BILAYER PLATES: MODEL REDUCTION, Γ-CONVERGENT FINITE ELEMENT APPROXIMATION AND DISCRETE GRADIENT FLOW

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1 BILAYER PLATES: MODEL REDUCTION, Γ-CONVERGENT FINITE ELEMENT APPROXIMATION AND DISCRETE GRADIENT FLOW SÖREN BARTELS, ANDREA BONITO, AND RICARDO H. NOCHETTO Abstract. Te bending of bilayer plates is a mecanism wic allows for large deformations via small externally induced lattice mismatces of te underlying materials. Its matematical modeling, discussed erein, consists of a nonlinear fourt order problem wit a pointwise isometry constraint. A discretization based on Kircoff quadrilaterals is devised and its Γ-convergence is proved. An iterative metod tat decreases te energy is proposed and its convergence to stationary configurations is investigated. Its performance, as well as reduced model capabilities, are explored via several insigtful numerical experiments involving large (geometrically nonlinear) deformations. 1. Introduction Te derivation of dimensionally reduced matematical models and teir numerical treatment is a classical and callenging scientific branc witin solid mecanics. Various models for describing te bending or membrane beavior of plates are available, eiter as linear models for te description of small displacements or nonlinear models wen large deformations are considered; see [11]. Te development of related numerical metods as mostly been concerned wit te treatment of second order derivatives and avoiding various locking effects. Te rigorous derivation of te geometrically nonlinear Kircoff model for te description of large bending deformations of plates from treedimensional yperelasticity in [14] as inspired various furter results, e.g., discussing oter energy regimes in [15], or te derivation of effective teories for prestressed multilayer materials in [21]. Bilayer plates consist of two films of different materials glued on top of eac oter. Te materials react differently to termal or electric stimuli, tereby canging teir molecular lattices. Tis mismatc allows for te development of large deformations by simple eat or electric actuation. Classical applications of tis effect include bimetal strips in termostats, wile modern applications use termally and electrically induced bending effect to produce nanorolls, microgrippers, and nano-tubes; see [5, 18, 19, 22, 23]. Preventing undesirable effects suc as dog-ears formation, as reported in [1, 24], motivates te matematical prediction of bilayer bending patterns via numerical simulation. Tis requires aving a model as simple as possible to be ameanable to numerical treatment and analysis, but sufficiently sopisticated to capture essential nonlinear geometric features associated wit large bending deformations. A two-dimensional matematical model for te bending beavior of bilayers as been rigorously derived from tree-dimensional yperelasticity in [21]. It consists of a nonconvex minimization problem wit nonlinear pointwise constraint. Te energy functional involves second order derivatives of deformations associated wit te second fundamental form of te mid-surface. Te pointwise Date: October 16, Matematics Subject Classification. 65N12, 65N30, 74K20. Key words and prases. Nonlinear elasticity, bilayer bending, finite element metod, iterative solution, Γ- convergence. Partially supported by NSF Grant DMS and AFOSR Grant FA Partially supported by NSF Grants DMS and DMS

2 constraint enforces deformations to be isometries, i.e., tat lengt and angle relations remain uncanged by te deformation as in te case of te bending of a piece of paper. A related numerical metod as been devised and analyzed for single layer plates in [3, 4]. It is our goal to develop a reliable numerical metod for te practical computation of large bilayer bending deformations. Our contributions in tis paper are: To present a formal dimension reduction model allowing for various effects not covered in te corresponding rigorous analysis in [20]; To propose a discretization of te matematical model and prove its Γ-convergence as discretization parameters tend to zero; To construct a gradient flow metod to compute stationary configurations; To carry out several numerical experiments to illustrate te performance of our numerical metod and explore te nonlinear geometric effects captured by te matematical model. In te remainder of tis introduction we discuss te matematical model and our main ideas to deal wit te ensuing strong nonlinearities. Description of bilayer plates. We consider a geometrically nonlinear Kircoff plate model tat allows for bending but not for stretcing or sear. Tis selection is related to te coice of a particular energy scaling, namely tat te elastic energy is proportional to te tird power of te plate tickness t. Given a domain R 2 tat describes te middle surface of te plate, te model formally derived in Section 2 consists of minimizing te dimensionally reduced elastic energy (1.1) E[y] = 1 H + Z 2 f y 2 witin te set of isometries y : R 3, i.e., mappings satisfying te identities (1.2) [ y] T y = I 2 i y j y = δ ij, i, j = 1, 2, in and wit prescribed values y = y D and y = Φ D on te Diriclet portion D of te boundary. Hereafter, I d denotes te identity matrix in R d for d = 2, 3, and H stands for te second fundamental form of te surface γ = y() parametrized by y wit unit normal ν, H i,j = ν i j y, ν = 1 y 2 y. Te symmetric matrix Z is given and can be viewed as a spontaneous curvature so tat in te absence of body forces f te plate is already pre-stressed. Given te identity for isometries H 2 = D 2 y 2, we rewrite te energy E[y] in (1.1) as follows: (1.3) Ẽ[y] := 1 2 ( D 2 y 2 1 y + i j y 2 1 y ) 2y Z ij y 2 i,j=1 Z 2 f y. It is tempting to simplify tis expression furter because i y = 1, i = 1, 2, for isometries. However, we refrain from doing so for stability purposes anticipating tat te isometry constraint will be later relaxed numerically. In particular, te normalization will enable us to prove various bounds for te variational derivative of te energy. In fact, notice tat te energy Ẽ[y] is finite for y H 2 () 3 W () 1 3 suc tat i y 1, i = 1, 2, as well as f L 2 () 3 and Z L 2 () 2 2. Te condition i y 1, i = 1, 2, will play a crucial role trougout tis paper. We encode boundary conditions and te isometry constraint in te set of admissible deformations (1.4) A := { y H 2 () 3 : y D = y D, y D = Φ D, [ y] T y = I 2 a.e. in } and define te tangent space of A at a point y A via (1.5) F[y] := { w H 2 () 3 : w D = 0, w D = 0, [ w] T y + [ y] T w = 0 a.e. in }. 2

3 Note tat it is always possible to extend te functional Ẽ to H1 () 3 as follows: Ẽ[y] := +, y H 1 () 3 \ A. Minimizing movements. To find stationary points of Ẽ in A, we propose a gradient flow in H2 (), i.e. according to te H 2 -scalar product: if s (0, ) is a pseudo-time, we formally seek a family y L 2 (0, ; A) wit s y(s) F[y(s)] for s (0, ) and D 2( s y(s) ) : D 2 w = δẽ[y(s), w] w F[y(s)]. Te expression δẽ[y, w] stands for te variational derivative of Ẽ at y A in te direction w F[y(s)], wic we make explicit below. We note tat upon taking w = s y(s) F[y(s)], we obtain formally d dsẽ[y(s)] = D 2 ( s y(s) ) 2 0; te energy tus decreases along trajectories. Te formal gradient flow is igly nonlinear and requires an appropriate interpretation. We adopt te concept of minimizing movements and consider an implicit first order time-discretization of te H 2 -gradient flow via successive minimization of y 1 2τ D2 (y y k ) 2 L 2 () + Ẽ[y] in te set of all y A to determine y k+1. Our motivation for te use of te H 2 metric to define te gradient flow is treefold. First, it simplifies te implementation since it leads to te same system matrix as te main part of te bilinear form associated wit te bending energy (1.3). Second, it is sufficiently strong to provide control over discrete time derivatives and discretization errors, suc as te isometry constraint, witout imposing severe step size restrictions. Tird, it may be regarded as a damping term modeling te deceleration of a bilayer witin a viscous fluid, wic in turn gives some pysical interpretation. Since te nonlinear isometry constraint is treated exactly, tis nonconvex minimization problem is of limited practical value. We tus propose instead a linearization of te isometry constraint wic yields a practical sceme. Algoritm 1 (minimizing movement). Let τ > 0 be te time-step size and set k = 0. Coose y 0 A. Compute v k+1 F[y k ] wic is minimal for te functional v τ 2 D2 v 2 L 2 () + Ẽ[yk + τv], set y k+1 = y k + τv k+1, increase k k + 1 and repeat. Te linearized isometry condition included in te set F[y k ] implies tat for every admissible vector field v F[y k ] we ave tat te constraint residual of te corresponding update y k + τv satisfies [ (y k + τv)] [ (y k + τv)] I 2 = [ y k ] [ y k ] I 2 + τ 2 [ v] [ v] [ y k ] [ y k ] I 2, were te inequality A B for square symmetric matrices A, B means tat A B is semi-positive definite; in particular te diagonal entries of A and B satisfy a ii b ii for all i. Applying tis formula inductively wit [ y 0 ] [ y 0 ] = I 2, we see tat [ y k+1 ] y k+1 I 2 wence j y k+1 1, j = 1, 2. Terefore, Ẽ[y k+1 ] is well defined and te minimization problem in Algoritm 1 admits a minimizer. Moreover, te space F[y k+1 ] is well defined, even toug y k+1 may not belong to A. Hence, te iteration of Algoritm 1 can be repeated. Every iteration of te algoritm requires computing a solution v k+1 F[y k ] of te following Euler Lagrange equation τ D 2 v k+1 : D 2 w + δẽ[yk + τv k+1, w] = 0 w F[y k ]. 3

4 In terms of te new iterate y k+1 = y k +τv k+1 tis is equivalent to te nonlinear system of equations 1 D 2 (y k+1 y k ) : D 2 w + D 2 y k+1 : D 2 w τ 2 ( 1 y k+1 + i j w 1 y k+1 2y k+1 ) 2 y k+1 Z ij (1.6) + + i,j=1 2 i,j=1 2 i,j=1 {[ i j y k+1 1 w 1 y k+1 1y k+1 ( 1 y k+1 1 w) ] 1 y 3 2y k+1 } 2 y k+1 Z ij { i j y k+1 1 y k+1 [ 1 y k+1 2 w 2 y k+1 2y k+1 ( 2 y k+1 2 w) ]} 2 y k+1 3 Z ij = f w, for all w F[y k ]. We sow in tis paper ow to discretize tis system in space and present an iterative algoritm for its approximation. We study tese algoritms and employ tem to compute several equilibrium configurations. Outline of te paper. Te paper is organized as follows. In 2 we discuss a formal derivation of (1.1) from tree dimensional yperelasticity, following a suggestion of S. Conti. In 3 we introduce Kircoff quadrilaterals, wic is a nonconforming finite element specially taylored for tis application. It is an extension of te Kircoff triangles [8, 3], and possesses te degrees of freedom for te function and its gradient at te vertices of te underlying partition T ; tis facilitates imposing te isometry constraint at te vertices. It turns out tat te space of deformations as well as tat of discrete gradients are subspaces of H 1 (), wic is extremely convenient to discretize (1.6). We next introduce space discretizations Ẽ of Ẽ and derive in 4 teir Γ-convergence to Ẽ in H1 (). As a consequence, we deduce convergence properties of discrete almost absolute minimizers. Ten, we study a practical iterative algoritm for te solution of te space discretization of (1.6). We sow convergence of suc an iterative sceme, tereby proving existence and uniqueness of our fully discrete problem. We also prove several important properties of tis gradient flow, suc as a precise control of te deviation from te isometry constraint (1.2). We conclude in 6 wit insigtful numerical experiments. In fact, we compute several configurations (suc as cylinders, dog ears, corkscrews) tat are observed in experiments wit micro- and nano-devices [1, 24]. We empasize tat some of tese effects, e.g. corkscrew sapes, are obtained wit anisotropic spontaneous curvatures Z and are terefore outside te framework developed and analyzed in [20]. 2. Dimension Reduction: Bilayer Plate Model We consider a plate t := ( t/2, t/2) R 3 of tickness t > 0 and wose middle surface is given by R 2 as illustrated in Figure 1 (left). Te plate is clamped on te left edge D and free on te rest of te boundary, and its lengt perpendicular to D is L. Te upper and lower layers are composed of materials wit different molecular lattices, for instance differing by a factor δ t > 0; tis could be acieved by termal or electric actuation in practice. To understand te natural scaling between t and δ t, we assume tat te middle surface of te upper layer contracts to lengt L(1 δ t ) wereas te middle surface of te lower layer expands to lengt L(1 + δ t ), so tat te plate t bends upwards as in Figure 1. Due to te clamped boundary condition along one side we imagine tat for small deformations te lower and upper middle surfaces are given by R ± = θ 1 L(1 ± δ t ), as depicted in Figure 1 (rigt). Since we aim at capturing bending effects in te limit t 0, we impose te condition lim t = lim = θ R + t 0 R L = κ, 4

5 κ > 0 being te curvature. Since t = R + R = κ 1( (1 + δ t ) (1 δ t ) ), we deduce t (2.1) lim = 2 t 0 δ t κ. It is tus natural to impose a linear scaling between t and δ t wic involves te curvature tat is expected in te limit of vanising tickness for a pure bending problem. θr = L(1 δ t ) D L(1 + δ t ) = θr + t L y x θ R R + = R + t Figure 1. Two layers of tickness t/2 are stacked on eac oter and form te bilayer plate. Te undeformed middle surface is denoted by and deforms into te surface γ. We consider te energy density W : R 3 3 t R (2.2) W (F, x) = 1 F F (I 3 ± δ t N(x )) 4 (I 3 ± δ t N(x )) 2 ± x 3 > 0, were x = (x, x 3 ), I 3 denotes te identity matrix in R 3 3, and A stands for te norm associated to te Frobenius scalar product A : B := d i,j=1 A ijb ij = tr (A T B). Hereafter δ t is a parameter only depending on te tickness t, describing te lattice mismatc of te two layers and satisfying δ t t, wereas N : R 3 3 is a symmetric matrix wic encodes inomogeneities (dependence on x ) and anisotropy (rectangular molecular lattice rater tan cubic and preferred directions) of te underlying materials. Togeter δ t and N(x ) describe te pre-stressed bilayer {x t : ±x 3 > 0}. Wen δ t = 0 te two materials composing te bilayers reduce to one, te reference configuration is stationary in te absence of a force f t and tus stress-free, and te energy density becomes (2.3) W (F ) = 1 F T 2 F I 3. 4 Tis function is asymptotically equivalent to te simplest energy density tat obeys te principles of frame indifference and isotropy, namely W (F ) = W (QF R) for all Q, R SO(3) and, see [15], (2.4) W (F ) dist 2 ( F, SO(3) ), were dist is given by te Frobenius metric. To see te relation between (2.3) and (2.4) we argue as follows. Let F be close to SO(3), wic is to say F = F 0 +ɛf 1 wit F 0 SO(3) and F 1 perpendicular to te tangent space T F0 SO(3) to SO(3) at F 0 and ɛ 1; we tus deduce dist 2 (F, SO(3)) = ɛ 2 F 1 2. Te space T F0 SO(3) can be written as T F0 SO(3) = { Z : F T 0 Z + Z T F 0 = 0 } ; tis follows by differentiation of te condition F T 0 F 0 = I. Consequently, te normal space N F0 SO(3) to T F0 SO(3) is N F0 SO(3) = { Y : F T 0 Y Y T F 0 = 0 }, as can be easily seen because T F0 SO(3) N F0 SO(3) = R 3 3 and Z : Y = tr (Z T Y ) = tr ( (Z T F 0 )(F T 0 Y ) ) = tr ( (Y T F 0 )(F T 0 Z) ) = tr (Y T Z) = Z : Y, 5

6 wence Z : Y = 0 and te subspaces are ortogonal. Since F 1 N F0 SO(3) we infer tat F T F I 3 2 = (F 0 + ɛf 1 ) T (F 0 + ɛf 1 ) I 3 2 = 2ɛF T 1 F 0 + ɛ 2 F T 1 F 1 2 = 4ɛ 2 F o(ɛ 2 ), wic sows te asserted relation between (2.3) and (2.4) for small ɛ. We are interested in te bending regime of te bilayer, wic corresponds to energies comparable to te tird power of te plate tickness, cf. [15, 21]. To a deformation u : t R 3 of te plate we tus associate te scaled yperelastic energy ( ) (2.5) I t [u] = t 3 W ( u, x) f t u dx. 12 M 22 t Te function f t is a body force, wereas te energy density W is written in (2.2) and reads W (F, x) = 1 F F M 2, 4 wit symmetric matrices M = M(x), N = N(x) R 3 3 given by [ ] [ ] M11 M 12 M := := I M 3 ± 2δ t N + δt 2 N 2 N11 m, N:=. m n Here and in wat follows alternating signs correspond to te upper and lower layers in wic we ave x 3 > 0 and x 3 < 0, respectively, i.e., ±x 3 > 0. Moreover, N 11 R 2 2 is symmetric, m R 2, n R is constant (to keep te formal discussion simple), and M 11 = I 2 ± 2δ t N 11 + δ 2 t (N mm T ), M 12 = ±2δ t m + δ 2 t (N 11 m + nm), M 22 = 1 ± 2δ t n + δ 2 t (n 2 + m 2 ). To derive a dimensionally reduced model we assume tat te actual deformation u of te plate, subject to boundary conditions and outer forces, as te form u(x, x 3 ) = y(x ) + x 3 b(x ) wit a vector field b : R 3 tat is perpendicular to te surface γ parametrized by y, i.e., we ave i y b = 0 for i = 1, 2. In oter words, fibers ortogonal to te middle surface in te reference configuration remain normal to γ and deform linearly. Tis special form of u is consistent wit [14, 21] for energy densities wit a vanising bulk modulus. In general, a more general expansion including quadratic terms in x 3 as to be used. Wit tis ansatz we find tat u = [ i u] 3 i=1 R3 3 can be written as u = [ y, b] + x 3 [ b, 0], were stands for te gradient wit respect to x, and deduce tat I t [u] = 1 ( 1 ( u) t 3 u M ) 2 f t u dx = 1 { [ ] 1 ( y) ( y) M 11 M 12 t 4 t 3 4 M12 T b 2 M 22 + x 3 [ ( b) y + ( y) b ( b) b ( b) 0 t ] [ ] } b + x 2 ( b) b f 0 0 t u dx. In order for tis integral to be bounded as t 0 we need tat te term b 2 M 22 be at least of order t 2. Since we ave δ t t, tis is guaranteed if we enforce tat b 2 (1 ± 2δ t n) δ 2 t (n 2 + m 2 ) = b 2 (1 ± δ t n) 2 δ 2 t m 2 = δ 2 t m 2, i.e., we impose te constraint b = 1 ± δ t n ± x 3 > 0. 6

7 Since b(x ) = β(x )ν(x ), were ν(x ) := 1y(x ) 2 y(x ) 1 y(x ) 2 y(x ) is te unit normal to te surface γ at y(x ), we obtain tat β(x ) = 1 ± δ t n wic is for simplicity assumed to be independent of x. Tis in turn implies tat ( b) b = 0, b = (1 ± δ t n) ν. Recalling tat te first and second fundamental forms of γ are given by (2.6) G = ( y) and introducing we infer tat G t := t 1( ( y) K t := ( b) I t [u] = 1 t 3 t y, H = ( ν) y, ( y) M 11 ) = t 1 ( G I 2 2δ t N 11 δ 2 t (N mm b = (1 ± δ t n) 2 ( ν) { 1 4 ν, [ tgt 2x 3 (1 ± δ t n)h + x 2 3 K ] } t M 12 2 M12 T δt 2 m 2 f t u dx. Retaining only te terms of order δt 2 or lower, because te iger order terms vanis in te limit t 0, we obtain I t [u] 1 { 1 ( t 3 t 2 G t 2 + 4x 2 t 4 3(1 ± δ t n) 2 H 2 + x 4 3 K t 2 4tx 3 (1 ± δ t n)g t : H + 2tx 2 3G t : K t 4x 3 3(1 ± δ t n)h : K t + 8δt 2 m 2) } f t u dx. To ensure tat te first term in te integral remains bounded as t 0 we require tat G = I 2, tat is te parametrization y of γ is an isometry. Consequently, we obtain G t = 2t 1 δ t N 11 t 1 δ 2 t P, P := N mm Since te quantities N 11, P, H, K, m are independent of x 3, we carry out te integration over x 3 ( t/2, t/2) and deduce tat t/2 t/2 t/2 t/2 t/2 t/2 t/2 t 2 G t 2 dx 3 = 4tδ 2 t N tδ 4 t P 2, 4x 2 3(1 ± δ t n) 2 H 2 dx 3 = t3 3 (1 + δ2 t n 2 ) H 2, t/2 t/2 x 4 3 K 2 dx 3 = t5 80 K 2, 4tx 3 (1 ± δ t n)g t : H dx 3 = 2t 2 δ t N 11 : H + t 2 δ 3 t np : H, t/2 t/2 t/2 2tx 2 3G t : K dx 3 = t3 δ 2 t 6 P : K, 4x 3 3(1 ± δ t n)h : K dx 3 = t4 δ t nh : K, 8 t/2 t/2 8δ 2 t m 2 dx 3 = 8tδ 2 t m ) ),

8 It remains to examine te forcing term f t, wic we assume to be of te form to give a nontrivial limit. In fact, if we let f(x ) := 1 t t/2 t/2 f t (x, x 3 ) = t 2ˆf(x, x 3 ) f(x, x 3 ) dx 3, g(x ) := 1 t t/2 t/2 x 3 f(x, x 3 ) dx 3 ten te contribution to te energy due to te body force becomes 1 ( ) t 3 f t u dx = f(x ) y(x ) + g(x ) b(x ) dx. t Inserting tese expressions back into I t [u], setting δ t λ := lim t 0 t R and keeping only terms of order one in t, we readily obtain lim I t[u] 1 ( ) ( H 2 + 6λN 11 : H dx + λ 2 N m 2) dx t 0 12 If we furter denote (2.7) Z := 3λN 11 f y dx. and ignore te second integral, wic is constant and so independent of te surface γ, we see tat te dimensionally reduced model is governed by te energy E[y] = 1 H + Z 2 dx f y dx, 12 were te parametrization y : R 3 of te surface γ is an isometry, namely it satisfies (1.2). We remark tat te derivation of te dimensionally reduced model can be carried out rigorously in te sense of Γ-convergence for a large class of isotropic energy densities [21]. Te only required assumptions are te cubic energy scaling (2.5) and te proportionality δ t t of (2.1). Te quantity Z acts as a spontaneous curvature for te bending energy E[y] and specifies properties of te bilayer material. If te material is omogeneous and isotropic, ten Z = αi 2 wit α R; we refer to [20] for a discussion of te qualitative properties of minimizers. On te oter and, te material could possess inomogeneities and anisotropies wic are x -dependent and are encoded in N 11 (x ); we discuss some options togeter wit numerical experiments in 6. We observe tat te components n and m of N play no role in te reduced energy. We assume tat te plate is subject to clamped boundary conditions on a portion D of y = y D, y = Φ D on D, were y D : R 3, Φ D : R 3 2 are sufficiently smoot, and Φ D = y D is an isometry in, i.e. Φ D (x ) T Φ D (x ) = I 2 for x. Te variational formulation of te reduced plate model consists of finding y A, defined in (1.4), suc tat (2.8) E[y] = 1 H + Z 2 dx f y dx. 2 is minimized, were H is te second fundamental form defined in (2.6) and Z te spontaneous curvature of (2.7). Te new scaling 1 2 is immaterial and just set for convenience. Existence of solutions of te constrained minimization problem is a consequence of te direct metod in te calculus of variations. 8

9 3. Kircoff Elements on Quadrilaterals Te fourt order nature of (2.8) and te pointwise constraint (1.2) on gradients in te bilayer bending problem reveal tat a careful coice of finite element spaces for spatial discretization is mandatory. To avoid C 1 -elements, wic are natural in H 2 but difficult to implement, we employ a nonconforming metod tat introduces a discrete gradient operator and wic allows us to impose te constraint (1.2) at te vertices of elements. Te components of te discrete deformation y belong to an H 1 conforming finite element space W and its discrete gradients to anoter H 1 conforming finite element space G. Te degrees of freedom of our numerical metod are te deformations and te deformation gradients at te nodes of te partition T of into rectangles wic are te vertices of elements. Definition 3.1. For a conforming partition T of into sape-regular, closed rectangles wit vertices N and edges E we define te midpoints of elements and edges, te diameters of elements, and te maximal messize by z T := 1 z, z E := 1 z, T := diam(t ), = max T 4 2 T T z N T z N E for all T T and all E E. For every E E we let n E, t E R 2 be unit vectors suc tat n E is normal to E and t E is tangent to E. We denote by z 1 E, z2 E N E te end-points of E so tat E = conv{z 1 E, z2 E }. Te following definition modifies te well known Kircoff triangles [6, 8] to quadrilaterals and is related to [7]. We let Q r (T ) and P r (T ) denote te set of polynomials on T of partial degree r on eac variable and of total degree r, respectively. Definition 3.2. Let T = {T } be a partition of R 2 into rectangles as in Defintion 3.1. (i) Discrete spaces: Define W := { w C() : w T Q 3 (T ) T T, w continuous in N, w (z E ) n E = 1 2( w (z 1 E) + w (z 2 E) ) n E E E, }, G := { ψ C() 2 : ψ T Q 2 (T ) 2 T T }. (ii) Interpolation operator: Let I 2 : H2 () 2 G be defined by I 2 ψ(z) = ψ(z) for all z N, I 2 ψ(z E) = ψ(z E ) for all E E, I 2 ψ(z T ) = 1 4 ψ(z) for all T T. z N T Te operator I 2 is also well-defined for discrete vector fields ψ W. (iii) Discrete gradients: Define : H 3 () G by w := I 2 [ w ]. Te operator is also well-defined for discrete functions w W. Since te set of vertices, midpoints of edges, and element midpoint is unisolvent for te polynomial space Q 2 (T ), te interpolation operator I 2 is well-defined. However, I2 differs from te canonical nodal interpolation operator because of te condition at te element midpoints. Te latter is imposed for practical purposes but makes I 2 inexact over Q 2(T ). Neverteless, I 2 is exact over Q 1 (T ) so tat te Bramble-Hilbert lemma implies (3.1) ψ I 2 ψ L p (T ) + T ψ I 2 ψ L p (T ) c 2 2 T D 2 ψ L p (T ) 9

10 for all ψ Wp 2 (T ) 2 and 2 p. Te operator I 2 is also well-defined on W since for every w W we ave tat w is continuous at te nodes N and at te midpoints of edges. We will also need te canonical nodal interpolation operator I 3 : H3 () W, wic is defined by evaluating function values and derivatives at vertices of elements and normal derivatives at midpoint of edges by averaging. Since I 3 is exact for w P 2(T ), te Bramble-Hilbert lemma yields (3.2) w I 3 w L p (T ) + T w I 3 w L p (T ) + 2 T D 2 w D 2 I 3 w L p (T ) c 2 3 T D 3 w L p (T ) for all w W 3 p (T ) and 2 p. A less obvious but useful stability bound reads (3.3) D 3 I 3 w L p (T ) c D 3 w L p (T ) for all w W 3 p (T ) and 2 p. To see tis, we first write I 3 ( w q ) = ( w q ) + ( I 3 w w ) for all q P 2 (T ). Terefore, invoking an inverse estimate togeter wit D 3 q = 0, we use (3.2) to obtain D 3 I 3 w L p (T ) c 2 T I3 (w q) L p (T ) c 2 T (w q) L p (T ) + c 2 T (I3 w w) L p (T ) c D 3 w L p (T ), provided tat q is appropriately cosen, e.g., as a suitable Lagrange interpolant of w over P 2 (T ). Hereafter, c > 0 indicates a generic geometric constant tat may cange at eac occurrence, depends on mes sape regularity, but is independent of te functions and parameters involved. Remark 3.1 (nodal degrees of freedom). Te degrees of freedom in W are only te function values at te vertices (w (z) : z N ), and te gradients at te vertices ( w (z) : z N ). In fact, te remaining four degrees of freedom of te finite element Q 3 (T ) are te normal components w (z E ) of te gradients at te midpoints z E of edges E wic are fixed as te averages of directional derivatives w (z i E ) n E at te endpoints z i E of te edges. Te values w (z E ) t E can be written in terms of w (z i E ) and w (z i E ) t E for i = 1, 2. Te matrix realizing te operator : W G elementwise is required for te implementation of Kircoff elements. Figure 2. Scematic description of te discrete gradient operator. Filled dots represent values of functions, circles of gradients, arrows of normal components, and boxes of vector fields. Te normal derivatives in te cubic space on te left are eliminated via linearity. Remark 3.2 (subspaces of H 1 ()). Enforcing degrees of freedom of W at vertices z N for bot function values and gradients implies global continuity; tus W H 1 (). Likewise, te degrees of freedom of G at vertices and midpoints of edges guarantee global continuity; ence G H 1 () 3. We collect important properties of te discrete gradient operator in te following proposition. Proposition 3.1 (properties of ). Let 2 p. Tere are constants c i, i = 1,..., 4, independent of suc tat te following properties of te discrete gradient are valid: (i) For all w W we ave (3.4) c 1 1 w L p () w L p () c 1 w L p (); 10

11 (ii) For all w W and T T we ave (3.5) c 1 2 D2 w L p (T ) w L p (T ) c 2 D 2 w L p (T ); (iii) For all w W 3 p (T ) and T T we ave (3.6) w w L p (T ) + T D 2 w w L p (T ) c 3 2 T D 3 w L p (T ); (iv) For all w W and T T we ave (3.7) w w L p (T ) c 4 T w L p (T ). Proof. (i) Given w W te function ψ = w G is well-defined and te operator : W G is linear, wence w = 0 implies w = 0. Conversely, if w = 0 ten we ave tat w (z) = 0 for all z N and w (z E ) = 0 for all E E. Since te tangential derivatives of w vanis at te endpoints and midpoints of E E, and w is cubic on E, we deduce tat w is constant on E. Te fact tat functions in W are globally continuous implies tat w is constant over te skeleton E of T. Let T T and note tat tere are four remaining degrees of freedom in Q 3 (T ). Since w (z E ) n E = 0 for all E E T, we see tat w is constant in T, wence w = 0. Te equivalence of te identities w = 0 and w = 0 implies te asserted norm equivalence because Q 3 (T ) is finite dimensional. (ii) We proceed as in (i). If D 2 w = 0, ten w is constant and so is w according to its definition; tus w = 0. Conversely, if w = 0, ten w is constant in T and tus w is te same constant at te vertices and midpoints of edges of T. Tis matces te 16 degrees of freedom of Q 3 (T ), wence w is constant in T and D 2 w = 0. (iii) Estimate (3.6) follows from te interpolation estimate (3.1) wit ψ = w upon noting tat I 2[ w] = w. (iv) Te estimate (3.7) is a consequence of (3.6), an inverse inequality, and (3.5). Te independence of all constants of te element-size T follows from scaling arguments. Remark 3.3 (bases of W and G ). We anticipate tat our discrete algoritms (Algoritms 2 and 3 below) do not require te coice of a particular basis for W. Instead, we apply vertex based quadratures requiring only te values of te approximate deformation and its gradient at te vertices. In contrast, a basis for G is required but standard. In our implementation we use te (nodal) Lagrange basis. 4. Discrete Energies and Γ-Convergence of te Discretization We employ te Kircoff elements on quadrilaterals W 3 H1 () 3 and te discrete gradient operator : W 3 G3, wose components are denoted j, j = 1, 2, to approximate te energy Ẽ given by (1.3). For practical purposes, we also impose a relaxed isometry constraint at te vertices of elements, but we introduce a parameter δ 0 to control its violation. We will sow in Section 5 tat in te context of an H 2 -gradient flow, δ is proportional to te gradient flow pseudo-timestep and can terefore be made arbitrary small. We next give a discrete version of (1.4) and (1.5). Definition 4.1. For δ 0, y D, W 3 and Φ D, G 3 D Ω let te discrete admissible set be A δ := { y W 3 : y D = y D, D, y D = Φ D, D, [ y (z)] y (z) I 2 z N, [ y ] Te (pseudo) tangent space of A δ at y A δ is defined by F [y ] := { w W 3 : w D = 0, w D = 0, y I 2 L 1 () δ}. [ w (z)] T y (z) + [ y (z)] T w (z) = 0 z N }. 11

12 Notice tat F [y ] would be te tangent space to A δ at y if [ y (z)] T y (z) were constant for vector fields in A δ at every node z N, wic explains te terminology. Notice also te use of te discrete norms φ L p (), wic for 1 p < are defined by φ p T L p := () 4 T T z N T φ T (z) p, and satisfy te equivalence relation v L p () v L p () for piecewise bilinear functions v C(). We also define te discrete inner product (, ) for piecewise continuous functions φ, ψ Π T T C 0 (T ) (φ, ψ) := T φ T (z)ψ T (z) 4 T T z N T and note tat φ p L p () = ( φ p, 1). Te finite element discretization Ẽ of te energy functional Ẽ in (1.3) is given by (4.1) Ẽ[y ] := 1 2 ( [ y 2 + i I 1 2 [ j y ] 1 y 1 y 2 y ) ], 2 y Z ij (Z, Z) (f, y ), i,j=1 for y A δ and Ẽ[y ] = oterwise, were I 1 is te canonical Lagrange interpolation operator into te continuous piecewise Q 1 elements, and bot Z and f are piecewise continuous in. Te latter enables te use of quadrature for te last tree terms, wereas te first term can be integrated exactly because y is piecewise Q 2. Te energy (4.1) is tus practical. Remark 4.1 (discrete isometry relation). Te nodal isometry relation [ y (z)] T y (z) I 2 for y A implies tat j y (z) 1 for j = 1, 2 and all z N. Hence, te normalization j y (z)/ j y (z) in te discrete energy functional Ẽ [y ] is well-defined. allows for suitable energy bounds and gives rise to a coercivity property. We start by sowing tat te family {Ẽ} 0 is (equi-)coercive. We will see tat it Proposition 4.1 (coercivity). Let te Diriclet boundary data satisfy y D H 3 () 3 and Φ D H 2 () 3 2, and let y D, := I 3 y D, Φ D, := I 2 Φ D. Let te data satisfy Z Π T T C 0 (T ) 2 2 and f Π T T C 0 (T ) 3. Let {y } >0 be a sequence of displacements in H 1 () 3 suc tat for a constant C independent of tere olds Ẽ [y ] C. Ten y A δ and tere exists a constant C depending on Z L (), f L (), y D H 3 (), and Φ D H 2 (), but independent of, suc tat (4.2) y L 2 () C. Proof. We first argue tat y A δ since oterwise Ẽ[y ] = +. As a consequence we ave j y (z) 1 for j = 1, 2 and all z N, wence tere exists a constant c independent of suc tat (4.3) Ẽ [y ] 1 2 y 2 L 2 () c( I 1 [ y ] L 2 () Z L () + y L 2 () f L ()). Since y = y D, and y = Φ D, on D, we can apply te Poincaré inequality twice and bound y L 2 () in terms of y L 2 (), y D, H 1 (), and Φ D, H 1 (). In view of (3.2) and (3.1), te latter two quantities are bounded by a constant times y D H 3 () and Φ D H 2 (), respectively. We observe tat for all T T (4.4) I 1 [ y ] L 2 (T ) c T I 1 [ y ] L (T ) c T y L (T ) c y L 2 (T ), 12

13 were te last step is an inverse inequality for y G. Tis implies te asserted bound. Remark 4.2 (coercivity and gradient flows). In te (energy decreasing) gradient flow setting adopted in Section 5, te assumptions of Proposition 4.1 are automatically satisfied provided te initial state as finite energy; see Proposition 5.2. We now sow Γ-convergence of Ẽ to Ẽ in H1 () 3 and deduce te accumulation of almost global minimizers of Ẽ at global minimizers of te continuous problem. For tis, we assume tat te discrete boundary conditions are obtained by interpolation of te continuous ones wit strong convergence in L 2 ( D ). We also assume for simplicity tat Z and f are piecewise constant. Teorem 4.1 (Γ-convergence). Let te Diriclet boundary data satisfy y D H 3 () 3 and Φ D H 2 () 3 2, and let y D, := I 3y D, Φ D, := I 2Φ D. If Z and f are piecewise constant over te partition T, ten te following two properties old: (i) Attainment. For all y A, tere exists a sequence {y } wit y A 0 Aδ for all > 0 suc tat y y in H 1 () 3 and lim sup Ẽ [y ] Ẽ[y]. (,δ) 0 (ii) Lower bound property. Assume tat δ 0 as 0. For all y H 1 () 3 and all sequences {y } H 1 () 3 suc tat y y in H 1 () 3, we ave Ẽ[y] lim inf 0 Proof. We prove properties (i) and (ii) separately. Ẽ [y ]. (i) Since y A H 2 () 3, for every ɛ > 0 te density of smoot isometries among isometries in H 2 () 3, cf. [17], implies te existence of an isometry y ɛ H 3 () 3 suc tat (4.5) y y ɛ H 2 () ɛ. Tis, in conjunction wit te isometry property of bot y and y ɛ, yields Ẽ[y] Ẽ[y ɛ] Cɛ. Terefore, we assume from now on tat y H 3 () 3 and do not write te subscript ɛ for simplicity. For > 0 let y = I 3y W3 be te nodal interpolant of y, i.e., we ave y (z) = y(z) and y (z) = y(z) for all z N. Te latter yields [ y ] y = I 2 at te nodes in N, wence y A 0. Convergence of y to y in H 1 () directly follows from te interpolation estimate (3.2) y y H 1 () c 2 2 y H 3 (). It tus remains to prove te convergence of te discrete energies Ẽ[y ] to convergence of te first term in (4.1), we write Ẽ[y]. To derive te (4.6) y y = ( y I 3 y) + ( I 3 y I2 [ I3 y]) and use (3.6) in conjunction wit (3.3) to get ( I 3 y I2 [ I3 y]) L 2 (T ) c T D 3 I 3 y L 2 (T ) c T D 3 y L 2 (T ). Combining tis wit (3.2) yields (4.7) y D 2 y L 2 () c D 3 y L 2 (). 13

14 For te second term in Ẽ[y ], we first note tat j y (z) = 1 for j = 1, 2 and all z N, and by nodal interpolation estimates ( i I 1 [ j y ] [ 1 y ] 2 y, Zij ) ( i I 1 [ j y ] [ 1 y ] ) 2 y, Zij c 2 T D ( i I 1 (4.8) [ j y ]) L 2 (T ) D [ 1 y ] 2 y L 2 (T ) T T + c i I 1 [ j y ] L 2 (T ) D 2[ 1 y 2 ] y L 2 (T ), T T 2 T because D 2 i I 1[ j y ] = 0 for every T T and Z is piecewise constant over T ; recall tat c denotes a generic constant independent of. Terefore, employing inverse estimates for bot terms on te rigt-and side of te preceding estimate, and recalling (4.4), we deduce ( i I 1 [ j y ] [ 1 y 2 ] y, Zij ) ( i I 1 [ j y ] [ 1 y 2 ] ) y, Zij c T T T y 2 L 2 (T ) y L (T ) c y 2 L 2 () y L (). We furter observe tat y L () is bounded uniformly because y = I 3y wit y H3 () 3 C 1 () being an isometry, and (3.4) wit p =. Tis, togeter wit (4.7), implies tat te quadrature term above is bounded by c y 2 H 3 (). It tus remains to examine ( i I 1 [ j y ] [ 1 y 2 ] ) ( y, Zij i j y [ 1 y 2 y ] ), Z ij. Invoking again (4.6), we infer tat y y L 2 () c 2 D 3 y L 2 () along wit [ 1 y 2 y ] [ 1 y 2 y ] L 2 () [ 1 y 1 y ] 2 y L 2 () + [ 1 y [ 2 y 2 y ] L 2 () c 2 D 3 y L 2 () because y L (), y L () C. In addition, we see tat along wit I 1 [ y ] 2 y = I 1 [ y y] + ( I 1 [ y] y), ( I 1 [ y] y) L 2 (T ) c T D 3 y L 2 (T ). Using an inverse estimate and stability of I 1 in L (T ), we get I 1 [ y y] L 2 (T ) c 1 T I1 [ y y] L 2 (T ) Moreover, we furter write y y L (T ) = I 2 [ I3 y] y L (T ) c I 1 [ y y] L (T ) c y y L (T ). I 2 [ I3 y] I3 y L (T ) + (I 3 y y) L (T ), and obtain, according to (3.1) and (3.3) wit p = and an inverse estimate, I 2 [ I3 y] I3 y L (T ) c T D 3 I 3 y L 2 (T ) c T D 3 y L 2 (T ). Since (I 3y y) L (T ) c T D 3 y L 2 (T ), we deduce I 1[ y ] 2 y L 2 (T ) c T D 3 y L 2 (T ). Tis, togeter wit te preceding bound for [ 1 y 2 y [ ] 1 y 2 y ] L 2 (), implies (4.9) ( i I 1 [ j y ] [ 1 y 2 ] ) ( y, Zij i j y [ 1 y 2 y ] ), Z ij c D 3 y L 2 (). 14

15 Collecting te preceding estimates, we obtain Ẽ [y ] Ẽ[y] c D 3 y L 2 () were y is an abbreviation for y ɛ. Selecting = (ɛ) to be sufficiently small so tat D 3 y ɛ L 2 () ɛ yields Ẽ [y ] Ẽ[y] cɛ. and te convergence of Ẽ[y ] to Ẽ[y] wen 0 follows. (ii) We may assume tat y A δ and Ẽ[y ] C uniformly in (peraps for a subsequence not relabeled) for oterwise lim inf 0 Ẽ [y ] = + and tere is noting to prove. Hence Proposition 4.1 implies tat te sequence { y } >0 is uniformly bounded in H 1 () 3 2. Tis guarantees te existence of Φ H 1 () 3 2 suc tat after extraction of a subsequence (not relabeled) we ave Φ = y Φ in H 1 () 3 2 and Φ Φ in L 2 () 3 2 as 0. Te discrete approximation estimate (3.7) yields y Φ L 2 () y y L 2 () + y y L 2 () y y L 2 () + c y L 2 (), wence taking te limit wen 0 we deduce Φ = y and y H 2 () 3 because y y in L 2 () 3 2 by assumption. Owing to te assumptions on te boundary data we ave tat y D = y D and y D = Φ D. To sow tat y is an isometry we utilize discrete interpolation estimates and y A δ Φ Φ I 2 L 1 () Φ Φ I 1 [Φ Φ ] L 1 () + I 1 [Φ Φ ] I 2 L 1 () c [Φ Φ ] L 1 () + c 0 δ. Te rigt-and side converges to zero as (, δ) 0 because of te uniform bound (4.2) of Φ in H 1 () 3 2. Hence, Φ Φ I 2 pointwise almost everywere in for an appropriate subsequence and, since Φ Φ Φ Φ pointwise almost everywere in, we deduce tat y is an isometry a.e. in, i.e., y A. Since te H 1 -seminorm is weakly lower semicontinuous we get D2 y 2 = Φ 2 lim inf 0 Φ 2. It remains to prove tat te following tree terms tend to 0: ( [ I = i I 1 [ j y ] 1 y 1 y 2 y ], 2 y Z ij ( ) i I 1 [ j y ] [ 1 y 2 ] ) y, Zij, II = ( i I 1 [ j y ] [ 1 y 2 ] y, Zij ( ) i I 1 [ j y ] [ 1 y 2 ] ) y, Zij, III = ( i I 1 [ j y ] [ 1 y 2 ] ) y, Zij ( i j y [ 1 y 2 y ] ), Z ij, for all 1 i, j 2. We first note tat I i I 1 [ j y ] L 2 () Z ij L () 1 y 1 y 2 y 2 y 1 y 2 L y. 2 () Te first factor on te rigt-and side is bounded as 0 according to (4.4) and (4.2), and te second one by assumption. Since i y 1, we estimate te last factor as follows: 1 y 1 y 2 y 2 y 1 y 2 L y 2 () ( y ) 1 1 y 1 y 2 y ( 2 y + L 2 1 y 2 y ) () 2 y L 2y 2 () 1 y 1 L 2 () + 1 y L 4 () 2 y 1 L 4 (). By te approximate isometry property and j y (z) 1 for all z N we obtain for y A δ (4.10) j y 1 L 1 () j y 2 1 L 1 () δ. 15

16 Moreover, since y is uniformly bounded in H 1 () 3 and R 2, we ave by Sobolev embeddings for all 1 q < ( ) (4.11) y L q () c q y L q () c q y L 2 () + y L 2 () cq, wence j y 1 L q () c q. Interpolating wit discrete Hölder inequalities between tis discrete L q -estimate and te discrete L1 -estimate in (4.10), we deduce tat j y 1 L p 0, j = 1, 2 () for p = 2, 4 as δ 0. Tis sows tat I 0. Te second term II accounts for te effect of quadrature and is te same as (4.8), wence II c y 2 L 2 () y L (). Since y A δ is not an exact nodal isometry, we do not ave direct control of y L (). We invoke instead te two-dimensional discrete Sobolev inequality y L () c log 1/2 y L 2 () [10, p.123], to infer tat II c log 1/2 y 3 L 2 () 0 0. Te last term III is te same as (4.9) except tat we do not ave y H 3 () 3. We split III as follows: ( ( i III = I 1 [ j y ] i j y ) [ 1 y 2 y ] ), Z ij + ( i I 1 [ j y ] {[ 1 y 2 y ] [ 1 y 2 y ]} ), Z ij. We observe tat i I 1 [ j y ] i j y in L 2 () 3, wence te first term tends to 0 as 0. In fact, te uniform bound (4.4) on I 1 [ y ], in conjunction wit (4.2), implies te asserted weak convergence, and te limit is found via ( i ( I 1 [ j y ] j y] ), ψ ) = ( I 1 [ j y ] j y ], i ψ ) ( j y j y, i ψ ) 0 0, wic olds for every ψ H 1 0 ()3 because I 1 [ y ] y L 2 () c y L 2 () c and y y in L 2 (). For te second term in III we resort again to te uniform L 2 -bound on I 1 [ y ] and write [ 1 y 2 y ] [ 1 y 2 y ] L 2 () y y L 4 ()( y L 4 () + y L 4 ()). By compactness of te embedding H 1 () L 4 (), we ave y y L 4 () 0 as 0. Finally, using (4.11) for q = 4 along wit y L () c because y is an isometry, we see tat te preceding term tends to 0 and tus conclude te proof. Teorem 4.1 extends easily to piecewise constant approximations to L 2 -data Z and f and to piecewise Lipscitz data over T ; we do not carry out te details. Te following result is a consequence of standard abstract Γ-convergence teory [12, 9] combined wit Teorem 4.1 and Proposition 4.1. Corollary 4.1 (convergence of absolute minimizers). Let δ 0 as 0. Let C > 0 be a constant independent of and {y } be a sequence of almost absolute discrete minimizers of Ẽ, namely (4.12) Ẽ [y ] inf Ẽ [w ] + ɛ C, w A δ 16

17 were ɛ 0 as 0. Ten {y } is precompact in H 1 () 3, and every cluster point y of y is an absolute minimizer of Ẽ, namely (4.13) Ẽ[y] = inf Ẽ[w]. w A Moreover, tere exists a subsequence of {y } (not relabeled) suc tat (4.14) lim 0 y y H 1 () = 0 and lim 0 Ẽ [y ] = Ẽ[y]. Proof. Te uniform bound for te discrete energies and te coercivity property of Proposition 4.1 imply tat te sequence { y } is precompact in L 2 () 3 2. Due to te norm equivalence (3.4) and a Poincaré inequality we ave tat {y } is bounded in H 1 () 3. Moreover, because of te estimate (3.6) te differences y y converge strongly to zero in L 2 () 3 2 as 0. Hence, tere exists y H 1 () 3 suc tat, up to te extraction of a subsequence, we ave y, y y in L 2 (). Te lower bound assertion of Teorem 4.1 implies tat y A and (4.15) Ẽ[y] lim inf 0 Ẽ [y ]. It remains to sow tat y is a global minimizer of Ẽ. To prove tis, let η > 0 be arbitrary and z A suc tat Ẽ[z] inf Ẽ[w] + η/2. w A Te attainment property stated in Teorem 4.1 implies tat tere exist > 0 and z A δ so tat Ẽ [z ] Ẽ[z] + η/2. On combining te previous two estimates and incorporating te fact tat y is a minimizer for Ẽ in A δ up to te value ε, we ave tat Ẽ [y ] Ẽ[z ] + ε Ẽ[z] + η/2 + ε inf Ẽ[w] + η + ε. w A Tis togeter wit (4.15) and te arbitrariness of η > 0 prove (4.14). Remark 4.3 (local minimizers). Statements about almost local minimizers of Ẽ are not available in general. However, if Ẽ as an isolated local minimizer y, ten tere exist local minimizers {y } of Ẽ converging to y provided is sufficiently small [9, Teorem 5.1]. We defer te discussion of almost local discrete minimizers of Ẽ to Section Fully Discrete Gradient Flow Corollary 4.1 guarantees tat every accumulation point of almost absolute minimizers of {Ẽ} is an absolute minimizer of E. We introduce and study in tis section a practical gradient flow algoritm to minimize Ẽ on A δ for > 0 and were δ is proportional to te gradient flow pseudotime parameter. Our fully discrete gradient flow gives rise to an energy decreasing iterative sceme tat converges to stationary points satisfying te isometry constraint up to a small error. However, like every gradient descent metod, weter te algoritm reaces an almost absolute minimizer, a saddle point, or a local minimizer is not possible to discern. We discuss tis furter in Section 6. Algoritm 2 (discrete H 2 -gradient flow). Let τ > 0 and set k = 0. Coose y 0 A0. (1) Compute y k+1 y k + F [ y k ] wic is minimal for te functionals y 1 2τ (y y k ) 2 L 2 () + Ẽ[y ] 17

18 in te set of all y y k + F [ y k ]. (2) increase k k + 1 and continue wit (1). Every step of te gradient flow requires solving a nonconvex minimization problem. primary variables of interest are te discrete gradients (5.1) Φ := y k+1, Φ := y k, Ψ := w, Since te [ ] wit w F y k, we let teir columns be Φ,j, Φ,j, Ψ,j for j = 1, 2, and write te corresponding Euler Lagrange equations as follows: 1 τ ( [Φ Φ 2 ( [ ], Ψ ) + ( Φ, Ψ ) + i I 1 [Ψ Φ,1,j] Φ,1 Φ ),2 ], Z ij Φ,2 + 2 i,j=1 i,j=1 ( [ i I 1 [Φ (PΦ,1,j] Ψ ) Φ,2,1 Φ,2 + Φ,1 Φ,1 ( ) ] ) P Φ,2 Ψ,2, Z ij = (f, w ) [ ] for all w F y k. Hereafter, to ave a simple and compact notation, we let Pa be te operator P a := 1 ( I 3 a a ) a a 2, for any given a R 3 and observe tat P a 1 provided a 1. Notice tat we omit writing te time steps k and k + 1 in (5.1). Existence of a locally unique solution to (5.1) follows from a local contraction property of te fixed-point iteration defined in te next algoritm, in wic we write y l for yk,l. Algoritm 3 (fixed-point iteration). Let ỹ A, define y0 = ỹ, and set l = 0. (1) Compute Φ l+1 := y l+1 wit y l+1 ] ỹ + F [ỹ suc tat 1 τ ( [Φl+1 Φ 2 ( [ Φ l ], Ψ ) + ( Φ l+1, Ψ ) = i I 1 [Ψ,1,j] Φ l,1 Φl ] ),2 Φ l,2, Z ij (5.2) 2 i,j=1 for all w F [ỹ ] wit Ψ := w. (2) increase l l + 1 and continue wit (1). i,j=1 [ ( i I 1 [Φl,j ] P Φ l Ψ,1 Φl,2,1 Φ l,2 + Φl ],1 Φ l,1 P Φ l Ψ,2, Z ij,2 ) + (f, w ) Note tat (5.2) is a linear system for Φ l+1. Under a moderate condition on te step size τ > 0 te iterates are uniformly bounded and te map Φ l+1 Φ l is a contraction in a suitable H1 -ball. In particular, te limiting discrete Euler Lagrange equations (5.1) ten admit a locally unique solution. Tis, and related properties, are discussed in te next two propositions. Proposition 5.1 (local contraction property). Let te mappings f, Z be elementwise continuous, let Φ = ỹ, and set C := max{1, Φ L 2 ()}. If B := { y A : y L 2 () 2 C }, ten te nonlinear map Φ l Φl+1 in (5.2) is well-defined from B into itself and is a contraction wit constant 1 2 provided tat τ C 0, wit C 0 > 0 depending explicitly on C, a Poincaré constant c P 1 of, f L () and Z L (). Consequently, if Algoritm 3 is initialized wit te k-t iterate of Algoritm 2, i.e. ỹ = y k, ten te solution Φl of Algoritm 3 converges to te unique solution of te Euler-Lagrange equation (5.1) witin B. 18

19 Proof. We proceed in tree steps and simplify te notation upon writing = L 2 (). (i) Existence and isometry relation: Given y l B, and in particular y l A, we see tat [Φ l (z)]t Φ l (z) I 2 for all z N. We tus ave j yl 1 for j = 1, 2, so tat te rigt-and side of (5.2) is well-defined ] and te Lax Milgram lemma ] gives te existence of a unique solution ỹ + F [ỹ. Due to te definition of F [ỹ we infer tat y l+1 [ Φ l+1 (z) Φ (z) ] for all z N. Tis implies y l+1 [Φ l+1 (z)] Φ l+1 (z) = [ Φ (z)] for all z N, because ỹ A. Φ (z) + Φ (z) A, namely Φ (z) + [ Φ l+1 (z) Φ (z) ] [ Φ l+1 (z) Φ (z) ] = 0 [ Φ l+1 (z) Φ (z) ] I 2 (ii) Uniform bound: We next sow tat te iterates satisfy Φ l 2 C. For tis, we coose Ψ = Φ l+1 Φ [ ] = y l+1 ỹ in (5.2). Tis leads to 1 τ (Φl+1 Φ ) (Φl+1 Φ ) Φl Φ 2 2 = (f, y l+1 2 i,j=1 ỹ ) i,j=1 ( i I 1 [Φl+1,j Φ [ Φ l,1,j ] Φ l,1 Φl ] ),2 Φ l,2, Z ij [ ( i I 1 [Φl,j ] P Φ l (Φ l+1,1,1 Φ,1 ) Φl,2 Φ l,2 + Φl,1 Φ l,1 P Φ l (Φ l+1,2,2 Φ ),2 ) ], Z ij Since P Φ l (z) 1 for all z N, we deduce 1 τ (Φl+1 Φ ) Φ 2 + c(f, Z) ( y l+1 ỹ + (Φ l+1 Φ ) + Φ l Φl+1 Φ ), wit c(f, Z) > 0 only depending on f and Z. We now apply te Poincaré inequality to y l+1 ỹ in conjunction wit (3.4), namely y l+1 ỹ c P Φ l+1 Φ, were we let c P be te product of te Poincaré constant of and te constant c 1 1 in (3.4) and take it to be c P 1. Applying te Poincaré inequality again, tis time to Φ l+1 Φ, we obtain 1 τ (Φl+1 Φ ) Φ 2 + c(f, Z)c 2 ( P 2 + Φ l ) (Φ l+1 Φ ). To prove tat Φ l+1 2 C we assume by induction tat Φ l 2 C, wic is valid for l = 0. Using Φ C yields Since C 1, coosing (Φ l+1 Φ ) 2 τ C 2 + 4τ 2 c(f, Z) 2 c 4 P ( 1 + C) 2. { 1 τ C 1 := min 2, 1 } 16c(f, Z) 2 c 4 P 1 C 2 4c(f, Z) 2 c 4 P (1 + C) 2 implies (Φ l+1 Φ ) 2 2τ C 2 C 2, wence Φ l+1 2 C as asserted. 19.

20 (iii) Contraction property: It remains to sow tat te map Φ l Φl+1 given in (5.2) is a contraction on B. For tis, we subtract te equations tat define Φ l+1 and Φ l in (5.2) and verify tat 1 τ ( (Φl+1 Φ l ), Ψ ) + ( (Φ l+1 Φ l ), Ψ ) 2 ( [ Φ l = i I 1 [Ψ,1,j] Φ l,1 Φl ] ),2 Φ l,2, Z ij + + i,j=1 2 i,j=1 2 i,j=1 2 i,j=1 [ ( i I 1 [Φl,j ] P Φ l Ψ,1 Φl,2,1 Φ l,2 + Φl ] ),1 Φ l,1 P Φ l Ψ,2, Z ij,2 ( [ Φ l 1 i I 1 [Ψ,1,j] Φ l 1 ( i I 1 [Φl 1,j ] [P Φ l 1,1 Φ,1 l 1,2 Φ l 1,2 ], Z ij ) Ψ,1 Φl 1,2 Φ l 1,2 + Φ l 1,1 Φ l 1,1 P Φ l 1,2 Ψ,2 ], Z ij ). Bounding separately te sum of te first and tird terms and tat of te second and fourt terms, and using te admissible coice Ψ = Φ l+1 Φ l, we deduce te estimate 1 τ (Φl+1 Φ l ) C 1 2 (Φl+1 Φ l ) (Φl 1 Φ l ), were we ave used Φ l 2 C sown in step (ii), tat Φ l,j (z) 1 for all l 0, j = 1, 2, and z N, and tat for a, b R 3 wit a, b 1 te following estimates old a a b a b, b P a P b = 3 a b. Tis implies te asserted contraction property wit constant 1 2 in B for τ C 0 := min{c 1, C 2 }. Finally, if ỹ = y k is te k-t iterate of Algoritm 2, ten any fixed point of (5.2) is a solution of (5.1) and conversely. Tis implies uniqueness of (5.1) witin B. Remark 5.1 (time step). If f = 0, ten te preceding proof sows tat τ as to be sufficiently small so tat τ ( 4c(Z) 2 c 2 P ) 1, were c(z) = Z L () and c P 1 is te Poincaré constant of wit vanising Diriclet condition on D. Due to a repeated application of te Poincaré inequality, te case f 0 requires a stringent condition on τ. Te following proposition sows tat te discrete H 2 gradient flow of Algoritm 2 is energy decreasing and becomes stationary, its iterates are uniformly bounded, and te violation of te isometry constraint is controlled by te pseudo-timestep size. Proposition 5.2 (properties of iterates). Let Z and f be piecewise continuous over te partition T. Let {y k} k=0 A be iterates of Algoritm 2. We ten ave tat for all k 0 (5.3) Ẽ [y k+1 ] + 1 2τ and in particular k l=0 (5.4) y k C 20 (y l+1 y l ) 2 Ẽ[y 0 ],

21 for a constant C > 0 depending on y 0, f and Z, but independent of k. In addition, if [ y 0(z)] [ y 0(z)] = I 2 for all z N ten y k+1 A c 0τ, i.e. (5.5) [ y k+1 ] were c 0 depends only on y 0. [ y k+1 ] I 2 L 1 () c 0τ k 0, Proof. Te proof splits into tree steps. (i) Energy decay: Tis is a direct consequence of te minimizing properties of te iterates, i.e., Ẽ [y k+1 ] + 1 2τ (y k+1 y k ) 2 L 2 () Ẽ[y k ]. (ii) Coercivity: For every y A we ave j y (z) 1 for j = 1, 2 and all z N. Tis, in conjunction wit (4.3) and (4.4), yields Ẽ [y ] 1 4 y 2 L 2 () c( Z 2 L () + f 2 L ()). Since Ẽ[y k] Ẽ[y 0 ] from (i), tis implies te asserted bound (5.4) of te iterates. (iii) Isometry violation: Abbreviating Φ k+1 := y k+1 and Φ k := y k, and noting tat yk+1 y k F[yk ], we ave for all z N [ Φ k+1 (z) Φ k (z)] wence [Φ k+1 (z)] [Φ k+1 (z)] = [Φ k (z)] Φ k (z) + [ Φ k (z)] [Φ k (z)] + [(Φk+1 A repeated application of tis identity along wit [ y 0 (z)] [Φ k+1 ] [Φ k+1 ] I 2 L 1 () c k l=0 [ Φ k+1 (z) Φ k (z)] = 0, Φ k )(z)] [(Φ k+1 Φ k )(z)]. [ y 0 (z)] = I 2 for all z N yields (y l+1 y l ) 2. Wit te elp of a Poincaré inequality for (y l+1 y l ) and (5.3), we deduce (5.5). We end tis section by pointing out tat te stationary state y reaced by te gradient flow migt not be a discrete (almost) absolute minimizer. However, assuming tat y is an (almost) absolute minimizer, ten Corollary 4.1 guarantees tat te accumulation points wen 0 are absolute minimizers of te exact energy Ẽ. 6. Numerical Experiments: Performance and Model Exploration Algoritms 2 and 3 are implemented using te deal.ii library [2]. All systems of linear equations arising in te iteration of Algoritm 3 were solved directly using UMFPACK [13] leaving te discussion on developing efficient solvers open. Te resulting deformations are visualized wit paraview [16]. Note tat only te displacement degrees of freedom at te vertices of T are used for tis purpose and te plates are tus displayed as continuous piecewise bi-linear elements. In addition, we say tat a plate as reaced numerically an equilibrium state parametrized by y k+1 wen Ẽ[y k+1 ] (6.1) Ẽ[y k] 10 6, τ were τ is te gradient flow timestep. We set y := yk+1, were K is te smallest k tat satisfies te above constraint. Eac pseudo-time iteration k of te gradient flow consists of a fixed-point iteration (Algoritm 3). Tis inner loop stops wen te (l + 1)-t inner iterate satisfies (y k,l+1 y k,l ) δ stop 21

22 for some δ stop > 0. It turns out tat te number of subiterations experienced in practice is low (see for instance Figure 12). For ease of computation te discrete energies are approximated according to Ẽ [y ] 1 H +Z 2, 2 were H is te approximation of te second fundamental form given by H,i,j := ( ( 1 y ) 1 ( 2 y ) 2 ) i j y, witout normalization of te vectors i y, i = 1, 2, and Z is given and symmetric. Te absence of space dependence in Z corresponds to omogeneous materials. Te purpose of te following numerical experiments is twofold. We first document te performance of Algoritms 2 and 3 by investigating teir beavior for decreasing values of messize and pseudo-timestep τ and examining te violation of te isometry constraint. We also study various qualitative properties of te dimensionally reduced model, te existence of local discrete minimizers oter tan cylinders, wic are for appropriate data and boundary conditions global minimizers according to [20], as well as te pseudo-evolution process (sometimes exibiting self-intersections). We remark tat our underlying nonlinear matematical model allows for te description of large deformations wic may be nonunique and cannot be expected to admit ig regularity. Terefore, a meaningful error analysis seems out of reac. For small displacements te model leads to a linear bending problem for wic best-approximation and interpolation results imply convergence rates. An experimental convergence analysis for te case of a simple exact solution reported below indicates a linear convergence rate. We document our quantitative findings in Tables 1, 2, and 3 and use te symbol N/A to indicate special combinations of and τ for wic we did not perform computations as tese appeared to be irrelevant for te discussion and in some cases computationally expensive. Bilayer bending as tecnological applications in design and fabrication of micro-switces and micro-grippers as well as nano-tubes [5, 18, 19, 22, 23]. In tese cases it is essential tat te bilayer plate undergoes a complete folding to a cylinder witout exibiting dog-ears or a corkscrew sape, wic may affect or impede te complete folding [24]. Better understanding and control of tis penomenon is wat motivated tis work. We describe below several equilibrium configurations oter tan cylinders Bencmark. We display in Figure 3 te pseudo-evolution of a bilayer plate = ( 5, 5) ( 2, 2), clamped on te left-side D = {x = 5} [ 2, 2], i.e., y = 0, y = I 3 2 on D, wit a spontaneous curvature Z = I 2. Te finite element partition consists of 5 uniform refinements of te rectangle. Te parameters for Algoritms 2 and 3 are te pseudo-timestep τ = and te sub-iteration stopping tolerance δ stop = Te discrete equilibrium state is a cylinder Relaxation Process. We consider te clamped plate described in Section 6.1 for different spontaneous curvatures Z as well as different discretization parameters, τ. Te subiterations stopping tolerance for Algoritm 3 is δ stop = We examine te relaxation process towards equilibrium for two different spontaneous curvatures, namely Z = I 2, 5I 2. According to [20], te absolute energy minimizers are cylinders of eigt 4 (lengt of te clamped side) and of radius 1 and 1/5 (reciprocal of te eigenvalues of Z) wit an energy of 20 and 500, respectively. Table 1 documents te influence of messize and pseudo-timestep τ on te equilibrium sape and corresponding energy. Te cases Z = I 2 and Z = 5I 2 are strikingly different. For Z = I 2 te equilibrium sape is a cylinder (absolute minimizer) provided τ C 0 for a sufficiently small constant C 0 ; see Figure 4 (left for τ = and middle for τ = 0.005). For Z = 5I 2, and 22

23 Figure 3. Pseudo-evolution (clockwise) towards te equilibrium of a clamped rectangular plate wit spontaneous curvature Z = I 2. Te bilayer plate is depicted (clockwise) for 0.0, 0.1, 2.4, 60.0, 100.0, 130.0, 135.0, times 10 3 iterations of Algoritm 2. Te bilayer plate reaces a cylindrical sape asymptotically (limit of discrete gradient flow). Tis is an absolute minimizer. regardless of te size of τ, te plate never reaces a cylinder but oter equilibrium configurations (local minimizers) wit muc iger energy tan 500; see Figure 4 (rigt). A plausible explanation is tat te relatively large spontaneous curvature in te direction of te clamped side favors bending in suc a direction, tereby creating a geometric obstruction to reacing a cylindrical sape. Table 1 also provides information about te tresold of τ needed for convergence of te subiterations of Algoritm 3. Tis value, being sensitive to Z L (), is more stringent for Z = 5I 2. In fact, suc iterations fail to converge for τ = 0.02 and Z = 5I 2, wereas for Z = I 2 give rise to a local minimizer. Tis is consistent wit Remark 4.3, wic establises te pessimistic tresolds τ 0 = for Z = I 2 and τ 0 = 10 4 for Z = 5I 2 if we consider a Poincaré constant c P = 10. In addition, we note tat te case τ = on te mes #6 and wit Z = I 2 yields a cylindrical equilibrium sape wit energy of 17.2 (not reported in Table 1). Tis, in conjunction wit te energies wen τ = on te mes resulting from four refinements and τ = on te mes resulting from five refinements, illustrates tat Ẽ[y ] E[y] = 20 as 0, = cτ and c is sufficiently small to obtain cylindrical sapes. Tis is in accordance wit Corollary Asymptotics. In tis section, we illustrate te predicted convergence rate of O(τ) for te violation of te isometry constraint in (5.5). We consider again te clamped plate = ( 5, 5) ( 2, 2) described in Section 6.1 wit a spontaneous curvature Z = I 2. Te space discretizations are subordinate to 4, 5, 6 and 7 uniform refinements of te initial partition of te plate consisting of 1 rectangle and are referred to as mes #4, #5, #6 and #7, respectively. Te sub-iterations stopping tolerance of Algoritm 3 is δ stop = Te equilibrium isometry defect is defined to be ID (y ) := [ y ] 23 [ y ] I 2 L 1 ().

24 Z = I 2 Z = 5I 2 Mes τ #4 #5 #6 #7 #4 #5 #6 # n n N/A N/A NoC NoC N/A N/A n n n n y y n N/A N/A y y y y N/A N/A Table 1. Equilibrium energies Ẽ[y ] for spontaneous curvatures Z = I 2, 5I 2 and different messizes and pseudo-timesteps τ. Te meses correspond to 4, 5, 6, and 7 uniform refinements of te plate = ( 5, 5) ( 2, 2). Te symbol next to te energy values for Z = I 2 indicates weter te equilibrium sape is a cylinder (y) or not (n) after te stopping test (6.1) is met. Typical equilibria are displayed in Figure 4 (left and middle, te latter corresponding to a local discrete minimizer). Te numerical experiments indicate tat te cylindrical sape (absolute minimizer) is reaced for Z = I 2 wen τ and satisfy τ C 0 for a sufficiently small constant C 0. Tis experimental condition is more restrictive tan te teoretically derived condition for mere convergence of our numerical sceme. Te symbol NoC for Z = 5I 2 indicates tat te sub-iterations of Algoritm 3 did not converge. Te cylindrical sape is never reaced for Z = 5I 2 ; see Figure 4 (rigt) for a typical equilibrium configuration. Figure 4. Equilibrium configurations of clamped rectangle plates = ( 5, 5) ( 2, 2) described in Section 6.1 for different spontaneous curvatures and numerical parameters. Left: cylinder sape wen Z = I 2, mes refinement 6 and τ = ; Middle: local minimum wen Z = I 2, mes refinement 6 and τ = 0.005; Rigt: local minimizer wen Z = 5I 2, mes refinement 7 and τ = If te spontaneous curvature is 1 or smaller, ten te cylinder (absolute minimizer) is reaced wen te pseudo-timestep and te messize satisfy τ C 0 for a sufficiently small constant C 0. For relatively large spontaneous curvatures, te curvature in te direction of te clamped side prevents te plate from bending completely in te ortogonal direction, tereby creating a geometric obstruction and leading to a (discrete) local minimizer for all numerical parameters tried. Te results reported in Table 2 indicate tat te predicted rate of convergence O(τ) in Proposition 5.2 is recovered numerically. Tis appens for bot decreasing time steps τ on a fixed mes (columns 1, 2, and 3) as well as simultaneous reduction of and τ wile keeping τ (diagonal). We stress tat, according to te discussion of Section 6.2, not all equilibrium configurations are cylinders but 24

25 te experimentally observed linear rate applies to all of tem. Tis is consistent wit Proposition 5.2 wic is not specific to absolute minimizers. Mes τ #4 #5 #6 # N/A N/A N/A Table 2. Isometry defect ID (y ) for te clamped plate = ( 5, 5) ( 2, 2) at equilibrium wit spontaneous curvature Z = I 2. A sequence of time steps τ = i, i = 0, 1, 2, 3, is considered for different space resolutions. A decay rate O(τ) is observed wen te space discretization remains uncanged (columns 1, 2 and 3) as well as wen te messize is reduced to satisfy τ (diagonal). Notice tat te isometry defect for a fixed time step is little affected by te space resolution (rows 2 and 4). For an experimental convergence test for te deformations we coose = (0, 2π) 2, define ( ) 1 0 Z =, 0.5 and consider clamped boundary conditions on te left side D = {x = 0} [0, 2π]. An exact solution is given by y(x, y) = ( sin(x), y, 1 cos(x) ). Table 3 sows te scaled L 2 and H 1 errors for te stationary configurations computed wit Algoritms 2 and 3; norms were computed wit a one-point Gaussian quadrature rule on every element. Te stopping criterion for te fixed-point iteration is δ stop = Te underlying meses correspond to l = 3, 4, 5, 6 refinements of our coarse mes so tat l = 2π2 l. We coose a step size proportional to te mes size, i.e., we set τ l = 2 l /25 for l = 3, 4, 5, 6. Te obtained errors indicate a nearly linear experimental convergence rate. Error Mes #3 #4 #5 #6 e / 1/ e / 1/ Table 3. Scaled approximation errors e = y y in an experimental convergence test wit exact solution y given by a cylinder of radius 1. Te approximations y are obtained wit Algoritms 2 and 3 from a flat initial configuration and timestep sizes proportional to messizes. Te numbers indicate a nearly linear experimental convergence rate Effect of Aspect Ratio and Spontaneous Curvature. Tis section investigates numerically te influence of (i) te spontaneous curvature (i.e. difference in material properties between te two plates) and (ii) te plates aspect ratio. Te numerical parameters in Algoritm 3 are τ = 0.005, δ stop = 10 4, and te partition corresponds to 5 uniform refinements of te initial plate. We consider plates := ( L, L) ( 2, 2) wit different lengts L > 0 and define te aspect ratio to be ρ := L/2. Te plates are clamped on te side D = {x = L} [ 2, 2]. It turns out tat te tendency to bend in te clamped direction (accentuated for relatively large spontaneous curvatures in te clamped direction according to Section 6.3) is attenuated for small aspect ratios. We illustrate 25

26 tis in Figure 5, wic displays almost equilibrium configurations for ρ = 5/2, 3/2, 1, 1/2 and spontaneous curvatures Z = ri 2 wit r = 1, 3, 5. For large spontaneous curvatures, Algoritm 2 did not always reac geometric equilibrium before te stopping test (6.1) was met. In addition, some pseudo-evolutions lead to severe folding and exibit self-intersections, in wic case tey are no longer representative from te pysics standpoint (500) 305.3(300) 220.4(100) 96.7(50) 188.6(180) 103.8(108) 67.9(36) 33.6(18) 16.55(20) 9.8(12) 6.5(8) 3.3(4) Figure 5. Equilibrium sapes of bilayer plates for several aspect-ratios ρ (from left to rigt ρ = 5/2, 3/2, 1, 1/2) and spontaneous curvatures Z = ri 2 (from top to bottom r = 5, 3, 1). Decreasing te aspect ratio restores te ability for te plate to fold into a cylindrical sape for larger spontaneous curvatures. For instance, tis is te case for plates wit parameters r = 3 and ρ = 3/2 or r = 5 and ρ = 1/2. Notice, owever, tat small regions around te free corners ave not completely relaxed to equilibrium. Tis effect is due to te violation of te isometry constraint and reduced upon decreasing te discretization parameters as well as te stopping criteria. Te numbers below eac stationary configuration are te corresponding approximate energies. For comparison, te energies of corresponding plates wit principal curvatures of 1 r and 0 are given between parentesis Boundary Conditions and Sapes. We consider now different boundary conditions and plate sapes. We intend to examine te robustness of our numerical sceme in different situations and investigate plate sapes wic are not studied in [20] and for wic we do not know te absolute minimizers. Boundary conditions: We start wit te plate = ( 3, 3) ( 2, 2) clamped in a neigborood of te bottom left corner, namely D = {x = 3} ( 2, 0) ( 3, 0) {y = 2}. We impose te spontaneous curvature Z = I 2, coose te numerical parameters τ = 0.005, δ stop = 10 4 in Algoritm 3, and use a partition of wit 5 uniform refinements. Several intermediate sapes of te discrete gradient flow are depicted in Figure 6. Te equilibrium configuration consists of a flat and a cylindrical part separated by a free boundary tat connects points on te boundary at wic boundary conditions cange. Sapes: We now consider te I-saped and te O-saped plates depicted in Figure 7. Te finite element meses contain 7168 and 8192 quasi-uniform rectangles respectively. We set Z = 5I 2, τ = and δ stop = Relaxation toward numerical equilibrium sapes, wic are not 26

27 Figure 6. Different snapsots of te deformed corner-clamped plate wit spontaneous curvature Z = I 2. Te equilibrium sape as energy and is not a cylinder. It is wort comparing wit te side-clamped plate discussed in Section 6.4, wic leads to a cylinder of smaller approximate energy cylinders, is depicted for bot plates in Figure 8. We stress tat for te I-saped plate te curvature in te clamped direction dominates te oter, tereby leading to a cigar sape tat opens up at te bottom to accomodate te boundary condition. In contrast, te O-saped plate is more rigid to bending and develops dog-ears at te free corners wic prevent furter bending Figure 7. Geometry of te I-saped and O-saped plates: te numbers indicate te lengt of te sides. In bot cases, te clamped edge is te far left vertical edge Anisotropic Spontaneous Curvature. We now turn our attention to anisotropic spontaneous curvatures, namely to matrices Z wit different eigenvalues. In te first two examples te eigenvectors are aligned wit te coordinate axis, but not in te tird example. Te spontaneous curvature is given by eiter ( ) ( ) (6.2) Z =, Z =, 0 a 2 3 wit a = 1 or 5. Te plate is = ( 2, 2) ( 3, 3) and te numerical parameters in Algoritm 3 are τ = 0.005, δ stop = Dominant curvature: Wit a = 1 being te curvature in te clamped direction, we notice a rater minimal bending effect in suc a direction. Te plate pseudo-evolutions are displayed in Figure 9 wic sows an almost perfect rolling to a cylinder of energy Curvatures wit opposite signs: We take a = 5 to be te curvature in te clamped direction. Tis coice models te tendency of te plate to bend equally in opposite directions along te coordinate 27

28 Figure 8. Different snapsots of te deformed I-saped plate (left) and O-saped plate (rigt). Te corresponding stationary energies wit spontaneous curvatures Z = 5I 2 are and respectively or about and relative to te plate areas. For comparison, te numerical stationary energy of a plate = ( 5, 5) ( 2, 2) under te same boundary condition and spontaneous curvature is or once divided by te plate area (see Figure 5). It turns out tat compared to te full plate, te stationary numerical energy per unit area is smaller for te I-saped plate and greater for te O-saped plate. Figure 9. Deformation of a plate wit anisotropic curvature given by (6.2) wit a = 1. Te spontaneous curvature is 1 in te clamped direction, its effect being barely noticeable, wereas it is 5 in te ortogonal direction. Te equilibrium sape is a cylinder (absolute minimizer) wit an energy of Compare wit te case a = 5, presented in Section 6.4, for wic te cylindrical sape is not acieved before te stopping test (6.1) is met. axes (principal directions). Tis is noticeable in te second and tird snapsots in Figure 10, te latter also exibiting self-crossing of te free corners. After tree complete rotations, te plate relaxes to a cylindrical sape (absolute minimizer). Surprisingly, a cylindrical sape is reaced before te stopping test (6.1) is met, unlike te case a = 5 (see first row - second column in Figure 5). Corkscrew sape: We consider now te second anisotropic spontaneous curvature Z in (6.2), wic as eigenpairs µ 1 = 5 e 1 = [1, 1] T, µ 2 = 1 e 2 = [1, 1] T. Tis means tat we still ave principal curvatures 5 and 1 but wit principal directions e 1 and e 2 forming te angle π/4 wit te coordinate axes. Te deformation of tis plate towards its equilibrium sape is displayed in Figure 11. Te plate exibits a corkscrew sape before self-intersecting and continuing its deformation to a conic sape. In fact, a cylindrical sape is not reaced before te stationarity test (6.1) is met. We empasize tat tis is not in contradiction wit [20], were scalar spontaneous curvatures are considered, and seds some ligt on equilibrium configurations 28

29 Figure 10. Deformation of a plate wit anisotropic curvature given by (6.2) wit a = 5. Te spontaneous curvatures are 5 in te clamped direction and 5 in te perpendicular direction, wic eventually dominates te former and leads to a cylindical sape after tree full rotations. Snapsots are displayed counterclockwise, starting from te bottom rigt, for 0.0, 0.3, 2.0, 4.0, 6.0, 10.0, 15.0, 18.0, 25.0, times 10 3 iterations of Algoritm 2. Te arrows indicate te clamped side. wen te two principal spontaneous curvature directions are not parallel and ortogonal to te clamped side. Notice, owever, tat te equilibrium energy obtained is 61.31, wic is larger tan te cylindrical sape obtained wen te principal direction is aligned wit te coordinate axes (see Figure 9). Figure 11. Deformation of a plate wit te second anisotropic curvature Z in (6.2). Te principal curvatures are 5 and 1 but te principal directions form an angle π/4 wit te coordinate axes. Te snapsots are displayed clockwise starting at te top left, for 0.0, 0.1, 0.4, 1.9, 3.0, times 10 3 iterations of Algoritm 2. Te plate adopts a corkscrew sape before self-intersecting Energy Decay and Time Scales. One critical aspect missing in tis study is te design of (pseudo)-time adaptive algoritms able to cope wit te disparate time scales inerent to te 29