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New advances in the study of generalized Willmore surfaces and flow. (English) Zbl 1345.35125

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 17th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 5–10, 2015. Sofia: Avangard Prima. 133-142 (2016).
Summary: We study a generalized Willmore flow for graphs and its numerical applications. First, we derive the time dependent equation which describes the geometric evolution of a generalized Willmore flow in the graph case. This equation is recast in divergence form as a coupled system of second-order nonlinear PDEs.
Furthermore, we study finite element numerical solutions for steady-state cases obtained with the help of the FEMuS (Finite Element Multiphysics Solver) library. We use automatic differentiation (AD) tools to compute the exact Jacobian of the coupled PDE system subject to Dirichlet boundary conditions.
For the entire collection see [Zbl 1330.53003].

MSC:

35R01 PDEs on manifolds
49Q20 Variational problems in a geometric measure-theoretic setting
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53Z05 Applications of differential geometry to physics
58J32 Boundary value problems on manifolds
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Software:

FEMuS
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Full Text: Euclid