MaplePortal/EngineeringOptimization - Maple Help

 Engineering Optimization

 Introduction

Maple lets you minimize or maximize objective functions with respect to constraints.

 >
 $\left[{-0.980004457937920792}{,}\left[{x}{=}{0.0469482872699630}{,}{y}{=}{4.47652585636502}\right]\right]$ (1)

The objective function can be a sum-of-squares error for parameter estimation, or the

 • weight of a mechanical device
 • cost of production
 • or energy required for a process

The constraints can be limits on the

 • dimensions of a mechanical device, or the allowable stresses
 • minimum and maximum process temperatures
 • or amount of base materials

Units can be employed in the objective function or the constraints.

You can use Maple's built-in linear, nonlinear, and quadratic optimizers, or the optional Global Optimization Toolbox.

 Example - Fuel Pod Design Optimization

You are designing a fuel pod with a hemispherical cap, cylindrical mid-section and conical cap.

What are values of L, H and R that minimize the surface area while maintaining the volume V at 3 m3?

 > $\mathrm{restart}:\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{with}\left(\mathrm{Optimization}\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$

Objective function - surface area of pod

 > $\mathrm{obj}≔\frac{1}{2}\cdot 4\cdot \mathrm{\pi }\cdot {\mathrm{R}}^{2}+2\cdot \mathrm{\pi }\cdot \mathrm{R}\cdot \mathrm{L}+\mathrm{\pi }\cdot \mathrm{R}\cdot \sqrt{{\mathrm{H}}^{2}+{\mathrm{R}}^{2}}:\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}$

Constraint on the volume area of pod

 >

All dimensions must be greater than 0

 > $\mathrm{cons2}≔0\le \mathrm{R},0\le \mathrm{L},0\le \mathrm{H}:$

Hence the optimized dimensions are

 > $\mathrm{dimensions}≔\mathrm{Minimize}\left(\mathrm{obj},\left\{\mathrm{cons1},\mathrm{cons2}\right\},\mathrm{initialpoint}=\left\{\mathrm{H}=1⟦\mathrm{m}⟧,\mathrm{L}=1⟦\mathrm{m}⟧,\mathrm{R}=1⟦\mathrm{m}⟧\right\}\right)$
 $\left[{10.2533536615869920}{}⟦{{\mathrm{m}}}^{{2}}⟧{,}\left[{\mathrm{H}}{=}{0.785093823049978}{}⟦{\mathrm{m}}⟧{,}{\mathrm{L}}{=}{0.392546902492684}{}⟦{\mathrm{m}}⟧{,}{\mathrm{R}}{=}{0.877761593519080}{}⟦{\mathrm{m}}⟧\right]\right]$ (2)

Check that the constraint on the pod volume is satisfied

 > $\mathrm{eval}\left(\left[\mathrm{cons1}\right],\mathrm{dimensions}\left[2\right]\right)$
 $\left[{3.00000000039170}{}{⟦{\mathrm{m}}⟧}^{{3}}{=}{3}{}⟦{{\mathrm{m}}}^{{3}}⟧\right]$ (3)



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