The metric-restricted inverse design problem

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by  Amit Acharya

THE METRIC-RESTRICTED INVERSE DESIGN PROBLEM
AMIT ACHARYA, MARTA LEWICKA AND MOHAMMAD REZA PAKZAD
6 Abstract. Westudyaclassofdesignproblemsinsolidmechanics,leadingtoavariationonthe
1 classicalquestionofequi-dimensionalembeddabilityofRiemannianmanifolds. Inthisgeneralnew
0
context, we derive a necessary and sufficient existence condition, given through a system of total
2
differential equations, and discuss its integrability. In the classical context, the same approach

r yields conditions of immersibility of a given metric in terms of the Riemann curvature tensor.
p
In the present situation, the equations do not close in a straightforward manner, and successive
A
differentiationofthecompatibilityconditionsleadstoanewalgebraicdescriptionofintegrability.
We also recast the problem in a variational setting and analyze the infimum of the appropriate
2
1 incompatibility energy, resembling the non-Euclidean elasticity. We then derive a Γ-convergence
result for dimension reduction from 3d to 2d in the Kirchhoff energy scaling regime.

]
P
A
h. 1. The metric-restricted inverse design problem
t
a Assume T is a manifold of material types, differentiated by their structure, density, swelling-
m
shrinkage rates and other qualities. We let any material type T ∈ T be naturally endowed with a
[ prestraing¯(T), whereg¯: T → R2sy×m2,pos isagivensmoothmapping, takingvaluesinthesymmetric
3 positive definite tensors.
v Suppose now that we need to manufacture a 2-dimensional membrane S ⊂ R3, where at any
8
given point p ∈ S a material of type T(p) ∈ T must be used for a given T : S → T. The question
3
7 is how to print a thin film U ⊂ R2 in a manner that the activation u : U → R3 of the prestrain in
1 the film would result in a deformation leading eventually to the desired surface shape S.
0
. Theabovedescribedproblemisnaturalasadesignquestioninvariousareasofsolidmechanics,
1
0 even though the involved tensors are not intrinsic geometric objects. For example, it includes the
5 subproblems and extensions to higher dimensions:
1
: (a) Given the deformed configuration of an elastic 2-dimensional membrane and the rect-
v
angular Cartesian components of the Right Cauchy-Green tensor field of a deformation,
i
X mapping a flat undeformed reference of the membrane to it, find the flat reference config-
r uration and the deformation of the membrane1.
a
(b) Given the deformed configuration of a 3-dimensional body and the rectangular Carte-
sian components of the Right Cauchy-Green tensor field of the deformation, mapping a
reference configuration to it, find the reference configuration and the deformation.
(c) Suppose the current configuration of a 3-dimensional, plastically deformed body is given,
andonitisspecifiedtherectangularCartesiancomponentsofaplasticdistortionF . Find
p
a reference configuration and a deformation ζ, mapping this reference to the given current
configuration, such that the latter is stress-free. Assume that the stress response of the
material is such that the stress vanishes if and only if (∇ζ(F )−1)T(∇ζ(F )−1) = Id .
p p 3
Date: today.
1We thank Kaushik Bhattacharya for bringing this problem to our attention.
1
2 AMITACHARYA,MARTALEWICKAANDMOHAMMADREZAPAKZAD
In view of [16, 9, 19], the activation u must be an isometric immersion of the Riemannian
manifold (U,G) into R3, where G is the prestrain in the flat (referential) thin film. Our design
problem requires hence that we find an unknown reference configuration U ⊂ R2, an unknown
material distribution T : U → T and an unknown deformation u : U → R3 such that
(i) S = u(U),
(ii) For any x ∈ U, the point u(x) carries a material of type T(u(x)), i.e. T(u(x)) = T(x),
(iii) ∇u(x)T∇u(x) = G(x) := g¯(T(x)).
If the membrane S ⊂ R3 is a smooth surface, then letting g := g¯◦T : S → R2×2 , the conditions
sym,pos
(i)-(iii) simplify to finding a domain U ⊂ R2 and a bijection u : U → S, such that:
(1.1) (∇u)T(∇u)(x) = g(u(x)) ∀x ∈ U.
The smoothness of g is determined by the regularity of T and of the mappings g¯ and T.
Essentially, in all of the applications defined above (e.g. membrane, 3-d), we are dealing with a
general class of nonlinear elastic constitutive assumptions involving pre-strain, with the require-
ment that the stored energy density evaluated at the Identity tensor (of appropriate dimension-
ality) attain the value zero; in other words, we look for stress-free deformations of a prestrained
body. Given a prestrain field specified on the target configuration, we explore the question of exis-
tence of deformations that allow such a minimum energy state to be attained pointwise, as well as
the characterization of the constraints on the pre-strain field that allows such attainment. In the
language of mechanics, note that the question (1.1) may be rephrased as looking for deformations
u of the reference U such that:
(cid:104) (cid:16)(cid:112) −1(cid:17)(cid:105)T (cid:104) (cid:16)(cid:112) −1(cid:17)(cid:105)
∇u g(u) ∇u g(u) = Id,
where the expression on the left-hand-side of the above equality is the sole argument of the frame-
(cid:112) −1
indifferent nonlinear elastic energy density function of the material. Thus, g(u) needs to be
capable of annihilation by the right stretch tensor of a deformation, a condition that is expressed
in terms of spatial derivatives of g(u) on U; the main difficulty is that both u and U are unknown,
so this differential condition cannot simply be written down and a more sophisticated idea than
the standard vanishing of the Riemann-Christoffel tensor of a metric is needed.
The inverse design problem that we study in this paper, can be further rephrased as follows.
Let y : Ω → R3, be a smooth parametrization of S = y(Ω). Find a change of variable ξ : Ω → U
so that the pull back of the Euclidean metric on S through y is realized by the following formula:
(1.2) (∇y)T∇y = (∇ξ)T(g◦y)∇ξ in Ω.
Clearly, once a solution ξ of (1.2) is found, the material type distribution T, which is needed for
the construction of the printed film U, can be calculated by:
T := T ◦y◦ξ−1 : U = ξ(Ω) → T,
since u = y ◦ξ−1 satisfies (1.1) and consequently the properties (i)-(iii) hold. Any ξ satisfying
(1.2) is an isometry between the Riemannian manifolds (Ω,G˜) and (U,G◦ξ−1), with metrics:
(1.3) G˜ = (∇y)T∇y and G = g◦y on Ω.
For convenience of the reader, we gather some of our notational symbols in Figure 1.1.
The same problem can be set up for a three dimensional shape S ⊂ R3. In that case, the pre-
strain mapping g¯ must take values in R3×3 and the printed prestrained reference configuration
sym,pos
THE METRIC-RESTRICTED INVERSE DESIGN PROBLEM 3
is modeled by U ⊂ R3. The equation to be solved is still given by (1.2), now posed in Ω ⊂ R3.
Equivalently, a solution u : U → S to (1.1) is obtained as in (iii) above for T := T ◦u and it is
the absolute minimizer of the prestrain elastic energy (see e.g. [19]):
ˆ
(1.4) E(u,U,G) := dist2(cid:0)(∇u)G−1/2,SO(3)(cid:1) dx = 0.
U
Here, dist(F,SO(3)) stands for the distance of a matrix F from the compact set SO(3), with
respecttotheHilbert-Schmidtnorm. Whatdistinguishesourproblemsfromtheclassicalisometric
immersion problem in differential geometry, where one looks for an isometric mapping between
two given manifolds (Ω,G˜) and (U,G), is that the target manifold U = ξ(Ω) and its Riemannian
metric G = G◦ξ−1 are only given a-posteriori, after the solution is found. Note that only when G
is constant, the target metric becomes a-priori well defined and can be extended over the whole
of Rn, as it is independent of ξ, and then the problem reduces to the classical case (see Example
5.4 and a few other similar cases in Examples 5.5 and 5.6).
g
( )
pre-strain :  = g u

S
construct u,U such that
g u =(u)Tu

u
y
n
UÌ z
metrics :, WÌn
x metrics :G,G
Figure 1.1. Geometry of the problem.
Remark 1.1. Note that the minimizing solution in the equidimensional problem (1.4) is unique
uptorigidmotions. Hence, Iftheconfiguration(U,G)isprinted, theelasticbodyisboundtotake
the required shape S as requested in the design problem. Note that in case the required shape S
is two dimensional, uniqueness fails due to one more degree of freedom as the deformation u of
U ⊂ R2 takes values in a higher dimensional space R3. In this case, other restrictions on u have
to be imposed, to solve the original design problem. One particular remedy is to consider the
membrane as a thin three dimensional body; we implement this approach in Section 3. Note that
solving (1.1) directly in two dimensions implies that a compatible reference configuration can be
found, as formulated in (a) above.
Another approach in the two dimensional case would be to use a parametrization y : Ω → S
which is bending energy minimizing among all other mappings y˜ : Ω → S that induce the same
4 AMITACHARYA,MARTALEWICKAANDMOHAMMADREZAPAKZAD
metric (∇y)T∇y. This would necessitate a study of the bending energy effects, which is beyond
thescopeofthepresentpaper. Forthesakeofcomparison, wementionaparallelproblem, studied
in[15], wheretheauthorscalculatetheprestraintobeprintedinthethinfilmbyminimizingafull
three dimensional energy of deformation, consisting of both stretching and bending. The nature
oftheproblemin[15]isdifferentthanours, inasmuchasthematerialconstraintsarenotpresent.
Thepaperisorganizedasfollows. InSections2and3,westudytheabovementionedvariational
formulation of (1.2) and analyze the infimum value of the appropriate incompatibility energy,
resembling the non-Euclidean elasticity [19]. We derive a Γ-convergence result for the dimension
reduction from 3d to 2d, in the Kirchhoff-like energy scaling regime, corresponding to the square
of thickness of the thin film. In Sections 4-9, we formulate (1.2) as an algebraically constrained
system of total differential equations, in which the second derivatives of ξ are expressed in terms
of its first derivatives and the Christoffel symbols of the involved metrics. The idea is then
to investigate the integrability conditions of this system. When this method is applied in the
context of the standard Riemannian isometric immersion problem, the parameters involving ξ
can be removed from the conditions and the intrinsic conditions of immersibility will be given
in terms of the Riemann curvature tensors. In our situation, the equations do not close in a
straightforward manner, and successive differentiation of the compatibility conditions leads to a
more sophisticated algebraic description of solvability. This approach has been adapted in [1] for
deriving compatibility conditions for the Left Cauchy-Green tensor.
Acknowledgments. This project is based upon work supported by, among others, the Na-
tional Science Foundation. A.A. acknowledges support in part from grants NSF-CMMI-1435624,
NSF-DMS-1434734, and ARO W911NF-15-1-0239. M.L. was partially supported by the NSF
grants DMS-0846996 and DMS-1406730. M.R.P. was partially supported by the NSF grant DMS-
1210258. A part of this work was completed while the second and the third authors visited the
Forschungsinstitut fu¨r Mathematik at ETH (Zurich, Switzerland). The institute’s hospitality is
gratefully acknowledged.
2. A variational reformulation of the problem (1.3)
In this section, we recast the problem (1.3) in a variational setting, similar to that of non-
Euclideanelasticity[19]. Usingthesameargumentsasin[11,19], wewillanalyzethepropertiesof
theinfimumvalueoftheappropriateincompatibilityenergy,overthenaturalclassofdeformations
of W1,2 regularity.
Wewillfirstdiscusstheproblem(1.2)inthegeneraln-dimensionalsettingandonlylaterrestrict
to the case n = 2 (or n = 3). Hence, we assume that Ω is an open, bounded, simply connected
and smooth subset of Rn. We look for a bilipschitz map ξ : Ω → U := ξ(Ω), satisfying (1.3) and
which is orientation preserving:
(2.1) det∇ξ > 0 in Ω.
We begin by rewriting (1.3) as:
(2.2) G˜ = (cid:0)G1/2∇ξ(cid:1)T(cid:0)G1/2∇ξ(cid:1).
Note that, in view of the polar decomposition theorem of matrices, a vector field ξ : Ω → Rn is a
solution to (2.2), augmented by the constraint (2.1), both valid a.e. in Ω, if and only if:
(2.3) ∀a.e. x ∈ Ω ∃R = R(x) ∈ SO(n) G1/2∇ξ = R G˜1/2,
THE METRIC-RESTRICTED INVERSE DESIGN PROBLEM 5
where G1/2(x) denotes the unique symmetric positive definite square root of G(x) ∈ Rn×n ,
sym,pos
while SO(n) stands for the set of special orthogonal matrices. Define:
ˆ
(2.4) E(ξ) = dist2(cid:0)G1/2(∇ξ)G˜−1/2,SO(n)(cid:1) dx ∀ξ ∈ W1,1(Ω,Rn),
loc
Ω
where dist(F,SO(n)) is the calculated distance of a matrix F from the compact set SO(n), with
respect to the Hilbert-Schmidt norm of matrices. It immediately follows that E(ξ) = 0 if and only
if ξ is a solution to (2.2) and hence to (1.3), together with (2.1). Also, note that E(ξ) < ∞ if and
only if ξ ∈ W1,2(Ω,Rn), as can be easily deduced from the inequality:
(2.5) ∀F ∈ Rn×n |F|2 ≤ C|G1/2FG˜−1/2|2,
valid with a constant C > 0 independent of x and F.
Finally, observe that, due to the uniform positive definiteness of the matrix field G:
n n
(cid:88) (cid:88)
(2.6) |F|2 = |Fe |2 ≤ C (cid:104)Fe ,GFe (cid:105) ≤ Ctrace (FTGF).
i i i
i=1 i=1
Proposition 2.1. (i) Assume that the metrics G,G˜ are C(Ω¯,Rn×n) regular. Let ξ ∈ W1,1(Ω,Rn)
loc
satisfy (1.3) for a.e. x ∈ Ω. Then ξ ∈ W1,∞(Ω,Rn) must be Lipschitz continuous.
(ii) Assume additionally that for some k ≥ 0 and 0 < µ < 1, G,G˜ ∈ Ck,µ(Ω,Rn×n). If (1.3),
(2.1) hold a.e. in Ω (so that E(ξ) = 0), then ξ ∈ Ck+1,µ(Ω,Rn).
Proof. The first assertion clearly follows from the boundedness of G˜ and positive definiteness of
G, through (2.6). To prove (ii), recall that for a matrix F ∈ Rn×n, the matrix of cofactors of F is
cof F, with (cof F) = (−1)i+jdetFˆ , where Fˆ ∈ R(n−1)×(n−1) is obtained from F by deleting
ij ij ij
its ith row and jth column. Then, (1.3) implies that:
(cid:16)detG˜(cid:17)1/2
det∇ξ = =: a ∈ C(Ω¯,R+) and cof∇ξ = aG(∇ξ)G˜−1.
detG
Since div(cof ∇ξ) = 0 for ξ ∈ W1,∞ (where the divergence of the cofactor matrix is always taken
row-wise), we obtain that ξ satisfies the following linear system of differential equations, in the
weak sense:
div(cid:0)aG(∇ξ)G˜−1(cid:1) = 0.
Writing in coordinates ξ = (ξ1...ξn), and using the Einstein summation convention, the above
system reads:
∀i = 1...n ∂ (cid:0)aG G˜βα∂ ξj(cid:1) = 0.
α ij β
Theregularityresultisnowanimmediateconsequenceof[13,Theorem3.3]inviewoftheellipticity
of the coefficient matrix Aαβ = aG G˜αβ.
ij ij
We now prove two further auxiliary results.
Lemma 2.2. There exist constants C,M > 0, depending only on (cid:107)G(cid:107) and (cid:107)G˜(cid:107) , such that
L∞ L∞
for every ξ ∈ W1,2(Ω,Rn) there exists ξ¯∈ W1,2(Ω,Rn) with the properties:
(cid:107)∇ξ¯(cid:107) ≤ M, (cid:107)∇ξ−∇ξ¯(cid:107)2 ≤ CE(ξ) and E(ξ¯) ≤ CE(ξ).
L∞ L2(Ω)
6 AMITACHARYA,MARTALEWICKAANDMOHAMMADREZAPAKZAD
Proof. Use the approximation result of Proposition A.1. in [11] to obtain the truncation ξ¯= ξλ,
for λ > 0 having the property that if a matrix F ∈ Rn×n satisfies |F| ≥ λ then:
|F|2 ≤ Cdist2(G1/2FG˜−1/2(x),SO(n)) ∀x ∈ Ω.
Then (cid:107)∇ξλ(cid:107) ≤ Cλ := M and further, since ∇ξ = ∇ξλ a.e. in the set {|∇ξ| ≤ λ}:
L∞ ˆ ˆ
(cid:107)∇ξ−∇ξλ(cid:107)2 = |∇ξ|2 ≤ c dist2(G1/2∇ξG˜−1/2,SO(n)) dx ≤ CE(ξ).
L2(Ω)
{|∇ξ|>λ} {|∇ξ|>λ}
The last inequality of the lemma follows from the above by the triangle inequality.
Lemma 2.3. Let ξ ∈ W1,∞(Ω,Rn). Then there exists a unique weak solution φ : Ω → Rn to:
(cid:40)div(cid:0)aG(∇φ)G˜−1(cid:1) = 0 in Ω,
(2.7)
φ = ξ on ∂Ω.
Moreover, there is constant C > 0, depending only on G and G˜, and (in a nondecreasing manner)
on (cid:107)∇ξ(cid:107) , such that:
L∞
(cid:107)∇(ξ−φ)(cid:107)2 ≤ CE(ξ).
L2(Ω)
Proof. Consider the functional:
ˆ ˆ
I(ϕ) := (cid:104)G(∇ϕ)G˜−1(x) : ∇ϕ(x)(cid:105) dx = |a1/2G1/2(∇ϕ)G˜−1/2|2 dx ∀ϕ ∈ W1,2(Ω,Rn).
Ω Ω
The formula (2.5), in which we have implicitly used the coercivity of G and G˜, implies that:
(cid:107)∇ϕ(cid:107)2 ≤ CI(ϕ).
L2(Ω)
Therefore, in view of the strict convexity of I, the direct method of calculus of variations implies
that I admits a unique critical point φ in the set:
(cid:8)ϕ ∈ W1,2(Ω,Rn); ϕ = ξ on ∂Ω(cid:9).
By the symmetry of G and G˜, (2.7) is precisely the Euler-Lagrange equation of I, and hence it is
satisfied, in the weak sense, by φ.
Further, for the correction ψ = ξ−φ ∈ W1,2(Ω,Rn) it follows that:
ˆ 0 ˆ ˆ
∀η ∈ W1,2(Ω,Rn) aG(∇ψ)G˜−1 : ∇η dx = aG(∇ξ)G˜−1 : ∇η− aG(∇φ)G˜−1 : ∇η
0
Ω ˆΩ Ω
= aG(∇ξ)G˜−1 : ∇η
ˆΩ ˆ
= aG(∇ξ)G˜−1 : ∇η− cof∇ξ : ∇η.
Ω Ω
Indeed, the last term above equals to 0, since the row-wise divergence of the cofactor matrix of
∇ξ is 0, in view of ξ being Lipschitz continuous. Use now η = ψ to obtain:
ˆ
(cid:107)∇ψ(cid:107)2 ≤ CI(ψ) = C (aG(∇ξ)G˜−1−cof∇ξ) : ∇ψ dx
L2(Ω)
Ω
ˆ
(cid:18) (cid:12) (cid:12)2(cid:19)1/2
≤ C(cid:107)∇ψ(cid:107) (cid:12)aG(∇ξ)G˜−1−cof∇ξ(cid:12)
L2(Ω) (cid:12) (cid:12)
Ω
≤ C(cid:107)∇ψ(cid:107) E(ξ)1/2.
L2(Ω)
THE METRIC-RESTRICTED INVERSE DESIGN PROBLEM 7
The last inequality above follows from:
∀|F| ≤ M ∀x ∈ Ω |aGFG˜−1(x)−cofF|2 ≤ C dist2(cid:0)G1/2FG˜−1/2,SO(n)(cid:1),
M
because when G1/2FG˜−1/2 ∈ SO(n) then the difference in the left hand side above equals 0.
Theorem 2.4. Assume that the metrics G,G˜ ∈ C(Ω¯,Rn×n) are Lipschitz continuous. Define:
(2.8) κ(G,G˜) = inf E(ξ).
ξ∈W1,2(Ω,Rn)
Then, κ(G,G˜) = 0 if and only if there exists a minimizer ξ ∈ W1,2(Ω,Rn) with E(ξ) = 0. In
particular, in view of Proposition 2.1, this is equivalent to ξ being a solution to (1.3) (2.1), and ξ
is smooth if G and G˜ are smooth.
Proof. Assume, by contradiction, that for some sequence of deformations ξ ∈ W1,2(Ω,Rn), there
k
holds lim E(ξ ) = 0. By Lemma 2.2, replacing ξ by ξ¯ , we may without loss of generality
k→∞ k k k
request that (cid:107)∇ξ (cid:107) ≤ M.
k L∞
The uniform boundedness of ∇ξ implies, via the Poincar´e inequality, and after a modification
k
by a constant and passing to a subsequence, if necessary:
(2.9) lim ξ = ξ weakly in W1,2(Ω).
k
k→∞
Consider the decomposition ξ = φ +ψ , where φ solves (2.7) with the boundary data φ = ξ
k k k k k k
on ∂Ω. By the Poincar´e inequality, Lemma 2.3 implies for the sequence ψ ∈ W1,2(Ω):
k 0
lim ψ = 0 strongly in W1,2(Ω).
k
k→∞
In view of the convergence in (2.9), the sequence φ must be uniformly bounded in W1,2(Ω), and
k
hence by [23, Theorem 4.11, estimate (4.18)]:
∀Ω(cid:48) ⊂⊂ Ω ∃C ∀k (cid:107)φ (cid:107) ≤ C (cid:107)φ (cid:107) ≤ C.
Ω(cid:48) k W2,2(Ω(cid:48)) Ω(cid:48) k W1,2(Ω)
Consequently, φ convergetoφstronglyinW1,2(Ω). RecallingthatE(ξ )convergeto0, wefinally
k loc k
conclude that:
E(ξ) = 0.
This proves the claimed result.
3. A dimension reduction result for the energies (2.4)
The variational problem induced by (2.4) is difficult due to the lack of convexity. One way
of reducing the complexity of this problem is to assume that the target shapes are thin bodies,
described by a “thin limit” residual theory, which is potentially easier to analyze. We concentrate
onthecasen = 3andtheenergyfunctional(2.4)relativetoafamilyofthinfilmsΩh = ω×(−h, h)
2 2
withthemidplateω ⊂ R2 givenbyanopen,smoothandboundedset. Thisset-upcorrespondstoa
scenariowhereatarget3-dimensionalthinshellistobemanufactured,ratherthana2-dimensional
surface as in Figure 1.1. As we shall see below, an approximate realization of the ideal thin shell
is delivered by solving the variational problem in the limit of the vanishing thickness h.
Assume further that the given smooth metrics G,G˜ : Ω¯h → R3×3 are thickness-independent:
h h
G,G˜(x(cid:48),x ) = G,G˜(x(cid:48)) ∀x = (x(cid:48),x ) ∈ ω×(− , ),
3 3
2 2
8 AMITACHARYA,MARTALEWICKAANDMOHAMMADREZAPAKZAD
and denote:
ˆ
(3.1) Eh(ξh) = 1 W(cid:0)G1/2(∇ξh)G˜−1/2(cid:1) dx ∀ξh ∈ W1,2(Ωh,R3).
h
Ωh
The energy density W : R3×3 → R¯ is assumed to be C2 regular close to SO(3), and to satisfy
+
the conditions of normalisation, frame invariance and bound from below:
∃c > 0 ∀F ∈ R3×3 ∀R ∈ SO(3) W(R) = 0, W(RF) = W(F),
W(F) ≥ c dist2(F,SO(3)).
Following the approach of [11], which has been further developed in [12, 19, 3, 18] (see also [18]
foranextensivereviewoftheliterature), weobtainthefollowingΓ-convergenceresults, describing
in a rigorous manner the asymptotic behavior of the approximate minimizers of the energy (3.1).
Theorem 3.1. For a given sequence of deformations ξh ∈ W1,2(Ωh,R3) satisfying:
(3.2) ∃C > 0 ∀h Eh(ξh) ≤ Ch2,
thereexistsasequenceofvectorsch ∈ R3, suchthatthefollowingpropertiesholdforthenormalised
deformations yh ∈ W1,2(Ω1,R3):
yh(x(cid:48),x ) = ξh(x(cid:48),hx )−ch.
3 3
(i) There exists y ∈ W2,2(ω,R3) such that, up to a subsequence:
yh → y strongly in W1,2(Ω1,R3).
The deformation y realizes the compatibility of metrics G and G˜ on the midplate ω:
(3.3) (∇y)TG ∇y = G˜ .
2×2
The unit normal N(cid:126) to the surface y(ω) and the metric G-induced normal M(cid:126) below have
the regularity N(cid:126),M(cid:126) ∈ W1,2∩L∞(ω,R3):
√
∂ y×∂ y detG
N(cid:126) = 1 2 M(cid:126) = G−1(∂ y×∂ y),
(cid:113) 1 2
|∂ y×∂ y|
1 2 detG˜
2×2
where we observe that (cid:104)∂ y,GM(cid:126) (cid:105) = 0 for i = 1,2 and (cid:104)M(cid:126) ,GM(cid:126) (cid:105) = 1.
i
(ii) Up to a subsequence, we have the convergence:
1
∂ yh →(cid:126)b strongly in L2(Ω1,R3),
3
h
where the Cosserat vector(cid:126)b ∈ W1,2∩L∞(ω,R3) is given by:
(cid:112)
(cid:20) G˜ (cid:21) detG˜
(3.4) (cid:126)b = (∇y)(G˜ )−1 13 + M(cid:126) .
2×2 G˜ (cid:113)
23 detG˜
2×2
(iii) Define the quadratic forms:
Q (F) = D2W(Id)(F,F),
3
(cid:110) (cid:16) (cid:17) (cid:111)
Q (x(cid:48),F ) = min Q G˜(x)−1/2F˜G˜(x(cid:48))−1/2 ; F˜ ∈ R3×3 with F˜ = F .
2 2×2 3 2×2 2×2
THE METRIC-RESTRICTED INVERSE DESIGN PROBLEM 9
The form Q is defined for all F ∈ R3×3, while Q (x(cid:48),·) are defined on F ∈ R2×2. Both
3 2 2×2
forms Q and all Q are nonnegative definite and depend only on the symmetric parts of
3 2
their arguments. In particular, when the energy density W is isotropic, i.e.:
∀F ∈ R3×3 ∀R ∈ SO(3) W(RF) = W(F),
then Q (x(cid:48),·) is given in terms of the Lam´e coefficients λ,µ > 0 by:
2
(cid:12) (cid:12)2 λµ (cid:12) (cid:16) (cid:17)(cid:12)2
Q (x(cid:48),F ) = µ(cid:12)(G˜ )−1/2F (G˜ )−1/2(cid:12) + (cid:12)tr (G˜ )−1/2F (G˜ )−1/2 (cid:12) ,
2 2×2 (cid:12) 2×2 2×2 2×2 (cid:12) λ+µ (cid:12) 2×2 2×2 2×2 (cid:12)
for all F ∈ R2×2.
2×2 sym
(iii) We have the lower bound:
ˆ
1 1 (cid:16) (cid:17)
(3.5) liminf Eh(ξh) ≥ I (y) := Q x(cid:48),(∇y)TG ∇(cid:126)b dx(cid:48).
h→0 h2 G,G˜ 24 Ω 2
Proof. The convergences in (i) and (ii) rely on a version of an approximation result from [11];
there exists matrix fields Qh ∈ W1,2(ω,R3×3) and a constant C uniform in h, i.e. depending only
on the geometry of ω and on G,G˜, such that:
ˆ ˆ
(cid:18) (cid:19)
1 |∇ξh(x(cid:48),x )−Qh(x(cid:48))|2 dx ≤ C h2+ 1 dist2(cid:0)G1/2∇ξhG˜−1/2,SO(3)(cid:1) dx ,
3
h h
ˆ Ωh ˆ Ωh
(cid:18) (cid:19)
|∇Qh(x(cid:48))|2 dx(cid:48) ≤ C 1+ 1 dist2(cid:0)G1/2∇ξhG˜−1/2,SO(3)(cid:1) dx .
h3
Ω Ωh
Further ingredients of the proof follow exactly as in [3], so we suppress the details.
Theorem 3.2. For every compatible immersion y ∈ W2,2(Ω,R3) satisfying (3.3), there exists a
sequence of recovery deformations ξh ∈ W1,2(Ωh,R3), such that:
(i) The rescaled sequence yh(x(cid:48),x ) = ξh(x(cid:48),hx ) converges in W1,2(Ω1,R3) to y.
3 3
(ii) One has:
1
lim Eh(ξh) = I (y),
h→0 h2 G,G˜
where the Cosserat vector(cid:126)b in the definition (3.5) of I is derived by (3.4).
G,G˜
Proof. Let y ∈ W2,2(Ω,R3) satisfy (3.3). Define(cid:126)b according to (3.4) and let:
(cid:104) (cid:105)
Q = ∂ y ∂ y (cid:126)b ∈ W1,2∩L∞(ω,R3×3).
1 2
By Theorem 3.1, it follows that:
G1/2QG˜−1/2 ∈ SO(3) ∀a.e. x(cid:48) ∈ ω.
Define the limiting warping field d(cid:126)∈ L2(Ω,R3):
  (cid:104)∂ (cid:126)b,G(cid:126)b(cid:105) 
1
d(cid:126)(x(cid:48)) = G−1QT,−1c(cid:0)x(cid:48),(∇y)TG ∇(cid:126)b(cid:1)− (cid:104)∂ (cid:126)b,G(cid:126)b(cid:105) ,
2
0
where c(x(cid:48),F ) denotes the unique minimizer of the problem in:
2×2
(cid:110) (cid:111)
∀F ∈ R2×2 Q (x(cid:48),F ) = min Q (cid:0)G˜−1/2(F∗ +sym(c⊗e ))G˜−1/2(cid:1); c ∈ R3 .
2×2 sym 2 2×2 3 2×2 3
10 AMITACHARYA,MARTALEWICKAANDMOHAMMADREZAPAKZAD
Let {dh} be a approximating sequence in W1,∞(Ω,R3), satisfying:
(3.6) dh → d(cid:126) strongly in L2(ω,R3), and h2(cid:107)dh(cid:107) → 0.
W1,∞
Note that such sequence can always be derived by reparametrizing (slowing down) a sequence
of smooth approximations of d(cid:126). Similiarly, consider the approximations yh ∈ W2,∞(ω,R3) and
bh ∈ W1,∞(ω,R3), with the following properties:
yh → y strongly in W2,2(ω,R3), and bh →(cid:126)b strongly in W1,2(ω,R3)
(cid:16) (cid:17)
h (cid:107)yh(cid:107) +(cid:107)bh(cid:107) ≤ (cid:15)
(3.7) W2,∞ W1,∞
1 (cid:110) (cid:111)
|ω\ω | → 0, where ω = x(cid:48) ∈ ω; yh(x(cid:48)) = y(x(cid:48)) and bh(x(cid:48)) =(cid:126)b(x(cid:48))
h2 h h
for some appropriately small (cid:15) > 0. Existence of approximations with the claimed properties
follows by partition of unity and truncation arguments, as a special case of the Lusin-type result
for Sobolev functions (see Proposition 2 in [12]).
We now define the recovery sequence ξh ∈ W1,∞(Ωh,R3) by:
x2
ξh(x(cid:48),x ) = yh(x(cid:48))+x bh(x(cid:48))+ 3dh(x(cid:48)).
3 3
2
Consequently, the rescalings yh ∈ W1,∞(Ω1,R3) are:
h2
yh(x(cid:48),x ) = yh(x(cid:48))+hx bh(x(cid:48))+ x2dh(x(cid:48)),
3 3 2 3
and so in view of (3.6) and (3.7), Theorem 3.2 (i) follows directly. The remaining convergence in
(ii) is achieved via standard calculations exactly as in [3]. We suppress the details.
It now immediately follows that:
Corollary 3.3. Existence of a W2,2 regular immersion satisfying (3.3) is equivalent to the upper
bound on the energy scaling at minimizers:
∃C > 0 inf Eh(ξ) ≤ Ch2.
ξ∈W1,2(Ωh,R3)
The following corollary is a standard conclusion of the established Γ-convergence (see e.g. [4]).
It indicates that sequences of approximate solutions to the original problem on a thin shell are in
one-one correspondence to the minimizers of the thin limit variational model I .
G,G˜
Corollary 3.4. Assume that (3.3) admits a W2,2-regular solution y. Then any sequence of ap-
proximate minimizers ξh of (3.1), satisfying the property:
lim 1 (cid:0)Eh(ξh)−infEh(cid:1) = 0,
h→0 h2
converges, after the proper rescaling and up to a subsequence (see Theorem 3.1), to a minimizer
of the functional I . In particular, I attains its minimum. Conversely, any minimizer y of
G,G˜ G,G˜
I is a limit of approximate minimizers ξh to (3.1).
G,G˜
Onafinalnote,observethatthedefectκ(G,G˜),asdefinedin(2.8)),isoftheorderh2minI +
G,G˜
o(h2). It would be hence of interest to discuss the necessary and sufficient conditions for having
minI = 0, which in case of G = Id have been precisely derived in [3].
G,G˜ 3