By Chia-Shung Yih
This hugely acclaimed sequence presents survey articles at the current kingdom and destiny course of analysis in vital branches of utilized mechanics
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Additional info for Advances in Applied Mechanics, Vol. 12
We shall show that in the case of two-dimensional flow or axial symmetric flow, a different finite element approach can be used. 48) where m = 0 and 1, respectively, for two-dimensional flow and axial symmetric flow. 46) is satisfied identically. 50) Theodore H . H . Pian and Pin Tong 44 and x1 is the axis of symmetry in the case of axisymmetric flow. 51) = -X2mvU, on aA, where v v and vs are the normal and tangential components of v, A is the cross section area of the domain considered, and aA is the boundary curve of A.
26) A finite element solution can be constructed for this version by using the variational statement ”! 27) for arbitrary ST and ah,. T h e generalized coordinates and the interpolation functions for both T and h, can be assumed independently. T h e only restriction is that either T or h , must be continuous over the entire domain in order t o have the integral defined. 5) must also be satisfied if a2 = 0 on aV. Some examples of the one-dimensional problem, assuming T to be constant and hi to be linear within each element, are given by Thompson and Chen (1970).
However, 7rm, becomes a maximum with respect to u, when ~ 7 are , ~ expressed in terms of u , . In the finite element solution, the increase of the number of the stress parameters /3 will tend to decrease the value of r m c , while the increase of the number of the boundary displacement parameters Thus there exists an optimum relation between will tend to increase rm,. the number of Is’s and 9’s. In practice this optimum is not known. However, 20 Theodore H . H . Pian and Pin Tong it is advisable to choose the stress variables and the displacement variable so that they maintain the same order of approximation (Pian and Tong, 1969a).