The extended adjoint method

THE EXTENDED ADJOINT METHOD

Stanislas Larnier

and Mohamed Masmoudi

Abstract.Searching for the optimal partitioning of a domain leads to the use of the adjoint method in topological asymptotic expansions to know the influence of a domain perturbation on a cost function.

Our approach works by restricting to local subproblems containing the perturbation and outperforms the adjoint method by providing approximations of higher order. It is a universal tool, easily adapted to different kinds of real problems and does not need the fundamental solution of the problem; fur- thermore our approach allows to consider finite perturbations and not infinitesimal ones. This paper provides theoretical justifications in the linear case and presents some applications with topological perturbations, continuous perturbations and mesh perturbations. This proposed approach can also be used to update the solution of singularly perturbed problems.

Mathematics Subject Classification. 49Q10, 49Q12, 74P10, 74P15.

Received November 18, 2011. Revised April 18, 2012.

Published online July 31, 2012.

1. Introduction

The optimal partitioning of a domain Ω has important real-life applications like shape optimal design, detection of inclusions, image classification and segmentation. Topological derivative methods have been used to solve this kind of problems [10,11,23,25,26,30,31,35,36]. These approaches have some drawbacks:

– the asymptotic topological expansion is not easy to obtain for complex problems;

– it needs to be adapted for many particular cases like the creation of a hole on the boundary of an existing one or on the original boundary of the domain;

– we don’t know how to calculate the variation of a cost function when a hole is to be filled;

– in real applications of topology optimization, a finite perturbation is performed and not an infinitesimal one such as an element deletion in a mesh.

Certain issues arise here, as for example the question on how large the topological change should be. In the present paper, we propose an extension of the adjoint method to overcome these problems.

The problem of optimal partitioning of a domain Ω is equivalent to the problem of looking for an optimal functioncwhich takes a finite number of values 0,1, . . . , m−1. Ifm= 2,cis the characteristic function of an unknown optimal subdomain ofΩ. For the sake of simplicity we will limit the study to the case wherem= 2.

Keywords and phrases.Adjoint method, topology optimization, calculus of variations.

1 Universit´e Paul Sabatier, Institut de Math´ematiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France.

stanislas.larnier@math.univ-toulouse.fr; mohamed.masmoudi@math.univ-toulouse.fr

Article published by EDP Sciences c EDP Sciences, SMAI 2012

Even if we are dealing with a 0−1 optimization problem, it is possible to use variational methods. For a cost function

j:L^p(Ω)→R, 1≤p <∞ c →j(c),

the topological asymptotic expansion consists in calculating the variation of j when c switches from 1 to 0 or from 0 to 1 in a small area. Generally, j depends on c via the resolution of a set of partial differential equations. Explicit regularization terms related to the volume, or the measure of the boundary, as well as the mean curvature, could be added to j and a more regular space thanL^p could be considered.

It is possible to use differential calculus tools [27] for estimating the variation of j when c switches from 1 to 0 or from 0 to 1 in a small region ω_ε =x₀+εω, where x₀ is a point ofΩ, ε is a small scalar, andω is a given domain. The perturbation δc =±χωε obtained when c switches from 1 to 0 or from 0 to 1 in a small regionω_ε, is small inL^p(Ω) even if not small in magnitude. We will refer toδcas a singular perturbation. Ifj is differentiable, we have

j(c+δc)−j(c)−

Ωg₁δc

=O(||δc||²_Lp),

whereg₁∈L^q(Ω),¹_p+¹_q = 1 is the gradient ofj. Moreover, ifg₁is regular, we obtain the following expansion:

|j(c+δc)−j(c)− |ωε| g₁(x₀)|=O

|ωε|^2p

+◦(|ωε|). (1.1)

There are two cases:

– when p < 2, the rest in equation (1.1) is of higher order and the topological gradient is equal to g₁, the gradient ofj. We refer to [27] for more details;

– whenp≥2, the quantityg₁is the first term of the topological gradient, the “rest”O(|ωε|^2p) could contribute to the topological gradient by a second (or hidden) termg₂.

The goal of this paper is to provide a simple, efficient and general way to calculate the second termg₂. In the casep≥2, the asymptotic expansion (1.1) is technically correct, but it may lead to wrong conclusions.

The adjoint method is the classical way to calculateg₁, the gradient of j, at each point of the domainΩand at a lower computing cost. In fact, at least for the casep <2, there is no need to calculate the variation of the state with respect toc, because its contribution is of higher order. And whenp≥2, the variation of the solution gives the second termg₂of the topological gradient. In simple cases, this second termg₂is quite easy to estimate using the explicit knowledge of the fundamental solution of the problem and there is no need to calculate the unknown variation of the state with respect to the considered perturbation [1,10,11,23,25,26,30,31,35,36].

In this case, the formal application of the chain rule gives only the first term. In some cases, such as the insertion of infinitesimal cracks [11], since |ωε| = 0, the first term is equal to zero and the variation of the cost function is equal to the hidden contribution. Currently, some authors such as Bonnet are interested by asymptotic topological expansions of higher-order [17,18].

Like we said before, these methods have many limitations. The fundamental solution is not known explicitly for more realistic models. The asymptotic expansion needs to be adapted for too many particular cases like the creation of a hole on the boundary of an existing one or on the original boundary of the domain. The number of particular cases increases if the type of boundary conditions on the hole is considered. In [34], the case of the Laplace equation with the creation of a hole on a polygonal boundary has been studied. In topology optimization, we need to know how to fill an existing small hole, a problem studied by Guillaume and Hassine [24].

In most contributions, the elementary solution is used to calculate the local variation of the state around the singular perturbation. The key idea of this paper is to take into account this variation by solving a local perturbed problem defined in a small fixed domain D containing the singular perturbation. The state of the initial uniform problem is imposed as a Dirichlet condition on the boundary ofD. The size of the domainD varies with the studied problem.

From the application point of view, this method is non invasive and can be easily adapted to any solver. It could be used with parallel computing algorithms when multiple evaluations are needed.

From the theoretical point of view, this method could be seen as an improvement of the domain truncation method, where the variation of the Dirichlet to Neumann operator is needed [10,23,30]. Here we just need to study the variation of a local solution. Thus, our method could be a tool in theoretical investigations. This paper presents theoretical justifications and examples in the linear case. In [29], the method has been used in a nonlinear case with mesh perturbations in fluids dynamics.

This extension of the adjoint method is called the numerical vault for two reasons. The imposed Dirichlet condition on the boundary ofD makes our solution very stable and recalls the vault in architecture which is supported by its surrounding walls. Moreover, similarly to the vault, our method is very simple and has great potential for applications.

Section 2recalls the adjoint technique and presents the basic concept of the vault method. In Section3, we consider the linear case and present a theoretical justification for the numerical vault as an extension to the adjoint method. Some simple examples are also considered to illustrate the theoretical study. In Section4, we show that the numerical vault is not limited to the estimation of the variation of a cost function, but can be applied to update the solution of a singularly perturbed problem. In Section5, we present some applications of the numerical vault to topological perturbations and we show that the hidden termg₂is not small in comparison tog₁. Moreover, in the case of a continuous perturbation, the numerical vault allows us to obtain a higher order behavior. We applied our method to continuous material properties and mesh perturbations. An application to elastography is presented in Sections6and7, the numerical vault is applied to update the solution in the case of an image restoration problem.

2. The adjoint method

2.1. The adjoint technique

We first recall the adjoint method in a formal way. Consider the following steady state equation

F(c, u) = 0 inΩ, (2.1)

wherec is a distributed parameter in a domainΩ. The aim is to minimize a cost functionj(c) :=J(u_c) where u_c is the solution of equation (2.1) for a givenc.

Let us suppose that every term is differentiable. We are considering a perturbationδcof the parameter c.

Since c is a distributed parameter, its discretization leads to a huge vector. A fast gradient computation is therefore of high importance.

Equation (2.1) can be seen as a constraint, and as a consequence, the Lagrangian is considered:

L(c, u, p) =J(u) + (F(c, u), p),

wherepis a Lagrange multiplier and (·,·) denotes the scalar product in a well-chosen Hilbert space.

To compute the derivative of j, one can remark that j(c) = L(c, u_c, p) for all c, if u_c is the solution of equation (2.1). The derivative ofj is then equal to the derivative ofL with respect toc:

d_cj(c)δc=∂_cL(c, u_c, p)δc+∂_uL(c, u_c, p)∂_cu δc.

All these terms can be calculated easily, except∂_cu δc, the solution of the linearized problem:

∂_uF(c, u_c)(∂_cu δc) =−∂cF(c, u_c)δc.

To avoid the resolution of this equation for eachδc, the term∂_uL(c, u_c, p) is cancelled by solving the following adjoint equation inp. Letp_c be the solution of the adjoint equation:

∂_uF(c, u_c)^Tp_c=−∂_uJ^T.

So the derivative ofj is explicitly given by

d_cj(c)δc=∂_cL(c, u_c, p_c).

This method works only when the problem is differentiable and δc is an admissible perturbation. Several counter-examples, where one of these conditions is not satisfied, are given in [27].

2.2. Generalized adjoint technique

Note that if the LagrangiansL(c+δc, . . . , . . .) andL(c, . . . , . . .) are defined on the same space, we have j(c+δc)−j(c) =L(c+δc, u_c+δc, p_c)− L(c, u_c, p_c),

it can be split in two terms:

j(c+δc)−j(c) = (L(c+δc, u_c+δc, p_c)− L(c+δc, u_c, p_c)) + (L(c+δc, u_c, p_c)− L(c, u_c, p_c)).

In the case of a regular perturbationδc, the second term gives the main variation and the first term is of higher order. In the case of a singular perturbation, the first term is of the same order as the second one and cannot be ignored. Then the variation ofu_c has to be estimated.

In the topology optimization papers [10,11,13,23,25,26,30], the local variation ofu_c can be expressed thanks to the elementary solution of the problem to solve. This solution is singular atx₀and gives the local variation ofu_c aroundx₀. In general, it is not always simple to get an explicit expression of the elementary solution. This is particularly true for problems with discontinuous coefficients, if the discontinuity is located atx₀.

Due to this difficulty, a numerical estimation of this variation is presented. The basic idea of the numerical vault is to update the solutionu_c by solving a local problem defined in a small domain aroundx₀.

If the LagrangiansL(c+δc, . . . , . . .) andL(c, . . . , . . .) are not defined on the same space, we have j(c+δc)−j(c) =L(c+δc, u_c+δc, p_c+δc)− L(c, uc, p_c).

In this case, the direct solutionu_c and also the adjointp_c are updated with the numerical vault.

3. Estimation of the variation of a cost function with the numerical vault

For the sake of simplicity, only the linear case is studied. Consider the variational problem depending on a parameterε

a^ε(u, v) =^ε(v) ∀v∈ V^ε, (3.1)

where V^ε is a Hilbert space, a^ε is a bilinear, continuous and coercive form and ^ε is a linear and continuous form. Typically,V^εis such that H¹₀⊂ V^ε⊂H¹.

In this study, a singular perturbation means a finite variation located in an infinitesimal area: the size of the perturbation goes to zero when εgoes to zero. Depending on the problem,εcan be the thickness of a coating layer around a variety of dimension≤ n−1 in the space of dimension n. In the case of a hole, ε can be the diameter of a tube around a curve in the space of dimension 3. And we do not forget the classical case, where εis the diameter of a hole created around a pointx₀.

Let u^ε be the solution of the problem (3.1) and u⁰ be the solution of the initial problem without any perturbation atε= 0.

The objective is to calculate the variation, with respect toε, of j(ε) :=J^ε(u^ε).

The cost functionJ^εis of classC¹, the adjoint problem associated to the problem (3.1) and the cost function J^εis defined as follows

a^ε(w, p^ε) =−∂_uJ^ε(u^ε)w ∀w∈ V^ε, (3.2) wherep^εis the solution of this problem.

Suppose thata^ε, ^εandJ^εare integrals over a domainΩ.

The domainΩis split into two parts, a partDcontaining the perturbation, and its complementaryΩ₀=Ω\D.

The perturbation can be located on the boundary ofΩ. For this reason, the domainDis not necessarily included in the domainΩ.

The formsa^ε, ^ε, and the cost functionJ^εare decomposed in the following way:

– a^ε=a_Ω0+a^ε_D; – ^ε=_Ω0+^ε_D; – J^ε=J_Ω0+J_D^ε,

wherea_Ω0,l_Ω0 et J_Ω0 are independent ofε.

The constantsαandM are positive constants used in respectively the minoration and the majoration of the bilinear forms.

Similarly, we consider

– VΩ0, the space consisting of functions ofV^εandV⁰restricted toΩ₀; – V_D^ε, the space consisting of functions ofV^εrestricted toD;

– V_D⁰, the space consisting of functions ofV⁰ restricted toD;

– V_D,0^ε , the subspace ofV_D^ε, with null trace on∂D;

– V_∂D, the space of traces on∂D ofV_Ω0. 3.1. Updating the direct solution

In this section, we assume that V⁰ ⊂ V^ε. When the perturbation consists in inserting a hole in the domain with a Dirichlet condition on the boundary of the hole, this condition does not hold. This case will be considered in Section3.2.

Let us consideru^ε_D, the local update ofu⁰:

⎧⎨

⎩

Findu^ε_D∈ V_D^ε solution of a^ε_D(u^ε_D, v) =^ε_D(v) ∀v∈ V_D,0^ε , u^ε_D=u⁰ on∂D,

(3.3)

and ˜u^εthe update ofu⁰is given by:

˜

u^ε= u^ε_D in D,

u⁰ in Ω₀. (3.4)

We assume the following hypothesis.

Hypothesis 3.1. There exist three positive constantsη, C andC_u independent ofεand a positive real valued function f defined onR+ such that

ε→0limf(ε) = 0,

J^ε(v)−J^ε(u)−∂_uJ^ε(u)(v−u)V^ε ≤Cv−u²_V^ε, ∀v, u∈B(u⁰, η), u^ε−u⁰_VΩ0 ≤C_uf(ε),

ε→0limp^ε−p⁰V^ε = 0.

Proposition 3.2. Under Hypothesis 3.1, we have

u^ε−u˜^ε_V^ε =O(f(ε)).

Proof. Hypothesis3.1tells us that

u^ε−u⁰_VΩ0 =O(f(ε)).

The trace properties allows us to write:

u^ε−u⁰V∂D =O(f(ε)).

We can write it under the following form

u^ε−u⁰_V∂D = min

φ=u^ε−u⁰on∂Dφ_VD^ε.

As a consequence, we defineϕ∈ V_D^ε as the minimum ofϕV_D^ε under the constraintϕ=u^ε−u⁰on∂D, then ϕ_V^ε_D =u^ε−u⁰_V∂D =O(f(ε)).

We haveϕ−(u^ε−u^ε_D) = 0 on∂D, we can use the coercivity ofa^ε_D: αϕ−(u^ε−u^ε_D)²_V^ε

D ≤a^ε_D(ϕ−(u^ε−u^ε_D), ϕ−(u^ε−u^ε_D))

≤a^ε_D(ϕ, ϕ−(u^ε−u^ε_D))−a^ε_D(u^ε, ϕ−(u^ε−u^ε_D)) +a^ε_D(u^ε_D, ϕ−(u^ε−u^ε_D)), and the continuity of the bilinear forma^ε_D:

αϕ−(u^ε−u^ε_D)²_VD^ε ≤MϕVDϕ−(u^ε−u^ε_D)VD −^ε_D(ϕ−(u^ε−u^ε_D)) +^ε_D(ϕ−(u^ε−u^ε_D))

≤Mu^ε−u⁰V∂Dϕ−(u^ε−u^ε_D)V_D^ε.

UsingϕV_D^ε =O(f(ε)),ϕ−(u^ε−u^ε_D)V_D^ε =O(f(ε)) and the triangular inequality, we obtain:

u^ε−u^ε_DV_D^ε =O(f(ε)).

With the equality

u^ε−u˜^ε2_Vε =u^ε−u⁰²_VΩ

0+u^ε−u^ε_D²_Vε D,

we obtain the final result.

Theorem 3.3. Under Hypothesis 3.1, we have

j(ε)−j(0) =L^ε(˜u^ε, p⁰)− L⁰(u⁰, p⁰) +o(f(ε)).

According to litterature [10,11,23,25,26,30], the differenceL^ε(˜u^ε, p⁰)− L⁰(u⁰, p⁰) is of orderf(ε).

Proof. We just split this variation into two terms j(ε)−j(0) =L^ε(u^ε, p⁰)− L⁰(u⁰, p⁰) =

L^ε(u^ε, p⁰)− L^ε(˜u^ε, p⁰) +

L^ε(˜u^ε, p⁰)− L⁰(u⁰, p⁰) , and prove that

L^ε(u^ε, p⁰)− L^ε(˜u^ε, p⁰) =o(f(ε)). Using Hypothesis3.1, we have

L^ε(u^ε, p⁰)− L^ε(˜u^ε, p⁰) =J^ε(u^ε)−J^ε(˜u^ε) +a^ε(u^ε, p⁰)−a^ε(˜u^ε, p⁰)

=∂_uJ^ε(u^ε)(u^ε−u˜^ε) +o(f(ε)) +a^ε(u^ε−u˜^ε, p⁰), and with equation (3.2), we obtain

L^ε(u^ε, p⁰)− L^ε(˜u^ε, p⁰) =−a^ε(u^ε−u˜^ε, p^ε) +a^ε(u^ε−u˜^ε, p⁰) +o(f(ε))

=a^ε(u^ε−u˜^ε, p⁰−p^ε) +o(f(ε)).

Using the continuity ofa^ε, Hypothesis3.1and Proposition3.2, we obtain the final result.

Remark 3.4. Generally, for topological perturbationsp^ε−p⁰_V^ε=O (f(ε))¹²

andj(ε)−j(0) =L^ε(˜u^ε, p⁰)−

L⁰(u⁰, p⁰) +O (f(ε))³²

[30]. We will see in the following section that if we update locallyuand p, the rest could be of orderO

f(ε)² .

3.2. Updating the direct and adjoint solutions

In this section V⁰ is not necessary a sub-space of V^ε. This is the case when the perturbation is a hole of radiusεwith a Dirichlet boundary condition on its boundary. The definition of ˜u^εremains unchanged and we updatep⁰ in the same way.

Let us considerp^ε_D the local update ofp⁰

⎧⎨

⎩

Find p^ε_D∈ V_D^ε solution of

a^ε_D(w, p^ε_D) =−∂_uJ_D^ε(u^ε_D)w, ∀w∈ V_D,0^ε ,

p^ε_D=p⁰ on∂D,

(3.5)

and ˜p^εthe update ofp⁰is given by:

˜ p^ε=

p^ε_D inD,

p⁰ inΩ₀. (3.6)

We assume the following hypothesis.

Hypothesis 3.5. There exist four positive constants η, C, C_u and C_p independent of ε and a positive real valued functionf defined onR₊ such that

ε→0limf(ε) = 0,

J^ε(v)−J^ε(u)−∂_uJ^ε(u)(v−u)_V^ε ≤Cv−u²_Vε, ∀v, u∈B(u⁰, η), u^ε−u⁰VΩ0 ≤C_uf(ε),

p^ε−p⁰VΩ0 ≤C_pf(ε).

Proposition 3.6. Under Hypothesis 3.5, we have

p^ε−p˜^ε_V^ε =O(f(ε)).

Proof. Hypothesis3.5tells us that

p^ε−p⁰VΩ0 =O(f(ε)).

The trace properties allows us to write:

p^ε−p⁰_V∂D =O(f(ε)).

We can write it under the following form

p^ε−p⁰_V∂D = min

φ=p^ε−p⁰on∂Dφ_VD^ε.

As a consequence, we defineϕ∈ V_D^ε as the minimum ofϕV_D^ε under the constraintϕ=p^ε−p⁰ on∂D, then ϕ_VD^ε =p^ε−p⁰_V∂D =O(f(ε)).

We haveϕ−(p^ε−p^ε_D) = 0 on∂D, we can use the coercivity ofa^ε_D: αϕ−(p^ε−p^ε_D)²_VD^ε ≤a^ε_D(ϕ−(p^ε−p^ε_D), ϕ−(p^ε−p^ε_D))

≤a^ε_D(ϕ−(p^ε−p^ε_D), ϕ)−a^ε_D(ϕ−(p^ε−p^ε_D), p^ε) +a^ε_D(ϕ−(p^ε−p^ε_D), p^ε_D).

Equations (3.2) and (3.5) give us:

αϕ−(p^ε−p^ε_D)²_V^ε

D≤a^ε_D(ϕ−(p^ε−p^ε_D), ϕ) +∂_uJ_D^ε(u^ε) (ϕ−(p^ε−p^ε_D))−∂_uJ_D^ε(u^ε_D) (ϕ−(p^ε−p^ε_D))

≤a^ε_D(ϕ−(p^ε−p^ε_D), ϕ) +∂_uuJ_D^ε(u^ε) (ϕ−(p^ε−p^ε_D), u^ε−u^ε_D) +ϕ−(p^ε−p^ε_D)_VD^εo(f(ε)).

The continuity of the bilinear forma^ε_Dand the continuity of the linear form∂_uJ_D^ε lead to the following result:

αϕ−(p^ε−p^ε_D)V_D^ε ≤M

ϕV_D^ε +u^ε−u^ε_DV_D^ε +o(f(ε))

≤M

p^ε−p⁰_V∂D +u^ε−u^ε_DVD^ε +o(f(ε)) .

Thanks top^ε−p⁰_V∂D =O(f(ε)) and Proposition3.2u^ε−u^ε_DVD^ε =O(f(ε)), we haveϕ−(p^ε−p^ε_D)_VD^ε = O(f(ε)).

Usingϕ_VD^ε =O(f(ε)),ϕ−(p^ε−p^ε_D)_VD^ε =O(f(ε)) and the triangular inequality, we obtain:

p^ε−p^ε_DVD^ε =O(f(ε)).

With the equality

p^ε−p˜^ε2_Vε =p^ε−p⁰²_VΩ0+p^ε−p^ε_D²_VD^ε,

we obtain the final result.

Theorem 3.7. Under Hypothesis 3.5, we have

j(ε)−j(0) =L^ε(˜u^ε,p˜^ε)− L⁰(u⁰, p⁰) +O(f(ε)²).

According to litterature [10,11,23,25,26,30], the differenceL^ε(˜u^ε,p˜^ε)− L⁰(u⁰, p⁰) is of orderf(ε).

Proof. We just split this variation into two terms

j(ε)−j(0) =L^ε(u^ε, p^ε)− L⁰(u⁰, p⁰)

= [L^ε(u^ε, p^ε)− L^ε(˜u^ε, p^ε)] +

L^ε(˜u^ε, p^ε)− L⁰(u⁰, p⁰) , and prove that

L^ε(u^ε, p^ε)− L^ε(˜u^ε,p˜^ε) =O(f(ε)²).

Using Hypothesis3.5, we have

L^ε(u^ε, p^ε)− L^ε(˜u^ε,p˜^ε) =J^ε(u^ε)−J^ε(˜u^ε) +a^ε(u^ε, p^ε)−a^ε(˜u^ε,p˜^ε)−^ε(p^ε) +^ε(˜p^ε)

=∂_uJ^ε(u^ε)(u^ε−u˜^ε) +O(f(ε)²) +a^ε(u^ε−u˜^ε, p^ε) +a^ε(˜u^ε, p^ε−p˜^ε)−^ε(p^ε−p˜^ε) and with equations (3.1) and (3.2), we obtain

L^ε(u^ε, p^ε)− L^ε(˜u^ε,p˜^ε) =a^ε(˜u^ε, p^ε−p˜^ε)−^ε(p^ε−p˜^ε) +O(f(ε)²)

=a^ε(˜u^ε−u^ε, p^ε−p˜^ε) +O(f(ε)²).

Using the continuity ofa^ε, Propositions3.2and3.6, we obtain the final result.

Remark 3.8. Updating pis not necessary when V⁰ ⊂ V^ε, but according to Theorem3.7, we obtain a better estimation of the variation ofj whenpis updated.

In the case of a Dirichlet condition on the boundary of the hole, V⁰ is not a subspace ofV^εand we need to update the adjoint.

3.3. Examples

3.3.1. Examples in literature

The most important hypothesis is

u^ε−u⁰_VΩ0 =O(f(ε)). (3.7)

This assumption (3.7) is generally satisfied. See – the elasticity case [23];

– the Poisson equation [25];

– the Navier-Stokes equation [26];

– the Helmholtz equation [10].

These results have been obtained in the frame of the domain truncation method.

3.3.2. The capacity case

This section presents an application of the previous results to the capacity problem. LetΩ∈R²be a domain.

Δu⁰= 0 inΩ,

u⁰= 1 on∂Ω. (3.8)

This equation implies thatu⁰= 1 inΩis solution.

The cost function is the following

J^ε(v) = 1 2

Ωε

|∇v|²dx, ∀v∈V. (3.9)

LetB(x₀, ε) be a hole in the domainΩ, the perturbed domain isΩ_ε.

⎧⎪

⎨

⎪⎩

Δu^ε= 0 in Ω_ε, u^ε= 0 on∂B(x₀, ε), u^ε= 1 on∂Ω.

(3.10)

LetD be equal toB(x₀, R)\B(x₀, ε) withR > εandB(x, R)⊂Ω.

The update ofu⁰is defined by

˜ u^ε=

u^ε_D inD,

u⁰ inΩ\D. (3.11)

whereu^ε_D is the solution of the following equation

⎧⎪

⎨

⎪⎩

Δu^ε_D= 0 in D, u^ε_D= 0 on∂B(x₀, ε), u^ε_D= 1 on∂B(x₀, R).

(3.12)

The hypothesisu^ε−u⁰VΩ0 =O(f(ε)) is satisfied [30].

The elementary solution of the Laplace equation in two dimensions is−^ln(r)_2π wherer=|x−x₀|.

The update is

˜ u^ε=

ln(|x−x0|)−ln(ε)

ln(R)−ln(ε) in D,

1 in Ω\D. (3.13)

LetΓbe the outer boundary of the domainDandnthe normal oriented towards the exterior on the boundary.

With the approximation ˜u^ε, the cost function (3.9) can be written as follows J^ε(˜u^ε) =π

Γ

∂u˜^ε

∂nu˜^εrdr. (3.14)

Taking into account Theorem3.3we obtain

j(ε)−j(0) =J^ε(u^ε)−J⁰(u⁰) =J^ε(˜u^ε)−J⁰(u⁰) +o(f(ε)), and given the update definition (3.13), we have

J^ε(˜u^ε)−J⁰(u⁰) =π _R

ε |∇u^ε|²rdr

⇔J^ε(˜u^ε)−J⁰(u⁰) = π (ln(R)−ln(ε))²

r=R

1

r(ln(r)−ln(ε))rdr

⇔J^ε(˜u^ε)−J⁰(u⁰) = π (ln(R)−ln(ε))· The final result is

j(ε)−j(0) =J^ε(u^ε)−J⁰(u⁰) = −π ln(ε)+o

1 ln(ε)

·

This result is exactly the same as the one obtained in [30].

4. Approximation of a perturbed solution

In this section, we assume that V⁰ ⊂ V^ε. The aim of the previous section was to estimate the variation of a cost function with respect to a local perturbation. In this part, the vault method is used to construct an approximation of the perturbed solutionu.

We assume the following hypothesis.

Hypothesis 4.1. There exists a constantC_u independent ofεand a positive real valued function f defined on R₊ such that

ε→0limf(ε) = 0, u^ε−u⁰VΩ0 ≤C_uf(ε), a^ε_D−a⁰_DL²(V_D^ε)=O(f(ε)).

The norm ._L²_(VD^ε₎ means the norm of the bilinear form.

Proposition 4.2. Under Hypothesis 4.1, we have

u^ε−u˜^εV^ε =O(f(ε)), a^ε−a⁰_L²_(V^ε₎=O(f(ε)).

Proof. Thanks to Proposition3.2, we have

u^ε−u˜^εV^ε =O(f(ε)).

Asa⁰=a_Ω0+a⁰_Dand a^ε=a_Ω0+a^ε_D, we have

a^ε−a⁰_L²_(V^ε₎=O(f(ε)).

A singular local perturbation has two effects: a local high frequency effect and a global low frequency effect.

The local effect is caught by ˜u^ε. In the previous section, the global effect is taken into account by the adjoint. In this section, we are considering the variation of the solution, the adjoint cannot be used, and we need to solve an auxiliary problem in order to calculate the global low frequency variation.

Let us considerδu^ε the solution of

Find δu^ε∈ V⁰ solution of

a⁰(δu^ε, v) =a^ε(u^ε−u˜^ε, v), ∀v∈ V⁰. (4.1) The approximation of u^ε with the numerical vault is noted ˜u˜^εand defined as follows

˜˜

u^ε= ˜u^ε+δu^ε. (4.2)

Proposition 4.3. Under Hypothesis 4.1, we have

δu^ε_V⁰ =O(f(ε)).

Proof. Thanks to the coercivity ofa⁰, we have

αδu^ε2_V0 ≤a⁰(δu^ε, δu^ε) =a^ε(u^ε−u˜^ε, δu^ε). Thanks to the continuity ofa^ε, we have

a^ε(u^ε−˜u^ε, δu^ε)≤Mu^ε−u˜^ε_V^εδu^ε_V⁰.

Hypothesis4.1and Proposition4.2allow us to conclude.

Theorem 4.4. Under Hypothesis 4.1and with the definition ofu˜˜^ε, we have u^ε−u˜˜^ε

V^ε =O((f(ε))²).

Proof. Thanks to the coercivity ofa^ε, we have αu^ε−u˜˜^ε2

V^ε ≤a^ε

u^ε−u˜˜^ε, u^ε−˜˜u^ε

=a^ε

u^ε−u˜^ε, u^ε−u˜˜^ε

−a^ε

δu^ε, u^ε−˜˜u^ε .

Thanks to the definition of ˜u˜^ε, we have αu^ε−u˜˜^ε2

V^ε ≤a⁰

δu^ε, u^ε−u˜˜^ε

−a^ε

δu^ε, u^ε−u˜˜^ε . Thanks to Proposition4.2, we have

a^ε−a⁰_L²_(V^ε₎=O(f(ε)), so

|a⁰

δu^ε, u^ε−u˜˜^ε

−a^ε

δu^ε, u^ε−u˜˜^ε

| ≤Cf(ε)δu^ε_V⁰u^ε−u˜˜^ε

V^ε. Thanks to Proposition4.3, we have

δu^ε_V⁰ =O(f(ε)).

This implies that

u^ε−u˜˜^ε

V^ε =O((f(ε))²).

Algorithm 1.Algorithm to approximate the perturbed solution

1: Solve the global problem usinga⁰to obtainu⁰ (3.1).

2: Solve the local problem usinga^ε_Dto obtainu^ε_D (3.3).

3: Solve the global problem usinga⁰to obtainδu^ε(4.1).

4: The solution is ˜u˜^ε= ˜u^ε+δu^ε.

We solved a local problem and two global problems atε= 0. The proposed method is interesting when the problem is easy to solve atε= 0. This the case when the initial problem can be solved using spectral methods with constant coefficient.

It is also interesting when the matrix arising froma⁰ is factorized. In the case of iterative method, when the same matrix is used to solve many problems, it is possible to build efficient preconditioners.

This method can be considered by researchers who do not use the adjoint method [2,3,6,7,19,20]. The low frequency effect can be taken into account by solving several time the same initial systems.

5. Application to topological and continuous variations

5.1. Problem statement

The aim of this section is to approximate the variation of a cost function in the following cases: topological pertubation, mesh perturbation, continuous variation of stiffness perturbation.

We model an experiment as follows: the domain Ω is a rectangle filled with an elastic material, the left vertical side is fixed and a vertical force of intensity μin the downward direction is applied on the middle of the right vertical side, see Figure1. The other sides are free.

Let Ω be a rectangular bounded domain ofR² and Γ be its boundary, composed of two parts Γ₁ and Γ₂. The considered problem has Dirichlet boundary conditions on Γ₁ and Neumann boundary conditions on Γ₂. The rectangle is subject to a displacementusolution of the following equation:

⎧⎪

⎨

⎪⎩

− ∇ ·σ(u) = 0 inΩ, u= 0 onΓ₁, σ(u)n=g onΓ₂,

(5.1) with

φ(u) = 1

2(Du+Du^T), σ(u) =hH₀φ(u),

whereH₀is the Hooke tensor andh(x) = 1 andρrepresents the material presence or absence.

The optimization problem consists in minimizing the following cost function j(h) =

Γ2

g·udx, (5.2)

with the following constraint

Ωh(x)dx=C, whereC is the material quantity to retain.

For each element, the functionhis modified, then the exact variation is computed using the cost function (5.2) withu⁰the solution of equation (5.1) andu^εthe solution of

⎧⎪

⎨

⎪⎩

− ∇ ·σ^ε(u^ε) = 0 inΩ^ε, u^ε= 0 onΓ₁, σ^ε(u^ε)n=g onΓ₂.

(5.3) For each element, the numerical vault variation is computed using Theorem3.7which is equivalent to

j(h^ε)−j(h) =

Γ2

(g·u˜^ε−g·u) dx−

Ω^εσ(˜u^ε)· ∇(˜u^ε) dx+

Ωσ(u)· ∇(u) dx, (5.4) with

˜

u^ε= u^ε_D in D,

u⁰ in Ω₀, (5.5)