# Goal-Oriented A-Posteriori Error Estimation and Adaptivity for Fluid-Structure Interaction:

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## Abstract

Numerical simulation of fluid-structure interaction generally requires vast computational resources. Paradoxically, the computational work is dominated by the complexity of the subsystem that is of least practical interest, viz. the fluid. The resolution of each of the many small-scale features in the fluid is prohibitively expensive. However, quantitative concern is generally restricted to the structural response. Goal-oriented adaptation strategies provide a paradigm to bypass this paradox. The crucial point is that in order to reliably determine the structural response, it is generally not necessary to fully resolve all the small-scale features in the fluid subsystem. Over-all computational cost can therefore be significantly reduced by resolving only those features that bear a pronounced influence on a specific structural output quantity of interest. In the context of fluid-structure interaction, meaningful output quantities are, for instance, the space-time average of the structural displacement and the nett energy that is transferred from the fluid to the structure. To enable the design of finite element meshes specifically tailored to the efficient computation of one such target quantity, an a posteriori error estimate must be obtained relating the error in the quantity of interest to local errors in the computed solution. This is accomplished by approximately solving an appropriate dual problem. The solution to this problem can be used to identify the sensitivity of the target-quantity error to local discretization residuals. Using this information, it is then possible to construct goal-oriented error indicators, providing the basis for an optimal-adaptive mesh refinement strategy, capable of delivering efficient and reliable control of the error in the target quantity to within a user-defined tolerance. In this thesis, we apply the existing framework for duality-based a posteriori error estimation and goaloriented adaptivity to a prototypical fluid-structure-interaction model problem. Specifically, we consider the two-dimensional panel problem concerning the aeroelastic response of a flexible panel with infinite aspect ratio immersed in an inviscid fluid flow. For general linear and nonlinear output quantities of interest, we formulate an appropriate dual problem and derive dual-weighted Type I a posteriori error estimates. The sharpness of these estimates is demonstrated through a series of numerical experiments for physically stable as well as physically unstable test conditions in both the subsonic and supersonic regime. On the basis of the derived a posteriori error estimates, we then design and implement an adaptive algorithm capable of producing space-time meshes specifically tailored to the efficient computation of a certain target quantity of interest. Numerical results are presented, highlighting the superiority of the proposed duality-based approach over a more traditional mesh refinement algorithm employing a residual-based indicator. Furthermore, comparisons between h- and hp-refinement strategies are made to illustrate the extra increase in efficiency, which can be gained from the use of hp-refinement techniques. See more