Approximation of self-avoiding inextensible curves

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1 Approximation of self-avoiding inextensible curves Sören Bartels Department of Applied Mathematics University of Freiburg, Germany Joint with Philipp Reiter (U Duisburg-Essen) & Johannes Riege (U Freiburg) 8th Workshop on Numerical Methods for Evolution Equations Heraklion, Crete, September 23-24, / 15

2 Motivation Energy for large bending deformations of thin elastic sheets: I bend (y) = 1 II 2 dx f y dx, I = id 2 ω with fundamental forms I, II : ω R 2 2 associated with y : ω R 3 flat isometries: angle/length relations preserved nonlinear constraint implies linearity: II 2 = D 2 y 2 Kirchhoff s bending model derived by Friesecke, James & Müller 02 Nanotechnology simulation (B., Bonito & Nochetto 15): bilayer bending ω Problem: Large deformations ok but injectivity not guaranteed 2 / 15

3 Curves Simpler setting: inextensible curves describing elastic rods I bend (u) = 1 u (x) 2 dx, u (x) 2 = 1 2 curves parametrized by arc-length well posed minimization problems on H 2 (I; R 3 ) derivation from hyperelasticity: Müller & Mora 04 I I u EL eq s: stationary u H 2 (I; R 3 ) with u (x) 2 = 1 characterized by (u, v ) = u (x) v (x) dx = 0 for admissible v H 2 (I; R 3 ) with u v = 0 in I. I 3 / 15

4 Iteration B 12: Practical minimization via gradient flow ( t u, v) + (u, v ) = 0, u(0) = u 0, u (t, x) 2 = 1 for admissible v H 2 (I; R 3 ) with v u (t, ) = 0 Observation: relation t u u = 0 implies constraint preservation, u (t) 2 u (0) 2 = t 0 d ds u (s) 2 ds = 2 t 0 t u u ds = 0 Algorithm: given u n 1 compute d t u n with [d t u n ] [u n 1 ] = 0 and (d t u n, v) + ([u n 1 + τd t u n ], v ) = 0 for v with v [u n 1 ] = 0 and update u n = u n 1 + τd t u n Unconditional stability and controlled constraint violation: n [u n ] 2 = [u n 1 ] 2 +τ 2 [d t u n ] 2 =... = [u 0 ] 2 +τ 2 [d t u l ] 2 = 1+O(τ) l=1 4 / 15

5 Discretization Use H 2 conforming piecewise cubic C 1 curves DOFs: positions and tangents good approximation properties flexibility for boundary conditions u h (z) u (z) h Impose constraint/orthogonality at nodes z, i.e., u h(z) 2 = 1, d t u h(z) u h(z) = 0 constraint satisfied in limit h 0 if H 2 bounds available 5 / 15

6 Self-avoidance Maddocks & Gonzales 99: Injectivity enforced via potential TP(u) = 2 q 1 q r q( ) dx dy u(y), u(x) I I with r = radius of circle tangential at u(y) and intersecting in u(x) u(x) l(y) α r u(y) α u(x) u(y) u(y) r 0 as u(x) u(y) u(x) Using u (y) = 1 and sin α = u (y) [u(y) u(x)] / u(y) u(x) r ( u(y), u(x) ) = and 1/r u (x) /2 for x y u(y) u(x) 2 2 u (y) [u(y) u(x)] 6 / 15

7 Properties Tangent-point potential TP(u) = 1 q I I u (y) [u(y) u(x)] q u(y) u(x) 2q dx dy Sobolev Slobodeckii spaces defined via seminorm, 0 < s < 1, ( f (x) f (y) p ) 1/p [f ] W s,p (I;R 3 ) = x y 1+sp dx dy I I Class of embedded and arc-length parametrized curves: { } C q = u W 2 1/q,q (I; R 3 ) : u embedded and u (x) = 1 Basic properties (Blatt 13, Blatt & Reiter 15) for q > 2: u C q implies 1 bil(u) = Lip(u 1 ) < TP finite on C q and TP(u) M implies bil(u) C M,q TP is knot energy: TP(u k ) if pointw. lim u k C q 7 / 15

8 Variation Blatt & Reiter 15: TP is C 1 on embedded W 2 1/q,q curves with where M(u; v, w) = A(u; v, w) = with δtp(u) = M(u; u, ϕ) + M(u; ϕ, u) 2A(u; u, ϕ) I I I I Φ(x, y) (v (y) ( w(x) w(y) )) dx dy Ψ(x, y) ( v(x) v(y) ) (w(x) w(y) ) dx dy Φ(x, y) = u (y) ( u(x) u(y) ) q 2 u(x) u(y) 2q ( u (y) ( u(x) u(y) )), Ψ(x, y) = u (y) ( u(x) u(y) ) q u(x) u(y) 2q+2 Note: A(u;, ) positive semidefinite and M(u; u, u) 0 8 / 15

9 Approximation (A) Removal of singular diagonal u (y) [u(y) u(x)] q TP ε (u) = u(y) u(x) 2q dx dy leads to error TP(u) TP ε (u) (B) Use of quadrature TP ε,h (u) = x y ε { c δ,q ε 2 2/q δ bil(u) 2q u H 1 u H 2 x y ε c q ε bil(u) 2q u 2 L q u W 2,q Q h [ u (y) [u(y) u(x)] q u(y) u(x) 2q ] dx dy with node average on rectangles I j I k gives TPε (u) TP ε,h (u) { c q h 1/2 ε ( (q+1) u L ) u H 2 c q h ( u W 3, + 1 ) u W 3, 9 / 15

10 Proofs With S ε = {(x, y) I I : x y < ε} we have q ( TP(u) TP ε (u) ) u (y) ( u(y + z) u(y) zu (y) ) q S ε u(y + z) u(y) 2q dz dy 1 ( bil(u) 2q 0 u (y + ϑz) u (y) ) dϑ q z q S ε z 2q dz dy 1 u bil(u) 2q (y + ϑz) u (y) q S ε 0 z q dϑ dz dy 1 u bil(u) 2q ϑ q 1 (y + z) u (y) q 0 S ϑε z q dz dy dϑ u bil(u) 2q (y + z) u (y) q ( ) z q dz dy... For quadrature estimate show that integrand is in C 0,1/2 S ε 10 / 15

11 Gradient flow Weighted sum of bending energy and self-avoidance potential leads to gradient flow dynamics I tot (u) = I bend (u) + ϱtp ε (u) ( t u, v) = δi tot (u)[v] = B(u, v) 2ϱM sym ε (u; u, v) + 2ϱA ε (u; u, v) with 2M sym ε (u; v, w) = M ε (u; v, w) + M ε (u; w, v) Semi-implicit discretization (d t u n, v) + B(u n, v) + 2ϱM sym ε (u n 1 ; u n, v) = 2ϱA ε (u n 1 ; u n 1, v) constrained linear system of equations fully populated system matrices parameter ϱ defines length scale 11 / 15

12 Experiment I With I = [0, 2π] define planar circle cos(y) u(y) = sin(y). 0 Relative error quantities: δ h TP = TPε,h (I h u) TP(u) /TP(u), δ h M = Mε,h (I h u; I h u, I h u) M(u; u, u) /M(u; u, u), Parameters: q = 3, ε = 2h, uniform partitions h j δ j TP δ j M 2 1 / / / / / / 15

13 Experiment II With Ĩ = [0, 1] define trefoil knot (2 + cos(6πỹ)) cos(4πỹ) ũ(ỹ) = (2 + cos(6πỹ)) sin(4πỹ), sin(6πỹ) and discrete transformation gives arc-length parametrized u : I R 3 Movie TP eps,h (u h ) (M=20) -"- (M=40) -"- (M=80) I tot h (u h ) (M=20) -"- (M=40) -"- (M=80) Stationary states 1/h 20, 40, τ = h/4, ε = 2h, ϱ = h 1/ / 15

14 Experiment III With Ĩ = [0, 1] define planar spiral 20 cos(2π(2ỹ + 1))/(2π(2ỹ + 1)) u(ỹ) = 20 sin(2π(2ỹ + 1))/(2π(2ỹ + 1)) 0 and discrete transformation gives arc-length parametrized u : I R 3 Movie TP eps,h (u h ) (M=20) -"- (M=40) -"- (M=80) I tot h (u h ) (M=20) -"- (M=40) -"- (M=80) τ = h/4, ε = 2h, ϱ = h 1/ / 15

15 Summary Numerical scheme for self-avoiding elastic curves Error bounds for approximation and discretization Good experimental stability properties Future aspects Stability analysis for time-stepping Treatment of surfaces, efficiency More information 15 / 15