20-06-2012, 12:20 PM
OPTIMIZATION
[attachment=24884]
INTRODUCTION
Optimization is a critical and challenging activity in structural design. Designers are able to produce better designs while saving time and money through optimization. The development and use of optimization models is well established. However, the use of many models has been restricted in some fields of economic analysis where the problem is large in size and there are a large number of non-linear interactions. In most cases, the use of linear approximations or simplification of the model has been necessary in order to derive a solution. Genetic algorithm (GA) is an evolutionary optimization approach which is an alternative to traditional optimization methods. GA’s are most appropriate for complex non-linear models where location of the global optimum is a difficult task. It may be possible to use GA techniques to consider problems which may not be modeled as accurately using other approaches. Therefore, GA appears to be a potentially useful approach.GA follows the concept of solution evolution by stochastically developing generations of solution populations using a given fitness statistic. They are particularly applicable to problems which are large, non-linear and possibly discrete in nature, features that traditionally add to the degree of complexity of solution. Due to the probabilistic development of the solution, GA does not guarantee optimality even when it may be reached. However, they are likely to be close to the global optimum. This probabilistic nature of the solution is also the reason they are not contained by local optima.
THE GOALS OF OPTIMIZATION
Before examining the mechanics and power of a simple genetic algorithm one must be clear about the goals to optimize a function or a process. Men’s longing for perfection finds expression in the theory of optimization. It studies how to describe and attain what is best. Once one know how to measure and alter what is good or bad. Optimization theory encompasses the quantitative study of optima and methods for finding them.
Thus optimization seeks to improve performance towards some optimal points. That the definition has two parts:
(1) Improvement to approach some.
(2) Optimal point.
There is a clear distinction between the process of improvement and destination or optimum itself. Yet in judging optimization procedures commonly focus solely upon convergence and forget entirely about interim performance. This emphasis stems from the origins of optimization in the calculus. It is however a natural emphasis. Attainment of the optimum is much less important for complex systems. It would be nice to be perfect meanwhile. We can only strive to improve.
OPTIMIZATION OF STRUCTURAL DESIGN
Design is one of the primary functions in engineering. In design the engineer creates a method, device, process or more broadly, objective being to satisfy a performance requirement while minimizing those factors which reduce the efficiency of the system.
Finite, often large, members of variables of the discrete type generally characterize the process of structural design. Universal steel beams available to designer are discrete in dimensions and properties even the thickness of a concrete slab and cross-section of transmission line are discrete variable in that practical dimensions will vary by discrete intervals. Nothing is lost in generation if it is assumed that all design variables can be described in discrete intervals. If we meet genuinely continuous variables then these can always be discredited into intervals. If we meet genuinely continuous variables then these can always be discredited to our assumption.
An optimum design is one which minimizes the objective function. One would usually base an objective function on cost but the best structure is not necessarily the cheapest and may sometimes be important and may be used conveniently in studies in optimum design.
The available methods of optimization may be divided into two different categories:
1 Analytical Method
2. Numerical Method
Analytical Method
These methods usually employ the mathematical theory of calculus, variation methods, etc. in studies of optimal layouts or geometrical forms of simple structural elements, such as beams, columns, plates and trusses. Analytical methods are most suited for such fundamental studies of single structural components, but are not able to handle larger structural systems. In analytical optimization problems the structural design is represented by a number of unknown functions and the goal is to find the form of these functions. The optimal design is theoretically found exactly through the solution of system of equations expressing the condition for optimality. Analytical solutions, when they can be found, provide valuable insight and facilitate the comparison between forms. An example for this approach is the theory of layout, which seeks the arrangement of uniaxial structural members that, produces a minimum volume structure for specified loads and materials.
Numerical Method
They usually employ a branch in the field of numerical mathematics called programming methods. The recent developments in this field are closely related to the rapid growth in computing capacities affected by the development of computers. In the numerical methods a near optimal design is automatically generated in an iterative manner. The search is terminated when certain criteria is satisfied. Problems solved by numerical methods are called finite optimization problems. This is due to the fact that they can be formulated by a finite number of variables.
TRANSMISSION LINE TOWERS
INTRODUCTION
A power transmission tower is a structure that plays an important role in bulk energy transfer systems. The basic role of this structure is to safely and effectively accommodate transmission lines. Such a structure, which is generally made of a metal such as galvanized steel, may also be referred to as a pylon. Power can be transported overhead or underground. Overhead transport is generally considered the better option because maintenance and repair is easier and the costs are lower. Overhead transport can usually be identified when electrical wires are seen running between tall metal structures.
Transmission towers are generally rectangular in cross-sectional plan. The number of circuits and clearances for the transmission line dictates the tower configuration, and the clearance required for ground and other obstructions. Various tower configurations can be developed and these configurations should satisfy all the electrical and code constraints. Transmission towers are designed to support one or two circuits, although some have been designed to support three or four. Towers are classified as single circuit, double circuit or multiple circuits. Each circuit consists of three phases. In all areas, except those that have no incidence of lighting, ground wires must shield the lines.
In the design of transmission towers, three items should be considered: cost of material, cost of erection and cost of foundation. The cost of material directly related to the number of splices and bolts to be installed. The cost of foundation directly related to the spread of the tower legs and soil conditions.
To arrive at the configuration of tower, vertical and horizontal spacing of the conductors need to be determined, this is influenced by type of conductors and earth wire chosen, temperature variations, and wind climate.
TOWER OPTIMIZATION
The optimal design of structural system can be classified as sizing optimal design, shape, optimal design, or topology optimal design. The nature of the design variable determines the type of the optimal design problem.
Sizing Optimal Design
In sizing optimal design problems, the design variables are sizing parameters associated with the finite element model such as the cross sectional areas and are assumed to be available as discrete values. In this problem the objective is to find optimal cross sections of all elements in order to achieve a minimum cost of materials and construction.
Shape Optimal Design
When the shape of the structure is allowed to vary, usually an improved design can be obtained. In shape optimal design problems, the shape of the structure is allowed to vary either through the use of parameters that describe the shape of the boundary, such as nodal co-ordinates of key nodes taken as design variables or indirectly by choosing fictitious loads or displacement that act on an auxiliary structure as design variables. The choice of the design variables dictates how the internal nodes move when the shape of the boundary changes. A small of design variables can bring about large changes in the shapes.
Topology Optimal Design
The problem of finding optimal layout or topology of a structure is another important optimization problem. The topology means not only how nodes are connected to each other, but also how many nodes are to be placed, and also how they are to be supported. The choice of member connectivity and support conditions as design variables leads to a non-convex and discontinuous design space.
The cross-sectional areas called sizing variables in truss i.e. structural optimization problems are usually assumed to be continuous. However, in most practical design tasks, the sizing variables have to be chosen from a list of discrete values, which are commercially available in prefabricated sizes. In this thesis the topology and configuration of trusses are held fixed and only cross-sectional areas are to be optimized. This leads to a discrete optimization problem, which is combinatorial in nature. The fact that the sizing variables are discrete makes the problem somewhat complex and difficult.
DIMENSIONAL OPTIMIZATION
In dimensional optimization the domain of the problem is maintained unaltered in other words there is no alteration of the finite element mesh. In this case, the design variable describes structural characteristic as for example, the cross section area, moments of inertia, etc. In using dimensional optimization, the goal is to obtain the best of the area by achieving the minimum or maximum of the objective function (weight, flexibility, stress, etc.) and concomitant attending the respective constraints of the problem. The constraints depend on the type of problem, but basically refer to an allowable limit for displacements, stresses, frequencies, and buckling, etc.
Codification
One of the advantages of genetic algorithms is found in its search structure that is based on genotypic codification. As such, representation of the individuals is crucial in the elaboration of the genetic algorithm, since it is on this that all the efficiency of the algorithm is based. The genetic operators, responsible for the evolution strategy, are applied to the codified individuals (genotypic space). Various representations for the genotypes exist, going from a simple chain of 0s and 1s (binary alphabet) to complex data structures.
MODELING
Each tower member has distinct structural behavior. For example, the legs of tower function as the beam elements, the bracing as truss elements, etc. However, the towers are generally modeled with spatial truss elements. Gabriella points out that when modeling with truss elements, the secondary bracing should not be considered, as they generate hypostatic mechanisms leading to a singular equations system. The presence of these mechanisms does not necessarily indicate an actual unstable structure, but indicates the need for their automatic identification into the computational analysis software, preventing in this way the stop of analysis.
[attachment=24884]
INTRODUCTION
Optimization is a critical and challenging activity in structural design. Designers are able to produce better designs while saving time and money through optimization. The development and use of optimization models is well established. However, the use of many models has been restricted in some fields of economic analysis where the problem is large in size and there are a large number of non-linear interactions. In most cases, the use of linear approximations or simplification of the model has been necessary in order to derive a solution. Genetic algorithm (GA) is an evolutionary optimization approach which is an alternative to traditional optimization methods. GA’s are most appropriate for complex non-linear models where location of the global optimum is a difficult task. It may be possible to use GA techniques to consider problems which may not be modeled as accurately using other approaches. Therefore, GA appears to be a potentially useful approach.GA follows the concept of solution evolution by stochastically developing generations of solution populations using a given fitness statistic. They are particularly applicable to problems which are large, non-linear and possibly discrete in nature, features that traditionally add to the degree of complexity of solution. Due to the probabilistic development of the solution, GA does not guarantee optimality even when it may be reached. However, they are likely to be close to the global optimum. This probabilistic nature of the solution is also the reason they are not contained by local optima.
THE GOALS OF OPTIMIZATION
Before examining the mechanics and power of a simple genetic algorithm one must be clear about the goals to optimize a function or a process. Men’s longing for perfection finds expression in the theory of optimization. It studies how to describe and attain what is best. Once one know how to measure and alter what is good or bad. Optimization theory encompasses the quantitative study of optima and methods for finding them.
Thus optimization seeks to improve performance towards some optimal points. That the definition has two parts:
(1) Improvement to approach some.
(2) Optimal point.
There is a clear distinction between the process of improvement and destination or optimum itself. Yet in judging optimization procedures commonly focus solely upon convergence and forget entirely about interim performance. This emphasis stems from the origins of optimization in the calculus. It is however a natural emphasis. Attainment of the optimum is much less important for complex systems. It would be nice to be perfect meanwhile. We can only strive to improve.
OPTIMIZATION OF STRUCTURAL DESIGN
Design is one of the primary functions in engineering. In design the engineer creates a method, device, process or more broadly, objective being to satisfy a performance requirement while minimizing those factors which reduce the efficiency of the system.
Finite, often large, members of variables of the discrete type generally characterize the process of structural design. Universal steel beams available to designer are discrete in dimensions and properties even the thickness of a concrete slab and cross-section of transmission line are discrete variable in that practical dimensions will vary by discrete intervals. Nothing is lost in generation if it is assumed that all design variables can be described in discrete intervals. If we meet genuinely continuous variables then these can always be discredited into intervals. If we meet genuinely continuous variables then these can always be discredited to our assumption.
An optimum design is one which minimizes the objective function. One would usually base an objective function on cost but the best structure is not necessarily the cheapest and may sometimes be important and may be used conveniently in studies in optimum design.
The available methods of optimization may be divided into two different categories:
1 Analytical Method
2. Numerical Method
Analytical Method
These methods usually employ the mathematical theory of calculus, variation methods, etc. in studies of optimal layouts or geometrical forms of simple structural elements, such as beams, columns, plates and trusses. Analytical methods are most suited for such fundamental studies of single structural components, but are not able to handle larger structural systems. In analytical optimization problems the structural design is represented by a number of unknown functions and the goal is to find the form of these functions. The optimal design is theoretically found exactly through the solution of system of equations expressing the condition for optimality. Analytical solutions, when they can be found, provide valuable insight and facilitate the comparison between forms. An example for this approach is the theory of layout, which seeks the arrangement of uniaxial structural members that, produces a minimum volume structure for specified loads and materials.
Numerical Method
They usually employ a branch in the field of numerical mathematics called programming methods. The recent developments in this field are closely related to the rapid growth in computing capacities affected by the development of computers. In the numerical methods a near optimal design is automatically generated in an iterative manner. The search is terminated when certain criteria is satisfied. Problems solved by numerical methods are called finite optimization problems. This is due to the fact that they can be formulated by a finite number of variables.
TRANSMISSION LINE TOWERS
INTRODUCTION
A power transmission tower is a structure that plays an important role in bulk energy transfer systems. The basic role of this structure is to safely and effectively accommodate transmission lines. Such a structure, which is generally made of a metal such as galvanized steel, may also be referred to as a pylon. Power can be transported overhead or underground. Overhead transport is generally considered the better option because maintenance and repair is easier and the costs are lower. Overhead transport can usually be identified when electrical wires are seen running between tall metal structures.
Transmission towers are generally rectangular in cross-sectional plan. The number of circuits and clearances for the transmission line dictates the tower configuration, and the clearance required for ground and other obstructions. Various tower configurations can be developed and these configurations should satisfy all the electrical and code constraints. Transmission towers are designed to support one or two circuits, although some have been designed to support three or four. Towers are classified as single circuit, double circuit or multiple circuits. Each circuit consists of three phases. In all areas, except those that have no incidence of lighting, ground wires must shield the lines.
In the design of transmission towers, three items should be considered: cost of material, cost of erection and cost of foundation. The cost of material directly related to the number of splices and bolts to be installed. The cost of foundation directly related to the spread of the tower legs and soil conditions.
To arrive at the configuration of tower, vertical and horizontal spacing of the conductors need to be determined, this is influenced by type of conductors and earth wire chosen, temperature variations, and wind climate.
TOWER OPTIMIZATION
The optimal design of structural system can be classified as sizing optimal design, shape, optimal design, or topology optimal design. The nature of the design variable determines the type of the optimal design problem.
Sizing Optimal Design
In sizing optimal design problems, the design variables are sizing parameters associated with the finite element model such as the cross sectional areas and are assumed to be available as discrete values. In this problem the objective is to find optimal cross sections of all elements in order to achieve a minimum cost of materials and construction.
Shape Optimal Design
When the shape of the structure is allowed to vary, usually an improved design can be obtained. In shape optimal design problems, the shape of the structure is allowed to vary either through the use of parameters that describe the shape of the boundary, such as nodal co-ordinates of key nodes taken as design variables or indirectly by choosing fictitious loads or displacement that act on an auxiliary structure as design variables. The choice of the design variables dictates how the internal nodes move when the shape of the boundary changes. A small of design variables can bring about large changes in the shapes.
Topology Optimal Design
The problem of finding optimal layout or topology of a structure is another important optimization problem. The topology means not only how nodes are connected to each other, but also how many nodes are to be placed, and also how they are to be supported. The choice of member connectivity and support conditions as design variables leads to a non-convex and discontinuous design space.
The cross-sectional areas called sizing variables in truss i.e. structural optimization problems are usually assumed to be continuous. However, in most practical design tasks, the sizing variables have to be chosen from a list of discrete values, which are commercially available in prefabricated sizes. In this thesis the topology and configuration of trusses are held fixed and only cross-sectional areas are to be optimized. This leads to a discrete optimization problem, which is combinatorial in nature. The fact that the sizing variables are discrete makes the problem somewhat complex and difficult.
DIMENSIONAL OPTIMIZATION
In dimensional optimization the domain of the problem is maintained unaltered in other words there is no alteration of the finite element mesh. In this case, the design variable describes structural characteristic as for example, the cross section area, moments of inertia, etc. In using dimensional optimization, the goal is to obtain the best of the area by achieving the minimum or maximum of the objective function (weight, flexibility, stress, etc.) and concomitant attending the respective constraints of the problem. The constraints depend on the type of problem, but basically refer to an allowable limit for displacements, stresses, frequencies, and buckling, etc.
Codification
One of the advantages of genetic algorithms is found in its search structure that is based on genotypic codification. As such, representation of the individuals is crucial in the elaboration of the genetic algorithm, since it is on this that all the efficiency of the algorithm is based. The genetic operators, responsible for the evolution strategy, are applied to the codified individuals (genotypic space). Various representations for the genotypes exist, going from a simple chain of 0s and 1s (binary alphabet) to complex data structures.
MODELING
Each tower member has distinct structural behavior. For example, the legs of tower function as the beam elements, the bracing as truss elements, etc. However, the towers are generally modeled with spatial truss elements. Gabriella points out that when modeling with truss elements, the secondary bracing should not be considered, as they generate hypostatic mechanisms leading to a singular equations system. The presence of these mechanisms does not necessarily indicate an actual unstable structure, but indicates the need for their automatic identification into the computational analysis software, preventing in this way the stop of analysis.