A Shape-Newton approach to the problem of covering with identical balls∗

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A Shape-Newton approach to the problem of covering with identical balls

^∗

Ernesto G. Birgin^† Antoine Laurain^‡ Rafael Massambone^† Arthur G. Santana^† June 4, 2021

Abstract

The problem of covering a region of the plane with a fixed number of minimum-radius identical balls is studied in the present work. An explicit construction of bi-Lipschitz mappings is provided to model small perturbations of the union of balls. This allows us to obtain ana- lytical expressions for first- and second-order derivatives using nonsmooth shape optimization techniques under appropriate regularity assumptions. Singular cases are also studied using asymptotic analysis. For the case of regions given by the union of disjoint convex polygons, algorithms based on Voronoi diagrams that do not rely on approximations are given to compute the derivatives. Extensive numerical experiments illustrate the capabilities and limitations of the introduced approach.

Keywords: covering problem, nonsmooth shape optimization, Augmented Lagrangian, New- ton’s method.

AMS subject classification: 49Q10, 49J52, 49Q12

1 Introduction

The problem of covering a region of the plane with a fixed number of minimum-radius identical balls is studied in the present work by expanding the nonsmooth shape optimization approach introduced in [6]. The main challenge in this previous work was the the first-order shape sensitivity analysis with respect to perturbations of the balls’ centers and radii. Therefore, investigating the second-order shape sensitivity is a natural albeit challenging extension of [6].

Shape optimization is the study of optimization problems where the variable is a geometric object; see [13, 17, 34]. One of the key concepts in this discipline is the notion ofshape derivative, that measures the sensitivity of functions with respect to perturbations of the geometry. The theoretical study of second-order shape derivatives is a difficult topic in shape optimization. There exists an abundant literature on the shape Hessian in the smooth setting [10, 11, 13, 34]; while in the

∗This work has been partially supported by FAPESP (grants 2013/07375-0, 2016/01860-1, 2018/24293-0, and 2019/25258-7) and CNPq (grants 302682/2019-8, 304258/2018-0, and 408175/2018-4).

†Department of Computer Science, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010, Cidade Universitária, 05508-090, São Paulo, SP, Brazil. e-mails: egbirgin@ime.usp.br, rmassam- bone@ime.usp.br, and ags@ime.usp.br

‡Department of Applied Mathematics, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão, 1010, Cidade Universitária, 05508-090, São Paulo, SP, Brazil. e-mail: laurain@ime.usp.br

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nonsmooth setting it is still an active research topic [24, 25]. Numerical methods based on second- order shape derivative are rarely used in shape optimization due to several difficulties. First of all, the second-order shape derivative is often difficult to compute and costly to implement numerically, especially when partial differential equations are involved. Second, the shape Hessian presents several theoretical issues, such as thetwo norms-discrepancy and lack of coercivity, that have been extensively studied in control problems; see [1, 10] and the references therein. There exist only few attempts at defining numerical methods based on second-order information in shape optimization.

In [14], a regularized shape-Newton method is introduced to solve an inverse problem for star- shaped geometries. Second order preconditioning of the shape gradient has been used in [19] for image segmentation and in [3, 32] for aerodynamic optimization. Automatic shape differentiation has also been successfully employed to compute first- and second-order shape derivatives [16, 31].

We also observe that the numerical investigations using Newton-type algorithms [14, 19] are set in a relatively smooth setting. In [25], the shape Hessian was calculated for nonsmooth geometries and polygons in a form that was convenient for numerical experiments, but no numerical investigations were performed. To the best of our knowledge, the present paper is the first attempt at designing and analyzing a shape-Newton algorithm in a genuinely nonsmooth setting.

From a theoretical perspective, the main achievement of [6] was to build bi-Lipschitz transformations to model the geometry perturbations corresponding to covering with identical balls. In the present work, these transformations are key elements for the calculation of the second-order shape derivative, which, unlike the first-order shape derivative, differs from the expression that would be obtained in a smooth setting. Indeed, for the piecewise smooth shapes considered in the covering problem, various terms with a support at singular boundary points, typically circles intersection, appear in the shape Hessian.

Due to the generality of the regions to be covered considered in [6], in the presented numerical experiments, the function that measures the covering and its first-order derivatives were approxi- mated with discretization strategies that may by very time consuming if high precision is required.

In the present work, by restricting the region to be covered to be the union of disjoint convex polygons, algorithms based on Voronoi diagrams to compute the covering function and its first- and second-order derivatives without relying on approximations are given.

The problem of covering a two-dimensional region with identical balls has already been considered in the literature. Covering equilateral triangles and squares was considered in [29] and [30], respectively; while covering the union and difference of polygons was considered in [35]. The covering of rectangles, triangles, squares and arbitrary regions was considered in [18], [27], [28] and [37], respectively. However, the problem addressed in [37] actually consists of covering an arbitrary set of points, which is substantially different from the problem of covering an entire region. All of these papers approach the problem as an optimization problem. In [18, 27, 28] a simulated annealing approach with local search in which the centers of the balls are chosen as points on an adaptive mesh is considered. In [29, 30], a discrete rule is used to define the radius; while a BFGS method is used to solve subproblems in which the radius is fixed. A feasible direction method that requires solving a linear programming problem at each iteration was proposed in [35]. None of the men- tioned works addresses the problem in a unified way as a continuous optimization problem, nor do they present first- or second-order derivatives of the functions that define the problem. In [5], the problem of covering an arbitrary region is modeled as a nonlinear semidefinite programming problem using convex algebraic geometry tools. The introduced model describes the covering problem without resorting to discretizations, but it depends on some polynomials of unknown degrees whose

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coefficients are difficult to compute, limiting the applicability of the method.

The rest of this paper is organized as follows. Section 2 presents a formal definition of the problem, the formula for the first-order derivative introduced in [6], and the formula for the second-order derivative being introduced in the present work. Section 3 presents the derivation of the second- order derivatives for non-degenerate cases; while degenerate cases are considered in Section 4.

Algorithms based on Voronoi diagrams for the exact calculation of the covering function and its first- and second-order derivatives are introduced in Section 5. Extensive numerical experiments are given in Section 6. Final considerations are given in Section 7.

Notation: Given x, y ∈ Rⁿ, x·y = x^>y ∈ R; while x⊗y = xy^> ∈ R^n×n. The divergence of a sufficiently smooth vector field R² 3 (x, y) 7→ V(x, y) = (V₁(x, y), V₂(x, y)) ∈ R² is defined by divV := ^∂V_∂x¹ +^∂V_∂y², and its Jacobian matrix is denotedDV. Given an open setS∈Rⁿ,S denotes its closure, ∂S=S\S its boundary, and Vol(S) its volume. LetB(xi, r) denote an open ball with centerxi∈R² and radiusr. For a sufficiently smooth setS ⊂R²,νS(z) denotes the unitary-norm outwards normal vector to S atz and τ_S(z) the unitary-norm tangent vector to∂S atz (pointing counter-clockwise). In the particular case S =B(xi, r) we use the simpler notation νi := ν_B(x_i_,r) and τi := τ_B(x_i_,r), and we have νi(z) = (cosθz,sinθz)^> and τi(z) = (−sinθz,cosθz)^>, where θz is the angular coordinate ofz−xi. For intersection pointsz∈∂S∩B(xi, r), we also use the notation ν−i(z) :=ν_S(z).

2 The shape optimization problem

LetA⊂R² and Ω(x, r) =∪^m_i=1B(xi, r) withx:={x_i}^m_i=1. We consider the problem of covering A using a fixed number m of identical balls B(x_i, r) with minimum radius r, i.e., we are looking for (x, r)∈R^2m+1 such thatA⊂Ω(x, r) with minimumr. The problem can be formulated as

Minimize

(x,r)∈R^2m+1 r subject to G(x, r) = 0, (1)

where

G(x, r) := Vol(A)−Vol(A∩Ω(x, r)). (2)

Note that G(x, r) = 0 if and only if A ⊂ Ω(x, r) up to a set of zero measure, i.e., when Ω(x, r) covers A.

The derivatives of G can be computed using techniques of shape calculus [13, 17, 26, 25, 34].

In particular it was shown in [6] that, under suitable assumptions,

∇G(x, r) =− Z

A1

ν₁(z)dz,· · · , Z

Am

ν_m(z)dz, Z

∂Ω(x,r)∩A

dz >

, (3)

where

A_i =∂B(x_i, r)∩∂Ω(x, r)∩A (4)

fori= 1, . . . , m.

In the present work, we show that

∇²G(x, r) =

∇²_xG(x, r) ∇²_x,rG(x, r)

∇²_x,rG(x, r)^> ∇²_rG(x, r)

, (5)

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where ∇²_xG(x, r) ∈ R^2m×2m, ∇²_x,rG(x, r) ∈ R^2m, and ∇²_rG(x, r) = ∂_r²G(x, r) ∈ R are described below. Their description is based on the fact that each setA_ican be represented by a finite number mi ≥0 of arcs of the circle∂B(xi, r). Note that, since (∪^m_i=1∂B(xi, r))∩∂Ω(x, r) =∂Ω(x, r), by (4),

m

[

i=1

A_i=∂Ω(x, r)∩A, (6)

i.e., the union of all A_i represents a partition of ∂Ω(x, r)∩A; see Figure 1. Each arc in A_i can be represented by a pair of points (v, w), named starting and ending points, in counter-clockwise direction, i.e., such that the angular coordinatesθ_v andθ_wofv−x_i andw−x_i, respectively, satisfy θv ∈ [0,2π) and θw ∈ (θv, θv + 2π]; see Figure 2. If A_i is not a full circle, we denote by Ai the set of pairs (v, w) that represent the arcs in A_i; otherwise, we define Ai = ∅. In addition, if A_i is a full circle, then we set Circle(Ai) equal to true; otherwise, we set Circle(Ai) equal to false.

We say a configuration (x, r) is non-degenerate if, for every i= 1, . . . , m, every (v, w) ∈ Ai, and everyz∈ {v, w}, there exists one and only oneν−i(z) andν−i(z)·τ_i(z)6= 0. A characterization of non-degenerate configurations, which satisfy Assumptions 1 and 2, is given in the next section.

(a) (b)

Figure 1: (a) represents a region A to be covered and an arbitrary configuration of balls Ω(x, r).

(b) represents, in red, ∂Ω(x, r)∩A. Each A_i corresponds to the red arcs that intersect ∂B(x_i, r).

Note that, in this example, most sets A_i contain two or three maximal arcs; and there is only one set A_i with four maximal arcs.

Assuming (x, r) is non-degenerate, we have that ∇²_rG(x, r) in (5) is given by

∇²_rG(x, r) =−Per(∂Ω(x, r)∩A)

r −

m

X

i=1

X

(v,w)∈Ai

s|L(z)| −ν−i(z)·ν_i(z) ν−i(z)·τ_i(z)

{w v

, (7)

where, for an arbitrary expression Φ(z),JΦ(z)K

wv := Φ(w)−Φ(v), Per(S) denotes the perimeter of the setS, and, for an extremezof an arc represented by (v, w)∈Ai,L(z) ={`∈ {1, . . . , m} \ {i} |z∈

∂B(x_`, r)}.

Matrix ∇²_xG(x, r) in (5) is given by the 2×2 diagonal blocks

∂_x²_i_x_iG(x, r) = 1 r

Z

A_i

−ν_i(z)⊗νi(z) +τi(z)⊗τi(z)dz+ X

(v,w)∈Ai

sν−i(z)·νi(z)

ν−i(z)·τi(z)νi(z)⊗νi(z) {w

v

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xi

x`

u v

w

A

ν_`(u)

ν_−`(u) =νA(u) ν_i(v)

ν_i(w) ν_−i(w) =ν_`(w)

ν_−i(v) =νA(v)

Figure 2: The setA_i =∂B(x_i, r)∩Ω(x, r)∩Ais composed of two arcs (in red). Ifz∈∂B(x_i, r)∩

∂B(x`, r) for some ` 6= i, as for z = w, then ν−i(z) = ν`(z), while if z ∈ ∂B(xi, r)∩∂A, as for z∈ {u, v}, thenν−i(z) =ν_A(z).

and the 2×2 off-diagonal blocks

∂_x²_i_x_`G(x, r) = X

v∈I_i`

νi(v)⊗ν`(v)

ν_`(v)·τi(v) − X

w∈O_i`

νi(w)⊗ν`(w)

ν_`(w)·τi(w) , (9) where I_i` = {v ∈ ∂B(x_`, r) | (v,·) ∈ Ai} and O_i` = {w ∈ ∂B(x_`, r) | (·, w) ∈ Ai}. (Note that I_i`=O_i`=∅for all`6=iifAi =∅.) Finally, array∇²_x,rG(x, r) in (5) is given by the 2-dimensional arrays

∂_x²_i_rG(x, r) =−1 r

Z

A_i

νi(z)dz+ X

(v,w)∈Ai

u v

ν−i(z)·νi(z)

ν−i(z)·τ_i(z)νi(z)− X

`∈L(z)

νi(z) τ_i(z)·ν_`(z)

}

~

w

v

. (10)

3 Proof of second-order differentiability of G

In this section, we prove that the second-order derivatives ofG, as defined in (2), are given by (5, 7, 8, 9, 10). In [6] we have built appropriate bi-Lipschitz mappingsT_tin order to use integration by substitution for the differentiation of G(x+tδx, r) and G(x, r+tδr). Some of the more technical aspects of these constructions were related to the fact that G(x, r) is an area functional, which required defining T_t on Ω(x, r)∩A and on ∂(Ω(x, r) ∩A). Since ∇G only involves boundary integrals that in addition can be decomposed into integrals on arcs, this facilitates the construction of the mappings Tt required for the calculation of ∇²G(x, r), as Tt only needs to be defined on

∂Ω(x, r)∩A.

We consider two types of transformations for the shape sensitivity analysis. First, in the case of fixed radius and center perturbations one needs a mappingTtbetween the reference set∂Ω(x, r)∩A

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and the perturbed set ∂Ω(x+tδx, r)∩A; see Theorem 2. Second, in the case of fixed centers and radius perturbation one needs a mapping T_t between the reference set ∂Ω(x, r)∩A and the perturbed set ∂Ω(x, r+tδr)∩A; see Theorem 3. The shape sensitivity analysis of ∇G is then achieved through integration by substitution using T_t. The construction of these mappings T_t is similar to the constructions in [6]; however the results are presented in a different way as we need specific properties ofTtto compute the derivatives of∇G. One of the main differences with respect to [6] appears in Theorem 2, where one considers a simultaneous perturbations of all the balls’

center, which allows us to simplify the calculations of the Hessian of G. On the one hand,T_t was used in [6] mainly to prove first-order shape differentiability and its unusual structure did not affect the expression of the first-order shape derivative, in the sense that a similar formula would have been obtained in a smooth setting. On the other hand, the expression of the second-order shape derivative of G at a nonsmooth reference domain Ω differs significantly from the expression that would be obtained for a smooth Ω, as it involves terms with a support at singular boundary points of Ω, and the particular structure of T_t now plays an important role in the calculation of those singular terms. This can be understood by considering that, unlike the first-order derivative, the second-order shape derivative depends on the tangential component of ∂tTt|_t=0 on the boundary of the reference domain.

In [6], we have described detailed conditions to avoid degenerate situations and we also discussed various examples of such degeneracies and how they may affect the numerical algorithm. In the present paper we use the same conditions to prove second-order differentiability ofG. To summarize, the main issues when studying the differentiability ofGarise when two balls are tangent or exactly superposed, when the boundaries of more than two balls intersect at the same point, or when Ω(x, r) and Aare not compatible in the sense of Definition 1. The role of Assumptions 1 and 2 is to avoid these singular cases, which allows us to prove second-order differentiability of G. We emphasize that these assumptions only exclude a null-measure set of balls’ configurations in R^2m+1, and in Section 4 we show via the study of several singular cases that the second-order differentiability ofG fails when these assumptions are not satisfied.

Assumption 1. The centers {x_i}^m_i=1 satisfy kx_i−x_jk ∈ {0,/ 2r} for all 1 ≤i, j ≤ m, i 6= j and

∂B(xi, r)∩∂B(xj, r)∩∂B(xk, r) =∅ for all1≤i, j, k ≤m with i, j, k pairwise distinct.

Definition 1. Let ω₁, ω₂ be open subsets of R². We callω₁ and ω₂ compatible ifω₁∩ω₂ 6=∅, ω₁ andω2 are Lipschitz domains, and the following conditions hold: (i)ω1∩ω2 is a Lipschitz domain;

(ii) ∂ω1∩∂ω2 is finite; (iii) ∂ω1 and ∂ω2 are locally smooth in a neighborhood of ∂ω1∩∂ω2; (iv) τ₁(x)·ν₂(x) 6= 0 for all x∈∂ω₁∩∂ω₂, where τ₁(x) is a tangent vector to∂ω₁ atx and ν₂(x) is a normal vector to ∂ω2 atx.

Assumption 2. Sets Ω(x, r) and A are compatible.

We observe that Ω(x, r) is Lipschitz under Assumption 1, and if, in addition, the intersection of∂Ω(x, r) and∂A is empty, then Assumption 2 holds. Hence, in this particular case we can drop Assumption 2 in Theorems 2 and 3.

We also recall the following basic results, which are key ingredients for the calculation of the shape Hessian ofG.

Theorem 1 (Tangential divergence theorem). Let Γ ⊂ R² be a C^k open curve, k ≥ 2, with a parameterizationγ, and denote(v, w) the starting and ending points ofΓ, respectively, with respect toγ. Letτ be the unitary-norm tangent vector to Γ,ν the unitary-norm normal vector toΓ, andH

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the mean curvature of Γ, with respect to the parameterizationγ. Let F ∈W^1,1(Γ,R²)∩C⁰(Γ,R²), then we have

Z

Γ

divΓ(F) = Z

Γ

HF ·ν+F(w)·τ(w)−F(v)·τ(v) = Z

Γ

HF·ν+JF(z)·τ(z)K

w v, where div_Γ(F) := div(F)−DF ν·ν is the tangential divergence of F onΓ.

Proof. The result follows from [33, § 7.2] and [13, Ch. 9,§ 5.5].

Lemma 1 (Integration by substitution for line integrals). Let Γ⊂R² be a C^k open curve, k≥2, and ν a unitary-norm normal vector toΓ. LetF ∈C⁰(Γ,R²) and Tt: Γ→Tt(Γ) be a bi-Lipschitz mapping. Then

Z

Tt(Γ)

F(z)dz = Z

Γ

F(T_t(z))ω_t, where

ω_t(z) :=kM(z, t)ν(z)k (11)

and M(z, t) := det(DT_t(z))DT_t(z)^−> is the cofactor matrix of DT_t(z). Furthermore, we have

∂_tω_t|_t=0 = div_ΓV with V :=∂_tT_t|_t=0 on Γ. (12) Proof. See [17, Prop. 5.4.3].

3.1 Construction of a perturbation field for center perturbations

Theorem 2 below employs several ideas from [6, Thm. 3.2 & Thm. 3.6]. However, an important difference is that we consider simultaneous center perturbations for all balls instead of just one, which is more convenient for the calculation of ∇²G. Theorem 2 provides an appropriate mapping T_t for the differentiation of ∂_x_iG(x+tδx, r) that will be used in Sections 3.4 and 3.5 and for the differentiation of∂_rG(x+tδx, r) in Section 3.6.

Theorem 2. Suppose that Assumptions 1 and 2 hold. Then there exists t0 >0 such that for all t∈[0, t₀]we have the following decomposition

∂Ω(x+tδx, r)∩A=

k¯

[

k=1

S_k(t), (13)

where ¯kis independent oft, S_k(t) are arcs parameterized by an angle aperture[θ_k,v(t), θ_k,w(t)], and t7→θ_k,v(t),t7→θ_k,w(t) are continuous functions on[0, t₀].

Also, for all t ∈ [0, t₀] there exists a bi-Lipschitz mapping T_t : ∂Ω(x, r)∩A → R² satisfying Tt(∂Ω(x, r)∩A) =∂Ω(x+tδx, r)∩A and Tt(S_k(0)) =S_k(t) for all k= 1, . . . ,¯k. Furthermore, we have

V := ∂_tT_t|_t=0 =δx_i+∂_tξ(0, θ)rτ_i onS_k(0)⊂∂B(x_i, r), (14) where ξ is defined in (20) and

V(z) =δx_i− ν_A(z)·δx_i

τ_i(z)·ν_A(z)τ_i(z) if z∈∂B(x_i, r)∩∂A, (15) V(z) =δxi−ν`(z)·(δxi−δx`)

τi(z)·ν_`(z) τi(z) if z∈∂B(xi, r)∩∂B(x`, r), i6=`. (16)

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Proof. The decomposition (13) relies on Assumptions 1 and 2 and is obtained in a similar way as in [6, Thm. 3.2]. Therefore, in this proof we focus on the construction of the mapping T_t. We observe that each extremity of the arcsS_k(t) in the decomposition (13) is either a point belonging to∂B(x_i+tδx_i, r)∩∂A or a point in∂B(x_i+tδx_i, r)∩∂B(x_`+tδx_`, r).

We first provide a general formula for the angle ϑ(t), in local polar coordinates with the pole xi +tδxi, describing an intersection point between the circle ∂B(xi +tδxi, r) and ∂A. Let z ∈

∂B(x_i, r)∩∂Aand φbe the oriented distance function toA, defined asφ(x) :=d(x, A)−d(x, A^c), where d(x, A) is the distance from x to the set A. Since Ω(x, r) and A are compatible due to Assumption 2, it follows that ∂Ais locally smooth around the points ∂B(xi, r)∩∂A, hence there exists a neighborhood U_z of z such that the restriction of φ to U_z is smooth, φ(x) = 0 and k∇φ(x)k= 1 for all x∈∂A∩U_z.

Let (r, θz) denote the polar coordinates of z, with the pole xi. Introduce the function ψ(t, ϑ) =φ

xi+tδxi+r cosϑ

sinϑ

.

We compute

∂_ϑψ(0, θ_z) =r

−sinθ_z cosθ_z

· ∇φ

x_i+r

cosθ_z sinθ_z

=rτ_i(z)· ∇φ(z).

Since Ω(x, r) andAare compatible,B(x_i, r) is not tangent to∂Aand usingk∇φ(z)k= 1 we obtain τi(z)· ∇φ(z)6= 0. Thus, we can apply the implicit function theorem and this yields the existence of a smooth function [0, t₀]3t7→ ϑ(t) withψ(t, ϑ(t)) = 0 andϑ(0) =θ_z. We also compute, using

∇φ(z) =k∇φ(z)kν_A(z) since φis the oriented distance function to ∂A, ϑ⁰(0) =−∂_tψ(0, ϑ(0))

∂ϑψ(0, ϑ(0)) =− ∇φ(z)·δx_i

rτi(z)· ∇φ(z) =− ν_A(z)·δx_i

rτi(z)·νA(z). (17) We now consider the second case of an intersection point in∂B(xi+tδxi, r)∩∂B(xj+tδx`, r), i6=`. Introduce the functions

ψ(t, ϑ) =kζ(t, ϑ)k²−r² with ζ(t, ϑ) =x_i+tδx_i−x_`−tδx_`+r cosϑ

sinϑ

.

Observe thatϑ7→ζ(t, ϑ) is a parameterization of the circle∂B(xi+tδxi, r) in a coordinate system of center x_`, which means that the solutions of the equation ψ(t, ϑ) = 0 describe the intersections between∂B(x_i+tδx_i, r) and ∂B(x_`+tδx_`, r). We compute ∂_ϑψ(0, ϑ) = 2ζ(0, ϑ)·∂_ϑζ(0, ϑ) with

ζ(0, ϑ) =x_i−x_`+r cosϑ

sinϑ

and ∂_ϑζ(0, ϑ) =r

−sinϑ cosϑ

.

Now letz∈∂B(xi, r)∩∂B(x`, r) and letθzbe the corresponding angle in a polar coordinate system with polexi. Since Assumption 1 is satisfied, it is easy to see that

∂_ϑψ(0, θ_z) = 2ζ(0, θ_z)·∂_ϑζ(0, θ_z)6= 0.

Hence, the implicit function theorem can be applied to (t, ϑ) 7→ ψ(t, ϑ) in a neighbourhood of (0, θ_z). This yields the existence, for t₀ sufficiently small, of a smooth function t7→ ϑ(t) in [0, t₀] such thatψ(t, ϑ(t)) = 0 in [0, t0] and ϑ(0) =θz. We also have the derivative

ϑ⁰(t) =−∂tψ(t, ϑ(t))

∂_ϑψ(t, ϑ(t)) =−ζ(t, ϑ(t))·∂tζ(t, ϑ(t)) ζ(t, ϑ(t))·∂_ϑζ(t, ϑ(t)),

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and in particular, using ν_i = (cosθ_z,sinθ_z)^> and τ_i = (−sinθ_z,cosθ_z)^>, ϑ⁰(0) =−(x_i−x_`+rν_i)·(δx_i−δx_`)

(xi−x`+rνi)·(rτi) =−ν_`·(δx_i−δx_`) rν`·τi

. (18)

We are now ready to build the mapping Tt. Let S(t) ⊂ ∂B(xi +tδxi, r) be one of the arcs parameterized by the angle aperture [θ_v(t), θ_w(t)] in the decomposition (13); we have dropped the index k for simplicity. Then, θv(t) and θw(t) are given by ϑ(t) with either θz = θv(0) or θz=θw(0), andϑ(t) either corresponds to an intersection∂B(xi+tδxi, r)∩∂Aor to an intersection

∂B(x_i+tδx_i, r)∩∂B(x_`+tδx_`, r). Thus we defineT_t on the arcS(0) as Tt(x) :=xi+tδxi+r

cosξ(t, θ) sinξ(t, θ)

with x=xi+r cosθ

sinθ

∈ S(0), (19) where

ξ(t, θ) :=α(t)(θ−θ_w(0)) +θ_w(t) for (t, θ)∈[0, t₀]×[θ_v(0), θ_w(0)] andα(t) := _θ^θ^w^(t)−θ^v^(t)

w(0)−θv(0). (20) The bi-Lipschitz property ofTton ∂Ω(x, r)∩A is obtained as in the proof of [6, Thm. 3.3].

Finally, differentiating in (19) with respect to t and using ξ(0, θ) = θ we get (14). Then (20) yields ξ(t, θv(0)) = θv(t), ξ(t, θw(0)) = θw(t), ∂tξ(0, θv(0)) = θ⁰_a(0), ∂tξ(0, θw(0)) = θ_b⁰(0), consequently using (17) we obtain (15) and using (18) we obtain (16).

3.2 Construction of a perturbation field for radius perturbations

Theorem 3 below relies on several ideas from [6, Thm. 3.3 & Thm. 3.8], and provides an appropriate mappingTt for the differentiation of∂rG(x, r+tδr) that will be used in Section 3.3.

Theorem 3. Suppose that Assumptions 1 and 2 hold. Then there exists t0 >0 such that for all t∈[0, t₀]we have the following decomposition

∂Ω(x, r+tδr)∩A=

¯k

[

k=1

S_k(t), (21)

where ¯kis independent oft, S_k(t) are arcs parameterized by an angle aperture[θ_k,v(t), θ_k,w(t)], and t7→θk,v(t),t7→θk,w(t) are continuous functions on[0, t0].

Also, for all t ∈ [0, t₀] there exists a bi-Lipschitz mapping T_t : ∂Ω(x, r)∩A → R² satisfying T_t(Ω(x, r)∩A) = ∂Ω(x, r+tδr)∩A and T_t(S_k(0)) = S_k(t) for all k = 1, . . . ,k. In addition, we¯ have

V := ∂_tT_t|_t=0 =δrν_i+∂_tξ(0, θ)rτ_i onS_k(0)⊂∂B(x_i, r), (22) where ξ is defined in (20) and

V(z) =δrν_i(z)−δrν_A(z)·ν_i(z)

τi(z)·νA(z)τ_i(z) ifz∈∂B(x_i, r)∩∂A, (23) V(z) =δrν_i(z) +δr1−ν_`(z)·ν_i(z)

τ_i(z)·ν_`(z) τ_i(z) if z∈∂B(x_i, r)∩∂B(x_`, r), i6=`. (24)

(10)

Proof. The proof has the same structure as the proof of Theorem 2, i.e., we separate the two cases of a point belonging to∂B(x_i, r+tδr)∩∂Aand a point in ∂B(x_i, r+tδr)∩∂B(x_`, r+tδr).

The decomposition (21) relies on Assumptions 1 and 2 and is obtained in a similar way as in [6, Thm. 3.2].

First we consider the case of a point in ∂B(x_i, r+tδr)∩∂A. We provide a general formula for the angle ϑ(t), in local polar coordinates with pole xi, describing such an intersection point.

Let z∈ ∂B(x_i, r)∩∂Aand (r, θ_z) denote the polar coordinates of z with centerx_i. Let φ be the oriented distance function toA defined as in the proof of Theorem 2. Introduce the function

ψ(t, ϑ) =φ

x_i+ (r+tδr) cosϑ

sinϑ

.

We compute

∂_ϑψ(0, θ_z) =r

−sinθ_z cosθ_z

· ∇φ

x_i+r

cosθ_z sinθ_z

=rτ_i(z)· ∇φ(z).

Since Ω(x, r) and Aare compatible due to Assumption 2, B(x_i, r) is not tangent to∂A and using k∇φ(z)k= 1 we obtain τi(z)· ∇φ(z)6= 0. Thus, we can apply the implicit function theorem and this yields the existence of a smooth function [0, t0]3t7→ϑ(t) with ψ(t, ϑ(t)) = 0 andϑ(0) =θz. We also compute the derivative

ϑ⁰(0) =−∂_tψ(0, ϑ(0))

∂_ϑψ(0, ϑ(0)) =−∂_tψ(0, ϑ(0))

∂_ϑψ(0, ϑ(0)) =−δr∇φ(z)·ν_i(z)

rτ_i(z)· ∇φ(z) =−δrν_A(z)·ν_i(z)

rτ_i(z)·ν_A(z) , (25) where we have used ∇φ(z) =k∇φ(z)kν_A(z) since φis the oriented distance function to∂A.

Now we provide a general formula for the angle ϑ(t), in local polar coordinates with pole xi, describing an intersection point of two circles∂B(xi, r+tδr) and∂B(x_`, r+tδr),i6=`. Introduce

ψ(t, ϑ) =kζ(t, ϑ)k²−(r+tδr)² with ζ(t, ϑ) =xi−x_`+ (r+tδr) cosϑ

sinϑ

.

Observe thatϑ7→ζ(t, ϑ) is a parameterization of the circle∂B(xi, r+tδr) in a coordinate system of center x_`, which means that the solutions of ψ(t, ϑ) = 0 describe the intersections between

∂B(x_i, r+tδr) and ∂B(x_`, r+tδr). We compute ∂_ϑψ(0, ϑ) = 2ζ(0, ϑ)·∂_ϑζ(0, ϑ) with ζ(0, ϑ) =x_i−x_`+r

cosϑ sinϑ

and ∂_ϑζ(0, ϑ) =r

−sinϑ cosϑ

.

Now letz∈∂B(x_i, r)∩∂B(x_`, r) and letθ_zbe the corresponding angle in a polar coordinate system with polexi. Since the conditions of Assumption 1 hold, it is easy to see that

∂_ϑψ(0, θz) = 2ζ(0, θz)·∂_ϑζ(0, θz)6= 0.

Hence, the implicit function theorem can be applied to (t, ϑ) 7→ ψ(t, ϑ) in a neighbourhood of (0, θz). This yields the existence, for t0 sufficiently small, of a smooth function t7→ ϑ(t) in [0, t0] such thatψ(t, ϑ(t)) = 0 in [0, t0] and ϑ(0) =θz. We also have the derivative

ϑ⁰(t) =−∂tψ(t, ϑ(t))

∂_ϑψ(t, ϑ(t)) =−ζ(t, ϑ(t))·∂tζ(t, ϑ(t))−(r+tδr)δr ζ(t, ϑ(t))·∂_ϑζ(t, ϑ(t)) ,

(11)

and in particular, using ν_i = (cosθ_z,sinθ_z)^> and τ_i = (−sinθ_z,cosθ_z)^>, ϑ⁰(0) =−(xi−x`+rνi)·(δrνi)−rδr

(x_i−x_`+rν_i)·(rτ_i) = δr r

1−ν`·νi

ν_`·τ_i

. (26)

We are now ready to build the mapping Tt. Let S(t) ⊂ ∂B(xi, r+tδr) be one of the arcs parameterized by the angle aperture [θv(t), θw(t)] in the decomposition (21); we have dropped the indexkfor simplicity. Then,θ_v(t) andθ_w(t) are given byϑ(t) with eitherθ_z =θ_v(0) orθ_z=θ_w(0), and ϑ(t) either corresponds to a point in ∂B(xi, r+tδr)∩∂A or to a point in ∂B(xi, r+tδr)∩

∂B(x_`, r+tδr).

Next, defineξ(t, θ) andα(t) as in (20). Then, forθ∈[θ_v(0), θ_w(0)] we haveξ(t, θ)∈[θ_v(t), θ_w(t)]

and ξ(t, θ) is a parameterization ofS(t). A pointx∈ S(0) may be parameterized by x=xi+r

cosθ sinθ

, and define Tt(θ) :=xi+ (r+tδr)

cosξ(t, θ) sinξ(t, θ)

. (27)

Writingξ(t, θ) =θ+β(t, θ) withβ(t, θ) := (α(t)−1)(θ−θ_b(t)), we observe that cosξ(t, θ)

sinξ(t, θ)

=R(xi, β(t, θ)) cosθ

sinθ

=R(xi, β(t, θ))νi,

where R(x_i, β(t, θ)) is a rotation matrix of center x_i and angle β(t, θ). Also, thanks to θ_v(0) <

θw(0) < θv(0) + 2π and θ ∈ [θv(0), θw(0)], there exists a smooth bijection θ : A 3 x 7→ θ(x) ∈ [θv(0), θw(0)]. Thus, using (27) we can define the mapping

Tt(x) :=Tt(θ(x)) =x−rνi(x) + (r+tδr)R(xi, β(t, θ(x)))νi(x) for all x∈ S(0)⊂∂B(xi, r). (28) The bi-Lipschitz property ofT_ton ∂Ω(x, r)∩A can be obtained as in the proof of [6, Thm. 3.3].

Finally, differentiating in (27) with respect to t and using ξ(0, θ) = θ we get (22). Then (20) yields ξ(t, θ_a(0)) = θ_a(t), ξ(t, θ_b(0)) = θ_b(t), ∂_tξ(0, θ_a(0)) = θ⁰_a(0), ∂_tξ(0, θ_b(0)) = θ⁰_b(0), consequently using (25) we obtain (23) and using (26) we obtain (24).

3.3 Second-order derivative of G with respect to the radius

The first-order derivative of Gwith respect to the radius is given by

∂rG(x, r) =− Z

∂Ω(x,r)∩A

dz,

see (3) and [6, §3.3] for the detailed calculation. As in [6], the calculation is achieved through integration by substitution using the mapping Tt given by Theorem 3, which requires that As- sumption 1 and Assumption 2 hold. According to Theorem 3, there exists a bi-Lipschitz mapping Ttsatisfying Tt(∂Ω(x, r)∩A) =∂Ω(x, r+tδr)∩A, and this yields, using Lemma 1 on each arc of

∂Ω(x, r)∩A,

∂_rG(x, r+tδr) =− Z

∂Ω(x,r+tδr)∩A

dz =− Z

Tt(∂Ω(x,r)∩A)

dz =− Z

∂Ω(x,r)∩A

ω_t(z)dz.

(12)

Thus, using Lemma 1 and the decomposition (6), we compute d

dt∂_rG(x, r+tδr) t=0

=− Z

∂Ω(x,r)∩A

div_ΓV(z)dz =−

m

X

i=1

Z

Ai

div_ΓV(z)dz.

Applying Theorem 1 for each arc inA_i, we obtain d

=−

m

X

i=1

Z

A_i

HV ·ν_idz−

m

X

i=1

X

(v,w)∈Ai

JV(z)·τ_i(z)K

wv. (29)

To get a more explicit formula we need to determine V(v), V(w) andV ·νi on A_i. For this we apply Theorem 3 to two different cases. On the one hand, if v ∈ ∂B(x_i, r)∩∂B(x_`, r) for some i6=`, then applying (24) we obtain

V(v)·τi(v) =δr1−ν`(v)·νi(v)

ν_`(v)·τ_i(v) . (30)

On the other hand, if v∈∂B(x_i, r)∩∂A, then applying (23) we get V(v)·τi(v) =−δrνA(v)·νi(v)

τi(v)·ν_A(v). (31)

Then, recalling that L(z) = {` ∈ {1, . . . , m} \ {i} | z ∈ ∂B(x_`, r)} for z ∈ {v, w}, and that ν−i(z) := ν`(z) if z ∈∂B(xi, r)∩∂B(x`, r), `6= i, and ν−i(z) := νA(z) if z ∈∂B(xi, r)∩∂A, we can merge (30) and (31) into a unique formula:

V(v)·τ_i(v) =δr|L(v)| −ν−i(v)·ν_i(v)

ν−i(v)·τ_i(v) . (32)

In a similar way, we also obtain

V(w)·τ_i(w) =δr|L(w)| −ν−i(w)·ν_i(w) ν−i(w)·τi(w) . Gathering these results we get

d

=−δrPer(∂Ω(x, r)∩A)

r −δr

m

X

i=1

X

(v,w)∈Ai

s|L(z)| −ν−i(z)·ν_i(z) ν−i(z)·τi(z)

{w v

,

where we have used HV ·ν_i = ^δr_r on A_i ⊂ ∂B(x_i, r) due to (22) and H = 1/r. Thus we have obtained (7).

3.4 Second-order derivative of G with respect to the centers The first-order derivative of Gwith respect to the centerxi is given by

∂xiG(x, r) =− Z

A_i

νi(z)dz.

(13)

see (3) and [6, § 3.4] for the detailed calculation. As in [6], the calculation is achieved through integration by substitution using the mapping T_t from Theorem 2 with the specific perturbation δx = (0, . . . ,0, δxi,0, . . . ,0), which requires that Assumption 1 and Assumption 2 hold. Using Lemma 1 yields

∂_x_iG(x+tδx, r) =− Z

∂B(xi+tδxi,r)∩∂Ω(x+tδx,r)∩A

ν_t(z)dz=− Z

Tt(Ai)

ν_t(z)dz

=− Z

Ai

νt(Tt(z))ωt(z)dz,

whereν_tis the outward unit normal vector to ∂B(x_i+tδx_i, r)∩∂Ω(x+tδx, r)∩A andω_t is given by (11).

To obtain the derivative of ∂_x_iG(x+tδx, r) with respect to t at t = 0 we need the so-called material derivative of the normal vector given by

d

dtνt(Tt(z))|_t=0 =−(D_ΓV)^>νi on A_i,

with V := ∂_tT_t|_t=0; see [Walker, Lemma 5.5, page 99]. Here, D_ΓV := DV −(DV)ν_i⊗ν_i denotes the tangential Jacobian ofV onA_i. Then, using (12) we obtain

d

dt∂_x_iG(x+tδx, r) t=0

=− Z

Ai

−(D_ΓV)^>ν_i+ν_idiv_Γ(V)dz.

This expression can be further transformed using the following tensor relations:

div_Γ(ν_i⊗V) = div_Γ(V)ν_i+ (D_Γν_i)V and∇_Γ(V ·ν_i) =D_Γν_i^>V +D_ΓV^>ν_i on A_i. (33) We show thatD_Γν_i^>V =D_Γν_iV on A_i. Indeed, letW ∈R² and denote V_τ and W_τ the tangential components of V and W on A_i. Differentiating νi·νi = 1 onA_i we get (DΓνi)^>νi = 0 and then

(D_Γν_i)^>V ·W = (D_Γν_i)^>V_τ ·W = (D_Γν_i)^>V_τ·W_τ = (D_Γν_i)V_τ·W_τ,

where we have used the well-known fact that the second fundamental form (V_τ, W_τ)7→(D_Γν_i)V_τ·W_τ is symmetric. Further,

(D_Γν_i)^>V ·W = (D_Γν_i)^>W_τ·V_τ = (D_Γν_i)^>W ·V = (D_Γν_i)V ·W on A_i. (34) Now, using (33), (34) we obtain

d

dt∂xiG(x+tδx, r) t=0

=− Z

A_i

divΓ(νi⊗V)− ∇_Γ(V ·νi)dz.

Applying Theorem 1 to the integral of divΓ(νi⊗V) on each arc in A_i we get d

=− Z

Ai

H(ν_i⊗V)·ν_i− ∇_Γ(V ·ν_i)dz− X

(v,w)∈Ai

J(ν_i(z)⊗V(z))·τ_i(z)K

w v,

(14)

and then, usingH= 1/r on A_i, d

dt∂xiG(x+tδx, r) t=0

=−1 r

Z

A_i

(V ·νi)νi−r∇_Γ(V ·νi)dz− X

(v,w)∈Ai

J(V(z)·τi(z))νi(z)K

w v. (35) Applying (14) yields V ·ν_i =δx_i·ν_i on A_i.Considering thatδx_`= 0 for `6=isince we use the specific perturbation δx= (0, . . . ,0, δxi,0, . . . ,0), (15) and (16) actually provide the same formula in this particular case:

V(z)·τi(z) =δxi·

τi− ν−i

τ_i·ν−i

(z)

=−

ν−i·ν_i ν−i·τ_iδx_i·ν_i

(z) for z∈ {v, w} and (v, w)∈Ai.

(36)

We also have ∇_Γ(V ·νi) =∇_Γ(δxi·ν) = (DΓν)^>δxi and D_Γν_i=D_Γ

cosθ sinθ

=

∇_Γ(cosθ)^>

∇_Γ(sinθ)^>

= 1 r

∂_θ(cosθ)τ_i^>

∂_θ(sinθ)τ_i^>

= 1

rτ_i⊗τ_i on A_i. Gathering these results and usingV ·νi =δxi·νi onA_i we get

d

=−1 r

Z

A_i

(δx_i·ν_i)ν_i−(τ_i⊗τ_i)δx_idz+ X

(v,w)∈Ai

sν−i·ν_i ν−i·τ_iν_i⊗ν_i

{w v

δx_i,

which yields (8).

3.5 Second order derivative with respect to x_i and x_` of G

As in Section 3.4 we use the mapping T_t from Theorem 2, which requires that Assumptions 1 and 2 hold, but now with the specific perturbation δx = (0, . . . ,0, δx`,0, . . . ,0). This yields a transformation Tt satisfying in particular Tt(A_i) = ∂B(xi, r)∩∂Ω(x+tδx, r)∩A. Then, using Lemma 1 we obtain

∂_x_iG(x+tδx, r) =− Z

∂B(xi,r)∩∂Ω(x+tδx,r)∩A

ν_t(z)dz=− Z

Tt(Ai)

ν_t(z)dz=− Z

Ai

ν_t(T_t(z))ω_t(z)dz, where νt is the outward unit normal vector to ∂B(xi, r)∩∂Ω(x+tδx, r)∩A and ωt is given by (11). Applying (14) and considering that δxi = 0 since we are using the specific perturbation δx= (0, . . . ,0, δx_`,0, . . . ,0), we get

V ·ν_i= 0 on A_i. (37)

Then, applying (16) withδxi = 0 we get V(z)·τi(z) = δx_`·ν_`(z)

τ_i(z)·ν_`(z) ifz∈∂B(x_`, r)∩∂B(xi, r), i6=`. (38) Next, the derivative of∂_x_iG(x+tδx, r) with respect totatt= 0 is already calculated in (35), but the terms (V ·νi)νi and ∇_Γ(V ·νi) in (35) vanish due to (37). We also observe thatV(z) = 0

(15)

ifz ∈ {v, w} with (v, w) ∈Ai and z /∈∂B(x_`, r). Finally, usingI_i` ={v∈∂B(x_`, r)|(v,·)∈Ai}, O_i`={w∈∂B(x_`, r)|(·, w)∈Ai}and (38) we get

d

= X

v∈I_i`

V(v)·τ_i(v)ν_i(v)− X

w∈O_i`

V(w)·τ_i(w)ν_i(w)

=



 X

v∈I_i`

νi(v)⊗ν`(v)

ν_`(v)·τi(v) − X

w∈O_i`

νi(w)⊗ν`(w) ν_`(w)·τi(w)



δx`,

which yields (9).

3.6 Second order derivative with respect to x_i and r of G

In a similar way as in Sections 3.4 and 3.5, we use the mappingT_tfrom Theorem 2 with the specific perturbationδx= (0, . . . ,0, δxi,0, . . . ,0). This yields, using Lemma 1,

∂_rG(x+tδx, r) =− Z

∂Ω(x+tδx,r)∩A

dz=− Z

Tt(∂Ω(x,r)∩A)

dz=− Z

∂Ω(x,r)∩A

ω_t(z)dz.

Proceeding as in the calculation leading to (29), we get d

dt∂_rG(x+tδx, r) _t=0

=−

m

X

`=1

Z

A_`

HV ·ν_`dz−

m

X

`=1

X

(v,w)∈A`

JV(z)·τ_`(z)K

w

v. (39)

Considering thatδx_`= 0 for`6=i, since we use the specific perturbationδx= (0, . . . ,0, δxi,0, . . . ,0), (15) and (16) actually provide the same formula in this particular case:

V(z)·τ_i(z) =δx_i·

τ_i− ν−i

τi·ν−i

(z)

=−

ν−i·νi

ν−i·τi

δxi·νi

(z) for z∈ {v, w} and (v, w)∈Ai,

(40)

and also

V(z)·τ_`(z) =δxi·

τ_`− ν`

τ_i·ν_`(τi·τ_`)

(z)

=δx_i· µ

τ_i·ν_`

(z) if z∈∂B(x_i, r)∩∂B(x_`, r) and `6=i.

withµ:=τ_`(τ_i·ν_`)−(τ_i·τ_`)ν_`. This yieldsµ·τ_i = 0 and

µ·ν_i= (τ_`·ν_i)(τ_i·ν_`)−(τ_i·τ_`)(ν_`·ν_i) =−(τ_i·ν_`)²−(τ_i·τ_`)² =−1,

where we have used the geometric properties τ_`·ν_i =−τ_i·ν_` and τ_i ·τ_` =ν_`·ν_i. Thus µ =−ν_i and we get

V(z)·τ_`(z) =− δx_i·ν_i(z)

τi(z)·ν`(z) ifz∈∂B(x_i, r)∩∂B(x_`, r) and`6=i. (41)

(16)

In (39), we observe thatV(z) = 0 wheneverz∈ {v, w}and z /∈∂B(x_i, r); this can be seen from (15)-(16) and the fact that we use the specific perturbation δx = (0, . . . ,0, δx_i,0, . . . ,0). Hence, recalling that L(z) ={`∈ {1, . . . , m} \ {i} |z∈∂B(x`, r)},

m

X

`=1

X

(v,w)∈A`

JV(z)·τ_`(z)K

w

v = X

(v,w)∈Ai

JV(z)·τ_i(z)K

w v +

m

X

`=1`6=i

X

(v,w)∈A`

JV(z)·τ_`(z)K

w v

= X

(v,w)∈Ai

JV(z)·τ_i(z)K

w

v − X

(v,w)∈Ai

u v

X

`∈L(z)

V(z)·τ_`(z) }

~

w

v

.

Note that the negative sign in front of the last sum is due to the fact that if an ending point of an arc in A` belongs to some arc in Ai, then it is a starting point for this arc in Ai, and vice versa.

Using (14) we have V ·ν_`≡0 on A_` for all `6=i. Since H= 1/r, we may write (39) as d

dt∂_rG(x+tδx, r) _t=0

=−1 r

Z

A_i

V ·ν_idz− X

(v,w)∈Ai

u

vV(z)·τ_i(z)− X

`∈L(z)

V(z)·τ_`(z) }

~

w

v

. (42)

Using (14) we getV ·ν_i =δx_i·ν_i on A_i. Finally, using (40)-(41) we get d

dt∂_rG(x+tδx, r) t=0

=−1 r

Z

A_i

δx_i·ν_idz

+ X

(v,w)∈Ai

u v

ν−i(z)·νi(z)

ν−i(z)·τ_i(z)δxi·νi(z)− X

`∈L(z)

δxi·νi(z) τ_i(z)·ν_`(z)

}

~

w

v

,

which yields (10).

4 Analysis of singular cases

The gradient∇Gand Hessian∇²Gwere obtained under Assumptions 1 and 2, and in this section we investigate several singular cases where these assumptions are not satisfied. On the one hand, it is shown in [6,§ 3.5] that Gis often differentiable even when Assumptions 1 and 2 do not hold, and in the few cases where Gis not differentiable it is at least Gateaux semidifferentiable. On the other hand, Gis never twice differentiable in any of the singular geometric configurations studied in this section. Nevertheless, Gateaux semidifferentiability of the components of∇Gcan often be proven.

We recall here that f :Rⁿ→R is Gateaux semidifferentiable at xin the direction v if

t&0lim

f(x+tv)−f(x)

t exists inRⁿ, and thatf has a derivative in the direction v atx if

t→0lim

f(x+tv)−f(x)

t exists inRⁿ.