variational.pdf

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16.21 Techniques of Structural Analysis andDesign

Spring 2005Unit #9 - Calculus of Variations

Raul Radovitzky

March 28, 2005

Let u be the actual configuration of a structure or mechanical system. usatisfies the displacement boundary conditions: u = u on Su. Define:

u = u + v, where: : scalar

v : arbitrary function such that v = 0 o n SuWe are going to define v as u, thefirst variationofu:

u = v (1)

Schematically:

u1

u2

u

a b

u(a)

u(b)

v

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As a first property of thefirst variation:du du dv= + dx dx dx

so we can identify dvdx

with the first variation of the derivative ofu:du dv =

dx dx

But:

dv dv d

dx=

dx=

dx(u)

We conclude that:

du d = (u)

dx dx

Consider a function of the following form:

F= F(x, u(x), u(x))

It depends on an independent variable x, another function ofx (u(x)) and itsderivative (u(x)). Consider the change in F, when u (therefore u) changes:

F= F(x, u + u, u+ u) F(x, u, u)= F(x, u + v, u+ v) F(x, u, u)

expanding in Taylor series:

F F 1 2F 1 2FF= F+ v + v+

u2(v)2 + (v)(v) + F

u u 2! 2! uuF F

= v + v+ h.o.t.u u

First total variation of F:

FF= lim

0 F(x, u + v, u+ v) F(x, u, u)= lim

0 Fv + Fv F F= lim u u = v + v 0 u u

F FF= u + u

u u

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Note that:

dF(x, u + v, u+ v)F= d =0

since:

dF(x, u + v, u+ v) F(x, u + v, u+ v)=

d uevaluated at = 0

F(x, u + v, u+ v)v + v

u

dF(x, u + v, u+ v)=F(x, u, u)v + F(x, u, u)v

d =0 u uNote analogy with differential calculus.

(aF1 + bF2) = aF1 + bF2 linearity

(F1F2) = F1F2 + F1F2

etc

The conclusions for F(x, u, u) can be generalized to functions of severaluiindependent variables xi and functions ui, xj :

F xi, ui,uixj

We will be making intensive use of these properties of the variational operator:

d d dv du(u) = (v) = =

dx dx dx dx

udx = vdx = vdx = udx

Concept of a functional

b

F(x, u(x), u

(x))dxI(u) =a

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First variation of a functional:

I= F(x, u(x), u(x))dx = F(x, u(x), u(x)) dx

F FI= u + u dx

u u

Extremum of a functional

u0 is the minumumof a functional if:

I(u) I(u

0

)uA necessary condition for a functional to attain an extremum at u0 is:

dII(u0) = 0, or (u0 + v,u0

+ v) = 0d =0

Note analogy with differential calculus. Also difference since here we requiredF = 0 at = 0.d

b

F FI= u + u dx

u u

a

Integrate by parts the second term to get rid ofu.

b F d F d F I= u + u u dx

u dx u dx ua b F bF d F = udx + u

u

dx u u aaRequire u to satisfy homogeneous displacement boundary conditions:

u(b) = u(a) = 0

Then: b F d F I= udx = 0,

u

dx ua

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u that satisfy the appropriate differentiability conditions and the homoge-

neous essential boundary conditions. Then:

F d F = 0

u

dx u

These are the Euler-Lagrange equations corresponding to the variationalproblem of finding an extremum of the functional I.

Natural and essential boundary conditions A weaker condition onu also allows to obtain the Euler equations, we just need:

F bu = 0u a

which is satisfied if:

u(a) = 0 and u(b) = 0 as before u(b) = 0 and F(b) = 0

uFu(a) = 0 and u(b) = 0

F(a) = 0 and F(b) = 0u u

Essential boundary conditions: u = 0, or u = u0 on SuSuNatural boundary conditions: F = 0 on S.u

Example: Derive Eulers equation corresponding to the total po-tential energy functional = U+ V of an elastic bar of length L, Youngsmodulus E, area of cross section A fixed at one end and subject to a load Pat the other end. L EA du2

(u) = dxPu(L)2 dx0

Compute the first variation:

EA du du = 2 dxPu(L)

2

dx dx

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Integrate by parts

d du d du = EA u EA u dxPu(L)dx dx

dx dx L du Ld du

= u EA dx + EA u0

Pu(L)0 dx dx dx

Setting = 0, u / u(0) = 0:d du

EA = 0dx dx

du

P= EA

dx L

Extension to more dimensions

I= F(xi, ui, ui,j)dV V F FI= ui + ui,j dV

V ui ui,j

F F F

= ui + ui ui dVV ui xj ui,j

xj ui,j

Using divergence theorem:

F F F I= uidV+ uinjdS

V ui

xj ui,j Sui,j

Extremum of functional Iis obtained when I= 0, or when:

F F = 0 , and

ui

xj ui,j

ui = 0 on Su

Fui,j

nj onSSu = StThe boxed expressions constitute the Euler-Lagrange equations correspond-ing to the variational problem of finding an extremum of the functional I.

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