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Stiffness Optimization of Distributed Parameter Systems
In previous chapters we have been concerned with the solution of discrete structural optimization problems: either the structures have been naturally discrete, like trusses, or we have made them discrete by a finite element discretization. In this chapter, on the other hand, we will look at some techniques of mathematics, from an area usually referred to as calculus of variations, that can handle some continuous optimization problems such as those of distributed parameter systems, without the need for a discrete approximation. Basic facts from this area will be applied to two types of optimization problems. Firstly, we will discuss linear elastic systems without introducing any design variables. It will be shown that the state variables of such systems are minimizers of the potential energy of the systems. Next, we look at design problems of a particular type: the design variable enters linearly in the potential energy and we seek to make the structure as stiff as possible in the sense previously considered in Chap. 5. It is shown that optimal structures of this type have the property that a particular specific strain energy is constant throughout the structure, which is to be compared to the fully stressed designs of Sect. 5.2.2. We treat mainly simple problems of beams and bars, but the general structure of this stiffness optimization problem will be used in the next chapter that treats topology optimization problems.
Stiffness Optimization of Distributed Parameter Systems
In previous chapters we have been concerned with the solution of discrete structural optimization problems: either the structures have been naturally discrete, like trusses, or we have made them discrete by a finite element discretization. In this chapter, on the other hand, we will look at some techniques of mathematics, from an area usually referred to as calculus of variations, that can handle some continuous optimization problems such as those of distributed parameter systems, without the need for a discrete approximation. Basic facts from this area will be applied to two types of optimization problems. Firstly, we will discuss linear elastic systems without introducing any design variables. It will be shown that the state variables of such systems are minimizers of the potential energy of the systems. Next, we look at design problems of a particular type: the design variable enters linearly in the potential energy and we seek to make the structure as stiff as possible in the sense previously considered in Chap. 5. It is shown that optimal structures of this type have the property that a particular specific strain energy is constant throughout the structure, which is to be compared to the fully stressed designs of Sect. 5.2.2. We treat mainly simple problems of beams and bars, but the general structure of this stiffness optimization problem will be used in the next chapter that treats topology optimization problems.
Stiffness Optimization of Distributed Parameter Systems
Christensen, Peter W. (author) / Klarbring, Anders (author)
2009-01-01
31 pages
Article/Chapter (Book)
Electronic Resource
English
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