Moreover, for higher-order approximations, a single exact analysis is sufficient for the whole design process.

Interactive optimal design of truss structures

Abstract A CAD system based on combining structural optimization methods and graphical interaction is presented. The optimization methods implement the automated decisions and algorithms while the interaction provides the means to implement the designers decisions. A new interactive optimization procedure for optimal truss design is proposed. The structural topology, geometry and member sizes are treated as design variables. Results show that the system provides a powerful tool to obtain a practical optimum design.