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Unformatted text preview: ational statement.
Smooth solutions of the Galerkin problem satisfy the original partial di erential
equation(s) and natural boundary conditions, and conversely.
Galerkin problems arising from self-adjoint di erential equations also satisfy extremal problems. In this case, approximate solutions found by Galerkin's method
are best in the sense of (2.6.5), i.e., in the sense of minimizing the strain energy of
the error. 8 Multi-Dimensional Variational Principles Problems
1. Prove Theorem 3.1.1 and its Corollary.
2. Prove Theorem 3.1.2 and aslo show that smooth solutions of (3.1.13) satisfy the
di erential system (3.1.9) - (3.1.11).
3. Consider an in nite solid medium of material M containing an in nite number of
periodically spaced circular cylindrical bers made of material F . The bers are
arranged in a square array with centers two units apart in the x and y directions
(Figure 3.1.3). The radius of each ber is a (< 1). The aim of this problem is to
nd a Galerkin problem that can be used to determine the e ective conductivity
of the composite medium. Because of embedded symmetries, it su ces to solve a y
M F a
1 Figure 3.1.3: Composite medium consisting of a regular array of circular cylindrical bers
embedded in in a matrix (left). Quadrant of a Periodicity cell used to solve this problem
problem on one quarter of a periodicity cell as shown on the right of Figure 3.1.3.
The governing di erential equations and boundary conditions for the temperature 3.1. Galerkin's Method and Extremal Principles 9 (or potential, etc.) u(x y) within this quadrant are r (pru) = 0 (x
(x ux(0 y) = ux(1 y) = 0
u(x 0) = 0
u(x 1) = 1
u 2 C0
pur 2 C 0 y) 2 F M
y) 2 x2 + y2 = a2 : (3.1.14) The subscripts F and M are used to indicate the regions and properties of the ber
and matrix, respectively. Thus, letting
:= f(x y)j 0 x 1 0 y 1g
F := f(r )j 0 r a 0 =2g and := ; F :
The conductivity p of the ber and matrix will generally be di erent and, hence, p
will jump at r = a. If necessary, we can write
p(x y) = pF iif x2 + y2 < a2 :
M f x +y >a
Although the conductivities are discontinuous, the last boundary condition con rms
that the temperature u and ux pur are continuous at r = a.
M 3.1. Following the steps leading to (3.1.6), show that the Galerkin form of this
problem consists of determining u 2 HE as the solution of ZZ F p(uxvx + uy vy )dxdy = 0 8v 2 H01: M 1
De ne the spaces HE and H01 for this problem. The Galerkin problem appears
to be the same as it would for a homogeneous medium. There is no indication
of the continuity conditions at r = a.
3.2. Show that the function w 2 HE that minimizes I w] = ZZ F 2
p(wx + wy )dxdy M is the solution u of the Galerkin problem, and conversely. Again, there is little
evidence that the problem involves an inhomogeneous medium. 10 Multi-Dimensional Variational Principles 3.2 Function Spaces and Approximation
Let us try to formalize some of the considerations that were raised about the properties
of function spaces and their smoothness requirements. Consider a Galerkin problem in
the form of (3.1.6). Using Galerkin's method, we nd approximate solutions by solving
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- Spring '14