Abstract:Abstract. We discuss an algorithm for the numerical solution of the Obstacle Problem in which the coincidence set is considered as the prime unknown. Domain functionals are defined for which the coincidence set serves as the minimizing element. Their gradients are computed (in the sense of the material derivative), and the gradient descent method employed to minimize these functionals. Numerical example is given.
“…We find in each of the four examples considered in (A) to (D) that the correct free boundary radius η = R Γ is the unique minimizer of J τ [ η ]. This observation is a manifestation of the results proven in Bogomolny and Hou for general loadings and general shapes of domains, obstacles, and free boundaries…”
Section: Choosing a Shape Functionalsupporting
confidence: 72%
“…The qualitative features of J 0 and revealed with the aid of simple examples serves as a reminder that choosing a shape functional for computing the free boundary in the obstacle problem is a rather nontrivial task. The functional J τ that we adopt is proposed in Bogomolny and Hou . Denoting an admissible solution to the free boundary by γ and the corresponding membrane deflection by u γ , we consider …”
Section: Choosing a Shape Functionalmentioning
confidence: 99%
“…C γ and D γ are the coincidence and noncoincidence sets corresponding to γ . The rationale for choosing J τ comes from the analysis in the work of Bogomolny and Hou, where it is proven to have a unique minimizer at the exact solution provided that some sufficient conditions on the problem data hold. Evidently, J 0 coincides with J τ for the choice τ = 0.…”
Section: Choosing a Shape Functionalmentioning
confidence: 99%
“…In Section 2, we demonstrate using simple examples that seemingly natural choices for domain functionals may have multiple stationary points or an inflection at the correct solution and are therefore unsuitable in practice. Our choice for the domain functional, which we label as J τ , is based on the work in Bogomolny and Hou . They show that certain conditions on the problem data, which are not overly restrictive, suffice to guarantee that Γ is the unique minimizer of J τ .…”
Summary
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 Γ 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 Γ then u, and not u then Γ. 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.
“…We find in each of the four examples considered in (A) to (D) that the correct free boundary radius η = R Γ is the unique minimizer of J τ [ η ]. This observation is a manifestation of the results proven in Bogomolny and Hou for general loadings and general shapes of domains, obstacles, and free boundaries…”
Section: Choosing a Shape Functionalsupporting
confidence: 72%
“…The qualitative features of J 0 and revealed with the aid of simple examples serves as a reminder that choosing a shape functional for computing the free boundary in the obstacle problem is a rather nontrivial task. The functional J τ that we adopt is proposed in Bogomolny and Hou . Denoting an admissible solution to the free boundary by γ and the corresponding membrane deflection by u γ , we consider …”
Section: Choosing a Shape Functionalmentioning
confidence: 99%
“…C γ and D γ are the coincidence and noncoincidence sets corresponding to γ . The rationale for choosing J τ comes from the analysis in the work of Bogomolny and Hou, where it is proven to have a unique minimizer at the exact solution provided that some sufficient conditions on the problem data hold. Evidently, J 0 coincides with J τ for the choice τ = 0.…”
Section: Choosing a Shape Functionalmentioning
confidence: 99%
“…In Section 2, we demonstrate using simple examples that seemingly natural choices for domain functionals may have multiple stationary points or an inflection at the correct solution and are therefore unsuitable in practice. Our choice for the domain functional, which we label as J τ , is based on the work in Bogomolny and Hou . They show that certain conditions on the problem data, which are not overly restrictive, suffice to guarantee that Γ is the unique minimizer of J τ .…”
Summary
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 Γ 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 Γ then u, and not u then Γ. 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.
“…The shape optimization approach for investigation of obstacle problems is used in [30,71,186,198]. The following results generalize the work by A. Dervieux [71] (see also [168]), in which the outer obstacle problem is considered in view of E. Zehnder's local implicit function theorem for dimension n = 2.…”
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