Solution Using the Graphical Method for Continuous Variables Matlab

Optimum Design with MATLAB®

Jasbir S. Arora , in Introduction to Optimum Design (Third Edition), 2012

Solution

The exact solution for the problem is given as

(d) ${\frac{{\sigma }_{xz}}{p_{\max }}|}_{\max }=\frac{1}{2}\left(\frac{1-2v}{2}+\frac{2}{9}(1+v)\sqrt{2(1+v)}\right)\text{at}\alpha =\sqrt{\frac{2+2v}{7-2v}}$

This is a single-variable optimization problem with only lower and upper bounds on the variable. Therefore, the function fminbnd in the Optimization Toolbox can be used to solve the problem. Table 7.10 shows the script m-file that invokes the function fminbnd, and Table 7.11 shows the function m-file that evaluates the function to be minimized. The script m-file also contains commands to plot the shear stress as a function of z, which is shown in Figure 7.2. The optimum solution matches the exact solution for ν=0.3 and is given as

Table 7.10. m-file to invoke function fminbnd for spherical contact problem

% File name: sphcont_opt.m

% Design variable: ratio of the max shear stress location to

% size of the contact patch

% Find location of the maximum shear stress along the z-axis

clear all

% Set lower and upper bound for the design variable

Lb=0; Ub=5;

% Plot normalized shear stress distribution along the z-axis in spherical contact

z=[Lb: 0.1: Ub]';

n=size (z);

for i=1: n

outz(i)=−sphcont_objf(z(i));

end

plot(z, outz); grid

xlabel ('normalized depth z/a');

ylabel ('normalized shear stress');

% Invoke the single-variable unconstrained optimizer

[alpha, FunVal, ExitFlag, Output]=fminbnd ('sphcont_objf', Lb, Ub)

Table 7.11. m-file for evaluation of objective function for the spherical contact problem

% File name=sphcont_objf.m

% Location of max shear stress along z-axis for spherical contact problem

function f=sphcont_objf(alpha)

% f=−shear stress/max pressure

nu=0.3; % Poisson's ratio

f=−0.5* ((1−2* nu)/2 + (1+nu)* alpha/sqrt(1+alpha^2)− …

1.5* (alpha/sqrt(1+alpha^2))^3);

Figure 7.2. Normalized shear stress along the z-axis.

(e) $\text{alpha}=0.\text{6374}\text{,}\text{FunVal}=-0.3329[{\alpha }^{*}=0.6374\text{,}f({\alpha }^{*})=-0.3329]$

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780123813756000073

Discrete Variable Optimum Design Concepts and Methods

Jasbir S. Arora , in Introduction to Optimum Design (Third Edition), 2012

15.10.1 Continuous Variable Optimization

To compare solutions, the continuous variable optimization problem is solved first. To use the program IDESIGN, user subroutines are developed using the material of Sections 14.1 and 14.2 Section 14.1 Section 14.2 to evaluate the functions and their gradients. These subroutines contain analysis of trusses and evaluation of gradients of constraints. The optimum solution, using a very strict stopping criterion and a uniform starting design of 1.62 in², is obtained as

Design variables: 28.28, 1.62, 27.262, 13.737, 1.62, 4.0026, 13.595, 17.544, 19.13, 1.62

Optimum cost function: 5396.5   lb

Number of iterations: 19

Number of analyses: 21

Maximum constraint violation at optimum: (8.024E−10)

Convergence parameter at optimum: (2.660E−05)

Active constraints at optimum and their Lagrange multipliers:

Frequency: 392.4

Stress in member 2: 38.06

Displacement at node 2 in the y direction: 4967

Lower bound for member 2: 7.829

Lower bound for member 5: 205.1

Lower bound for member 10: 140.5

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780123813756000152

Design Optimization Applications with Implicit Functions

Jasbir S. Arora , in Introduction to Optimum Design (Second Edition), 2004

14.7 Discrete Variable Optimum Design

In many practical applications of optimization, design variables for a problem must be selected from a given set of values. For example, structural elements must be chosen from those that are already commercially available. This is called discrete variable optimization, which is essential to economize on fabrication costs for the design. The subject is briefly discussed in Section 2.11.4. We shall demonstrate the procedure described there for a simple design problem.

The application area that we have chosen is the optimum design of aerospace, automotive, mechanical, and structural systems, by employing finite element models. The problem is to design a minimum weight system with constraints on various performance specifications. As a sample application, we shall consider the 10-bar cantilever structure shown in Fig. 14-3. The loading and other design data for the problem are given in Table 14-11. The set of discrete values taken from the American Institute of Steel Construction (AISC) Manual are also given there. The final design for the structure must be selected from this set. The cross-sectional area of each member is treated as a design variable giving a total of 10 variables. Constraints are imposed on member stress (10), nodal displacement (8), member buckling (10), vibration frequency (1), and explicit bounds on the design variables (20). This gives a total of 49 constraints. In imposing the member buckling constraint, the moment of inertia is taken as I = βA ², where β is a constant and A is the member cross-sectional area. The formulation for the problem is quite similar to the one for the three-bar structure discussed in Section 2.10. The only difference is that the explicit form of the constraint function is not known. Therefore, we must use the finite element procedures described in Sections 14.1 and 14.2 for structural analysis and the gradient evaluation of constraints.

FIGURE 14-3. Ten-bar cantilever truss.

TABLE 14-11. Design Data for a 10-Bar Structure

Modulus of elasticity:	E = (1.00E+07) psi
Material weight density:	γ = (1.0E-01) lb/in3
Displacement limit:	±2.0 in
Stress limit:	25,000 psi
Frequency limit:	22 Hz
Lower limit on design variables:	1.62 in2
Upper limit on design variables:	none
Constant β(I = βA 2):	1.0
Loading Data:
Node no.	Load in y-direction (lb)
1	50,000
2	−150,000
3	50,000
4	−150,000

Available member sizes (in²): 1.62, 1.80, 1.99, 2.13, 2.38, 2.62, 2.63, 2.88, 2.93, 3.09, 3.13, 3.38, 3.47, 3.55, 3.63, 3.84, 3.87, 3.88, 4.18, 4.22, 4.49, 4.59, 4.80, 4.97, 5.12, 5.74, 7.22, 7.97, 11.50, 13.50, 13.90, 14.20, 15.50, 16.00, 16.90, 18.80, 19.90, 22.00, 22.90, 26.50, 30.00, 33.50.

14.7.1 Continuous Variable Optimization

To compare solutions, the continuous variable optimization problem is solved first. To use the program IDESIGN, USER subroutines are developed using the material of Sections 14.1 and 14.2 to evaluate the functions and their gradients. The optimum solution using a very severe convergence criteria and a uniform starting design of 1.62 in², is obtained as

Design variables: 28.28, 1.62, 27.262, 13.737, 1.62, 4.0026, 13.595, 17.544, 19.13, 1.62

Optimum cost function: 5396.5 lb

Number of iterations: 19

Number of analyses: 21

Maximum constraint violation at optimum: (8.024E-10)

Convergence parameter at optimum: (2.660E-05)

Active constraints at optimum and their Lagrange multipliers

frequency, 392.4

stress in member 2, 38.06

displacement at node 2 in the y direction, 4967

lower bound for member 2, 7.829

lower bound for member 5, 205.1

lower bound for member 10, 140.5

14.7.2 Discrete Variable Optimization

We use the adaptive numerical optimization procedure described in Section 2.11.4 to obtain a discrete variable solution. The procedure is to use the program IDESIGN in an interactive mode. Design conditions are monitored and decisions made to fix design variables that are not changing. The interactive facilities used include design variable histories, maximum constraint violation, and the cost function.

Table 14-12 contains a snapshot of the design conditions at various iterations and the decisions made. It can be seen that for the first five iterations the constraint violations are

TABLE 14-12. Interactive Solution for a 10-Member Structure with Discrete Variables

Iteration no.	Maximum violation (%)	Cost function	Algorithm used	Variables fixed to value shown in parentheses
1	1.274E+04	679.83	CC	All free
2	4.556E+03	1019.74	CC	All free
3	1.268E+03	1529.61	CC	All free
4	4.623E+02	2294.42	CC	All free
5	1.144E+02	3441.63	CC	All free
6	2.020E+01	4722.73	CC	5 (1.62), 10 (1.62)
7	2.418E+00	5389.28	CCC	2 (1.80)
11	1.223E-01	5402.62	SQP	1 (30.0), 6 (3.84), 7 (13.5)
13	5.204E-04	5411.13	SQP	3 (26.5), 9 (19.9)
14	1.388E+00	5424.69	—	4 (13.5), 8 (16.9)

CC, constraint correction algorithm; CCC, constraint correction at constant cost; SQP, sequential quadratic programming.

very large, so the constraint correction (CC) algorithm is used to correct the constraints. At the sixth iteration, it is determined that design variables 5 and 10 are not changing, so they are fixed to their current value, Similarly, at other iterations, variables are assigned values from the available set. At the 14th iteration, variables have discrete values, the constraint violation is about 1.4 percent and the structural weight is 5424.69, which is an increase of less than 1 percent from the true optimum. This is a reasonable final solution.

It should be noted that with the discrete variables, several solutions near the true optimum point are possible. A different sequence of fixing variables can give a different solution. For example, starting from the optimum solution with continuous variables, the following acceptable discrete solutions are obtained interactively:

1.

30.0, 1.62, 26.5, 13.9, 1.62, 4.18, 13.5, 18.8, 18.8, 1.62; cost = 5485.6, max. viol. = 4.167 percent for stress in member 2.

2.

Same as (1) except the eighth design variable is 16.9; cost = 5388.9 and max. viol. = 0.58 percent.

3.

Same as (1) except design variables 2 and 6 are 2.38 and 2.62; cost = 5456.8, max. viol. = 3.74 percent for stress in member 2

4.

Same as (3) except design variable 2 is 2.62; cost = 5465.4; all constraints are satisfied.

It can be seen that the interactive facilities described in Chapter 13 can be exploited to obtain practical engineering designs.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780120641550500148

Discrete Variable Optimum Design Concepts and Methods

Jasbir S. Arora , in Introduction to Optimum Design (Second Edition), 2004

15.9 Selection of a Method

Selection of a method to solve a particular mixed variable optimization problem depends on the nature of the problem functions. Features of the methods and their suitability for various types of MV-OPT problems are summarized in Table 15-4. It is seen that branch and bound, simulated annealing, and genetic algorithms (discussed in Chapter 16) are the most general methods. They can be used to solve all the problem types. However, these are also the most expensive ones in terms of computational effort. If the problem functions are differentiable and discrete variables can be assigned nondiscrete values during the iterative solution process, then there are numerous strategies for their solution that are more efficient than the three methods just discussed. Most of these involve a combination of two or more algorithms.

TABLE 15-4. Characteristics of Discrete Variable Optimization Methods

	MV-OPT problem type solved	Can find feasible discrete solution?	Can find global minimum for convex prob.?	Need gradients?
Branch and bound	1 − 5	Yes	Yes	No/Yes
Simulated annealing	1 − 5	Yes	Yes	No
Genetic algorithm	1 − 5	Yes	Yes	No
Sequential linearization	1	Yes	Yes	Yes
Dynamic round-off	1	Yes	No Guar.	Yes
Neighborhood search	1	Yes	Yes	Yes

Huang and Arora (1997a,b) have evaluated the discrete variable optimization methods presented in this chapter using 15 different types of test problems. Applications involving linked discrete variables are described in Huang and Arora (1997), Arora and Huang (1996), and Arora (2002). Applications of discrete variable optimization methods to electric transmission line structures are described in Kocer and Arora (1996, 1997, 1999, 2002). Discrete variable optimum solutions for the plate girder design problem formulated and solved in Section 10.6 are described and discussed in Arora and coworkers (1997).

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B978012064155050015X

One Variable Optimization

Mark M. Meerschaert , in Mathematical Modeling (Fourth Edition), 2013

Abstract

This chapter discusses on the optimizationmodels and a wide variety of practical applications to achieve a desired goal for production and delivery at a minimum cost. It elaborates how one-variable optimization problems can be used to determine the values of the control variables that lead to the optimal outcome, given the constraints of the problem. One-variable optimization problems which is also called as maximum-minimum problems can be used to solve problems using mathematical modeling through a sequence of procedures such as five-step method, sensitivity and its analysis, and robustness. The five-step method derives solution when all the values are constant, whereas sensitivity analysis process can be used for finding alternating assumptions. Consideration of robustness is a necessary ingredient in any mathematical modeling project. A mathematical model is said to be robust if its conclusions remain true even though the model is not completely accurate.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780123869128500014

Inductive fault current limiters: A review

Amir Heidary , ... Gevork B. Gharehpetian , in Electric Power Systems Research, 2020

3.10 Decoupled open saturated core saturated core fault current limiter (DO-SCFCL)

A single-phase dual open core FCL is presented in [58–67], Fig. 17 . The design of a saturated core FCL is a multi-variable optimization problem. This FCL needs a core with high voltage dielectric, the DC biasing coil, and the high voltage AC coil. Furthermore, an optimal design depends on the network line voltage, fault current, desirable fault current reduction, steady-state load current, and other variable network conditions [58].

Fig. 17. (a) Magnetic Structure of decoupled saturated core FCL with dual open core and (b) magnetic equivalent circuit.

The results are compared with close dual saturated core FCL shows the disadvantage of the open core arrangement is the flux density variation along with the AC coils height. During the pre-fault period, the core is well saturated beneath the center of the AC coils (with low flux variation shown in Fig. 13), but not so saturated at the top of the AC coils (with significant flux variation shown in Fig. 17) [56]. The FCL can have a significant voltage drop under the pre-fault condition. This characteristic of the open core arrangement can be improved using more than one DC biasing coil and spacing the coils to achieve a more uniform flux distribution along with the height of the AC coils.

Considering Fig. 17, the voltage drop of introduced FCL in normal and faulty states is presented by Eqs. (15) and (16), respectively. where, R_m1 and R_m2 are the reluctance of each of cores, N_AC1 and N_AC2 are windings turns and R_sat1 and R_sat2 are saturated reluctance of each core.

(15) $V_{FCL}=j\left(\frac{\omega N_{AC1}}{(R_{a1}+R_{sat1})}+\frac{\omega N_{AC2}}{(R_{a2}+R_{sat2})}\right)I_{line}$

(16) $v_{FCL}\left(t\right)=\left(\frac{N_{AC1}}{(R_{m1}+R_{a1})}+\frac{N_{AC2}}{(R_{m2}+R_{a2})}\right)\frac{d{i}_{line}}{dt}$

Read full article

URL:

https://www.sciencedirect.com/science/article/pii/S0378779620303023

sorensonguir1996.blogspot.com

Source: https://www.sciencedirect.com/topics/mathematics/variable-optimization-problem

Share :