T Zhou

J.-D Chazot

E Perrey-Debain

L Cheng
email: li.cheng@polyu.edu.hk

Performance of the Partition of Unity Finite Element Method for the modeling of Timoshenko beams

Keywords: Partition of Unity Finite Element Method, Timoshenko beam, Wave propagation, Lagrange multiplier PACS

The partition of unity finite element method (PUFEM) is developed and applied to compute the vibrational response of a Timoshenko beam subject to a uniformly distributed harmonic loading. In the proposed method, classical finite elements are enriched with three types of special functions: propagating and evanescent wave functions, a Fourier-type series and a polynomial enrichment. Different formulations are first assessed through comparisons on the frequency response functions at a specific point on the beam. The computational performance, in terms of both accuracy and data reduction, is then illustrated through convergence analyses. It is found that, by using a very limited number of degrees of freedom, the wave enrichment offers highly accurate results with a convergence rate which is much higher than other formulations. Evanescent waves and the constant term in the wave basis are also shown to play an important role. In all cases, the proposed PUFEM formulations clearly outperform classical finite element method in terms of computational efficiency.

Introduction

The numerical simulation of mechanical waves in the so-called mid-frequency range has been the subject of intensive research in the past two decades and continues to be a very challenging topic for many research engineers and applied mathematicians (see [START_REF] Desmet | Final report summary -mid-frequency[END_REF], [START_REF]Boundary and finite element methods for high frequency scattering problems[END_REF] and [START_REF] Tanner | Special issue of wave motion -"innovations in wave modelling[END_REF]). This mid-frequency gap in modelling capabilities separates the low frequency range for which standard Finite Element Method (FEM) are applicable and largely used and the high-frequency range which is normally dealt with by statistical methods such as the very popular Statistical Energy Analysis (SEA). To better tackle short-wave simulation problems, enriched methods have been developed in recent decades. These numerical techniques have been tailored to incorporate a prior knowledge of the propagating waves in the formulation. A rather complete survey on the topic can be found in a recent review paper [START_REF] Deckers | The wave based method: An overview of 15 years of research[END_REF]. Among these methods the Partition of Unity Finite Element Method (PUFEM) has the advantage of possessing high similarities with the classical FEM (see Refs. [START_REF] Melenk | The partition of unity finite element method: Basic theory and applications[END_REF][START_REF] Babuska | The partition of unity method[END_REF]). It can be easily implemented for numerical analysis using the existing finite element meshes and simulation codes. The PUFEM has been applied to simulate the acoustic and elastic wave propagation (see Refs. [START_REF] Mohamed | Some numerical aspects of the pufem for efficient solution of 2d helmholtz problems[END_REF][START_REF] Laghrouche | Locally enriched finite elements for the helmholtz equation in two dimensions[END_REF][START_REF] Laghrouche | Wave interpolation finite elements for helmholtz problems with jumps in the wave speed[END_REF] and Refs. [START_REF] Kacimi | Numerical modelling of elastic wave scattering in frequency domain by the partition of unity finite element method[END_REF][START_REF] Kacimi | Improvement of pufem for the numerical solution of high-frequency elastic wave scattering on unstructured triangular mesh grids[END_REF] respectively). In particular, numerical simulations of acoustic waves propagating in air, porous and poro-elastic media [START_REF] Chazot | Performances of the partition of unity finite element method for the analysis of two-dimensional interior sound fields with absorbing materials[END_REF][START_REF] Chazot | The partition of unity finite element method for the simulation of waves in air and poroelastic media[END_REF][START_REF] Yang | Development of 3d pufem with linear tetrahedral elements for the simulation of acoustic waves in enclosed cavities[END_REF] have also been attempted in our previous work, which constitutes a natural extension of the method for noise control applications.

Up to now, there are few works on the modelling of vibrations of beams and plates with the PUFEM. The first paper relating to this topic is probably the static analysis of Timoshenko beams with elastic supports presented by Babuska in Ref. [START_REF] Babuška | The partition of unity method for the elastically supported beam[END_REF], which shows that shear locking disappears with PUFEM enrichments, contrary to classical FEM. Vibrational modes of a cantilever beam have been studied independantly by Arndt et al. [START_REF] Arndt | The generalized finite element method applied to free vibration of beams[END_REF][START_REF] Arndt | The generalized finite element method applied to free vibration of framed structures[END_REF] and Shang-Hsu [START_REF] Hsu | Enriched finite element methods for timoshenko beam free vibration analysis[END_REF] with an enrichment based on trigonometric sine expansions, hierarchical polynomial functions and modal expansion (this latter is based on an idea presented by Craig [START_REF] Craig | Structural dynamics: an introduction to computer methods[END_REF]). This type of enrichment, though failing to capture the essential wave characters of the solution, except maybe for the modal expansion which somehow contains the geometry and material properties of the beam in the formulation, has the advantage of being frequency-independent thus allowing the use of standard algebraic modal analysis solvers. Polynomial functions in PUFEM have also been used to the development of enriched Mindlin plate elements [START_REF] Hazard | Structural dynamics of viscoelastic sandwich plates by the partition of unity finite element method[END_REF] and in this context, the method shares similarities with p-FEM technique. De Bel et al. [START_REF] Bel | Forced vibrations in the medium frequency range solved by a partition of unity method with local information[END_REF] used flexural waves propagating in different directions as the enrichment functions. The originality of the approach is that the propagation angle is generated iteratively at each node of the PUFEM mesh. However, shear deformations and rotary inertia effects are neglected in their analysis. Finally, though the method does not formally fit in with the PUFEM approach, we can cite the early work of Hashemi et al. [START_REF] Hashemi | A new dynamic finite element (dfe) formulation for lateral free vibrations of euler-bernoulli spinning beams using trigonometric shape functions[END_REF] who developed a Dynamic Finite Element for the vibrational analysis of spinning beams, by including frequency dependent trigonometric shape functions in their formulation.

Motivated by the above analyses, the aim of this paper is to develop and investigate the applicability of PUFEM to the dynamic analysis of thin vibrating structures. To this end, a simply supported beam under a distributed harmonic loading is chosen as a benchmark for further developments involving vibro-acoustic coupling in one and two dimensions. Timoshenko beam theory is adopted to ensure a correct description of the vibration behaviour at high frequencies, when the wavelength is comparable to the thickness of the beam. Furthermore, as opposed to the classical fourth-order wave equation of the Euler-Bernoulli beam whose numerical treatment requires the use of specific C 1 elements [START_REF] Arndt | The generalized finite element method applied to free vibration of framed structures[END_REF], Timoshenko theory leads to a coupled system of second order partial differential equations for the translational and rotational displacements which permits to employ conventional piecewise-continuous Lagrangian finite element shape functions for the partition of unity. As one of the key ingredients of the method, particular attention is paid to choosing an appropriate function space for the finite element enrichment, which should have good approximation properties for the solutions to a given differential equation [START_REF] Melenk | The partition of unity finite element method: Basic theory and applications[END_REF]. In the present work, exact solutions of an unloaded infinite beam including both the propagating and evanescent waves [START_REF] Guyader | Vibration in Continuous Media[END_REF] are exploited, with the addition of additional terms to account for the pressure loading. Comparisons are made with classical type of enrichments such as Fourier-type series and polynomials. Finally, the treatment of the boundary conditions needs particulate attention. Although classical finite element procedures can be followed for some particular types of enrichment functions, penalty or Lagrange's multiplier technique [START_REF] Bel | Forced vibrations in the medium frequency range solved by a partition of unity method with local information[END_REF][START_REF] Arndt | The generalized finite element method applied to free vibration of framed structures[END_REF] is adopted to accommodate all types of enrichment functions. Numerical analyses are conducted with comparisons among different types of enrichment functions, in terms of computational accuracy and data reduction. It is concluded that the wave basis in the PUFEM is the best approach leading to the best convergence rate.

The paper is organized as follows. After recalling the classical Timoshenko beam theory as well as its associated variational formulation in Section 2, PUFEM Timoshenko beam ele-ments, with three type of enrichments, are constructed and explained in Section 3. In Section 4, classical FEM formulations including linear elements with exact and reduced integration and an enhanced formulation based on a cubic and quadratic interpolation for the transverse displacement and the rotation are presented. This should serve as a reference solution and allows to identify, wherever necessary, shear-locking effects. In the last section, performance of PUFEM elements is shown and compared with classical FEM. In all cases, a reference solution, is obtained using linear interpolation calculated on a very refined mesh. Convergence curves, corresponding to h-refinement', i.e. by reducing the element size, and 'p-refinement', i.e. by increasing the number of enrichment functions, are given and analyzed. The role of evanescent waves in the PUFEM wave basis is also discussed.

Timoshenko beam theory

The flexural vibration of beams is under investigation. Figure 1 presents a schematic representation associated with the two main theories dedicated to beams: Euler-Bernoulli and Timoshenko theories. For the latter the displacements of the beam are denoted by u(x, z) = zβ(x) and w(x, z) = w(x) where β is the total angle of rotation of the section, and w is the displacement of the mid-surface in the z-direction. These two independant variables obey the equations of motion:

f z + κGSγ ,x = ρS ẅ, (1)
EIβ ,xx -κGSγ = ρI β, (2)
where γ = β + w ,x is the shear deformation angle and f z is the distributed load. The material properties are the Young's modulus E, the shear modulus G and the density ρ. The geometrical parameters are the shear correction factor κ = 5/6, the second moment of area I, the cross section area S and the beam length L. With simply supported boundary conditions, the displacement and the bending moment vanish at the locations of the two supports , i.e. at x = 0, L.

In this case, the associated variational formulation writes

L 0 δwρS ẅ + δβρI β + δβ ,x EIβ ,x + δγ κGSγ -δwf z dx -δw 0 λ 0 -δw L λ L = 0, (3)
where δ(•) donates the virtual quantity and w 0 and w L are the displacement at x = 0, L, respectively. In formulation (3), the transverse shear forces appear naturally as Lagrange multipliers λ 0 and λ L . Though it is common to discard these terms by simply choosing δw 0 = δw L = 0, the best way to handle the boundary terms with PUFEM is to weakly enforce the essential conditions as:

δλ 0 w 0 = δλ L w L = 0, ∀(δλ 0 , δλ L ). (4)
This has the advantage of preserving the symmetry of the linear sytem and permits to handle efficiently the coupling conditions between two media (see for instance [START_REF] Laghrouche | Wave interpolation finite elements for helmholtz problems with jumps in the wave speed[END_REF][START_REF] Chazot | Performances of the partition of unity finite element method for the analysis of two-dimensional interior sound fields with absorbing materials[END_REF][START_REF] Chazot | The partition of unity finite element method for the simulation of waves in air and poroelastic media[END_REF]).

Application of the PUFEM

In this work, we only investigate the Timoshenko beam vibration subject to a harmonic loading at an angular frequency ω and the time-dependent term e -iωt is omitted hereafter. As done in classical FEM, the beam is partitioned into non-overlapping elements and the degrees of freedom are interpolated over each elements with nodal unknowns. The key ingredient of the PUFEM relies on the enrichment of the conventional finite element approximation by including special functions in order to enhance the convergence of the numerical solution. For an infinite beam, the two propagating and two evanescent waves characterized by the wavenumbers write (see Ref. [START_REF] Guyader | Vibration in Continuous Media[END_REF] for more details):

k p = ρIω 2 (1 + E/(κG)) + √ δ 2EI , (5)
k e = -ρIω 2 (1 + E/(κG)) + √ δ 2EI , (6)
with δ = (ρIω 2 ) 2 (1 -E/(κG)) 2 + 4EIρSω 2 . Each PUFEM element of length l e = x 2 -x 1 is given by the geometric mapping x(ξ) = N 1 x 1 + N 2 x 2 where x i are the nodes and ξ is the coordinate in the reference frame ξ ∈ [0, 1]. Here, N 1 = ξ and N 2 = 1 -ξ are the classical linear shape functions. The transverse displacement and the rotation are then expanded as:

w = 2 i=1 N i (ξ) N n=1 A n i Ψ n i , (7)
β = 2 i=1 N i (ξ) N n=1 B n i Ψ n i . (8)
For wave enrichment, we consider N = 5 functions Ψ n i defined as:

Ψ n i ∈ {1, cos[k p (x -x i )], sin[k p (x -x i )], cosh[k e (x -x i )], sinh[k e (x -x i ))]}, (9)
where the constant term Ψ 1 i = 1 has been added in the enrichment in order to capture contributions of the distributed load, i.e. the particular solutions of the governing equations [START_REF] Leissa | Vibrations of Continuous Systems[END_REF]. Two other kinds of enrichment are also considered in the present work. The first one is a polynomial enrichment:

Ψ n i ∈ {1, η i , η 2 i , η 3 i , η 4 i , . . .} , (10)
where η i = (x -x i )/l e . For N = 2, there are four enrichment terms associated with one polynomial-enriched element whilst the highest order of the corresponding bases ( 7) and ( 8) is two. Therefore, these basis functions are linearly dependent since only three polynomial terms form a complete quadratic basis. The second one is a Fourier-type series

Ψ n i ∈ {1, cos(πη i ), sin(πη i ), cos(2πη i ), sin(2πη i ), . . .} . (11)
Note that (i) Fourier and polynomial enrichments can be built with an arbitrary order N whereas, by construction, the wave enrichment is necessarily limited to N = 5. (ii) Since the PUFEM element can contain many wavelengths, the elementary mass and stiffness matrices associated with the PUFEM expansion ( 7) and ( 8) must be constructed using sufficient Gaussian integration points in order to ensure convergence.

Classical FEM

In order to evaluate the PUFEM efficiency in comparisons with classical FEM formulations, two beam finite elements are reminded here: a linear element used as a reference, and an enhanced element which is also commonly used.

Linear element

The beam is discretized with linear shape functions:

w = 2 i=1 A i N i (ξ) and β = 2 i=1 B i N i (ξ). (12)
The associated elementary stiffness matrix can be evaluated with exact integration method. However, this formulation over-emphases the effect of shear deformation in comparison with the bending effect, which would generate shear-locking effects for the cases where the Euler-Bernoulli or thin beam model is applicable. To tackle the problem, a reduced integration technique is usually employed [START_REF] Hughes | A simple and efficient finite element for plate bending[END_REF][START_REF] Petyt | Introduction to finite element vibration analysis[END_REF][START_REF] Bang | The finite element method using MATLAB[END_REF]. The linear element with reduced integration serves as a reference solution and permits to identify, wherever needed, the shear-locking effects. Details of linear elements using reduced and exact integration schemes are given in the Appendix.

Enhanced element

A specific timoshenko beam element is also often encountered in the litterature (see Refs. [START_REF] Thomas | Timoshenko beam finite elements[END_REF][START_REF] Przemieniecki | Theory of matrix structural analysis[END_REF][START_REF] Petyt | Introduction to finite element vibration analysis[END_REF]). This enhanced element is also tested in this work, and compared with the PUFEM. It is based on a cubic and a quadratic interpolation for the transverse displacement w and the rotation β, respectively, with an added constraint between w and β in order to satisfy the static equilibrium equation. This type of enhanced element is also free of shear locking. The displacement and rotation are expanded as

w = 4 i=1 C i Ni (ξ) and β = 8 i=5 C i Ni (ξ), (13)
where Ni (ξ) is the shape function of the enhanced element and the Appendix gives their detailed expressions.

Geometrical parameters Material parameters

L = 1 m E = 70 GPa h = 0.01 m ρ = 2780 kg/m 3 b = 0.01 m G = 27 GPa
Table 1: Parameters used in our computations.

PUFEM performance

The tested configuration is a simply supported beam subject to a uniformly distributed harmonic loading with a unit amplitude. The geometrical and material parameters of the beam are tabulated in Table 1.

The performance of the PUFEM with different enrichment functions is evaluated through the comparisons of their Frequency Response Function (FRF), as shown in Figure 2. The reference solution is obtained using classical linear elements with reduced integration and with 50'000 elements (in grey). The calculations of the FRF curves with PUFEM using different enrichment methods, i.e. wave enrichment, Fourier and polynomial enrichment, are all carried out using 2 elements and 5 enrichment functions (N = 5) which corresponds to a total number of degrees of freedom of 3 × 2 × 5 = 30. It can be seen from Figure 2 that PUFEM can provide accurate predictions up to a certain frequency limit, depending on the enrichment function. Clearly, the wave enrichment (blue solid line) offers best performance and a good agreement with the reference solution up to 3000 Hz, above which small, but growing, discrepancies start to appear. The other two enrichments, using the Fourier series (green dotted line) and the polynomial functions (red mixed line), are only accurate up to a reduced frequency range, around 1400 and 500 Hz, respectively. Of course, the frequency band can be extended by applying either a h-refinement or a p-refinement, as evidenced by the following convergence analyses. Figure 3 shows the deformed shape along the beam close to the upper limit frequency for each enrichment. It can be seen that the PUFEM with waves can capture multiple wavelenghs per element (up to 3 for the present case), which is a typical feature of wave enriched elements. Figure 4 compares the convergence of the different formulations obtained using a h-refinement for two specific frequencies: 1000 and 3500 Hz while keeping the same enrichment order (N = 5) with PUFEM. The L 2 errors are plotted versus the number of degrees of freedom N dof . Here, errors are estimated via L 2 -norm as where w ref is the reference solution. The superiority of the PUFEM with wave enrichment can be clearly seen: the highest convergence rate and very good accuracy with a very small number of degrees of freedom even at high frequencies. It should be noted that the error is limited here by the accuracy of the reference solution. This explains the visible plateau by the wave enrichment at a very low error level (around 10 -5 % for 1000Hz and 10 -4 % for 3500Hz). The PUFEM with the polynomial enrichment is also very efficient with a high convergence rate. Indeed, results indicate that the error behaves like ε ∼ Cl α e where α ≈ 8 (recall that l e is the element length and the total number of degrees of freedom N dof is inversely proportional to l e ). The convergence law of the classical FEM with complete polynomial expansions does not apply to the PUFEM with polynomial enrichment. It is remarkable to see that classical linear elements, with and without reduced integration, as well as the enhanced element formulation give the same convergence rate and ε ∼ C l 2 e , which is line with the classical linear interpolation. What differentiates the three formulations is that (i) results obtained with the enhanced element are 100 times more accurate than the linear formulation with reduced integration and (ii) classical linear FEM with exact integration suffers from slow convergence due to shear-locking effects which can be avoided at the expense of a very refined mesh. Finally, the Fourier-type enrichment performs similarily to classical FEM once the length of the element is sufficiently small, this is because the mesh spacing is decreased and the oscillating nature of the solution within a single element is lost and the Fourier series, with a fixed order of approximation (here N = 5), does not show any advantage with respect to classical FEM [START_REF] Tveito | Introduction to partial differential equations: a computational approach[END_REF]. The fact that the exact solution has a strong wave component with wavenumber k p = 2π/λ p explains the peculiar behaviour clearly observed when finite elements are larger. Since the Fourier enrichment is chosen to capture half of a wavelength up to one wavelength per element, the formulation is expected to yield best results around 0.5 ≤ l e /λ p ≤ 1 and this is confirmed in Table 2 where numerical errors are shown with respect to that criteria. Finally, none of the PUFEM formulations suffers from shear-locking and this is consistent with observations made in Ref. [START_REF] Hsu | Enriched finite element methods for timoshenko beam free vibration analysis[END_REF].

ε = L 0 |w computed -w ref | 2 dx L 0 |w ref | 2 dx × 100% , (14)
As mentioned before, a p-refinement analysis is possible with Fourier and polynomial enrichments. Results are shown in Figure 5 for two selected frequencies, 1000 and 3500 Hz. For the sake of comparison, the previous results using h-refinement are also reported. In the present case, the beam is meshed with 2 elements (same as in Figure 2 and 3) while the approximation order N is increased. As expected, p-refinement performs better than h-refinement does. The Fourier enrichment behaves nearly as well as the wave enrichment. However, if the polynomial shows similar trends for low and moderate appproximation order N , results quickly deteriorate as soon as the exponent in the polynomial exceeds a certain value. The reason for this probably stems from the linear dependence and the loss of orthogonality properties of the polynomial bases, and the occurrence of very ill-conditioned matrices [START_REF] Hazard | Structural dynamics of viscoelastic sandwich plates by the partition of unity finite element method[END_REF]. In an attempt to clarify this, the 4) (in bold are values below 0.5% at the cup). associated conditioning numbers of the system matrices are shown in Figure 6. As opposed to classical FEM, PUFEM formulations clearly produce matrices with a higher condition number, a well-known feature which is inherent to the method [START_REF] Hazard | Structural dynamics of viscoelastic sandwich plates by the partition of unity finite element method[END_REF]. This, however, does not necessarily impede on the quality of the results (some explanations are given by one of the present authors in [START_REF] Perrey-Debain | Plane wave interpolation in direct collocation boundary element method for radiation and wave scattering: numerical aspects and applications[END_REF] in a BEM context). For instance, wave and Fourier enrichments show very good stability despite a growing condition number wich is comparable, though smaller, with that of the polynomial enrichment. An alternative would be to employ orthogonal polynomials instead, in which case the method would share some similarities with hierachical FEM [START_REF] Beslin | A hierarchical functions set for predicting very high order plate bending modes with any boundary conditions[END_REF].

Since the wave enrichement offers best performance. It is relevant to assess the influence of each term in the wave basis Figure 7 shows the convergence at 1000 Hz and 3500 Hz using hrefinement with the complete wave basis, and the one with certain terms removed, in comparison with the classical linear FEM results. When the constant term is removed from the wave bais, i.e. Ψ 1 i = 1, the PUFEM is only enriched with free vibration solutions. With a uniform loading, this has noticeable effects on the convergence rate and on the number of degrees of freedom required to produce accurate results. When the evanescent waves are removed, the nearfield × h-refinement with classical FEM, × h-refinement with linear FEM and exact integration, + h-refinement with enhanced FEM, h-refinement for PUFEM with wave enrichment, p-refinement for PUFEM with Fourier enrichment, p-refinement for PUFEM with polynomial enrichment. effects of the decaying waves near the beam supports cannot be properly modelled and this, again, has noticeable effects on the convergence rate. In all scenarios, however, all PUFEM formulations clearly outperforms classical FEM to various extent. × Linear FEM with reduced integrations, PUFEM with complete wave enrichment, PUFEM wave enrichment without the constant term, PUFEM wave enrichment without the envanescent waves.

Conclusions

In this study, PUFEM Timoshenko beam elements are developed for solving forced vibration problems. Three types of enrichment are investigated: the wave enrichment based on the solutions of the governing equations, Fourier series and the polynomials. The performance of different enrichment functions is numerically evaluated in terms of frequency response functions and convergence properties.

Analyses lead to the prevailing conclusion that the wave enrichment, through the embodiment of specific information based on physical features, offers the best performance in terms of both computational accuracy and data reduction. In all cases, all three PUFEM formulations outperform the classical finite element discretization and the best convergence is obtained using a p-refinement strategy. However, it is found that, due to a lack of orthogonality property, the polynomial basis is recommended to adopt a h-refinement strategy instead. Finally, the constant term in the wave basis shows its importance to account for the loading effects.

As a final remark, one direction of particular interest is to further extend the method to the numerical prediction of complex vibrating structures involving vibro-acoustic coupling. In this regard, it would be interesting to analyse more specifically the type of enrichment needed to correctly capture the spatially oscillating pattern of the loading due to surface acoustic waves.

with φ = 12EI/GκSl 2 e . This formulation is known to suffer from shear-locking effects and a reduced integration technique is usually employed [START_REF] Petyt | Introduction to finite element vibration analysis[END_REF], this gives

K e = EI l 3 e φ    

12

-6l e -12 -6l e l 2 e (3 + φ) 6l e l 2 e (3 -φ) 12 6l e sym.

l 2 e (3 + φ)

    , (16)
More details of the stiffness matrix can be found Refs. [START_REF] Hughes | A simple and efficient finite element for plate bending[END_REF][START_REF] Petyt | Introduction to finite element vibration analysis[END_REF][START_REF] Bang | The finite element method using MATLAB[END_REF] (here some signs can change depending on the convention for the rotation angle). The associated mass matrix is

M e = ρSl e     1/3 0 1/6 0 r 2 /3 0 r 2 /6 1/3 0 sym. r 2 /3     , (17)
where r = I/S is the radius of gyration. This matrix can also be found with more details in Refs. [START_REF] Bhashyam | The second frequency spectrum of timoshenko beams[END_REF][START_REF] Petyt | Introduction to finite element vibration analysis[END_REF]. Note the differences between the two resulting stiffness matrices appears in the components K 22 , K 24 , K 42 and K 44 . This difference is due to the different integration points adopted for evaluating the shear modulus matrix. When using exact integrations, the shear locking effects appear for the cases where the Euler-Bernoulli or thin beam model is applicable.

The shear-locking effects can be overcome by h-refinement, which makes the stiffness matrix obtained by exact and reduced integrations consistent.

Enhanced element

A specific timoshenko beam element is also often used to solve beam problems in the litterature [START_REF] Przemieniecki | Theory of matrix structural analysis[END_REF][START_REF] Petyt | Introduction to finite element vibration analysis[END_REF][START_REF] Davis | A timoshenko beam element[END_REF][START_REF] Thomas | Timoshenko beam finite elements[END_REF][START_REF] Corn | Transverse vibrations of short beams: Finite element models obtained by a condensation method[END_REF][START_REF] Lees | Unified timoshenko beam finite element[END_REF]. This enhanced element is based on a cubic and a quadratic interpolation for the transverse displacement w and the rotation β, respectively, with an added constraint between w and β in order to satisfy the static equilibrium equation. The shape functions Ni in Eq. ( 13) are given below N1 = 1 1 + φ 1 + φ -φξ -3ξ 2 + 2ξ

This matrix can be found with all necessary details in [START_REF] Thomas | Timoshenko beam finite elements[END_REF][START_REF] Przemieniecki | Theory of matrix structural analysis[END_REF][START_REF] Petyt | Introduction to finite element vibration analysis[END_REF]. This enhanced element is also free of shear locking. The the mass matrix writes :

where coefficients m i are given below and this mass matrix can also be found with more details in Refs. [START_REF] Thomas | Timoshenko beam finite elements[END_REF][START_REF] Przemieniecki | Theory of matrix structural analysis[END_REF][START_REF] Petyt | Introduction to finite element vibration analysis[END_REF].

m

Figure 1 :

 1 Figure 1: Beam description in the x-z plane.

FrequencyFigure 2 :

 2 Figure 2: FRF comparison wo/Fe (at L/4 from left end): reference solution with classical FEM, PUFEM with the wave enrichment, PUFEM with the Fourier enrichment, PUFEM with the polynomial enrichment.

6 xFigure 3 :

 63 Figure 3: Illustration of the deformed shape at 3000Hz, 1200Hz and 400Hz from left to right. reference solution with classical FEM, PUFEM with the wave enrichment, PUFEM with Fourier enrichment, PUFEM with polynomial enrichment.

Figure 4 :

 4 Figure 4: Convergence curves obtained with a h-refinement at 1000Hz (left) and 3500Hz (right).× classical FEM, × linear FEM with exact integration, + enhanced FEM, PUFEM with wave enrichment, PUFEM with Fourier enrichment, PUFEM with polynomial enrichment.

Figure 5 :

 5 Figure 5: Comparison of convergence at 1000Hz (left) and 3500Hz (right).× h-refinement with classical FEM, × h-refinement with linear FEM and exact integration, + h-refinement with enhanced FEM, h-refinement for PUFEM with wave enrichment, p-refinement for PUFEM with Fourier enrichment, p-refinement for PUFEM with polynomial enrichment.

Figure 6 :

 6 Figure 6: Comparison of conditioning number with h and p refinement at 1000Hz on the left and 3500Hz on the right. × h-refinement for classical FEM, h-refinement for PUFEM with wave enrichment, p-refinement for PUFEM with Fourier enrichment, p-refinement for PUFEM with polynomial enrichment.

Figure 7 :

 7 Figure 7: Convergence curves obtained with h-refinement at 1000Hz on the left and 3500Hz on the right.× Linear FEM with reduced integrations, PUFEM with complete wave enrichment, PUFEM wave enrichment without the constant term, PUFEM wave enrichment without the envanescent waves.

Table 2 :

 2 ). Convergenve of Fourier-type enrichment (corresponding to in Figure

	× classical

  GκSl 2 e . The resultant stifness matrix remains of the same size as they correpond to nodal values and

					3 , N2 =	l e /2 1 + φ	(2 + φ)ξ -(4 + φ)ξ 2 + 2ξ 3 ,
	N3 =	1 1 + φ	φξ + 3ξ 2 -2ξ 3 ,		N4 =	l e /2 1 + φ	-φξ -(2 -φ)ξ 2 + 2ξ 3 ,
	N5 =	2/l e 1 + φ	-3ξ + 3ξ 2 ,		N6 =	1 1 + φ	1 + φ -(4 + φ)ξ + 3ξ 2 ,
	N7 =	2/l e 1 + φ	3ξ -3ξ 2 ,			N8 =	1 1 + φ	-(2 -φ)ξ + 3ξ 2 ,
	with φ = 12EI/K e =	EI e (1 + φ) l 3	 12   	-6l e sym. l 2 e (4 + φ) 6l e l 2 -12 e (4 + φ) -6l e  l 2 e (2 -φ) 12 6l e    .

Appendix A

Linear element

The stiffness matrix of the classical linear element using exact integrations has the form

12

-6l e -12 -6l e l 2 e (4 + φ) 6l e l 2 e (2 -φ) 12 6l e sym.

l 2 e (4 + φ)


Nathalie Eisenbaum

On local times of Ornstein-Uhlenbeck processes

We establish expressions of the local time process of an Ornstein-Uhlenbeck process in terms of the local times on curves of a Brownian motion.

Introduction and main result

The purpose of this note is to provide expressions of the local time process of an Ornstein-Uhlenbeck process in terms of the local times of a Brownian motion. Indeed we have recently been confronted to the lack of such expression. More precisely, we have recently solved a stochastic differential equation modeling macrophage dynamics in atherosclerotic plaques [START_REF] Eisenbaum | A lipid structured model for macrophage dynamics in atherosclerotic plaques[END_REF]. The solution is a functional of the local time process of an Ornstein-Uhlenbeck process and surprisingly we could not find in the literature a tractable expression of this local time. Our contribution below (Theorem 1.1) is based on the notion of local time on curves for Brownian motion. Various definitions of this notion (see [START_REF] Bass | The supremum of Brownian local times on hölder curves[END_REF], [START_REF] Föllmer | Quadratic covariation and an extension of Itô's formula[END_REF], [START_REF] Eisenbaum | Integration with respect to local time[END_REF]) exist and all coincide. We remind in section 2, the ones that we need to establish Theorem 1.1. Theorem 1.1 is established in section 3. Finally in section 4, we make some remarks on the asymptotic behavior of the local time of the Ornstein-Uhlenbeck process. Fix λ > 0. Let Y be an Ornstein-Uhlenbeck process solution of

Y t = y 0 + W t - λ 2 t 0 Y s ds, (1.1)
where (W t , t ≥ 0) is a real Brownian motion starting from 0 and y 0 a fixed real number. Doob [START_REF] Doob | The Brownian movement and stochastic equations[END_REF] has established the following expression for Y

(Y t , t ≥ 0) = (e -λt/2 (Z u(t) + y 0 ), t ≥ 0), (1.2)
where Z = (Z t , t ≥ 0) is a real Brownian motion starting from 0 and u(t) = e λt -1 λ .

1

We will denote the local time process of Z by (L x t (Z), x ∈ IR, t ≥ 0).

As a continuous semimartingale Y admits a local time process (L y t (Y ), y ∈ IR, t ≥ 0), satisfying the following occupation time formula ( [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF] Chap VI -Corollary 1.6)

t 0 f (Y s )ds = IR f (y)L y t (Y )dy. (1.3)
Under the assumption that the Ornstein-Uhlenbeck process Y starts at 0, the expression of its local time at 0 has already been noticed in [START_REF] Gradinaru | The laws of Brownian local time integrals[END_REF] (

L 0 t (Y ), t ≥ 0) = ( e λt -1 λ 0 1 √ 1 + λs dL 0 s (Z), t ≥ 0).
In [START_REF] Gradinaru | The laws of Brownian local time integrals[END_REF], the result is given without proof. In their introduction, the authors first mention that the local time at 0 of a continuous semimartingale M = (ϕ(t)Z t ) t≥0 with Z real Brownian motion starting from 0, is given by (L 0 t (M )) t≥0 = ( t 0 ϕ(s)d s L 0 s (Z)) t≥0 . This can be actually obtained as a consequence of the extended occupation formula

t 0 h(s, Z s )ds = IR t 0 h(s, x) d s L x s (Z) dx, (1.4)
where d s L x s (Z) denotes integration with respect to the time variable. Using (1.2), Y can be written under the form (M u(t) ) t≥0 , with u(t) = e λt -1 λ and ϕ(t) = 1 √ 1+λt . One then obtains immediately the expression of its local time at 0

L 0 t (Y ) = L 0 u(t) (M ) = u(t) 0 ϕ(s)d s L 0 s (Z), t ≥ 0.
The problem is that this argument does not lead to a tractable expression of L a t (Y ) for a distinct from 0. The reason is that one has to deal with more general functional of Z than functionals of type t 0 h(s, Z s )ds. More precisely, the key notion to handle L a t (Y ) is stochastic integration over the plane with respect to the local time process (L x t (Z), x ∈ IR, t ≥ 0) as a doubly-indexed process. We give a brief exposure of that notion in section 2.

Theorem 1.1 Fix a real number y 0 . The Ornstein-Uhlenbeck process Y starting from y 0 , admits a local time process (L y t (Y ), y ∈ IR, t ≥ 0), related to the local time process of Z, as follows

(L a t (Y ), a ∈ IR, t ≥ 0) = ( e λt -1 λ 0 1 √ 1 + λs d s L Fa(.) s (Z), a ∈ IR, t ≥ 0) (1.5)
where F : a → F a is a functional from IR into the continuous path from IR + into IR, defined as follows

F a (t) = a √ 1 + λt -y 0 ,
and (L Fa(.) t (Z), t ≥ 0) denotes the local time process of Z along the curve F a . In particular, one has

(L 0 t (Y ), t ≥ 0) = ( e λt -1 λ 0 1 √ 1 + λs d s L -y 0 s (Z), t ≥ 0). Remark 1.2 Note that (1.5) is equivalent to L a t (Y ) = a u(t) 0 1 Z s + y 0 d s L Fa(.) s (Z).
Besides, to avoid integration with respect to (L Fa(.) t (Z), t ≥ 0), one can make an integration by parts, and obtain

L a t (Y ) = L Fa(.) u(t) (Z) (1 + λu(t)) 1/2 + λ/2 u(t) 0 L Fa(.) s (Z)(1 + λs) -3/2 ds, a ∈ IR, t ≥ 0. (1.6)
Remark 1.3 As a diffusion, the Ornstein-Uhlenbeck process Y admits a bicontinuous local time = ( x t (Y ), x ∈ IR, t ≥ 0) with respect to its speed measure m (m(dy) = 2e -λ 2 y 2 dy). One obviously has

( x t (Y ), x ∈ IR, t ≥ 0) = ( e λ 2 x 2 2 L x t (Y ), x ∈ IR, t ≥ 0).
But can also be related to the local time process of the Brownian motion W . Indeed denote by (L x t (W ), x ∈ IR, t ≥ 0) the local time process of W . For simplicity assume that Y starts from 0. Following Orey [START_REF] Orey | Conditions for the absolute continuity of two diffusions[END_REF], one obtains

( x t (Y ), x ∈ IR, t ≥ 0) = (L ρ(x) β(t) (W ), x ∈ IR, t ≥ 0),
where ρ is the inverse of S the scale function of Y (S(x) =

x 0 e λ 2 y 2 dy, x ∈ IR) and β is the inverse of the function α defined by

α(t) = IR L x t (W )(m • ρ)(dx) = IR L S(x) t (W )m(dx).
One finally obtains

(L a t (Y ), a ∈ IR, t ≥ 0) = (2e -λ 2 a 2 L ρ(a) β(t) (W ), a ∈ IR, t ≥ 0). (1.7)
Some questions (as questions of stochastic structure) on the local time of Y might be easier to handle using (1.7) than (1.5).

Local times on curves

Let Z be a real Brownian motion starting from 0. If one needs a local time of Z along a deterministic mesurable curve b = (b(t), t ≥ 0), the first idea that comes to mind is to define it as follows

L b(.) t (Z) = lim ε→0 1 2ε t 0 1 [b(s)-ε,b(s)+ε] (Z s )ds, (2.1)
which is precisely the definition introduced by Bass and Burdzy in [START_REF] Bass | The supremum of Brownian local times on hölder curves[END_REF]. They show that the limit is uniform in t on compact sets, in L 2 .

When b is such that (Z t -b(t)) t≥0 is a semi-martingale, then it admits a local time process and in particular a local time process at 0 which is given by Tanaka's formula

L 0 t (Z . -b(.)) = |Z t -b(t)| -|b(0)| - t 0 sgn(Z s -b(s))d(Z s -b(s)). (2.2)
As noticed in [START_REF] Bass | The supremum of Brownian local times on hölder curves[END_REF], in that case the two definitions (2.1) and (2.2) coincide a.s.

L b(.) t (Z) = L 0 t (Z . -b(.)), t ≥ 0.
In [START_REF] Eisenbaum | Integration with respect to local time[END_REF], we have defined a local time for Z along any mesurable curve by using integration with respect to local time over the plane. We first remind this notion of integration of deterministic functions with respect to local time over the plane. Consider the space H H = {F :

IR + × IR → IR : ||F || < ∞},
where the norm ||.|| is defined by

||F || = 2( ∞ 0 IR F 2 (s, x) exp(- x 2 2s ) dsdx √ 2πs ) 1/2 + ∞ 0 IR |xF (s, x)| exp(- x 2 2s ) dsdx s √ 2πs .
For F elementary function i.e. such that there exist a finite sequence of IR + , (s i ) 1≤i≤n , and a finite sequence of IR, (x j ) 1≤j≤n , with

F (s, x) = 1≤i,j≤n F (s i , x j )1 [s i ,s i+1 ) (s)1 [x j ,x j+1 ) (x), one sets t 0 IR F (s, x)dL x s = 1≤i,j≤n F (s i , x j )(L x j+1 s i+1 -L x j+1 s i -L x j s i+1 + L x j s i ).
We have shown in [START_REF] Eisenbaum | Integration with respect to local time[END_REF] that for every F in H, and every sequence of elementary functions converging to F for the norm ||.||, (F n ) n≥0 , the sequence (

t 0 IR F n (s, x)dL x s ) n≥0
converges in L 1 uniformly in t on compact sets. The limit does not depend on the choice of the sequence (F n ) and represents t 0 IR F (s, x)dL x s .

Moreover

IE[| t 0 IR F (s, x)dL x s |] ≤ ||F ||. (2.3)
We remind that for every t > 0, the process (Z t-s , 0 ≤ s ≤ t) is a semimartingale. Extending a result of [START_REF] Föllmer | Quadratic covariation and an extension of Itô's formula[END_REF], we have established in [START_REF] Eisenbaum | Integration with respect to local time[END_REF], that

v 0 IR F (s, x)dL x s = v 0 F (s, Z s )dZ s + t t-v F (t -s, Z t-s )dZ t-s , 0 ≤ v ≤ t. (2.4)
We have also shown (Corollary 3.2 (ii) in [START_REF] Eisenbaum | Integration with respect to local time[END_REF]) that if ∂F ∂x (s, x) exists, then

t 0 IR F (s, x)dL x s = - t 0 ∂F ∂x (s, Z s )ds. (2.5)
Finally, it has been established in [START_REF] Eisenbaum | Integration with respect to local time[END_REF] that for any measurable curve (b(s), s ≥ 0), one has a.s.

(L b(.) t (Z), t ≥ 0) = ( t 0 IR 1 {x<b(s)} dL x s (Z), t ≥ 0). (2.6)
Using (2.4), one obtains

(L b(.) v (Z), 0 ≤ v ≤ t) = ( v 0 1 {Zs<b(s)} dZ s + t t-v 1 {Z t-s <b(t-s)} dZ t-s , 0 ≤ v ≤ t).
3 Proof of Theorem 1.1

As a continuous semimartingale, the process Y admits a local time process (L y t (Y ), y ∈ IR, t ≥ 0) which satisfies the occupation time formula (1.3). This local time process admits a bicontinuous modification. Indeed, according to Theorem 1.7 in [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF], the local time process of Y is a.s. continuous in t and cadlag in y with the following discontinuities

L y t (Y ) -L y- t (Y ) = - λ 2 t 0 1 (Ys=y) Y s ds = - λy 2 t 0 1 (Ys=y) ds = 0, using (1.
3). We work with a bicontinuous modification of the local time process of Y . Using (1.2), we have

t 0 f (Y s )ds = t 0 f (e -λs/2 (Z u(s) + y 0 ))ds = e λt -1 λ 0 f ( 1 √ 1 + λu (Z u + y 0 )) du 1 + λu . (3.1)
We now choose the function f :

f (x) = f a,ε (x) = 1 2ε 1 [a-ε,a+ε] (x)
. On one hand, using the occupation time formula and the continuity of the local time process of Y , a.s. for every t ≥ 0 , every real a

lim ε→0 t 0 f a,ε (Y s )ds = L a t (Y ). (3.2)
On the other hand, defining the function F a,ε on IR + × IR by

F a,ε (s, x) = ∞ x 1 2ε 1 [a-ε,a+ε] ( y + y 0 √ 1 + λs ) 1 1 + λs dy,
one notes that for every T > 0, the function F a,ε (s, x)1 [0,T ] (s) belongs to H. Using (2.5), one obtains

e λt -1 λ 0 f a,ε ( 1 √ 1 + λs (Z s + y 0 )) ds 1 + λs = e λt -1 λ 0 IR F a,ε (s, x)dL x s (Z),
where dL x s (Z) refers to integration over the plane IR + × IR wrt local times (see section 2). For every T > 0, as ε tends to 0, the function F a,ε (s, x)1 [0,T ] (s) converges for the norm One easily shows that for any measurable bounded function h from

||.|| to 1 √ 1+λs 1 [x,+∞) (a √ 1 + λs -y 0 )1 [0,T ] (s)
IR + into IR T 0 h(s)d s L b(.) s (Z) = T 0 IR h(s)1 {x<b(s)} dL x s (Z).
Consequently,

e λt -1 λ 0 IR F a,ε (s, x)dL x s (Z) tends in L 1 to e λt -1 λ 0 1 √ 1 + λs d s L Fa(.) s (Z).
(Z) = |Z t + y 0 -a √ 1 + λt| -|a -y 0 | - t 0 sgn(Z s + y 0 -a √ 1 + λs)dZ s + aλ 2 t 0 sgn(Z s + y 0 -a √ 1 + λs) ds √ 1 + λs . ( 4

  . Hence, thanks to (2.3), one obtainse λt -1 λ 0 IR F a,ε (s, x)dL x s (Z)tends in L 1 as ε tends to 0, to 0 } dL x s (Z).

Hence, thanks to ( 3

 3 in L 1 uniformly in t on compact sets. This is still true finite-dimensionnaly in a. Using (3.2), one obtains then (1.5) by continuity arguments.

4

  Some asymptotics of local times Proposition 4.1 For Y Ornstein-Uhlenbeck process starting from any fixed real number, the following properties hold for every real a.

	(i) lim sup t→∞	e	L a t (Y ) λt 2 √ log t	≤ 2 2 λ a.s.
	and in particular: lim sup t→∞	e	L 0 t (Y ) λt 2 √ log(t)	≤ 2 λ a.s.
	(ii) lim sup t→∞	L a t (Y ) √ log t ≥ 2 λ a.s.
					Consequently
	(2.2) leads to
	Fa(.) L t

Proof: Denote by y 0 the starting point of Y . Since F a (s) is an increasing continuous function of s, M = (Z t -F a (t), t ≥ 0) is a continuous semimartingale.

  (ii) Using the left hand inequality of (4.2) one has:

	denoting by B the Brownian motion (-	L a t (Y ) √ 2 λ √ log t 0 sgn(Z s + y 0 -a t √ ≥ 1 + λs)dZ s , t ≥ 0), one L Fa(.) u(t) (Z) √ 2 λ e λt 2 . Besides, √ log t
	obtains with (4.1)
	lim sup t→∞	L Fa(.) u(t) (Z) 2 λ e λt 2 √ log t	≥ lim sup t→∞	B u(t) H(u(t))	+	|Z u(t) | H(u(t))	≥ lim sup t→∞	B u(t) H(u(t))	= 1 a.s.,
	which leads to the result.
										.1)
	Besides, Theorem 1.1 gives the following bounds
				L 1 + λu(t) Fa(.) u(t) (Z)	≤ L a t (Y ) ≤ L	Fa(.) u(t) (Z).	(4.2)
	Set: H(t) = 2t log 2 t. One has: H(u(t)) ∼ t→∞	2 λ e	λt 2	√	log t.
	(i) According to the well-known Strassen law of iterated logarithm [9], one has for B
	real Brownian motion	lim sup t→∞	B t H(t)	= 1 a.s.	(4.3)
	Applying it to the two Brownian motions Z and ( one obtains immediately from (4.1): lim sup t→∞ L Fa(.) t 0 sgn(Z s + y 0 -a u(t) e λt 2 √ log t ≤ 2 2 λ , which with the right √ 1 + λs)dZ s ) t≥0 ,
	hand inequality of (4.2) leads to (i).
	From (4.3) one easily deduces the following law of iterated logarithm for local time:
	lim sup t→∞	L -y 0 t H(t) (Z)	= 1 a.s. It leads to: lim sup t→∞	e	L -y 0 u(t) (Z) λt 2 √ log(t)	= 2 λ a.s. Using the right
	hand inequality of (4.2) for a = 0, one obtains: lim sup t→∞	e	L 0 t (Y ) λt 2 √ log(t)	≤ 2 λ a.s.


Keywords: mechanobiology, mechanosensing, focal adhesions, cytoskeleton, cell stretching, superresolution microscopy, single particle tracking

A micromechanical cell stretching device compatible with super-resolution microscopy and single protein tracking

Introduction

Growing evidence demonstrates that macromolecular assemblies driving critical cellular functions are regulated by mechanical forces. However, the exact molecular mechanisms of force-sensing within most macromolecular assemblies in cells are still unknown. Several innovative techniques allow to measure and generate forces on proteins in vitro or within cells [START_REF] Goldyn | Force-induced cell polarisation is linked to RhoA-driven microtubule-independent focal-adhesion sliding[END_REF][START_REF] Grashoff | Measuring mechanical tension across vinculin reveals regulation of focal adhesion dynamics[END_REF][START_REF] Polacheck | Measuring cell-generated forces: A guide to the available tools[END_REF][START_REF] Iskratsch | Appreciating force and shape -the rise of mechanotransduction in cell biology[END_REF] , but they cannot probe protein mechanical responses within crowded macromolecular structures inside the cell or con ned at the interface with the extracellular environment.

Super-resolution microscopy (SRM) and single protein tracking (SPT) techniques are ideal to study the dynamic organization of proteins at the nanoscale. There are two major classes of SRM: coordinate stochastic based on Single Molecule Localization Microscopy (SMLM: PALM, STORM, PAINT) [START_REF] Liu | Imaging Live-Cell Dynamics and Structure at the Single-Molecule Level[END_REF][START_REF] Sahl | Fluorescence nanoscopy in cell biology[END_REF][START_REF] Huang | Super-resolution uorescence microscopy[END_REF] ; or coordinate targeted (STED, RESOLFT) [START_REF] Liu | Imaging Live-Cell Dynamics and Structure at the Single-Molecule Level[END_REF][START_REF] Sahl | Fluorescence nanoscopy in cell biology[END_REF][START_REF] Huang | Super-resolution uorescence microscopy[END_REF] . SRM and SPT con rmed, together with uorescent tensionsensors, that proteins could be stretched under mechanical tension 2,8-10 in mechanosensitive structures.

In parallel, stretchable substrates of Polydimethylsiloxane (PDMS) have been coupled with optical imaging to investigate cell responses to external forces [START_REF] Goldyn | Force-induced cell polarisation is linked to RhoA-driven microtubule-independent focal-adhesion sliding[END_REF][START_REF] Sinha | Cells respond to mechanical stress by rapid disassembly of caveolae[END_REF][START_REF] Chen | Orientation-speci c responses to sustained uniaxial stretching in focal adhesion growth and[END_REF] . However, the simultaneous combination of SRM or SPT with cell stretching is extremely challenging, since it requires to combine glass-like optical properties with mechanical stability of the imaged plane during substrate deformation.

SMLM/SPT require the optimal signal to noise ratio of single molecules emission to attain the best spatial resolution (typically 10-50 nm) [START_REF] Liu | Imaging Live-Cell Dynamics and Structure at the Single-Molecule Level[END_REF][START_REF] Sahl | Fluorescence nanoscopy in cell biology[END_REF][START_REF] Huang | Super-resolution uorescence microscopy[END_REF] . SMLM/SPT techniques are thus ideally performed in the total internal re ection uorescence (TIRF) or oblique illumination modes using high numerical aperture (NA) short working distance oil immersion objectives matching the index of refraction of glass slides [START_REF] Liu | Imaging Live-Cell Dynamics and Structure at the Single-Molecule Level[END_REF][START_REF] Sahl | Fluorescence nanoscopy in cell biology[END_REF][START_REF] Huang | Super-resolution uorescence microscopy[END_REF][START_REF] Giannone | Dynamic superresolution imaging of endogenous proteins on living cells at ultrahigh density[END_REF][START_REF] Rossier | Integrins β1 and β3 exhibit distinct dynamic nanoscale organizations inside focal adhesions[END_REF] .

In addition, SMLM/SPT techniques rely on object reconstruction or tracking from thousands of imaging planes, which implies perfect mechanical stability of the sample while imaging. This is incompatible with large deformations and displacements of the substrate in the axis (Z) and plane (XY) of observation during stretching.

Similarly, coordinate targeted STED/RESOLFT nanoscopy techniques perform better using high numerical aperture (NA) short working distance oil immersion or glycerol immersion objectives. Compared to SMLM/SPT techniques, STED like techniques will be less sensitive to drift, as they have low pixel dwell times [START_REF] Vangindertael | An introduction to optical super-resolution microscopy for the adventurous biologist[END_REF][START_REF] Vicidomini | STED super-resolved microscopy[END_REF] . However, mechanical drift stemming from multiple factors (e.g., motorized and piezoelectric stages) can greatly compromise the performance of the STED system, degrading signal-tonoise ratio and spatial resolution [START_REF] Vangindertael | An introduction to optical super-resolution microscopy for the adventurous biologist[END_REF][START_REF] Vicidomini | STED super-resolved microscopy[END_REF] . Thus, perfect mechanical stability of the sample has to be ensured throughout acquisitions. Once again, this could be incompatible with large XYZ deformations during stretching.

In most commercial con gurations, cell stretching is performed using macroscopic devices and images are acquired after xation, or thick elastomeric substrates are stretched in combination with low NA objectives and upright microscopes [START_REF] Sawada | Force transduction by Triton cytoskeletons[END_REF][START_REF] Giannone | Talin1 is critical for forcedependent reinforcement of initial integrin-cytoskeleton bonds but not tyrosine kinase activation[END_REF] . These low magni cation con gurations are quite permissive to slight defocusing [START_REF] Jungbauer | Two characteristic regimes in frequencydependent dynamic reorientation of broblasts on cyclically stretched substrates[END_REF] . Various custom-made devices could potentially enable simultaneous stretching and live cell imaging [START_REF] Goldyn | Force-induced cell polarisation is linked to RhoA-driven microtubule-independent focal-adhesion sliding[END_REF][START_REF] Sinha | Cells respond to mechanical stress by rapid disassembly of caveolae[END_REF][START_REF] Jungbauer | Two characteristic regimes in frequencydependent dynamic reorientation of broblasts on cyclically stretched substrates[END_REF][START_REF] Kosmalska | Physical principles of membrane remodelling during cell mechanoadaptation[END_REF][START_REF] Wang | Tissue-speci c mechanical and geometrical control of cell viability and actin cytoskeleton alignment[END_REF][START_REF] Camelliti | Micropatterned cell cultures on elastic membranes as an in vitro model of myocardium[END_REF] . However, they are either limited to low-magni cation imaging or are incompatible with continuous automatic focusing during stretching

Here, we present a cell stretching device compatible with SRM and SPT which enables to study the nanoscale reorganizations and deformations of protein assembly or individual proteins inside mechanosensitive structures [START_REF] Massou | Cell stretching is ampli ed by active actin remodeling to deform and recruit proteins in mechanosensitive structures[END_REF] (Fig. 1a). This device is composed of an ultra-thin Polydimethylsiloxane (PDMS) layer (10 µm) providing glass-like optical properties compatible with SRM and SPT (Fig. 1b,c). To simultaneously enable substrate stretching and ensure atness upon deformation, this ultra-thin PDMS layer is deposited on a glycerol-lubricated glass cover-slip (Fig. 1b,c). Glycerol allows the deformable substrate to oat freely while avoiding PDMS adhesion and refractive index mismatch. A drop of glycerol is sandwiched between the plasma cleaned PDMS sheet and glass coverslip forming a glycerol layer of ∼ 0.7 µm in thickness (Fig. 1c) To manipulate the thin PDMS substrate and avoid any distortions along the optical path, the size of the observation chamber is kept as small as possible (∼9 mm2 (Fig. 1b)), and its mechanical stability is reinforced by adding a thicker (40 µm) Gel-Pak frame on top of the thin PDMS sheet (Fig. 1c). To generate uniaxial stretches on the glass-PDMS assembly, we design a 3D-printed micro-device (Fig. 1a,b). The micromechanical device consists of a xed (holding) arm and a mobile (stretching) arm, connected to a piezo motor, positioned on opposite sides of the observation chamber on the PDMS frame (Fig. 1a,b). A clamp xes the glass-PDMS slide to the base of the device and to the holding arm (Fig. 1a). The observation chamber or the whole microchip can be lled with culture or observation medium.

Our micromechanical stretching device is compatible with the two major classes of SRM: coordinate stochastic (e.g. PALM, STORM, PAINT) and coordinate targeted (e.g. STED) [START_REF] Liu | Imaging Live-Cell Dynamics and Structure at the Single-Molecule Level[END_REF][START_REF] Sahl | Fluorescence nanoscopy in cell biology[END_REF][START_REF] Huang | Super-resolution uorescence microscopy[END_REF][START_REF] Schnitzbauer | Super-resolution microscopy with DNA-PAINT[END_REF] (Fig. 2). Experiments can be divided into two main approaches: 1) live stretching and SRM/SPT imaging in live cells (Fig. 2a-c) and 2) live stretching followed by rapid xation, labelling and SRM imaging (Fig. 2d,e).

Concerning the rst approach, we can perform SPT acquisitions in cells before and after large (10-50%)

or small (2-5%) stretches to study the effect of external stress on protein dynamics and diffusive properties (Fig. 2a). In addition, we can also perform simultaneous live cell stretching (2-5%) and SPT to study acute mechanical response of individual proteins: 1) force-dependent protein unfolding or deformations (Fig. 2b); or 2) force-dependent protein recruitments and reorganizations. Finally, the device can also be used to acquire STED SRM images of live cells that experience stretching (Fig. 2c). Regarding the second approach, the device can be used to perform SRM (e.g. DNA-PAINT, STED, STORM) in xed cells after live stretching (2-50 %) followed by rapid xation and labelling for SRM (Fig. 2d,e).

Reagents

Reagents for assembling the micromechanical device • Inverted confocal microscope (Leica SP8 WLL2) equipped with a HC PL APO CS2 motCORR 93X Glycerol, NA 1.3 objective. The confocal microscope is equipped of a white light laser 2 (WLL2) with freely tunable excitation from 470 to 670 nm (1 nm steps) and is also equipped with a STED module tunable to STED microscopy. STED module is equipped with 3 depletion lasers: 592 nm, 660 nm and 775 nm.

•

Inverted motorized microscope (Nikon Ti) equipped with a CFI Apo TIRF 100x oil, NA 1.49 objective and a perfect focus system PFS-2. Microscope is equipped with 4 continuous wave (cw) lasers (405 nm, 488 nm, 561 nm, 643 nm). Assembling the micromechanical device Glass-PDMS assembly 1. Plasma clean 24x40 mm glass coverslips.

2. Spin coat each slide with glycerol to form a uniform glycerol layer of ∼0.7 µm thickness.

3. Plasma clean the PDMS substrate for 1 minute, on the side that will glide on the glycerol layer.

4. Immediately after, lay the PDMS substrate onto the glycerol-coated coverslip.

Printing the 3D micro-device 1. Design the xed (holding) arm, the mobile (stretching) arm and the clamp composing the device on a CAD software.

a. Note: For experiments that require sustained stretching after xation, the design of the holding arm is modi ed to include a threaded hole, while the stretching arm is enlarged to include two grooves. Two screws allow to clamp the stretching arm onto the holding arm and sustain the stretching.

2. Export the les (.obj or any 3D printing compatible format) and upload them onto PreForm software. Layer thickness should be intermediate or small to avoid imperfections in the nal structure. As for the resin, we recommend using Grey Pro resin because it offers high precision, moderate elongation, and low creep (https://formlabs.com/eu/materials/engineering/#grey-pro-resinGrey Pro Resin). This material is ideal for concept modeling and functional prototyping, especially for parts that will be handled repeatedly.

a. Note: Ensure that the support material is correctly placed to avoid collapsing of the piece during the printing process.

3. After nishing the printing, wash and cure the pieces according to the requirements of the resin. 3Dprinted micro devices can be re-used several times, especially if they are composed of a resin conceived for engineering.

a. Note: Although we use SLA 3D printing for our micro-devices, we have also tested 3D micro-devices printed with fused deposition modelling (FDM). It requires polylactic acid (PLA) laments and it can be easily implemented in a lab, besides having shorter printing times. However, we found that the devices are less resistant and less durable than the ones printed with SLA.

Attaching glass-PDMS assembly to 3D-printed micro-device 1. Cut glass coverslips in small parts with high precision knife.

2. Using superglue or Dow Corning™ High-Vacuum Grease, attach small glass parts to holding and stretching arm.

3. Stick double sided tape to the glass parts on the holding and stretching arm.

4. Place the glass-PDMS assembly inside the clamp and attach them to the holding arm 5. Attach the stretching arm to the glass-PDMS assembly using double-sided tape.

Cell preparation

Cell electroporation (24 to 48h before imaging)

Actively dividing immortalized MEFs are cultured in DMEM supplemented medium (see solutions for cell culture) in 75 cm2 asks. b. Note: For STED experiments, if necessary, cells transfected with a GFP-coupled reporter of the structure of interest (1-2 μg per condition, e.g., tubulin-GFP for microtubules). Presence of a GFP-coupled reporter is only required for low-resolution images of the cells before and after stretching, especially when performing large stretches (e.g. , 30%) c. Note: For DNA-PAINT microscopy, vimentin-Halo can be visualized with Cy3B-labelled DNA imager strands, added to the stretching chamber at variable concentrations (2-5 nM), as previously described [START_REF] Schnitzbauer | Super-resolution microscopy with DNA-PAINT[END_REF] .

4. For all cases, electroporate the cells with the Nucleofector™ 2b Device using the MEF T-020 program (Lonza Nucleofactor protocol) 5. After electroporation, replate the cells in a 6-well plate (about 0.3 million cells/well) in preheated culture medium and place them in a 37 °C incubator with humidi ed air containing 5% CO2

Coating the micromechanical stretching device (same day of the experiment)

1. Cover the micromechanical device with 10 μg/ml bronectin solution (500-750 µl per device) and incubate at 37 °C for 1 to 1.5 hours.

2. Discard the bronectin solution and wash 4-5 times with PBS.

Sample preparation

1. Wash cells twice with PBS after removing the culture medium 2. Incubate with trypsin-EDTA solution (0.3 ml per well) for 1-3 min at 37 °C for detaching cells.

3. Inactivate trypsin with trypsin inactivation medium (1 ml per well) and count the cells (use any conventional cell counting method).

4. Centrifuge at 300 ×g for 5 min and resuspend cells in warm Ringer medium (1-2 ml). Allow cells to recover for 20-30 min inside the incubator at 37 °C with 5% CO2

5. Spread the cells on the device under a density of 70000-80000 cells per device.

Mounting the stretching device for live stretching or large stretches followed by rapid xation

The following steps are common for mounting and preparing the micromechanical device for stretching experiments in live cells or xed cells combined with super resolution microscopy and single protein tracking.

1. Prepare a 1:500 solution of TetraSpeck™ 0.1 µm uorescent beads in warm Ringer solution and add 200-300 µL to each device to adsorb uorescent beads on the substrate.

2. Mount the piezoelectric motor on a custom holder adapted to the microscope stage. Then, mount the holder on the microscope.

a. Note: Custom-made holders should be designed according to speci c stage dimensions and can be either 3D-printed or 3D-milled. In this case, we have 3D-milled a PMMA holder for mounting our device on the motorized stage, since 3D-milled parts are often more resistant.

3. Connect the piezoelectric motor controller and launch the PI Mikro Motor software.

4. Mount the micromechanical device on the stage-adapted holder.

5. Attach the stretching arm to the linear stage with a screw 6. Adjust focus to the 3x3 mm observation 10 µm PDMS chamber.

7. Before acquiring any cells, test whether the device is working properly and de ne parameters for speci c stretching percentages. For that, select a region with a good density of uorescent beads and acquire a snapshot. Measure the distance between two beads in the same horizontal axis (D before ). Apply a test stretching by displacing the linear stage of the piezoelectric motor while keeping the same region in focus, either manually or with a custom-written Metamorph routine. Take another snapshot after the stretching and measure again the distance between the same two beads (D after ). If D after is larger than D before , then the device is working properly. After that, cell stretching can be performed.

a. Note: Stretching percentage is calculated by , a formula which can be applied for all stretching experiments in order to determine stretching percentage.

b. Note: By knowing the displacement and the percentage for the test stretching, it is possible to determine with reasonable precision the necessary displacement to obtain a desired stretching percentage when performing actual cell stretching.

c. Note: Step size and speed of motor displacement should be kept consistent throughout all acquisitions.

Live cell stretching combined with super-resolution microscopy and single protein tracking

Stretching and live sptPALM

Cells are imaged at 37°C in the micromechanical device. Here, an inverted motorized microscope (Nikon Ti) was used, equipped with a CFI Apo TIRF 100x oil, NA 1.49 objective and a perfect focus system PFS-2), allowing long acquisition in TIRF illumination mode. For photoactivation localization microscopy, cells expressing mEos2/tdEos tagged constructs were photoactivated using a 405 nm laser (Omicron) and the resulting photoconverted single molecule uorescence was excited with a 561 nm laser (Cobolt Jive).

Both lasers illuminated the sample simultaneously. Their respective power was adjusted to keep the number of the stochastically activated molecules constant and well separated during the acquisition. Fluorescence was collected by the combination of a dichroic and emission lters (dichroic: Di01-R561, emission: FF01-617/73, Semrock) and a sensitive EMCCD (electron-multiplying charge-coupled device, Evolve, Photometric). The acquisition was steered by Metamorph software (Molecular Devices) in streaming mode at 50 Hz. GFP-paxillin was imaged using a conventional GFP lter cube (excitation:

FF01-472/30, dichroic: FF-495Di02, emission: FF02-520/28, Semrock).

1. For simultaneous stretching and sptPALM with trapeze like patterns: select a cell and launch a PALM acquisition at high frequency (50 Hz) for the entire eld of observation. The duration of the acquisition should comprise the entire trapeze pattern (at least 2500 frames). 1. For sequential large stretching and sptPALM (Before vs After): select a cell and, after acquiring an image of the GFP reporter, launch a PALM acquisition at high frequency (50 Hz) for the entire eld of observation throughout 4000 frames.

2. Perform large stretching (10-50%) while following the cell displacement with the combination of a dichroic and emission lters (dichroic: Di01-R561, emission: FF01-617/73, Semrock). After stretching is nished, acquire an image of the GFP reporter (to have a perspective of the morphological changes) and launch another sptPALM acquisition.

Stretching and live STED

Cells are imaged at 37°C in the micromechanical device. Here, an inverted confocal microscope (Leica SP8 WLL2) was used, equipped with a HC PL APO CS2 motCORR 93X Glycerol, NA 1.3 objective. The confocal microscope was equipped of a white light laser 2 (WLL2) with freely tuneable excitation from 470 to 670 nm (1 nm steps). Scanning was done using a conventional scanner (10Hz to 1800 Hz). The confocal microscope was equipped with the STED module tunable to STED microscopy. A twodimensional (2D) STED donut was generated using a vortex phase plate. This STED microscope was equipped with 3 depletion lasers: 592 nm, 660 nm and 775 nm For STED microscopy, cells were imaged with a combination of a WLL2 laser and a 775 nm depletion laser. Fluorescence was collected with an internal hybrid detector. The acquisition was steered by LASX Software (Leica).

1. Perform live labelling of the target protein on the micromechanical device. For actin or tubulin labelling, use SiR-Actin or SiR-Tubulin compounds, according to previous studies and manufacturer's instructions [START_REF] Lukinavičius | Fluorogenic probes for live-cell imaging of the cytoskeleton[END_REF] .

2. After the labelling, wash the staining solution and incubate the device in warm Ringer solution until the experiment.

3. After bead incubation and mounting of the sample as previously described, select a cell and acquire a confocal image followed by a STED image on a sub region of the cell (pixel size has to be inferior to 20 µm).

4. Stretch the cell according to the desired percentage (e.g. 4 or 30%) and maintain the cell on the eld of observation by observation by compensating for XY displacements using manual repositioning (Leica stage steered by a joystick) 5. After stretching, acquire a new confocal image followed by a STED image for the same sub region.

Super-resolution microscopy in xed cells with stretches

The following steps are required to perform large and sustained stretching followed by rapid cell xation.

1. Warm 4% PFA with 0.25% Glutaraldehyde at 37°C.

2.

After mounting the micromechanical device on the microscope, acquire several low resolution images of GFP markers for different cells.

3. Remove the entire module (device and motor in the stage-adapted holder) and stretch the cells outside the microscope. Immediately after stretching, remove the Ringer solution and x the cells in warm 4% PFA with 0.25% Glutaraldehyde.

4. After xation, rinse 3-4X with PBS.

5. Clamp the stretching arm to the holding arm using the thread and groove system and two M3 screws. Afterwards, remove the screw that connects the stretching arm to the motor. With this, stretching is sustained throughout all the subsequent labelling steps and super-resolution imaging.

6. Label target proteins according to the imaging technique.

7. After labelling, and before performing super-resolution imaging, acquire again several low resolution images of GFP markers for the same cells.

DNA-PAINT on xed cells with large stretches

Cells are imaged at 25°C in Buffer C in the same microscope used for live sptPALM. Cy3B-labelled strands were visualized with a 561 nm laser (Cobolt Jive). Fluorescence was collected by the combination of a dichroic and emission lters (dichroic: Di01-R561, emission: FF01-617/73, Semrock) and a sensitive sCMOS (scienti c CMOS, ORCA-Flash4.0, Hammatasu). The acquisition was steered by Metamorph software (Molecular Devices) in streaming mode at 6.7 Hz. Vimentin-GFP was imaged using a conventional GFP lter cube (excitation: FF01-472/30, dichroic: FF-495Di02, emission: FF02-520/28, Semrock). Super-resolution DNA-PAINT reconstruction and drift correction were carried out as described before, using the software package Picasso [START_REF] Schnitzbauer | Super-resolution microscopy with DNA-PAINT[END_REF] .

1. Dilute the desired imager strand in Buffer C. For instance, Vimentin-Halo was visualized with Cy3Blabelled DNA imager strands.

2. Incubate the micromechanical device with 200-300 µL of 90 nm gold nanoparticles, diluted in PBS with a 1:5 ratio, for 15 min at RT. Gold nanoparticles serve as ducial markers 3. Wash 3X with PBS and 1X with Buffer C.

4. Add imager strands to the stretching chamber until reaching the ideal density of blinking events. Vimentin-Halo was visualized with Cy3B-labelled DNA imager strands, added to the stretching chamber at variable concentrations (2-5 nM), as previously described [START_REF] Schnitzbauer | Super-resolution microscopy with DNA-PAINT[END_REF] .

STED on xed cells with large stretches

Cells are imaged at 25°C in PBS 1X in the same microscope used for live STED, with the same combination of a WLL2 laser and a 775 nm depletion laser.

Troubleshooting

Auto uorescence of the medium inside the micromechanical device This is likely caused by the 3D-printed micro-device itself. Since resins used for SLA are photoactivable at UV light, they can also be excited by the 405 nm laser used for sptPALM. Detachment of particles from the 3D-printed micro-device will therefore be visible and might interfere with the mEos /tdEos signal.

There is no clear information about uorescence of different SLA resins so, if auto uorescence persists, two possibilities exist. First, the entire holding arm can be covered with Dow Corning™ High-Vacuum Grease, which prevents release of auto uorescent particles. Alternatively, other resins can be tested, such as Black (Formlabs).

PDMS is not stretched

Case 1: During PDMS elastic substrate production, air bubbles formed when Gel-Pak frame was brought in contact with 10 µm PDMS layer on the silicon wafer. Usually this will generate irregularities in the substrate which, if close to the 3x3 mm observation chamber, will affect the stretching.

Case 2: Glycerol coating was incomplete and thus PDMS adhered to the glass.

Case 3: Plasma cleaning of PDMS elastic substrate was not effective (either too long or too short) and thus PDMS is not gliding on glycerol.

Case 4: Stretching arm was not properly attached to the glass-PDMS assembly Sustained stretching causes the micromechanical device to rupture or leak

The groove and thread system is still rather preliminary and requires further optimization, since it causes pressure on the micro-device, leading to leakage of liquid from inside of the device. Different designing strategies can be developed to sustain stretching, such as two 3D-printed clamps from the side and an adjustable screw in the middle, or simply a 3D-printed rod that blocks the recoil of the arm.

Time Taken

Before experiment:

Assembling the micromechanical device

Successful PDMS gliding on glycerol layer can be observed later in the microscope during stretching, by calculating the actual displacement after a test stretching. The device will be properly fully assembled if no medium is leaking from the chamber and the arm remains well attached throughout all subsequent steps.

Cell

the device is well coated and cells are viable after electroporation, detachment and seeding, they should start to spread brie y after seeding. 1h later, spread cells should be visible in the observation chamber of 3x3 µm.

4. Preparing the stretching device for live stretching or large stretches followed by rapid xation Bead can be quickly con rmed using GFP lter cube. Successful stretching and viability of the device can be con rmed by performing a test stretch and assessing that the same two beads are more distant between each other after stretching. If the device is assembled correctly, stretches closer to the holding arm will produce smaller displacements.

5. Live cell stretching combined with super-resolution microscopy and single protein tracking

Filipe

  Nunes Vicente, Sophie Massou, Franziska Wetzel, Amine Mehidi, Dan Strehle, Cecile Leduc, Raphaël Voituriez, Olivier Rossier, Pierre Nassoy, Grégory Giannone To cite this version: Filipe Nunes Vicente, Sophie Massou, Franziska Wetzel, Amine Mehidi, Dan Strehle, et al.. A micromechanical cell stretching device compatible with super-resolution microscopy and single protein tracking. 2021. hal-03402139

•

  Nucleofector™ 2b device (Lonza, Cat. No. AAB-1001) • 37 °C incubator with humidi ed air containing 5% CO2 • Softwares: o Computer-Assisted Design (CAD) software, such as Inventor (Autodesk) or open source versions (FreeCAD) o PreForm (Formlabs) o Metamorph (Molecular Devices) o Leica Application Suite X (Leica Microsystems) o PI Mikro Motor (Physik Instrumente) Procedure Fabricating a 10 µm PDMS elastic substrate 1. Using the Graphtec Cutting Plotter, pre-cut an elastomer Gel-Pak frame (40 µm) to the size of the glass coverslip with a squared (3x3 mm) observation chamber. a. Note: Layout for cutting can be drawn on the software provided by the company or using a plug-in for Adobe Illustrator® and CorelDraw®. 2. Mix PDMS and curing agent in a 10:1 ratio. 3. Centrifuge for 5 min at 2500 rpm to remove air bubbles 4. Spin the PDMS on a silicon wafer to a nal thickness of 10 µm. 5. Pre-cure the PDMS for 25 min at 70°C 6. Place the pre-cut Gel-Pak frame in contact with the 10 µm PDMS 7. Curate the whole assembly overnight at 70°C 8. Cut around the Gel-Pak frame with a blunt pair of and gently detach the PDMS substrate from the wafer.

1 . 2 .

 12 Detach cells with trypsin/EDTA solution (1.5 mL). Inactivate trypsin immediately after detachment by adding serum-containing DMEM (5 ml). Count cells and adjust cell suspension volume to keep 1-2 million cells per tube per experimental condition. Centrifuge cells at 300 × g for 5 min 3. Resuspend the cell pellet in transfection reagent and mix with the DNA plasmids. a. Note: For sptPALM experiments, cells are usually co-transfected with DNAs encoding for the protein of interest, (3-5 μg per condition, e.g., Talin-C-tdEos), and for a GFP-coupled reporter of the structure of interest (1-2 μg per condition, e.g., GFP-paxillin for adhesive structures).

2 .

 2 Several hundred frames after, stretch by displacing the linear stage of the piezoelectric motor. Maintain the cell in the eld of observation by compensating for XY displacements using manual repositioning (Nikon stage steered by a joystick) or automated stage repositioning (custom plugin developed in Metamorph). After the stretching has stopped, allow the plateau phase to last for 8-12 seconds before relaxation. a. Note: When looking at focal adhesions (FAs), select cells with most of their FAs aligned almost parallel to the stretching axis in the eld of observation. b. Note: Imaging cells closer to the holding arm requires smaller XY repositioning while allowing to reach 6 % stretching.

Figures

  Figures

Figure 1 Developing

 1 Figure 1

  Harrick Plasma, Cat. No. PDC-002-(230V)) with PlasmaFlo gas mixer (Harrick Plasma, Cat. No. PDC-FMG-2 (230V))

	•	NaCl (Sigma-Aldrich, Cat. No. S5886-5KG)
	•	KCl (Sigma-Aldrich, Cat. No. P5405-1KG)
	•	MgCl 2 (Sigma-Aldrich, Cat. No. M4880-100G)
	•	Glucose (Sigma-Aldrich, Cat. No. G7021-1KG)
	•	3D printer: We recommend the Form series (Form 2 or 3, Formlabs), for SLA 3D printing
	•	Fibronectin (Sigma-Aldrich, Cat. No. 10838039001)
	•	Piezoelectric motor (M-663 Linear Positioning Stage, 19 mm, Linear Encoder, 0.1 μm resolution, PI)
	•	Phosphate-buffered saline (PBS), pH 7.4
	•	CharlyRobot 3D milling machine (Mécanuméric)
	•	Nucleofector™ transfection kit for MEF-1 (Lonza, Cat. No. VPD-1004)
	•	Poly(methyl methacrylate) (PMMA) plates (Evonik)
	•	0.1 µm uorescent beads (TetraSpeck™ Microspheres, 0.1 µm, ThermoFisher, Cat. No. T7279)
	•	90 nm Stabilized Gold Nanoparticles (Cytodiagnostics, CG-90-500)
	•	PDMS (Sylgard 184, Samaro, Cat. No. DE9330)
	•	PF lm XO 1.5 mil (Gel-Pak®)
	•	Glycerol for uorescence microscopy (Merck, Cat. No. 1040950250)
	•	Resin for Stereolitography (SLA) 3D printing. We recommend Grey Pro resin (Formlabs)
	•	Dow Corning™ High-Vacuum Grease (Fisher Scienti c, Cat. No. 14-635-5D)
	Reagents for cell culture, sample preparation and imaging
	•	DMEM High Glucose with Sodium Pyruvate (Biowest, Cat. No.L01606)
	•	Fetal bovine serum (FBS, Eurobio, Cat. No. CVFSVF00-01)
	•	GlutaMAX (GIBCO, Cat. No. 35050038)
	•	Penicillin-Streptomycin (GIBCO, Cat. No. 15140-122)
	•	HEPES (GIBCO, Cat. No. 15630-056).
	•	Trypsin (GIBCO, Cat. No. 25300-054)
	•	Soybean trypsin inhibitor (Sigma, Cat. No. T9003-1G)

• Paraformaldehyde, 16% aquous solution (Sigma, Cat. No. P6148) • Glutaraldehyde, 10% aquous solution (Fisher Scienti c, Cat. No. 50-262-01) Solutions for cell culture, sample preparation and imaging • Cell culture medium: DMEM High Glucose with Sodium Pyruvate, supplemented with 10% FBS , 2 mM GlutaMAX, 100 U/ml penicillin-streptomycin and 15 mM HEPES. • Trypsin inactivation medium: DMEM High Glucose with Sodium Pyruvate, 1 mg/ml soybean trypsin inhibitor, 2 mM GlutaMAX, 100 U/ml penicillin-streptomycin, 15 mM HEPES Trypsin inhibitor. • Ringer solution: 150 mM NaCl, 5 mM KCl, 2 mM CaCl2, 2 mM MgCl2, 10 mM HEPES ,11 mM glucose at pH 7.4. • Buffer C: 1× PBS at pH 7.2 supplemented with additional 500 mM NaCl, and it can be stored at RT for up to 6 months 24 . Equipment • Silicon wafers CZ , 4" thickness 525 µm type/dopant , n-P, resistance 10-20 ohm.cm, 1 polished side, Ra < 0.5 nm (Neyco, Cat. No. WAS4P1020) • Spin Coater SPIN150i-PTFE (SPS Europe, Cat. No. 42024) • Graphtec cutting plotter (Graphtec Craft ROBO pro, CE5000-40-CRP) • Heating and drying lab oven • Glass coverslips no. 1.5H 24x40 mm (Marienfeld, Cat. No. 0107242) • Expanded plasma cleaner PDC-002 (• M3 screws, 20 cm (RS, Cat. No. 849-423)

Acknowledgements

We thank Z. Karatas, M. Fabre, R. Sterling, J. Carrere for technical assistance; T. Orré for helpful discussions. C. Poujol, S. Marais (Bordeaux Imaging Center, BIC) for technical help; F. Cordelières. (BIC) for support in kymograph analysis (Kymo Tool Box). We thank J. B. Sibarita (IINS, Bordeaux, France) for his support in sptPALM analysis; V. Studer (IINS, Bordeaux, France) and P.O. Strale (alvéole) for important discussions on microfabrication; M. Cabilic (IINS, Bordeaux, France) for the repositioning plugin developed in Metamorph; T. Schlichthärle, L. Fischer, R. Jungmann (MPI Biochemistry, Munich, Germany) and C. Grashoff (Institute for Molecular Cell Biology, Münster, Germany) for providing materials, support and important discussions on DNA-PAINT; P. Kanchanawong (MBI, NUS, Singapore) for providing the talin1-22-tdEOS (Talin-C), tdEOS-18-Talin1 (Talin-N), mEos2-7-TalinABS (THATCH) constructs. We acknowledge nancial support from the French Ministry of Research and CNRS, ANR grant Integractome (GG), ANR grant FastNano (GG), ANR grant IntegrinNanoPlan (GG), ANR Decav-Recav (PN, ANR MecanoCav (PN), Fondation pour la Recherche Médicale (GG), and the Institut National du Cancer (PN).

Successful combination of live sptPALM or STED with stretching is achieved if 1) focus is maintained throughout stretching, 2) Live XYZ repositioning is possible throughout stretching and 3) quality of either the single-molecule signal or the STED effect is assured throughout the entire acquisition.

Super-resolution microscopy in xed cells with large stretches

Sustaining the stretching after xation can be veri ed immediately after xation by assessing morphology of the 3x3 chamber on a bench top microscope. If the chamber remains deformed, stretching is maintained. This can also be con rmed when imaging the sample after labelling, by re-measuring the distance between the same two beads used to test the stretching before xation. For DNA-PAINT acquisitions to be successful, drift correction is a key aspect and has to be ensured.