Adaptive Path Following Primal Dual Interior Point Methods for Shape Optimization of Linear and Nonlinear Stokes Flow Pr
We are concerned with structural optimization problems in CFD where the state variables are supposed to satisfy a linear or nonlinear Stokes system and the design variables are subject to bilateral pointwise constraints. Within a primal-dual setting, we s
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University of Houston, Department of Mathematics http://www.math.uh.edu/∼ rohop/ 2 University of Augsburg, Institute for Mathematics http://scicomp.math.uni-augsburg.de
Abstract. We are concerned with structural optimization problems in CFD where the state variables are supposed to satisfy a linear or nonlinear Stokes system and the design variables are subject to bilateral pointwise constraints. Within a primal-dual setting, we suggest an allat-once approach based on interior-point methods. The discretization is taken care of by Taylor-Hood elements with respect to a simplicial triangulation of the computational domain. The efficient numerical solution of the discretized problem relies on adaptive path-following techniques featuring a predictor-corrector scheme with inexact Newton solves of the KKT system by means of an iterative null-space approach. The performance of the suggested method is documented by several illustrative numerical examples.
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Introduction
Simplified problems in shape optimization have already been addressed by Bernoulli, Euler, Lagrange and Saint-Venant. However, it became its own discipline during the second half of the last century when the rapidly growing performance of computing platforms and the simultaneously achieved significant improvement of algorithmic tools enabled the appropriate treatment of complex problems (cf. [1,3,6,9,13,14,15] and the references therein). The design criteria in shape optimization are determined by a goal oriented operational behavior of the devices and systems under consideration and typically occur as nonlinear, often non convex, objective functionals which depend on the state variables describing the operational mode and the design variables determining the shape. The state variables often satisfy partial differential equations or systems thereof representing the underlying physical laws. Technological aspects are taken into account by constraints on the state and/or design variables which may occur both as equality and inequality constraints in the model. Shape optimization problems associated with fluid flow problems play an important role in a wide variety of engineering applications [13]. A typical setting is the design of the geometry of the container of the fluid, e.g., a channel, a I. Lirkov, S. Margenov, and J. Wa´ sniewski (Eds.): LSSC 2007, LNCS 4818, pp. 259–266, 2008. c Springer-Verlag Berlin Heidelberg 2008
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R.H.W. Hoppe, C. Linsenmann, and H. Antil
reservoir, or a network of channels and reservoirs such that a desired flow velocity and/or pressure profile is achieved. The solution of the problem amounts to the minimization of an objective functional that depends on the state variables (velocity, pressure) and on the design variables which determine the geometry of the fluid filled domain. The state variables are supposed to satisfy the underlying fluid mechanical equations, and there are typically constraints on the design variables which restrict the shape of the fluid filled domain to that what is technologically feasible. The typical approa
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