Gergely Molnár
email: gergely.molnar@insa-lyon.fr

Nawfal Blal

Topology optimization of periodic beam lattices using Cosserat elasticity

Keywords: Cosserat elasticity, Topology optimization, Lattice structures, Gradient elasticity, Metamaterial

published or not. The documents may come

Introduction

Historically, when creating structures with large spans (e.g., bridges, roofs, or towers), lattice and truss elements were used to reduce the total mass and increase the global stiffness. However, early in history, structural engineers realized that by increasing the space between bent components, the added tensile and compression forces would increase the bending inertia quadratically with the distance (Huygens-Steiner theorem). As a consequence, the additional bending momentum would magnify not only the stiffness but also the global stability of the structures. This way, structures can span considerable distances that are not accessible to solid forms.

A similar phenomenon is seen in nature. A structural hierarchy is found when observing the microstructure of diverse load-bearing components. Examples range from cork [START_REF] Chen | Fibre reinforced cellular microstructure of cork wood[END_REF], through many Diatom species [START_REF] Jang | Fabrication and deformation of three-dimensional hollow ceramic nanostructures[END_REF], honeycombs [START_REF] Mousanezhad | Hierarchical honeycomb auxetic metamaterials[END_REF] and the trabecular bone [START_REF] Lakes | Materials with structural hierarchy[END_REF][START_REF] Ritchie | The conflicts between strength and toughness[END_REF]. In all cases, smaller beam-like elements form an intricate network based on the characteristic loads.

In recent years, with the advancements in additive manufacturing, metamaterials with custom-made microstructures have been prepared [START_REF] Askari | Additive manufacturing of metamaterials: A review[END_REF]. In addition to an increased stiffness, this small-scale procedure makes the load-bearing elements stronger. Moreover, as the material approaches its microscopic length scale, the manufacturing defects' size and effect are reduced. As a result, macroscopically brittle materials might behave in a ductile manner [START_REF] Zheng | Multiscale metallic metamaterials[END_REF][START_REF] Bauer | Approaching theoretical strength in glassy carbon nanolattices[END_REF] thereby giving rise not only to a lightweight and stiff material but also a resistant one.

Thanks to the advanced manufacturing processes, developing new products with specific optimized properties is possible by modifying their shape or topology. Topology optimization provides a suitable mathematical framework to optimize the material distribution, i.e., the spatial distribution of the material in a design domain. After defining a cost function to be minimized, the sensitivity analysis provides an updating scheme for the design variables. The cost function can be defined for various quantities of interest (compliance, maximal stress, target shape, displacements, etc.). Topology optimization problems can be considered constrained as Partial Differential Equation (PDE) design problems, requiring a combination of optimization solvers and numerical discretization schemes to solve the PDEs physical equations (mainly the finite element method). Two types of optimization solvers can be used: (i) meta-heuristic approaches needing only a back run call of different cost function computations [START_REF] Di Cesare | A new hybrid topology optimization method based on i-pr-pso and eso. application to continuum structural mechanics[END_REF] and (ii) gradient-based solvers where the derivative of the cost function is required. Gradient-based topology optimization is usually based on the adjoint state method.

The literature cites several approaches for topology optimization. Density-based techniques pioneered by the standard Solid Isotropic Material with Penalization (SIMP) method [START_REF] Bendsøe | Generating optimal topologies in structural design using a homogenization method[END_REF][START_REF] Bendsøe | Topology Optimization: Theory[END_REF] have proven their efficiency in structural topology design for a broad range of applications. SIMP considers density variables defined at each element of the finite element discretization as the optimized topology's design parameters (design variables). The main idea is to consider the elastic behavior at each element designated by a power law introducing a density variable that has a value between 0 (no material) and 1 (material) and a power law parameter (larger than 1) aiming to penalize intermediate densities. Consequently, when the penalization parameter is adequately chosen, the spatial material distribution leads to an optimized structure with only two material types: void regions (elements with no stiffness) and solid regions (elements whose stiffnesses equal the bulk behavior). Empirically, the penalization parameters are taken to be greater than 3. However, the works of Bendsøe and Sigmund [START_REF] Bendsøe | Material interpolation schemes in topology optimization[END_REF] propose a rigorous methodology for their calibration.

One other important issue when using such approaches is the checkerboard pathology. It is due to the apparition of neighboring elements with alternating void and solid materials. This problem can be addressed with the use of filter solutions. Usually, one can employ density filters to impose limitations on the density variation by a fixed length scale in the stiffness distribution. Another option is to introduce sensitivity filtering so that the design sensitivity is maintained in a fixed neighborhood. Filtering techniques are efficient for limiting mesh-dependency in density-based topology optimization solvers [START_REF] Bourdin | Filters in topology optimization[END_REF].

The other class of topology optimization is based on the level-set function as a parametrization of the design domain topology [START_REF] Sethian | Level set methods, evolving interfaces in geometry, fluid mechanics comuputer vision, and materials sciences[END_REF][START_REF] Peng | A pde-based fast local level set method[END_REF]. The level-set function represents the boundaries between void and solid phases, and the spatial evolution of the void-solid interfaces is usually governed by a shape derivative [START_REF] Sokolowski | Introduction to shape optimization[END_REF] or a topological derivative [START_REF] Jackowska-Strumillo | The topological derivative method in shape optimization[END_REF] to find the gradient direction toward the optimized topology. In recent years, numerous investigations have been proposed for level-set-based topology optimization approaches [START_REF] Allaire | Structural optimization using sensitivity analysis and a level-set method[END_REF][START_REF] Allaire | Structural optimization using topological and shape sensitivity via a level set method[END_REF][START_REF] Yamada | A topology optimization method based on the level set method incorporating a fictitious interface energy[END_REF][START_REF] Otomori | Matlab code for a level set-based topology optimization method using a reaction diffusion equation[END_REF].

Thanks to the recent advent of manufacturing technologies, the design of advanced materials, incorporating small scales with complex topologies, has become possible. However, simulations for structural optimization describing the finest scales are computationally expensive. Therefore, multi-scale topology optimization techniques have emerged as a macro-to-micro optimization technique with noteworthy performances. The optimized structure results from the macro topology design as the optimized representation of ascribed microstructures (which can be spatially varying). Such two-scale optimization bridges the underlying scales with less expansive effort while capturing relevant microstructural effects. However, the optimization process considers the microstructure effect exclusively by its effective (homogenized) behavior. Various previous studies have investigated the potential macro-micro design optimization [START_REF] Li | Topology optimization for concurrent design of structures with multipatch microstructures by level sets[END_REF][START_REF] Kato | Micro-macro concurrent topology optimization for nonlinear solids with a decoupling multiscale analysis[END_REF][START_REF] Groen | Homogenization-based topology optimization for high-resolution manufacturable microstructures[END_REF][START_REF] Zheng | Robust topology optimization for cellular composites with hybrid uncertainties[END_REF][START_REF] Wang | Multiscale reliability-based topology optimization methodology for truss-like microstructures with unknown-but-bounded uncertainties[END_REF][START_REF] Gaynor | Multiple-material topology optimization of compliant mechanisms created via polyjet three-dimensional printing[END_REF][START_REF] Sanders | Multi-material continuum topology optimization with arbitrary volume and mass constraints[END_REF][START_REF] Zhang | Multimaterial topology optimization with multiple volume constraints: Combining the zpr update with a ground-structure algorithm to select a single material per overlapping set[END_REF][START_REF] Tozoni | A low-parametric rhombic microstructure family for irregular lattices[END_REF][START_REF] Lu | Structural optimization of metamaterials based on periodic surface modeling[END_REF]], and most of the corresponding papers are based on scale separation assumptions and periodic homogenization theory.

In order to accelerate multi-scale processes, approaches merging data-driven techniques have recently been developed to accelerate multi-scale topology optimization solvers [START_REF] Ferrer | Two-scale topology optimization in computational material design: An integrated approach[END_REF][START_REF] Watts | Simple, accurate surrogate models of the elastic response of three-dimensional open truss micro-architectures with applications to multiscale topology design[END_REF][START_REF] Kumar | Inverse-designed spinodoid metamaterials[END_REF][START_REF] Zheng | Data-driven topology optimization of spinodoid metamaterials with seamlessly tunable anisotropy[END_REF][START_REF] Djourachkovitch | Multiscale topology optimization of 3d structures: A micro-architectured materials database assisted strategy[END_REF].

The main explorations deal with the framework of the Cauchy theory, and the methods are thus suitable for large-scale transitions but limited to exhibiting the microstructure size effects. Moreover, such first-order frameworks fail when the scale separation assumption does not prevail (e.g., when topology cells are kinematically coupled as in some compliant mechanisms). Enriched kinematic homogenization schemes are needed to efficiently design architected materials and capture local microstructural effects, mutual interactions, non-localities, or instabilities.

One of the first higher-order theories was proposed by the Cosserat brothers [START_REF] Cosserat | Théorie des corps déformables[END_REF]. They enriched the simple Cauchy model with an independent rotation field gradient. This way, the model had three displacement and three rotational degrees of freedom (in 2D: two displacements DoFs and one rotation DoF). This theory was the first to define couple stresses and to render the Cauchy stress tensor non-symmetric. Since its first introduction, Toupin [START_REF] Toupin | Elastic materials with couple-stresses[END_REF] formulated the energy density function using the classic displacement and rotation gradient terms. Finally, Mindlin [START_REF] Mindlin | Microstructure in linear elasticity[END_REF] gave the linearized mathematical theory, serving as the basis of the variational solution.

Since its first introduction, the Cosserat theory has been used in numerous fields such as granular materials [START_REF] Li | A micro-macro homogenization approach for discrete particle assemblycosserat continuum modeling of granular materials[END_REF], masonry structures [START_REF] Addessi | Cosserat model for periodic masonry deduced by nonlinear homogenization[END_REF], composites [START_REF] Lakes | Experimental Micro Mechanics Methods for Conventional and Negative Poisson's Ratio Cellular Solids as Cosserat Continua[END_REF], or even human bone [START_REF] Lakes | Cement line motion in bone[END_REF][START_REF] Park | Cosserat micromechanics of human bone: strain redistribution by a hydration sensitive constituent[END_REF]. However, most importantly, it was shown that Cosserat elasticity efficiently captures the effect of the intrinsic length scale in cellular structures [START_REF] Rueger | Experimental cosserat elasticity in open-cell polymer foam[END_REF]. The constitutive equations of the model can be written in several forms ranging from a single added constant to Hooke's law [START_REF] Aifantis | On the microstructural origin of certain inelastic models[END_REF] to the entirely redefined stiffness matrix [START_REF] Zhang | Analysis of Cosserat materials with Voronoi cell finite element method and parametric variational principle[END_REF]. However, one of the main disadvantages of the Cosserat theory is that is has too many material parameters.

Bottom-up homogenization methods are usually applied to determine the elastic Cosserat constants of lattice structures.

Two main approaches exist: (i) the micro-scale is represented by an inhomogeneous Cauchy continuum [START_REF] Forest | Cosserat overall modeling of heterogeneous materials[END_REF][START_REF] Forest | Mechanics of generalized continua: construction by homogenizaton[END_REF][START_REF] Forest | Asymptotic analysis of heterogeneous cosserat media[END_REF][START_REF] Forest | Homogenization methods and mechanics of generalized continua-Part 2[END_REF], and (ii) the lattices are modeled with either Euler-Bernoulli [START_REF] Pradel | Cosserat modelling of elastic periodic lattice structures[END_REF][START_REF] Sab | Homogenisation of periodic cosserat media[END_REF][START_REF] Reis | Construction of micropolar continua from the asymptotic homogenization of beam lattices[END_REF] or Timoshenko-Ehrenfest beams [START_REF] Liebenstein | Determining cosserat constants of 2D cellular solids from beam models[END_REF]. The important difference is that in the latter case, the rotational degrees of freedom are already present at the microscale. On the other hand, with beam theory, the solid volume fraction cannot be represented. This paper focuses on analytical results obtained using the asymptotic method on Euler-Bernoulli beams [START_REF] Pradel | Cosserat modelling of elastic periodic lattice structures[END_REF][START_REF] Sab | Homogenisation of periodic cosserat media[END_REF], and the limitations of the model are discussed.

In recent years, various studies have been devoted to obtaining the optimal topology with a characteristic length scale by using the Cosserat theory. Most works have focused on fictitious materials with only a few elastic constants.

Parameter studies have been conducted to investigate the effect of the intrinsic length [START_REF] Gei | Effect of internal length scale on optimal topologies for cosserat continua[END_REF][START_REF] Rovati | Optimal topologies for micropolar solids[END_REF][START_REF] Liu | Topology optimization of couple-stress material structures[END_REF][START_REF] Arimitsu | Optimal topologies in structural design of micropolar materials[END_REF][START_REF] Li | Topology optimization of structures with length-scale effects using elasticity with microstructure theory[END_REF], the fundamental eigenfrequencies [START_REF] Bruggi | Maximization of the fundamental eigenfrequency of micropolar solids through topology optimization[END_REF][START_REF] Su | Topology design for maximization of fundamental frequency of couple-stress continuum[END_REF][START_REF] Su | Size-dependent microstructure design for maximal fundamental frequencies of structures[END_REF], the Cosserat shear modulus [START_REF] Chen | Parameterized level set method for structural topology optimization based on the cosserat elasticity[END_REF], and even 3D non-centrosymmetricity [START_REF] Veber | Topology optimization of three-dimensional non-centrosymmetric micropolar bodies[END_REF]. A majority of the investigations were based on the use of the SIMP and the level set methods to obtain the results.

Among the various models proposed in the literature, second gradient elasticity (such as a Cosserat medium) and micromorphic [START_REF] Yvonnet | Computational second-order homogenization of materials with effective anisotropic strain-gradient behavior[END_REF][START_REF] Rokoš | Micromorphic computational homogenization for mechanical metamaterials with patterning fluctuation fields[END_REF] computational homogenization methods seem to be adequate tools to be integrated into topology optimization simulations. Indeed, encouraging recent results for bottom-up multiscale modeling of microstructural materials with enriched kinematics revive their use in design optimization. However, this topic remains an open research field [START_REF] Calisti | Sensitivity of the second order homogenized elasticity tensor to topological microstructural change[END_REF].

Most studies focusing on the topology optimization of the Cosserat medium use fictitious material parameters, whereas papers presenting the optimization of lattice structures calculate only the Cauchy coefficients [START_REF] Watts | Simple, accurate surrogate models of the elastic response of three-dimensional open truss micro-architectures with applications to multiscale topology design[END_REF]. A comparative analysis using real microstructures and a homogenized second-order continuum has yet to be performed.

In the present paper, we present an new method to optimize the topology of slender lattice structures with small local volume fractions. We compare the optimal topology of discrete Euler-Bernoulli beam lattices to topologies obtained using the Cosserat theory. The local stiffness is calculated based on real geometrical properties, such as the beam height or the slenderness. We finally highlight the importance of the enriched model in the optimization of lattice structures and present optimal microstructures for a variety of macroscopic mechanical problems.

The paper is structured as follows. First, section 2 introduces the basic concept of topology optimization, after which section 3 lays out the different mechanical descriptions. Subsequently section 4 discusses the effect of both numerical and lattice parameters on the optimal topology. This is followed by a discussion on the optimal microstructure in section 5.

Finally, section 6 concludes the paper.

Topology optimization

Topology optimization is a design method in which the material is distributed inhomogeneously in the design domain to maximize a specific property. We chose to use the algorithm published by Sigmund [START_REF] Sigmund | A 99 line topology optimization code written in matlab[END_REF] and replaced the arbitrary design variable with the height of the beams. Among its advantages, the modified algorithm is easy to implement, robust, and converges rapidly. Despite that Bendsøe and Sigmund [START_REF] Bendsøe | Material interpolation schemes in topology optimization[END_REF] showed that, for the SIMP method, the cubic penalty function is physically permissible, we decided to use the constitutive analytic relations. Based on the Bernoulli beam theory, the Cosserat stiffness constants used in this paper were deduced from asymptotic approaches [START_REF] Pradel | Cosserat modelling of elastic periodic lattice structures[END_REF][START_REF] Sab | Homogenisation of periodic cosserat media[END_REF]. In addition, constraints on the maximum height and volume ratios are discussed in section 3.

The overall aim of the topology optimization procedure is to keep the compliance of the model at a minimum by minimizing the potential energy as a function of the local design variables:

h = Arg inf h c (h) = 1 2 U T KU = 1 2 N e=1 u T e k e (h e ) u e . (1)
This was done by respecting the equilibrium constraint:

KU = F, (2)
and the volume constraint:

V (h) V 0 = f, (3)
with a lower and an upper limit on the design variables (beam heights):

0 < h min ≤ h e ≤ h max . (4)
In eq. ( 1), c is the total potential energy, U and u e are the vectors of the global and elementary degrees of freedom (translation, rotation), K and k e are the global end elementary stiffness matrices, N is the number of finite elements in the design domain, and h is the vector containing the elementary design variables (h e ). In our case, this is the height of the beam cross sections. In this paper, we assume rectangular beam sections with a unitary thickness in the out-of-plane (z) direction. In the equilibrium equation ( 2), F is the global force vector, V is the total volume of the model, V 0 is the volume of the design domain and f is called the relative density or prescribed volume fraction.

The optimization problem is formulated based on the algorithm proposed by Sigmund [START_REF] Sigmund | A 99 line topology optimization code written in matlab[END_REF]. Apart from that work, the penalty exponent was replaced by the exact stiffness as a function of the beam height (the design variable); as in lattice structures, this variable comes with a physical meaning. The optimality criteria method [START_REF] Bendsøe | Optimization of structural topology, shape, and material[END_REF] with a sensitivity filter [START_REF] Sigmund | On the design of compliant mechanisms using topology optimization[END_REF] was utilized to update the design variables for a given equilibrium problem.

As shown in Fig. 1, we kept the original structure of Ref. [START_REF] Sigmund | A 99 line topology optimization code written in matlab[END_REF]. However, we replaced each block with our own features.

As the paper discusses different mechanical descriptions (Cauchy, Cosserat, Euler-Bernoulli beam) for the same mechanical problem, these blocks varied accordingly.

First, the finite element model was generated, after which the first equilibrium was obtained. The analytic solutions were applied to calculate the stiffness, energy, and sensitivity functions. For the Cosserat medium, the results of references [START_REF] Pradel | Cosserat modelling of elastic periodic lattice structures[END_REF] and [START_REF] Sab | Homogenisation of periodic cosserat media[END_REF] were utilized. Finally, the filter was applied directly to the array containing the sensitivity functions, as initially done by Sigmund [START_REF] Sigmund | A 99 line topology optimization code written in matlab[END_REF] where rmin was the effective filter distance. Next, the design variables were updated iteratively to satisfy the volume constraint. Finally, the convergence was verified: if the maximum change in the design variable was smaller than (h max -h min )/1000, the algorithm was stopped. Otherwise, a new equilibrium was determined with the new design variable distribution.

Mechanical description

This section briefly summarizes the basic mechanical descriptions used in the paper. All simulations assumed a static equilibrium, wherefore, each model was governed by a set of equations describing equilibrium, the kinetic constraints, and the material model.

Cauchy continuum

In continuum mechanics, probably one of the most frequently used assumptions is proposed by Cauchy [START_REF] Cauchy | Recherches sur l'équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques[END_REF]. The equilibrium of the differential object is depicted in Fig. 2(a) and can be described with the following set of equations: ∇σ

+ b = 0 in Ω, σ • n = t on Γ N , u = ū on Γ D , (5)
where σ is the stress tensor, and b represents body forces (in our case zero). The second and third rows describe respectively Neumann and Dirichlet boundary conditions. The bar symbol represents external forces ( t), and prescribed displacements (ū).

The material is considered to be linear elastic:

σ = Cε, ( 6
)
where C is the stiffness matrix, and ε the strain tensor. Due to the local equilibrium (σ xy = σ yx = τ xy ) the Voight notation can be used, thus simplifying the stress tensor to a vector with 3 components in 2D: σ = σ x σ y τ xy T .

Consequently, C becomes a 3 × 3 matrix. The strain tensor is also simplified to the three elementary deformations shown in Fig. 2(a). Assuming small deformations, the displacement-strain relationship can be expressed as:

ε x = ∂u x ∂x , ε y = ∂u y ∂y , γ xy = ∂u x ∂y + ∂u y ∂x . (7)
To solve the mechanical problem, the following energy is minimized with respect to the degrees of freedom by following the kinematically admissible set given in eq. ( 5) 3 . In this first case, the displacements are obtained by solving the following minimization problem:

u = Arg inf u    Ω 1 2 ε T Cε -b • udΩ - Γ t • udΓ    . (8)

Euler-Bernoulli model

To model slender beam structures, the Euler-Bernoulli beam theory provides the most approachable framework. The classical beam theory is adapted to describe the behavior of elongated load-bearing elements with one side significantly larger than the others. The three primary assumptions that have to be made are that (i) the plane sections remain plane, (ii) the plane sections are perpendicular to the neutral axis, and (iii) the deformed beam angles are small. Thus, the equilibrium shown in Fig. 2(b) can be described by the following equations:

d 2 M dt 2 + p n = 0, dN dt + p t = 0. (9)
Here, M and N represent the bending moment and the normal force, and p n and p t are distributed loads perpendicular and parallel to the neutral axis. These equations are completed with Neumann and Dirichlet boundary conditions similar to that of eq. ( 5) 2,3 . The Euler-Bernoulli beam theory neglects the effect of shear deformation, and as a result, it is adapted to describe the response of slender beams. The linear elastic response can be obtained by:

M = EIκ t , N = EAε t , (10)
where κ t and ε t are the two types of deformations present beams can be subjected to: (i) curvature/bending and (ii) elongation/compression. E is Young's modulus, I is the bending moment of inertia around axis z, and A is the area of the cross-section.

To obtain the relationship between the degrees of freedom and the deformations, the following assumptions are used:

κ t = ∂ϕ ∂t = ∂ 2 u n ∂t 2 , ε t = ∂u t ∂t , ( 11
)
where ϕ is the rotation, u n is the perpendicular displacement, and u t is the parallel displacement in relation to the neutral axis.

To obtain equilibrium, the following energy functional is minimized by following the kinematically admissible set:

(u, ϕ) = Arg inf u,ϕ    L 1 2 ε 2 t EA + κ 2 t EI dL -p • udL -t • u -M • ϕ    . (12)
The bar symbol represents external forces ( t), and moments ( M ).

Cosserat continuum

The classical description of continuum mechanics is ill-suited to characterize the response of materials with an inhomogeneous microstructure, i.e., a characteristic microscopic length scale. However, the mechanical behavior of architected materials (e.g., lattice structures) is often determined by their specific micro-scale configurations. Therefore, the Cosserat theory (or micropolar elasticity) incorporates rotational degrees of freedom (ϕ) into the mechanical description.

The Cauchy model is completed with an additional set of equations describing momentum equilibrium as shown in Fig. 2(c):

∇σ + b = 0 in Ω, ∇µ + σε = 0 in Ω, σ • n = t on Γ N , µ • n = M on Γ N , u = ū on Γ D , ϕ = φ on Γ D . (13)
In this equation, σ is now a non-symmetric (σ xy ̸ = σ yx ) force-stress tensor, µ is the moment or couple-stress tensor, and ε is the Levi-Civita symbol. The bar symbol represents external forces ( t), moments ( M ), prescribed displacements (ū), and rotations ( φ).

The literature [START_REF] Forest | Asymptotic analysis of heterogeneous cosserat media[END_REF] recounts various ways to define linear elastic behavior. In this paper, we chose to correlate the complete stress tensor to the deformation components using the following model: As a result, the first three elements of the stress vector correspond to the Cauchy stress components, the fourth element provides the difference between σ xy and σ yx , finally, the last components are the couple stresses.

        σ x σ y τ xy = σxy+σyx 2 ϑ xy = σxy-σyx 2 µ x µ y         =   [C] 3×3 0 0 0 G c 0 0 0 [D] 2×2           ε x ε y γ xy ω xy κ x κ y         . ( 14
)
The corresponding deformations are depicted in Fig. 2(c), with the missing kinematic constraints described by the following:

ω xy = ∂u x ∂y - ∂u y ∂x -2ϕ, κ x = ∂ϕ x ∂x , κ y = ∂ϕ y ∂y . ( 15
)
The equilibrium is then obtained by minimizing the following functional equations:

(u, ϕ) = Arg inf u,ϕ    Ω 1 2 ε T Cε + ω 2 xy G c + κ T Dκ -b • udΩ - Γ t • u + m • ϕdΓ    . (16)
By incorporating rotational degrees of freedom into the continuum description, the Cosserat theory proposes an analogous description to the discrete beam model with a significantly lower computational cost.

Numerical examples

Fig. 3(a) depicts the geometry of the optimization problem. In this section, a cantilever geometry of length (L) 100 and height (H) 50 was chosen. The concentrated force (F ) applied at the end was 10 -3 unless otherwise specified. This paper presents the model and the results in dimensionless quantities. Young's modulus was set to 1. Both translational and rotational degrees of freedom were constrained on the nodes on the left side.

For a beam structure, the volume constraint defined in eq. ( 3) can be expressed by summing up all the individual elements and dividing the sum by the overall area of the design domain:

f = V (h) V 0 = N e=1 h e l e LH , (17)
where V 0 is the total volume of the design domain, l e represents the individual length of the beams, and h e is their height. There are N beams in the structure. In this case, h e , The design variable has a geometrical significance wherefore its bounds can be defined as:

0 < h min ≤ h e ≤ h max , (18)
with h min as a lower bound responsible for numerical stability (h min = h max /5000), and h max as an upper bound:

h max ≤ V 0 N i=1 l i . ( 19
)
Lattice type The characteristic quantities, upper bounds, and the correlation between local volume fraction and beam height are summarized in Table 1.

l x l y l e V RVE h (f RVE ) f RVE (h) f RVE (l e /h = 5) Rectangular l m l m 2l m l 2 m lmfRVE 2 2h lm 0.4 Honeycomb √ 3l m 3l m 6l m 3 √ 3l 2 m √ 3 2 l m f RVE 2h √ 3lm 2 5 √ 3 = 0.23 Reinforced honeycomb √ 3l m 3l m l m 6 + 6 √ 3 3 √ 3l 2 m √ 3lmfRVE 2+2 √ 3 h lm 2+2 √ 3 √ 3 2+2 √ 3 5 √ 3 = 0.63
When using h max , the RVE should be considered solid (f RVE = 1). This limit is unsuitable for beam theory, and consequently, in the following examples, h max is set to l e /5, at which value the Euler-Bernoulli beam theory still gives a fairly precise outcome (the difference to the Timoshenko-Ehrenfest theory is 3.06%).

Part (b) of Fig. 3 shows the optimization results of a rectangular lattice modeled using the Cauchy continuum (with SIMP) and Euler-Bernoulli beams. When employing the Cauchy description, the design variable was penalized using a cubic function, and the maximum value of the design variable was limited to 0.4, which corresponds to the volume ratio of the rectangular grid RVE when h e = l m /5. It can be clearly seen that Cauchy's description with the SIMP technique is unable to correctly determine the optimal topology.

In the present paper, we discuss three microstructures: square, hexagonal, and reinforced honeycomb lattices. The reason for this is that the analytic Cosserat stiffness components for these structures are available in the literature [START_REF] Pradel | Cosserat modelling of elastic periodic lattice structures[END_REF][START_REF] Sab | Homogenisation of periodic cosserat media[END_REF].

Tables 2 and3 The aim of section 4 is to prove that the Cosserat medium is necessary and sufficient to optimize the topology of light and slender beam lattices. Therefore, each material and numerical parameter was tested using the Euler-Bernoulli beam and the Cosserat theory, after which the optimal topology and the final displacement (u F y ) measured at the concentrated force F were compared. It should be noted, that the objective function (potential energy) was equal to c = F u F y 2, and Lattice type

K s G s G c D 11 , D 22 √ 3 6
Eth lm

Eth 3 √ 3lm(l 2 m +h 2 ) Eth 3 2 √ 3l 3 m Eth 3 12 √ 3lm Honeycomb 3+ √ 3 6

Eth lm

t 2 h 6 3 + 4 3 ( √ 3+1)t 2 h 4 l 2 m +l 4 m t 4 h 4 4l 3 m (l 2 m th+th 3 ) E 3+ √ 3 6 √ 3
Eth 3 l 3 m 1+ √ 3 12 √ 3
Eth 3 lm the potential energy was thus not an independent measure of optimality.

Reinforced honeycomb

Square lattice

The homogenized stiffness of a rectangular grid lattice is orthotropic. The two principal orientations are parallel to the two main beam directions. Due to the lack of normal interaction, C 12 = 0, which means that extension in one direction causes no contraction in the other. This property signifies that the homogenized material's Poisson's ratio is zero (ν = 0).

The rectangular beam lattice was used to demonstrate the effect of the volume fractions on the optimal topology. We distinguish between local f RVE and global f values, where f RVE represents the allowed maximum local volume fraction, while f is the global volume fraction defined in eq. ( 17). These two quantities are independent, with f ≤ f RVE , and f RVE is mostly controlled by prescribing a lower beam height in the model based on the correlation shown in Table 1. Due to the difference in local geometry, various lattice structures have different f RVE for the same l e /h e ratio.

The beam model was constructed from 300 × 150 RVEs. The beam segments were unified, which resulted in 90 450 beam finite elements with l m = 1/3. The load was applied on a node situated on the bottom line furthest to the right.

The Cosserat model was divided into 300 × 150 4-node quadrilateral elements. We used 45000 elements to capture similar topology details as in the beam model. The advantage of the Cosserat model is that l m reduces to a material parameter, and the element size does not affect it. The material and the boundary conditions can be found in the main part of Section 4. A filter of r min = 1 was used in both cases.

The first two rows of Fig. 4(a) and (b) show the optimal topology for two global volume fractions obtained using a beam model (left) and the Cosserat continuum (right). The third row portrays the change if thicker beams are allowed.

The results displayed in the two columns are almost identical. However, they differed significantly from the outcome obtained using the Cauchy continuum and the SIMP technique shown in Fig. 3(b).

Similar to the topologies, the displacement results were also in good correspondence. Fig. 4(c) shows the effect of the local slenderness. The maximum difference between the deflection values was 2.16%, which was within the precision of the built models. Furthermore, we observed a tendency to favor thicker beams. When f RVE increased, the maximum deflection decreased. Without a global stability analysis, the optimal topology favors thinner elements with higher local volume fractions. When h max was reduced, the size of the global elements grew, raising the moment of inertia but lowering the leverage-resulting in a less rigid structure. This effect was demonstrated but not discussed by Watts [START_REF] Watts | Simple, accurate surrogate models of the elastic response of three-dimensional open truss micro-architectures with applications to multiscale topology design[END_REF], where the optimal topology was fairly independent of the chosen structure, and local maxima gave mostly solid parts (with f RVE = 1).

Of course, this solution defeats the purpose of creating an architected material.

Finally, Fig. 4(d) presents a plot of the deflection results as a function of the global volume fraction with h max = l m /5

(f RVE = 0.4). The results were fitted perfectly using a hyperbolic function. Moreover, the maximum difference was 3.8%, which demonstrated how well the Cosserat theory reproduced the mechanical response of these periodic beam structures.

Honeycomb lattice

In contrast, the homogenization of honeycomb lattices gives an isotropic response with a Poison's ratio depending on the slenderness varying between 0.3-0.5 (for l e /h e = 5, it is 0.46). Therefore, as shown in Fig. 5, the optimal topologies are much closer to the results produced by SIMP. This section thus describes the use of the honeycomb structure to demonstrate the effect of the filter width (r˙min on the topology and the maximum deflection at the concentrated force as a measure of optimality.

The beam model was constructed from 300 × 87 RVEs. The beam segments were unified, which resulted in 156 387 beam finite elements with l m = 0.192. The load was applied on a node situated on the bottom line furthest to the right.

The Cosserat model was divided into 300×150 4-node quadrilateral elements (45000 elements), similarly to the rectangular case. The material and the boundary conditions can be found in the main part of Section 4. The filter width was varied through the analysis.

Fig. 5(a) and (b) show the results obtained using Euler-Bernoulli beams and the Cosserat continuum, respectively. An agreement was observed, highlighting the Cosserat theory's versatile potential. When the filtering was switched off, the two theories started to deviate, as shown in Fig. 5(d). However, it is our opinion that these results bear little practical importance.

Finally, we plotted the maximum deflection in Fig. 5(c). Introducing the filter reduced the optimality and the overall stiffness of the structure. By increasing the filter width, the displacement increased linearly. However, when r m in was smaller than the shortest distance between the beam elements, the filter could no longer complete its task, and the results started to fluctuate. Until that point, the maximum difference between the two methods was 1.51%.

Reinforced honeycomb lattice

Our third example focuses on the effect of the characteristic micro length scale on the reinforced honeycomb network.

In this case, the classic hexagonal structure was fortified with a triangular grid. While the hexagonal structure was mostly a bending-dominated lattice, triangular cells provided normal rigidity. The homogenized Cauchy stiffness was isotropic with a Poisson's ratio varying between 0.3-0.34 (for l e /h e = 5, it is 0.325).

Different beam structures were created to test the effect of the micro-length. However, only whole RVEs were used to create the beam models. Due to the irrational size of the RVE, not many configurations had precisely the same size.

Therefore, the following sample sizes were considered: 142 × 41, 239 × 69, 284 × 82, 336 and 426 × 123, which corresponded to l m = 0.4, 0.24, 0.2, 0.17 and 0.14, respectively. The advantage of the Cosserat model is that there is no need to change the mesh for a different l m value, as it is only a material parameter. However, both models were tested with a mesh following the change of l m and one independent constant of 300 × 150 elements.

The finite element mesh was generated from 4 node quadrilateral elements. The material and the Dirichlet boundary conditions can be found in the main part of Section 4. A filter of r min = 1 was used in both cases.

After initial testing, the results showed that the concentrated force and the changing mesh had more significant effects on the deflection than l m . Therefore, a model was developed to distribute the loading on a unit length. Practically, the force was divided and applied on the beam and the continuum model as shown in Fig. 6. as a function of l m . Its effect was found to be much smaller than the increased deflection originating from the singularity.

When the load was distributed, a small increase in rigidity was observed in favor of larger motifs.

Discussion

In Section 4 we demonstrated that the Cosserat description was sufficient not only qualitatively but also quantitatively to describe the mechanical behavior of beam lattice structures. Moreover, the response as well as the optimal topologies was found to be in agreement.

Here, we take a step back to understand the critical components of the Cosserat description, ultimately investigating different structures (design domains) with varying L/H ratios to identify an "ideal" microstructure. Fig. 7(a) presents, the optimal topology using the SIMP method with a ν = 0 material and a cubic penalty. In this first case, no limit was applied to the local volume ratio (f RVE = 1), and as a result, the final topology differed significantly from the results obtained as compared to the beam lattice. Furthermore, the deflection was much smaller because of the higher local volume ratio. As a result, the SIMP method with an isotropic material was inadequate to model slender lattice structures. In part (b) the maximum stiffness was calculated with f RVE = 0.4 based on Tables 1 and2. To scale between empty and solid states, a cubic penalty was applied. This modification was the first step when the form started to resemble the results obtained with the beam model. The global form was recovered, however, the details remained dissimilar. Part (c) portrays the results when the penalty was omitted, and the stiffness was calculated based on 2. The details started to emerge, however, the deflection values overestimated the beam results (u F y = 2.5) by 7.2%. Finally, part (d) presents the optimal topology obtained using the Cosserat theory. The details were fully recovered with a 1.2% difference in the deflation value.

The introduction of the rotational degree of freedom and the Cosserat theory clearly helped recover the finer details and the precise mechanical response of the underlying beam structures. Compared to the classical continuum, the finite 2 and3. The maximum displacement for the reference beam model at the concentrated load was u F y = -2.50 (see topology result in Fig. 3). element implementation of the theory was not much more complicated. The only difficulty lies in determining the Cosserat constants for arbitrary beam structures. However, this can be done using various homogenization methods [START_REF] Forest | Cosserat overall modeling of heterogeneous materials[END_REF][START_REF] Kouznetsova | Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme[END_REF][START_REF] Liebenstein | Determining cosserat constants of 2D cellular solids from beam models[END_REF].

The Cosserat optimization code used in this paper is included in Supplementary Materials.

The aspect ratio of the design space was varied to compare the three lattice structures. We were interested in determining whether an optimal microstructure could be found as a function of the geometry. According to Fig. 4(c), the locally allowed maximum volume fraction had a significant effect on the deflection results. Therefore, in Fig. 8(a) we first compared results using the same local filling ratio, f RVE = 0.23. Interestingly, there was a competition between the honeycomb and the rectangular grid. However, the reinforced honeycomb was consistently more rigid. When the local volume fraction of the rectangular grid was increased to 0.4, it became superior to the honeycomb, even in the initial interval. The results obtained using the Cosserat theory repeatedly corresponded well to the beam models. It could thus be stated that, depending on the design problem, the optimal RVE can be appropriately chosen using Cosserat elasticity.

Conclusion

This paper presents a topology optimization algorithm based on the Cosserat theory. The continuum results were compared and verified using Euler-Bernoulli beam models, and the stiffness of the enriched medium was determined from analytic calculations.

We observed an excellent quantitative correspondence between continuum Cosserat and discrete beam results. We showed that the Cosserat theory was necessary and sufficient to optimize slender, lightweight designs with beam microstructures and that the maximum local volume ratio (thus the maximum beam height) significantly affected the optimal deflection values. The effect of the filter width and the local length scale were also tested.

The addition of the rotational degree of freedom allowed us to capture the equivalent behavior of the beam theory.

The advantage of the Cosserat model is that l m reduces to a material parameter wherefore, the element size does not

Figure 1 :

 1 Figure 1: Basic structure of the topology optimization algorithm.

Figure 2 :

 2 Figure 2: Equilibrium and admissible deformations of the elementary volume of the (a) Cauchy, (b) Euler-Bernoulli beam, and (c) Cosserat theories.

Figure 3 :

 3 Figure 3: Optimized topology of a cantilever: (a) geometry (L = 100, H = 50), (b) optimal topology using the Cauchy and the beam model for f = 0.12 (r min = 1).

Lattice type C 11 ,Table 2 :

 112 C 22 C 12 , C 21 C 33 G c D 11 , Cosserat constants for the rectangular beam lattice.

  summarize each constant of the matrices C, D and the component G c . The remaining components of the matrices were zero.

Table 3 :

 3 Cosserat constants for honeycomb and reinforced honeycomb beam lattices. C 11 = C 22 = Ks + Gs, C 12 = C 21 = Ks -Gs, C 33 = Gs.

Figure 4 :

 4 Figure 4: Topology and deflection results on a rectangular grid as a function of local and global volume fractions.

Figure 5 :

 5 Figure 5: Topology and deflection results on a honeycomb structure as a function of the filter width.

Figure 6 :

 6 Figure 6: Topology and deflection results on a reinforced honeycomb structure as a function of lm. The results for the concentrated force are shown with solid symbols, whereas for the empty ones, the load was distributed on a unit length.

Fig. 6 (

 6 Fig. 6(a) and (b) show that neither l m nor the Neumann boundary affected the optimal topology. Both the beam and Cosserat models gave a solid-like outer shell (with f RVE ) and a less filled interior. Fig. 6(c) displays the deflection results

Fig. 7

 7 Fig. 7 shows the optimization problem introduced in Section 4. The elementary grid is a rectangular lattice with 300 × 150 RVEs, and the global volume fraction was set to f = 0.12 with an r min = 1 filter width. The results were compared with those of the beam model shown Fig. 3(b).

Figure 7 :

 7 Figure 7: Different approximations of the rectangular grid with the global volume ratio of f = 0.12: (a) isotropic Cauchy model with a maximum of f RVE = 1 and a cubic penalty; (b) Cauchy model with a maximum of f RVE = 0.4 and a cubic penalty; (c) Cauchy model with a maximum of f RVE = 0.4 based on the beam heights; (d) full Cosserat description. The rigidity was calculated based on Tables2 and 3. The maximum displacement for the reference beam model at the concentrated load was u F y = -2.50 (see topology result in Fig.3).

Figure 8 :

 8 Figure 8: Maximum displacement as a function of the design space's aspect ratio for a global volume ratio of f = 0.075 and lm = 1/3: (a) with f RV E = 0.23 for all lattice geometries; (b) with their maximum volume ratios by respecting le/he ≥ 5.

Table 1 :

 1 Lattice geometries and correlation between volume ratio and beam height.

affect the microstructure. Consequently, the calculations can be significantly accelerated compared to beam models.

While the finite element implementation of the Cosserat theory is relatively simple, the stiffness components are available analytically only for a few periodic microstructures. Consequently, developing and publishing a universal homogenization algorithm is crucial in order to establish a database with various lattice structures used in practice. Furthermore, the algorithm should be employed to optimize the beam heights and, for example, the microscopic length scale.


Florence Padovani
email: florence.padovani@beijing-cfc.org

When a marginal area is transformed into a touristic high spot: Tianzifang in Shanghai

Robert Park's concept of Marginal Man has a great longevity. In 2012 Chad Alan Goldgerg explained that it is still relevant today. He underlines that marginality suggests a potential maladjustment and disorganization but at the same time creativity and innovation. Today the notion of marginality is extended to a broad range of social phenomena well beyond race and ethnic relations, which were the main points developed by Robert Park when he published his article in 1928. This paper analyses different forms of marginality through the case of Tianzifang community in downtown Shanghai. Different actors are considered legitimate to raise their voices in favour or against safeguarding this area. In addition to residents, some administrative cadre, developers, academics and artists played an important role in giving birth to Tianzifang. We will analyse how these actors confident in their legitimacy negotiate together. The link between marginality and legitimacy is a key to understand this particular context. Marginality is not only about human beings (their origin and status), it is also about buildings and heritage. Tianzifang is marginal in many ways and I would argue this saved it from destruction. A Chinese proverb states that the tallest the tree the more chances it has to be cut. Benefits can be taken out of marginality This is how being a small area quite dilapidated made it possible to create a new model of urban development at a micro level.

Description of Tianzifang and Shanghai's Lilong

The case study of Tianzifang(田子坊)focuses on a small community in the heart of Shanghai. It is localized in the former French concession where many old buildings remain (dating from the end of the 19 th century to the beginning of the 20 th ) and whose popularity with tourists is unquestionable. The analysis is based on long-term fieldwork developed through participant observation and interviews from 2005 to 2012 1 . It also relies on official documents and articles published in Chinese and English.

Tianzifang is close to the city centre. It is under the authority of the Luwan district. The main entrance is located on the 210 Taikang road but several other entrances were opened during the development of the area. All of them can be spotted on the figure 1 above. The 1 For this paper I used twenty five interviews, five with people from the administration and twenty with people living in Tianzifang. community of Tianzifang is squeezed in between Jianguo Zhong Road to the north, a relatively narrow one-way street with two lanes, lined with plane trees and usually busy, the small Taikang Road to the south, a one-way street that was not very popular before Tianzifang's success, Ruijin Nan Road to the west, an important thoroughfare (also one-way, with two lanes and plane tree-lined), and finally the small Sinan Road to the East, also oneway, and comparatively less busy. Sinan road is famous in the communist history has there is Sun Yat-Sen house as well as Zhou Enlai's house. Both are still a place of pilgrimage. Further south is Zhaojiabang road one of the municipality's main thruways with six lanes. It was an old waterway used in the past for the transportation of goods produced in the small factories that were bordering it. Now covered, it continues to play an important role in Shanghai's transport system.

In the late nineteenth century and early twentieth, along the Ruijin street were some residences and villas, along the Jianguo street were some offices and administrative department for the French representatives. On the southern part, was the river then called Zaochar Bang. On its banks were shanties and garbage yards. Between the shanties and the administrative buildings, the area was step by step filled with lilongs, among them was Tianzifang.

It was built during the last stage in the 1930s. It consists of a set of small three-floor buildings made of grey and red bricks and a dozen of houses called Shikumen 2 made of two to three floors. Its original size was 7.2 hectares. There was only one lane, but since the 2000's the perimeter has expanded first to the east and then westward, incorporating other lilong under the label Tianziang, as it can be seen on figure1.

As Demgenski underlines in his analysis of Qingdao's heritage preservation, there have been a different evolution in architectural style. It reflects in Shanghai and Qingdao an influence of Western architecture but also an influence of Chinese migrants bringing their own traditions plus the Foreign administration regulations as some foreign concessions 3 were established in Qingdao and Shanghai alike. All these influences modelled the Lilong in Shanghai and the Dabaodao in Qingdao (Demgenski 2018:232). These buildings were demolished, rebuilt, transformed, subdivided up to what we can see today.

2 They have a front yard protected from outside by a wall and high wooden gates. On each side and above there are craved stones which gave its name to this style of house. 3 At the end of the series of opium wars some unequal treaties as they are called were signed between Western countries (plus Japan) and China. This how France gained a concession in Shanghai in 1849 and Germany got one in Qingdao in 1898.

Due to political instability, wars, rebellions the situation was very instable in China at the end of the 19 th century and foreign concessions in big cities like Shanghai appeared as a safe refuge. With the rapid increase of population there was a housing crisis. Some housings where erected quickly in underdeveloped areas of foreign settlements in order to accommodate the influx of refugees. Until the mid-twentieth century people flowed in Shanghai according to the military and economic situation. Ho is even describing some housing which were merely grass huts (Ho 2010:39).

The lilongs were built for basic living standards. It is typical local architecture. "The word, as an abbreviation of Lilong housing neighborhood, is a rich concept that not only refers to the materiality of the dwelling form, but also the vivid social life within and around it, which can be characterized in at least three ways: 1/ the physical forms of housing (...) which had shared a similar neighborhood structure, being organized along small alleys; 2/ the social community; 3/ a particular dwelling culture created by the inhabitants" (Zhao C.L 2004:50-51).

There are different styles of Lilong architectural design according to the period when they were erected and the material used for construction. It changed a little but the important characteristics remained (Rutcosky 2007, Fan 2004).

The buildings are usually made of three floors. They are linked together through alleys and lanes leading to the main gates, which were guarded. This gave a feeling of safety to the inhabitants and allowed them to create strong social ties within a closed perimeter. As a matter of fact, the intensive use of commune areas and a vivid life in the alleys generated a strong sense of community4 .

In 1949 when the communists took power and founded the People's Republic of China, the lilong accounted for 69%5 of the urban dwelling (Fan W.B 2004). These constructions were stopped in 1950 as the new government changed the policy towards city dwellers. There was no more private property and the State was the only owner. Industries and businesses were nationalized as well as housing. The State was providing people working in the cities with accommodation for a very low rent. Part of Tianzifang was then converted into factories and workers were crammed into Lilong housing nearby.

Since the beginning, a variety of residents lived in Tianzifang. In addition to workers, they included minor officials, artists, and teachers who took advantage of the low prices of these buildings to move in. The organizational configuration of the lilong induced very important spatial as well as social constraints for their inhabitants. As noted above, it developed a strong feeling of belonging inside a group of people protected from outside. In the early twentieth century -one extended family could own one building (especially the Shikumen style). During the pre-communist period housing was relatively comfortable for an average size family, but after 1950 it was crowding when several families had to use the same space [START_REF] Laurans | Shanghai: Modern Conveniences as an Argument for Displacing Residents[END_REF].

Each building was re-divided creating small flats which were rented inexpensively to poor families and workers. The kitchen and the bathroom became common spaces. Moreover, there were no private toilets, so residents had to go to public toilets in the morning, carrying their chamber pots. Another consequence was the degradation of the premises as none of the residents was willing to take responsibility for upkeep of the buildings. In the 1950s, in addition to redefining the space used by individual buildings, several lanes in the lilong were demolished to make room for six small factories, which produced spare timepieces single parts, canned food, etc. The food factory, which housed a photo gallery, Deke-Erh from the 1990's until 2012 (Zhou P. 2012), has three floors, and the tallest building has six floors. At the turn of the 21 st century, because of its location in the historical centre of Shanghai, the area attracted property developers who wanted to benefit from the latest urban transformation policies. When private developers collaborating with local authorities took advantage of the opportunity to launch large-scale modernization projects in the district, Tianzifang was part of them.

Implementation of urban renovation in today's Shanghai

Urban renovation is quite a common phenomenon in major cities vying for international recognition. They all had to face similar problems in relation to urban transformation. The analysis of Istanbul's renovation is very interesting in that regard (Pérousse. J.F 2007). Just like in Shanghai, districts that underwent the largest renovations were located in the historical centre or its immediate outskirts. There too, urban policies were actively promoted by the authorities, who presented them as necessary for modernizing the city, enhancing its image, and improving the living conditions of the inhabitants (Berry-Chikhaoui. I 2007).

According to Shanghai municipality leaders, the main objective was to eliminate old industrial buildings, dilapidated housings to give a beautiful image of the city connected to the world. Their goal was to transform Shanghai so that it could compete with other global cities (Diglio. S 2006). As a consequence, the resident population changed and a phenomenon of gentrification followed the morphological transformations made to the city.

In this context of high-speed transformations, the issue of historical sites' preservation needs to be addressed. What is to be preserved? How to preserve? Although Shanghai was officially recognized by the Chinese central government as a "national historic city" in December 1986, concerns for historic preservation seemed quite weak 6 compared to the dominant modern architecture multiplying business centres, commercial malls etc. At this point preservation or conservation of cultural heritage in its original stage was not the main goal to be achieved.

Heritage should be understood as a process in perpetual change. As Ashworth points out "heritage is the usage of the past in the present and that new present will constantly imagine new pasts to satisfy changing needs." (Ashworth 2011-10). The municipality listed places to be preserved and succeeded to have heritage sites listed at the national level (Yue 2014). By 2006, Shanghai had already: 16 key state protective relics, 110 key city protective relics, 21 historic memorial spots and 15 protective sites, all together there are 400 buildings of historical value (Diglio. S 2006).

Like many countries in the West which went through urban development and modernisation of the cities, it is difficult to find the balance between growth and fast development on one hand and cultural heritage preservation on the other hand (Zukin 2009). An additional point might be that old buildings mainly go back from the concession period and one can wonder to what extend remembering this period of Chinese history is not too shameful (this idea is also shared by Diglio. S 2006). The ambiguity between the image of a dynamic international city even at an early stage that the Chinese government is promoting and the presence of western administration in the concessions which is sidelined make it complicated to preserve the remains of this period of history.

Lastly, the residents of old buildings have themselves a negative perception of these constructions or at least are very ambivalent, feeling they are old fashioned, not comfortable, too densely populated etc. None of the people I interviewed raised the issue that the buildings 6 Article 14 §1-2 "The development of new urban areas and the redevelopment of old urban areas must be integrated in a unified plan, must be planned and arranged rationally, must pursue a course of comprehensive development, and must give priority to the construction of auxiliary, complementary projects and of infrastructure works, and construction work must be carried out with a relative concentration of resources. The redevelopment of old urban areas must be integrated with the restructuring and redistribution of industries, must rationally readjust the use of land, must keep a check on the building of high-rises, must lower building density, must increase public green space, must help improve urban traffic, must further improve urban infrastructure, and must strengthen the multifunctionality of the city. (…)" Regulations of Shanghai Municipality on City Planning, 1995 they are living in could be considered as historical heritage. The problem in Shanghai, but it is also valid for other cities in China, is that the massive and rapid destruction, which happened in the 1990s, left small room for discussion and elaboration of a good protection plan (Demgenski 2018[START_REF] Ho | The transition Study of Postsocialist China -An Ethnographic Study of a Model Community[END_REF], Yin 2012, Wang and Li 2009, Evans 2014). Basically it is just a decade later that the issue raised concern and then academics started to publish a lot (Wu FL 2000, 2002[START_REF] Deljana | Borderland Urbanism: seeing between enclaves[END_REF]Ren.X.F 2008, Tsai W.L 2008, Yao. P 2009), just to quote the main Chinese authors who published in English.

To borrow the expression from Lu Duanfang's book: China today is remaking its urban form (Lu D.F 2006) and of course while doing so, social relations are also re-ordered while a process of gentrification is taking place. For sure, the expectations from the new group of residents are very different. While some old residents wanted a wet market nearby and some daily use shops, richer residents preferred some leisure activities, luxurious shops etc. After being centred on industrial production (during the Maoist era) Chinese cities are now connected to market economy (during the post-reform era). Modernization is the main motto.

The transformations that happened in Shanghai are the consequences of this new definition of the city.

Here appears a contradiction between what the central State (in Beijing) and even the Shanghai municipality's leaders aim at and what the other social actors dream of. Here we understand social actors as the people who act -or more often react -to policy that they are concerned by. This includes residents who don't want to move or who wish to stay in the same area for example. It includes also some groups like the artists in Tianzifang who decided to defend their working place. It also includes the no-voice people, the ones who have no space to speak, they are the migrants who come for a non-specified period of time and then are supposed to go back to their rural areas so they have no say. As a consequence, they are the marginalized one. In a nutshell, urbanization is closely linked to socialization and the creation of new identities (Dubar C, 2000). In Tianzifang, social actors made use of the urban planning to their own advantage and they readapted the plan in their own way of imagining sustainable development meaning that through daily activities they transformed what was decided from above. This way of analysing social actor is in line with Erving Goffman's work (Goffman E 1959). I underline a few elements out of his work. Individuals may have different roles in the play. They can change and make choices. The interaction with other actors is led by the sense of a common good to defend.

Two groups of people plus one saving Tianzifang

The three groups of people who were actors in Tianzifang transformation were the artists, the residents and the local administration. In addition, a fourth group was present but voiceless: the economic migrants.

All over China, at the end of the 1980s, unprofitable state-owned production units were closed. Some were then converted into galleries and artist studios and gradually some restaurants and souvenir shops got there. This phenomenon can be observed in Beijing (eg.798 district) but also in several locations in Shanghai like Red house, Suzhou Creek etc. (Wang 2009, Zheng 2011[START_REF] He | State-sponsored Gentrification Under Market Transition The Case of Shanghai[END_REF]. The six small factory work units in Tianzifang filed for bankruptcy in the 1990s. Soon after they were closed. Some artists were then able to rent from the municipality the former industrial sites at very low prices; here they installed their workshops. A community of painters, sculptors, photographers, designers slowly developed. Some of them had studied in the West and were aware of places like Soho, in New York City.

They considered this a model to be followed and they viewed Tianzifang as a "Chinese Soho". In Shanghai at that time, the idea of converting old factory buildings into artistic venues was brand new.

After 1991, the employees of the state-owned enterprises were given the opportunity to buy their flats at a discounted price. Most of Tianzifang's older residents, who chose to buy their flats, moved in during the 1950s they were workers or small shop owners; the majority of them had low revenue.

In addition to artists and the residents there was also the group of the economic migrants. In the Lilong, different people get alongside. The artists and the older residents did not have much in common and did not have many opportunities to interact in daily life 7 . In fact, the artists did not live in Tianzifang, but only worked there. They paid rent for their studios directly to a government office.

The residents' community in Tianzifang underwent deep transformations in the 1990's. At that time, long-time residents started to look for a new housing, with modern facilities. The ones who were rich enough left Tianzifang. Economic migrants then started to rent available rooms and establish themselves there. They were usually from nearby provinces (Anhui, Jiangsu, Zhejiang), and they came in the hope of earning a better living in Shanghai. The 7 MS Li (summer 2010) « When Chen Yifei died many famous people came to his studio and they wanted to see what Tianzifang looked like but nobody spoke to us, they stayed together. TV was there. (…) afterwards some artists came back here but it was their business. » residents and the economic migrants interacted on a daily basis. They jointly used the common facilities and buy their goods in the same wet market. Their income level was not very different, and their level of education being rather similar. In consequence, the two groups had more common interests than with the artists. The specificity of the economic migrants is that they are the "no-voice", the ones that are tolerated but who have no rights.

Although since 2003 the government abolished the system of forced repatriation, they were never given proper rights to stay in the cities (Zhao & Padovani 2014). So, they were not involved in the creation of Tianzifang even though they were living there during the time of the negotiations.

The group that we call the artists is made of different types of actors. Some of them have a certain degree of notoriety, like the aforementioned Huang Yongyu, or Chen Yifei, a renowned photographer, one of those with a long stay in North America where he discovered the practice of converting old factories into art galleries. His contribution was particularly significant for the artists that settled in Tianzifang. Some others were just attracted by low rents. May be due to their higher level of education, compared with the long-time residents, may be because they have been abroad and witnessed foreign artists using industrial ruins, may be because they made use of their fame to speak aloud; in any case the artists became the defenders of Tianzifang. All these elements gave them legitimacy to defend the preservation of Tianzifang. As Demgenski states in the case of Qingdao they are the "old-town protectors" (Demgenski 2016), being very skilful to make use of media and personal connections to pressure the administration to protect Tianzifang.

By 2004, an estimated 1,8 hectares of industrial space had been converted into studios, galleries, artists' studios or interior design offices, occupying almost all of the previously available factory space (Wang W.H 2011). From the start, Tianzifang stood out as different because artists, the residents and economic migrants coexisted in the same limited space. In other places where artists have settled in Shanghai, such as Suzhou Creek or Red House, there were often only abandoned industrial structures, thus no inhabitants. The relative harmonious relationship between artists, residents and entrepreneurs is part of the good image on which Shanghai leaders have tried to build. Moreover, there were no protests against eviction and no negative media coverage about the displacement of residents. So, harmonious Tianzifang was put forth as a model to be followed. However, as we will show in the next section the picture is not that rosy.

Fighting for the Preservation of Tianzifang

We just mentioned two groups of people -plus one-who tried to save Tianzifang. In addition, the three layers of Chinese administration the central, the municipal and the district level governments were involved in the fight for preservation. They were all interested in issues of renovation. The last group was made of private investors, responsible for financing renovation works. All the groups had their own agenda. The administration and the investors had to negotiate an agreement before any project could carry on. Developers obtained urban land use right and then they carry on the project. In this case, in an effort to build a more modern image of Shanghai and to take advantage of an attractive financial opportunity,8 city authorities made the deal with a Hong Kong real estate company. This company, ShuiOn Land was the one that designed Xintiandi and several other lilong. Its plan was to build large luxury residences and a shopping centre.

As a consequence of this deal, the economic migrants, the ones who were renting rooms only had to prepare to move further away, they are the silent ones. So when the decision to destroy the area was taken, the occupants could only prepare to move sooner or later.

Amongst the two other groups were the old residents and the artists. The first to react were the artists, they became an important group as the influx of artists seeking inspiration and affordable rents had increased. The mobilization of artists, academics9 and even political executives took place in various forms to denounce the massive destruction of local heritage.

The debate, although sometimes violent, did not receive significant media coverage. It was held in the Luwan district government offices and was not public. Interlocutors suggest that the main argument was that a modern international metropolis needs to keep some historical enclaves. Second, according to people who wanted to preserve Tianzifang from destruction, the experiment of mixing different groups was going on fine; moreover, the public seemed to appreciate this mix of residential life for the popular classes and artistic flavour. Eventually, Luwan leaders decided to use Tianzifang as an experiment in a new kind of historic preservation. The Asian economic crisis (1997) forced the developers to reduce their project because of the lack of funds and the fall of real estate prices. So, the Luwan district government took the decision to give Tianzifang some time to show what could be done. It is important to note that the decision was made at the lowest level of the administrative hierarchy. It was a real risk for the leaders because if the experiment failed, they would be made responsible for that.

Despite being the first to be affected by the development project, the degree of involvement of long-time residents was minimal. They appeared to be caught in a dynamic that they were not able to stop. Sidelined from the discussions, they were to be evicted for the purpose of adding value to the district. Initiated by the government and by private developers, this huge demolition and development project in Tianzifang, following the model of Xintiandi, was to completely transform the district by reallocating its space. Some people I interviewed said that they were not strong enough to fight against a government decision. They also said they did not want to get into trouble. A very fatalistic feeling was shared by many.

The new Tianzifang

When it was sure that Tianzifang would not be destroyed soon that negotiations began between the artists, the businessmen and the residents-owners. It was particularly uneasy because on the one hand, state regulations prohibited changing the zoning of land use. 10 On the other hand, because of the large number of owners in such a small area, reaching an agreement thus became a major challenge. One may need the agreement of several private owners in order to get a big enough space to accommodate goods and consumers. Because lilong are small buildings with a maximum of three floors, they do not have elevators. It is easier to do business on the first floor or the second one. The third floor is not easily accessible and often not visible from the alley. Therefore, it is more difficult to rent thirdfloor buildings, and those residents are often likely to stay, even as apartments on the lower floors are rented. So, the social fabric has changed under the pressure of urban renewal.

Tianzifang as we know it today was born out of these challenges.

Tianzifang was gaining notoriety amongst Shanghai people and it became quickly a trendy place for young people. But still the situation remained precarious for many of its residents.

However, as real estate prices collapsed during the Asian crisis of 1997, all major construction projects were either postponed or scaled back. Tianzifang benefited from the 10 Article 45 "The use of an architectural structure must conform with the function designated and approved for it in the Permit for Planning of Construction Project. Any change in the designated function of an architectural structure must be reported to the Municipal or Area/District/County Administrative Department in Charge of City Planning that originally handled the examination and approval procedures for approval." Regulations of Shanghai Municipality on City Planning, 1995 situation and, thus, many of its historic buildings, and the people who lived in them were spared. In 2004, an agreement was sealed between, the artists, the government officials and newcomer entrepreneurs, to save Tianzifang. But in return new management structures were created to oversee its development. So far, it was informal and small-scale committee. The owners are still private ones but most of them left to make room for shops or galleries. They receive a monthly rent. For the government in order to keep an eye on these transactions they decided to create a management committee to supervise. The main role of the committee was to prevent disputes given the complexity of establishing a lease. The involvement of municipal authorities in the management of Tianzifang marked a turning point in the evolution of the site. One year later, in 2005, Tianzifang was named "Shanghai's most creative industrial zone." This was the final victory for people who wanted to protect the place from destruction because gaining this title meant that it has been formerly recognized as a creative and touristic area that could not be destroyed easily.

If the residents were not fighting on the front line to save their living space, they were the ones who gained economic benefits (at least for the ones living on the ground or the first floor. In 2004, they began renting their property to newcomer artists who did not find room in former factories. They also started to rent to entrepreneurs who opened cafes, restaurants and souvenir boutiques. Shops started to be seen everywhere and this was a good chance for the residents who wished to move to a less confined environment, the opportunity was too good to pass up. For those living on the third floor, it wasand still isquite difficult to find a tenant. Thus, there is the rare situation where some residents voluntarily leave their homes and even receive a significant monthly rent and others who wished that the government had removed them from their homes. These people usually living on the second or the third floor would like to move but cannot afford to do so if they cannot rent their flat. For M. Zhang, living on the third floor (summer 2010) "If I move out now without any compensation from the government and without renting my place, how am I going to survive with my little retirement pension? My kids don't want me to stay with them and I don't have anywhere to go. This is really unfair." They are a marginalised group in Tianzifang. Even if they are part of the good image of Tianzifang as a place of social diversity.

Shanghai newspapers reported that the first resident in Tianzifang who rented his flat did a bit of renovation work and, before he has even finished, he was offered 4,000 Yuan per month by a fashion designer, who also ended up hiring him as an employee for a salary of 1,500 Yuan, which represented more than ten times the worker's former monthly income. Today, prices are continuously increasing, and a 40 m 2 apartment in the main aisle can be rented for up to 40,000 Yuan per month. 11 . The creativity of Tianzifang and its official recognition brought great vitality to the local community and a substantial increase in income for the residents. Initially, Tianzifang was only one lane located at 210, Taikang Road; subsequently, it was extended to a wider area, and today it encompasses three lanes (210, 248 and 274 Taikang Road). In 2011, the proportion of rented premises was 75% at number 210, 78% at number 248, and only 28% in the third and final section, one that was developed more recently and is located further west 12 .

In total, 45% of the families that were surveyed chose to remain there, 37% have rented their apartments, and 17% have become entrepreneurs (1% were not able to answer the questionnaire). Out of the 301 families living there, 114 would like to find a tenant, 34 wanted to leave, 97 wanted to stay, and 75 were not sure. These figures show that the development of Tianzifang has a very significant impact on the residents. Residents, who did not have much 11 This figure was given by several people during interviews in May 2014. There is no set maximum price; rather prices are set by negotiation between parties. 12 Results of a public poll ordered by the local committee from Luwan district in 2010. The survey interviewed 671 families in Tianzifang. The whole survey has not been published, but some abstracts are available Xu Y.C 2011.

to say about the transformation of their community, nevertheless want to take advantage of the development. As people decide to leave their apartment and rent it out, this dynamic will lead to a very different Tianzifang, which likely will look more like Xintiandi. Residents whose income has increased find themselves in a position of choice. However, those with the most choice are those who had ground floor units.

Alongside this transformation, because the resident population density is decreasing; one might expect that the structures should be better maintained. However, the increase in the tourist population and the turnover of entrepreneurs as the rent prices are rocketing, have an impact on the maintenance of the buildings. In addition, construction was not planned for shops and restaurants, few inspections are performed so some fear that the buildings might collapse.

During interviews conducted between 2005 and 2012, some residents, especially those who lived on the second or third floors, complained about not finding tenants. Many were also distressed by the onslaught of tourists, who seemed to consider them as curiosities. Another subject of recrimination was the disappearance of small shops useful to residents. For example, the large market at the end of Taikang Road, small hairdressers, shoemakers, glaziers, etc. have all disappeared to give way to more luxurious boutiques. The retired Mrs.

Li thus complained in 2010: "Back in the days, I used to go to the market every morning and we would all meet up, chatting while buying vegetables. Now the new market is smaller, some services are no longer available and then it's much more expensive. Only fashionable shops are left; it is good for young people and foreigners, but not for people like us!" For Mr.

Huang, a retired man, the situation is not easier "I can only go out early in the morning, otherwise I would have no respite; they all want to take my picture and they make so much noise that my bird becomes frightened. At night, they are shouting and making noise until the late hours. If I could, I would move as quickly as possible. This is not a life!" MS Yao (May 2014), who has been living in a close by Lilong since her childhood and who is selling tea in Tianzifang is more critical of residents "people are not sensible; they all think that their small flat on the second or third floor is gold. They are too greedy, so my neighbours underneath are condemned to cope with noise and daily stress". Disputes between residents, bar owners and even customers happen from time to time, forcing the neighbourhood committee to intervene.

Although residents did not carry out formal demonstrations in front of Luwan's District Committee, some third-floor residents set up banners denouncing the changes they faced (witnessed by the author in April, 2008). This shows sporadic resistance, with little impact on the development of Tianzifang.

The departure of residents has only accelerated. There used to be 670 households living in Tianzifang eight years ago; now only 80 remain, most of whom are elderly. Since 2011, street signs in Chinese and English indicating that visitors are not allowed to penetrate further or that pictures are forbidden beyond a certain limit have started to appear at the end of the shopping aisles. Small booths have been installed at the entrances of the alleys for the security guards 13 on duty; barriers were set to prevent vehicles and even bicycles from entering.

13 They are paid by the management committee. In China one can see those at the entrance of gated communities and in most public spaces. In that sense Tianzifang is not an exception. residents, they don't have any more private space where they can meet within themselves 15 .

The businessmen are not satisfied either because the rent is now too high, there is a harsh competition. Some artists left already for more attractive places. As an example, Deke-Erh, one of the first famous artist to have moved to Tianzifang left for Suzhou where the municipality provided him with a large space for a low rent. After eight years of development, there were mixed results. Miss Wu, a Taiwanese, (summer 2011) thus explains her feeling: "During my first visit, while having coffee in an alley, we could see some residents cycling about, carrying the vegetables they had just purchased. This passage of the recent past was quite fascinating. (...) During the past two years, the number of daily visitors has greatly increased. Inside Tianzifang, people have to push and bump into each in order to move forward." 15 The role of the Lilong traditionally was to get a community life inside and shops outside with a few alleys opening to the outside. Now it is the opposite inside the Lilong is full of outsiders and there is no more community life. The co-residence of artists and other residents in a single neighbourhood now appears to be a victim of its success. Flat owners are divided among those who enjoy a comfortable income and usually left Tianzifang, and others who have to stay and whose quality of life has deteriorated. As for the artists, who allowed Tianzifang to exist, there are also different situations and some left as shown above with Deke-Erh. Rents have become too high, tourists often pass by quickly without making a purchase, and competition has become increasingly tough. In this sense, Tianzifang can be said to be victim of its success.

When a marginal area is transformed into a touristic high spot

As was bitterly pointed out by some residents, the lilong's surroundings have changed. Until the early 2000s, Taikang Road was a quiet, narrow, one-way road, lined with plane trees. On both sides of this road, from west to east, there were "ordinary" stalls, a glazier, a mattressmaker, a hairdresser who could only take care of a single customer at a time, a covered twofloor market with fruits and vegetables on the first floor, and poultry and live fish on the second, a female vendor of small steamed bread buns, a watchmaker, a small grocery store, etc. All of these merchants lived at or near Tianzifang. All of them knew each other, and there were strong ties between them, amidst an intense community life. It was a lilong, like many others in Shanghai on the margins of the modern city. As Tianzifang evolved, the daily use shops have given way to increasingly upmarket boutiques to please new costumers: the visiting tourists. These include many shops selling tea, stylish clothing shops, galleries, etc. This kind of development did not satisfy the residents. The irruption of new actors in their daily life transformed Tianzifang's residents into marginal men in their living space.

Opposite Tianzifang, still on Taikang Road, another lilong (Xinxili) was completely destroyed to make way for several luxury buildings and a shopping mall with restaurants, banks, cinemas, various boutiques, etc. Here, the residents were evicted by the police and sent to the outer suburbs. At the place of this former Lilong, a subway station (line 9) was created just before the opening of the World Expo in 2010, with an exit right in front of Tianzifang's main entrance.

The Lilong of Tianzifang had to invent its own and to fight for its existence. The whole process involved creativity and a particular integration of residents, artists and merchants which is unique to Tianzifang and contributes to its attractiveness. Tourists, who are mostly Chinese, come to walk through the streets and make a selective reading of the local history through a sanitized reconstruction of a place that we have previously described as lacking any form of modern comfort. However, this success is not without its problems. At a certain point, sustainable development, one that would allow social and economic development as well as historic preservation, appears to have become untenable.

Tianzifang is different from the other Shanghai redevelopment projects in many regards: first because of the long and difficult struggle for survival that took place between the different actors, second because of the manner in which it escaped demolition16 , and finally because of the fact that it later became a model in terms of urban renewal. After 2005, when Tianzifang was proclaimed "Shanghai most creative industrial zone", its status changed and it is from then on that it was seen as a model to be emulated. But, to my knowledge, up to this date no area in China claims to have emulated it. It was initially saved by a spontaneous congregation of artists and other actors who felt the need to get involved, but the artists are starting to leave now, joining the residents who already left; both have been progressively replaced by shopkeepers. The alleys have become saturated with waves of tourists gradually altering the unique character of the area. Buildings have undergone a number of changes intended to allow the creation of galleries and boutiques. Housing have been transformed into restaurants, galleries, any kind of shops. Today, Tianzifang is a victim of its own success, and it is becoming, step by step, similar to other "historic villages" and enclaves in China, a reconstructed leisure area with an obsolete fragrance where you can buy the same souvenirs as elsewhere. What was making the uniqueness of Tianzifang: the cohabitation of residents, artists, shops and tourists is fading away.

In other words, the ultimate goal of the leaders of the Luwan district, with the approval of higher authorities, which was to reshape the urban landscape and to modernize it while at the same time protecting Tianzifang as an enclave seems to work beyond expectation. At the same time, taking advantage of the opportunities available to them, the artists and the residents reorganized the space in terms of function, thus reshaping its social fabric as well. In Tianzifang, residents were not evicted but gradually moved away, and gentrification inexorably continues. This enables the fabrication of citizens seemingly corresponding to the standards of a truly international city, and it is used to display a flattering image of Shanghai.

In a way one might say that Tianzifang's uniqueness is fading away and it seems to be marginalized as it did not manage to become a model for urban development in China.

Bibliography

Figure 1

 1 Figure 1 evolution of building usage from 1997 to 2009

Figure 2 :

 2 Figure 2: Tianzifang's main entrance 210 Taikang lu

Figure 3

 3 Figure 3 Summer 2010At the end of 2012, a survey 14 of the tourists visiting Tianzifang commissioned by Luwan district, showed that it was ranked last among the Shanghainese sites. According to the journalist who analysed the reports, visitors complain that there are too many tourists in the narrow alleys. They also complain about inflated prices for average quality goods. As for the

Figure

  Figure 4: Summer 2006

FigureFigure 6

 6 Figure 5: summer 2015

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This is well described by the Shanghainese writer QiuXiaolong (2010)

Even in the 2000's before the urban renovation, Lilongs were the most common type of housing in Shanghai.

"In China, in 2010, revenue from trade in real estate accounted for 1/3 of national revenue"(Yu H 2011: 24).

Like Chen Yingfang, then Professor of Sociology at East China Normal University. She published in Chinese two books about slums in Shanghai and social problems with urban development(2008,2009).

Some other projects were put on hold but the difference is that the Lilong were first demolished before a new allotment would be defined.

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