Efficient Methods for Topology Optimisation of Fluid flow
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We consider topology optimisation of two-dimensional Stokes flow. The objectiveis to distribute a certain amount of solid material in a given domain, such thatthe total power dissipation is minimised. A generalised Stokes problem acts asthe governing PDE, and both existence of solutions to the state equations and theoptimisation problem is shown. The optimisation problem is solved using the optimalitycriteria method, and the MINRES method is used for solving the algebraiclinear system arising from discretising the governing PDE with the finite elementmethod. Residual estimates are used in order to prematurely stop MINRES, andthe results indicate that the residual related to the momentum part of the Stokesequations is sufficient for formulating a meaningful stopping criterion. Throughtesting the algorithm on several different numerical examples, we propose a tolerancesuch that the total number of MINRES iterations is low, while largest observederror in the resulting objective value is 3.5% compared to when the linear system issolved exactly. Adaptive mesh refinement based on the elementwise residual estimatesis performed during the course of the optimisation, and it was found that thisapproach yields a significant reduction in the residuals when compared to startingwith a fine mesh.