TY - JOUR
T1 - Three-dimensional shape optimization in stokes flow problems
AU - Zabarankin, Michael
AU - Molyboha, Anton
PY - 2010
Y1 - 2010
N2 - An integral equation constrained optimization approach to finding minimum-drag shapes for solid bodies of revolution in Stokes flows subject to constraints on body's volume and shape has been developed. An axially symmetric Stokes flow problem has been formulated in the form of an integral equation (state equation), and finding an optimal shape has been reduced to an integral equation constrained optimization problem. The total variation of the Lagrangian functional of the problem has been reduced to the variation only with respect to the shape by the adjoint equation-based method, and the steepest descent direction for the coefficients in the function series representing the shape has been found analytically. It has been shown that solutions to the state and adjoint integral equations are related by a simple algebraic formula, which eliminates the need for solving the adjoint equation. The approach has been illustrated in finding the minimum-drag shape for a solid unit volume particle and in finding the optimal shape for the fixed-volume nose of a solid torpedo-shaped body encountering minimum drag along its axis of revolution. In the later problem, the dependence of the nose's optimal shape on the torpedo's length has been investigated.
AB - An integral equation constrained optimization approach to finding minimum-drag shapes for solid bodies of revolution in Stokes flows subject to constraints on body's volume and shape has been developed. An axially symmetric Stokes flow problem has been formulated in the form of an integral equation (state equation), and finding an optimal shape has been reduced to an integral equation constrained optimization problem. The total variation of the Lagrangian functional of the problem has been reduced to the variation only with respect to the shape by the adjoint equation-based method, and the steepest descent direction for the coefficients in the function series representing the shape has been found analytically. It has been shown that solutions to the state and adjoint integral equations are related by a simple algebraic formula, which eliminates the need for solving the adjoint equation. The approach has been illustrated in finding the minimum-drag shape for a solid unit volume particle and in finding the optimal shape for the fixed-volume nose of a solid torpedo-shaped body encountering minimum drag along its axis of revolution. In the later problem, the dependence of the nose's optimal shape on the torpedo's length has been investigated.
KW - Adjoint equation-based method
KW - Drag minimization
KW - Generalized analytic functions
KW - Integral equation
KW - Optimal shape
KW - Steepest descent method
KW - Stokes flows
UR - http://www.scopus.com/inward/record.url?scp=77951195665&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=77951195665&partnerID=8YFLogxK
U2 - 10.1137/090751578
DO - 10.1137/090751578
M3 - Article
AN - SCOPUS:77951195665
SN - 0036-1399
VL - 70
SP - 1788
EP - 1809
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
IS - 6
ER -