Consistent and Convergent Discretizations of Helfrich-Type Energies on General Meshes

Abstract We analyze discrete approximations of the second fundamental form on graphs of functions that are piecewise affine on irregular meshes. Being related with the Morley finite element, the approximation in this precise form has first been suggested in [E. Grinspun, Y. Gingold, J. Reisman and D. Zorin, Computing discrete shape operators on general meshes, Comput. Graphics Forum 25 2006, 3, 547–556]. We show how to use this framework to variationally approximate functionals of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>E</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>:=</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mi>M</m:mi> </m:msub> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mrow> <m:msub> <m:mi>n</m:mi> <m:mi>M</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>n</m:mi> <m:mi>M</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace="4.2pt" stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>d</m:mo> <m:msup> <m:mi mathvariant="script">ℋ</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> {E_{0}(M):=\int_{M}f(x,n_{M}(x),Dn_{M}(x))\,\mathrm{d}{\mathscr{H}}^{2}(x)} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>n</m:mi> <m:mi>M</m:mi> </m:msub> </m:math> {n_{M}} denotes the normal of the surface M . Here the integrand f is not necessarily quadratic. This corresponds to nonlinear Euler–Lagrange equations. Our approximation is rigorously formulated in the framework of Γ-convergence: We combine an ansatz-free asymptotic lower bound for any uniform approximation and a recovery sequence consisting of any regular triangulation of the limit sequence and an almost optimal choice of edge director. We give numerical examples showing the efficiency and accuracy of the algorithm in nonlinear problems.