\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 181, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/181\hfil Optimal design of a bar]
{Optimal design of a bar with an attached mass for maximizing the heat transfer}

\author[B. P. Belinskiy, J. W. Hiestand, M. L. McCarthy \hfil EJDE-2012/181\hfilneg]
{Boris P. Belinskiy, James W. Hiestand, Maeve L. McCarthy }  % in alphabetical order

\address{Boris P. Belinskiy \newline
Department of Mathematics\\
University of Tennessee at Chattanooga\\
615 McCallie  Avenue\\
Chattanooga, TN 37403-2598, USA}
\email{Boris-Belinskiy@utc.edu}

\address{James W. Hiestand \newline
College of Engineering\\
University of Tennessee at Chattanooga\\
615 McCallie  Avenue\\
Chattanooga, TN 37403-2598, USA}
\email{James-Hiestand@utc.edu}

\address{Maeve L. McCarthy \newline
Department of Mathematics \& Statistics\\
Murray State University\\
6C Faculty Hall\\
Murray, KY 42071-334, USA}
\email{mmccarthy@murraystate.edu}

\thanks{Submitted August 15, 2012. Published October 19, 2012.}
\subjclass[2000]{74A15, 74P10, 49K15, 34B24}
\keywords{Optimal design; heat transfer; heat equation; least eigenvalue;
\hfill\break\indent Sturm-Liouville problem;
  Helly's principle; calculus of variations}

\begin{abstract}
 We maximize, with respect to the cross sectional area, the rate of heat transfer
 through a bar of given mass. The bar serves as an extended surface to enhance
 the heat transfer surface of a larger heated known mass to which the bar is
 attached. In this paper we neglect heat transfer from the sides of the bar and
 consider only conduction through its length. The rate of cooling is defined by
 the first eigenvalue of the corresponding Sturm-Liouville problem.
 We establish existence of an optimal design via rearrangement techniques.
 The necessary conditions of optimality admit a unique optimal design.
 We compare the rate of heat transfer for that bar with the rate for the bar
 of the same mass but of a constant cross-section area.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\def \pd#1#2{\frac{\partial#1}{\partial#2}}

\section{Introduction}\label{intro_sect}

Materials are often cooled by convection to a surrounding ambient medium such as the atmosphere. For example, the heat generated in an automobile engine is transferred first to the cooling water that circulates through the engine and then to the atmosphere through the radiator. Convective heat transfer is described by the equation
\begin{equation}
\dot Q=h A_s (T-T_\infty)
\label{1.1}
\end{equation}
where $\dot Q$ is the heat transfer rate, $h$ is an empirical heat transfer 
coefficient, $A_s$ is the surface area, $T$ is the temperature of the surface 
and $T_\infty$ is the temperature of the surrounding medium. 
We want to maximize the surface area since the convective heat transfer
rate is proportional to this area. On the other hand, we want to minimize 
the volume of the heat transfer region, in order to keep its weight and hence
material cost as low as possible. Thus we seek to maximize the surface to volume 
ratio of the heat transfer surface.

Extended surfaces attached to a given base mass, $M_0$, are frequently used in
commercial applications to increase the heat transfer surface area without 
significantly increasing the associated mass and hence the material cost of 
the device. The additional surface might be in the form of thin donuts around a 
central pipe, parallel plates attached to the surface as in small engines or 
automobile radiators, or fins extending outward like hairs from a surface. 
The mass of the added extended surface is small compared to the base mass $M_0$. 
For this reason a high surface-to-volume ratio for the extended surface is sought. 
Such mass additions to enhance heat transfer are referred to as fins.

Nature also ``designs" according to this criterion. The ears of an elephant have 
large surface area compared to their volume which allows the blood passing through 
them to be efficiently cooled. Likewise members of species like deer that live 
near the equator must be able to dissipate heat efficiently. They achieve this 
desirable ratio by being smaller than their counterparts that live towards the 
poles (think of a sphere where the surface to volume is inversely proportional to $r$).

Engineering heat transfer texts sometimes consider fins of variable 
cross-section \cite[pp. 124-126]{Incropera} but the cross-section is assumed 
to be for a regular shape, e.g. a cylinder with variable radius. 
Or the optimization of a fin with a given cross-section (e.g. a rectangle) 
is optimized with respect to the length and thickness \cite[pp. 74]{Schneider}. 
However, a general variation of shape to maximize the heat transfer from the fin 
is not considered.


Heat transfer within the surface is by conduction and the rate is given by 
the equation
\begin{equation}
\dot Q=-{{kA \Delta T}\over l}
\end{equation}
where $k$ is the thermal conductivity of the material, $A$ is the cross-sectional 
area, and $l$ is the length of the material. Here $\Delta T$ is the temperature 
difference between the end points of the heat transfer.  The equations above, 
along with the corresponding physical background, may be found in
 \cite[pp. 110-114]{Incropera}.

If an energy balance is performed for the region of the bar  between $x$ and 
$x+\Delta x$, energy enters by conduction at $x$ and leaves by conduction
at $x+\Delta x$ and also from the side by convection (see Equation \eqref{1.1}).
The difference is the rate of change of the energy content of that region of the bar
$$
\text{rate of change of energy}=\text{energy in}-\text{energy out}\quad \text{or}
$$
\begin{equation}
\rho c A \Delta x {{\Delta T}\over {\Delta t}}=-kA{{\Delta T}\over {\Delta x}}
\Big \vert_x+kA{{\Delta T}\over {\Delta x}}\Big \vert_{x+\Delta x}-hA_s(T-T_{\infty}).
\label{1.3}
\end{equation}
Here $T(x,t)$ is the temperature distribution. The surface area is 
$A_s=P\Delta x$, where $P(x)$ is the perimeter of the cross-section at the point $x$. 
The ratio ${{\Delta T}\over {\Delta t}}$ is the rate of change of temperature with 
time and ${{\Delta T}\over {\Delta x}}$ is the local temperature gradient. 
The following bar material parameters are introduced, the density $\rho$, 
the specific heat capacity $c$, the thermal conductivity $k$, and the convective 
heat transfer coefficient $h$. It is assumed that $\rho,c,k,h$ are positive constants.

Dividing by $\Delta x$, and taking the limit as $\Delta x$ and $\Delta t$ approach
 zero, yields the partial differential equation
\begin{equation}
A{\pd {T}{t}}={k\over {\rho c}}{\pd {}{x}}\Big(A{\pd {T}{x}}\Big)
-{{hP}\over {\rho c}}(T-T_{\infty}),  (x,t)\in (0,l)\times (0,\infty).
\label{1.4}
\end{equation}
We discuss a particular case of this general equation when convective heat transfer 
from the side of the bar is neglected, i.e., the limiting case $h\to 0$ is considered.
It is the purpose of this paper to find the optimal distribution of the 
cross-section area, $A$, of a surface of revolution of a given mass such that 
the heat transfer rate is a maximum. This will produce a maximum cooling per 
unit mass and may be considered the optimum.

Detailed discussion of techniques and results in structural optimization can 
be found in \cite{Dym,Atanackovic} and the references therein.
We mention in particular the maximization of a column's buckling load
 \cite{Taylor,TaylorLiu}, the minimization of the mass of an oscillating 
bar \cite{Taylor2,Turner} or a rotating rod \cite{Atanackovic2001,Atanackovic2004}, 
the maximization of a column's height \cite{KellerNiordson} and the minimization 
of the moment of inertia of an oscillating turbine \cite{BelinskiyMcCarthy}.
If the design variable tapers too rapidly, the eigenvalue being optimized is not 
isolated from the remainder of the spectrum 
\cite{KellerNiordson,CoxMcCarthy,Atanackovic2001}.  In these cases optimality 
conditions can be derived using non-smooth analysis in conjunction with  the more 
classical Calculus of Variations techniques \cite{McCarthy,McCarthy2}.

The complexity of the Sturm-Liouville problem also increases if the boundary
conditions contain an eigenparameter.
This is due to the fact the the Sturm-Liouville operator is not self-adjoint with  
respect to the usual $L^2(0,l)$ inner product.
The spectral properties of the Sturm-Liouville problems that arise from diverse
 mechanical models and contain the spectral parameter in the boundary condition(s) 
have been studied in \cite{Walter,Fulton,Hinton2,BBS,Belinskiy}. Numerical schemes for the inverse problem were developed by \cite{McCarthyRundell}. Design problems of this type have been considered by  \cite{Turner,BelinskiyMcCarthy}.
In the latter, existence of an optimal design was treated seriously, 
as it will be here.

In the design problem considered here, we encounter a spectral parameter in a
boundary condition.
In Section \ref{sl_sect} we give the mathematical description of the model, 
apply separation of variables and formulate the spectral properties of the 
corresponding Sturm-Liouville problem. We also give the solution for an elementary 
case of the problem when the cross-section area is constant, which we need later 
for comparison with the solution in case of variable cross-section area. 
In Section \ref{cov_sect} we derive the necessary conditions of optimality and 
hence find an optimal form of the bar.
In Section \ref{rearr_sect} we use a rearrangement technique to prove that the 
optimal design is increasing and maximizes the first eigenvalue of 
the Sturm-Liouville problem.
In Section \ref{num_sect} we give the numerical comparison of cooling properties 
for the bar of optimal shape and a bar having the same mass but with the 
constant cross-section area. In the Appendix, we prove that the rate of cooling 
for the bar with the optimal cross-section area is greater than for a bar of the 
same mass but with constant cross-section area.

\section{Heat transfer of a bar of a variable cross-section area: separation 
of variables and the Sturm-Liouville problem}
\label{sl_sect}

We consider the heat transfer in a bar $\{0<x<l\}$ with a base mass $M_0$ 
attached at the end point $x=0$. The temperature distribution 
$T:[0,l]\times [0,\infty)\to R$ satisfies the transient one-dimensional 
conduction equation
\begin{equation}
A{\pd {T}{t}}={k\over {\rho c}}{\pd {}{x}}\Big(A{\pd {T}{x}}\Big),\quad
(x,t)\in (0,l)\times (0,\infty)
\label{2.1}
\end{equation}
that is the limiting case of Equation \eqref{1.4} as $h\to 0$.
Here the cross-section area $A(x):[0,l]\to R_+$ is a continuous differentiable 
positive function. As was mentioned in the Introduction, parameters 
$k, \rho, c$ are positive constants. The end point $x=l$ is kept at the (constant) 
temperature of the surrounding medium,
\begin{equation}
T(l,t)=T_{\infty},  t\in [0, \infty).
\label{2.2}
\end{equation}
The rate of change of the energy content of the base is given by the difference 
between the energy flow into and out of it, as in the derivation of  \eqref{1.3}. 
For energy flow only by conduction outward at $x = 0$ this becomes
$$
cM_0 {{\Delta T}\over {\Delta t}}=kA {{\Delta T}\over {\Delta x}}\bigl|_{x=0}.
$$
In the limit as $\Delta t$ and $\Delta x \to 0$ this becomes
\begin{equation}
cM_0{\pd{T}{t}}(0,t)=kA(0){\pd{T}{x}}(0,t)\quad t\in [0, \infty).
\label{2.3}
\end{equation}
The initial distribution $T_0:[0,l]\to R$ of the temperature is given,
\begin{equation}T(x,0)=T_0(x).
\label{2.4}
\end{equation}
It is well known that the initial boundary value problem \eqref{2.1}-\eqref{2.4}
 has a unique solution \cite{Powers,Ladyzhenskaia}. 
It is convenient to extract the term $T_{\infty}$ from the solution,
\begin{equation}
\tau(x,t)\equiv T(x,t)-T_{\infty}. \label{2.5}
\end{equation}
The new unknown function $\tau:[0,l]\times [0,\infty)\to R$ is the unique solution 
of the initial boundary value problem
\begin{gather}
A{\pd {\tau}{t}}={k\over {\rho c}}{\pd {}{x}}\Big(A{\pd {\tau}{x}}\Big),\quad
(x,t)\in (0,l)\times (0,\infty), \label{2.6}\\
\tau(l,t)=0,\quad t\in [0, \infty),\label{2.7} \\
cM_0{\pd{\tau}{t}}(0,t)=kA(0){\pd{\tau}{x}}(0,t),\quad t\in [0, \infty),\label{2.8}\\
\tau(x,0)=\tau_0(x),\quad \in [0, l];\text{ where }\tau_0(x)\equiv T_0(x)-T_{\infty}.
\label{2.9}
\end{gather}
If we use the standard procedure of separation of variables
\begin{equation}\tau(x,t)\equiv e^{-\sigma t} u(x) \label{2.10}
\end{equation}
and introduce the notation
\begin{equation}
{{\rho c}\over k} \sigma\equiv \lambda\text{ so that }
{{c \sigma}\over k} ={{\lambda}\over {\rho}},
\label{2.11}
\end{equation}
then the function $u:[0,l]\to R$ satisfies the following Sturm-Liouville problem
\begin{equation}
\begin{gathered}
(Au')'+\lambda Au=0,\quad x\in (0,l);\\
u(l)=0,\quad A(0)u'(0)+{{M_0}\over{\rho}} \lambda u(0)=0.
\end{gathered}\label{2.12}
\end{equation}
As we see, the spectral parameter $\lambda$ appears in the second boundary condition. 
The general theory for Sturm-Liouville problems of this type developed 
in \cite{Walter,Fulton,Hinton2,BBS,Belinskiy} may be used. 
It can be verified that the conditions of the corresponding theorems are satisfied.
In particular, by Walter \cite[Theorem 1]{Walter}, we know that
the eigenparameter dependent Sturm-Liouville problem \eqref{2.12}
 has a pure discrete positive real spectrum with the only point of accumulation 
at $+\infty$.
The set of eigenfunctions satisfies the orthogonality relation
\begin{equation}
\int_0^l Au_n u_j dx+{{M_0}\over {\rho}} u_n(0)u_j(0)=0\quad\text{if }n\ne j
\label{2.13}
\end{equation}
which allows us to define an inner product over which our Sturm-Liouville problem 
is self-adjoint.
The Rayleigh quotient
\begin{equation}
\lambda_n={{\rho\int_0^l Au_n'^2 dx}\over {\rho\int_0^l Au_n^2 dx+M_0 u_n^2(0)}}
\label{2.14}
\end{equation}
immediately follows.


Existence and uniqueness of the solution of the initial boundary value 
problem \eqref{2.6}-\eqref{2.9} follow from known techniques, \cite{Ladyzhenskaia}.
Its series representation is given by
\begin{equation}
\tau(x,t)\equiv \sum_{n\ge 1} c_ne^{-\sigma_nt}u_n(x)
\label{3.1}
\end{equation}
where
$$
\sigma_n\equiv {k\over {\rho c}}\lambda_n
$$
and
\begin{equation}c_n={{\rho\int_0^l A\tau_0 u_n dx+M_0 
\tau_0(0)u_n(0)}\over {\rho\int_0^l
Au_n^2 dx+M_0 u_n^2(0)}}.
\label{3.2}
\end{equation}

We note that if the cross-section area is constant, $A(x)=A$, the mass of the bar is $M=\rho Al$, and
the exact solution of the problem \eqref{2.6}-\eqref{2.9} is given by
$$
\tau(x,t)\equiv \sum_{n\ge 1} c_ne^{-\sigma_n t} \sin \sqrt{\lambda_n}(x-l).
$$
Here
$\lambda_n$ are the positive solutions of the transcendental equation
\begin{equation}
\tan (\sqrt{\lambda_n} l)=\frac{M}{M_0 \sqrt{\lambda_n} l}, \quad n=1,2,\dots
\label{4.7}
\end{equation}
and
\begin{equation}c_n={{\rho A\int_0^l \tau_0(x)\sin\sqrt{\lambda_n}(x-l)dx
-M_0 \tau_0(0)\sin \sqrt{\lambda_n} l}\over {\rho A\int_0^l \sin^2 
\sqrt{\lambda_n}(x-l)dx + M_0 \sin^2 \sqrt{\lambda_n} l}}.
\label{4.8}
\end{equation}

\section{Heat transfer of a bar of a variable cross-section area: 
optimality conditions}
\label{cov_sect}

The representation \eqref{3.1}-\eqref{3.2}, and \eqref{2.5} for the solution 
shows that the temperature $T(x,t)$ approaches the level $T_{\infty}$ exponentially fast,
 and the rate of approach is determined by the first eigenvalue $\lambda_1$.
 We now formulate the problem of optimal design and consider the variational problem:

{\it Find the form of the cross--section $A(x)\in(0,\infty)$ that yields the maximum 
to the functional}
\begin{equation} \label{B11}
\lambda_1=\min_{u\in H^1[0,l]} \frac{\int_0^l A(x)(u'(x))^2dx}{\int_0^l A(x)(u(x))^2dx+\frac{M_0}{\rho}u^2(0)}
\end{equation}
{\it given}
\begin{equation} \label{B2}
\int_0^l A(x)=\frac{M}{\rho}.
\end{equation}

We note here that for any $\hat{u}\in H^1[0,l]$
\begin{equation}
\lambda_1(A) \leq \frac{\int_0^l A(x)(\hat{u}'(x))^2dx}{\int_0^l A(x)
(\hat{u}(x))^2dx+\frac{M_0}{\rho}\hat{u}^2(0)}
\end{equation}
In particular, if we choose $u(x)=l-x$, it follows that $\lambda_1(A)$
is bounded because
\begin{equation}
\label{lambda-bdd}
\lambda_1(A) \leq \frac{\int_0^l A(x)dx}{\int_0^l A(x)(l-x)^2dx
+\frac{M_0}{\rho}l^2}
\leq \frac{\rho M}{M_0 l^2}.
\end{equation}

It is well-known that the variational problem of the minimization of
 the ratio above is equivalent to the following problem: 
{\sl Find the function $u\in H^1[0,l]$ that yields the minimum to the functional
\begin{equation} \label{B31}
L=\int_0^l A(x)(u'(x))^2dx
\end{equation}
subject to the constraint
\begin{equation} \label{B41}
\int_0^l A(x)(u(x))^2dx+\frac{M_0}{\rho}u^2(0)=1.
\end{equation}
}
Hence, we come to the following variational problem:

{\sl Find the form of the cross--section $A(x)\in(0,\infty)$ that yields the maximum 
to the functional
\begin{equation} \label{B51}
L_1=\min_{u\in H^1[0,l]}\int_0^l A(x)(u'(x))^2dx
\end{equation}
subject to the constraints
\begin{equation} \label{B61}
\int_0^l A(x)(u(x))^2dx+\frac{M_0}{\rho}u^2(0)=1,  \int_0^l A(x)=\frac{M}{\rho}.
\end{equation}
}

Using the Lagrange method, we introduce the new functional
\begin{equation}
\begin{split}
F[A;u] &=\int_0^l A(x)(u'(x))^2dx-\mu_1\Big(\int_0^l A(x)u^2(x)dx
 +\frac{M_0}{\rho} u^2(0)\Big)  \\
&\quad -\mu_2\Big(\int_0^l A(x)dx-\frac{M}{\rho}\Big)
\label{F}
\end{split}
\end{equation}
where $\mu_1, \mu_2$ are Lagrange multipliers. The necessary condition of the
extremum in terms of the first variation of the functional $F[A;u]$ has the form
$\delta F[A;u]=0$.
Using the boundary condition
\begin{equation} \label{XXX}
u(l)=0,
\end{equation}
this can be written as
\begin{align*}
&\int_0^l (u')^2\delta A  dx  + \int_0^l 2Au'\delta u' dx
-\mu_1\int_0^l u^2\delta A dx\\
&-\mu_1\int_0^l 2Au\delta u  dx
-\mu_1\frac{M_0}{\rho} 2u(0)\delta u(0)-\mu_2\int_0^l \delta A dx =0.
\end{align*}


The variations $\delta u(0), \delta u(x), \delta A(x)$ are independent. 
Equating the corresponding parts of the variation of $F[A;u]$ to zero 
yields a boundary condition and two differential equations
\begin{gather}
\label{var-u-0}
(Au')(0)+\mu_1\frac{M_0}{\rho} u(0)=0, \\
 \label{var-u}
(Au')'+\mu_1Au=0, 0<x<l, \\
 \label{var-A}
(u')^2-\mu_1u^2-\mu_2=0.
\end{gather}
We observe that the differential equation \eqref{var-u} subject to the
 boundary conditions \eqref{XXX}, \eqref{var-u-0} yields $u(x)$ to be the 
first eigenfunction and $\mu_1$ to be the first eigenvalue, $\mu_1=\lambda_1$, 
of the original Sturm-Liouville problem \eqref{2.12}.
The optimality conditions represented by the nonlinear differential 
equation \eqref{var-A} subject to the boundary condition \eqref{XXX} 
may be solved explicitly,
\begin{equation} \label{u-sol}
u(x)=\frac{\sqrt{\mu_2}}{\lambda_1} \sinh \sqrt{\lambda_1}(l-x).
\end{equation}
Substituting $u(x)$ from \eqref{u-sol} into the Sturm-Liouville equation
\eqref{var-u} yields the following differential equation for $A(x)$
\begin{equation} \label{A}
\big(A\cosh \sqrt{\lambda_1}(l-x)\big)'-\lambda_1A\frac{1}{\sqrt{\lambda_1}} \sinh \sqrt{\lambda_1}(l-x)=0
\end{equation}
which also may be solved explicitly
\begin{equation} \label{A-1}
A(x)=\frac{C}{\cosh^2 \sqrt{\lambda_1}(x-l)}.
\end{equation}
The boundary condition \eqref{var-u-0} yields
\begin{equation} \label{C}
C=\frac{M_0\sqrt{\lambda_1}}{2\rho} \sinh(2\sqrt{\lambda_1}l).
\end{equation}
Using Conditions \eqref{B2} yields finally
\begin{equation} \label{transc}
M_0\sinh^2(\sqrt{\lambda_1}l)=M.
\end{equation}

Solving \eqref{transc} for $\lambda_1$ finally yields the optimal rate of cooling 
for the bar with the given mass
\begin{equation}
\lambda_1= \Big( {1\over l} \ln\Big(\sqrt{{M\over M_0}}
+\sqrt{{M\over M_0}+1} \Big) \Big)^2.
\label{6.17}
\end{equation}

We introduce the dimensionless parameter
\begin{equation}z^{\rm opt}\equiv \sqrt{\lambda_1} l
=\ln\Big(\sqrt{{M\over {M_0}}}+\sqrt{{M\over {M_0}}+1} \Big).
\label{6.18}
\end{equation}
It is of interest to compare it with the similar parameter $z$ for the 
constant cross-section.
From the transcendental equation \eqref{4.7}, our dimensionless parameter satisfies
\begin{equation}
z \tan z= {M\over {M_0}}. \label{6.19}
\end{equation}
In the Appendix, we show that the inequality
\begin{equation}z^{\rm opt} >z
\label{6.20}
\end{equation}
holds for any positive ratio $M/M_0$. This is confirmed by our 
numerical results in Section \ref{num_sect}.


\section{The Optimal Design is Increasing}
\label{rearr_sect}

We prove here that the heat transfer rate $\lambda_1(A)$ can be increased 
through the use of increasing rearrangements of the cross-sectional 
area $A(x)$. This is achieved through the use of an alternative characterization 
of $\lambda_1(A)$. We begin by defining decreasing and increasing rearrangements, 
and stating some of their relevant properties.

\begin{definition}  \rm
The  decreasing rearrangement of a nonnegative function, $f$, on $(a,b)$ is simply
$$
f^*(x) \equiv \sup \{ t > 0 : \mu_f(t) > x \},
$$
where $\mu_f$ is the distribution function of $f$,
$$\mu_f(t) = |\{x\in (a,b):f(x)>t\}| \quad t \ge 0.
$$
The  increasing rearrangement of $f$ is $f_*(x) \equiv f^*(b-x)$.
\end{definition}

If $g$ and $h$ are nonnegative functions on $(a,b)$, with $g$  increasing 
and $h$ decreasing, then
\begin{equation}
\int_a^b f  dx = \int_a^b f^* dx  = \int_a^b f_*  dx ,
\label{5.7}
\end{equation}
By \eqref{5.7}, if we replace a particular design $A\in ad$ by either its increasing
or decreasing rearrangements $A_*$ or $A^*$ then the new design has the same integral.
 Furthermore,
\begin{equation}
\int_a^b f^* g dx  \leq \int_a^b fg dx , \qquad \int_a^b f_* h dx  
\leq \int_a^b fh dx.
\label{5.8}
\end{equation}
These results are a special case of those established in \cite[pp. 153]{PolyaSzego}.

\begin{theorem} \label{rearr_thm}
For any cross-sectional area $A$ satisfying
$$
0<A(x)<\infty,  x\in [0,l], \int_0^l A(x)   dx = {M\over {\rho}},
$$
its increasing rearrangement $A_*$ satisfies
$$
\lambda_1(A) \leq \lambda_1(A_*).
$$
%\label{lemma3}
\end{theorem}

\begin{proof}
Using variation of parameters, as in \cite{Hinton2,BBS} we find that if $u(x)$ 
is a solution of \eqref{2.12} corresponding to $A(x)$, then 
$v(x)=\sqrt{A(x)}u(x)$ satisfies
$$
 v(x) = \lambda  [ \phi_A(x)  + \left(G_Av\right)(x) ] 
$$
for $0< x< l$, where
\begin{gather} 
\phi_A(x) = \sqrt{A(x)} {{M_0}\over{\rho}} \Big(\int_x^l {{dx}\over{A(x)}} \Big) ,
\label{5.3} \\
(G_Av)(x)=  \int_0^l g_A(x,t)v(t) dt, \\
g_A(x,s) =  \sqrt{A(x)} \sqrt{A(s)} \int_{x\wedge s}^l {{dy}\over{A(y)}}
\label{5.4}
\end{gather}
and $x\wedge s =\max{\{x,s\}}$.
If $\langle u,v\rangle $ denotes the
$L^2(0,l)$ inner product and $\| \cdot \|$ denotes its associated norm, 
then  this can be written as
$$ 
\|v \|^2= \lambda  [ \langle \phi_A,v\rangle  + \langle G_Av,v\rangle ] . 
$$
Thus, our second characterization is a variational characterization similar 
to that of Porter and Stirling,
\cite[Lemma 5.1]{PorterStirling},
\begin{equation}
{{1}\over{\lambda_1(A)}} = \max_{\|v\|=1} [ \langle \phi_A,v\rangle
+ \langle G_A v,v \rangle ]. \label{5.5}
\end{equation}
The maximum is attained at $v_1(x) = \sqrt{A(x)}u_1(x)$ where $u_1(x)$ 
is the first eigenfunction of the Sturm-Liouville Problem \eqref{2.12} 
associated with $A(x)$ for which $\int_0^l u_1^2(x)A(x) dx =1$.


Let $u_1$ be the first eigenfunction of
the Sturm-Liouville Problem \eqref{2.12} associated with $A_*$.
Using $v(x) = \sqrt{A(x)}u(x)$ in \eqref{5.5} and integrating by parts, we find
\begin{equation*}
{{1}\over{\lambda_1(A)}} = \max_{\|u\|_A=1}
\Big[ {{M_0}\over{\rho}} \int_0^l   \Big( \int_0^x A(y) u(y) dy \Big) {{dx}\over{A(x)}}
 +  \int_0^l   \Big( \int_0^x A(y) u(y) dy \Big)^2 {{dx}\over{A(x)}} \Big]
%\label{5.6}
\end{equation*}
where $\|\cdot\|_A$ denotes the norm associated with the $L^2(0,l; A(x) )$
inner product.
Integrating \eqref{2.12} with $u_1$ and $A_*$ between $0$ and $x$,
 and using the boundary condition at $x=0$
yields
$$
u_1'(x) = {{-\lambda_1(A_*)}\over{A_*(x)}}
\Big[ \int_0^x A_*u_1   dr + {{M_0}\over{\rho}} u_1(0) \Big].
$$
Binding, Browne and Seddighi established oscillation results
for Sturm-Liouville problems with eigenparameter dependent boundary conditions
in \cite{BBS}. In particular, their Corollary 5.2 implies that our
first eigenfunction has no interior zeros. We assume without loss of
generality that $u_1>0$. Since $\lambda_1(A_*), M_0, \rho,  u_1(x), A(x) >0$,
 this implies that $u_1$ is decreasing.
The second part of \eqref{5.8} implies that
$\int_0^{x} Au_1 dy \geq \int_0^{x} A_* u_1 dy $ and so
$$
 {{1}\over{\lambda_1(A)}} \geq  {{M_0}\over{\rho}} \int_0^l
\left( \int_0^x A_*(y) u_1(y) dy \right) {{dx}\over{A(x)}}
+\int_0^l   \left( \int_0^x A_*(y) u_1(y) dy \right)^2 {{dx}\over{A(x)}}.
$$
Clearly, the functions $( \int_0^x A_* u_1 dy)$ and
$( \int_0^x A_*u_1 dy)^2$
are nonnegative increasing functions of $x$. Once again \eqref{5.8} yields
\begin{align*}
{{1}\over{\lambda_1(A)}}
& \geq  {{M_0}\over{\rho}} \int_0^l   \Big( \int_0^x A_*(y) u_1(y) dy \Big)
 \Big( {{1}\over{A(x)}} \Big)^* dx \\
&\quad +\int_0^l   \Big( \int_0^x A_*(y) u_1(y) dy \Big)^2
\Big( {{1}\over{A(x)}} \Big)^* dx.
\end{align*}
If $f$ is decreasing on the range of $g$ then the composition
$(f\circ g)^*=f\circ g_*$, see \cite{Cox}, which implies that
\begin{align*}
{{1}\over{\lambda_1(A)}}
&\geq  {{M_0}\over{\rho}} \int_0^l   \Big( \int_0^x A_*(y) u_1(y) dy \Big)
 {{dx}\over{A_*(x)}}
 + \int_0^l   \Big( \int_0^x A_*(y) u_1(y) dy \Big)^2 {{dx}\over{A_*(x)}}\\
& =  {{1}\over{\lambda_1(A_*)}}. \qedhere
\end{align*}
\end{proof}

\begin{theorem} \label{thm4.3}
The design satisfying the first order optimality conditions 
\eqref{XXX}-\eqref{var-A}
$$
A(x)=\frac{M\sqrt{\lambda_1}\coth(\sqrt{\lambda_1}l)}{\rho \cosh^2 (\sqrt{\lambda_1}(x-l))}
$$
maximizes the functional $A\mapsto\lambda_1(A)$ on the set
$$
ad = \big\{ A : 0<A(r)<\infty, x\in [0,l], \int_0^l A(r)   dr = \frac{M}{\rho} \big\}
$$
\end{theorem}

\begin{proof}
By \eqref{lambda-bdd} we know that $\lambda_1(A)$ is bounded.
$A$ is the unique solution of the optimality system given by 
\eqref{XXX}-\eqref{var-A}.
Suppose that $A$ is not a maximizer. By Theorem \ref{rearr_thm}, the associated 
design can be improved and the first eigenvalue increased by replacing $A$ 
with its increasing rearrangement $A_*$. Since $A$ is already an increasing 
function of $x$, the design cannot be improved. Hence
$A$ maximizes the functional.
\end{proof}

\section{Numerical comparison of cooling properties}
\label{num_sect}

The product $z=\sqrt{\lambda} l$ is a function of the ratio $M/M_0$ in both the 
constant area case \eqref{6.19} and the optimal case \eqref{6.17}.
The equation \eqref{6.19} was solved numerically.
Similar to the presentation of \cite{Atanackovic2004}, a comparison of constant
 and variable  cross-section is shown in Figure \ref{bhm} as a function of $M/M_0$.
 Extended surfaces with more mass, $M$, than the base mass, $M_0$, are not 
used in engineering practice, and hence values of $M/M_0>1$ in the graph are 
shown for illustrative purposes only.


\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
\end{center}
\caption{Comparision of optimal design and constant area design}
\label{bhm}
\end{figure}

Numerical results show that the advantage of the optimum cross-section over 
the constant cross-section is small and becomes less so as the base mass, 
$M_0$, increases. This is physically reasonable. Indeed, recall that 
convective heat transfer from the side of the area has been neglected. 
Hence addition of the extended surface does little but move the boundary 
condition at $x=l$ that distance from $M_0$. Furthermore, as $M$ becomes 
small compared to $M_0$, its very presence becomes negligible and hence 
its shape does not matter.

In each case $z^{\rm opt}\ge z$, as shown  in the Appendix. 
This numerical observation is certainly in agreement with the general 
results of Sections \ref{cov_sect} and \ref{rearr_sect}. However, 
the effect is not large because of the physical reasons explained above.
 Moreover, for $M/M_0 \to 0$ the optimum and constant area results merge. 
This numerical observation is in agreement with the asymptotic formula \eqref{B.5}.

\section{Conclusion}

We have found the optimal distribution of the cross-section area of a bar in 
the form of a surface of revolution of a given total mass with a point mass 
attached at the end such that the heat transfer rate is a maximum. 
That rate is defined by the least eigenvalue of the corresponding 
Sturm-Liouville problem. This is of independent interest because the spectral 
parameter appears not only in the differential equation but also in the 
boundary condition. The bar will produce the maximum cooling per unit mass 
and may be considered the optimum. The optimal distribution coincides with 
one found by  Taylor \cite{Taylor} and M.J. Turner \cite{Turner} for the 
design of a bar having a maximum lowest eigenfrequency with the given mass.
Numerical results show that the advantage of the optimal design over the
 constant cross-section is small and decreases as the base mass increases. 
We believe this to be a result of the fact that our model neglects heat 
transfer from the side of the bar.

We should emphasize that we have considered a special case of the heat 
transfer assuming that convective heat transfer from the side of the bar is neglected
and only conduction through the length of the bar is considered
(see Section \ref{intro_sect}).

We expect that the solution of the optimal design problem for
the more general problem will show a more noticeable difference between the 
optimal design and the constant case.
If we were to include the heat transfer phenomenon from the sides of the bar, 
we would have to consider the partial differential equation
\begin{equation}
a^2(x){\pd {T}{t}}={k\over {\rho c}}{\pd {}{x}}\Big(a^2(x){\pd {T}{x}}\Big)
-{{ha(x)\sqrt{1+(a'(x)^2}}\over {\rho c}}(T-T_{\infty}), 
\end{equation}
where $ (x,t)\in (0,l)\times (0,\infty)$, and  $a(x)$ would be the radius of the body 
of revolution that represents the bar.
The corresponding Sturm-Liouville problem has a discrete spectrum and a complete 
set of eigenfunctions. We could derive a Rayleigh-Ritz ratio for the least 
eigenvalue similar to expression \eqref{2.14}. 
But the technique of the Calculus of Variations used in Section \ref{cov_sect} 
will not lead to an explicit form of the cross-section area and the least eigenvalue.
 For that problem, we had hoped to use a numerical approach based on 
the discretization of our bar that would reduce the problem of the optimal 
design to the problem of optimization for a function of several variables
 (this idea was developed for the optimal design of mechanical systems 
in \cite{Cardou}). We have recently learned that an equation with similar 
appearance of the function $a(x)$ is optimized in \cite{Henrot-Privat}. 
The techniques used there may be applicable when we consider the more general
 heat transfer model.
 This will be will be considered in a future paper.

\section{Appendix}

Having derived an explicit formula for the optimal rate of cooling and 
an equation for the rate for the bar with the constant cross-section area, 
we may compare them directly. We prove below that the optimal rate is greater 
than the rate for the bar of the same mass but with the constant cross-section area. 
This inequality clearly is demonstrated in our numerical results above, 
but is proven here for completeness.

\begin{lemma}
The inequality
\begin{equation}
z^{\rm opt} >z \label{B.1}
\end{equation}
where
\begin{equation}
z^{\rm opt}=\ln\Big(\mu+\sqrt{\mu^2+1} \Big) \label{B.2}
\end{equation}
and $z$ is the minimal positive root of the equation
\begin{equation}
z \tan z= \mu^2 \label{B.3}
\end{equation}
holds for any positive quantity
\begin{equation}
\mu\equiv \sqrt{{M\over {M_0}}}. \label{B.4}
\end{equation}
\end{lemma}

\begin{proof}
We consider both $z^{\rm opt}$ and $z$ as functions of $\mu> 0$. 
Note first that
$z^{\rm opt}(\mu)\asymp \mu,  z(\mu)\asymp \mu$, and  hence
\begin{equation}
 \lim_{\mu\to 0} z^{\rm opt}(\mu)=\lim_{\mu\to 0} z(\mu)=0. \label{B.5}
\end{equation}
Hence the inequality \eqref{B.1} holds if the derivative of $z^{\rm opt}(\mu)$ 
is greater or equal to the derivative of $z(\mu)$. 
It is easy to prove that the first positive solution of \eqref{B.3} is a 
uniquely defined function on $z\in (0,\pi/2)$  with the derivative
\begin{equation}
z'(\mu)={{2\mu}\over {\tan z+{z\over {\cos^2 z}}}}={{2\mu}\over {{{\mu^2}\over z}
+z+{{\mu^4}\over z}}}
\label{B.6}
\end{equation}
where we used \eqref{B.3} to get the final form. The derivative of $z^{\rm opt}$ 
can be easily found from \eqref{B.2}. We finally come up with the 
necessity to prove the following inequality
\begin{equation}
F(\mu)\equiv {1\over {\sqrt{1+\mu^2}}}-{{2\mu z}\over {\mu^2+\mu^4+z^2}}\ge 0.
\label{B.7}
\end{equation}
We find first
\begin{equation}
\mu^2+\mu^4+z^2\ge 2\sqrt{\mu^2+\mu^4} z=2\mu \sqrt{1+\mu^2} z.
\label{B.8}
\end{equation}
Hence
\begin{equation}
F(\mu)\ge {1\over {\sqrt{1+\mu^2}}}-{{2\mu z}\over {2\mu \sqrt{1+\mu^2} z}}=0
\label{B.9}
\end{equation}
which proves \eqref{B.7} and, along with \eqref{B.5}, proves \eqref{B.1}.
\end{proof}


\subsection*{Acknowledgments}
The first author was supported in part by a University of Tennessee
 at Chattanooga Faculty Research Grant. The third author was supported
in part by NSF DMS \#  0209562 and \# 0531865.
The authors are grateful to the referee for pointing out the
work of Henrot and Privat.


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