By Marnaghan F. D., Wintner A.

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I f z is the solution of Equation 17, which depends on the shape of ~ ( t ) , then z depends on t z = z(x, t ) . both through the position x = X + tV(X) and e x p l i c i t l y ; i . e . , Under certain regularity hypothesis on ~ and the vector f i e l d V(X) [7,8], one can define ~(x) ~ lim [z(X+tV~- z(X)] t+o = z'(X) + vz(X) (26) • A(X) where ~ is the material derivative and z' is the partial derivative, defined as lim [zlX, t,,),,T,,z(X, 0)] z'(X) ~ t-~O t (27) I f z ~ H l ( ~ ) , with smoothness assumptions on the domain and velocity f i e l d V(X) [7], then z ' E Hl(fl) [9 ],and z#H1(fl) F7, 8].

27 13. K. J. Haug, Optimization of Structures with Repeated Eigenvalues, Ibid, p. 219-277. 14. N. Olhoff and J. Taylor, Designing Constinuous Columns for Minimal Cost of Material and of Interior Supports, J. of Structural Mechanics, Vol. 6, (1978), p. 367-382. IS. Z. N. Rozvany, Optimal Design of Structures with Variable Support Conditions, J. Optimization Theory and Applications, Vol. 15, #i, (1975), p. 85-i01. 16. F. Masur and Z. Mroz, Singular Solutions in Structural Optimization Problems, Proceedings IUTAM Sumposium.

70] S. Drobot and A. Rybarski, A variational principle in hydrodynamics, Arch. Rational Mech. Anal. 2, No. 5 (1958), 393-410. K. Knowles and E. Sternberg, On a class of conservation laws in linearized, and f i n i t e e l a s t i c i t y , Arch. Rational Mech. Anal. 44 (1972), 187. [72] B. VujanoviE, Int. J. Non-linear Mech. 13~(1978), 185-197. SHAPE OPTIMIZATION OF ELASTIC BARS IN TORSION Jean W. Hou, Edward J. Haug, and Robert L. Benedict Center for Computer Aided Design College of Engineering The University of Iowa Iowa City, Iowa 52243 ABSTRACT The problem of shape optimal design for multiply-connected elastic bars in torsion is formulated and solved numerically.

### A Canonical Form for Real Matrices under Orthogonal Transformations by Marnaghan F. D., Wintner A.

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