text: [14] K. Schmidt, A. Thöns-Zueva, and P. Joly. High order asymptotic expansion for the acostics in viscous gases close to rigid walls. , 24(9):1823–1856, 2014. Math. Models Methods Appl. Sci. [15] G. Searby, A. Nicole, M. Habiballah, and E. Laroche. Prediction of the eﬃciency of acoustic damping cavities. , 24(3):516–523, 2008. J. Propul. Power. [16] H. Tijdeman. On the propagation of sound waves in cylindrical tubes. , 39(1):1–33, J. Sound Vibration 1975. [17] Comsol Multiphysics version 5.2a. Acoustics Module, User’s Guide. 2016. [18] Comsol Multiphysics version 5.2a. Reference Manual. 2016. [19] J. Wloka. . Cambridge University Press, 1987. Partial Diﬀerential Equations 25

label: reference

text: K.

text: Schmidt,

text: A.

text: Thöns-Zueva,

text: and

text: P.

text: Joly.

text: High

text: order

text: asymptotic

text: expansion

text: for

text: the

text: acostics

text: in

text: viscous

text: gases

text: close

text: to

text: rigid

text: walls.

text: Math.

text: Models

text: Methods

text: Appl.

text: Sci.

text: G.

text: Searby,

text: A.

text: Nicole,

text: M.

text: Habiballah,

text: and

text: E.

text: Laroche.

text: Prediction

text: of

text: the

text: eﬃciency

text: of

text: acoustic

text: damping

text: cavities.

text: J.

text: Propul.

text: Power.

text: H.

text: Tijdeman.

text: On

text: the

text: propagation

text: of

text: sound

text: waves

text: in

text: cylindrical

text: tubes.

text: J.

text: Sound

text: Vibration

text: Comsol

text: Multiphysics

text: version

text: Acoustics

text: Module,

text: User’s

text: Guide.

text: Comsol

text: Multiphysics

text: version

text: Reference

text: Manual.

text: J.

text: Wloka.

text: Cambridge

text: University

text: Press,

text: Partial

text: Diﬀerential

text: Equations


text: can also be carried out in the case of a smooth non-ﬂat surface. In that case, the normal ﬁeld vector can 𝒏 be extended into the inside of the domain in the vicinity of the wall using the deﬁnition ∇𝑑(𝒙) (27) 𝒏(𝒙) = , |∇𝑑(𝒙)| where is the [6]. Taking derivatives in the directions of this extended 𝑑(𝒙) wall distance function normal ﬁeld, we can in the vicinity of the wall split the gradient and divergence operators in their normal and tangential parts, analogously as in the case of a ﬂat wall, ∇𝑇 = ∇T𝑇 + 𝒏𝜕𝑇 , (28a) 𝜕𝑛 ∇ ⋅ 𝑼 = ∇T ⋅ 𝒖 + 𝒏 ⋅ 𝜕𝑼 , (28b) 𝜕𝑛 where and the tangential operators simply are deﬁned through the expressions above. 𝒖 = 𝑼 − (𝑼 ⋅ 𝒏)𝒏 However, the splitting of the Laplacian operator, needed in the derivation of the boundary-layer equations, is more complicated in the curved-wall case, 𝜕2𝑇 𝜕𝑇 (29) Δ𝑇 = ΔT𝑇 + + 𝐻 , 𝜕𝑛2 𝜕𝑛 involving an extra term with , the sum of the principal curvatures of the level surface to 𝐻 = ∇T ⋅ 𝒏 𝑑 that passes through the point of interest. However, in many practical situations with a smooth wall, it is reasonable to assume that the minimal radius of the principal curvatures is much larger than the boundary layer thicknesses, which is of the order of 20–400 µm in the audio range. We will therefore here apply boundary conditions (24c) or (26b) also for nonplanar smooth boundaries, interpreting the tangential operators as in deﬁnitions (28). Due to the likely scale separation between boundary-layer thickness and the wall’s radii of curvature, we conjecture that taking wall curvature into account in the boundary conditions would constitute in many cases at most a second-order correction to the boundary conditions derived above. 4 Example: a cavity with lossy walls Here we exemplify the use of wall boundary condition (26b) in the context of an acoustic cavity problem. Let the domain of the cavity, conceptually illustrated in ﬁgure 1, be denoted . The cavity boundary Ω ⊂ ℝ3 consists of a solid-wall part and a part where sound waves can enter and exit. We assume 𝜕Ω Γw Γio that both boundary parts are smooth and, as discussed in § , that the radii of the principal curvatures of the surface everywhere is much larger than and . Possible edges and corners of the cavity can Γw 𝛿 𝛿 therefore only be located at the interfaces between𝑉 an𝑇d . Γio Γw The acoustic pressure amplitude in the cavity is then modeled by the boundary-value problem −𝑘2𝑝 − Δ𝑝 = 0 in Ω, (30a) 0 i 𝜕𝑝 i on . (30b) 𝑘 𝑝 + = 2 𝑘 𝑔 Γio 0 0 𝜕𝑛 i i −𝛿 − 1ΔT𝑝 + 𝛿 𝑘2 ( − 1)(𝛾 − 1)𝑝 + 𝜕𝑝 = 0 on Γw, (30c) 𝑉 2 𝑇 0 2 𝜕𝑛 on , (30d) 𝒏T ⋅ ∇T𝑝 = 0 𝜕Γw Boundary condition (30b) is a simple radiation (or impedance) condition, in which function supplies an 𝑔 incoming acoustic wave and where outgoing planar waves are absorbed. Since wall boundary condition (30c) in itself constitutes a diﬀusion problem on the bounded surface , an extra condition—“a boundary Γw 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: can

label: paragraph

font: ZNEFIL+STIXGeneral-Regular

text: also

text: be

text: carried

text: out

text: in

text: the

text: case

text: of

text: a

text: smooth

text: non-ﬂat

text: surface.

text: In

text: that

text: case,

text: the

text: normal

text: ﬁeld

text: vector

text: can

text: 𝒏

font: MKFGCQ+STIXMath-BoldItalic

text: be

text: extended

text: into

text: the

text: inside

text: of

text: the

text: domain

text: in

text: the

text: vicinity

text: of

text: the

text: wall

text: using

text: the

text: deﬁnition

text: ∇𝑑(𝒙)

label: equation

font: LEXSTI+STIXMath-Regular

text: 𝒏(𝒙)

font: PGZLFD+STIXMath-Italic

text: |∇𝑑(𝒙)|

text: where

text: is

text: the

text: Taking

text: derivatives

text: in

text: the

text: directions

text: of

text: this

text: extended

text: 𝑑(𝒙)

text: wall

text: distance

text: function

text: normal

text: ﬁeld,

text: we

text: can

text: in

text: the

text: vicinity

text: of

text: the

text: wall

text: split

text: the

text: gradient

text: and

text: divergence

text: operators

text: in

text: their

text: normal

text: and

text: tangential

text: parts,

text: analogously

text: as

text: in

text: the

text: case

text: of

text: a

text: ﬂat

text: wall,

text: ∇𝑇

text: ∇T𝑇

text: 𝒏𝜕𝑇

text: 𝜕𝑛

text: 𝑼

text: ∇T

text: 𝒖

text: 𝒏

text: 𝜕𝑼

text: 𝜕𝑛

text: where

text: and

text: the

text: tangential

text: operators

text: simply

text: are

text: deﬁned

text: through

text: the

text: expressions

text: above.

text: 𝒖

text: 𝑼

text: (𝑼

text: 𝒏)𝒏

text: However,

text: the

text: splitting

text: of

text: the

text: Laplacian

text: operator,

text: needed

text: in

text: the

text: derivation

text: of

text: the

text: boundary-layer

text: equations,

text: is

text: more

text: complicated

text: in

text: the

text: curved-wall

text: case,

text: 𝜕2𝑇

text: 𝜕𝑇

text: Δ𝑇

text: ΔT𝑇

text: 𝐻

text: 𝜕𝑛2

text: 𝜕𝑛

text: involving

text: an

text: extra

text: term

text: with

text: the

text: sum

text: of

text: the

text: principal

text: curvatures

text: of

text: the

text: level

text: surface

text: to

text: 𝐻

text: ∇T

text: 𝒏

text: 𝑑

text: that

text: passes

text: through

text: the

text: point

text: of

text: interest.

text: However,

text: in

text: many

text: practical

text: situations

text: with

text: a

text: smooth

text: wall,

text: it

text: is

text: reasonable

text: to

text: assume

text: that

text: the

text: minimal

text: radius

text: of

text: the

text: principal

text: curvatures

text: is

text: much

text: larger

text: than

text: the

text: boundary

text: layer

text: thicknesses,

text: which

text: is

text: of

text: the

text: order

text: of

text: µm

text: in

text: the

text: audio

text: range.

text: We

text: will

text: therefore

text: here

text: apply

text: boundary

text: conditions

text: or

text: also

text: for

text: nonplanar

text: smooth

text: boundaries,

text: interpreting

text: the

text: tangential

text: operators

text: as

text: in

text: deﬁnitions

text: Due

text: to

text: the

text: likely

text: scale

text: separation

text: between

text: boundary-layer

text: thickness

text: and

text: the

text: wall’s

text: radii

text: of

text: curvature,

text: we

text: conjecture

text: that

text: taking

text: wall

text: curvature

text: into

text: account

text: in

text: the

text: boundary

text: conditions

text: would

text: constitute

text: in

text: many

text: cases

text: at

text: most

text: a

text: second-order

text: correction

text: to

text: the

text: boundary

text: conditions

text: derived

text: above.

text: Example:

text: a

text: cavity

text: with

text: lossy

text: walls

text: Here

text: we

text: exemplify

text: the

text: use

text: of

text: wall

text: boundary

text: condition

text: in

text: the

text: context

text: of

text: an

text: acoustic

text: cavity

text: problem.

text: Let

text: the

text: domain

text: of

text: the

text: cavity,

text: conceptually

text: illustrated

text: in

text: ﬁgure

text: be

text: denoted

text: The

text: cavity

text: boundary

text: Ω

text: ℝ3

text: consists

text: of

text: a

text: solid-wall

text: part

text: and

text: a

text: part

text: where

text: sound

text: waves

text: can

text: enter

text: and

text: exit.

text: We

text: assume

text: 𝜕Ω

text: Γw

text: Γio

text: that

text: both

text: boundary

text: parts

text: are

text: smooth

text: and,

text: as

text: discussed

text: in

text: that

text: the

text: radii

text: of

text: the

text: principal

text: curvatures

text: of

text: the

text: surface

text: everywhere

text: is

text: much

text: larger

text: than

text: and

text: Possible

text: edges

text: and

text: corners

text: of

text: the

text: cavity

text: can

text: Γw

text: 𝛿

text: 𝛿

text: therefore

text: only

text: be

text: located

text: at

text: the

text: interfaces

text: between𝑉

text: an𝑇d

text: Γio

text: Γw

text: The

text: acoustic

text: pressure

text: amplitude

text: in

text: the

text: cavity

text: is

text: then

text: modeled

text: by

text: the

text: boundary-value

text: problem

text: −𝑘2𝑝

text: Δ𝑝

text: in

text: Ω,

text: i

text: 𝜕𝑝

text: i

text: on

text: 𝑘

text: 𝑝

text: 𝑘

text: 𝑔

text: Γio

text: 𝜕𝑛

text: i

text: i

text: −𝛿

text: 1ΔT𝑝

text: 𝛿

text: 𝑘2

text: 𝜕𝑝

text: on

text: Γw,

text: 𝑉

text: 𝑇

text: 𝜕𝑛

text: on

text: 𝒏T

text: ∇T𝑝

text: 𝜕Γw

text: Boundary

text: condition

text: is

text: a

text: simple

text: radiation

text: (or

text: impedance)

text: condition,

text: in

text: which

text: function

text: supplies

text: an

text: 𝑔

text: incoming

text: acoustic

text: wave

text: and

text: where

text: outgoing

text: planar

text: waves

text: are

text: absorbed.

text: Since

text: wall

text: boundary

text: condition

text: in

text: itself

text: constitutes

text: a

text: diﬀusion

text: problem

text: on

text: the

text: bounded

text: surface

text: an

text: extra

text: condition—“a

text: boundary

text: Γw

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: |𝑝 − 𝑔|2 = |𝑝|2 + |𝑔|2 − 2 Re 𝑝̄𝑔, (34) into equality (33) and dividing by , to obtain terms in units of power, we obtain 2𝜌 𝑐 0 ∫ ∫ ∫ ∫ 1 |𝑔|2 = 1 |𝑝 − 𝑔|2 + (𝛾 − 1) 𝛿𝑇 𝜔 |𝑝|2 + 𝛿𝑉 |∇T𝑝|2. (35) 2𝜌0𝑐 2𝜌0𝑐 4𝜌0𝑐2 4𝜔𝜌0 Γio Γio Γw Γw Expression (35) expresses that the incoming power equals the sum of the reﬂected power and the power losses due to the thermal and viscous boundary layers. 5 Comparisons with classical results 5.1 Boundary-layer theory Searby et al. [15] consider the problem of calculating the visco–thermal boundary-layer damping in acoustic cavities. The authors review the boundary-layer theory and propose a two-step procedure in which a isentropic Helmholtz solver ﬁrst calculates the pressure distribution on the solid surfaces. In a second step, the total power loss from the visco-thermal boundary layers is calculated using boundary-layer theory. The loss terms (6) and (10) given by Searby et al. [15] agree with the two last terms in the power balance law (35). Note, however, that the procedure of Searby et al. does not predict, for instance, phase shift eﬀects of the boundary layers, nor does it provide, as here, an explicit locally-reacting boundary condition, coupled to the interior problem. Another approach is to consider a one-dimensional analysis, as presented by Rienstra & Hirschberg [11, § 4.5], who analyze the thermal boundary layer in the case of a plane wave at normal incidence towards an isothermal wall and the viscous boundary layer in the case of a plane wave propagating parallel to the wall. The so-called that their analysis yield are then applied to the case of a plane wave displacement thicknesses propagating in a wave guide. The ﬁnal result is an expression for the complex wavenumber that governs 𝑘 the pressure amplitude, assumed to be of the form e i , where the coordinate is along the axis of 𝑝(𝑧) = − 𝑘𝑧 𝑧 the wave guide, and in which the imaginary part of represents the visco–thermal damping. We will now 𝑘 show that if boundary-value problem (30) in its variational form (31) is applied to such a case of a narrow waveguide, we obtain the same expression as Rienstra & Hirschberg. We consider the setup illustrated in ﬁgure , where is a cylindrical wave guide of length Ω = 𝑆 × (0, 𝓁) . The wave guide’s cross section is ﬁxed, of area , and has a smooth boundary of circumference 𝓁 𝑆 |𝑆| 𝜕𝑆 . In order for the boundary-layer approximations to be valid, we assume that and are small |𝜕𝑆| 𝛿 𝛿 compared to √ , and that √ is of the same order as , precluding overly ﬂa𝑉ttened 𝑇geometries. |𝑆| |𝑆| |𝜕𝑆| Otherwise, the shape of can be arbitrary. Let the waveguide be oriented along the -axis and let be 𝑆 𝑧 Γio the two cross section planes located at and , respectively. Furthermore, let and 𝑥 = 0 𝑥 = 𝓁 𝑔(𝑥, 0) = 𝑔 for a given number . According to boundary-layer theory and since the driving signal0 is 𝑔(𝑥, 𝓁) = 0 𝑔 𝑔 constant at the inlet, we may mak0e the ansatz that the pressure ﬁeld is constant in each cross section, t0hat is, that . 𝑝 = 𝑝(𝑧) 5.1 The assumption that also constitutes a of variational Remark . 𝑝 = 𝑝(𝑧) Galerkin approximation form (31) such that and are constant over each cross section. 𝑝 𝑞 Under this approximation, , where is a unit vector in the direction, and variational ∇𝑝 = ∇T𝑝 = 𝒆 𝑝′ 𝒆 𝑧 form (31) reduces to 𝑧 𝑧 ∫ 𝓁 [ i − 1 ] ∫ 𝓁 [ i − 1 ] 𝑞′ |𝑆| + 𝛿 |𝜕𝑆| 𝑝′ dz − 𝑘2 𝑞 |𝑆| − 𝛿 (𝛾 − 1)|𝜕𝑆| 𝑝 dz (36) 𝑉 2 0 𝑇 2 0 [ ] 0 i i + 𝑘 |𝑆| 𝑞(𝓁)𝑝(𝓁) + 𝑞(0)𝑝(0) = 2 𝑘 𝑞(0)𝑔 . 0 0 0 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: |𝑝

font: EIMKHM+STIXMathExtensions-Regular

text: Re

text: 𝑝̄𝑔,

text: into

text: equality

text: and

text: dividing

text: by

text: to

text: obtain

text: terms

text: in

text: units

text: of

text: power,

text: we

text: obtain

text: 2𝜌

text: 𝑐

font: DRIPPK+STIXMathCalligraphy-Regular

text: |𝑝

text: (𝛾

text: 𝛿𝑇

text: 𝜔

text: 𝛿𝑉

text: 2𝜌0𝑐

text: 2𝜌0𝑐

text: 4𝜔𝜌0

text: Γio

text: Γio

text: Γw

text: Γw

text: Expression

text: expresses

text: that

text: the

text: incoming

text: power

text: equals

text: the

text: sum

text: of

text: the

text: reﬂected

text: power

text: and

text: the

text: power

text: losses

text: due

text: to

text: the

text: thermal

text: and

text: viscous

text: boundary

text: layers.

text: Comparisons

text: with

text: classical

text: results

text: Boundary-layer

text: theory

text: Searby

text: et

text: al.

text: consider

text: the

text: problem

text: of

text: calculating

text: the

text: visco–thermal

text: boundary-layer

text: damping

text: in

text: acoustic

text: cavities.

text: The

text: authors

text: review

text: the

text: boundary-layer

text: theory

text: and

text: propose

text: a

text: two-step

text: procedure

text: in

text: which

text: a

text: isentropic

text: Helmholtz

text: solver

text: ﬁrst

text: calculates

text: the

text: pressure

text: distribution

text: on

text: the

text: solid

text: surfaces.

text: In

text: a

text: second

text: step,

text: the

text: total

text: power

text: loss

text: from

text: the

text: visco-thermal

text: boundary

text: layers

text: is

text: calculated

text: using

text: boundary-layer

text: theory.

text: The

text: loss

text: terms

text: and

text: given

text: by

text: Searby

text: et

text: al.

text: agree

text: with

text: the

text: two

text: last

text: terms

text: in

text: the

text: power

text: balance

text: law

text: Note,

text: however,

text: that

text: the

text: procedure

text: of

text: Searby

text: et

text: al.

text: does

text: not

text: predict,

text: for

text: instance,

text: phase

text: shift

text: eﬀects

text: of

text: the

text: boundary

text: layers,

text: nor

text: does

text: it

text: provide,

text: as

text: here,

text: an

text: explicit

text: locally-reacting

text: boundary

text: condition,

text: coupled

text: to

text: the

text: interior

text: problem.

text: Another

text: approach

text: is

text: to

text: consider

text: a

text: one-dimensional

text: analysis,

text: as

text: presented

text: by

text: Rienstra

text: Hirschberg

text: who

text: analyze

text: the

text: thermal

text: boundary

text: layer

text: in

text: the

text: case

text: of

text: a

text: plane

text: wave

text: at

text: normal

text: incidence

text: towards

text: an

text: isothermal

text: wall

text: and

text: the

text: viscous

text: boundary

text: layer

text: in

text: the

text: case

text: of

text: a

text: plane

text: wave

text: propagating

text: parallel

text: to

text: the

text: wall.

text: The

text: so-called

text: that

text: their

text: analysis

text: yield

text: are

text: then

text: applied

text: to

text: the

text: case

text: of

text: a

text: plane

text: wave

text: displacement

text: thicknesses

text: propagating

text: in

text: a

text: wave

text: guide.

text: The

text: ﬁnal

text: result

text: is

text: an

text: expression

text: for

text: the

text: complex

text: wavenumber

text: that

text: governs

text: 𝑘

text: the

text: pressure

text: amplitude,

text: assumed

text: to

text: be

text: of

text: the

text: form

text: e

text: i

text: where

text: the

text: coordinate

text: is

text: along

text: the

text: axis

text: of

text: 𝑝(𝑧)

text: 𝑘𝑧

text: 𝑧

text: the

text: wave

text: guide,

text: and

text: in

text: which

text: the

text: imaginary

text: part

text: of

text: represents

text: the

text: visco–thermal

text: damping.

text: We

text: will

text: now

text: 𝑘

text: show

text: that

text: if

text: boundary-value

text: problem

text: in

text: its

text: variational

text: form

text: is

text: applied

text: to

text: such

text: a

text: case

text: of

text: a

text: narrow

text: waveguide,

text: we

text: obtain

text: the

text: same

text: expression

text: as

text: Rienstra

text: Hirschberg.

text: We

text: consider

text: the

text: setup

text: illustrated

text: in

text: ﬁgure

text: where

text: is

text: a

text: cylindrical

text: wave

text: guide

text: of

text: length

text: Ω

text: 𝑆

text: 𝓁)

text: The

text: wave

text: guide’s

text: cross

text: section

text: is

text: ﬁxed,

text: of

text: area

text: and

text: has

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text: 12 A. Gomez Ramirez @auth, C. Lara @auth, L. Betev @auth, D. Bilanovic @auth, U. Kebschull @auth, for the ALICE Collaboration CNN accuracy SVM accuracy 1.00 0.98 0.99 0.96 0.98 0.94 0.97 0.92 0.90 0.96 0.88 0.95 Training Training 0.86 Validation Validation 0 20 40 60 80 100 0 20 40 60 80 100 Epoch Epoch Fig. 9 Classiﬁcation accuracy of the CNN applied to system calls Fig. 10 Classiﬁcation accuracy of SVM applied to system calls CNN vs SVM accuracy CNN vs SVM false positive rate 1.00 CNN SVM 0.10 0.98 0.96 0.08 0.94 0.06 0.92 0.04 0.90 0.88 0.02 CNN 0.86 SVM 0.00 0 20 40 60 80 100 0 20 40 60 80 100 Epoch Epoch Fig. 11 Accuracy comparison of CNN vs SVM applied to system calls Fig. 12 False Positive Rate comparison of CNN vs SVM applied to system calls CNN accuracy SVM accuracy 0.9 1.0 0.8 0.9 0.7 0.8 0.6 0.7 0.5 0.6 Training 0.4 Training Validation Validation 0 20 40 60 80 100 0 20 40 60 80 100 Epoch Epoch Fig. 13 Classiﬁcation accuracy of CNN applied network data Fig. 14 Classiﬁcation accuracy of SVM applied to network data ycaruccA ycaruccA ycaruccA ycaruccA etar 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text: 6 sellation [29], and setting the direction of the anisotropy -2.00 vector randomly from 0◦ to 8◦ from the + -axis for each z No interpolation grain. The ﬁlms were initialized to a stripe pattern sim- Direct interpolation -2.02 ilar to Ref. [9], with approximately 80 nm wide stripes Scalar potential method and let relax (Resulting in a system similar to what was shown in Fig. 1), after which the upper ﬁlm was driven -2.04 in the + -direction by a spring moving at a constant ve- x locity = 2 m/s for 300 ns. The friction force was mea- vd − -2.06 sured from the spring elongation = ( ), Fspring k vdt xs where is the position of the slider. The spring con- xs stant was chosen as = 0 005 N/m. The simulations -2.08 k . were carried out in zero temperature. We also included a viscous damping term in the equation of motion, -2.10 0 20 40 60 80 100 − F = ˙ (13) d γmxs, t [ns] Figure 5. Energies for diﬀerent interpolation methods. where m is the mass of the slider, x˙s its velocity and γ Though the dissimilar time evolution of the magnetization is a viscous damping coeﬃcient. The complete equation is reﬂected by the energy of the system in the various cases, of motion for the slider is then the trend in the total energy is similar in all three cases. The drop in energy is the result of a magnetization reconﬁguration in response to motion of the slider. ¨ = ( − ) − ˙ + s (14) mxs k vdt xs γmxs Fm, 1.2 where s is the -directional component of the force d = 20 nm Fm x 1.0 d = 80 nm exerted by the base on the slider deﬁned in Equation (3). The mass and the damping coeﬃcient in these numerical 0.8 examples are chosen such that the spring-slider system is critically damped and the resulting viscous damping force is roughly an order of magnitude smaller than s . 0.6 Fm To see whether the ﬁlm distance would aﬀect the friction behavior in our tests as it did in Ref. [9], we ran sim- 0.4 ulations for various ﬁlm distances, ranging from 20 nm to 80 nm. We started the simulation with the spring al- 0.2 ready ahead of the slider by 40 nm, so that the system is already near the steady state dragging at the beginning 0.0 of the simulation. A snapshot of the system total energy for the diﬀer- 0 50 100 150 200 250 300 ent interpolation methods in an example simulation can time [ns] be seen in Fig. 5. For both the direct componentwise interpolation and the scalar potential interpolation, the Figure 6. The friction forces with inter-ﬁlm distance of 20 nm, in which the stripe domains in the slider deform but are held system becomes continuously driven instead of the pe- in place by the stripes of the base, and 80 nm, where the slider riodic jumps of the non-interpolated case and thus the experiences stick-slip motion, and the stripes experience only discontinuities in the total energy are eliminated, as ex- minor changes. The diminishing force in the 80 nm case is due pected. to the stripes having slightly diﬀerent widths, and the initial For a relatively long simulation time, both the direct conﬁguration having better matching stripes in the slider and interpolation and the scalar potential method predict the base. same magnetization dynamics and hence the same en- ergy. When driven for long enough, small discrepancies in the predicted ﬁeld values can lead to slightly diﬀerent the magnetic ﬁlms, the interaction between the stripe do- time evolution of domains. However, the eﬀect this has mains of the base and slider are stronger than the pinning on quantities such as the average friction force is mini- eﬀect of the disorder in the slider, and thus the stripes mal. are locked in place. Due to the movement of the slider, Considering the fact that we used diﬀerent material the stripe domains deform, but they don’t move with the and way of realizing disorder along with smaller ﬁlms, the slider and thus the slider is actually dragged through its qualitative behavior and the numerical results are com- own stripe domains. In this case, the friction force is at parable to those of Ref. [9]. With little distance between its strongest. 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