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Inhaltsverzeichnis

1. Introduction

Abstract
The early work in structural optimization was based on the the optimality criteria and in some instances on the calculus of variations. In recent decades, considerable research has been directed toward the use of mathematical programming methods in the design of structures. This research has dealt with the problem of member sizing for a given structural geometry and topology as well as the problem of overall shape optimization (geometry) for a given topology.
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2. Geometric Programming with Equality Constraints

Overview
As outlined in chapter 1, integrated optimum structural design represents an optimization problem with both equality constraints and inequality constraints. In this method the analysis and optimal design of the structure are carried out simultaneously. The disadvantage of this approach, however, is that the formulation results in a large size problem. Generalized geometric programming is a powerful mathematical tool that can be used to solve the multidimensional optimization problems with nonlinear equations, Burns (1985). In this chapter we will present the generalized geometric programming (GGP) and its extensions to the solution of problems with equality constraints (GGPE). We will also describe the IGGP solver, Burns (1982), and the added extensions for the optimal solution.
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3. Decomposition and Reduction Techniques for Large Scale Structural Optimization

Overview
In Chapter 1 the integrated approach for analysis and optimal design of a structure was presented. It was shown that in this approach the problem is formulated in such a way that no distinction is made between analysis and design operations. The constraints include the equilibrium and compatibility conditions for each load case, which are equations, as well as the design and behavior requirements, which are inequalities. In the member sizing of structural systems, the variables consist of the section properties of the members of the structure and the behavior variables that are stresses and displacements for each load case. The objective function is the volume of the structure.
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4. Parallel Decomposition for Alternative Loads in Structural Optimization

Overview
Structural optimization based on the integrated optimum structural design approach often gives rise to large optimization problems. This is because the optimization problem must include as design variables all the design shape variables (i.e the areas or thicknesses of the various elements of the members of the structure) as well as the response variables (i.e the nodal displacements and stresses). The difficulty of the solution of large optimization problem is increased when the structure is subject to more than one load condition, which is the case in most engineering design problems. Here, the number of response variables, of equality constraints of analysis, and of behavior constraints are multiplied by the number of load conditions considered. The computer memory may become insufficient in such problems. In order to alleviate this difficulty and in order to understand the behavior of a structure during the design for alternative loads, a decomposition technique based on the load conditions is presented. In this chapter we develop a new parallel decomposition method based on each load condition when the goal is optimization of structures subject to alternative loads.
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5. Application to Truss Type Structures

Abstract
In the preceding chapter we developed a parallel computation technique for alternative loads (move coordination) in the integrated optimal design of structures. The technique was illustrated by the three bar truss example. The optimization problem was briefly outlined without a detailed description of the formulation and generation of the constraints. In this section we will apply the move coordination method to the solution of two truss structures. We will also describe the different equality and inequality constraints as well as variables and their respective generation from nodal and structural element information for the move coordination algorithm as well as the global solution.
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6. Summary, Conclusions and Plans for Future Work

Summary
This work presents an investigation of a new parallel decomposition method specifically intended for alternative systems of loads in an integrated optimum structural design formulation. The method developed, called the move coordination, consists of casting the large optimization problem into the solution of smaller subproblems based on the load conditions. This is accomplished by allowing the structural design variables to differ from one load condition to another. The objective function of the global optimization problem for all load cases is rewritten in separately additive terms involving the variables related to one load condition only. These terms are considered as objective functions of the different subproblems related to the load conditions. Similarly, the constraints (inequality as well as equality constraints) are partitioned into different groups, where each group is related to a separate load condition. This leads to the partition of the global optimization problem into the solution of smaller subproblems, and where the structural optimization is performed for each load condition in one subproblem.
The coupling among the load conditions is accomplished by the introduction of additional constraints in the solution of each subproblem. These constraints, called the coordinating constraints, require that the structural design variables obtained from the solution of one load condition must be greater than those obtained from other load cases. The coordinating constraints have the effect of penalty-relaxation on the subproblems, and they ensure that the final design is the same for all subproblems at the end of the optimization process. The solution is obtained in a cyclic way, where all subproblems are solved at the same time in a parallel way. The coordinating constraints are updated dynamically in each cycle. In contrast to other multilevel decompositions techniques where the optimization problems are solved in two levels, in this method no optimization problem is solved in the second level.
Two algorithms are proposed to get the numerical solution by the move coordination method. The second algorithm (algorithm b) is an improvement of the first algorithm (algorithm a) to accelerate convergence. This is accomplished by restricting the move in the coordinating constraints.
The method developed is applied to the solution of some examples of truss type structures. The optimization solution uses geometric programming with equality constraints. The generation of the variables, equality and inequality constraints is carried out automatically from the structural layout information through some computer routines that we developed. The method developed presents an advantage of reducing the size of the optimization problem. The algorithm is also very suitable for implementation on computers using parallel processing.
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