As with 316 and 3113 inhomogeneous conditions add

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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 1 M F a r θ x 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 (right). 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 0 0 (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 y1 x1 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 we have 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 2 2 2 p 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 1 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. 1 3.2. Show that the function w 2 HE that minimizes I w] = ZZ F 2 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 (3.1.6) i...
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