V Rey

S Krumscheid

F Nobile

Quantifying uncertainties in contact mechanics of rough surfaces using the Multilevel Monte Carlo method

We quantify the eect of uncertainties on quantities of interest related to contact mechanics of rough surfaces. Specically, we consider the problem of frictionless non adhesive normal contact between two semi innite linear elastic solids subject to uncertainties. These uncertainties may for example originate from an incomplete surface description. To account for surface uncertainties, we model a rough surface as a suitable Gaussian random eld whose covariance function encodes the surface's roughness, which is experimentally measurable. Then, we introduce the multilevel Monte Carlo method which is a computationally ecient sampling method for the computation of the expectation and higher statistical moments of uncertain system output's, such as those derived from contact simulations. In particular, we consider two dierent quantities of interest, namely the contact area and the number of contact clusters, and show via numerical experiments that the multilevel Monte Carlo method oers signicant computational gains compared to an approximation via a classic Monte Carlo sampling. 1

Introduction

Contact between interfaces is omnipresent in both nature and engineering applications, so that understanding the mechanical response to such contacts is a major scientic challenge of current interest. In fact, these contact problems can be found in virtually any engineering application characterized by interactions between separate parts and involving, for example, friction or wear.

Despite its fundamental importance, many aspects of contact mechanics remain poorly understood. One of the key challenges stems from the multiscale nature of the contact interfaces. Indeed, both natural and manufactured surfaces are not perfectly at but are made of multiple asperities that exist at all length scales. The rough surfaces are usually modeled as semi-ane fractal surfaces [START_REF] Nayak | Random process model for rough surfaces[END_REF], which are characterized through their height distribution (see [START_REF] Persson | On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion[END_REF] for experimental measurements). In spite of the fact that some surface properties can be measured, there remain uncertainties e.g. due to nite precision measurement and incomplete information. Therefore, surfaces are considered to be random and are usually characterized only through their statistical properties.

Describing the roughness of a surface via statistical properties for every engineering application would imply prohibitive computational costs, many numerical studies typically aim at considering semi-innite volume (bulk with a rough surface) and obtaining mechanical responses, which will afterwards be used in a macroscopic model. More precisely, for a surface dened by its statistical properties, the contact area or the number of contact clusters under an imposed pressure is computed and this result is then used for more advanced mechanical problems. For example, being able to know the morphology of contact clusters for some class of rough surfaces is of primary importance to be able to study sealing technologies [START_REF] Hl | Hydrodynamic lubrication of textured steel surfaces under reciprocating sliding conditions[END_REF], the adherence of tires on roads [START_REF] Lorenz | Rubber friction for tire tread compound on road surfaces[END_REF][START_REF] Wang | Tireroad contact stiness[END_REF], reliability considerations for nano electromechanical systems (NEMS) and micro electromechanical systems (MEMS) appliances [START_REF] Corigliano | Microsystems and mechanics[END_REF], or tribology in general.

Despite its clear practical importance, a systematic and reliable quantication of the eects of surfaces uncertainties on contact mechanics is often disregarded in the relevant literature. Furthermore, even works that acknowledge model uncertainties (e.g. [START_REF] Putignano | The inuence of the statistical properties of self-ane surfaces in elastic contacts: A numerical investigation[END_REF][START_REF] Pohrt | Normal contact stiness of elastic solids with fractal rough surfaces[END_REF][START_REF] Prodanov | On the contact area and mean gap of rough, elastic contacts: Dimensional analysis, numerical corrections, and reference data[END_REF]) quantify their eects on mechanical outputs often in a somewhat ad-hoc way. Moreover, since the numerical studies are typically done on a nite size surface with periodic boundary conditions, an error due to homogenization (with nite size of the representative surface element) is introduced.

Since the contact problem is often computationally challenging due to the presence of multiple length scales, quantifying the eect of a random rough surface, by e.g. the classic Monte Carlo method, becomes quickly computationally prohibitive. Recently, the multilevel Monte Carlo (MLMC) method has been established as a computationally ecient sampling method that is applicable to a wide range of random models [START_REF] Giles | Multilevel Monte Carlo path simulation[END_REF][START_REF] Clie | Multilevel monte carlo methods and applications to elliptic PDEs with random coecients[END_REF][START_REF] Teckentrup | Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coecients[END_REF][START_REF] Giles | Multilevel monte carlo methods[END_REF][START_REF] Pisaroni | A Continuation Multi Level Monte Carlo (C-MLMC) method for uncertainty quantication in compressible inviscid aerodynamics[END_REF]. In this work, we will investigate the benets of these multilevel Monte Carlo techniques applied to random rough contact mechanics problems, for which it will be key to carefully balance statistical errors and discretization errors.

Somewhat related works that have applied the multilevel Monte Carlo method to a similar mechanical problem include [START_REF] Bierig | Convergence analysis of multilevel monte carlo variance estimators and application for random obstacle problems[END_REF][START_REF] Bierig | Approximation of probability density functions by the multilevel Monte Carlo maximum entropy method[END_REF][START_REF] Bierig | Estimation of arbitrary order central statistical moments by the multilevel Monte Carlo method[END_REF]. In these works, simplied random obstacle problems governed by the Laplacian (i.e. ignoring elasticity) have been considered, for which the surface of the random obstacle is considered to be rough and given in terms of a Fourier cosine series with random shift, or with random shift and random amplitude. Recently, the MLMC method has also successfully been applied in the context of (stochastic) numerical homogenization of randomly heterogeneous material [START_REF] Efendiev | Multilevel monte carlo approaches for numerical homogenization[END_REF].

The rest of this paper is organized as follows. In Sect. 2 we introduce the deterministic contact mechanics model and its discretization, which is the basis of this work. Then, in Sect. 3, we introduce the random contact model, which includes the denition of a class of suitable rough random surfaces and a description of how to sample these surfaces in practice. In Sect. 4 we introduce the multilevel Monte Carlo sampling technique, which is subsequently used in Sect. 5 to estimate the average contact area or the average number of contact clusters. Finally, conclusions and an outlook on future works are oered in Sect. 6.

2

The deterministic mechanic contact model

In this section, we briey describe the deterministic contact model and its discretization for a given surface. We focus on the non-adhesive frictionless normal elastic contact. It is noteworthy that the deterministic model described below is not novel, but is standard and well-established in the literature [START_REF] Bemporad | Optimization algorithms for the solution of the frictionless normal contact between rough surfaces[END_REF][START_REF] Wriggers | Computational contact mechanics[END_REF].

Semi-innite continuous model

As initial idealized model (denoted by P), we consider two semi-innite continuous solids. The rst solid is rigid (i.e. innitely sti) and rough, in the sense that this solid's surface is characterized by the presence of asperities on a wide range of length scales. In fact, the roughness of a surface can be described via its surface prole. We denote a surface's prole by s, so that every point on the surface can be written as x, s(x

) ∈ R 3 , x ∈ R 2 .
The second solid is assumed to be perfectly at and deformable. Throughout this paper we will work under the small strain assumption and consider the solid's behavior to be linear homogeneous isotropic elastic. Furthermore, we denote the solid's Young's modulus by E and its Poisson ratio by ν.

A mathematically convenient representation of the contact problem is in terms of an integral representation; see, e.g., [START_REF] Alberto Brebbia | Boundary element techniques: theory and applications in engineering[END_REF]. In fact, this integral formulation provides a representation in which the only unknown that needs to be determined is the normal displacement eld of the elastic solid's surface when the two solids come into contact. The advantage of this formulation thus is a dimension reduction, since the problem becomes eectively two dimensional. Moreover, the displacement eld inside the bulk material does not need to be resolved as it can simply be recovered from the displacement eld at the surface of the elastic solid; see [START_REF] Putignano | A new ecient numerical method for contact mechanics of rough surfaces[END_REF] for details.

Truncated continuous model

Instead of considering the two solids as semi-innite, we will consider both solids on a bounded domain. Specically, we consider them on a square D, say, with edge length L > 0, i.e. D = [0, L] 2 ⊂ R 2 . The only unbounded direction is therefore the inward normal at the surface. To represent semiinnite solids using the bounded domain D, we impose periodic boundary conditions (PBCs) on the displacement eld on the boundary of D. The resulting model is denoted by P L . Notice that P L is still a fully continuous contact model.

The link between the normal displacement u and the normal pressure p at the surface is given by

u = K * p . (1)
Here, the kernel K depends on the material properties E and ν as well as on the fundamental solution that is chosen in the integral formulation; see [START_REF] Alberto Brebbia | Boundary element techniques: theory and applications in engineering[END_REF] for details. In order to formulate the contact problem, it is furthermore convenient to introduce the so-called gap function g as the distance between the two contacting surfaces, so that g = u -s. The well-known HertzSignorini Moreau orthogonality condition then characterizes the non-adhesive contact problem and reads:

p(x)g(x) = 0 , p(x) ≥ 0 , g(x) ≥ 0 ∀x ∈ D . (2)
In other words, the condition states that interpenetration is forbidden (g(x) ≥ 0), that contact pressure can only be compressive (p(x) ≥ 0), and that each point is either in contact with the rigid surface (g(x) = 0) or free of forces (p(x) = 0). For clarity, we give a schematic representation of the problem in Figure 1.

The mechanic contact problem depends on a non-negative prescribed loading p, which is related to the average spatial pressure, in the sense that

1 |D| D p(x) dx = p . (3)
Combining everything together, problem P L can be stated as:

Find p ∈ A such that E c (p) is minimized. ( 4
)
Figure 1: A 2-D sketch of two semi innite solids coming into contact

Here, E c is the so-called complementary energy functional, which is given by

E c (p) = 1 2 D u(x)p(x) dx - D s(x)p(x) dx , (5)
and A denotes the space of admissible periodic pressure elds

A ≡ A(p) := p : D → R periodic such that p ≥ 0 and 1 |D| D p(x) dx = p . (6)
Note that there may be a model discrepancy due to the truncation of the domain and the PBCs. However, this model discrepancy vanishes as L → ∞.

Discretization of the truncated model

We discretize the model P L via the boundary element method (BEM) and denote the resulting model by P L h . To apply the BEM, we rst introduce the spatial discretization parameter h > 0 that is used to construct a grid. Here we consider a uniform grid on the square domain D consisting of n 2 grid points (nodes), where n ≥ 2 with h = L/n (opposite edges are considered only once). An example of such uniform grid is illustrated in Figure 3 (left). We represent both the surface of the at deformable solid and the rough surface of the rigid solid on said grid by their nodal values to obtain discrete (i.e. nite dimensional) surface representations. Moreover, we denote by u ∈ R n 2 the vector containing the n 2 nodal values of the displacement function u and, analogously, by p ∈ R n 2 the vector containing the n 2 nodal values of the pressure p. That is, the component p i (respectively u i ) of p (resp. u) corresponds to the value of the pressure p (resp. the displacement u) at node i.

For the continuous model P L , equation [START_REF] Argatov | The contact problem for a periodic cluster of microcontacts[END_REF] implies that u = F -1 F(K)F(p) in view of the convolution theorem for the Fourier transform F. For the discrete case, we thus relate the displacement vector u and the elastic pressure vector p via u = FFT -1 ( K FFT(p)) ,

where FFT and FFT -1 denote the 2D fast Fourier transform (2D-FFT) and the inverse 2D-FFT, respectively. Here we use the fast Fourier transform because of its eciency, which is also one of the main reasons making the BEM appealing for contact problems. Moreover, note that the FFT implicitly enforces periodic boundary conditions on the surface, since displacements computed from (7) will be L-periodic. In equation ( 7), K ∈ C n 2 ×n 2 is a diagonal matrix that contains the so-called inuence coecients for the fundamental solution in Fourier space. Here, we use the Westergaard's reference solution [START_REF] Westergaard | Bearing pressures and cracks[END_REF]; see also [START_REF] Stanley | An FFT-based method for rough surface contact[END_REF] for the expression of the inuence coecients. We note that the coecient in the matrix K associated to the mean value (i.e. the zero frequency) is set to zero. Indeed, the displacement of the surface is only known up to a rigid-body motion, since the problem is ill-posed in the x 3 direction. This choice will force u to have zero average, i.e. 1 n 2 n 2 i=1 u i = 0. Equivalently, one can, of course, also obtain a pressure from a displacement vector u using

p = FFT -1 ( K-1 FFT(u)) + p . (8)
Here, we slightly abuse notation and assume that the coecient of K-1 associated to the mean value (zero frequency) vanishes. Analogously to the continuous model, we then dene the space of admissible (discrete) pressure vectors by

A ≡ A(p) := p ∈ R n 2 , p i ≥ 0 1 n 2 n 2 i=1 p i = p . (9)
Eventually, the discrete problem P L h reads: Find p ∈ A such that E c (p) is minimized, [START_REF] Hl | Hydrodynamic lubrication of textured steel surfaces under reciprocating sliding conditions[END_REF] where the discrete complementary energy is

E c (p) = 1 2 n 2 i=1 u i p i - n 2 i=1 s i p i . (11)
The minimization problem above is then solved using a constrained conjugate gradient approach; see, e.g., [START_REF] Polonsky | A numerical method for solving rough contact problems based on the multi-level multi-summation and conjugate gradient techniques[END_REF][START_REF] Rey | Normal adhesive contact on rough surfaces: ecient algorithm for t-based bem resolution[END_REF]. Finally, we recall that model P L h is an approximation of model P L , so that solutions to these problems may dier due to the discretization error. However, this error vanished as h tends to zero [START_REF] Alberto Brebbia | Boundary element techniques: theory and applications in engineering[END_REF]. [START_REF] Bemporad | Optimization algorithms for the solution of the frictionless normal contact between rough surfaces[END_REF] Random rough surfaces for contact mechanics As mentioned in the introduction, both natural and manufactured surfaces are rough, in the sense of being composed of multiple asperities. In fact, experimental data indicate that surfaces are characterized by a wide range of length scales, from sample size to nanometer. To accurately model contact between two rough solids, one has to take this surface roughness into account because it can drastically aect the mechanical response of the structure. For example, the true contact area is actually signicantly smaller than the nominal contact area obtained when neglecting surface roughness.

To this end, we consider the deterministic contact models described in Sect. 2 for a class of random rough surfaces, which are modeled as Gaussian random elds. The considered class of surfaces is thereby dened by means of a characterization of its height distribution or of its power spectrum, which may be available through experimental results. As an illustration, Figure 2 shows the pressure distributions under a specic load p = 10 and the two associated dierent realizations of a rough random surface. Each surface is one realization dened by the same xed power spectrum. One can observe that the contact pattern strongly depends on the given surface realization and that it is necessary to perform simulations on various realizations to obtain statistically meaningful estimates. In what follows, we will discuss how to sample appropriate random surfaces that can then be used to produce the plots shown in Figure 2.

A characterization of rough surfaces

In this subsection, we dene the class of rough surfaces that we consider in this work. Specically, we model a random surface s as an isotropic Gaussian random eld with mean zero [START_REF] Lord | An Introduction to Computational Stochastic PDEs[END_REF]. That is, the surface heights in any m points x 1 , . . . , x m ∈ D are jointly normally distributed with mean zero and the entries Σ i,j of the covariance matrix Σ are given by

Σ i,j = c |x i -x j | ≡ Cov s(x i ), s(x j ) . (12)
Here, the function c is called isotropic covariance, which only depends on the euclidean norm r := |x| of any point x ∈ R 2 . Notice that the isotropy also implies the stationarity of the eld s. The (power) spectral density Φ of the stationary Gaussian eld s is given as the Fourier transform of its covariance function, in the sense that

Φ(k) = 1 4π 2 R 2 e -ik•x c(|x|) dx.
Furthermore, the isotropy of s implies that the spectral density only depends on the euclidean norm k := |k| of any Fourier point k ∈ R 2 and is therefore called the radial spectral density, which we denote by Φ r .

Experimental results [START_REF] Persson | On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion[END_REF] suggest that rough surfaces can be characterized by a (radial power) spectral density of the form

Φ r |k| =        C , if k l ≤ |k| ≤ k r , C |k| kr -2-2H , if k r ≤ |k| ≤ k s , 0 , otherwise. (13)
Here, H ∈ [0, 1] denotes the Hurst exponent, C ≥ 0 is the roughness amplitude, and the wave numbers 0 ≤ k l ≤ k r ≤ k s control the dierent roughness regimes. From the wave numbers controlling the dierent regimes, one can, as is common practice, dene the associated wave lengths λ l = 2π k l , λ s = 2π ks and λ r = 2π kr . Finally we note that there exist alternative approaches based on modeling the rough surface as a set of densely situated punches [START_REF] Argatov | Homogenization of a contact problem for a system of densely situated punches[END_REF][START_REF] Argatov | The contact problem for a periodic cluster of microcontacts[END_REF] before analytically solving the frictionless normal contact problem.

Generating random rough surfaces on a grid

In this section, we present an algorithm to generate a (discretized) rough surface on the domain D = [0, L] 2 respecting a prescribed radial power spectrum Φ r .

In view of the periodicity inherent in the truncated problem (cf., Sect. 2.2) and its discretization it is natural to also generate random surfaces that are periodic. We will denote the radial power spectrum corresponding to approximate periodic surfaces by Φ r ; see also appendix A for further details.

In Figure 3 (center) we illustrate the frequential grid used to discretize a power spectrum density Φ, which is also sketched in the same gure. We observe the isotropy in Figure 3 (right).

The wave-vector k has n 2 components. As a consequence of Shanon's theorem, on a n×n frequential grid one can only consider generating surfaces with k s ≤ n 2L . Therefore, rening the spatial discretization (i.e. increasing n) for a xed size L allows representing larger wave numbers (that is to say smaller wave lengths).

Algorithm 1, which is a suitably modied version of the sampling method presented in [START_REF] Hu | Simulation of 3-D random rough surface by 2-D digital lter and fourier analysis[END_REF], details the procedure to generate one realization of a rough surface on the domain [0, L] 2 with prescribed spectral density Φ r (that is to say for the given parameters: k l , k r , k s , C, and H). The algorithm's main input is the prescribed radial spectral density Φ r . Its output is a surface in Algorithm 1: Surface generator Inputs: Φ r , n, L ; Give the values for the two rst rows of S corresponding to the discretization of [0, L] 2 ; Generate the complex random vector ξ = η + iζ of size n 2 , where η and ζ are indep. N (0, I); Compute Φ r as the evaluation of Φ r on the frequential grid;

Compute the elementwise product q = Φ 1 2 r ξ; Compute q = 1 n 2 fftw2 -1 (q); Compute the real part R (q) of q; Complete the third column of S with R (q); Return S; R 3 evaluated on a spatial grid, which is encoded as the matrix S of size n 2 ×3 (n 2 nodes with 3 coordinates). That is, rst two rows of S are composed of the coordinates of the nodes on the grid, while the last row is lled at the end of the algorithm with the generated surface heights.

In appendix A, we give further details on this sampling methodology and verify that the used sampling routine does indeed produce samples with the desired statistical properties.

Monte Carlo methods for quantifying uncertainties in rough contact mechanics

As discussed in the introduction, it is crucial to understand how the surface uncertainties aect the average outputs of the contact problem (for example the contact area, the morphology of contact clusters, ...).

Let u denote the solution to the continuous problem associated with the random surface s. It is worthwhile to emphasize here that the surface's randomness implies that also the solutions to the contact problems (as introduced above) are random, such as the displacement, the pressure, the gap function, and other derived quantities. The goal then is to quantify the eect of the random surface s by computing the expected value E(Q) of an appropriate quantity of interest Q that is computed from the displacement eld u. For example, the quantity of interest Q may be the contact area or the maximum pressure under a given load. However, as one cannot solve the continuous problem exactly, computing the expected value E(Q) is unfeasible. Instead, we can only use the discretized model to obtain an approximation Q h .

In this Section, we will describe two Monte Carlo based sampling methods, which are based on generating many surface realizations and computing the corresponding approximate quantity of interest Q h . The appeal of these sampling methods is that they are non-intrusive, in the sense that an existing deterministic numerical solver does not need to be modied and can thus be used as a black box.

In what follows we briey review the classic Monte Carlo method for the approximation of E(Q) based on simulations. Afterwards we introduce the multilevel Monte Carlo method [START_REF] Giles | Multilevel Monte Carlo path simulation[END_REF][START_REF] Clie | Multilevel monte carlo methods and applications to elliptic PDEs with random coecients[END_REF][START_REF] Teckentrup | Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coecients[END_REF][START_REF] Giles | Multilevel monte carlo methods[END_REF] as a variant that oers drastic computational savings for a wide range of applications. See also [START_REF] Heinrich | Multilevel Monte Carlo methods[END_REF] for some related earlier works. Although the focus of this work is on scalar quantities of interest, it is noteworthy that the multilevel Monte Carlo method can also be applied to multidimensional quantities or reliability studies [START_REF] Krumscheid | Multilevel Monte Carlo approximation of functions[END_REF]. We will leave the application of these options in the context of contact problems for future works.

Classic Monte Carlo method 4.1.1 Denition of the estimator

The classic Monte Carlo (MC) estimator is dened as the mean of a sample of N independent and identically distributed (i.i.d.) realizations of Q h , that is

E MC N (Q h ) := 1 N N j=1 Q h ω (j) , (14)
where ω (j) are i.i.d. random events (each one corresponding to a generated realization of the random surface). The accuracy of the Monte Carlo estimator E MC N (Q h ) as an approximation of E(Q) can be characterized via the mean squared error, viz.

E E(Q) -E MC N (Q h ) 2 = Var(Q h ) N + E(Q -Q h ) 2 . (15)
Here, the rst term on the right-hand side (the so-called statistical error) quanties the estimator's variance and is inversely proportional to the number of realizations N . The second term (the so-called squared bias) accounts for the expected error due to the spatial discretization frequency of the contact problem and depends on the grid size h and on the domain truncation

L.
In what follows, we assume that the domain D = [0, L] 2 has been taken large enough so that the truncation error is negligible with respect to the discretization error due to the grid size h. Henceforth we focus only on the latter. Clearly, for E MC N to be an ecient approximation of E(Q) both terms should be balanced and small.

To further apply the Monte Carlo method, it is necessary to quantify the accuracy and the computational cost of the solution computed with the deterministic contact solver. As the discretization length h decreases, it is desired that the approximation converges to the exact solution, which is related to properties of the boundary element method for each deterministic problem; cf. [START_REF] Alberto Brebbia | Boundary element techniques: theory and applications in engineering[END_REF]. Similarly, we expect that the cost of the iterative scheme used to approximate the solution to the contact problem is inversely proportional to a power law with respect to the grid size h, since the grid size controls both the accuracy and the dimension of the discretized problem. Finally, we also make the assumption that the variance of Q h is nite for h 1, which is natural in view of [START_REF] Haji-Ali | Optimization of mesh hierarchies in multilevel Monte Carlo samplers[END_REF]. In a more formal way, it leads to the three following assumptions:

MC-1 the cost to compute each i.

i.d. realization of Q h is bounded : cost(Q h ) ≤ C γ h -γ for some constants C γ , γ > 0,
MC-2 the bias decays with order α > 0, in the sense that

|E(Q -Q h )| ≤ C α h α for some constant C α > 0, MC-3 the variance of Q h is nite for h 1.

Application of the MC method

To apply the MC method, one has to rst choose a root mean square error tolerance ε. Then to balance the bias and the statistical error and achieve an overall mean squared error of order ε 2 in (15), it is thus sucient to choose 1

h ε 1/α and N ε -2 . ( 16
)
Thanks to [START_REF] Heinrich | Multilevel Monte Carlo methods[END_REF], the number of realizations to perform is directly computable from the tolerance ε. However, to estimate the grid size h, it is necessary to know α, or in other words, how the expected discretization error decays as a function of h. If analytical convergence results are available for the specic discretization method, α can be deduced directly from these results. Otherwise, one can estimate this convergence rate through a screening procedure that will be described in subsection 4.2.2.

Multilevel Monte Carlo method 4.2.1 Denition of the estimator

The underlying idea of the multilevel Monte Carlo (MLMC) method is to use realizations of approximations to Q with varying accuracy. Specically, we consider a hierarchy of M grids, called levels, with grid sizes

h 0 > h 1 > • • • > h M .
As we will see below, the grid size h M of the nest discretization level takes over the role of the grid size used in the classic MC method above. We denote by Q h the approximation of Q on a grid with size h . Using the linearity of the expectation operator, one can then write the expectation of the approximation of Q on the nest level h M as a telescoping sum. That is, E Q h M can be written as the expectation of the approximation of Q on the coarsest level h 0 plus a sum of correction terms that are expectations of approximations on consecutive levels:

E Q h M = E Q h 0 + M =1 E Q h -Q h -1 ≡ M =0 E ∆ , (17)
where

∆ 0 := Q h 0 and ∆ := Q h -Q h -1 for > 0. The MLMC estimator E MLMC (Q h M )
is then obtained by approximating each of the expectations E(∆ ) by a MC estimator using N i.i.d. realizations:

E MLMC (Q h M ) := M =0 1 N N j=1 ∆ ω (j) = 1 N 0 N 0 j=1 Q h 0 ω (j) 0 + M =1 1 N N j=1 Q h ω (j) -Q h -1 ω (j)
.

(18) Notice that the correction terms ∆ ω j) are computed using the same random realization on both levels and -1. Moreover, the ∆ terms are sampled independently on dierent levels. As for the MC estimator, the accuracy of the MLMC estimator is commonly assessed via the mean squared error, which can also be decomposed into the statistical error and the squared bias:

(j) := Q h ω (j) -Q h -1 ω (
E E(Q) -E MLMC (Q h M ) 2 = M =0 Var ∆ N + E(Q -Q h M ) 2 . ( 19
)
The identity above demonstrates the virtue of considering multiple levels instead of just one ne grid. In fact, if Var ∆ decreases as increases, then one may need many realizations only on the coarse levels, which are typically cheap to generate, but only very few on the ner, more expensive, levels. To make this intuition precise, we will assume that ML-1 the cost to compute each i.i.d. realization of Q h is bounded by cost(Q h ) ≤ C γ h -γ for all > and some constants C γ , γ > 0, ML-2 the bias decays with order α > 0, in the sense that

|E(Q -Q h )| ≤ C α h α for all > and some constant C α > 0, ML-3 the variance of ∆ = Q h -Q h -1
decays with rate β > 0, in the sense that Var ∆ ≤ C β h β for all > and some constant C β > 0.

Notice that assumptions ML-1 and ML-2 for the MLMC method are the same as assumptions MC-1 and MC-2 for the MC method.

In this work we will consider geometric grids, in the sense that h = δ -h 0 for some δ > 1. In practice the value δ = 2 is often used, which we also adopt here. Although a non-optimal choice of δ can aect the MLMC method's performance, this eect is minor [START_REF] Haji-Ali | Optimization of mesh hierarchies in multilevel Monte Carlo samplers[END_REF].

Application of the MLMC method

For the practical application of the MLMC method to the contact problem at hand, a few comments are in order. Firstly, the coarsest grid size h 0 should be chosen suciently small so that a basic problem resolution is provided and also such that L h 0 is an integer. That is, the coarsest grid size is related to the roughness of the surface since we aim at accurately representing the largest wavelengths of size L/k l . A heuristics would be to chose h 0 = 10 L k l in order to resolve the largest asperity, which is also used here.

Secondly, assembling the MLMC estimator [START_REF] Kroese | Handbook of Monte Carlo Methods[END_REF] requires to compute realizations of the dierences ∆ = Q h -Q h -1 on the dierent levels. Specically, one has to compute these dierences of approximations on consecutive grids for the same random event ω , i.e. for the same realization of the random surface s(ω ) on both level and level -1. This can, for example, be done by generating a realization of the discretized surface associated with the grid on level using Algorithm 1, which is then used to compute the corresponding realization of Q h . The version of the surface on level -1, required to compute Q h -1 , is then obtained by projecting the realization of discretized surface on level onto the coarser (nested) grid of level -1. This particular approach oers the additional advantage that one can use the solution to the contact problem on level , after projecting it on the grid on level -1, as initial condition for the iterative solver for the contact problem on the coarser level; see Sect. 2. Indeed, this procedure can signicantly reduce the computational eort for solving the problem on the coarser level -1.

Finally, the parameters dening the MLMC method need to be determined, namely the number of levels M and the number of samples on each level N l for l = 0, . . . , M . We begin with the level M corresponding to the nest grid h M . After choosing the desired tolerance ε, to obtain a squared bias error smaller than ε 2 2 it is sucient to choose:

h M ε 1/α ⇒ M log δ ε -1 α (20)
Observe that the grid size condition of the nest level is identical to the grid size condition for the MC method.

Next, the number of realizations per level is obtained by minimizing the computational cost under the constraint that the statistical error is smaller than ε 2 2 . It leads to

N l ε -2 Var ∆ l cost(∆ l ) M =0 Var ∆ cost(∆ ) , (21)
for l = 0, . . . , M (see [START_REF] Giles | Multilevel monte carlo methods[END_REF] for details).

That is, if one can determine the constants C γ , γ, C α , α, C β , β and use assumptions ML-1, ML-2 and ML-3 to express Var ∆ l and cost(∆ l ), then M and N l are fully computable in view of ( 20) and ( 21). The computation of the constants C γ , γ, C α , α, C β , β can be done using a screening procedure, as described in [START_REF] Giles | Multilevel monte carlo methods[END_REF]. The screening phase consists in estimating the quantities E(Q h ), Var(∆ ), cost(∆ ) for dierent grid sizes h . Here, the expectation E(Q h ) and the variance Var(∆ ) are estimated using the usual sample mean and variance estimates. Performing a linear regression for the three quantities E(Q h ), Var(∆ ), cost(∆ ) with respect to eventually yields estimates for the convergence rates (that is to say for the constants C γ , γ, C α , α, C β , β ). This procedure is the one carried out in the numerical experiments of Section 5.

Numerical Experiments

In this section, we consider two dierent quantities of interest that are derived from the solution to the contact problem. The rst one is the total contact area under a given load p. This is a global quantity, which is straightforward to estimate once the contact problem is solved, and it is of major interest in many engineering applications. For instance, the thermal conductivity between solids is proportional to the contact area. The second quantity is the number of contact clusters under load p. This quantity describes the morphology of the contact area and it is of interest, for example, in leakage problems.

The numerical experiments that follow were carried out in Python on a standard laptop computer with 7.7 GB of memory and an Intel R Core TM i7-5600U processor. Moreover, for the contact problems, we consider the wave numbers k l = 1, k r = 1, k s = 64, H = 0.8, and C is chosen such that c(0) = 1. Notice that the ratio λ l /λ s = k s /k l is large to model a surface's self-anity across a wide range of frequencies. We also note that the Hurst exponent H = 0.8 is motivated by the experimental results published in [START_REF] Persson | On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion[END_REF]. Furthermore, Young's modulus E and the Poisson ratio ν are such that the equivalent modulus is

E = E 1-ν 2 = 1.

The real contact area

To study the eects of the surface uncertainties to the real contact area, we consider the random quantity of interest Q given as the normalized contact area (in percent). For this quantity of interest, we set the applied load to be p = 0.1 √ m 2 , where m 2 is the second spectral moment of the rough surface s so that √ m 2 is the root mean square of the slopes (see [START_REF] Vladislav | From innitesimal to full contact between rough surfaces: Evolution of the contact area[END_REF] for its computation).

In the discretized problem, since we obtain piecewise constant solutions, the contact area is therefore computed as

Q h (ω) = 1 n 2 i∈Ic(ω) 1 , (22)
where the set of contact points I c is dened by: i ∈ I c (ω) ⇔ u i (ω)-s i (ω) = 0.

Choosing the surface size L

What is not considered yet is the error introduced by the domain truncation and the PBCs. To identify the size L of the truncated problem such that model error P -P L does not dominate the overall error in the the numerical study that follows, we numerically investigate this eect. Specically, we estimate the expectation of the contact area Of course, by changing the size of the surface, the number of grid points increases but h remains constant. The obtained results are summarized in the Table 1. We observe that E MC (Q L ) seems to monotonically approach the limit value Q∞ = 20 as L increases. In fact, the distance to the limit

Q L via E MC (Q L ) for dierent domain truncations L ∈ {1, 2,
value is |E MC (Q L ) -Q∞ | ≤ 0.1 for L ≥ 2.
Notice that an absolute error of 0.1 already corresponds to a relative error of 0.5%. Therefore, we decided to choose L = 2 for the numerical study that follows, for which we will then explore mean squared error tolerances ε 2 for 0.2 ≤ ε. Consequently, we can be condent that the model error is not dominating the overall error, in the sense that it is at most comparable to the prescribed mean squared error tolerance.

Estimating convergence rates

Before we use the MLMC method described in Sect. 4 to estimate the expectation of Q, we verify that the Assumptions ML-1ML-3 are satised. To the authors' knowledge, no theoretical convergence results concerning these hypotheses for the contact problem solved with the discretization method used in this paper are available. We thus estimate the characterizing rates using the techniques described in [START_REF] Giles | Multilevel monte carlo methods[END_REF], which is based on an initial screening phase where the rates are estimated via linear regression. Here, this screening phase was carried out for various grid sizes h = h 0 2 -, using 50 realizations for each resolution, which was sucient here. The coarsest grid considered was 128 × 128. Figures 4 and5 show the outcomes of this screening procedure. We nd that this screening procedure yields α = 1.32, C α = 20.5, β = 1.85, C β = 0.46, γ = 2.45, and C γ = 1.69. It is noteworthy that the coecient γ only depends on the implementation of the contact solver, but not on the machine used for the simulations; the machine dependence only aects the constant C γ .

E(Q h ) E(∆ ) model t (b) decay

Estimation of the real contact area

Motivated by the screening process for the rates (see Fig. 4), we set the coarsest level of the MLMC grid hierarchy to be a 256 × 256 grid. Based on the coarsest level grid size h 0 and the estimated rates, we can thus use the formulas in ( 20) and ( 21) to construct the MLMC estimator to approximate the mean real contact area for a given tolerance. Table 2 shows the estimated expectation of the true contact area computed via both the classic Monte

Carlo method E MC (Q h ) and the multilevel Monte Carlo method E MLMC (Q h )

for dierent tolerances ε. As the exact value of E(Q) is not known, we cannot exactly assess the approximation error. However, it can be seen in Table 2 that both estimators provide similar results. In fact, the estimated values always satisfy which conrms the theoretical mean squared error results discussed in Sect. 4. It is noteworthy that the mean squared error criterion is an absolute error criterion. To ease the interpretation of the used tolerance ε, we have also added the relative precision 2 compares the estimated mean contact area for various values of the tolerance ε, Table 3 summarizes the corresponding computational times required to obtain said estimators. Specically, in Table 3, we present the computational time in seconds for the classic Monte Carlo method (denoted by W MC ) and for the multilevel Monte Carlo method (W MLMC ) for dierent prescribed tolerance requirements. These results clearly demonstrates that the multilevel Monte Carlo method is highly ecient. Indeed, the smaller the tolerance ε, the more drastic the gain. The data presented in Table 3 are also illustrated in Figure 6. The plot shows the gain brought by the MLMC method compared to MC method. Figure 7 moreover presents the number of realizations required for two different prescribed tolerances for both the MC method and the MLMC method. For the tolerance ε = 0.7, Table 3 shows that the cost of the MC method is only marginally higher than the cost of MLMC method, while Figure 7a indicates that the distribution of required realizations (i.e. the work) across levels is very dierent. This is even more so for a tolerance of ε = 0.2, as can be seen in Figure 7b. In fact, the MLMC method spreads the number of realizations across various levels with requiring most realizations on the coarse (cheap) grids and only few on the ner (more expensive) grids, whereas the MC requires all of the many realizations on the nest grid (most expensive).

|E MC (Q h ) -E MLMC (Q h )| ≤ 2ε , (23)
ε r = ε E MLMC (Q h ) to the table. While Table

The number of contact clusters

The second quantity of interest we study here is the number of contact clusters (i.e. the number of contact zones) per square meters under the given load p = 0.1 √ m 2 . As an example, an illustration is given in Figure 8 that exhibits 11 contact clusters, which are the 11 connected black patches.

Choosing the surface size L

To identify the size L of the truncated problem such that the model error P -P L does not dominate the overall error in the the numerical study that follows, we proceed exactly as for the rst quantity of interest. The corresponding results are summarized in the Table 4. There, we see that E MC (Q L ) seems to approach the limit value Q∞ = 110 as L increases. More precisely, the distance to the limit value is that follows, for which we will then explore mean squared error tolerances ε 2 for 2 ≤ ε. Having to choose a dierent box size L than for the rst quantity of interest is not surprising since mechanical quantities are not aected the same way by the PBCs.

|E MC (Q L ) -Q∞ | ≤ 1 for L ≥ 3.

Estimation of convergence rates

We estimate the rates needed for the MLMC method using the same screening procedure as for the rst quantity of interest. This procedure yields α = 2.79, C α = 211, β = 2.41, C β = 42.6, γ = 2.53, and C γ = 318.

Estimation of the number of contact clusters

Guided by the screening procedure, we set the coarsest level to be a 128 × 128 grid and use both the MC method and MLMC method to approximate the mean number of contact clusters for various required tolerances ε ∈ {40, 30, 20, 15, 10, 9, 8, 7, 6, 5, 4, 3, 2}.

In Table 5, we then present the approximations of the expected number of contact clusters E(Q), computed with the MC method and the MLMC method, respectively, for the tolerances introduced above. As in the previous example, the exact value of E(Q) is not known. However, the results in Table 5 show that

|E MC (Q h ) -E MLMC (Q h )| ≤ 2ε , (24)
thus verifying the tolerance requirement. We also add in the same table the relative precision ε r = ε E MLMC (Q h ) for an easier interpretation. The results in Table 5 also reveal that both MC and MLMC estimators seem to overkill the desired tolerance. A possible explanation for this phenomenon is that the coarsest grid may not allow to accurately represent small wavelengths, which thus leads to a smaller variance in the number of contact clusters. Finally, in Table 6 we report the computational time in seconds for both the Monte Carlo method (W MC ) and multilevel Monte Carlo method (W MLMC ), each for dierent tolerance requirements ε. Once again we observe that the MLMC method is not only highly competitive, but actually superior for any tolerance demand larger than 15. Indeed, the smaller ε, the larger the gain.

Conclusion

In this work, we have applied the multilevel Monte Carlo method to estimate the average contact area and the average number of contact clusters for the problem of frictionless normal contact of rough surfaces. After describing the statistical properties of the random rough surfaces in detail, we rst veried that a variant of the surface generator introduced in [START_REF] Hu | Simulation of 3-D random rough surface by 2-D digital lter and fourier analysis[END_REF] generates samples of rough surfaces with the desired statistical properties. Then, we briey reviewed the principles of the multilevel Monte Carlo method. Afterwards, we detailed how to use the MLMC method eectively in the context of rough contact problem and demonstrated by means of numerical assessments the huge computational gain that the MLMC methods provides compared to the classic Monte Carlo method. Future work building upon the framework presented here, consists, for example, of studying other quantities of interest, such as the percentage of the surface subject to a pressure larger than a given critical pressure.

A

Generator of random rough surfaces

A.1 Further comments on the surface generator algorithm

The procedure described in Algorithm 1 generates random surfaces with a discrete radial spectral density Φ r made by a sum of Dirac deltas located in the frequency grid points. In the spatial domain, this leads to an approximate covariance function c that is the trigonometric interpolant on the spatial grid of the periodized covariance c per = z∈Z 2 c(|x + Lz|). In particular, all generated surfaces are continuous L-periodic functions. The presented Algorithm 1 takes, in essence, advantage of the fact that any ane transformation of a multivariate Gaussian random variable has a multivariate Gaussian distribution with explicitly known mean and covariance structure. In fact, the rst step of Algorithm 1 consists of generating the complex random vector ξ = η + iζ, with η and ζ being independent and normally distributed random vectors in R n 2 with mean zero and covariance matrix being the identity matrix of size n 2 ×n 2 . After multiplying this vector by Φ 1 2 r , which is the pointwise square root of Φ r , we obtain the complex vector ŝ, which is then transformed to the physical space via the two-dimensional inverse Fourier transform. Finally, Algorithm 1 returns the real part of this transformation. It is noteworthy that the scaling by the factor 1 n 2 in the third step of the algorithm is necessary since our implementation uses the fftw2 library [START_REF] Frigo | The design and implementation of FFTW3[END_REF], which has the property that fftw2 -1 (fftw2(x)) = n 2 x for any x ∈ R n 2 . Finally, we remark that one could also use the imaginary part of q as a return value for Algorithm 1 so that two independent samples are generated at the same time.

A.2 Verication of the sampling algorithm and illustrations

The purpose of this section is to illustrate the properties of the random surface generator described in Algorithm 1. Here, we consider the parameters k l = 4 = k r , k s = 32, H = 0.8, L = 32 and C is chosen so that the pointwise variances are equal to 1. Moreover, we consider the grid size h = 1 256 L, if not specied otherwise.

A.2.1 Gaussian distribution

First, we verify that Algorithm 1 produces samples that are centered Gaussian random elds. Specically, we consider four trial points in the domain D = [0, L] 2 with coordinates x 1 = (h, h), x 2 = (h, h+ L 2 ), x 3 = (h+ L 2 , h), and x 4 = (h+ L 2 , h+ L 2 ), and approximate the distribution of each random surface point s(x i , ω), i = 1, . . . , 4. To this end, we generate a sample of N = 50000 independent surface realizations using Algorithm 1, which is then used to approximate the probability density function (PDF) of each surface point s(x i ) via a kernel density estimator with Gaussian kernel and Silverman's bandwidth selection criterion; see, e.g., [START_REF] Kroese | Handbook of Monte Carlo Methods[END_REF]Ch. 8.5] for details. Figure 9a shows the approximated PDFs of the surface points s(x i , ω), i = 1, . . . , 4, in addition to the target N 0, 1) PDF. The fact that all curves are almost indistinguishable from the target PDF veries that Algorithm 1 generates surface realizations whose surface points have the desired distribution.

This conclusion is also conrmed by the QQ plot shown in Figure 9b. There, we plot the theoretical quantiles of the target N 0, 1) distribution against the quantiles obtained from the data for each surface point (color key of crosses is the same as in Figure 9a). For each of the four cases we observe an almost perfect linear trend (y = x), so that the data t the target distribution very well in each case. The only regions showing minor deviations from the straight line behavior are at the very low and very high end of the range, respectively. However, these regions correspond to rare outcomes (i.e. the probability of observing values in these regions is very small), so that only a small fraction of the N samples lies in these regions, which explains the marginally worse ts in there.

Autocorrelation

Next, we illustrate that Algorithm 1 also produces realizations with covariance function that approximates well c per , hence c when the period L is taken large enough. That is, we investigate the covariance matrix Σ ∈ R n 2 ×n 2 dening the n 2 -dimensional Gaussian random vector obtained when evaluating Gaussian random eld on the given grid. We estimate Σ via the sample covariance matrix, based on a sample of N = 20000 independent discrete surface realizations and we denote this estimator by ΣN . Inspecting the el- ements of ΣN reveals that ΣN is indeed a discretized version of an isotropic covariance.

Figure 10a shows the entries of three dierent estimated discretized covariance matrices as a function of r, where each matrix approximates Cov s(r+ L/2, L/2), s(L/2, L/2) and is obtained for dierent grid sizes h ∈ {2 -8 L, 2 -7 L, 2 -6 L}. Additionally, the plot contains the curve of the periodic target isotropic covariance c per . We observe that Algorithm 1 generates realizations with the correct isotropic covariance structure.

Although not directly related to the quality of Algorithm 1, we also address the dierence between the non-periodic isotropic covariance c and approximations to its periodic version c per . For this, Figure 10b shows two dierent estimated discretized covariance matrices, obtained for a xed mesh size h = 2 -8 L but dierent domain sizes L ∈ {32, 64}, in addition to the non-periodic isotropic covariance c. We nd that increasing the size L of the domain D, while keeping the grid size h xed, leads to a more accurate approximation of the non-periodic isotropic covariance c. Indeed, with the increasing size L, the discretized spectrum tends to the continuous spectrum dened in equation [START_REF] Giles | Multilevel Monte Carlo path simulation[END_REF]. We observe that improving the discretization quality reduces the distance to the exact autocorrelation.

Figure 2 :

 2 Figure 2: Typical pressure distributions (in color) obtained under a load p = 10 corresponding to two dierent realizations of a rough random surface (in gray) dened by the same power spectrum.

Figure 3 :

 3 Figure 3: Spatial (left) and frequential (center and right) grids used for the generation and the representation of the rough surface.

Figure 4 :

 4 Figure 4: Estimation of convergence rates.

Figure 5 :Table 3 :

 53 Figure 5: Estimation of the cost model.

Figure 6 :

 6 Figure 6: Measured costs for the MC and the MLMC estimators for dierent tolerances. The quantity of interest is the contact area.

  ε = 0.2

Figure 7 :

 7 Figure 7: Number of realizations required per level for both the Monte Carlo method and the multilevel Monte Carlo method for two tolerances.

Figure 8 :

 8 Figure 8: Binary contact map: contact zones are in black.

Figure 9 :

 9 Figure 9: Verication of surface height distribution in four points on the surface

  Eect of the size of the domain L

Figure 10 :

 10 Figure 10: Comparison between isotropic covariance approximations, the periodic target covariance c per , and the non-periodic version of c.

Table 1 :

 1 3, 4}. For each size L, we compute 50 i.i.d realizations, where the discretization parameter h is chosen such that the discretization error is negligible. Indeed, we choose h = 1/512 in order to properly describe the smallest wavelength λ s (8 nodes per wavelength). MC estimators of the contact area for various box sizes L.

		Box size L
	1	2	3	4
	E MC (Q L ) 19.52 19.90 19.94 19.99

Table 2 :

 2 19.94 20.26 19.90 20.11 19.48 19.55 19.64 19.97 20.01 E MLMC (Q h ) 19.42 19.57 19.53 19.53 19.74 19.59 19.65 20.09 20.03 MC and MLMC estimators of the contact area for various tolerances.

		.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2
	ε r (%)	5 % 4.5 % 4 % 3.5 % 3 % 2.5 % 2 % 1.5 % 1 %

E MC (Q h )

Table 4 :

 4 MC estimators of the mean number of contact clusters for various box sizes L.

	Note that

Table 5 :

 5 Estimations of the mean number of contact clusters for dierent tolerances with the MC method and the MLMC method.

	tolerance ε

Table 6 :

 6 Cost (in seconds) required to compute the estimation of the mean number of contact clusters for dierent tolerances with the MC method and the MLMC method.

We use the notation a b, if there exists a constant c>0, such that a ≤ cb; analogously for . If a b and a b, then we write a b.

Acknowledgments

Support for the rst author from the `EPFL Fellows' fellowship program cofunded by Marie Skªodowska-Curie, Horizon 2020 Grant agreement no. 665667 is gratefully acknowledged. The authors would like to thank Professor Jean-François Molinari, Vladislav Yastrebov, and Guillaume Anciaux for helpful discussions.


Hannah Goldstein

Bensalem, North Korea, and our inability to imagine a utopian alternative

INTRODUCTION

The genre of classical utopian fiction raises some skepticism in the postmodern world. We prefer to entertain ourselves with (post)apocalyptic narratives, suggesting that the only fathomable hope for a new social order is in the total demolition of democratic capitalism, and through such an escape from capitalist realism, followed by a period of rebuilding.

Perhaps this discomfort with the antiquated genre is to do with its structuring, the narrator being foreign and excluded from utopian life, and, as such, less trustworthy to a contemporary audience familiar with the unreliability of the exported, propaganda-based "reality" of totalitarian rule(s). The stranger must be kept at arm's length for the genre to be successful, his function being observational, rather than proactive. This is perhaps exemplified best in New Atlantis by Francis Bacon, in which the stranger is kept in a literal Strangers' House, with limits constraining his capacity to explore the utopian society of Bensalem, so that he is only able to experience it through dialogue with representatives of the utopia. A contemporary reader need only compare this to the eerie footage of visits to North Korea, available through Vice News, to be skeptical of the legitimacy of unsubstantiated claims of a perfect social order. What would happen if the stranger were invited in?

The rise of the dystopian genre implies that the answer to this question is, "nothing good," and points towards the unsustainability of the utopian genre from a closer vantage point.

In this paper, contemporary discomforts with the early modern utopia will be interrogated, and it will be theorized that within our globalized world it becomes nearly impossible to imagine a never-before-seen alternative world order. What's more, there is ample evidence that one should fear an upheaval of our modern democratic capitalism, be it fictionalized, or else rooted in reality, for New Faces essay collection, Hannah Goldstein, September 2019 http://www.new-faces-erasmusplus.fr/ 2 example the utterly isolated and regimented life within North Korea. As the world becomes more homogenous beneath a global rule, with those opting out of the global social order seemingly leaning into a totalitarian order, it is no wonder that the only imaginable escape from our current capitalist system seems to be through apocalypse and subsequent rebuilding.

NEW ATLANTIS AND THE STOCK CHARACTER OF THE STRANGER

The stock character of the stranger can be observed in classical utopian texts such as Thomas More's Utopia, Tommaso Campanella's La Citta del Sole and, perhaps most starkly, and as such most importantly for this essay, Francis Bacon's New Atlantis. The stranger is not meant to be strange to the early modern reader, but rather to the utopian society itself. An explorer, he encounters the utopia, but is never truly invited in, as there is always some sort of limitations imposed upon him. Take, for example, the stranger in New Atlantis. The stranger's experience of Bensalem is firstly mediated by a representative, in this instance the governor of the Strangers' House. Although the stranger is able to move around the utopian space, this movement is limited, and not reported upon in the narrative.

Rather, the conversation between the stranger and the governor is what the text concerns itself with.

The conversation that makes up the bulk of the text reveals the limitations of the stranger's stay. Along with his team of sailors, he must stay in The Strangers' House, a space serving none but this very function. Furthermore, the stranger cannot stay indefinitely. He is told by the governor that, "The state hath given you license to stay on land for the space of six weeks" (Bacon 157) and although there is mention that the stranger could likely stay longer, this sets the expectation that the visitors are being tolerated as outsiders, rather than welcomed into the society as potential future citizens.

During their stay, they are told that, "none of you must go above a karan, (that is with them a mile and an half) from the walls of the city, without especial leave." (Bacon 157-58) These restrictions serve the utopian genre, with the utopia meant to function, not as a place, rather as a foil for the stranger's local community to juxtapose their own societal shortcomings against. This juxtaposition is most evident in the ways in which the utopia is described. It is often not described in isolation, but in relation to what the stranger already knows. Upon his arrival, the stranger observes that, "The Strangers' House is a fair and spacious house, built of brick, of somewhat a bluer colour than our brick" (Bacon 159) and then later, after having tasted the food, that it was, "better than any collegiate diet that I have known in Europe." (Bacon 156) The use of comparative adjectives, such as "bluer" and "better" instead of "blue" or "good" gives a hierarchy to the different forms of governance, with the utopia unarguably in the more powerful place in this comparison. As such, these descriptors serve a duel purpose, both commenting on the exemplary nature of the utopian space, and also conversely giving attention to the inadequacy of the stranger's (familiar) world.

As the early-modern utopia served the purpose of commenting reflexively upon the reader's society, this distancing tactic of experiencing the text through dialogue rather than narrative prose makes it easier for this comparison to occur. Rather than concern himself with the quality of life of the utopian citizen, the stranger only wants to report back to his own society, in order to draw attention to those aspects of which he views as problematic.

Although the goal of the stranger is to explore, the exploration is constrained the moment the stranger enters the utopian space. Even the notion that the stranger is kept in a special guest-house reveals the totality of the control the utopia must impose upon its own narrative. However, to the contemporary reader, the utopia implies, rather than perfection, some sort of a propaganda state. What forms of governing benefit from strict propaganda in our contemporary world? Not capitalism, which, though flawed, allows at least for self-criticism through art and entertainment. Instead, totalitarianism comes to mind, casting a backwards facing shadow upon the naive trust of the stranger in utopian narratives. In New Atlantis Bensalem is able to improve itself by simultaneously looking externally at the world and keeping its own social structures hidden, revealed only in mediated and small doses to their occasional visitors. The governor explains, "that by the means of our solitary solution, and of the laws of secrecy which we have for our travelers, and our rare admission of strangers, we know well most part of the habitable world, and are ourselves unknown." (Bacon 159) Again, it is worth bearing in mind that this is a representative of the state that is relaying this information. It is easy to imagine that this knowledge of the external world may not permeate all of the social stratospheres of Bensalem, all the more so if Bensalem is compared to the propaganda state of North Korea. Dukalskis and Hooker note when considering North Korea that "without access to the country and its people, achieving analytical subtlety is extremely difficult." (53) Or that, because any access the rest of the world has to North Korea is in many ways mediated by the state itself, it is difficult not to have a simplified understanding of the structures within. In utopian fiction, this simplified understanding allows for an idealization of the unknown state to occur. However, here we see a perversion of the same interaction. Because we, as western media consumers, can only access two extremes, the idealized portrayal in what North Korean propaganda we can access through watching state-produced media, juxtaposed against the first-person testimonies of North Korean refugees, we must grow suspicious of any indication of an idealized utopian existence being constructed for us. When considering a contemporary reader's discomfort with the utopian genre, this again can be contextualized when taking into account what allows any totalitarian society to be successful. It's elaborate that, "a totalitarian regime [...] because it is concerned with political participation and allegiance to its all-encompassing ideology, will tolerate very little activity that is contrary to its worldview." (Dukalskis, Hooker 55) Or, merely stating that a political structure is perfect is no longer enough to prove perfection to the modern reader, as such statements are in actuality one indicator of a state that controls the movements, behaviors, and ideas of its populace.

When watching Vice's documentary series on North Korea there is much made of the distinction between what is being curated for the visitor's experience, and what is believed to actually be the quality of life based on a western understanding of the regime. Similarly to in New Atlantis, the visitor is kept in special quarters. What's more, their movements are monitored and controlled, and their access is limited. Even gaining initial access proves difficult, Shane Smith reflecting that, "We tried to get in for a year and a half, but couldn't because North Korea does not let anyone in.

They do not want anyone to corrupt their one-hundred-percent homogenous society." (Smith) Here, again, the similarity to Bensalem is evident in the self-isolating tactics of the walled-off society. When access is finally granted, it is evident that Smith is not having an authentic, unmediated experience.

Once granted access to North Korea, Smith visits the philosopher hotline located in The People's Library of North Korea, where philosophical Marxist dialectic problems can be asked to a professor who will then, "give them the correct answers immediately," according to Smith's North Korean tour guide. (Smith) The idea of the immediacy and totality of the knowledge resembles in some ways New Atlantis' Salomon's House, which is described as, "the noblest foundation (as we think) that ever was upon the earth; and the lanthorn of this kingdom," (Bacon 167) where all knowledge is being collected with some semblance of finality. Taking notice of the switch from comparative adjectives to superlatives, the description mirrors the tactics used by a totalitarian state, implying that the established order is undeniably and inarguably the best. Amongst the many things the father of Salomon's House tells the stranger about, he boasts, "We have also precious stones of all kinds, many of them of great beauty, and to you unknown," (Bacon 182) "We have also soundhouses, where we practise and demonstrate all sounds, and their generation," (Bacon 182) "We have also a mathematical-house, where are represented all instruments, as well of geometry as astronomy, exquisitely made," (183) and so on and so on. The use of "all" in these sentences represents a totality of knowledge that is unrealistic, and as such suspect to the contemporary reader. Much as the professor in The People's Library of North Korea can be viewed as suspect because the claim is made that he can give all correct answers immediately, and with no caveat, the concept of a utopian state must be questioned if it presents itself as having access to all knowledge. As such, and without unmediated access to the lived experience of the citizens in any utopian state, it is easy for a modern interpretation of classical utopian texts such as New Atlantis to be one of skepticism.

CAPITALIST REALISM AND OUR INABILITY TO IMAGINE A UTOPIAN ALTERNATIVE

Gone is the notion that one might be able to sail a ship to a faraway land and come into contact with a more advanced unknown. With the only large-scale political alternatives seemingly more dystopian than utopian, it becomes difficult to imagine anything that could function smoothly while also existing external to the globalized social order of democratic capitalism. This sentiment lends itself to capitalist realism, defined by Mark Fisher as, "the widespread sense that not only is capitalism the only viable political and economic system, but also that it is now impossible to even imagine a coherent alternative to it." (8) The death of the popularity of utopian fiction supports this idea. Reliant on the stranger, it is no wonder that, in a globalized world, it is difficult to imagine how such a narrative could possibly manifest. Who could possibly exist outside of this system, to be encountered and subsequently learned from? Paradoxically, capitalism gains its power through its ability to conquer, implying that there was once some other, but it has since been subsumed. The different system must always be found and then dominated, made to bend itself to the capitalist world order or else risk global exclusion. As such, perhaps early days of capitalism were defined by a similar mode of exploration as is envisioned in the utopian genre. However, when applied in our contemporary world, the symbolic "stranger" must never have recognized the potential utopia as such, but rather must have concluded time and time again that the only way for global progression was through subsuming the other until it fit into the pre-established world order. As capitalism makes itself more ubiquitous, the question becomes, "having all too-successfully incorporated externality, how can [capitalism] function without an outside it can colonize and appropriate?" (Fisher 12) As evidence of capitalism's global domination: only some 30 years after the Berlin Wall fell, symbolizing the fall of the Soviet Union, Moscow itself is saturated with craft beer bars and a McDonald's is situated 1.5 kilometers from Red Square. This total encapsulation helps to explain why it is so difficult to imagine something external from what is already known and established.

Even internal modes of resistance don't seem capable of withstanding the larger social order.

For example, the notion of "punk" became a commodity almost as soon as it was a form of resistance, most famously with Malcom McLaren's cultivation and marketing of the image of the Sex Pistols.

The punk music of the mid-to-late-1970s concerned itself with being a removal from the capitalist infrastructure that enabled most music production. Rather than affiliate itself with big and expensive labels, the movement found other cheaper and independent avenues to gain exposure. Instead of sign on with one of the Big Six record labels, "punks reverted to 'front-room studios' and recorded their music relatively cheaply, using four-track tape recorders" (Thompson 51). This was in line with the punk ideology, which fancied itself to be more interested in DIY, both in style and in music production, than commercial and fiscal success. However, by eventually signing with larger labels, early punk bands such as the Sex Pistols and The Clash gave away much of their agency, seemingly the price to be paid for inclusion in the capitalist market that the movement had initially eschewed.

In this way the concept of rebelling against the system was sold almost as quickly as it was established. In contemporary representations of UK culture, rather than the notion of "punk" being a radical reaction against commercialization, it has been almost entirely subsumed by the commodifying machine of capitalist realism. Consider, at the opening ceremony of the 2012 London Olympic Games, alongside Daniel Craig's James Bond delivering the Queen to the arena, Arctic Monkeys' rendition of "Come Together" by The Beatles, and Mr Bean playing the keyboard, can be heard "Pretty Vacant" by the Sex Pistols. Video footage of the lead singer Johnny Rotten leers behind masked dancers with cartoonish Mohawks atop their artificial bobble-heads. (Boyle) At the time of the performance, the Queen was in the audience watching. This is a far removal from the reaction in 1977, when the Sex Pistols track "God Save the Queen" was banned from play on the BBC. If the opening ceremony can be considered as representative of the idealized cultural heritage that the UK, and namely England, wants exported globally, then the Sex Pistols' inclusion in the opening ceremony reveals the totality of the declawing and commodification of the original punk sentiment of anarchy and upheaval. By 2012, the Sex Pistols were palatable enough to be tastefully performed in front of royalty, and accepted as something aesthetic, rather than anarchic.

Ruth Adams notes in regards to the Sex Pistols, "Arguably the band themselves have been complicit in the 'Pistols Heritage Industry,' staging their own 'Silver Jubilee' celebrations in the form of a(nother) reunion concert in 2002 and licensing numerous souvenir commodities from pencil cases to fridge magnets" (473) but one need not wait until the early 2000s to see this rapid commodification.

Nearly as soon as the punk movement was started, it was perverted, as the sudden global awareness of the movement seemingly sapped all authenticity from the image of the rebellious outsider it had previously strived for. By 1979, "Punk was history, finished; the full story could now be told," This quick turnaround experienced by the punk movement, from resistance against to product of a capitalist culture, can be illuminated by Fisher's concept of "precorperation". Fisher explains, "what we are dealing with now is not the incorporation of materials that previously seemed to possess subversive potentials, but instead, their precorperation: the pre-emotive formatting and shaping of desires, aspirations and hopes by capitalist culture." (12) Punk may have been the first instance of this phenomena, being subsumed by the consumeristic machine so that the concept of independence from said machine could be oxymoronically purchased and worn. In this way, the alternative became the mainstream, and the feelings of antipathy towards society became muted and controlled; the rebellion and dissent against commercial capitalism repackaged as something that would fit cleanly within the framework provided.

All of which leaves us with a dual problem. We cannot find the change externally, as any external system is demonized as either regressive or totalitarian, or else ultimately incorporated and forced to conform to the already established system, and yet we also can't seem to enact change within, as any attempt to do so is swallowed up by the machine, repackaged, and then sold as acceptable, and ultimately aesthetic, dissent. The only option that presents itself easily is to burn the world down and begin again. There, the utopian state can be glimpsed, although it is ultimately postapocalyptic. (421) This rise in popularity is problematic in that it supports Fisher's theory that we cannot imagine anything outside of our contemporary social order. Furthermore, if we cannot imagine a new global order as solution, how could we then implement it?

Capitalism continually refuses to solve the problem of impending disaster. Capitalism, and not capitalists. Returning to Mark Fisher, he makes the point that, "Instead of saying that everyonei.e. every one -is responsible for climate change, we all have to do our bit, it would be better to say that no-one is, and that's the very problem. The cause of eco-catastrophe is an impersonal structure which, even though it is capable of producing all manner of effects, is precisely not a subject capable of exercising responsibility." (70) Given that capitalism keeps itself powerful by instilling the belief that individual decisions drive good or bad outcomes, it is difficult to imagine a solution to large and difficult problems such as climate change arising from it. Because it fixates on the individual rather than the systematic, much as novels do, we see these individual narratives follow the same path that we ourselves feel pulling us, that of being subject to the tides of history, rather than proactive in changing them. Margaret Atwood's MaddAddam series exemplifies this inability within our global society to imagine a solution. In The Year of the Flood, we are introduced to the Gardeners, a radical environmental group that styles itself after the Garden of Eden. This is arguably a utopian state within the dystopian landscape. The society lives in near-total isolation, and does not let in just anyone.

What's more, the individual has been subsumed by the collective, so that all women in leadership roles are named Eve, the men named Adam. This functions so that the members identify with the group, rather than with their own egos, mirroring the representations of early-modern utopias, which are more concerned with the larger societal dynamics than individual agency.

The Gardeners' idealized ecological acts of resistance, such as living as "strict-vegetarians" and farming their own produce, are still not enough to stall the end of the world. Although the Gardeners are living an eco-friendly, utopian life, their bubble is too small to enact any real change.

They venture out in order to hold protests, described by Toby thusly, "The leader had a beard and was wearing a caftan that looked as if it had been sewn by elves on hash. Behind him came an assortment of children-various heights, all colors, but all in dark clothing-holding their slates with slogans printed on them: God's Gardeners for God's Garden! Don't Eat Death! Animals R Us!" (Atwood 66) but the Gardeners are dismissed immediately by the surrounding public. Utopian idealism, it seems, when attempting wider-scale impact, is met by distrust and skepticism. How, with a contemporary understanding of the world, can these people be trusted? With the knowledge of how closed groups oftentimes are riddled with their own internal power dynamics, resulting in them being much less idealistic than they present themselves to be, the Gardeners are working for change at a deficit, being forced to prove themselves as truly utopian, rather than just aesthetically so. As such, the Gardener's plead for change must be met by heckling, "'Shut the fuck up, ecofreak,"' (67) a bystander chides.

Compare these efforts to the current UK-based movement Extinction Rebellion. Arising in the UK circa 2018, they have three key demands: Tell the truth, zero emissions by 2025, and the instillation of a Citizens' assembly to address the crisis. Although it is inarguable that their efforts have made a global media ripple, the question remains: How effective can these protests be to enact change? As of September 2019, their successes have been mostly superficial, and based on rhetoric, rather than truly radical societal change. On May 1, 2019, members of the UK parliament declared a climate and environment emergency, followed by the convening of a citizen's assembly in June, with this later concession being only a partial victory, due to the fact that the recommendations deriving from the assembly will not be legally binding. (Knight) Notably, the demand that has been most ignored by those in power, being zero emissions by 2025, also requires the most concrete and systematic governmental action. Consider how capitalist realism is enabled by 'interpassivity,' or the performance of anti-capitalism which eventually functions as a cathartic act that ultimately allows the participants to remain within the capitalist system, all the while pacified by knowing in their hearts that they are not the problem. (Fisher 16) The fight is ongoing, and the next action seems to be a withholding of taxpayer money until demands are met. This opting out at least appears as something new, a refusal to participate in the system as long as it remains as it is. However, due to the fact that the demands are aimed rather broadly at one small aspect of capitalistic governing, and considering that the efforts of both the Gardeners and Extinction Rebellion exist within a system dominated by capitalist realism, it is suggested that these efforts, both fictional and non-fictional, will be ultimately aesthetic, rather than truly world-changing.

At the climate march in Berlin, Germany on September 20, 2019, the images evoked were those of apocalypse. Placards were illustrated with images of the Earth burning. As has been seen in other major cities throughout Europe, a girl stood before Brandenburger Tor on a block of ice with a noose around her neck, waiting for the ice to melt away and hang her where she stood. As one theorist observes, "Arguably, modern environmentalism has been infused with a strong current of apocalyptic sentiment from its very birth, being distinct from earlier forms of conservationist or preservationist activism -and from social movement activism in general -through its invocation of impending global doom as a tool to rouse action and mobilize support." (Cassegard, Tho rn 562) The debate seems to be whether this forward-looking pessimism is rallying, or debilitating.

Returning to the girl at the climate march who stood atop the block of ice with her hands positioned as if tied behind her back and a noose around her neck, the image becomes all the more poetic when considering the idea of someone having their hand's tied. Having one's hands tied meaning having the will to do something, but not the power, due to some invisible force which stops the person from acting freely. There is the disaster happening right in front of us, but there is also this hands-tied feeling of 'interpassivity'. We recognize the problems but, beyond building awareness of the depth and breadth of them, feel an inability to enact real societal change, which is keeping us from the revolutionary upheaval that is likely required to actually save the planet, and ourselves.

CONCLUSION

Long gone are the days when utopias could be used to instruct and warn the public. Where in the early modern period utopian texts might have been used as a foil to comment upon preexisting social structures, in our contemporary world they instead seem to share traits with totalitarian regimes. Furthermore, the hope that one might stumble upon some other society that is both exemplary and removed from our own, as the stock character of the stranger does, seems to have disappeared as capitalism tightens its global chokehold. Though still dissatisfied with the current social order of democratic capitalism, trends in fiction seem to indicate that we are unable to imagine anything that could challenge the accepted world order. In the place of utopian dreams, we are left with grim narratives centralized around our impending societal collapse.

The death of human society has been constructed and reconstructed, however today's speculation feels different. The antinuclear activism of the 60s and 70s, which concerned itself with warnings against a sudden apocalypse, has been replaced by the protest of today, described as, "Neither to be nourished by a strong sense of hope, nor of a future disaster, but a sense and an idea that the catastrophe is already ongoing." (Cassegard, Tho rn 562) There is an overarching feeling of being too late, without the capacity or the time to imagine an alternative global order. As such, the question becomes not how we could remove ourselves from this global rule, but rather what we can do once capitalism has finally run its course. The only hope left is in the imagining of an apocalypse with the capacity to wipe out the capitalistic beast. With any luck, this apocalypse might spare a few, and allow for rebuilding once there is a fresh slate to work with.

  TOTALITARIANISM: NORTH KOREA, ISOLATION AND PROPAGANDA An attempt to analyze utopian narratives such as New Atlantis from a contemporary vantage point is to see parallels between the utopian city portrayed in the text and present day totalitarian political systems. Given the necessity of the total isolation of the utopia from global political structures, a parallel can be drawn between Bensalem and the Democratic People's Republic of Korea.

(

  Adams 473) as evidenced by the biographies already published by that date, including: Sex Pistols: The Inside Story (1977) by Fred and Judy Vermorel, The Boy Looked at Johnny (1978) by Julie Burchill and Tony Parsons. (Adams 473) This "ending" of the movement, meaning its incorporation into the world of capitalist commodity exchange through "selling-out" as it has been criticized of doing, occurred a mere 5 years after what is understood to be its birth in New York City's CBGBs, circa 1974.

New

  Faces essay collection, Hannah Goldstein, September 2019 http://www.new-faces-erasmusplus.fr/ 10 POST-APOCALYPTIC FICTION: STARTING OVER AS THE ONLY MEANS OF RECONSTRUCTION One genre of fiction benefits from our inability to imagine ourselves out of the contemporary global world order of capitalism. Although exact and up-to-date figures are difficult to locate, a study on a the apocalyptic and post-apocalyptic cannon conducted by Jerry Määttä suggests that, "In future, a similar study would likely show a sharp rise in interesting apocalyptic and post-apocalyptic disaster stories form the years 2000-2015, following a decade of slightly less important work in the genre."

  BIBLIOGRAPHY Adams, Ruth. "The Englishness of English punk: Sex Pistols, subcultures, and nostalgia." Popular Music and Society, 20 Jun. 2018, pp. 469-488. Atwood, Margaret. The Year of the Flood. Toronto: McClelland & Stewart, 2009. Bacon, Francis, and Susan Bruce ed. New Atlantis. Oxford: Oxford University Press, 1999. Beilharz, Peter and Christine Ellem. "Planting Utopia: Some Classic Images." Globalization and Utopia, edited by Patrick Hayden and Chamsy El-Ojeili. Springer Nature, 2009, pp. 13-27. Boyle, Danny. "The complete London 2012 opening ceremony | London 2012 Olympic Games." Youtube, 27 Jul. 2012, https://youtu.be/4As0e4de-rI. Cassegard, Carl and Ha kan Tho rn. "Toward a postapocalyptic environmentalism? Responses to loss and visions of the future in climate activism." Environment and Planning E: Nature and Space, Vol. 1, 4, 2018, pp. 561-578. Dukalskis, Alexander, and Zachary Hooker. "Legitimating totalitarianism: Melodrama and mass politics in North Korean film." Communist and Post-Communist Studies, 2011, pp. 53-62. Fisher, Mark. Capitalist Realism: Is there no alternative? Winchester: Zero books, 2009.