Minimization of a Nonlinear Elasticity Functional Using Steepest Descent Page: 2
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McCabe, Terence W., Minimization of a Nonlinear
Elasticity Functional Using Steepest Descent. Doctor of
Philosophy (Mathematics), August, 1988, 76 pp.,
bibliography, 21 titles.
The method of steepest descent is used to minimize
typical functionals from elasticity. These functionals are
coercive and bounded below but are not defined everywhere on
an appropriate Sobolev space H®(fl) or product of such
spaces. For the case ft = [a,b], suppose w € H2(ft),
u € H2(ft) * H2(ft)HHq(0) w(a) < w(b), and u'+w' > 0 on [a,b].
The function u + w is called a deformation of the interval
[a,b]. The functional I is defined as follows:
I(u) - fb[[u'(x)+w'(x)]2 + * ldx
J a*- [u1 (x)+w* (x) | J
where W is the stored-energy, I(u) is the total
stored-energy and the potential energy is assumed to be
zero. We are interested in minimizing I while preserving
the boundary conditions of w. In the case ft = [a,bj and
s = 2, I is convex; but for bounded regions ft in IR2 with
appropriate smoothness on #ft and s ■ 3, the corresponding I
is not convex. The choice of s in both cases is made to
insure I is Frechet differentiable at many points in H2(ft)
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McCabe, Terence W. (Terence William). Minimization of a Nonlinear Elasticity Functional Using Steepest Descent, dissertation, August 1988; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc331296/m1/2/: accessed May 25, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; .