Stéphane Cotin

Michel Duprez

Vanessa Lleras

Alexei Lozinski
email: alexei.lozinski@univ-fcomte.fr

Killian Vuillemot

φ-FEM: an efficient simulation tool using simple meshes for problems in structure mechanics and heat transfer

One of the major issues in the computational mechanics is to take into account the geometrical complexity. To overcome this difficulty and to avoid the expensive mesh generation, geometrically unfitted methods, i.e. the numerical methods using the simple computational meshes that do not fit the boundary of the domain, and/or the internal interfaces, have been widely developed. In the present work, we investigate the performances of an unfitted method called φ-FEM that converges optimally and uses classical finite element spaces so that it can be easily implemented using general FEM libraries. The main idea is to take into account the geometry thanks to a level set function describing the boundary or the interface. Up to now, the φ-FEM approach has been proposed, tested and substantiated mathematically only in some simplest settings: Poisson equation with Dirichlet/Neumann/Robin boundary conditions. Our goal here is to demonstrate its applicability to some more sophisticated governing equations arising in the computational mechanics. We consider the linear elasticity equations accompanied by either pure Dirichlet boundary conditions or by the mixed ones (Dirichlet and Neumann boundary conditios co-existing on parts of the boundary), an interface problem (linear elasticity with material coefficients abruptly changing over an internal interface), a model of elastic structures with cracks, and finally the heat equation. In all these settings, we derive an appropriate variant of φ-FEM and then illustrate it by numerical tests on manufactured solutions. We also compare the accuracy and efficiency of φ-FEM with those of the standard fitted FEM on the meshes of similar size, revealing the substantial gains that can be achieved by φ-FEM in both the accuracy and the computational time.

Introduction

Taking the geometrical complexity into account is one of the major issues in the computational mechanics. Although some spectacular advances in mesh generation have been achieved in recent years, constructing and using the meshes fitting the geometry of, for example, human organs may still be prohibitively expensive in realistic 3D configurations. Moreover, when the geometry is changing in time or on iterations of an optimization algorithm, the mesh should be frequently adapted, either by complete remeshing (expensive) or by moving the nodes (may lead to a degradation of the mesh quality, impacting the accuracy and the stability of computations).

Geometrically unfitted methods, i.e. the numerical methods using the computational meshes that do not fit the boundary of the domain, and/or the internal interfaces, have been widely investigated in the computational mechanics for decades. Their variants come under the name of Immersed Boundary [START_REF] Mittal | Immersed boundary methods[END_REF] or Fictitious Domain [START_REF] Glowinski | A fictitious domain method for Dirichlet problem and applications[END_REF] methods. However, these classical approaches Figure 1: Left: Meshes and notations for a 2D domain Ω = {φ < 0}; the computational mesh T h is obtained from a structured background mesh and is represented by both white and yellow triangles forming the domain Ω h ; the yellow triangles constitute the submesh T Γ h occupying the domain Ω Γ h . Right: a more involved example of an active mesh in 3D that can be used in φ-FEM; a hexahedral mesh covering a brain geometry.

suffer from poor accuracy because of their rudimentary (but easy to implement) treatment of the boundary conditions, cf. [START_REF] Girault | Error analysis of a fictitious domain method applied to a Dirichlet problem[END_REF]. For example, in the case of the linear elasticity equations, these methods start by extending the displacement u, from the physical domain Ω to a fictitious domain (typically a rectangular box) O ⊃ Ω assuming that u still solves the same governing equations on O as on Ω. This creates an artificial singularity on the boundary of Ω (a jump in the normal derivative) so that the resulting numerical approximation is, at best, √ h-accurate in the energy norm with whatever finite elements (from now on, h denotes the mesh size).

The last two decades have seen the arrival of more accurate geometrically unfitted methods such as XFEM [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF][START_REF] Haslinger | A new fictitious domain approach inspired by the extended finite element method[END_REF], CutFEM [START_REF] Burman | Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method[END_REF][START_REF] Burman | Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method[END_REF][START_REF] Burman | Cutfem: discretizing geometry and partial differential equations[END_REF][START_REF] Hansbo | Cut finite element methods for linear elasticity problems[END_REF] and Shifted Boundary Method (SBM) [START_REF] Main | The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems[END_REF][START_REF] Atallah | The shifted boundary method for solid mechanics[END_REF]. We are citing here only the methods based on the finite element (FE) approach; the list would be much longer if the methods based on finite differences were included. In the case of XFEM/CutFEM, the optimal accuracy, i.e. the same convergence rates as those of the standard FEM on a geometrically fitted mesh, is achieved at the price of a considerable sophistication in the implementation of boundary conditions. The idea is to introduce the unfitted mesh (known as the active mesh) starting from the simple background mesh and getting rid of the cells lying entirely outside the physical domain, as illustrated at Fig. 1. The finite elements are then set up on the active mesh, the variational formulation is imposed on the physical domain, and an appropriate stabilization is added. In practice, one should thus compute the integrals on the actual boundary and on the parts of the active mesh cells cut by the boundary (the cut cells). To this end, one should typically construct a boundary fitted mesh, now only locally near the boundary and only for the numerical integration purposes, but the generation of a non trivial mesh is still not completely avoided.

On the other hand, the non trivial integration is completely absent from SBM. This method introduces again an active mesh as a submesh of the background mesh (unlike CutFEM, the active mesh here contains only the cells inside Ω) and then imposes the approximate boundary conditions on the boundary of the active mesh by a Taylor expansion around the actual boundary. The absence of non-standard numerical integration is an important practical advantage of SBM over XFEM/CutFEM. We note however that, to the best of our knowledge, SBM is readily available only for the lowest order FE. Moreover, in the case of Neumann boundary conditions, the original version of SBM [START_REF] Main | The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems[END_REF] needs an extrapolation of the second derivatives of the solution that makes its implementation rather tricky. This difficulty can be alleviated if the problem is recast in a mixed form introducing the secondary variables for the gradient [START_REF] Nouveau | High-order gradients with the shifted boundary method: an embedded enriched mixed formulation for elliptic PDEs[END_REF].

In this chapter, we present yet another unfitted FE-based method, first introduced in [15, 14] and baptised φ-FEM to emphasize the prominent role played in it by the level set (LS) function, traditionally denoted by φ. From now on, we suppose that the physical domain is characterized by a given LS function:

1 Ω = {φ < 0} . (1)
Similarly to CutFEM/XFEM/SBM, we suppose that Ω is embedded into a simple background mesh and we introduce the active computational mesh T h as in CutFEM, cf. Fig. 1. However, unlike CutFEM, we abandon the variational formulation on Ω. We rather introduce a non-standard formulation on the extended domain Ω h (slightly larger than Ω) occupied by the active mesh T h . The general procedure is as follows:

• Extend the governing equations from Ω to Ω h and write down a formal variational formulation on Ω h without taking into account the boundary conditions on ∂Ω.

• Impose the boundary conditions using appropriate ansatz or additional variables, explicitly involving the level set φ which provides the link to the actual boundary. For instance, the homogeneous Dirichlet boundary conditions (u = 0 on ∂Ω) can be imposed by the ansatz u = φw thus reformulating the problem in terms of the new unknown w (modifications for non-homogeneous conditions, mixed boundary conditions and other settings are introduced further in the text).

• Add appropriate stabilization, including the ghost penalty [START_REF] Burman | Ghost penalty[END_REF] as in CutFEM plus a least square imposition of the governing equation on the mesh cells near the boundary, to guarantee coerciveness/stability on the discrete level.

This approach allows us to achieve the optimal accuracy using classical FE spaces of any order and the usual numerical integration: all the integrals in φ-FEM can be computed by standard quadrature rules on entire mesh cells and on entire boundary facets; no integration on cut cells or on the actual boundary is needed. This is the principal advantage of φ-FEM over CutFEM/XFEM. Moreover, we can cite the following features of φ-FEM which distinguish it from both CutFEM/XFEM and SBM:

• FE of any order can be straightforwardly used in φ-FEM. The geometry is naturally taken into account with the needed optimal accuracy: it suffices to approximate the LS function φ by piecewise polynomials of the same degree as that used for the primal unknown. This should be contrasted to CutFEM where a special additional treatment is needed if one uses FEM of order ≥ 2. Indeed, a piecewise linear representation of the boundary is not sufficient in this case. One needs either a special implementation of the isoparametric method [START_REF] Lehrenfeld | High order unfitted finite element methods on level set domains using isoparametric mappings[END_REF] or a local correction by Taylor expansions [START_REF] Boiveau | Fictitious domain method with boundary value correction using penalty-free Nitsche method[END_REF]. The extension to higher order FE is not trivial for SBM either.

• Contrary to SBM, φ-FEM is based on a purely variational formulation so that the existing standard FEM libraries suffice to implement it. The geometry of the domain comes into the formulation only through the level set φ. We emphasize that φ is not necessarily the signed distance to the boundary of Ω. It is sufficient to give to the method any φ satisfying (1) which is the minimal imaginable geometrical input. This can be contrasted with SBM which assumes that the distance to the actual boundary in the normal direction is known on all the boundary facets of the active mesh.

Moreover, φ-FEM is designed so that the matrices of the problems on the discrete level are reasonably conditioned, i.e. their condition numbers are of the same order as those of a standard fitting FEM on a mesh of comparable size. φ-FEM shares this feature with both CutFEM/XFEM and SBM.

Up to now, the φ-FEM approach has been proposed, tested and substantiated mathematically only in some simplest settings: Poisson equation with Dirichlet boundary conditions [START_REF] Duprez | φ-FEM: a finite element method on domains defined by level-sets[END_REF], or with Neumann/Robin boundary conditions [START_REF] Duprez | A new φ-FEM approach for problems with natural boundary conditions[END_REF]. The goal of the present chapter is to demonstrate its applicability to some more sophisticated governing equations arising in the computational mechanics. In section 2, we adapt φ-FEM to the linear elasticity equations accompanied by either pure Dirichlet boundary conditions, or with mixed conditions (both Dirichlet and Neumann on parts of the boundary). In Section 3, we consider the interface problem (elasticity with material coefficients abruptly changing over an internal interface). Section 4 is devoted to the treatment of internal cracks. Finally, our method is adapted to the heat equation in Section 5. In all these settings, we start by deriving an appropriate variant of φ-FEM and then illustrate it by numerical tests on manufactured solutions. We also compare the accuracy and efficiency of φ-FEM with those of the standard fitted FEM on the meshes of similar size, revealing the substantial gains that can be achieved by φ-FEM in both the accuracy and the computational time.

All the codes used in the present work have been implemented thanks to the open libraries fenics [START_REF] Alnaes | Archive of numerical software: The fenics project version 1[END_REF] and multiphenics [START_REF] Ballarin | multiphenics[END_REF]. They are available at the link https://github.com/michelduprez/phi-FEM-an-efficient-simulation-tool-using-simplemeshes-for-problems-in-structure-mechanics.git

Linear elasticity

In this section, we consider the static linear elasticity for homogeneous and isotropic materials. The governing equation for the displacement u is thus

div σ(u) + f = 0, ( 2
)
where the stress σ(u) is given by

σ(u) = 2µε(u) + λ(div u)I, ε(u) = 1 2 (∇u + ∇u T )
is the strain tensor, and Lamé parameters λ, µ are defined via the Young modulus E and the Poisson coefficient ν by

µ = E 2(1 + ν) and λ = Eν (1 + ν)(1 -2ν) . (3)
Equation ( 2) is posed in a domain Ω, which can be two or three dimensional, and should be accompanied with Dirichlet and Neumann boundary conditions on Γ = ∂Ω. We assume that Γ is decomposed into two disjoint parts, Γ = Γ D ∪ Γ N with Γ D = ∅, and

u = u g on Γ D , (4) σ
(u)n = g on Γ N , (5)
with the given displacement u g on Γ D and the given force g on Γ N . Let us first recall the weak formulation of this problem (to be compared with forthcoming φ-FEM formulations): find the vector field u on Ω s.t. u| Γ D = u g and

Ω σ(u) : ∇v = Ω f • v + Γ N g • v, ∀v on Ω such that v| Γ D = 0. ( 6
)
This is obtained by multiplying the equation by a test function v, integrating over Ω and taking into account the boundary conditions. Formulation [START_REF] Burman | Ghost penalty[END_REF] is routinely used to construct conforming FE methods, which necessitate a mesh that fits the domain Ω in order to approximate the integrals on Ω and Γ N and to impose u = u g on Γ D . We now consider the situation where a fitting mesh of Ω is not available. We rather assume that Ω is inscribed in a box O which is covered by a simple background mesh T O h . We further introduce the computational mesh T h (also referred to as the active mesh) by getting rid of cells lying entirely outside Ω. In practice, Ω is given by the level-set function φ: Ω = {φ < 0}. Usually, the level set is known only approximately. Accordingly, we assume that we are given a FE function φ h , i.e. a piecewise polynomial function on mesh T O h , which approximate sufficiently well φ. The selection of the mesh cells forming the active mesh is done on the basis of φ h rather than φ:

T h := {T ∈ T O h : T ∩ {φ h < 0} = ∅} . ( 7
)
The domain occupied by T h is denoted by Ω h , i.e. Ω h = (∪ T ∈T h T ) o . In some of our methods, we shall also need a submesh of T h , referred to as T Γ h , consisting of the cells intersected with the curve (surface) {φ h = 0}, approximating Γ:

T Γ h := {T ∈ T O h : T ∩ {φ h = 0} = ∅} . (8)
The domain covered by mesh T Γ h will be denoted by Ω Γ h , cf. Fig. 1. The starting point of all variants of φ-FEM is a variational formulation of problem (2) extended to Ω h , in which we do not impose any boundary conditions since they are lacking on ∂Ω h . We thus assume that the right-hand side f is given on the whole Ω h rather than on Ω alone, and suppose moreover that u can be extended from Ω to Ω h as the solution to the governing equation ( 2), now posed on Ω h instead of Ω. In a usual manner, we take then any test function v on Ω h , multiply the governing equation by v and integrate it over Ω h . This gives the following formulation: find a vector field u on Ω h such that

Ω h σ(u) : ∇v - ∂Ω h σ(u)n • v = Ω h f • v, ∀v on Ω h (9)
We emphasize that this formulation is fundamentally different from the standard formulation [START_REF] Burman | Ghost penalty[END_REF]. First of all, no boundary conditions are incorporated in (9) so that we cannot expect it to admit a unique solution. Furthermore, if we add somehow the boundary conditions on ∂Ω to (9), which we shall do indeed when constructing our φ-FEM variants, the resulting formulation will still be ill posed, meaning that its solution (on the continuous level) either does not exist, or is not unique. However, we shall be able to turn these problems into well defined numerical schemes by adding an appropriate stabilization on the discrete level.

Dirichlet conditions

Let us first consider the case of pure Dirichlet conditions: Γ = Γ D . On the continuous level, we want thus to impose u = u g on Γ = Γ D = {φ = 0} on top of the general formulation (9) of the problem on Ω h . We consider here 2 options to achieve this: 1) direct Dirichlet φ-FEM, as proposed in [START_REF] Duprez | φ-FEM: a finite element method on domains defined by level-sets[END_REF], introducing a new unknown w and redefining u through the product φw which automatically vanishes on Γ; 2) dual Dirichlet φ-FEM, inspired by [START_REF] Duprez | A new φ-FEM approach for problems with natural boundary conditions[END_REF], keeping the original unknown u and imposing u = u g on Γ with the aid of an auxiliary variable p in a least-square manner. In more details, our two approaches can be described as follows:

• Direct Dirichlet φ-FEM (on continuous level). Supposing that u g is actually given on the whole Ω h rather than on Γ alone, we make the ansatz u = u g + φw, on Ω h [START_REF] Burman | Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method[END_REF] and substitute it into (9). To make the formulation more symmetric we also replace the test functions v by φz. This yields: find a vector field w on Ω h such that

Ω h σ(φw) : ∇(φz) - ∂Ω h σ(φw)n • φz = Ω h f • φz - Ω h σ(u g ) : ∇(φz) + ∂Ω h σ(u g )n • φz, ∀z on Ω h . ( 11
)
The idea is thus to work with the new unknown w on Ω h , discretize it by FEM starting from the variational formulation above, and to reconstitute the approximation to u by the ansatz [START_REF] Burman | Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method[END_REF].

• Dual Dirichlet φ-FEM (on continuous level). We now suppose that u g is defined on Ω Γ h , cf. [START_REF] Burman | Hybridized CutFEM for elliptic interface problems[END_REF], rather than on the whole of Ω h . We keep the primal unknown u in [START_REF] Burman | Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method[END_REF] and we want to impose

u = u g + φp, on Ω Γ h ( 12
)
on top of it, with a new auxiliary unknown p on Ω Γ h . The new variable p lives beside u inside a variational formulation that combines [START_REF] Burman | Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method[END_REF] with [START_REF] Carraro | On the implementation of the eXtended Finite Element Method (XFEM) for interface problems[END_REF]: find vector fields u on Ω h and p on Ω Γ h such that

Ω h σ(u) : ∇v - ∂Ω h σ(u)n • v + γ Ω Γ h (u -φp) • (v -φq) = Ω h f • v + γ Ω Γ h u g • (v -φq), ∀v on Ω h , q on Ω Γ h ( 13
)
with a positive parameter γ. Comparing the direct and dual variants, we observe that the expressions [START_REF] Burman | Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method[END_REF] and ( 12) are of course pretty similar, but their roles are quite different in the corresponding methods. The variable w replaces u in [START_REF] Burman | Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions[END_REF], while p lives alongside u in [START_REF] Claus | A stable and optimally convergent latin-cutfem algorithm for multiple unilateral contact problems[END_REF]. The introduction of the additional variable p makes the dual method only slightly more expensive than the direct one, since this new variable is introduced only on a narrow strip around Γ. On the other hand, a certain advantage of the dual variant over the direct one lies in the fact that both φ and u g should be here known only locally around Γ since they enter into equation ( 13) only on Ω Γ h . This can facilitate the construction of φ and u g in practice. More importantly, it is the dual method that we shall be able to adapt to various, more and more complicated settings below.

As mentioned above, both variational problems [START_REF] Burman | Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions[END_REF] and ( 13) are derived on a very formal level. They are not valid in any mathematically rigorous way: we cannot expect to have a meaningful boundary value problems on a domain Ω h with no boundary conditions on ∂Ω h , while prescribing some conditions on a curve (surface) Γ which is inside Ω h . However, both formulations can serve as starting problems to write down FE problems which become well-posed once an appropriate stabilization is added.

We start by introducing the FE spaces: fix an integer k ≥ 1 and let

V h := v h : Ω h → R d : v h|T ∈ P k (T ) d ∀T ∈ T h , v h continuous on Ω h . (14)
For future reference, we introduce the local version of this space for any submesh M h of T h and polynomial degree l ≥ 0

Q l h (M h ) := q h : M h → R d : q h|T ∈ P l (T ) d ∀T ∈ M h , q h continuous on M h if l ≥ 0 . (15)
In particular, we shall need the space

Q k h (Ω Γ h ) on the submesh Ω Γ h in the Dual version of Dirichlet φ-FEM.
The two variants of φ-FEM introduced above can now be written on the fully discrete level as:

• Direct Dirichlet φ-FEM: find w h ∈ V h such that Ω h σ(φ h w h ) : ∇(φ h z h ) - ∂Ω h σ(φ h w h )n • φ h z h + G h (φ h w h , φ h z h ) + J lhs h (φ h w h , φ h z h ) = Ω h f • φ h z h - Ω h σ(u g h ) : ∇(φ h z h ) + ∂Ω h σ(u g h )n • φ h z h , + J rhs h (φ h z h ), ∀z h ∈ V h (16) and set u h = u g h + φ h w h .
Here φ h , u g h are FE approximations for φ, u g on the whole Ω h , and G h , J lhs h , J rhs h stand for the stabilization terms

G h (u, v) := σ D h E∈F Γ h E [σ(u)n] • [σ(v)n] , (17)
J lhs h (u, v) := σ D h 2 T ∈T Γ h T div σ(u) • div σ(v) , J rhs h (v) := -σ D h 2 T ∈T Γ h T f • div σ(v) . ( 18
)
The stabilization G h (17) is known as the ghost penalty. σ D in ( 17) is a positive stabilization parameter which should be chosen sufficiently big (in a mesh independent manner). F Γ h stands for the set of internal facets of mesh T h which are also the facets of T Γ h (these are the facets either intersected by Γ, or belonging to the cells intersected by Γ). Stabilization [START_REF] Girault | Error analysis of a fictitious domain method applied to a Dirichlet problem[END_REF] was first introduced in [START_REF] Burman | Ghost penalty[END_REF] in the form of penalization of jumps in the normal derivatives of the FE solution. Here, we prefer to penalize the jumps of internal elastic forces, following [START_REF] Claus | A stable and optimally convergent latin-cutfem algorithm for multiple unilateral contact problems[END_REF], thus controlling appropriate combinations of the derivatives, rather than the normal derivatives themselves. We emphasize however that the original ghost penalty in [START_REF] Hansbo | Cut finite element methods for linear elasticity problems[END_REF] also involved the jumps of higher order derivatives of u (up to the highest order of polynomials present in the FE formulation), while our variant affects the first order derivatives only. We can allow ourselves to reduce the order of stabilized derivatives thanks to the presence of additional stabilization terms J lhs h [START_REF] Glowinski | A fictitious domain method for Dirichlet problem and applications[END_REF], as first suggested in [START_REF] Duprez | φ-FEM: a finite element method on domains defined by level-sets[END_REF] (a similar idea can also be found in [START_REF] Elfverson | A new least squares stabilized Nitsche method for cut isogeometric analysis[END_REF]). The combination of G h and J lhs h allows us indeed to get rid of possible spurious oscillations of the approximate solution on "badly cut" cells near Γ and to guarantee the coerciveness of the bilinear form in our FE formulation. Note that the terms J lhs h are not consistent by themselves but they are consistently compensated by their right-hand side counterpart J rhs h . Indeed, the exact solution satisfies div σ(u) = -f so that

J lhs h (u, v) = J rhs h (v) if u is the exact solution. • Dual φ-FEM-Dirichlet: find u h ∈ V h , p h ∈ Q k h (Ω Γ h ) such that Ω h σ(u h ) : ∇v h - ∂Ω h σ(u h )n • v h + γ h 2 Ω Γ h (u h - 1 h φ h p h ) • (v h - 1 h φ h q h ) + G h (u h , v h ) + J lhs h (u h , v h ) = Ω h f • v h + γ h 2 Ω Γ h u g h • (v h - 1 h φ h q h ) + J rhs h (v h ), ∀v h ∈ V h , q h ∈ Q k h (Ω Γ h ). ( 19
)
With respect to (13, we have added here the factors 1 h , 1 h 2 . They serve to control the condition numbers, cf. [START_REF] Duprez | A new φ-FEM approach for problems with natural boundary conditions[END_REF]. The stabilizations G h , J lhs h , J rhs h are again defined by ( 17) and (18). [START_REF] Hansbo | Nitsche's method for interface problems in computational mechanics[END_REF]. Left: active meshes for φ-FEM (with cells from T Γ h in yellow). Right: a fitted mesh for the standard FEM.

φ(x, y) = - 1 8 + (x -0.5) 2 + (y -0.5) 2 . ( 20
)
We take the elasticity parameters E = 2 and ν = 0.3, and the scheme parameters γ = σ D = 20.0. We use P 2 -Lagrange polynomials for both FE spaces V h and Q h , i.e. we set k = 2 in ( 14) and [START_REF] Duprez | φ-FEM: a finite element method on domains defined by level-sets[END_REF]. We finally choose a manufactured exact solution u = u ex := (sin(x) exp(y), sin(y) exp(x)) [START_REF] Hansbo | Cut finite element methods for linear elasticity problems[END_REF] giving the right hand side f by substitution to (2) and the boundary conditions u g = u ex on Γ.

In order to set up both φ-FEM schemes above, we should extend u g from Γ to Ω h (in the case of the direct method) or to Ω Γ h (in the case of the dual method). To mimic the realistic situation where u g is known on Γ only, we prefer not to extend u g by u ex everywhere. We rather set

u g = u ex (1 + φ),
on Ω h or on Ω Γ h adding to u ex a perturbation which vanishes on the boundary. The typical active meshes T h and T Γ h for φ-FEM are illustrated on Fig. 2 (left). Besides the direct φ-FEM [START_REF] Elfverson | A new least squares stabilized Nitsche method for cut isogeometric analysis[END_REF] and the dual φ-FEM [START_REF] Hansbo | An unfitted finite element method, based on Nitsche's method, for elliptic interface problems[END_REF], we shall present the numerical results obtained by the standard FEM with P 2 -Lagrange polynomials on fitted meshes for approximately the same values of h, as illustrated on Fig. 2 (right). The results obtained by both variants of φ-FEM and by the standard FEM are reported in Figs. 3 and4.

We first illustrate the numerical convergences order for the relative errors in L 2 and H 1 norms at Fig. 3. We observe that both variants of φ-FEM demonstrate indeed the expected optimal convergence orders: h 2 is the H 1 -seminorm and h 3 in the L 2 -norm, and the direct variant performs significantly better than the dual one. This can be attributed to a better representation of the solution near the boundary in the direct variant: indeed it is effectively approximated there by fourth-order polynomials (P 2 for w h times P 2 for φ h ). Moreover, both φ-FEMs, even the dual one, significantly outperform the standard FEM (the latter is even of a suboptimal order in the L 2 -norm). This can be partially attributed to a coarse geometry approximation. Indeed, we use triangular meshes so that the curved boundary of Ω is actually approximated by a collection of straight segments, i.e. the boundary facets of the fitted mesh, cf. Fig. 2 (right). The superior efficiency of φ-FEM with respect to the standard FEM is further confirmed by Fig. 4. We report there the computing times on different meshes for the 3 methods and set them against the relative L 2 error. These computing times include assembling of the FE matrices and resolution of the resulting linear systems. For a given relative error, the calculations are always much faster with φ-FEM than with the standard FEM. The advantage would be even more significant if the mesh generation times were included, since the construction of active meshes in φ-FEM only involves choosing a subset of cells according to a simple criterion, and some renumbering of the degrees of freedom. We do not dispose however of an efficient implementation of cell selection algorithm at the moment. All our computations are performed using the Python interface for the popular FEniCS computing platform, and the selection of active cells is done by a simple, non-optimized Python script.

Mixed boundary conditions

We now consider the much more complicated case of mixed conditions ( 4)-( 5) on the boundary Γ = Γ N ∪ Γ D with Γ D = ∅ and Γ N = ∅. This setting is challenging for any geometrically unfitted method since the junction between the Dirichlet and Neummann boundary parts can occur inside a mesh cell, so that approximating polynomials in this cell should account simultaneously for both boundary conditions. In [START_REF] Hansbo | Cut finite element methods for linear elasticity problems[END_REF], it is demonstrated that the linear elasticity with mixed boundary conditions can be successfully treated by CutFEM. A rigorous mathematical substantiation allowing of the low regularity of the solution is available in [START_REF] Burman | Low regularity estimates for CutFEM approximations of an elliptic problem with mixed boundary conditions[END_REF]. Here, we shall adapt φ-FEM (in the dual form) to the mixed boundary conditions by adopting a "lazy" approach: we choose to do not impose any boundary conditions on a mesh cell if the Dirichlet/Neumann junction happens to be inside it.

To set up the geometry of the problem, we recall that the domain Ω is given by the level set function φ, Ω = {φ < 0}, and assume furthermore that the boundary partition into the Dirichlet and Neumann parts is governed by a secondary level set ψ,

Γ D = Γ ∩ {ψ < 0}, Γ N = Γ ∩ {ψ > 0} .
Introducing the active meshes T h and T Γ h as above, cf. ( 7), ( 8), and Fig. 1, we want now further partition the submesh T Γ h into two parts:

T Γ D h around Γ D ,
T Γ D h := {T ∈ T Γ h : ψ 0 on T } and T Γ N h := {T ∈ T Γ h : ψ 0 on T } . ( 22
)
As before, we denote the domains occupied by meshes

T h ,T Γ h ,T Γ D h ,T Γ N h by Ω h ,Ω Γ h ,Ω Γ D h ,Ω Γ N h
respectively. Note that these definitions may leave a small number of cells of T Γ h out of both T Γ D h and T Γ N h . Indeed, there may be mesh cells, near the junction of Dirichlet and Neumann parts, where ψ changes sign inside the cell, so that ψ is neither everywhere positive not everywhere negative on such a cell. This is illustrated at Fig. 8 (left) where the Dirichlet/Neumann junction is supposed at x = 0.5, i.e. the secondary level set is ψ(x, y) = 0.5 -y, c.f. Fig. 5. The active mesh cells intersected by Γ on Fig. 8 or T Γ N h , and are colored in yellow). Assuming once more that u, the solution to ( 2)-( 4)-( 5), can be extended from Ω to Ω h as the solution to the same governing equation (2), we introduce a φ-FEM scheme, combining the Dual φ-FEM Dirichlet approach, as introduced in ( 13) and [START_REF] Hansbo | An unfitted finite element method, based on Nitsche's method, for elliptic interface problems[END_REF], with the indirect imposition of Neumann boundary condition as proposed in [START_REF] Duprez | A new φ-FEM approach for problems with natural boundary conditions[END_REF]. We thus keep u as the primary unknown on Ω h and recall that it satisfies the variational formulation [START_REF] Burman | Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method[END_REF]. The Dirichlet boundary condition affects the solution on Ω Γ D h through the introduction of the auxiliary variable p D there. We thus adapt [START_REF] Carraro | On the implementation of the eXtended Finite Element Method (XFEM) for interface problems[END_REF] from the pure Dirichlet case as

u = u g + φp D , on Ω Γ D h . (23)
We have assumed here that u g is extended from Γ D to Ω Γ D h . The Neumann boundary condition will affect u on Ω Γ N h through the introduction of two auxiliary variables there. We first introduce a tensor-valued variable y on Ω Γ N h setting y = -σ(u). It remains to impose yn = -g on Γ N . To this end, we note that the outward-looking unit normal n is given on Γ by n = 1 |∇φ| ∇φ so that the Neumann boundary condition is satisfied by setting y∇φ + g|∇φ| = -p N φ on Ω Γ N h where p N is yet another (vector-valued) auxiliary variable on Ω Γ N h . This can be summarized as

y + σ(u) = 0, on Ω Γ N h , (24a)
y∇φ + pφ = -g|∇φ|, on Ω Γ N h . (24b)
Note that the combination of ( 23) with (24a-b) does not impose the mixed Dirichlet/Neumann conditions on the whole of Γ since the latter may be not completely covered by Ω Γ D h ∪ Ω Γ N h . Fortunately, this defect of the formulation on the continuous level can be repaired on the discrete level by adding the appropriate stabilization to the FE discretization.

To describe the resulting FE scheme, we start by introducing the FE spaces. As before, we fix an integer k ≥ 1 and keep the space V h , as defined in [START_REF] Duprez | A new φ-FEM approach for problems with natural boundary conditions[END_REF], for the approximation u h of the primary variable u. We need also the spaces for the approximation of the auxiliary variables p h,D and p h,N , respectively

Q k h (Ω Γ D h ) and Q k-1 h (Ω Γ N h )
as defined in [START_REF] Duprez | φ-FEM: a finite element method on domains defined by level-sets[END_REF], as well as the space Z h (Ω Γ N h ) to approximate y, where for each submesh M h of T h , Z h (M h ) is defined by

Z h (M h ) := z h : M h → R (d×d) : z h|T ∈ P k (T ) (d×d) ∀T ∈ M h , z h continuous on M h . (25)
Now, combining the variational formulation [START_REF] Burman | Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method[END_REF] with the ( 23) and (24a-b) imposed in a leastsquares manner, we get the following scheme: find

u h ∈ V h , p h,D ∈ Q k h (Ω Γ D h ), y h ∈ Z h (Ω Γ N h ) and p h,N ∈ Q k-1 h (Ω Γ N h ) such that Ω h σ(u h ) : ∇v h - ∂Ω h \∂Ω h,N σ(u h )n • v h + ∂Ω h,N y h n • v h + γ u Ω Γ N h (y h + σ(u h )) : (z h + σ(v h )) + γ p h 2 Ω Γ N h y h ∇φ h + 1 h p h,N φ h • z h ∇φ h + 1 h q h,N φ h + γ h 2 Ω Γ D h (u h - 1 h φ h p h,D ) • (v h - 1 h φ h q h,D ) + G h (u h , v h ) + J lhs,D h (u h , v h ) + J lhs,N h (y h , z h ) = Ω h f • v h + γ h 2 Ω D h u g h • (v h - 1 h φ h q h,D ) - γ p h 2 Ω Γ N h g • |∇φ h |(z h • ∇φ h + 1 h q h,N φ h ) + J rhs,D h (v h ) + J rhs,N h (z h ), ∀v h ∈ V h , q h,D ∈ Q k h (Ω Γ D h ), z h ∈ Z h (Ω Γ N h ), q h,N ∈ Q k-1 h (Ω Γ N h ) . ( 26
)
We have added here the ghost stabilization G h defined by [START_REF] Girault | Error analysis of a fictitious domain method applied to a Dirichlet problem[END_REF] as in the pure Dirichlet case. The additional stabilizations terms J lhs h , J rhs h are now adapted from [START_REF] Glowinski | A fictitious domain method for Dirichlet problem and applications[END_REF] and separated into the terms acting on u h on the Dirichlet cells of T Γ h (and also those not marked), and the terms acting on y h on the Neumann cells:

J lhs,D h (u, v) := σ D h 2 T ∈T Γ h \T Γ N h T div σ(u) • div σ(v) , J rhs,D h (v) := -σ D h 2 T ∈T Γ h \T Γ N h T f • div σ(v) , J lhs,N h (y, z) = γ div Ω Γ N h div y • div z , J rhs,N h (z) = γ div Ω Γ N h f • div z. ( 27
)
These stabilizations are consistent with the governing equations div σ(u) = -f , rewritten as div y = f , using (24a), wherever possible, i.e. on Ω Γ N h . Note that a similar treatment is applied to the boundary integral terms on ∂Ω in [START_REF] Burman | Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method[END_REF]. In [START_REF] Main | The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems[END_REF], they are rewritten in terms of y, using (24a) and (24b), wherever possible. We thus introduce a part of the boundary ∂Ω h , referred to as ∂Ω h,N , formed by the boundary facets of T h belonging to the cells in T Γ N h . We replace σ(u h ) by -y h on ∂Ω h,N , while keeping the boundary term as is on the remaining part of the boundary. All this contributes to the coerciveness of the bilinear form in [START_REF] Main | The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems[END_REF] and good conditioning of the matrix as can be proven following the ideas of [START_REF] Duprez | A new φ-FEM approach for problems with natural boundary conditions[END_REF]. We emphasize again that neither Dirichlet nor Neumann boundary conditions are imposed in any way in scheme [START_REF] Main | The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems[END_REF]

on the cells in T

Γ h \ (T Γ D h ∪ T Γ N h ) (the
cells in yellow on Fig. 8). On the other hand, both stabilizations G h and J h are active on the whole T Γ h , comprising these cells not marked as Dirichlet or Neumann.

Test case:

We are now going to present some numerical results with method (26) highlighting the optimal convergence of φ-FEM and comparing it with a standard FEM. We use the same geometry [START_REF] Hansbo | Nitsche's method for interface problems in computational mechanics[END_REF], elasticity parameters and the exact solution [START_REF] Hansbo | Cut finite element methods for linear elasticity problems[END_REF] as for the case of pure Dirichlet conditions on page 9. We set furthermore the Dirichlet boundary conditions (4) for x > 0.5 and the Neumann boundary conditions (5) for x < 0.5, c.f. Fig. 5, i.e. we choose the secondary level set as ψ = 0.5 -x. The data u g and g are computed from the exact solution. In φ-FEM they should be extended from Γ to appropriate portion of the strip Ω Γ h . We choose these extensions as Again, both expressions are perturbed away from Γ to mimic the real-life situation where the data are available only on Γ. The stabilization parameters are set to γ div = γ u = γ p = 1.0, σ = 0.01 and γ = σ D = 20.0. We start by studying mesh configurations where the Dirichlet-Neumann junction line {x = 0.5} happens to be covered by the mesh facets both in the background mesh used by φ-FEM, and in the fitted mesh used by FEM, as illustrated in Fig. 6. All the boundary cells in T Γ h are marked in this case either as Dirichlet or as Neumann ones, according to the criterion [START_REF] Haslinger | A new fictitious domain approach inspired by the extended finite element method[END_REF], giving, respectively, red and blue cells on Fig. 6 (left). There is no ambiguity for the standard FEM fitted meshes: all the boundary facets are straightforwardly marked either as Dirichlet or as Neumann, cf. Fig. 6 (right) with the same color code as for the unfitted mesh. The results obtained by both φ-FEM (26) and the standard FEM, using P 2 -Lagrange polynomials for u h in both cases, are reported in Fig. 7. On the left, the relative errors are plotted with respect to the mesh step. We observe again the optimal convergence orders for φ-FEM, while the convergence of the standard FEM is sub-optimal in the L 2 -norm. The φ-FEM approach is again systematically more precise in both norms. On the right side of the same figure, we plot the computing times and notice again that φ-FEM is less expensive than the standard FEM.

Ω Γ D Γ N σn = g u = u g
Let us now turn to a less artificial mesh configuration where the Dirichlet/Neumann junction point can turn up inside a mesh cell of the background mesh, or inside a boundary facet of the fitted mesh. We study these situations on a series of meshes, as illustrated in Fig. 8. In the case of the background meshes used for φ-FEM, we ensure in particular that there are no vertical grid line with the abscissa x = 0.5 so that there are exactly 4 cells cells in T Γ h that are neither in

T Γ D h nor in T Γ N h
(yellow cells on the left side of Fig. 8). We recall that scheme [START_REF] Main | The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems[END_REF] does not impose any boundary conditions on these cells, but retains the stabilization there (in particular, the governing equation is still re-enforced on these cells in the least squares manner). Note that the fitted FEM is not straightforward to implement in this case either, since the Dirichlet boundary conditions cannot be strongly imposed on the boundary facets which turn up only partially on the Dirichlet side. We bypass this difficulty by treating the Dirichlet conditions by penalization, so that the "standard" FEM is now defined as: find u h in the P k FE space (without any restrictions on the boundary) such that

Ω σ(u h ) : ∇v h + 1 ε Γ D u h • v h = Ω f • v h + Γ N g • v h + 1 ε Γ D u g • v h ( 28
)
for all v h in the same FE space as u h , with a small parameter ε > 0.

The mesh refinement study in this case is reported at Fig. 9. Comparing the results with those of Fig. 7 (obtained on idealized unrealistic meshes without any unmarked cells), we observe that the behavior of φ-FEM [START_REF] Main | The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems[END_REF] is almost unaffected by the presence (or not) of the unmarked "yellow" cells, although the convergence curve for the L 2 relative error is now slightly less regular. In particular, the conclusions about the relative merits of φ-FEM and the fitted FEM, now in version [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF], remain unchanged: φ-FEM is more precise on comparable meshes and less expensive in terms of the computing times for a given error tolerance.

3 Linear elasticity with multiple materials.

We now consider the case of interfaces problems, i.e. partial differential equations with coefficients jumping across an interface, which can cut the computational mesh in an arbitrary manner. The simplest meaningful example in the realm of linear elasticity is given by structures consisting of multiple materials having different elasticity parameters. This situation has already been treated in XFEM [START_REF] Carraro | On the implementation of the eXtended Finite Element Method (XFEM) for interface problems[END_REF][START_REF] Annavarapu | A robust Nitsche's formulation for interface problems[END_REF][START_REF] Yuanming | High-order extended finite element methods for solving interface problems[END_REF][START_REF] Xiao | High-order finite element methods for interface problems: Theory and implementations[END_REF], CutFEM [START_REF] Burman | Hybridized CutFEM for elliptic interface problems[END_REF][START_REF] Hansbo | Nitsche's method for interface problems in computational mechanics[END_REF][START_REF] Hansbo | An unfitted finite element method, based on Nitsche's method, for elliptic interface problems[END_REF][START_REF] Lehrenfeld | Analysis of a high-order unfitted finite element method for elliptic interface problems[END_REF], and SBM [START_REF] Li | The shifted interface method: A flexible approach to embedded interface computations[END_REF] paradigms. We are now going to demonstrate the applicability of φ-FEM in this context.

Let us assume that the structure occupies a domain Ω and it consists of two materials that occupy two subdomains Ω 1 and Ω 2 separated by the interface Γ. To fix the ideas, we further assume that Ω 1 is surrounded by Ω 2 , so that the interface Γ can actually be described as Γ = ∂Ω 1 , as illustrated at Fig. 10. We also assume that the displacement u is given on the external boundary (these assumptions are not restrictive and the forthcoming method can be easily adapted to other situations, e.g. with Γ touching ∂Ω or with Neumann boundary conditions on the external boundary). We then consider the problem for the displacement u on Ω:

         -div σ(u) = f , on Ω\Γ , u = u g , on ∂Ω , [u] = 0 , on Γ , [σ(u)n] = 0 , on Γ , ( 29
)
where n is the unit normal pointing from Ω 1 to Ω 2 , and the brackets [•] stand for the jump across Γ. The elasticity parameters are assumed constant on each sub-domain, but different from each other. The stress tensor is thus given by

σ(u) = σ 1 (u) = 2µ 1 ε(u) + λ 1 (div u)I , on Ω 1 , σ 2 (u) = 2µ 2 ε(u) + λ 2 (div u)I , on Ω 2 ,
with the Lamé parameters λ i and µ i defined via the formulas (3) with given

E i , ν i , i = 1, 2.
Introducing the displacements u i = u| Ωi , i = 1, 2 on Ω 1 and Ω 2 separately, problem (29) can be rewritten as the system of two coupled sub-problems:

         -div σ i (u i ) = f , on Ω i , i = 1, 2, u 2 = u g , on ∂Ω , u 1 = u 2 , on Γ , σ 1 (u 1 )n = σ 2 (u 2 )n , on Γ . ( 30
)
We suppose that Ω is sufficiently simple-shaped so that a matching mesh T h on Ω is easily available (again, this assumption is not restrictive; we have seen that a complex-shape domain Ω can be also treated by φ-FEM). On the contrary, the mesh T h is not supposed to match the internal interface Γ and we are going to adapt φ-FEM to this situation. The starting point is the reformulation [START_REF] Xiao | High-order finite element methods for interface problems: Theory and implementations[END_REF]. We are thus going to discretize separately u 1 on Ω 1 and u 2 on Ω 2 . To this end, we introduce two active meshes T h,1 and T h,2 , sub-meshes of T h , constructed by retaining in T h,i the cells of T h having a non-empty intersection with Ω i . In practice, the sub-domains are defined through a level-set φ:

Ω 1 Γ [σ(u)n] = 0 [u] = 0 Ω 2 u = u g
Ω 1 = {φ > 0} ∩ Ω, Ω 2 = {φ < 0}, Γ = {φ = 0} ∩ Ω .
The sub-meshes T h,i are defined using a piecewise-polynomial approximation φ h of φ, rather than φ itself:

T h,1 := {T ∈ T h : T ∩ {φ h > 0} = ∅} and T h,2 := {T ∈ T h : T ∩ {φ h < 0} = ∅} . ( 31
)
We also introduce the sub-mesh T Γ h as the intersection T h,1 ∩ T h,2 and denote by Ω h,1 , Ω h,2 , Ω Γ h the domains covered by meshes T h,1 , T h,2 , T Γ h respectively. Similarly to the simpler settings considered above, the unknowns u 1 and u 2 , living physically on Ω 1 and Ω 2 , will be discretized on larger domains Ω h,1 and Ω h,2 , introducing artificial extensions on narrow fictitious strips near Γ. On the discrete level, the unknowns will be thus redoubled on the joint sub-mesh T Γ h . Several auxiliary unknowns will be introduced on Ω Γ h similar to the case of mixed boundary conditions above (indeed, we have to discretize both Dirichlet and Neumann conditions on the interface Γ in the current setting).

We now put the program above into the equations, first on the continuous level. Similarly to (9), the unknowns u i extended to larger domains Ω i h satisfy formally the variational formulations, cf. the first equation in [START_REF] Xiao | High-order finite element methods for interface problems: Theory and implementations[END_REF]:

Ω h,i σ i (u i ) : ∇v i - ∂Ω h,i σ i (u i )n i • v i = Ω h,i f • v i , ∀v i on Ω hi s.t.v i = 0 on ∂Ω . (32)
Here, with a slight abuse of notations, ∂Ω h,i denotes the component of the boundary of Ω h,i other than ∂Ω, and n i denotes the unit normal vector on ∂Ω h,i pointing outside Ω h,i . The boundary conditions on the external boundary ∂Ω, i.e. the second equation in [START_REF] Xiao | High-order finite element methods for interface problems: Theory and implementations[END_REF], will be imposed strongly. The remaining equations in [START_REF] Xiao | High-order finite element methods for interface problems: Theory and implementations[END_REF], i.e. the interface conditions on Γ, will be imposed by introduction of auxiliary variables on Ω Γ h : the vector-valued p (similar to the dual version of φ-FEM for the Dirichlet boundary conditions above) and matrix-valued y 1 ,y 2 (similar to φ-FEM for the Neumann boundary conditions). This gives, cf. the last two equations in [START_REF] Xiao | High-order finite element methods for interface problems: Theory and implementations[END_REF]:

u 1 -u 2 + pφ = 0 , on Ω Γ h , (33)
y i + σ i (u i ) = 0 , on Ω Γ h , i = 1, 2, (34)
y 1 ∇φ -y 2 ∇φ = 0 , on Ω Γ h . (35)
Equation ( 35) above extends the last equation in [START_REF] Xiao | High-order finite element methods for interface problems: Theory and implementations[END_REF] from Γ to Ω Γ h since the normal on Γ is colinear with the vector ∇φ there.

We are now going to discretize equations (32)-( 35). We fix an integer k ≥ 1 and introduce the FE spaces for the primary variables u i :

V h,i := v h : Ω h,i → R d : v h|T ∈ P k (T ) d ∀T ∈ T h , v h continuous on Ω h,i , and v h = I h u g on ∂Ω (36)
with the standard FE interpolation I h , and their homogeneous counterparts V 0 h,i with the constraint v h = 0 on ∂Ω, to be used for the test functions. We recall moreover the spaces Q h (Ω Γ h ) and Z h (Ω Γ h ) defined respectively by ( 15) and [START_REF] Li | The shifted interface method: A flexible approach to embedded interface computations[END_REF]. Combining (32) with ( 33)-( 35) taken in the least square sense, gives the following scheme:

find u h,1 ∈ V h,1 , u h,2 ∈ V h,2 , p h ∈ Q k h (Ω Γ h ), y h,1 , y h,2 ∈ Z h (Ω Γ h ) such that, 2 i=1 Ω h,i σ i (u h,i ) : ∇v h,i + 2 i=1 ∂Ω h,i y h,i n • v h + γ p h 2 Ω Γ h (u h,1 -u h,2 + 1 h p h φ h ) • (v h,1 -v h,2 + 1 h q h φ h ) + γ u 2 i=1 Ω Γ h (y h,i + σ i (u h,i )) : (z h,i + σ i (v h,i )) + γ y h 2 Ω Γ h (y h,1 ∇φ h -y h,2 ∇φ h ) • (z h,1 ∇φ h -z h,2 ∇φ h ) + 2 i=1 G h (u h,i , v h,i ) + J lhs,N h (y h,i , z h,i ) = 2 i=1 Ω h,i f • v h,i + 2 i=1 J rhs,N h (z h,i ) , ∀ v h,1 ∈ V 0 h,1 , v h,2 ∈ V 0 h,2 , q h ∈ Q k h (Ω Γ h ), z h,1 , z h,2 ∈ Z h (Ω Γ h ) . ( 37
)
Similarly to the previous settings, we have added here the ghost stabilization G h defined by [START_REF] Girault | Error analysis of a fictitious domain method applied to a Dirichlet problem[END_REF] and the additional stabilization J rhs,N h defined by [START_REF] Mittal | Immersed boundary methods[END_REF] with Ω Γ N h replaced by Ω Γ h and imposing div y i = f on Ω Γ h in the least squares sense.

Test case: Consider Ω = (0, 1)2 and Ω 1 , Ω 2 defined by the the level-set

φ φ(x, y) = -R 2 + (x -0.5) 2 + (y -0.5) 2 ,
with R = 0.3 as illustrated on Fig. 10. We want to solve [START_REF] Nouveau | High-order gradients with the shifted boundary method: an embedded enriched mixed formulation for elliptic PDEs[END_REF] with the manufactured radial solution

u = u ex = 1 E1 (cos(r) -cos(R))(1, 1) T if r < R, 1 E2 (cos(r) -cos(R))(1, 1) T else,
, where r = (x -0.5) 2 + (y -0.5) 2 . Thus

f = -div(σ 1 ((cos(r) -cos(R))(1, 1) T )/E 1 and u g = u ex .
The material parameters are given by E 1 = 7, E 2 = 2.28 and ν 1 = ν 2 = 0.3. The meshes used for φ-FEM and for the standard FEM are illustrated in Fig. 11. In the latter case, the mesh should resolve the interface r = R so that the solution u h ∈ V h is obtained by the straight-forward scheme where V h is the conforming P k FE space approximating u g on ∂Ω and V 0 h is its homogeneous analogue. The results obtained with φ-FEM (37) and FEM (38) using P 2 piecewise polynomials (k = 2) are reported in Fig. 12. The conclusions remain the same as in the previous setting: φ-FEM is more precise on comparable meshes and less expensive in terms of the computing times for a given error tolerance.

Linear elasticity with cracks

We now want to consider the linear elasticity problem posed on a cracked domain Ω \ Γ f with Γ f being a line (a surface) inside Ω:

     -div σ(u) = f , on Ω \ Γ f , u = u g , on ∂Ω , σ(u)n = g , on Γ f . ( 39
)
This problem is actually what XFEM was originally designed for, cf. [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF]. We are now going to adapt φ-FEM to it.

In practice, the crack geometry is given by the primary level set φ (to locate the line or surface of the crack) and the secondary level set ψ (to locate the tip or the front of the crack):

Γ f := Ω ∩ {φ = 0} ∩ {ψ < 0} .
To fix the ideas, let us suppose that the line (surface) Γ := {φ = 0} splits Ω into two sub-domains Ω 1 and Ω 2 , characterized by {φ < 0} and {φ > 0} respectively, as illustrated at Fig. 13. The interface Γ thus consists of the fracture location Γ f and the remaining (fictitious) part Γ int :

Γ int := Ω ∩ {φ = 0} ∩ {ψ > 0} .
In order to reuse the φ-FEM scheme (37) introduced for the interface problem above, we reformulate problem (39) in terms of two separate unknowns u i = u| Ωi , i = 1, 2: We are interested again in a situation where Ω is sufficiently simple-shaped so that a matching mesh T h on Ω is easily available, but this mesh does not match the internal interface Γ. As in the preceding section, we are thus going to discretize separately u 1 on Ω 1 and u 2 on Ω 2 starting from the reformulation (40). To this end, we introduce two active sub-meshes T h,1 , T h,2 as in [START_REF] Yuanming | High-order extended finite element methods for solving interface problems[END_REF], based on the piecewise polynomial approximation φ h of φ. We also introduce the interface mesh T Γ h = T h,1 ∩ T h,2 , which we further split into two sub-meshes with respect to the secondary level set ψ, similarly to our treatment of the mixed boundary conditions, cf. [START_REF] Haslinger | A new fictitious domain approach inspired by the extended finite element method[END_REF]:

               -div σ(u i ) = f , on Ω i , u i = u g , on ∂Ω , [u] = 0 , on Γ int , [σ(u)n] = 0 , on Γ int , σ(u)n = g , on Γ f . (40) 0 1 1 Γ int Γ f Ω 1 Ω 2 u = u g
T Γ f h := {T ∈ T Γ h : ψ 0 on T } and T Γint h := {T ∈ T Γ h : ψ 0 on T } .
Note that there may be some cells in T Γ h that are not marked as either T

Γ f h or T Γint h
. This is illustrated by the mesh example on the right of Fig. 14, where the cells in T Γ f h and T Γint h are painted in red and blue respectively, but there remain some cells T Γ h that are in neither of these categories. The are painted in yellow on the picture. These are the cells intersected by the line {ψ = 0}. The crack tip happens to be thus inside one of the yellow cells.

Everything is now set up to adapt the φ-FEM approaches of the two preceding sections to the equations (40). We choose an integer k ≥ 1 and introduce first the FE spaces V h,1 , V h,2 together with their homogeneous counterparts V 0 h,1 , V 0 h,2 as in (36) to approximate u 1 and u 2 . These will be used in the discretization of the variational formulation of the first equation in (40) together with the boundary conditions on ∂Ω. The remaining equations in (40), i.e. the relations on Γ int and Γ f will be treated by the introduction of auxiliary variables on the appropriate parts of Ω Γ h (the domain of the mesh T Γ h ):

• the vector-valued unknown p and the matrix-valued unknowns y 1 ,y 2 on Ω Γint h (the domain of the mesh T Γint h ). These will serve to impose the continuity of both the displacement and the normal force on Γ int thorough the equations

u 1 -u 2 + pφ = 0 , on Ω Γint h , y i = -σ(u i ) , on Ω Γint h , y 1 • ∇φ -y 2 • ∇φ = 0 , on Ω Γint h ,
which are exactly the same as (33)-(35) with the only exception that they are posed on the appropriate portion of Ω Γ h rather than on entire Ω Γ h . These variables will be discretized in FE spaces Q k h (Ω Γint h ) for p and Z h (Ω Γint h ) for y 1 ,y 2 , defined by ( 15) and ( 25) respectively.

• the vector-valued unknowns p N i and the matrix-valued unknown y N i , i = 1, 2 on Ω Γ f h (the domain of the mesh T Γ f h ). These will serve to impose the Neumann boundary conditions on both sides of Γ f thorough the equations

y N i = -σ(u i ) , on Ω Γ f h , y N i ∇φ + p N i φ + g|∇φ| = 0 , on Ω Γ f h ,
which are exactly the same as (24a-b) with the only exception that the domain is renamed to Ω

Γ f h from Ω Γ N h . These variables will be discretized in FE spaces Q k-1 h (Ω Γ f h ) for p N i and Z h (Ω Γ f h ) for y N
i , defined again by ( 15) and ( 25) respectively.

Note that the combination of equations above does not impose the appropriate interface conditions on the whole of Γ since the latter may be not completely covered by Ω

Γ f h ∪ Ω Γint h .
Fortunately, this defect of the formulation on the continuous level can be repaired on the discrete level by adding the appropriate stabilization to the FE discretization, similarly to what we have already seen in the setting with mixed boundary conditions. All this results in the following FE scheme: find

u h,1 ∈ V h,1 , u h,2 ∈ V h,2 , p h ∈ Q k h (Ω Γ D h ), y h,1 , y h,2 ∈ Z h (Ω Γ N h ), p N h,1 , p N h,2 ∈ Q k-1 h (Ω Γ N h ), y N h,1 , y N h,2 ∈ Z h (Ω Γ N h ) such that 2 i=1 Ω h,i σ(u h,i ) : ∇v h,i + ∂Ω h,i,int y h,i n • v h,i + ∂Ω h,i,f y N h,i n • v h,i - ∂Ω h,i \(∂Ω h,i,int ∪∂Ω h,i,f ) σ(u h,i )n • v h,i + γ p h 2 Ω Γ int h (u h,1 -u h,2 + 1 h p h φ h ) • (v h,1 -v h,2 + 1 h q h φ h ) + γ u 2 i=1 Ω Γ int h (y h,i + σ(u h,i )) : (z h,i + σ(v h,i )) + γ y h 2 Ω Γ int h (y h,1 ∇φ h -y h,2 ∇φ h ) • (z h,1 ∇φ h -z h,2 ∇φ h ) + γ u,N 2 i=1 Ω Γ f h (y N h,i + σ(u h,i )) : (z N h,i + σ(v h,i )) + γ p,N h 2 2 i=1 Ω Γ f h (y N h,i ∇φ h + 1 h p N h,i φ h ) • (z N h,i ∇φ h + 1 h q N h,i φ h ) + 2 i=1 G h (u h,i , v h,i ) + J lhs,int h y h,i , z h,i + J lhs,f h y N h,i , z N h,i = 2 i=1 Ω h,i f • v h,i - γ p,N h 2 2 i=1 Ω Γ f h g|∇φ h N h,i ∇φ h + 1 h q N h,i φ h ) + 2 i=1 J rhs,int h (z h,i ) + J rhs,f h z N h,i , ∀v h,1 ∈ V 0 h,1 , v h,2 ∈ V 0 h,2 , q h ∈ Q k h (Ω Γ D h ), z h,1 , z h,2 ∈ Z h , q N h,1 , q N h,2 ∈ Q k-1 h (Ω Γ N h ), z N h,1 , z N h,2 ∈ Z h (Ω Γ N h ) . (41)
As usual, we have added here the ghost stabilization G h [START_REF] Girault | Error analysis of a fictitious domain method applied to a Dirichlet problem[END_REF] and the additional stabilizations J lhs,int h , J lhs,f h (accompanied by the their counterparts on the right-hand side for the consistency) that are copied from the J lhs,N h in [START_REF] Mittal | Immersed boundary methods[END_REF] but adjusted to the corresponding sub-meshes:

J lhs,int h (y, z) = γ div Ω Γ int h div y • div z , J lhs,f h (y, z) = γ div Ω Γ f h div y • div z .
The boundary integrals are rewritten in terms of y i ,y N i wherever possible. We have here denoted by ∂Ω h,i the part of the boundary of Ω h,i other than ∂Ω and introduced ∂Ω h,i,int as the part of ∂Ω h,i formed by the boundary facets of T h,i belonging to the cells in T Γint h . The same for ∂Ω h,i,f . Test case: Let Ω = (0, 1) 2 and the interface Γ be given by the level set

φ(x, y) = y - 1 4 sin(2πx) - 1 2 .
We choose the crack tip to be at x = 0.5 so that

Γ int := {φ = 0} ∩ {x < 0.5} and Γ f := {φ = 0} ∩ {x > 0.5} .
This is the setting represented at Fig. 13. We use the φ-FEM (41) to solve (39) with the manufactured solution u = u ex = (sin(x) × exp(y), sin(y) × exp(x)) T which gives f , g, and u g by substitution. The force on the crack g should be extended to a vicinity of Γ f and we implement it by g = σ(u ex ) ∇φ ∇φ + φu ex .

We choose γ u = γ p = γ div = γ u,N = γ p,N = γ div,N = 1.0, σ p = 1.0 and σ D = 20.0.

We have conducted two series of numerical experiments using φ-FEM (41) with P 2 Lagrange polynomials (k = 2) on families of meshes presented at Fig. 14, either resolving the crack tip (the mesh on the left) or not (the mesh on the right). The results are reported on Fig. 15. We see that φ-FEM converges optimally, giving very similar results on both types of meshes.

Heat equation

We finally demonstrate the applicability of the φ-FEM approach to time-dependent problems. We take the example of the heat equation with Dirichlet boundary conditions: given a bounded domain Ω ∈ R d , the initial conditions u 0 on Ω, and the final time T > 0, find the scalar field u

= u(x, t) such that    u t -∆u = f in Ω × (0, T ), u = 0 on Γ × (0, T ), u(., 0) = u 0 in Ω. (42)
We are interested again in the situation where a fitting mesh of Ω is not available. We rather assume that Ω is inscribed in a box O which is covered by a simple background mesh T O h , and introduce the active mesh T h as in [START_REF] Burman | Cutfem: discretizing geometry and partial differential equations[END_REF]. We then follow the Direct Dirichlet φ-FEM approach (11), [START_REF] Elfverson | A new least squares stabilized Nitsche method for cut isogeometric analysis[END_REF] with the following modifications:

• We introduce the uniform partition of the time interval I = [0, T ] into time steps of length ∆t by the nodes t i = i∆t. We discretize then (42) in time using implicit Euler scheme. On the continuous level this is formally written as: find u n (the approximation to u at time t n ) in the form u n = φw n successively for n = 1, 2, . . . solving

φw n -φw n-1 ∆t -∆(φw n ) = f n (43)
where

f n (•) = f (t n , •).
• We extend (43) to Ω h , integrate by parts on Ω h , and discretize the resulting variational formulation using a FE space and adding appropriate stabilizations.

The φ-FEM for (42) reads thus as: find w n h ∈ V h for n = 1, 2, . . . with V h defined by ( 14) such that

Ω h φ h w n h ∆t φ h v h + Ω h ∇(φ h w n h ) • ∇(φ h v h ) - ∂Ω h ∂ ∂n (φ h w n h )φ h v h + σ D h E∈F Γ h E [∂ n (φ h w n h )] • [∂ n (φ h v h )] -σh 2 T ∈T Γ h T φ h w n h ∆t -∆(φ h w n h ) ∆(φ h v h ) = Ω h φ h w n-1 h ∆t + f n φ h v h -σh 2 T ∈T Γ h T φ h w n-1 h ∆t + f n ∆(φ h v h ). ( 44
)
We have added here the ghost stabilisation, similar to [START_REF] Girault | Error analysis of a fictitious domain method applied to a Dirichlet problem[END_REF] but in simpler scalar setting, and additional stabilization inspired by [START_REF] Glowinski | A fictitious domain method for Dirichlet problem and applications[END_REF]. The idea for the latter is to take the governing equation in the strong form, which is now (43), and to impose it in a least squares manner cell by cell. Test case: We consider again the geometry of Ω and of the surrounding box O as in our first test case on page 9. In particular, the level set is given by (20) so that Ω is the circle centered at (0.5, 0.5). Examples of meshes used both by φ-FEM and by the standard FEM are given in Fig. 2. We want to solve (42) with the manufactured solution u = u ex = exp(x) sin(2πy) sin(t) and extrapolated boundary conditions

u g = u ex (1 + φ).
We are going to compare the convergence of the φ-FEM (44) with that of the standard FEM using P 1 Lagrange polynomials in space and the implicit Euler scheme in time in both cases. The φ-FEM stabilization parameter is taken as σ = 20. The results are reported in Figs. 16 and17, for ∆t = h and ∆t = 10h 2 , respectively. Once again, φ-FEM converges faster than standard FEM. the test considered here, the predominant source of error seems to be in the time discretization. In particular, we observe only O(h) convergence in the L 2 -norm in space in the regime ∆t = h on Fig. 16. A cleaner 2nd order in time should be possible to achieve using the BDF2 marching scheme, but this remains out of the scope of the present paper.

6 Conclusions and perspectives φ-FEM is a relative newcomer to the field of unfitted FE methods. Up to now, it was only applied to scalar 2nd order elliptic equations with pure Dirichlet or pure Neumann/Robin boundary conditions in [START_REF] Duprez | φ-FEM: a finite element method on domains defined by level-sets[END_REF][START_REF] Duprez | A new φ-FEM approach for problems with natural boundary conditions[END_REF]. The purpose of the present contribution is to demonstrate its applicability to more sophisticated settings including the linear elasticity with mixed boundary conditions and material properties jumping across the internal interfaces, elasticity with cracks, and the heat transfer. In all the cases considered here, the numerical tests confirm the optimal accuracy on manufactured smooth solutions. φ-FEM is easily implementable in standard FEM packages (we have chosen FEniCS for the numerical illustration in this chapter). In particular, φ-FEM uses classical finite element spaces and avoids the mesh generation and any non-trivial numerical integration.

Interestingly, our methods systematically outperform the standard FEM on comparable meshes. This can be attributed to a better representation of the boundary and of the solution near the boundary, as opposed to the approximation of the domain by a polyhedron/polygon in standard FEM. We recall that the computing times, reported in some of our tests with φ-FEM and favourably compared with those of the standard FEM, only include assembling of the matrices and the resolution of the linear systems. It would be interesting to add the mesh generation time to the comparison, which should be even more in favour of φ-FEM (when efficiently implemented).

Admittedly, the test cases presented in this contribution do not comprise all the complexity of the real-life problems. We have restricted ourselves to simple geometries in 2D only. Even more importantly, we have tested the methods only on smooth solutions, which is not supposed to happen in practice in problems with cracks, for example. Taking accurately into account the singularity at the crack tip remains an important challenge for the future φ-FEM developments. A relatively easily implementable approach would be to combine φ-FEM with a local mesh refinement by quadtree/octree structures near the crack tip (front). We emphasize that such a refinement should be necessary only in the vicinity of the front, since the discontinuous solution along the crack should be efficiently approximated by φ-FEM on a reasonably coarse unfitted mesh.

The mathematical analysis of the schemes presented in this paper is in progress. We also plan to adapt φ-FEM to fluid-structure simulations starting by the creeping flow around of a Newtonian fluid (Stokes equations) in the presence of rigid particles.

Figure 2 :

 2 Figure 2: Circular domain given by[START_REF] Hansbo | Nitsche's method for interface problems in computational mechanics[END_REF]. Left: active meshes for φ-FEM (with cells from T Γ h in yellow). Right: a fitted mesh for the standard FEM.

Figure 3 : 1 L 2 Figure 4 : 2 4.

 31242 Figure 3: Test case with pure Dirichlet conditions. L 2 relative errors on the left, H 1 relative errors on the right.

  serving to impose the Dirichlet boundary conditions, and T Γ N h around Γ N for the Neumann ones. The natural choice for these is

  are either on the Dirichlet side (they form thus T Γ D h and are colored in red), or on on the Neumann side (they form thus T Γ N h and are colored in blue), or in between (they are then in T Γ h but not in T Γ D h

Figure 5 :

 5 Figure 5: Test case with mixed boundary conditions: the geometry of Dirichlet and Neumann boundary parts.

Figure 6 :L 2 Figure 7 :

 627 Figure 6: Test case with mixed boundary conditions, meshes resolving the Dirichlet/Neumann junction. Left: active meshes for φ-FEM, red for T Γ D h , blue for T Γ N h . Right: a mesh for standard FEM, red boundary facets on Γ D , blue boundary facets on Γ N .

Figure 8 :

 8 Figure 8: Test case with mixed boundary conditions, meshes not resolving the Dirichlet/Neumann junction. Left: active meshes for φ-FEM, red for T Γ D h , blue for T Γ N h , yellow for T Γ h otherwise unmarked. Right: a mesh for standard FEM, red boundary facets on Γ D , blue boundary facets on Γ N , note that some boundary facets contain both Dirichlet and Neumann parts.

L 2 Figure 9 :

 29 Figure 9: Test case with mixed boundary conditions, results on meshes as on Fig. 8. Left: L 2 and H 1 relative errors under the mesh refinement. Right: computing time vs. the L 2 relative error.

Figure 10 :

 10 Figure 10: Geometry with the interface Γ: elasticity with multiple materials.

Figure 11 :L 2 Figure 12 :

 11212 Figure 11: Linear elasticity with multiple materials. Left: a mesh used for φ-FEM (Ω Γ h painted in yellow); Right: a mesh matching the interface for standard FEM (yellow and white represent the two materials).

Figure 13 :

 13 Figure 13: Geometry notations to represent the crack. Γ int and Γ f represents the fictitious interface and the actual crack respectively.

Figure 14 :Figure 15 :

 1415 Figure 14: Test case with a crack, meshes used for φ-FEM. Left: a mesh resolving the crack tip; the cells in T Γint h

FigureFigure 17 :

 17 Figure Test case for the heat equation; ∆t = h. Left: L ∞ (0, T ; L 2 (Ω)) relative errors. Right: L 2 (0, T ; H 1 (Ω)) relative errors.

In some settings presented further, the level set φ will describe an interior interface inside Ω rather than the geometry of Ω itself.

u g = u ex (1 + φ), on Ω Γ h ∩ {x 0.5} , g = σ(u ex ) ∇φ ∇φ + u ex φ, on Ω Γ h ∩ {x < 0.5} .

i=1 Ω h,i σ i (u h ) : ∇v h = Ω f • v h , ∀ v h ∈ V 0 h ,(38)


Lucile Gelin

Morgane Daniel

Thomas Pellegrini

Julien Pinquier

Babble noise augmentation for phone recognition applied to children reading aloud in a classroom environment

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

BABBLE NOISE AUGMENTATION FOR PHONE RECOGNITION APPLIED TO CHILDREN READING ALOUD IN A CLASSROOM ENVIRONMENT

Lucile Gelin 1,2 , Morgane Daniel 1 , Thomas Pellegrini 2 , Julien Pinquier 2 1 Lalilo, 2 Institut de Recherche en Informatique de Toulouse (IRIT)

OBJECTIVES

Reading is a key step in a child's learning, but teachers do not have time to spend individually with each student to monitor progress. We are developing a tool that aims at automatically evaluating children's fluency and pronunciation as they read aloud.

CHALLENGES

R[ 1 ]

 1 Thiemann J. et al. (2013). DEMAND: a collection of multi-channel recordings of Acoustic Noise in Diverse Environments [2] Povey D. et al. (2018). Semi-Orthogonal Low-Rank Matrix Factorization for Deep Neural Networks [3] Shivakumar P.G. and Georgiou P. (2018). Transfer Learning from Adult to Children for Speech Recognition: Evaluation, Analysis and Recommendations.

Babble noise present in classroom environment CORPORA

		Lalilo French children corpus, composed of 5-7 year-old children read-
		ing aloud in a classroom environment
		Commonvoice French adult corpus
		DEMAND babble noise corpus [1]
		Corpus	Lalilo	Commonvoice DEMAND
	Lack of children speech data in French	Set	Train	Test	Train	Babble
	Characteristic non-reader children's prosody	Duration (h)	13.8	0.8	148.9	1.3
		SNR (dB)	24 ±13 22 ±11	35 ±16	-
		Information on corpora
	Babble noise
	augmentation (Aug):
	concatenating children
	data with babble noise
	data at different signal to
	noise ratios (SNR) to
	train the acoustic model
	→ makes the model
	more robust to classroom
	babble noise
	Transfer Learning (TL,
	[3]): transferring
	knowledge learned from
	a big corpus of adult data
	by re-training it with
	children data
	→ overcomes lack of
	children data
			Transfer learning brings overall relative improvement of 7%,
			with a greater efficiency on clean recordings.
			Babble noise augmentation brings further decrease of PER,
			especially on low and medium SNR ranges (respectively
			17% and 6% relative improvement over TL model). While
			the improvement over TL model is not significant for clean
			recordings, the PER is not degraded. While the improve-
			ment over TL model is not significant for clean recordings,
			the global relative PER improvements brought by the TL-
			Aug model are 6% over the TL model and 12% over the
			baseline model.

METHOD

The baseline model is a children-trained factorized time-delay neural network

[2]

. We combine two methods to improve our system's performance:

RESULTS

Phoneme Error Rate (PER,%) obtained with the different acoustic models, global (All files) and for different SNR ranges CONCLUSION


Marc Margalef-Rovira
email: marc.margalef-rovira@univ-grenoble-alpes.fr

Tadeu Mota Frutuoso

Abdelhalim A Saadi
email: ahmed-abdelhalim.saadi@univ-grenoble-alpes.fr

Loïc Vincent

Manuel J Barragan
email: manuel.barragan@univ-grenoble-alpes.fr

Emmanuel Pistono
email: emmanuel.pistono@univ-grenoble-alpes.fr

Christophe Gaquière

Philippe Ferrari
email: philippe.ferrari@univ-grenoble-alpes.fr

Loic Vincent
email: loic.vincent@grenoble-inp.fr

Christophe Gaquiere
email: christophe.gaquiere@univ-lille.fr

Tadeu Mota-Frutuoso
email: tadeu.mota-frutuoso@univ-grenoble-alpes.fr

Miniaturization of Transmission Lines: Meandered Slow-wave CPWs

Keywords: transmission line, slow-wave, CPW, mm-wave

I. INTRODUCTION

Current and future use of transceivers for diverse applications such as Internet of Things (IoT), automotive radar, 5G (future 6G) or medical applications require to operate in several frequency bands and the use of diverse communication standards. In the best scenario, the change of such telecommunication standard can be achieved through the operation of tunable elements in the transceiver. However, changing the telecommunication standard under use often means switching complete parts of the transceiver or even using a completely different transceiver.

In this scenario, the downscaling of the transistor size plays a major role in the miniaturization of complex transceivers. However, miniaturization of passive devices, needed to perform filtering, impedance matching or beamsteering functions, is a less technology-dependent issue. Passive devices are divided into two families: (i) lumped elements and (ii) distributed elements. Lumped elements (i.e. capacitors, inductors and transformers) tend to have a high Quality-factor (Q-factor) at RF frequencies (i.e. around 10 GHz) that decreases with the increase of frequency, while distributed elements (i.e. transmission lines) tend to have a low Q-factor at RF frequencies that increases with the frequency. Moreover, the physical length of a transmission line (TLine) with a given electrical length is inversely proportional to the frequency. For these reasons, lumped elements are preferred at low frequencies (below ~30-50 GHz in integrated technology). Above these frequencies, TLines are preferred thanks to their increased performance and acceptable size.

TLine footprint reduction has been addressed in the literature using two main approaches: (i) meandering [START_REF] Kuhn | HTSC inverted and conventional geometry microstrip resonator for UHF frequencies[END_REF][START_REF] Chen | Synthetic quasi-TEM meandered transmission lines for compacted microwave integrated circuits[END_REF] or (ii) using slow-wave concept [START_REF] Cheung | On-chip interconnect for mmwave applications using an all-copper technology and wavelength reduction[END_REF]. In 2003, the first proposal of the application of the slow-wave effect on CPWs was published [START_REF] Cheung | On-chip interconnect for mmwave applications using an all-copper technology and wavelength reduction[END_REF], presenting the so-called Slow-wave CPW structure (S-CPW). In 2013, the first proposal of the distributed electrical model for S-CPWs was published [START_REF] Franc | A lossy circuit model based on physical interpretation for integrated shielded slow-wave cmos coplanar waveguide structures[END_REF] and subsequently improved in [START_REF] Bautista | Accurate Parametric Electrical Model for Slow-Wave CPW and Application to Circuits Design[END_REF]. In [START_REF] Bautista | Accurate Parametric Electrical Model for Slow-Wave CPW and Application to Circuits Design[END_REF], authors provide tools for the analytical modeling of this structure, allowing the designer to accurately simulate the S-CPW in any technology, greatly reducing the effort to be devoted to time-consuming EM simulations.

In this work, innovative structures aiming to reduce the footprint of CoPlanar Waveguides (CPWs) are proposed. The proposed structures utilize the meandering and the slow-wave concept to reduce their physical length per unit of electrical length. Recently, meandering techniques together with the slow-wave concept was proposed for slow-wave microstrips, as in [START_REF] Acri | A Millimeter-Wave Miniature Branch-Line Coupler in 22-nm CMOS Technology[END_REF]. However, to the best of author's knowledge, this is the first time that an approach merging both techniques is proposed for CPW structures. This work is organized as follows: section II provides a brief explanation of the slow-wave concept. In section III, the proposed structures are presented. Section IV presents the results of the fabricated TLines. Finally, section V presents the main conclusions of this work.

II. THE SLOW-WAVE PRINCIPLE

The slow-wave concept has been applied to multiple TLine architectures such as: microstrip [START_REF] Acri | A Millimeter-Wave Miniature Branch-Line Coupler in 22-nm CMOS Technology[END_REF], CoPlanar Waveguides (CPWs) [START_REF] Cheung | On-chip interconnect for mmwave applications using an all-copper technology and wavelength reduction[END_REF][START_REF] Franc | A lossy circuit model based on physical interpretation for integrated shielded slow-wave cmos coplanar waveguide structures[END_REF][START_REF] Bautista | Accurate Parametric Electrical Model for Slow-Wave CPW and Application to Circuits Design[END_REF], and CoPlanar Striplines (CPS) [START_REF] Aziz | Shielded Coplanar Striplines for RF Integrated Applications[END_REF]. This technique utilizes some patterned structures to capture the electrical field of the TLine while allowing the magnetic field to flow freely. From a circuit point of view, this results in an increase of the capacitance per unit of length of the Tline, 𝐶 "#$ , and an do not modify inductance per unit of length, 𝐿 "#$ . The increase of 𝐶 "#$ leads to a reduction of the propagation velocity (𝑣 ' ), whose expression is given by:

𝑣 ' = ) *+ ,-. •0 ,-. (1)
Hence, the physical length (𝐿) of a TLine, with a given electrical length (𝜃), is reduced when the slow-wave concept is used since:

𝜃 = 2 3 4 • 𝐿, (2)
where 𝜔 is the considered angular frequency.

In addition, the Q-factor of a TLine can be expressed as:

𝑄 = 2 7•3 4 •8 , ( 3
)
where 𝛼 is the attenuation constant of the considered TLine.

Using this expression, it can be proved that the use of the slowwave concept can lead to an improved Q-factor as compared to the classical structures.

In practice, the slow-wave concept for coplanar structures is achieved through the implementation of a floating shield beneath the signal and ground strips. This floating shield is composed of narrow metal ribbons placed transversally to the propagation direction.

III. PROPOSED ARCHITECTURE

This work explores the effect of meandering the signal strip of S-CPWs. Note that two kinds of structures can be envisioned with this consideration: (i) a structure where most of the signal strips are placed transversally to the overall propagation direction, or (ii) a structure where most of the signal strips are placed in parallel to the overall propagation direction. In this paper, these kinds of Meandered S-CPWs (MS-CPWs) are named type 1 and 2, respectively.

Fig. 1(a) presents a classical meandered microstrip line and the geometrical parameters for its design. In this work, together with a classical microstrip, this structure will be used as a reference for the evaluation of the performance of the proposed MS-CPWs. Figs. 1(b) and 1(c) show 3D views of the so-called type 1 and type 2 MS-CPWs and their geometrical parameters.

In a traditional meandered microstrip line, whether the strip is meandered transversally or parallelly to the overall propagation direction has a negligible effect on the performance of the TLine. However, the behavior of the type 1 or type 2 MS-CPWs is slightly different.

Both of the proposed MS-CPW types present two propagation modes within the structure. A first mode where the propagation occurs as in a S-CPW. This mode is found for the regions where the signal strip is placed parallelly to the overall propagation direction. A second mode of propagation is found for the regions where the signal strip is placed transversally to the overall propagation direction. In this case, the propagation mode is similar to that of a microstrip line. Even though the electrical field is still captured by the shield below, the signal strip is parallel to the metallic ribbons. Hence, return-current path is created in the metallic ribbons. Thus, behaving like a degenerated microstrip line where the return path for the signal is done through the metallic ribbons, with a capacitance existing between the signal strip and the metallic ribbons, and an inductance given by the flux of the magnetic field around the strips. As a result, type 1 MS-CPWs are expected to show a lower propagation constant as compared to their type 2 counterparts.

IV. RESULTS

As a proof-of-concept, four different Tlines were designed, fabricated and measured on the AMS 0.35 µm CMOS technology. The selected architectures were: (i) a 50-Ω traditional microstrip line, (ii) a 50-Ω meandered microstrip, (iii) a 50-Ω type 1 MS-CPW, and (iv) a 50-Ω type 2 MS-CPW. Measurements were performed using an Anritsu VectorStar ME7838A4, from 70 kHz to 100 GHz, with a Line-Reflect-Reflect-Line (LRRM) [START_REF] Davidson | LRM and LRRM Calibrations with Automatic Determination of Load Inductance[END_REF] calibration on an Impedance Standard Substrate (ISS). Subsequently, the effect of the pads and the 200-µm access lines were de-embedded using the technique described in [START_REF] Mangan | De-embedding transmission line measurements for accurate modeling of IC designs[END_REF].

For a fair comparison, all the circuits were allocated in an area of 120 µm of length and 132 µm of width. In particular, meandered microstrip TLines were designed with a margin of 20 µm in each of their sides due to the fact that in a practical case any circuit placed inside this spatial margin would suffer from electromagnetic coupling. On the other hand, the confined electrical field of S-CPW-based TLines allows to place circuitry close to the TLine, while avoiding cross-talk issues.

The used technology features 3 metal layers and an optional additional thick layer, which was used for the strips design. In this technology, aluminum is the metal of choice for the Back-End-Of-Line (BEOL). It is well-known that the conductivity of aluminum is sensibly lower than the conductivity of copper. Hence, better results are expected in a technology featuring a copper BEOL.

The microstrip line was designed with a width of 5.5 µm using the top thick metal for the signal strip and metal 1 for the ground plane. The meandered microstrip line was designed using the same metal layers and width than the classical microstrip and a spacing between parallel lines, 𝐷, of 11 µm, to avoid self-coupling between meandered strips. The type 1 MS-CPW was designed using metal 4 for the signal and ground strips and metal 3 for the floating ribbons. The width of the ground, 𝑊 < , and signal strips, 𝑊 = , was set to 8 µm and 10 µm, respectively. The distance between signal and ground strips, 𝐺, was set to 13 µm. Finally, the spacing between parallel lines, 𝐷 , was designed using 19 µm. Concerning the type 2 MS-CPW, the floating ribbons were placed in the metal 3 layer and the signal and ground strips in the metal 4 layer. 𝑊 ? and 𝑊 < were set to 5.5 µm and 16 µm, respectively. The distance between the signal and ground strips, 𝐺, was designed using 13 µm. The distance between parallel strips, 𝐷, was set to 12 µm. Finally, for both MS-CPWs the width and spacing of the floating ribbons was set to 0.6 and 2.6 µm, respectively.

Measurement and simulation are shown in Fig. 2 using solid and dashed lines, respectively. The presented results show very good agreement throughout the considered band. Fig. 2(a) shows the attenuation constant, 𝛼, throughout the considered frequency band of the measured TLines. At 80 GHz, the traditional microstrip presents an attenuation constant of around 1.3 dB/mm, 𝛼 is equal to 4 dB/mm for the meandered microstrip, around 5 dB/mm for the MS-CPW type 1, and around 6.6 dB/mm for the MS-CPW type, respectively. presents a 𝛽 and an 𝜖 ABCC of 3.5 rad/mm and 4.2, respectively. The meandered microstrip has a 𝛽 and an 𝜖 ABCC of 12 rad/mm and 51, respectively. The type 1 and 2 MS-CPWs present a 𝛽 of around 11.5 and 16, respectively. Finally, these two TLines present an 𝜖 ABCC of 47 and 88, respectively.

Note that the values presented in Figs. 2(a) and 2(b) express magnitudes per unit of length. These values were obtained using the effective length of the structure, 120 µm.

Figure 2(c) presents the Q-factor of the considered Tlines. At 80 GHz, the microstrip and meandered microstrip present a Q-factor of 14 and 13, respectively. At the same frequency, the type 1 and type 2 MS-CPWs present Q-factors of 12 and 10, respectively.

Figure 2(d) presents the characteristic impedance, 𝑍 F , of the considered TLines throughout the analyzed frequency band, with all of them achieving the desired 50 Ω. Note that the type 2 MS-CPW shows a resonance at around 120 GHz. This is due to the fact that, at this frequency this TLine reaches 180 degrees of electrical length.

Finally, Table I summarizes the performance of the considered Tlines at 80 GHz. A Figure-of-Merit (FoM) is introduced, aiming to compare the miniaturization capabilities of the presented TLines. The considered FoM is obtained through the ratio between the electrical length of the TLine at the considered frequency and the occupied surface, which is equal to 0.016 mm 2 for all the considered TLines. For the microstrip and meandered microstrip, the FoM has a value of 26 and 91 rad/mm 2 , respectively. The FoM has a a value of 87 and 120 rad/mm 2 for the type 1 and type 2 MS-CPWs, respectively.

V. CONCLUSIONS

In this work, four 50-Ω transmission line architectures were compared: (i) a straight microstrip line, (ii) a meandered microstrip line, (iii) a type 1 MS-CPW and (iv) a type 2 MS-CPW. The presented measurements, from 70 kHz to 100 GHz, show the capability of the type 2 MS-CPW to achieve high miniaturization. At 80 GHz, this TLine shows an 𝜖 ABCC of 88, greater than the 𝜖 ABCC shown by the meandered microstrip or the type 1 M-SCPW, which is of 51 and 47, respectively. In addition, measurement results of the fabricated TLines show their equivalent performance in terms of Q-factor, with values ranging from 14 to 10 at 80 GHz.

Together with the presented TLines, other TLines were sent into fabrication to carry out a comparison between the synthesizable characteristic impedance in this technology by using these architectures. In addition, microstrip lines and MS-CPWs were fabricated in a 55-nm technology featuring a copper BEOL, aiming to show the Q-factor improvement due to the increased conductivity. Future work includes the characterization of these TLines as well as the development of a parametric electrical model.

Fig. 1 (

 1 Fig. 1 (a) Meandered microstrip. (b) Meandered S-CPW type 1. (c) Meandered S-CPW type 2.

Fig. 2 (Fig. 2

 22 Fig. 2 Measurement (solid lines) and simulation results (dashed lines) of the considered TLines. (a) Attenuation constant. (b) Propagation constant and effective dielectric constant (c) Qfactor. (d) Characteristic impedance.

Table I .

 I TLINES PERFORMANCE COMPARISON AT 80 GHZ

ACKNOWLEDGMENT

This work has been funded by the European Union (ECSEL JU GA 737454: TowARds Advanced BiCMOS NanoTechnology platforms for RF applicatiOns -Taranto), and the General Directorate for Enterprises (DGE) in France.