Numerical solution for the anisotropic Willmore flow of graphs |
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Institution: | 1. School of Mechanical Engineering, Shandong University of Technology, Zibo, Shandong 255000, PR China;2. Centre for Advanced Laser Manufacturing (CALM), Shandong University of Technology, Zibo, Shandong 255000, PR China;3. School of Mechanical Engineering, Shandong University, Jinan, Shandong 250061, PR China;4. Elementary education college, Zaozhuang College, Zaozhuang, Shandong 277160, PR China |
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Abstract: | The Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we prove an energy equality. We approximate the solution numerically by the complementary finite volume method. To show the stability, we re-formulate the resulting scheme in terms of the finite difference method. By using simple framework of the finite difference method (FDM) we show discrete version of the energy equality. The time discretization is done by the method of lines and the resulting system of ODEs is solved by the Runge–Kutta–Merson solver with adaptive integration step. We also show experimental order of convergence as well as results of the numerical experiments, both for several different anisotropies. |
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Keywords: | Anisotropy Willmore flow Curvature minimization Gradient flow Laplace–Beltrami operator Method of lines Complementary finite volume method Finite difference method |
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