Abstract
Purpose
The goal of this paper is to give a comprehensive and short review on how to compute the first- and second-order topological derivatives and potentially higher-order topological derivatives for partial differential equation (PDE) constrained shape functionals.
Design/methodology/approach
The authors employ the adjoint and averaged adjoint variable within the Lagrangian framework and compare three different adjoint-based methods to compute higher-order topological derivatives. To illustrate the methodology proposed in this paper, the authors then apply the methods to a linear elasticity model.
Findings
The authors compute the first- and second-order topological derivatives of the linear elasticity model for various shape functionals in dimension two and three using Amstutz' method, the averaged adjoint method and Delfour's method.
Originality/value
In contrast to other contributions regarding this subject, the authors not only compute the first- and second-order topological derivatives, but additionally give some insight on various methods and compare their applicability and efficiency with respect to the underlying problem formulation.
Keywords
Citation
Baumann, P. and Sturm, K. (2022), "Adjoint-based methods to compute higher-order topological derivatives with an application to elasticity", Engineering Computations, Vol. 39 No. 1, pp. 60-114. https://doi.org/10.1108/EC-07-2021-0407
Publisher
:Emerald Publishing Limited
Copyright © 2021, Phillip Baumann and Kevin Sturm
License
Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode.
1. Introduction
In this paper we provide a review of techniques for the computation of the first- and second-order topological derivatives. We compare and apply three techniques to the following model problem: Let
Let
The topological derivative was first introduced in Eschenauer et al. (1994) and later mathematically justified in Sokolowski and Zochowski (1999), Garreau et al. (2001) with an application to linear elasticity. Follow-up works of many authors studied the asymptotic behaviour of shape functionals for various partial differential equations (PDEs). For instance, for Kirchhoff plates Amstutz and Novotny (2010), electrical impedance tomography Hintermüller and Laurain (2008), Hintermüller et al. (2011), Maxwell's equation Masmoudi et al. (2005), Stokes' equation Hassine and Masmoudi (2004) and elliptic variational inequalities Hintermüller and Laurain (2011). We also refer to the monograph Novotny and Sokolowski (2013) for more applications and references therein.
The idea of the topological derivative is to perturb the design variable with a singular perturbation and study the asymptotic behaviour of the shape functional
Higher-order topological derivatives are less studied, but have been computed for several problems. For instance, in Hintermüller and Laurain (2008), second-order topological derivatives for an electrical impedance tomography problem are studied. In Bonnet (2018), higher-order topological derivatives in dimension two for linear elasticity using the method of Novotny et al. (2003) are established. In Bonnet and Cornaggia (2017), the expansion of higher-order topological derivatives for a least square misfit function for linear elasticity in dimension three exploiting a Green's function is established. In Bonnet (2018), a similar misfit function subject to a scattering problem is expanded.
The first ingredient to compute higher topological derivatives is the asymptotic behaviour of the solution of the state equation, in our concrete example this is Equation (1.2). The second ingredient is an expansion of the shape function and is mostly, although not necessary, done via the introduction of an adjoint variable. As is well known from optimal control and shape optimisation theory (see, e.g. Hinze et al., 2009; Ito and Kunisch, 2008), the advantage of using an adjoint variable is the numerically efficient computation of the topological derivative. First-order topological derivatives for ball inclusions and linear problems can be computed solely from the knowledge of the state variable and the adjoint state variable; see, for example, in Sokolowski and Zochowski (1999). For higher-order topological derivatives in most cases, additional exterior partial differential equations, so-called corrector equations, have to be solved, although in some cases these can also be solved explicitly; Hintermüller and Laurain (2008).
While most papers deal with linear partial differential equations, also nonlinear partial differential equations have been studied. We refer to Iguernane et al. (2009), Beretta et al. (2017), Sturm (2020), Amstutz (2006b) for the study of first-order topological derivatives for semilinear elliptic partial differential equations. To the authors’ knowledge, there is no research for higher-order topological derivatives for these equations and thus remains an open and challenging topic. Also, quasi-linear problems have been studied first in Amstutz and Bonnafé (2017) and more recently in Gangl and Sturm (2020a), Amstutz and Gangl (2019), Gangl and Sturm (2021). In particular in Gangl and Sturm (2020a), a projection trick is used to avoid the use of a fundamental solution, which is in contrast to most works on semilinear partial differential equations.
An established method to compute the topological derivative and higher derivatives is the method of Amstutz (2003). It amounts to study the asymptotic behaviour of a perturbed adjoint equation, which depends on the unperturbed state equations. It has been used in some of the papers mentioned above such as Masmoudi et al. (2005), Hassine and Masmoudi (2004) and also Amstutz (2006a, b), to only mention a few. The advantage of the method is that it simplifies the computation of the topological derivative compared to a direct computation of the topological derivative by expanding the cost function with Taylor's expansion.
A second method, which has been introduced in the context of shape optimisation and the computation of shape derivatives, was used in Sturm (2020) to compute topological derivatives for semilinear problems. It has been extended in Gangl and Sturm (2020a) to compute topological derivatives for quasi-linear problems. In contrast to Amstutz' method, the averaged adjoint variable also depends on the perturbed state equation, which makes the anaylsis of the asymptotic behaviour of the adjoint variable more challenging. However, the advantage is that it seems to be readily applicable to a wide range of cost functions, and also the computation of the final formula for higher-order topological derivatives is straight forward once the asymptotics of the averaged adjoint variable is known.
A third method was introduced in Delfour (2018) and uses the usual unperturbed adjoint variable. The advantage is that no analysis of a perturbed adjoint variable is required, but, as shown in Gangl and Sturm (2020a), it seems to be more difficult to apply this method to certain cost functions, such as the L2-tracking-type cost functions.
Finally, let us mention the method of Novotny et al. (2003), where a method to compute the topological derivatives is proposed as the limit of the shape derivative. This method is not always applicable, but it provides a fast method to compute also higher-order topological derivatives; see Silva et al. (2010).
In this paper we thoroughly study and review the first three mentioned methods and apply them to the model problem of linear elasticity introduced in (1.2). We first exam the asymptotic behaviour of (1.2) up to order two and then study the asymptotic behaviour of Amstutz' perturbed adjoint variable and the averaged adjoint variable. We then apply the three methods to compute first- and second-order topological derivatives for three types of cost functions, the compliance, a boundary tracking-type cost function and a tracking-type cost function of the gradient.
1.1 Structure of the paper
In Section 2, we discuss three different teqhniques to compute the topological derivative. This is done by introducing the Lagrangian setting, which simplifies the notation. In Section 3, we derive the complete asymptotic analysis for a linear elasticity model including remainder estimates. The section covers both the two-dimensional and three-dimensional cases, whose analysis differs since the fundamental solution of the linear elasticity equation has a different asymptotic behaviour. In Section 4, we derive the asymptotic analysis for the adjoint and averaged adjoint variable, respectively. This is done in a similar fashion to Section 3. In Section 5, we employ the previously derived results to compute the topological derivative. That is, we apply the theoretical background derived in Section 2 to our elasticity model and a versatile cost function.
1.2 Notation
In the whole paper we denote by
Then we define the Beppo-Levi space
The Euclidean norm on Rd will be denoted as |⋅| and the corresponding operator norm Rd×d will be also denoted as |⋅|. The Euclidean ball of radius r > 0 located at x0 ∈ Rd will be denoted as Br(x0). Additionally, for a domain Ω with sufficiently smooth boundary ∂Ω, we denote the outer normal vector as n. The Slobodeckij seminorm
For convenience we will later on use the abbreviated notation of the averaged integral defined as
2. Lagrangian techniques to compute the topological derivative
In this section we review Lagrangian techniques to compute topological derivatives. While it is well established in optimisation algorithms to compute derivatives of PDE constrained problems with the help of Lagrangians, it seems rather new to the topology optimisation community. However, we will show that actually Amustutz's method can be interpreted as a Lagrangian approach by introducing a suitable Lagrangian function and recasting his original result in terms of this Lagrangian. More recently, another Lagrangian approach was proposed in Delfour and Sturm (2016), where essentially an extra term appears when differentiating the Lagrangian function. Finally, we will review Delfour's approach of (Delfour, 2018, Thm.3.3) using only the unperturbed adjoint state variable.
2.1 Abstract setting
Let
2.2 Amstutz' method
We first review the approach of Amstutz (2003); see also (Amstutz, 2006a, Prop. 2.1). This approach has been proved to be versatile and has been applied to a number of linear and non-linear problems. For instance, in Amstutz (2006a) a linear transmission problem was examined and its first-order topological derivative was computed. In Amstutz et al. (2014), the topological derivative of elliptic differentiation equations with 2m differential operator was derived. In Amstutz (2006b), the topological derivative for a class of certain non-linear equations has been studied.
(Amstutz, 2006a, Prop. 2.1). Assume that the following hypotheses hold.
There exist numbers δa(1) and δf(1) and a function ℓ1 : R+ → R+ with
There exist two numbers
Then the following expansion holds
We will reformulate and generalise the previous result in terms of a Lagrangian function
Let ℓ1 : R+ → R+ be a function with
In particular,
Let ℓ2 : R+ → R+ be a function with
Proof. ad (1): Using that
Now, the result follows by dividing by ℓ1(ɛ) for ɛ > 0 and passing to the limit ɛ ↘ 0.
ad (2): This follows the same lines as the proof of item (1) and is left to the reader. □
The computation of the asymptotic expansions (2.11), (2.14) requires the study of the asymptotic behaviour of uɛ on the whole domain
2.3 Averaged adjoint method
Another approach to compute topological derivatives was proposed in Sturm (2020) and applied to non-linear problems in Gangl and Sturm (2020a), Sturm (2020), Gangl and Sturm (2021) and used for the optimisation on surfaces in Gangl and Sturm (2020b). Recall the Lagrangian function
We henceforth assume that for all
In addition, plugging φ = uɛ − u0 into (2.22), one obtains
Let ℓ1 : R+ → R+ be a function with
Let ℓ2 : R+ → R+ be a function with
Proof. ad (1): Recalling
Dividing by ℓ1(ɛ) for ɛ > 0 and passing to the limit ɛ ↘ 0 yields the result.
ad (2): Similar to item (1). □
The previous result can be readily generalised to compute the nth-order topological derivative as shown in the following proposition.
(nth topological derivative). Assume that the following hypotheses hold.
There exist numbers δa(i) and δf(i), i = 1, 2, …, n and a function ℓ1 : R+ → R+ with
There exist numbers δA(i) and δF(i), i = 1, 2, …, n, such that
Then the following expansion holds
Proof. Similar to the proof of Proposition 2.4, we write
The second term on the right-hand side reads
So using (2.28)–(2.30), we can expand each difference in this expression. As for the first difference on right-hand side, one has
Therefore, employing (2.31), (2.32), we can also expand these two differences and obtain the claimed formula (2.34). □
Checking the expansions (2.24), (2.27) in applications usually requires some regularity of the state u0 and adjoint state q0 = p0. However, the computation of this expansion is a straightforward application of Taylor's formula. The reader will find an application in Section 5.2
The computation of the asymptotic expansions (2.23), (2.26) requires the study of the asymptotic behaviour of qɛ and therefore also of uɛ. This is the most difficult part and can be done by the compounded layer expansion involving corrector equations (see for instance, Mazya et al., 2000b; Mazya et al., 2000a) as is presented in Section 4.2
2.4 Delfour's method
In this section we discuss a method proposed by M.C. Delfour in (Delfour, 2018, Thm.3.3). The definite advantage is that it uses the unperturbed adjoint equation and only requires the asymptotic analysis of the state equation, but it seems to come with the shortcoming that it is only applicable to certain cost functions; see Gangl and Sturm (2020a). As before, we let
Let ℓ1 : R+ → R+ be a function with ℓ1 ≥ 0 and
Let ℓ2 : R+ → R+ be a function with ℓ2 ≥ 0 and
Proof. ad (1): Firstly, note that by definition the unperturbed adjoint state p0 satisfies
Thus, we can write j(ɛ) − j(0) in the following way:
Now, dividing by ℓ1(ɛ), ɛ > 0 and passing to the limit ɛ ↘ 0 yield the result.
ad (2): This can be shown similarly to (1). □
Similarly to Amstutz' method and the averaged adjoint method, Delfour's method requires the asymptotic behaviour of uɛ on the whole domain to compute (2.37), (2.41). This may be challenging in the analysis in dimension two for some cost functionals. Additionally, (2.38), (2.42) can be checked by smoothness assumptions on p0 and u0 and the knowledge of the asymptotics of uɛ on a small subset of size ɛ. The remaining terms (2.39), (2.43) usually are computed making use of Taylor's expansion of u0 and p0, respectively.
2.4.1 Overview of the employed adjoint equations
The methods reviewed in the previous sections make use of three different adjoint equations. The method of Amstutz (2006a) uses an adjoint equation which depends on the unperturbed state variable:
Delfour's method uses the unperturbed adjoint equation:
Finally, there is the averaged adjoint method, which employs the averaged adjoint equation Sturm (2015) and Delfour and Sturm (2016):
3. Analysis of the perturbed state equation
Let Ω ⊂ D open, ω ⊂ Rd be a bounded domain containing the origin 0 ∈ ω and let
In the following sections we are going to derive the asymptotic expansion of uɛ using the compounded layer method; see, Mazya et al. (2000a, b). We note that this expansion has already been computed in Bonnet and Cornaggia (2017) by means of Green's function and earlier in Ammari et al. (2002) for fΩ = 0. In the following two sections we state some preliminary results regarding the scaling of inequalities and remainder estimates, which will be needed later on.
3.1 Scaling of inequalities
In this section we discuss the influence of a parametrised affine transformation Φɛ : Rd → Rd onto norms and the scaling behaviour of some well-known inequalities with respect to that parameter.
For ɛ > 0 we define the inflation of
For convenience, we denote the inflated boundary
For ɛ > 0 and
Let
For 1 ≤ p < ∞ and
For 1 ≤ p < ∞ and
For
Proof.
A change of variables yields
Taking into account that
This follows from item (1) and (2).□
Let
For 1 ≤ p ≤ q ≤ ∞, there exists a constant C > 0, such that
Let d ≥ 3 and 2* denote the Sobolev conjugate of 2. There exists a constant C > 0, such that
Let d = 2 and α > 0 small. There exists a constant C > 0 and δ > 0 small, such that
For
Given a smooth connected domain
Let
Proof.
This is a direct consequence of Lemma 3.3 item (1).
We use Lemma 3.3 item (1) and (2) and apply the Gagliardo–Nirenberg inequality (Evans, 2010, p. 265, Thm. 2) to the bounded domain
Now the result follows from
We apply the Gagliardo–Nirenberg inequality with respect to p≔2 − δ < 2 and use the continuous embedding
Since (2 − δ)* diverges to ∞ as δ ↘ 0, the result follows.
This follows from a change of variables and the continuity of the trace operator.
From (Wloka, 1987, p. 129, Thm. 8.8), we know there exists a continuous extension operator
Items (1) and (2) of Lemma 3.3 and an application of Friedrich's inequality yield the result.□
3.2 Remainder estimates
We begin this section with the following auxiliary result.
Let
Proof.
Let
Now taking the square root shows the result.
Let 0 < r1 < r2 such that
In order to compute the first term (3.18), we consider for each pair
Thus, by Hölder's inequality we conclude
Since this inequality holds for every smooth path φx,y connecting x and y, the estimate holds for
As a result, we conclude
The key here was to choose the set S such that
The second term (3.19) can be estimated by using a straight line connecting
Plugging this estimate into (3.19), yields
To finish our proof, we need to show that the integral on the right-hand side is finite. Therefore, let
Now we can split the inner integral into layers according to these sets:
Hence, combining (3.24) and (3.27) and using
The proof follows the lines of item (1) and is therefore left to the reader.□
3.3 First-order asymptotic expansion
Let
We henceforth assume that the u0 ∈ C3(Bδ(x0)) for a small radius δ > 0.
There is a constant C > 0, such that for all ɛ > 0 sufficiently small there holds
Proof. Subtracting (3.1) for ɛ > 0 and (3.29) yields
Therefore, testing with
Now, the result follows from
For almost every
The second variation of uɛ is defined by
More generally, we define the i-th variation of uɛ for i ≥ 2 by
By extending uɛ and u0 outside of
In the following, we show that the first variation of the state converges to a function
Let A : Rd×d → Rd×d be uniformly positive definite,
Furthermore, there exists a constant C > 0 such that
Proof. Let
Now consider
Hence, we conclude that Vɛ≔uɛ + Gɛ satisfies (3.37) and (3.38). Uniqueness is guaranteed by the ellipticity of aɛ. Applying the triangle inequality and using the continuity of
There exists a unique solution
Moreover, there exists a representative U(1) ∈ [U], which satisfies pointwise for |x| → ∞:
Proof. Unique solvability of (3.42) follows directly from the Lemma of Lax–Milgram. Thus, the only thing left to show is the asymptotic behaviour (3.43) of U(1). For this we first note that U(1) can be characterised by the following set of equations:
By (Ammari, 2008, p. 76, Thm. 3.3.8) there are f, g ∈ L2(∂ω)d, such that
Let
Proof. We start by deriving an equation for
Splitting the integral on the left-hand side of (3.42), integrating by parts and using Div(C2ϵ(U(1))) = 0 in
To finish our proof, we need to estimate the norms of
Let
At first, we consider ∫ω(C2 − C1)[ϵ(u0)◦Φɛ − ϵ(u0)(x0)] : ϵ(φ) dx. Since u0 ∈ C3(Bδ(x0)), we get
Together with an application of Hölder's inequality, we conclude
Next, we consider ɛ∫ω(f1 − f2)◦Φɛ ⋅ φ dx. Since we want to apply the Gagliardo–Nirenberg inequality, we need to distinguish between dimensions d = 2 and d = 3.
For d = 3, an application of Hölder's inequality with respect to p = 2* and Lemma 3.4, item (2) yield
For d = 2 we apply Hölder's inequality with respect to p = (2 − δ)* for δ > 0 sufficiently small and Lemma 3.4, item (3) to obtain
Finally, the last term can be estimated using Hölder's inequality and the scaled trace inequality (Lemma 3.4 item (4)):
Thus, Lemma 3.5, item (3) with m = d − 1 yields
Combining these estimates results in
Now plugging (3.62) and (3.63) into (3.54) finishes the proof. □
Rewriting
3.4 Second-order asymptotic expansion
As mentioned earlier, the boundary layer corrector U(1) introduces an error at the boundary of
There is a unique solution
There is a solution
There exists a solution
There is a unique solution
Note that the requirement for p to be greater than 2 in dimension two is necessary to guarantee that the gradient of
Proof. Unique solvability of (3.65) and (3.72) follows from the Lax–Milgram theorem. In order to show the existence and the desired representation formula of
Now consider the volume potential u(x)≔∫ωΓ1(x − y)[(f1(x0) − f2(x0))] dy, for x ∈ ω, which satisfies the inhomogeneous equation inside ω. By (Ammari, 2008, p. 76, Thm. 3.3.8) there are f, g ∈ L2(
Finally,
As a consequence of the equivalence relation defining the Beppo-Levi space, the function U(2) is defined up to a constant. Thus, we are allowed to add arbitrary constants to the boundary layer corrector U(2). As a result of the additive property of the leading term R(2)(x) = ln(x), we need to add the ɛ dependent constant c ln(ɛ), with a suitable constant c ∈ R in dimension d = 2. In dimension d = 3 this problem does not appear since the leading term |x|−1 is multiplicative and therefore can be compensated by the factor ɛd−2 found in Definition 3.7.
A possible approach to approximate the solution
Let
There exists a constant C > 0, such that
For d ∈ {2, 3}, there holds
Proof. ad (1): Similar to the estimation of the first-order expansion, we aim to apply Lemma 3.8 in order to handle the inhomogeneous Dirichlet boundary condition on Γɛ. Hence, we start by deriving an equation satisfied by
Since the bilinear form only depends on the symmetrised gradient of
Now we can apply Lemma 3.8 to
Hence, we get the apriori estimate
Due to great similarity between d = 2 and d = 3, we will discuss both cases together and only highlight the terms that have to be treated separately. Thus, if not further specified, let d = 2, 3. Again, we start by estimating ‖Fɛ‖. Let
A Taylor expansion of (f1(Φɛ(x)) − f2(Φɛ(x))) at x0, Hölder's inequality and Lemma 3.4, item (2), (3) yield
Since u0 is three times differentiable in a neighbourhood of x0, there is a constant C > 0, such that |ɛ−1(ϵ(u0)◦Φɛ − ϵ(u0)(x0)) − ∇ϵ(u0)(x0)x| ≤ Cɛ, for x ∈ ω. Hence, Hölder's inequality yields
Furthermore, by Hölder's inequality we get
Next we consider the boundary integral terms:
Here, we note that ϵ(U(1)) − ɛdϵ(R(1))(ɛx) cancels out the leading term of U(1) on
Similarly, we deduce from Lemma 3.5, item (3) with m = d − 1 that there is a constant C > 0, such that
Combining the previous estimates yields
Then, by Lemma 3.5, item (1), (2) with m = d and m = d − 1 respectively, there is a constant C > 0, such that
Now we can plug (3.94) and (3.95) into the a priori estimate (3.88), which shows (3.81).
ad (2): By the triangle inequality, we have
Note that by the triangle inequality one has
In order to give a better understanding of the scheme of the asymptotic expansion, we would like to point out the main difference between the first- and second-order expansion, which is the slower decay of the boundary layer corrector U(2) compared to U(1). As a result, there was no necessity to introduce the regular corrector u(1) in the first-order expansion, whereas u(2) was needed to obtain the desired order of at least ɛ1−α, for α > 0 small. Additionally, one should note that boundary layer correctors appearing in higher-order expansions have asymptotics similar to U(2) and therefore demand a correction of the associated regular correctors. Thus, the scheme of the asymptotic expansion of arbitrary order resembles the second-order expansion given in this chapter, rather than the first-order expansion.
4. Analysis of the perturbed adjoint equation
In this section we study the asymptotic analysis of the Amstutz' adjoint equation and the averaged adjoint equation for our elasticity model problem. We shall first exam Amstutz' adjoint and derive its asymptotic expansion up to order two.
4.1 Amstutz' adjoint equation
The adjoint state pɛ, ɛ ≥ 0 satisfies
With the cost function defined in (1.1), this equation reads explicitly
We now compute an asymptotic expansion of pɛ in a similar fashion to the direct state uɛ. Therefore we define the variation of the adjoint state
There exists a solution
Proof. Using the adjoint tensor
For α ∈ (0, 1) and ɛ > 0 sufficiently small there is a constant C > 0, such that
Proof. Similarly to the analysis of the direct state, we derive an equation of the form
A detailed derivation and estimation of the functional
Thus, considering (4.11) and (4.12), an application of Lemma 3.8 with
We now continue with the second-order expansion. Similar to the state variable expansion, we therefore introduce a number of correctors in the following Lemma, which approximate the first-order expansion inside ωɛ and on the boundary
There is a unique solution
There is a solution
There is a solution
There is a unique solution
Proof. Rewriting these equations with the help of the adjoint operator
Now we are able to state our main result regarding the second-order expansion of the adjoint state variable pɛ:
There exists a constant C > 0, such that
For d ∈ {2, 3}, there holds
Proof. ad (1): For the sake of clarity, we restrict ourselves to the case of d = 3. Dimension d = 2 can be treated in a similar fashion. In view of the auxiliary result Lemma 3.8, we seek a governing equation for
Hence, considering (4.24) and (4.25), Lemma 3.8 shows (4.21).
ad (2): By the triangle inequality we have
4.2 Averaged adjoint equation
The averaged adjoint state qɛ satisfies
With the cost function defined in (1.1), this equation reads explicitly
We now introduce the first terms of the asymptotic expansion:
There exists a solution
There exists a solution
Now let
Proof. Similar to Lemma 4.1, but due to the inhomogeneity in the exterior domain, we use a Newton potential to represent the solution. □
For α ∈ (0, 1) and ɛ > 0 sufficiently small there is a constant C > 0, such that
Proof. We only show the case of d = 3, since the proof for d = 2 follows the same lines. Again, we start by deriving an equation of the form
A detailed derivation and estimation of the functional
Thus, considering (4.38) and (4.39), an application of Lemma 3.8 shows (4.36). □
We now continue with the second-order expansion. Similar to the previous asymptotic expansions, we therefore introduce each component in the following lemma. Note that the regular correctors aim to approximate U(1) in addition to their approximation of the occurring boundary layer correctors Q(1), Q(2). This is a result of the appearance of
There is a unique solution
There is a solution
There is a solution
There is a unique solution
There is a unique solution
Furthermore, we define
Proof. Similar to Lemma 3.12 and Lemma 4.5. □
Now we are able to state our main result regarding the second-order expansion of the averaged adjoint state variable qɛ:
Let α ∈ (0, 1). There exists a constant C > 0, such that for d = 3 and d = 2, we have respectively:
For d ∈ {2, 3}, there holds
Proof. ad (1): Similar to the proof of the second-order expansion of the adjoint state variable, we restrict ourselves to the case of d = 3. The proof for dimension d = 2 follows the same lines and is therefore omitted. In view of the auxiliary result Lemma 3.8, we seek a governing equation for
In view of (4.52) and (4.53), Lemma 3.8 shows (4.49).
ad (2): Let d = 3. By the triangle inequality we have
5. Computation of the topological derivatives for linear elasticity problem
In this section we compute the first- and second-order topological derivatives of our elasticity problem introduced in (1.1), namely
For ɛ ≥ 0 let
Similarly, the second-order topological derivative is given as
More specifically, since we considered Ω =∅, we compute the topological derivative at a point
We compute the topological derivatives using Proposition 2.2 (Amstutz' method), Proposition 2.4 (averaged adjoint) and Proposition 2.7 (Delfour's method).
We would like to point out that, contrary to the setting in Section 2,
Note that the more general case Ω ≠ ∅,
5.1 Amstutz' method
In order to compute the first-order topological derivative, let ℓ1(ɛ)≔|ωɛ|. By Proposition 2.2, item (1), we have
Now, a change of variables leads to
Next, we consider
By Hölder's inequality, Lemma 3.4, item (2), (3) and Theorem 4.6 one readily checks that
Therefore, the first-order topological derivative is given by
In order to compute
Now, considering Theorem 4.4, item (2), we have
Thus, the second-order topological derivative is given by
5.2 Averaged adjoint method
We start with the first-order topological derivative. Therefore, let ℓ1(ɛ)≔|ωɛ|. By Proposition 2.4 item (1) we have
Since u0 ∈ C3(Bδ(x0)) for δ > 0 small and by Theorem 4.6
Furthermore, applying Hölder's inequality, Lemma 3.4 item (2), (3) and Theorem 4.6, one readily checks that
It follows that
Now, since u0, q0, f1, f2 are smooth in a neighbourhood of x0, we get
Hence, the first-order topological derivative is given by
An elegant way to represent the topological derivative is by the use of a polarisation tensor (see Novotny and Sokolowski, 2013; Ammari et al., 2005). For this, note that the mappings
Hence, there are tensors
Next we compute the second-order topological derivative. Therefore, let ℓ2(ɛ)≔ɛℓ1(ɛ). By Proposition 2.4 item (2) we have
5.3 Delfour's method
At last, we consider Delfour's method to compute the topological derivative. Therefore, recall that by Proposition 2.7 item (1) we have
Hence, passing to the limit ɛ ↘ 0 yields
Since
A similar computation yields
A more detailed derivation of
Similar to (5.15) it follows that
Furthermore, we have
Hence, passing to the limit ɛ ↘ 0, we deduce
We obtain
We would like to point out that using the defining equations of the boundary layer correctors, one can show that all three expressions of the second-order topological derivative coincide and therefore all methods lead to the same result. To get an idea, we show that the first-order topological derivative of Amstutz' approach and the averaged adjoint method are the same. Plugging in φ = Q(1) in (3.42) yields
Additionally, by choosing φ = P(1) in(4.5) and φ = U(1) in (4.30), (4.33) we get
Now using p0 = q0 it follows that both results (5.13), (5.25) coincide.
6. Conclusion
In the present work we review three different methods to compute the second-order topological derivative and illustrate their methodologies by applying them to a linear elasticity model. To give a better insight into the differences of these methods, the cost functional consists of three terms: the compliance, a L2 tracking type over a part of the Neumann boundary and a gradient tracking type over the whole domain, whereas the first one is linear and the latter two are quadratic.
Amstutz' method to compute the topological derivative requires besides the analysis of the direct state uɛ also the analysis of the adjoint state pɛ. Even though this seems to lead to additional work, we would like to point out that, due to the ɛ-dependence of the defining equation of the adjoint state variable, the analysis of pɛ resembles the analysis of the direct state and can be done in a similar way. The computation of the topological derivative for the compliance term is straightforward, whereas checking the occurring limits for the nonlinearities requires the asymptotic analysis of uɛ on the whole domain.
The averaged adjoint method shifts the work from the computation of the topological derivative to the asymptotic analysis of the averaged adjoint variable qɛ. Since the defining equation depends on the state variable uɛ, the asymptotic analysis of pɛ does not resemble the analysis of uɛ and therefore needs to be treated differently. In fact, we would like to mention that again the non-linearities of the cost functional are the reason for additional work during this process. When it comes to the computation of the topological derivative, the averaged adjoint method simplifies the procedure as it only requires convergence of qɛ on a small subdomain of size ɛ.
Finally, Delfour's method resembles Amstutz' method as it requires the asymptotic analysis of uɛ on the whole domain, yet it does not need the analysis of the adjoint state pɛ. This advantage seems to come with the shortcoming, that this method is only applicable to a selective set of cost functions.
To recapitulate, each method proposed in this work has some advantages and disadvantages over the others. The decision on which method fits the actual problem setting the best greatly depends on the actual cost function as well as the properties of the underlying partial differential equation.
Here we derive equations satisfied by the variations of the adjoint and average adjoint variable respectively.
A1 Derivation and estimation of Equation (4.10)
In order to compute
Now we can find a constant C > 0, such that the following estimates hold:
Combining the previous estimates yields
A2 Derivation and estimation of Equation (4.23)
We start by dividing (Section A3) by ɛ and subtract (4.13), (4.20), which can be formulated on the domain
Now we want to estimate the norm of
Since p0 is three times differentiable in a neighbourhood of x0, there is a constant C > 0, such that |ɛ−1(ϵ(p0)◦Φɛ − ϵ(p0)(x0)) − ∇ϵ(p0)(x0)x| ≤ Cɛ, for x ∈ ω. Hence, Hölder's inequality yields
A Taylor expansion of (f2 − f1)◦Φɛ at x0, Hölder's inequality and Lemma 3.4 item (2) yield
Furthermore, by Hölder's inequality we get
Combining the above results leaves us with
From Hölder's inequality, Lemma 3.5 item (3) with m = d and the scaled trace inequality we get
Similarly, we deduce from Lemma 3.5 item (3) with m = d − 1 that there is a constant C > 0, such that
Thus, these estimates result in
A3 Derivation and estimation of Equation (4.37)
In order to compute a governing equation for
Now let
Combining the above results leaves us with
A4 Derivation and estimation of Equation (4.51)
Due to the high number of terms, we derive the governing equation in more detail. Therefore, we formulate (4.40), (4.47), (4.48), (4.41) and (4.44) on the domain
Now dividing (A.16), (A.17) by ɛ and subtracting (A.18)–(A.21) leaves us with
In the following let
A Taylor's expansion and Hölder's inequality yield
A Taylor's expansion followed by an application of Hölder's inequality with respect to p = 2* and Lemma 3.4 item (2) yield
From Theorem 3.16 we deduce
Furthermore, from Hölder's inequality it follows
Similarly, one gets
Combining these estimates, we get
By smuggling in ɛ−1U(1) and U(2) we get
The first term on the right-hand side can be estimated by Hölder's inequality, the scaled trace inequality and Theorem 3.16, whereas the remaining terms can be estimated by Hölder's inequality, the scaled trace inequality and Lemma 3.5 item (1). Thus we conclude
A similar computation to Lemma 3.5 and the scaled trace inequality yield
A similar argument shows
Furthermore, the remaining terms can be estimated by Hölder's inequality, the scaled trace inequality and Lemma 3.5 item (3) with m = d − 1 and m = d respectively, to deduce
Hence, we conclude
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Acknowledgements
Phillip Baumann has been funded by the Austrian Science Fund (FWF) project P 32911.