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Optimization For Engineers By Kalyanmoy Deb - Download as PDF File .pdf), Text File .txt) VI Optimization for Engineering Design: Algorithms and Examples. kanpur optimization for engineering design by kalyanmoy deb pdf - optimization engineering design kalyanmoy hamhillfort.info free download here. introduction to design optimization - engineering - what are common aspects in optimization for engineering design phi learning pvt ltd solution ebook download. , free kalyanmoy deb optimization for engineering design phi learning pvt ltd.

Enter your mobile number or email address below and we'll send you a link to download the free Kindle App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required. To get the free app, enter your mobile phone number. Would you like to tell us about a lower price? This well-received book, now in its second edition, continues to provide a number of optimization algorithms which are commonly used in computer-aided engineering design. The book begins with simple single-variable optimization techniques, and then goes on to give unconstrained and constrained optimization techniques in a step-by-step format so that they can be coded in any user-specific computer language.

In this problem, the objective is to find the optimal cross-sections of all elements in order to achieve a minimum cost of materials and construction. Although both these problems attempt to achieve the same objective, the search space and the optimization algorithm required to solve each problem are different.

Here, we discuss the latter problem only. However, there exist certain algorithms which can be used to solve both the above problems simultaneously. We discuss more about these algorithms in Chapter 6. Consider the seven-bar truss structure shown in Figure 1. The loading is also shown in the figure. Once the connectivity of the truss is given, the cross-sectional area and the material properties of the members are the design parameters. Let us choose the cross-sectional area of members as the design variables for this problem.

There are seven design variables, each specifying the cross-section of a member AI to A 7.

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The next task is to formulate the constraints. For the given load, we can compute the axial force generated in each element Table 1. The positive force signifies tensile load and the negative force signifies compressive load acting on the member.

Thereafter, Table 1.

Ebook free for download deb design kalyanmoy engineering optimization

Algorithms and Examples Pcsco: Realizing that each of these members is connected by pin joints, we can write the Euler buckling conditions for the axial load in members AB and BD Shigley, By using Castigliano's theorem Timoshenko, , we obtain the deflection constraint: Once the constraints are formulated, the next task is to formulate the objective function.

In this problem, we are interested in minimizing the weight of the truss structure. Since we have assumed the same material for all members, the minimization of the total volume of material will yield the same optimal solution as the minimization of the total weight. Thus, we write the objective function as The fourth task is to set some lower and upper bounds for the four cross-sectional areas.

We may choose to make all four areas lie 2 between 10 and mm. Then, while moving downward, the nitrogen and hydrogen present in the feed gas undergo reaction to form ammonia in the presence of a catalyst placed in the reactor. The production of ammonia depends on the temperature of the feed gas, the temperature at the top of the reactor, the partial pressures of the reactants nitrogen and hydrogen , and the reactor length.

The optimal design problem requires to achieve of the optimal reactor length yielding maximum economic returns profits from the reactor operation corresponding to various top temperatures. First, the decrease in the feed gas temperature must be according to the heat loss to the reaction gas: Secondly, the change in the reaction gas temperature must be according to the heat gain from the feed gas and heat generated in the reaction: The parameter S2 denotes the cross-sectional area of the catalyst.

We use the following parameter values: Note that all the above three differential equations 1. In order to solve these equations, we use the following boundary conditions: The three constraints Equations 1. We choose to keep the reactor length X as the design variable. The objective of the reactor design problem is to achieve as much profit as possible in the production of ammonia.

As in the previous example, this problem also requires numerical solution of coupled differential equations.

Thus, the NLP problem is as follows: The solid lines Station B Figure 1. The problem is to determine schedules for the routes such that the transit system provides the best level of service LOS to its passengers, within the resources available.

One of the good measures of the LOS is the amount of time -passengers wait during their ou: On any transit network, passengers wait either to board the vehicle at the station of origin or they wait at a transfer station at which they transfer from one vehicle to another. For example, a passenger wishing to travel from station 18 Optimization for Engineering Design: Algorithms and Examples A to station B in the network shown in Figure 1. Thus, the optimization problem involves finding a schedule of vehicles on all routes arrival and departure times such that the total waiting time for the passengers is minimum.

The design variables in this problem are the arrival time af and departure time df for the k-th vehicle at i-th route. Thus, if in a problem, there are a total of M routes and each route has K vehicles, the total number of design variables is 2MK.

In addition, there are a few more artificial variables which we shall discuss later. The constraints in this problem appear from different service related limitations. Some of these constraints are formulated in the following: Minimum stopping time: A vehicle cannot start as soon as it stops; it has to wait at the stop for a certain period of time, or d7 - a7 2: Smin for all i and k. Maximum stopping time: A vehicle cannot stop for more than a certain period of time even if it means increasing the total transfer time on the network, or d7 - a Maximum allowable transfer time: No passenger on the transit network should have to wait more than a certain period of time T at any transfer station.

This can be enforced by checking all possible differences between departure and arrival times and limiting those values to T. Mathematically, this is difficult to achieve. We sim- plify the formulation of this constraint by introducing a new set of variables btl between the k-th vehicle of the i-th route and the l-th vehicle of the j-th route.

These variables can take either a zero or a one. A value of zero means that the transfer of passengers between those two vehicles is not feasible. A value of one means otherwise. Consider the arrival and departure times of vehicles in two different routes at a particular station, as shown in Figure 1.

J Figure 1. A passenger from the k-th vehicle in the i-th route can only transfer to a vehicle in the j-th route, which is arriving at the station after af. According to the figure, the transfer of a pas- senger from the k-th vehicle in the i-th route is not possible to the l-1 -th vehicle in the j-th route, because the departure time of the latter vehicle is earlier than af. Thus, the parameter k I takes a value zero, whereas the parameter 6 i ,j takes a value one.

In order to simplify the model, we assume that transfers to vehicles departing after l-th vehicle in the j-th route are also not possible. Thus, between any two vehicles, the following condition must be satisfied: Mathematically, I: I ',J The introduction of the artificial variables 6: Many optimization algorithms cannot handle discrete design variables efficiently.

Since the artificial design variables 6: Algorithms and Examples constraints is added to enforce the binary values: The above constraint ensures that the variable 8: This constraint is derived purely from the knowledge of the available optimization algorithms.

There may be other ways to formulate the concept of feasible transfers, but inclusion of such artjfidal design variables often make the understanding of the problem easier.

Maximum headway: The objective is to minimize the foUowing function: The parameter Wfj is the number of passengers transferring from the k-th vehicle of the i-th route to the j-th route. The parameter Vi,k t is the number of passengers arriving at the stop for the k-th vehicle in the i-th route at a given time t. For example, a passenger arriving at the stop just after the previous vehicle has left has to wait for the fuU headway time af - a7- 1 before the next vehicle arrives.

On the other hand, a passenger arriving at the stop later has to wait for a shorter time. The calculation of the second term assumes that passengers arrive at the stop during the time interval a7- 1 to af according to the known 21 Introduction time-varying function Vi,k t , where t is measured from af-l. Then the quantity gives the sum of the initial waiting times for all passengers who board the k-th vehicle of the i-th route. We then sum it over all the routes and vehicles to estimate the network total of the Thus, the complete NLP problem can be written as follows: I ',J In the above NLP problem, the variables btj are binary variables taking only a value zero or a one and other variables af and df are real-valued.

Thus, a mixed integer programming technique described in Chapter 5 or genetic algorithms described in Chapter 6 can be used to solve the above NLP problem Chakroborty, et al. The car body is usually supported by a suspension coil spring and a damper at each wheel Figure 1. In some cars, the, axle assembly is directly supported on the wheel.

Optimization Engineering Design Kalyanmoy hamhillfort.info - eBook and Manual Free download

The tyre of 22 Optimization for Engineering Design: Algorithms and Examples Figure 1. A two-dimensional dynamic model of a car suspension system is shown in Figure 1. In this model, only two wheels one each at rear and front are considered.

The sprung mass of the car is considered to be supported on two axles front and rear by means of a suspension coil spring and a shock absorber damper.

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Each axle contains some unsprung mass which is supported by the tyre. In order to formulate the optimal design problem, the first task qz Rear Front Figure 1. The above model has four degrees-of-freedom ql to Q4. Let us first identify all the design parameters that could govern the dynamic behaviour of the car vibration. In the following, we list all these parameters: We may consider all the above parameters as design variables, but the time taken for convergence of the optimization algorithm may be too much.

In order to simplify the formulation, we consider only four of the above parameters-front coil stiffness k j s, rear coil stiffness k rs ' front damper coefficient oj, and rear damper coefficient or-as design variables.

We keep the other design parameters as constant: Using these parameters, differential equations governing the vertical motion of the unsprung mass at the front axle ql , the sprung mass Q2 , and the unsprung mass at the rear axle Q4 , and the angular motion of the sprung mass Q3 are written Figure 1.

Algorithms and Examples where the forces F 1 to F 6 are calculated as follows: Figure 1. The relative deformations in springs and tyres can be written as follows: Any function can be used for h t. When a car is moving forward, the front wheel experiences. The coupled differential equations specified in Equations 1. After the design variables are chosen, the next task is to formulate the constraints associated with the above car suspension problem.

In order to simplify the problem, we consider only one constraint. The jerk the rate of change of the vertical acceleration of the sprung mass is a major factor concerning the comfort of the riding passengers. The guideline used in car industries suggests that the maximum jerk experienced by the passengers should not be more than about 18 mls 3.

Mathematically, max qz" t:: Certain problems involve linear terms for constraints and objective function but certain other problems involve nonlinear terms for them.

In some problems, the terms are not explicit functions of the design variables. Unfortunately, there does not exist a single optimization algorithm which will work in all optimization problems equally efficiently. Some algorithms perform better on one problem, but may perform poorly on other problems. That is why the optimization literature contains a large number of algorithms, each suitable to solve a particular type of problem.

The choice of a suitable algorithm for an optimization problem is, to a large extent, dependent on the user's experience in solving similar problems.

This book provides a number of optimization algorithms used in engineering design activities. Since the optimization algorithms involve repetitive application of certain procedures, they need to be used with the help of a computer. That is why the algorithms are presented in a step-by- step format so that they can be easily coded.

To demonstrate the ease of conversion of the given algorithms into computer codes, most chapters contain a representative working computer code. Further, in order to give a clear understanding of the working of the algorithms, they are hand-simulated on numerical exercise problems. Simulations are performed for two to three iterations following the steps outlined in the algorithm sequentially. Thus, for example, when the algorithm suggests to move from Step 5 to Step 2 in order to carry out a conditional statement, the exercise problem demonstrates this by performing Step 2 after Step 5.

For the sake of clarity, the optimization algorithms are classified into a number of groups, which are now briefly discussed. Single-variable optimization algorithms. Because of their sim- plicity, single-variable optimization techniques are discussed first. These algorithms provide a good understanding of the properties of the minimum and maximum points in a function and how op- timization algorithms work iteratively to find the optimum point in a problem. The algorithms are classified into two categories- direct methods and gradient- based methods.

Direct methods do not use any derivative information of the objective function; only ob- jective function values are used to guide the search process. Although engineering op- 27 Introduction timization problems usually contain more than one design variable, single-variable optimization algorithms are mainly used as unidirec- tional search methods in multivariable optimization algorithms.

Multi-variable optimization algorithms. A number of algorithms for unconstrained, multivariable optimization problems are discussed next. These algorithms demonstrate how the search for the optimum point progresses in multiple dimensions.

Depending on whether the gradient information is used or not used, these algorithms are also classified into direct and gradient-based techniques. Constrained optimization algorithms. Constrained optimization algorithms are described next. These algorithms use the single- variable and multivariable optimization algorithms repeatedly and simultaneously maintain the search effort inside the feasible search region.

Since these algorithms are mostly used in engineering opti- mization problems, the discussion of these algorithms covers most of the material of this book. Specialized optimization algorithms. There exist a number of structured algorithms, which are ideal for only a certain class of optimization problems.

Two of these algorithms-integer programming and geometric programming-are often used in engineering design problems and are discussed.

Integer programming methods can solve optimization problems with integer design variables. Geometric programming methods solve optimization problems with objective functions and constraints written in a special form. There exist quite a few variations of each of the above algorithms. These algorithms are being used in engineering design problems since sixties.

Because of their existence and use for quite some years, we call these algorithms as traditional optimization algorithms. Nontraditional optimization algorithms. There exist a number of other search and optimization algorithms which are comparatively new and are becoming popular in engineering design optimization problems in the recent past.

Two such algorithms-genetic algorithms and simulated annealing-are discussed in this book. We have put together about 34 different optimization algorithms. Over the years, researchers and practitioners have modified these algorithms to suit their problems and to increase the efficiency of the algorithms. However, there exist a few other optimization algorithms-stochastic programming methods and dynamic programming method-which are very different than the above algorithms.

Because of the space limitation and occasional 28 Optimization for Engineering Design: Algorithms and Examples use of these algorithms in engineering design problems, we have not.

A detailed discussion of these algorithms can be found elsewhere Rao, Many engineering optimization problems contain multiple optimum solutions, among which one or more may be the absolute minimum or maximum solutions. These absolute optimum solutions are known as global optimal solutions and other optimum solutions are known as local optimum solutions. Ideally, we are interested in the global optimal solutions because they correspond to the absolute optimum objective function value. Unfortunately, none of the traditional algorithms are guaranteed to find the global optimal solution, but genetic algorithms and simulated annealing algorithm are found to have a better global perspective than the traditional methods.

The global optimality issues are discussed in Chapter 6. Moreover, when an optimal design problem contains multiple global solutions, designers are not only interested in finding just one global optimum solution, but as many as possible for various reasons.

Firstly, a design suitable in one situation may not be valid in another situation. Secondly, it is also not possible to include all aspects of the design in the optimization problem formulation.

Thus, there always remains some uncertainty about the obtained optimal solution. Thirdly, designers may not be interested in finding the absolute global solution, instead may be interested in a solution which corresponds to a marginally inferior objective function value but is more amenable to fabrication. Thus, it is always prudent to know about other equally good solutions for later use. However, if the traditional methods are used to find multiple optimal solutions, they need to be applied a number of times, each time starting from a different initial solution and hoping to achieve a different optimal solution each time.

Genetic algorithms described in Chapter 6 allow an easier way to find multiple optimal solutions simultaneously in a single simulation. Another class of optimization problems deals with simultaneous optimization of multiple objective functions. In formulating an optimal design problem, designers are often faced with a number of objective functions.

For example, the truss structure problem described earlier should really be reformulated as the minimization of both the weight of the truss and the deflection at the point C. Multiobjective optimization problems give rise to a set of optimal solutions known as Pareto-optimal solutions Chankong and Haimes, , all of which are equally important as far as all objectives are concerned.

Thus, the aim in these problems is to find as many Pareto-optimal solutions as possible. Because of the complexity 29 Introduction involved in the multiobjective optimization algorithms, designers usually choose to consider only one objective and formulate other objectives as constraints.

Genetic algorithms described in Chapter 6 demonstrate one way to handle multiple objectives and help find multiple Pareto-optimal solutions simultaneously. At the end of the optimization process, one obvious question may arise: Is the obtained solution a true optimum solution? Unfortunately, there is no easy answer to this question for all optimization problems.

In problems where the objective functions and constraints can be written in simple, explicit mathematical forms, the Kuhn-Tucker conditions described in Chapter 4 may be used to check the optimality of the obtained solution. However, those conditions are valid only for a few classes of optimization problems.

In a generic problem, this question is answered in a more practical way. In many engineering design problems, a good solution is usually known either from the previous studies or from experience. After formulating the optimal problem and applying the optimization algorithm if a better solution is obtained, the new solution becomes the current best solution. The optimality of the obtained solution is usually confirmed by applying the optimization algorithms a number of times from different initial solutions.

The formulation process begins with identifying the important design variables that can be changed in a design. The other design parameters are usually kept fixed. Thereafter, constraints associated with the design are formulated. The constraints may arise due to resource limitations such as deflection limitations, strength limitations, frequency limitations, and others. Constraints may also arise due to codal restrictions that govern the design. The next task is to formulate the objective function which the designer is interested in minimizing or maximizing.

The final task of the formulation phase is to identify some bounding limits for the design variables. The formulation of an optimization problem can be more difficult than solving the optimization problem. Unfortunately, every optimization problem requires different considerations for formulating objectives, constraints, and variable bounds. Thus, it is not possible to describe all considerations in a single book.

However, many of these considerations require some knowledge about the 30 Optimiiation for Engineering Design: Algorithms and Examples problem, which is usually available with the experienced designers due to their involvement with similar other design problems. The rest of the book assumes that the formulation of an optimization problem is available. Chapters 2 to 6 describe a number of different optimization algorithms-traditional and nontraditional-in step-by-step format.

To demonstrate the working of each algorithm, hand-simulations on a numerical example problem are illustrated. Sample computer codes for a number of optimization algorithms are also appended to demonstrate the ease of conversion of other algorithms into similar computer codes.

Optimal scheduling of urban transit systems using genetic algorithms. Chankong, V. Multiobjective Decision Making Theory and Methodology. New York: Murase, A. Optimal thermal design of an autothermal ammonia synthesis reactor.

Process Des. Rao, S. Optimization Theory and Applications. New Delhi: Wiley Eastern. Shigley, J. Mechanical Engineering Design. McGraw- Hill. Timoshenko, S. Strength of Materials, Part 1: Elementary Theory and Problems. CBS Publishers. Upreti, S. Optimal design of an ammonia synthesis reactor using genetic algorithms. Flag for inappropriate content.

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