A shape optimization approach for simulating contact of elastic membranes with rigid obstacles

Sharma, Akriti and Rangarajan, Ramsharan (2019) A shape optimization approach for simulating contact of elastic membranes with rigid obstacles. In: INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, 117 (4). pp. 371-404.

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Abstract

The obstacle problem consists in computing equilibrium shapes of elastic membranes in contact with rigid obstacles. In addition to the displacement u of the membrane, the interface Gamma on the membrane demarcating the region in contact with the obstacle is also an unknown and plays the role of a free boundary. Numerical methods that simulate obstacle problems as variational inequalities share the unifying feature of first computing membrane displacements and then deducing the location of the free boundary a posteriori. We present a shape optimization-based approach here that inverts this paradigm by considering the free boundary to be the primary unknown and compute it as the minimizer of a certain shape functional using a gradient descent algorithm. In a nutshell, we compute Gamma then u, and not u then Gamma. Our approach proffers clear algorithmic advantages. Unilateral contact constraints on displacements, which render traditional approaches into expensive quadratic programs, appear only as Dirichlet boundary conditions along the free boundary. Displacements of the membrane need to be approximated only over the noncoincidence set, thereby rendering smaller discrete problems to be resolved. The issue of suboptimal convergence of finite element solutions stemming from the reduced regularity of displacements across the free boundary is naturally circumvented. Most importantly perhaps, our numerical experiments reveal that the free boundary can be approximated to within distances that are two orders of magnitude smaller than the mesh size used for spatial discretization. The success of the proposed algorithm relies on a confluence of factors- choosing a suitable shape functional, representing free boundary iterates with smooth implicit functions, an ansatz for the velocity of the free boundary that helps realize a gradient descent scheme and triangulating evolving domains with universal meshes. We discuss these aspects in detail and present numerous examples examining the performance of the algorithm.

Item Type:	Journal Article
Publication:	INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Publisher:	WILEY
Additional Information:	Copyright of this article belongs to WILEY
Keywords:	gradient descent; free boundary; obstacle problem; shape functional; universal meshes; variational inequalities
Department/Centre:	Division of Mechanical Sciences > Mechanical Engineering
Date Deposited:	29 Jan 2019 06:08
Last Modified:	29 Jan 2019 06:08
URI:	http://eprints.iisc.ac.in/id/eprint/61522

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