Design Optimization

A two-bar truss is a type of truss structure consisting of two bars connected at the ends by either pins or joints. This type of truss is commonly used in construction and engineering applications.

from gekko import GEKKO

m.Minimize(weight)

Python (GEKKO) Solution

GEKKO is optimization software for mixed-integer and differential algebraic equations. It is coupled with large-scale solvers for linear, quadratic, nonlinear, and mixed integer programming (LP, QP, NLP, MILP, MINLP).

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APM Python Tutorial

• APM Python Two Bar Files

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APM MATLAB Tutorial

• APM MATLAB Two Bar Files

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Python (GEKKO) Solution

GEKKO is optimization software for mixed-integer and differential algebraic equations. It is coupled with large-scale solvers for linear, quadratic, nonlinear, and mixed integer programming (LP, QP, NLP, MILP, MINLP).

GEKKO is optimization software for mixed-integer and differential algebraic equations. It is coupled with large-scale solvers for linear, quadratic, nonlinear, and mixed integer programming (LP, QP, NLP, MILP, MINLP).

Solution Contour Plot (Python)

Generate Contour Plot (Python)

(:sourceend:)

Python Generates Contour Plots

(:source lang=python:)

APM Python Tutorial

APM MATLAB Tutorial

Python (GEKKO) Solution

(:source lang=python:) import numpy as np

1. import gekko, pip install if needed

try:

    from gekko import GEKKO

except:

    # pip install gekko
    import pip
    pip.main(['install','gekko'])
    from gekko import GEKKO

1. create new model

m = GEKKO()

1. declare model parameters

width = m.Param(value=60) thickness = m.Param(value=0.15) density = m.Param(value=0.3) modulus = m.Param(value=30000) load = m.Param(value=66)

1. declare variables and initial guesses

height = m.Var(value=30.00,lb=10.0,ub=50.0) diameter = m.Var(value=3.00,lb=1.0,ub=4.0) weight = m.Var()

1. intermediate variables with explicit equations

leng = m.Intermediate(m.sqrt((width/2)**2 + height**2)) area = m.Intermediate(np.pi * diameter * thickness) iovera = m.Intermediate((diameter**2 + thickness**2)/8) stress = m.Intermediate(load * leng / (2*area*height)) buckling = m.Intermediate(np.pi**2 * modulus * iovera / (leng**2)) deflection = m.Intermediate(load * leng**3 / (2 * modulus * area * height**2))

1. implicit equations

m.Equation(weight==2*density*area*leng) m.Equation(weight < 24) m.Equation(stress < 100) m.Equation(stress < buckling) m.Equation(deflection < 0.25)

1. minimize weight

m.Obj(weight)

1. solve optimization

m.solve() # remote=False for local solve

print ('') print (-- Results of the Optimization Problem --) print ('Height: ' + str(height.value)) print ('Diameter: ' + str(diameter.value)) print ('Weight: ' + str(weight.value))

1. Generate a contour plot

1. Import some other libraries that we'll need
2. matplotlib and numpy packages must also be installed

import matplotlib import numpy as np import matplotlib.pyplot as plt

1. Constants

pi = 3.14159 dens = 0.3 modu = 30000.0 load = 66.0

1. Analysis variables

wdth = 60.0 thik = 0.15

1. Design variables at mesh points

x = np.arange(10.0, 30.0, 2.0) y = np.arange(1.0, 3.0, 0.3) hght, diam = np.meshgrid(x, y)

1. Equations and Constraints

leng = ((wdth/2.0)**2.0 + hght**2)**0.5 area = pi * diam * thik iovera = (diam**2.0 + thik**2.0)/8.0 wght = 2.0 * dens * leng * area strs = load * leng / (2.0 * area * hght) buck = pi**2.0 * modu * iovera / (leng**2.0) defl = load * leng**3.0 / (2.0*modu * area * hght**2.0)

1. Create a contour plot
2. Visit https://matplotlib.org/examples/pylab_examples/contour_demo.html
3. for more examples and options for contour plots

plt.figure()

1. Weight contours

CS = plt.contour(hght, diam, wght) plt.clabel(CS, inline=1, fontsize=10)

1. Stress<100

CS = plt.contour(hght, diam, strs,[100.0],colors='k',linewidths=[4.0]) plt.clabel(CS, inline=1, fontsize=10)

1. Deflection<0.25

CS = plt.contour(hght, diam, defl,[0.25],colors='b',linewidths=[4.0]) plt.clabel(CS, inline=1, fontsize=10)

1. Stress-Buckling<0

CS = plt.contour(hght, diam, strs-buck,[0.0],colors='r',linewidths=[4.0]) plt.clabel(CS, inline=1, fontsize=10)

1. Add some labels

plt.title('Two Bar Optimization Problem') plt.xlabel('Height') plt.ylabel('Diameter')

1. Save the figure as a PNG

plt.savefig('contour1.png')

1. Create a new figure to see more detail

plt.figure()

1. Weight contours

CS = plt.contour(hght, diam, wght) plt.clabel(CS, inline=1, fontsize=10)

1. Stress<100

CS = plt.contour(hght, diam, strs,[90.0,100.0],colors='k',linewidths=[0.5, 4.0]) plt.clabel(CS, inline=1, fontsize=10)

1. Deflection<0.25

CS = plt.contour(hght, diam, defl,[0.22,0.25],colors='b',linewidths=[0.5, 4.0]) plt.clabel(CS, inline=1, fontsize=10)

1. Stress-Buckling<0

CS = plt.contour(hght, diam, strs-buck,[-5.0,0.0],colors='r',linewidths=[0.5, 4.0]) plt.clabel(CS, inline=1, fontsize=10)

1. Add some labels

plt.title('Two Bar Optimization Problem') plt.xlabel('Height') plt.ylabel('Diameter')

1. Save the figure as a PNG

plt.savefig('contour2.png')

1. Show the plots

plt.show() (:sourceend:)

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• APM Python Two Bar Files

• APM MATLAB Two Bar Files

Objective Function Plot

One part of the assignment asks you to select width and load as variables for a 3d optimal surface plot and plot the solution of the optimization problem to minimize deflection at each of the width / load combinations. This tutorial example shows how to do this same activity but for the alternative problem of minimizing weight.

• Two Bar Objective Plot Files

• Two Bar FMINCON MATLAB Tutorial

• Two Bar Mathematica Tutorial

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This assignment can be completed in collaboration with others. Additional guidelines on individual, collaborative, and group assignments are provided under the Expectations link.

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Python Tutorial

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MATLAB Tutorial

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• Two Bar APM MATLAB Tutorial
• Two Bar APM Python Tutorial

• Two Bar Isight Tutorial

• Two Bar APM MATLAB Tutorial
• Two Bar FMINCON MATLAB Tutorial

This introductory assignment is designed as a means of demonstrating the optimization capabilities of a number of software packages. Below are tutorials for solving this problem with a number of software tools. Below are a few step-by-step tutorials.

• Two Bar APMonitor Tutorial
• Two Bar Excel Tutorial Δ
• Two Bar MATLAB Tutorial
• Two Bar OptdesX Tutorial

(:title Two Bar Truss Design:)

This introductory assignment is designed as a means of demonstrating the optimization capabilities of a number of software packages. Below are tutorials for solving this problem with a number of software tools.

This assignment can be completed as a collaborative assignment. Additional guidelines on individual, collaborative, and group assignments are provided under the Expectations link.

This assignment can be completed as a collaborative assignment. Additional guidelines on individual, collaborative, and group assignments are provided under the Expectations link.

Two Bar Assignment

This introductory assignment is designed as a means of demonstrating the optimization capabilities of a number of software packages. Below are tutorials for solving this problem with a number of software tools.

(:title Two Bar Optimization Problem:) (:keywords nonlinear, optimization, engineering optimization, two-bar optimization, engineering design, interior point, active set, differential, algebraic, modeling language, university course:) (:description Engineering design of a two-bar structure to support a load. Optimization principles are used to design the system.:)

A design of the truss is specified by a unique set of values for the analysis variables: height (H), diameter, (d), thickness (t), separation distance (B), modulus of elasticity (E), and material density (rho). Suppose we are interested in designing a truss that has a minimum weight, will not yield, will not buckle, and does not deflect "excessively,” and so we decide our model should calculate weight, stress, buckling stress and deflection.

Two Bar Assignment