Topological derivatives for thermo-mechanical semi-coupled system

The topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation. According to the literature, the topological derivative has been fully developed for a wide range of one single physical phenomenon modeled by partial differential equations. In addition, up to our knowledge, the topological asymptotic analysis associated to multi-physics problems has so far not been reported in the literature. In this work, we present the topological asymptotic analysis for the total potential mechanical energy associated to a thermo-mechanical system, when a small circular inclusion is introduced at an arbitrary point of the domain. In particular, we consider the linear elasticity system (modeled by the Navier equation) coupled with the steadystate heat conduction problem (modeled by the Laplace equation). The mechanical coupling term comes out from the thermal stress induced by the temperature field. Since this term is non-local, we introduce a non-standard adjoint state, which allows to obtain a closed form for the topological derivative. Finally, we provide a full mathematical justification for the derived formulas and develop precise estimates for the remainders of the topological asymptotic expansion.


Introduction
The topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation, such as the insertion of holes, inclusions, source-terms or even cracks ( [7]).The topological derivative was rigorously introduced by [19].Since then, this concept has proved to be extremely useful in the treatment of a wide range of problems, for instance, topology optimization ( [5,18]), inverse analysis ( [4,14]) and image processing ( [13,15]), and has became a subject of intensive research.See, for instance, applications of the topological derivative in the multi-scale constitutive modeling context ( [3,9]), fracture mechanics sensitivity analysis ( [10]) and damage evolution modeling ( [1]).Concerning the theoretical development of the topological asymptotic analysis, the reader may refer to the papers by [2] and [16], for instance.
In order to introduce these concepts, let us consider a bounded domain Ω ⊂ R 2 , which is subject to a non-smooth perturbation confined in a small region ω ε ( x) = x + εω of size ε, as shown in fig. 1.Here, x is an arbitrary point of Ω and ω is a fixed domain of R 2 .We introduce a characteristic function x → χ(x), x ∈ R 2 , associated to the unperturbed domain, namely χ = 1 Ω .Then, we define a characteristic function associated to the topologically perturbed domain of the form x → χ ε ( x; x), x ∈ R 2 .In the case of a perforation, for instance, χ ε ( x) = 1 Ω − 1 ωε( x) and the perforated domain is obtained as Ω ε = Ω \ ω ε .Then, we assume that a given shape functional ψ(χ ε ( x)), associated to the topologically perturbed domain, admits the following topological asymptotic expansion where ψ(χ) is the shape functional associated to the original (unperturbed) domain, f (ε) is a positive function such that f (ε) → 0, when ε → 0. The function x → D T ψ( x) is called the topological derivative of ψ at x. Therefore, this derivative can be seen as a first order correction of ψ(χ) to approximate ψ(χ ε ( x)).In fact, after rearranging (1) we have The limit passage ε → 0 in the above expression leads to Since we are dealing with singular domain perturbations, the shape functionals ψ(χ ε ( x)) and ψ(χ) are associated to topologically different domains.Therefore, the above limit is not trivial to be calculated.In particular, we need to perform an asymptotic analysis of the shape functional ψ(χ ε ( x)) with respect to the small parameter ε.In order to calculate the topological derivative, in this work we will apply the methodology developed in [17].The method is based on the following result: The derivative of ψ(χ ε ( x)) with respect to ε can be seen as the sensitivity of ψ(χ ε ( x)), in the classical sense [6,20], to the domain variation produced by an uniform expansion of the perturbation ω ε , namely, ω ε+t ( x) = ω ε ( x) + tω.In fact, we have where ψ(χ ε+t ( x)) is the shape functional associated to the perturbed domain, whose perturbation is given by ω ε+t .Therefore, since ψ(χ ε+t ( x)) and ψ(χ ε ( x)) are now associated to topologically identical domains, we can use the concept of shape sensitivity analysis as an intermediate step in the topological derivative calculation.We will see later that this procedure enormously simplifies the analysis.
According to the literature, the topological derivative has been fully developed for a wide range of one single physical phenomenon modeled by partial differential equations.In addition, up to our knowledge, the topological asymptotic analysis associated to multi-physics problems has so far not been reported in the literature.In this work, therefore, we derive the topological derivative in its closed form for the total potential mechanical energy associated to a thermo-mechanical semi-coupled system, when a small circular inclusion is introduced at an arbitrary point of the domain.In particular, we consider the linear elasticity system (modeled by the Navier equation) coupled with the steady-state heat conduction problem (modeled by the Laplace equation).The mechanical coupling term comes out from the thermal stress induced by the temperature field.Since this term is non-local, we introduce a non-standard adjoint state, which simplifies the analysis allowing to obtain a closed form for the topological derivative.Finally, we provide a full mathematical justification for the derived formula and develop precise estimates for the remainders of the topological asymptotic expansion.We note that this result can be applied in technological research areas such as multi-physic topology design of structures under mechanical and/or thermal loads.
This paper is organized as follows.Section 2 describes the model associated to a thermo-mechanical semi-coupled problem.The topological sensitivity analysis is presented in Section 3, where the main result of this work -the topological derivative in its closed form for the total potential mechanical energy associated to a thermo-mechanical semi-coupled system -is derived.The paper ends in Section 4 where concluding remarks are presented.

Formulation of the problem
As mentioned in previous section, in this work the topological derivative of the total potential energy associated to the mechanical problem submitted to thermal stresses is derived.As topological perturbation we consider a nucleation of a small circular inclusion, with a contrast in the elastic, thermal and thermal-expansion constitutive properties.Then, it is needed to formulate the problems associated to the original and topological perturbed domains.

Unperturbed problem
Consider an open and bounded domain Ω ∈ R 2 representing an elastic solid body subject to a linear thermomechanical deformation process.Assuming small deformation and variations of temperatures, the functional that represents the total potential energy of the mechanical system is written as: where u represents the displacement field and t is a external traction acting on boundary The Cauchy stress tensor σ(u) in ( 6) is defined as: where ∇u s is used to denote the symmetric part of the gradient of the displacement field u, i.e.
The induced thermal stress tensor Q(θ) in ( 6) is defined as: where θ is the temperature field.In addition, C denotes the four-order elastic tensor and B denotes the second-order thermo-elastic tensor.In the case of isotropic elastic body, theses tensors are given by: with µ and λ denoting the Lame's coefficients, and α the thermal expansion coefficient.In terms of the enginnering constant E (Young's modulus) and ν (Poisson's ratio) the above constitutive response can be written as: In addition, the field u is the solution of the following variational problem: with the tensor S(u, θ) representing the total stress, i.e. the contribution of the mechanical and thermal stresses, In the variational problem (12), the set U M and the space V M are defined as Finally, the temperature field of the body θ is solution of the following variational problem: find θ ∈ U T , such that where q is a prescribed heat flux on the Neumann boundary Γ N T .In the Dirichlet boundary Γ D T there is a prescribed temperature denoted as θ.Then, The heat flux operator q(θ) is defined as where K is an second order tensor representing the thermal conductivity of the medium.In the isotropic case, the tensor K can be written as being k the thermal conductivity coefficient.In the variational problem (15), the set U T and the space V T are defined as: In order to simplify further analysis, we introduce the following auxiliary problem: find φ ∈ V T , such that:

Perturbed problem
Considering the introduction of a circular inclusion, denoted as ω ε ( x) := B ε ( x), with radius ε and centered at point x in Ω, the total potential energy functional associated to the perturbed domain mechanical system can be written as: where u ε and θ ε denotes, respectively, the displacement and temperature fields, both associated to the perturbed system.In addition, σ ε (u ε ) and Q ε (θ ε ) are used to denote the mechanical and the induced thermal stresses tensors associated to the perturbed problem.These tensors are defined as: being γ M ε and γ C ε the contrast parameters in the constitutive properties, defined as with γ M and γ C used to denote the values of the contrast.In the perturbed configuration, the displacement field satisfies the variational problem: find where the total stress operator S ε (u ε , θ ε ) associated to the perturbed domain is given by The set U M ε and the space V M ε in the variational problem (23) are defined as where the operator (•) is introduced to denote the jump of (•) across the boundary of the perturbation.
In addition, the thermal equilibrium problem can be written in the variational form as: with the thermal flux in the perturbed domain being defined as: where γ T ε is the parameter that define the contrast between the thermal (constitutive) properties of the matrix and the inclusion, and is defined by: being γ T the value of the contrast.In the variational problem (26) the set U T ε and the space V T ε are defined as: Finally, the auxiliary problem (19), associated to the topologically perturbed domain is written as: find φ ε ∈ V T ε , such that:

Topological Sensitivity Analysis
In order to proceed, it is convenient to introduce an analogy to classical continuum mechanics [11] where by the shape change velocity field V is identified with the classical velocity field of a deforming continuum and ε is identified as a time parameter.Since we are dealing with an uniform expansion of the inclusion B ε , the shape velocity field V satisfies: V | ∂Ω = 0 and V | ∂Bε = −n.Then, the shape derivative of the functional (20) can be written as: where each term represents the derivative with respect to the parameter ε.Therefore, we can state the following propositions: Proposition 1 Let J χε (u ε , θ ε ) be the functional defined by (20).Then, its derivative with respect to the small parameter ε is given by where V is the shape change velocity field defined in Ω that satisfies V | ∂Ω = 0 and V | ∂Bε = −n; θε is the material derivative of the temperature field and Σ ε is a generalization of the classical Eshelby momentum-energy tensor [8], given -for this particular case -by (33) with u ε , θ ε and φ ε denoting the solutions to ( 23), ( 26) and to the auxiliary problem (30).
Proof.By making use of Reynolds' Transport Theorem [11,20] we obtain the identities Then, by considering the above results in (31), the shape derivative of the functional J χε (u ε , θ ε ) is given by Since uε ∈ U M ε , see [20], the terms in uε satisfy the state equation ( 23), then Now, adding the term ± ∫ Ω Q ε ( θε ) • ∇u s in the above result, the derivative Jχε (u ε , θ ε ) can be written alternatively as On the other hand, the deriative of the state equation ( 26) with respect to the parameter ε is given by Next, taking η = φ ε in the above expression, we obtain and tacking η = θε in the auxiliary problem (30), we obtain By using the definition of the heat flux operator (27) and comparing the two last expressions, the following identity holds From the above result, the derivative of the shape functional J χε (u ε , θ ε ) can be written equivalently in the following form: which leads to the result with Σ ε given by (33).
Proposition 2 Let J χε (u ε , θ ε ) be the functional defined by (20).Then, its derivative with respect to the small parameter ε is given by where V is the shape change velocity field defined in Ω that satisfies V | ∂Ω = 0 and V | ∂Bε = −n; θ ′ ε is the spatial derivative of the temperature field and Σ ε is a generalization of the classical Eshelby momentum-energy tensor presented in (33).
Proof.By making use of the Reynolds' Transport Theorem [11,20], we obtain the following identities: Introducing the above expressions in the definitions of the shape derivative (31) and taking into account that: (i) uε ∈ U M ε , see [20], the terms in uε satisfy the state equation ( 23 By using the relation between the material and spatial derivatives of the temperature field, the above expression can be written as, On the other hand, the derivative of the state equation ( 26) with respect to parameter ε is given by Next, tacking η = φ ε in the above expression, we obtain and tacking η = θε in the auxiliary problem (30), we obtain By using the definition of the heat flux operator (27) and comparing the two last expressions, the following identity holds From the above result, the derivative of the shape functional J χε (u ε , θ ε ) can be written equivalently in the following form, By integrating by parts the second term in the above expression and using the definition of the Eshelby's tensor Σ ε , we have Taking into account the state equation ( 26) and the auxiliary problem (30), we observe that the second term in the above expression satisfies the following identity Then, the lats two terms in (54) vanish, leading to the result.
Corollary 3 By considering the relation between the material and spatial derivative of the temperature field, (32) can be written as: By integrating by part the firt term of the above expression and using the restriccion of the velocity field V on the boundaries ∂Ω and ∂B ε , we obtain (57) By comparing (44) with (57) and recalling that both identities are valid for all V ∈ Ω, the follow result holds true ∫ Thus, the equation for the balance of the configurational forces [12] can be written as: To analytically solve the integrals expression of the derivative Jχε (u ε , θ ε ) it is necessary to perform an asymptotic analysis of the solutions of the PDE's involved in these coupled problems.In order to simplify the analysys, let us use the linearity property of the shape functional with respect to the solution of the thermal problem (26) and split the analysis in two cases: (i) γ T = 1 and (ii) γ M = γ C = 1.

Case γ T = 1
For this particular case, γ T = 1, we have that the temperature field is not perturbed by the presence of the inclusion B ε in the mechanical problem.Then, the temperature for the unperturbed and perturbed problems coincides, i.e. θ ε = θ.Thus, the derivative of the shape functional can be written as: (60) Considering a curvilinear coordinate system (t, n) defined on the boundary of the inclusion ∂B ε , the jump condition of the stress field S ε (u ε , θ) in the boundary ∂B ε can be written, tacking into account the orthogonality of the normal (n) and tangential (t) vectors, as: [[S ε (u ε , θ) n = S nn ε (u ε , θ) n + S tn ε (u ε , θ) t = 0 , (61) which leads to the following result, In the same way, the continuity condition of the displacement field defined on the boundary ∂B ε results in the following relations The above continuity relations implies the continuity of the tangential component of the deformation tensor ∇u s ε , In view of the above decomposition, it is possible to analyze each term of (60) separately: where ∂ n u t ε = (∇u ε ) tn .With the above continuity properties, the expression (65) can be written as: By using the same continuity properties, (66) can be re-written as: By considering an isotropic thermal expansion, i.e.Q nn (θ) = Q tt (θ) and Q nt (θ) = 0, from (67) we have that