text: 17 (cid:18) (cid:19) (cid:90) (cid:113) 4 xc(∞)−xc0 = cos(cid:18) θ10 +2 θ20(cid:19)(cid:90) ∞ dt (126) 2 Locus of convergence points, ρ = 1 0 (cid:113) 1+φ2e2Kt 0 yc(∞)−yc0 = sin θ10 +2 θ20 0∞ 1+dφt 2e2Kt . (127) −02 Centroid 90◦ 0 Initial centroid −4 Integrating (126) and (127), we get(cid:113)  (cid:18) (cid:19) Agent 1 −6 Agent 2 (cid:113)  −4 −2 0 2 4 6 8 xc(∞)−xc0 = 21K cos(cid:18) θ10 +2 θ20(cid:19) ln(cid:113) 1+φ02 +1 (128) Fig. 9. The locus of the convergencXe (mpo) ints for ρ = 1, and the trajectories (cid:113) 1+φ02 −1 oagfetnhtesaagreenntsorumndalertothtehecolnotcruosl o(2f3t)hewictohnKve1r=genKc2e=po1i.nTtsh.e trajectories of the yc(∞)−yc0 = 21K sin θ10 +2 θ20 ln 1+φ02 +1 , (129) 1+φ2 −1 case, the concept of heterogeneous controller gains can be used 0 to ensure that the applied control force does not violates the which implies that (cid:18) (cid:19) maximum allowable limit. This situation is already addressed in [15] in achieving synchronization, and can be equivalently stated in phase balancing. yc(∞)−yc0 = tan θ10 +θ20 . (130) xc(∞)−xc0 2 VI. CONCLUSIONS Therefore, for K1 = K2 = K, the locus of convergence point In this paper, we have investigated the phenomenon of phase is a straight line of slope m1 = tan((θ10 +θ20)/2). balancing in a group of heterogeneously coupled agents. It Also, from (85) and (86), it can be observed that the slopes has been shown that a desired reference direction, and hence, of the straight line trajectories of both the agents in balanced the desired orientations of the agents in balanced formation, formation, are same and is given by (cid:18) (cid:19) can be achieved by appropriately selecting the heterogeneous controller gains Kk,∀k, given according to the Assumption 3.1. Moreover, it has been illustrated through simulation that the θ θ m2 = tan(θ1 (t → ∞)) = tan(θ2 (t → ∞)) = −cot 10 +2 20 . reachable set of the reference direction further expands when both positive and negative values of the heterogeneous gains (131) are incorporated in the control scheme. In particular, it has Thus, m1m2 = −1, which is a condition when two straight been proved analytically for N = 2 that there exists a condition lines of slopes m and m are perpendicular. Hence, this can on the heterogeneous controller gains which allows them to 1 2 assume both positive and negative values, and hence, results be concluded that the trajectories of the agents are normal to in, further expansion of the reachable set of the reference the locus of the convergence point. Remark 5.1: Note that, for controller gains K1 = K2 = K > 0, direction. By obtaining the closed form expressions of the the ratio K1/K2(= ρ) is ﬁxed and unity. In this situation, velocity directions for N = 2, we have further shown that the locus of the convergence point, for various values of the the locus of the convergence is a straight line of slope tan((θ10 +θ20)/2). However, by using heterogeneous gains K1 heterogenous gains provided their ratio is ﬁxed, is a straight and K , we can get any desired convergence point in a two line passing through the initial centroid. Furthermore, it has 2 dimensional plane corresponding to different values of ρ (see been pointed out for realistic systems that an upper bound on the control force, applied to each agent, can be obtained by Fig. 8). Therefore, by using heterogeneous controller gains, in bounding the heterogeneous control gains. balanced formation, we can regulate the velocity directions as Simulation results show the effectiveness of using het- well as convergence point of the agents in balanced formation. Example 5.1: Two agents are considered as in Example 4.1. erogenous control gains in regulating the velocity directions Note that the initial centroid is located at (xc0,yc0) = (2,−2). of N agents in balanced formation. It would be interesting as a future research to ﬁnd out an analytical expression The locus of the convergence points for various values of the heterogeneous gains K1 and K2 provided √ρ = 1, is also relating angular separation between agents’ velocity vectors in balanced formation with the heterogeneous control gains for shown in Fig. 9, which is straight line of slope 3, and passes through the the point (xc0,yc0). The trajectories of the agents the general case of N agents. The consideration of issue of collision avoidance among agents is also an interesting future in balanced formation are also shown in the same ﬁgure for the heterogeneous gains K1 = K2 = 1. Clearly, trajectories of problem. the agents are perpendicular to the locus of the convergence VII. APPENDIX points. The splay phase (a special case of phase balancing) is Remark 5.2: In practical, autonomous vehicle can withstand an arrangement in which the agents are at equal angular a limited control force due to physical constraints. In such a separation, that is, their phases are separated by multiples of )m( Y ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: 5 (cid:68) (cid:69) term Tk(θθθ). However, in this paper, the idea of introducing heterogeneous gains is illustrated mainly for the restrictive N ∂U N 1 N N ∑ ∂θ = ∑ pθ,ieiθk = N ∑ ∑ sin(θj −θk) = 0 (17) sufﬁcient condition on Kk, that is, Kk > 0,∀k, since the analysis k=1 k k=1 k=1 j=1 for the set of gains Kk satisfying ∑Nk=1 KkTk(θθθ) > 0 is quite involved for N > 2. Moreover, it will be shown for N = 2 Using (17), (16) can be rewritten as that the reachable set of the velocity directions of the agents in balanced formation further expands for the controller gains N U˙ (θθθ) = − ∑ KkTk(θθθ), (18) Kk satisfying the condition ∑Nk=1 KkTk(θθθ) > 0. k=1 which is the same as (8). Therefore, the conclusions of III. REACHABLE VELOCITY DIRECTIONS Theorem 2.1 are unchanged under control (15). The velocity directions that are achievable in balanced Corollary 2.3: Under the conditions given in Theorem 2.1, formation of agents for different values of the heterogeneous the magnitude |pθ| of the phase order parameter pθ, given by controller gains are called the reachable velocity directions. (2), is strictly decreasing with time if ∑Nk=1 KkTk(θθθ) > 0, and In this section, we analyze the reachability condition of the non-increasing with time if Kk > 0,∀k. velocity directions of agents in balanced formation against The time derivative of (5) yields heterogeneous controller gains Kk > 0,∀k. In particular, it will be shown for N ∈ {2,3} that a desired arrangement of velocity U˙ (θθθ) = N|pθ|d|pθ|, (19) vectors of the agents in balanced formation can be obtained dt by suitably choosing the heterogeneous control gains. At ﬁrst, we discuss the results for ω0 = 0. Then, we extend which by using (9) can be rewritten as these results to ω0 (cid:54)= 0 by performing the analysis in a rotating frame of reference. |pθ|d|dptθ| = −N1 ∑N KkTk(θθθ). (20) A. Case 1: ω0 = 0 k=1 For ω0 = 0, the control law, given by (22), can be written as According to Theorem 2.1, since the agents asymptotically stabilize to a balanced formation, that is, pθ → 0 as t → ∞, K N pθ (cid:54)= 0 for all intermediate times, and hence, (20) can be θ˙k = − Nk ∑ sin(θj −θk). (23) rewritten as j=1 d|dptθ| = −N|1pθ| k∑=N1KkTk(θθθ), (21) tθi1a0Ll <ehteaθthd2ei0n<aggae.nn..gtsl<,eswθθθθiNt(h00.)dTy=hnia[sθm1ei0nc,ss.u.g.rie,vθseNnt0h]baTty∈t(h1(e)−,iπsnt,iatπirat)lNfr,voewmlohceinirteiy-, which implies that d|pθ|/dt < 0 (i.e, |pθ| is strictly decreas- vectors eiθ10,...,eiθN0 of the agents, can be arranged on the unit circle (in a complex plane) in a cyclic manner in which iinsgn)own-hinecnreevaesrin∑gNk)=i1fKKkkT>k(θθθ0),∀>k.0T, hainsdcdom|ppθl|e/tdets ≤the0 p(ir.oeo,f|.pθ| the successor of agent k is agent k +1 modulo N (counted in the anticlockwise direction). This is illustrated in Fig. 1(a) for the ﬁve agents. D. Problem Description For the initial heading angles θθθ(0) of the agents as deﬁned Now, we formally state the main objective of this paper. above, the initial phase order vector pθ(θθθ(0)), from (2), is The control law, given by (15), can be written as given by θ˙k = ω0 − KNk ∑N sin(θj −θk). (22) pθ(θθθ(0)) = N1 ∑N eiθk0 (cid:44) pθ0 = |pθ0|eiΨ0, (24) j=1 k=1 (cid:12) The term K is the control gain of the kth agent. Prior work which is the resultant of all equally(cid:12) scaled initial velocity k in [5] uses the same control gain Kk = K,∀k, whereas we vectors (1/N)eiθ10,...,(1/N)eiθN0. CSince 0 ≤ |pθ0| ≤ 1, the extend the analysis by using different gains Kk for different vector pθ0 ∈ Sz, where, Sz = {z ∈ |z| ≤ 1} is the set of agents. This is the heterogeneous control gains case of interest all the points residing in the interior and on the boundary of to us in this paper. Unlike the case of synchronization in a unit circle in the complex plane, and is shown in Fig. 1(b) [15], the analysis for N agents is quite involved in the case for the arrangement of unit vectors in Fig. 1(a). of balanced formation. Therefore, in this work, the analytical In order to obtain the results and for the ease of analysis in results for heterogeneous gains are given mainly for two and this paper, we divide the initial velocity vectors eiθ10,...,eiθN0 three-agent systems. However, the results for N agents, have of N agents in two subgroups with respect to the the initial been presented though simulations. phase order vector p . This is done by choosing a reference θ 0 Remark 2.1: Note that, in the Theorem 2.1, the conditions axis along the initial phase order parameter p , as shown in θ 0 ∑Nk=1 KkTk(θθθ) > 0 may be satisﬁed for both positive and Fig. 1(b), and measuring the angle of each unit vector eiθk0,∀k, negative values of gains Kk because of the involvement of the with respect to it. By doing so, let there be N < N agents in ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Kk

text: In

text: particular,

text: it

text: will

text: be

text: shown

text: for

text: N

text: that

text: a

text: desired

text: arrangement

text: of

text: velocity

text: U˙

text: (θθθ)

text: N|pθ|d|pθ|,

text: vectors

text: of

text: the

text: agents

text: in

text: balanced

text: formation

text: can

text: be

text: obtained

text: dt

text: by

text: suitably

text: choosing

text: the

text: heterogeneous

text: control

text: gains.

text: At

text: ﬁrst,

text: we

text: discuss

text: the

text: results

text: for

text: ω0

text: Then,

text: we

text: extend

text: which

text: by

text: using

text: can

text: be

text: rewritten

text: as

text: these

text: results

text: to

text: ω0

text: by

text: performing

text: the

text: analysis

text: in

text: a

text: rotating

text: frame

text: of

text: reference.

text: |pθ|d|dptθ|

text: ∑N

text: KkTk(θθθ).

text: A.

text: Case

text: ω0

text: For

text: ω0

text: the

text: control

text: law,

text: given

text: by

text: can

text: be

text: written

text: as

text: According

text: to

text: Theorem

text: since

text: the

text: agents

text: asymptotically

text: stabilize

text: to

text: a

text: balanced

text: formation,

text: that

text: is,

text: pθ

text: as

text: t

text: K

text: N

text: pθ

text: for

text: all

text: intermediate

text: times,

text: and

text: hence,

text: can

text: be

text: θ˙k

text: Nk

text: sin(θj

text: −θk).

text: rewritten

text: as

text: d|dptθ|

text: which

text: implies

text: that

text: d|pθ|/dt

text: (i.e,

text: |pθ|

text: is

text: strictly

text: decreas-

text: vectors

text: of

text: the

text: agents,

text: can

text: be

text: arranged

text: on

text: the

text: unit

text: circle

text: (in

text: a

text: complex

text: plane)

text: in

text: a

text: cyclic

text: manner

text: in

text: which

text: hainsdcdom|ppθl|e/tdets

text: p(ir.oeo,f|.pθ|

text: the

text: successor

text: of

text: agent

text: k

text: is

text: agent

text: k

text: modulo

text: N

text: (counted

text: in

text: the

text: anticlockwise

text: direction).

text: This

text: is

text: illustrated

text: in

text: Fig.

text: for

text: the

text: ﬁve

text: agents.

text: D.

text: Problem

text: Description

text: For

text: the

text: initial

text: heading

text: angles

text: θθθ(0)

text: of

text: the

text: agents

text: as

text: deﬁned

text: Now,

text: we

text: formally

text: state

text: the

text: main

text: objective

text: of

text: this

text: paper.

text: above,

text: the

text: initial

text: phase

text: order

text: vector

text: from

text: is

text: The

text: control

text: law,

text: given

text: by

text: can

text: be

text: written

text: as

text: given

text: by

text: θ˙k

text: ω0

text: KNk

text: ∑N

text: sin(θj

text: −θk).

text: ∑N

text: The

text: term

text: K

text: is

text: the

text: control

text: gain

text: of

text: the

text: kth

text: agent.

text: Prior

text: work

text: which

text: is

text: the

text: resultant

text: of

text: all

text: scaled

text: initial

text: velocity

text: k

text: in

text: uses

text: the

text: same

text: control

text: gain

text: Kk

text: K,∀k,

text: whereas

text: we

text: vectors

text: CSince

text: the

text: extend

text: the

text: analysis

text: by

text: using

text: different

text: gains

text: Kk

text: for

text: different

text: vector

text: Sz,

text: where,

text: Sz

text: {z

text: |z|

text: is

text: the

text: set

text: of

text: agents.

text: This

text: is

text: the

text: heterogeneous

text: control

text: gains

text: case

text: of

text: interest

text: all

text: the

text: points

text: residing

text: in

text: the

text: interior

text: and

text: on

text: the

text: boundary

text: of

text: to

text: us

text: in

text: this

text: paper.

text: Unlike

text: the

text: case

text: of

text: synchronization

text: in

text: a

text: unit

text: circle

text: in

text: the

text: complex

text: plane,

text: and

text: is

text: shown

text: in

text: Fig.

text: the

text: analysis

text: for

text: N

text: agents

text: is

text: quite

text: involved

text: in

text: the

text: case

text: for

text: the

text: arrangement

text: of

text: unit

text: vectors

text: in

text: Fig.

text: of

text: balanced

text: formation.

text: Therefore,

text: in

text: this

text: work,

text: the

text: analytical

text: In

text: order

text: to

text: obtain

text: the

text: results

text: and

text: for

text: the

text: ease

text: of

text: analysis

text: in

text: results

text: for

text: heterogeneous

text: gains

text: are

text: given

text: mainly

text: for

text: two

text: and

text: this

text: paper,

text: we

text: divide

text: the

text: initial

text: velocity

text: vectors

text: three-agent

text: systems.

text: However,

text: the

text: results

text: for

text: N

text: agents,

text: have

text: of

text: N

text: agents

text: in

text: two

text: subgroups

text: with

text: respect

text: to

text: the

text: the

text: initial

text: been

text: presented

text: though

text: simulations.

text: phase

text: order

text: vector

text: p

text: This

text: is

text: done

text: by

text: choosing

text: a

text: reference

text: θ

text: Remark

text: Note

text: that,

text: in

text: the

text: Theorem

text: the

text: conditions

text: axis

text: along

text: the

text: initial

text: phase

text: order

text: parameter

text: p

text: as

text: shown

text: in

text: θ

text: KkTk(θθθ)

text: may

text: be

text: satisﬁed

text: for

text: both

text: positive

text: and

text: Fig.

text: and

text: measuring

text: the

text: angle

text: of

text: each

text: unit

text: vector

text: negative

text: values

text: of

text: gains

text: Kk

text: because

text: of

text: the

text: involvement

text: of

text: the

text: with

text: respect

text: to

text: it.

text: By

text: doing

text: so,

text: let

text: there

text: be

text: N

text: N

text: agents

text: in

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: results [21, 23, 29] (see Table II). This DB transition draws interesting similarities with the BA-MF transition, since both of them exhibit a symmetry-breaking from D = d to D = 1, ≈ have critical points close to 4, and D 1.71, for q = 1 and η = 1, even though we are dealing with completely diﬀerent growth processes (see Fig. 4b and Table I). On the other hand, since the DB model is associated to a generalized Laplacian process ∝ |∇ | described by µ (φ) η, one would be tempted to associate the BA-MF model to a → similar processes where η q. However, it will not be correct because both processes would become equivalent and the numerical solution of the equation would result in the generation of equivalent morphological clusters, which is obviously incorrect. Moreover, another important diﬀerence between the DB and BA-MF description, is that the case η = 1 will not be associated with any inﬂection point in the dynamics, as it is the case for q = 1, since the condition χ > 1 is not satisﬁed, as can be seen from simple inspection of i the GD function. Interestingly enough, if the case η = 1 could have been directly related to the case q = 1, this would have indicated that the DLA process is directly associated to i an inﬂection point in the dynamics and not to a critical point as was suggested through a completely diﬀerent approach [30, 31]. Worthy of remark is that our approach provides the corrected form for all the previous mean-ﬁeld equation for the fractality of the generalized Laplacian processes, within the ﬁrst-order approximation of the general GD-Λ equation, given by, − − − D(d, d , η) = 1 + (d 1)e Λη(dw 1)/d, (11) w whose ﬁrst-order approximation, GD -Λ, is given by, 1 − d + Λη(d 1)/d D(d, d , η)(1) = w − , (12) w 1 + Λη(d 1)/d w ≈ where, Λ = 0.817 4/5, thus unifying all of the models. As such, the corrected GHTM equation for the DB model is recovered by setting d = 2; the corrected HTM equation for w the BA-DLA transition is recovered for η = 1; and the corrected DLA mean-ﬁeld equation is recovered for d = 2 and η = 1. Finally and importantly, under the reduced dimensionality w ∗ − transformation, equations GD-Λ and GD -Λ are respectively given by D(Φ(η)) = exp( Φ) 1 12 ##LTLine##

text: results

text: (see

text: Table

text: II).

text: This

text: DB

text: transition

text: draws

text: interesting

text: similarities

text: with

text: the

text: BA-MF

text: transition,

text: since

text: both

text: of

text: them

text: exhibit

text: a

text: symmetry-breaking

text: from

text: D

text: d

text: to

text: D

text: have

text: critical

text: points

text: close

text: to

text: and

text: D

text: for

text: q

text: and

text: η

text: even

text: though

text: we

text: are

text: dealing

text: with

text: completely

text: diﬀerent

text: growth

text: processes

text: (see

text: Fig.

text: and

text: Table

text: I).

text: On

text: the

text: other

text: hand,

text: since

text: the

text: DB

text: model

text: is

text: associated

text: to

text: a

text: generalized

text: Laplacian

text: process

text: described

text: by

text: µ

text: (φ)

text: η,

text: one

text: would

text: be

text: tempted

text: to

text: associate

text: the

text: BA-MF

text: model

text: to

text: a

text: similar

text: processes

text: where

text: η

text: q.

text: However,

text: it

text: will

text: not

text: be

text: correct

text: because

text: both

text: processes

text: would

text: become

text: equivalent

text: and

text: the

text: numerical

text: solution

text: of

text: the

text: equation

text: would

text: result

text: in

text: the

text: generation

text: of

text: equivalent

text: morphological

text: clusters,

text: which

text: is

text: obviously

text: incorrect.

text: Moreover,

text: another

text: important

text: diﬀerence

text: between

text: the

text: DB

text: and

text: BA-MF

text: description,

text: is

text: that

text: the

text: case

text: η

text: will

text: not

text: be

text: associated

text: with

text: any

text: inﬂection

text: point

text: in

text: the

text: dynamics,

text: as

text: it

text: is

text: the

text: case

text: for

text: q

text: since

text: the

text: condition

text: χ

text: is

text: not

text: satisﬁed,

text: as

text: can

text: be

text: seen

text: from

text: simple

text: inspection

text: of

text: i

text: the

text: GD

text: function.

text: Interestingly

text: enough,

text: if

text: the

text: case

text: η

text: could

text: have

text: been

text: directly

text: related

text: to

text: the

text: case

text: q

text: this

text: would

text: have

text: indicated

text: that

text: the

text: DLA

text: process

text: is

text: directly

text: associated

text: to

text: i

text: an

text: inﬂection

text: point

text: in

text: the

text: dynamics

text: and

text: not

text: to

text: a

text: critical

text: point

text: as

text: was

text: suggested

text: through

text: a

text: completely

text: diﬀerent

text: approach

text: Worthy

text: of

text: remark

text: is

text: that

text: our

text: approach

text: provides

text: the

text: corrected

text: form

text: for

text: all

text: the

text: previous

text: mean-ﬁeld

text: equation

text: for

text: the

text: fractality

text: of

text: the

text: generalized

text: Laplacian

text: processes,

text: within

text: the

text: ﬁrst-order

text: approximation

text: of

text: the

text: general

text: GD-Λ

text: equation,

text: given

text: by,

text: D(d,

text: d

text: η)

text: (d

text: Λη(dw

text: w

text: whose

text: ﬁrst-order

text: approximation,

text: GD

text: -Λ,

text: is

text: given

text: by,

text: d

text: Λη(d

text: D(d,

text: d

text: w

text: w

text: Λη(d

text: w

text: where,

text: Λ

text: thus

text: unifying

text: all

text: of

text: the

text: models.

text: As

text: such,

text: the

text: corrected

text: GHTM

text: equation

text: for

text: the

text: DB

text: model

text: is

text: recovered

text: by

text: setting

text: d

text: the

text: corrected

text: HTM

text: equation

text: for

text: w

text: the

text: BA-DLA

text: transition

text: is

text: recovered

text: for

text: η

text: and

text: the

text: corrected

text: DLA

text: mean-ﬁeld

text: equation

text: is

text: recovered

text: for

text: d

text: and

text: η

text: Finally

text: and

text: importantly,

text: under

text: the

text: reduced

text: dimensionality

text: w

text: transformation,

text: equations

text: GD-Λ

text: and

text: GD

text: -Λ

text: are

text: respectively

text: given

text: by

text: D(Φ(η))

text: exp(

text: Φ)

text: ##LTLine##


text: Conclusions In this work we present a set of dimensionality functions that provide an useful and power- ful description of the fractality in the anisotropy-driven DLA-MF and BA-MF transitions. Under a generalized anisotropy-force approach and a reduced dimensionality transforma- tion, these transitions follow universal curves, showing that they are independent of their initial fractal dimension, and only dependent on the anisotropy force acting upon them. Also, provided that the initial fractal dimension is a function of the dimensionality of the embedding space (as described by the HTM equation), these results reveal the universality of these fractal to non-fractal morphological transitions. As well, we also show that the DB model (the generalized Laplacian model) belongs to the same universality-class, sharing similarities with the BA-MF model such as critical points and close values of their fractal dimension at q = 1 or η = 1. Additionally, we introduce a correction to the well-established i GHTM mean-ﬁeld approximation, that leads to a solution capable to quantitatively re- produce with remarkable precision previous well-established observations. In summary, we present here, for the ﬁrst time, a comprehensive discussion on the dynamical origin of the fractality and, with this, the basis for the understanding its subjacent algebra originated in these far-from-equilibrium transitions as an emergent feature of the anisotropy eﬀects. These results represent an important unifying step towards a complete theory of growth, and they might provide important insights to understand pattern formation phenomena in many related areas of research [3, 6, 37–39]. METHODS Aggregation dynamics In all simulations, each particle has a diameter equal to one. This is the basic unit of distance for the system. For aggregates based on BA or MF (Fig. 1a and 1c), we follow a standard procedure in which particles are launched at random from the circumference of radius r = 2r + δ, with equal probability in position and direction of motion. Here, L max r is the distance of the farthest particle in the cluster with respect to the seed particle max placed at the origin. In our simulations we used δ = 1000 particle diameters to avoid undesirable screening eﬀects. In particular, for the MF model, particles always aggregate to the closest particle in the cluster, this is determined by the projected position of the 14

text: Conclusions

label: section

text: In

text: this

text: work

text: we

text: present

text: a

text: set

text: of

text: dimensionality

text: functions

text: that

text: provide

text: an

text: useful

text: and

text: power-

text: ful

text: description

text: of

text: the

text: fractality

text: in

text: the

text: anisotropy-driven

text: DLA-MF

text: and

text: BA-MF

text: transitions.

text: Under

text: a

text: generalized

text: anisotropy-force

text: approach

text: and

text: a

text: reduced

text: dimensionality

text: transforma-

text: tion,

text: these

text: transitions

text: follow

text: universal

text: curves,

text: showing

text: that

text: they

text: are

text: independent

text: of

text: their

text: initial

text: fractal

text: dimension,

text: and

text: only

text: dependent

text: on

text: the

text: anisotropy

text: force

text: acting

text: upon

text: them.

text: Also,

text: provided

text: that

text: the

text: initial

text: fractal

text: dimension

text: is

text: a

text: function

text: of

text: the

text: dimensionality

text: of

text: the

text: embedding

text: space

text: (as

text: described

text: by

text: the

text: HTM

text: equation),

text: these

text: results

text: reveal

text: the

text: universality

text: of

text: these

text: fractal

text: to

text: non-fractal

text: morphological

text: transitions.

text: As

text: well,

text: we

text: also

text: show

text: that

text: the

text: DB

text: model

text: (the

text: generalized

text: Laplacian

text: model)

text: belongs

text: to

text: the

text: same

text: universality-class,

text: sharing

text: similarities

text: with

text: the

text: BA-MF

text: model

text: such

text: as

text: critical

text: points

text: and

text: close

text: values

text: of

text: their

text: fractal

text: dimension

text: at

text: q

text: or

text: η

text: Additionally,

text: we

text: introduce

text: a

text: correction

text: to

text: the

text: well-established

text: i

text: GHTM

text: mean-ﬁeld

text: approximation,

text: that

text: leads

text: to

text: a

text: solution

text: capable

text: to

text: quantitatively

text: re-

text: produce

text: with

text: remarkable

text: precision

text: previous

text: well-established

text: observations.

text: In

text: summary,

text: we

text: present

text: here,

text: for

text: the

text: ﬁrst

text: time,

text: a

text: comprehensive

text: discussion

text: on

text: the

text: dynamical

text: origin

text: of

text: the

text: fractality

text: and,

text: with

text: this,

text: the

text: basis

text: for

text: the

text: understanding

text: its

text: subjacent

text: algebra

text: originated

text: in

text: these

text: far-from-equilibrium

text: transitions

text: as

text: an

text: emergent

text: feature

text: of

text: the

text: anisotropy

text: eﬀects.

text: These

text: results

text: represent

text: an

text: important

text: unifying

text: step

text: towards

text: a

text: complete

text: theory

text: of

text: growth,

text: and

text: they

text: might

text: provide

text: important

text: insights

text: to

text: understand

text: pattern

text: formation

text: phenomena

text: in

text: many

text: related

text: areas

text: of

text: research

text: METHODS

text: Aggregation

text: dynamics

text: In

text: all

text: simulations,

text: each

text: particle

text: has

text: a

text: diameter

text: equal

text: to

text: one.

text: This

text: is

text: the

text: basic

text: unit

text: of

text: distance

text: for

text: the

text: system.

text: For

text: aggregates

text: based

text: on

text: BA

text: or

text: MF

text: (Fig.

text: and

text: we

text: follow

text: a

text: standard

text: procedure

text: in

text: which

text: particles

text: are

text: launched

text: at

text: random

text: from

text: the

text: circumference

text: of

text: radius

text: r

text: δ,

text: with

text: equal

text: probability

text: in

text: position

text: and

text: direction

text: of

text: motion.

text: Here,

text: L

text: max

text: r

text: is

text: the

text: distance

text: of

text: the

text: farthest

text: particle

text: in

text: the

text: cluster

text: with

text: respect

text: to

text: the

text: seed

text: particle

text: max

text: placed

text: at

text: the

text: origin.

text: In

text: our

text: simulations

text: we

text: used

text: δ

text: particle

text: diameters

text: to

text: avoid

text: undesirable

text: screening

text: eﬀects.

text: In

text: particular,

text: for

text: the

text: MF

text: model,

text: particles

text: always

text: aggregate

text: to

text: the

text: closest

text: particle

text: in

text: the

text: cluster,

text: this

text: is

text: determined

text: by

text: the

text: projected

text: position

text: of

text: the


text: (cid:20) (cid:18) (cid:19)(cid:21) − D χ 1 1/χ 0 p = . (3) i Λ χ The ﬁrst-order approximation (GD ) of the general dimensionality function (2) can be 1 written as, D2 + f(p) D(1) = 0 , (4) D + f(p) 0 with an inﬂection point that must now satisfy (df/dp)2 = 2(d2f/dp2)/(D + f), and is 0 given by (cid:20) (cid:18) (cid:19)(cid:21) − D χ 1 1/χ p(1) = 0 . (5) i Λ χ + 1 → As shown, these expressions predict a continuous morphological transition from D D 0 → → → ∞ as f(p) 0 (disordered/fractal state) towards D 1 as f(p) (ordered state), with an expected change in growth dynamics at p . Additionally, we will also deﬁne the reduced i ∗ − − ∗ ∈ dimensionality (RD) function as the transformation D = (D 1)/(D 1), with D [0, 1], 0 that leaves the inﬂection-point of equations (3) and (5) invariant, given by ∗ − D = e f(p)/D0, (6) the ﬁrst-order approximation (RD ) becomes, 1 ∗ 1 D (1) = , (7) 1 + f(p)/D 0 where f(p) is given as before. In the following section we will use these expressions to analyze some important cases. RESULTS AND DISCUSSION For all of our numerically generated clusters, the fractal dimension was measured by means of two standard procedures: the two-point density radial correlation function C(r) 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: D

text: χ

text: p

text: i

text: Λ

text: χ

text: The

text: ﬁrst-order

text: approximation

text: (GD

text: of

text: the

text: general

text: dimensionality

text: function

text: can

text: be

text: written

text: as,

text: f(p)

text: D

text: f(p)

text: with

text: an

text: inﬂection

text: point

text: that

text: must

text: now

text: satisfy

text: f),

text: and

text: is

text: given

text: by

text: D

text: χ

text: i

text: Λ

text: χ

text: As

text: shown,

text: these

text: expressions

text: predict

text: a

text: continuous

text: morphological

text: transition

text: from

text: D

text: D

text: as

text: f(p)

text: (disordered/fractal

text: state)

text: towards

text: D

text: as

text: f(p)

text: (ordered

text: state),

text: with

text: an

text: expected

text: change

text: in

text: growth

text: dynamics

text: at

text: p

text: Additionally,

text: we

text: will

text: also

text: deﬁne

text: the

text: reduced

text: i

text: dimensionality

text: (RD)

text: function

text: as

text: the

text: transformation

text: D

text: (D

text: with

text: D

text: that

text: leaves

text: the

text: inﬂection-point

text: of

text: equations

text: and

text: invariant,

text: given

text: by

text: D

text: e

text: the

text: ﬁrst-order

text: approximation

text: (RD

text: becomes,

text: D

text: f(p)/D

text: where

text: f(p)

text: is

text: given

text: as

text: before.

text: In

text: the

text: following

text: section

text: we

text: will

text: use

text: these

text: expressions

text: to

text: analyze

text: some

text: important

text: cases.

text: RESULTS

text: AND

text: DISCUSSION

text: For

text: all

text: of

text: our

text: numerically

text: generated

text: clusters,

text: the

text: fractal

text: dimension

text: was

text: measured

text: by

text: means

text: of

text: two

text: standard

text: procedures:

text: the

text: two-point

text: density

text: radial

text: correlation

text: function

text: C(r)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: It is clear that assumptions (H1) and (H2) of the Theorem 3 are satisﬁed. On the other hand, for any t ∈ [0,1(cid:12)(cid:12)(cid:12)] an(cid:16)d(cid:112)x,y,x1,y1 (cid:112)∈ R, we (cid:17)have (cid:16)(cid:112) (cid:112) (cid:17)(cid:12)(cid:12)(cid:12) (cid:12) (cid:12) | (cid:12)(cid:112)− (cid:112) | 1(cid:12) (cid:12)(cid:112)| | (cid:112)| | − 1(cid:12) | | | | f(t,x(cid:12)(cid:12),y) f(t,x1,y1) = α(cid:12)(cid:12) 4 1(cid:12)(cid:12)+ x + 4 1+ y α(cid:12)(cid:12) 4 1+ x1 + 4 1+ y1 ≤ 1 (cid:16)(cid:112) | |− (cid:17) (cid:16)| (cid:112)| 1 (cid:17) | |−(cid:16)(cid:112) | | (cid:17) (cid:16)(cid:112) (cid:17) α 4 1+ x 4 1+ x1 + α 4 1+ y 4 1+ y1 ≤ 1 (cid:16)(cid:112) | |− − (cid:17) | (cid:16)|(cid:112)− 1 | |(cid:17)− − | |− α 4 1+ x 1 4 1+ x1 1 + α 4 1+ y 1 4 1+ y1 1 ≤ 1 (cid:16)(cid:112)|| |−| || − 1 ||(cid:17)|−| || − α 4 x x1 +1 1 + α 4 y y1 +1 1 ≤ 2 | − | | − | − α 4 max( x x1 , y y1 )+1 1 , where we have used Lemma 3. (cid:16)(cid:112) (cid:17) By using a similar argument, we can prove | − | ≤ 2 | − | | − | − g(t,x,y) g(t,x1,y1) β 3 max( x x1 , y y1 )+1 1 ∈ ∈ R for any t [0,1] and x,y,x1,y1 . ≥ Therefore, assumption (H3) of the Theorem 3 is satisﬁed when α,β 2 with k = 1/4 and r = 1/3. {| | ∈ } {| | ∈ }In our case, K1 = sup f(t,0,0) : t [0,1] = α2 and K2 = sup g(t,0,0) : t [0,1] = 2 . β (cid:18) (cid:19)(cid:18) (cid:19) The inequality appearing in (H4) of Theorem 3 has the expression − 2 − 2 ≤ (r0 +1)1/4 1+ α (r0 +1)1/3 1+ β r0Γ(3/2). (cid:18) (cid:19)(cid:18) (cid:19) For α = 4 and β = 3, we have − 1 − 12 ≤ (r0 +1)1/4 2 (r0 +1)1/3 3 r0Γ(3/2). (cid:18) (cid:19) This inequality is satisﬁed for r0 = 0.8. Moreover, √ − 1 − ∼ ≤ (r0 +1)1/3 3 = 3 1.8 1/3 = 0.8832 Γ(3/2) = 0.8862. ∈ Therefore, Problem (2) has at least one solution x(t) C[0,1] for α = 4 and β = 3 || || ≤ with x 0.8. An interesting question is the non-oscillatory character of the solutions of Problem (1), i.e., that the solutions of Problem (1) have a constant sign. Notice that if f(t,x,y) ≥ and g(t,x,y) have the same sign (this means that f(t,x,y) > 0 and g(t,x,y) 0 or ≤ ∈ ∈ R f(t,x,y) and g(t,x,y) 0 for any t [0,1] and x,y ) and assumptions of Theorem 12 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: It

text: is

text: clear

text: that

text: assumptions

text: and

text: of

text: the

text: Theorem

text: are

text: satisﬁed.

text: On

text: the

text: other

text: hand,

text: for

text: any

text: t

text: R,

text: we

text: x

text: y

text: α

text: x

text: α

text: y

text: α

text: x

text: α

text: y

text: α

text: x

text: α

text: y

text: α

text: max(

text: x

text: y

text: where

text: we

text: have

text: used

text: Lemma

text: By

text: using

text: a

text: similar

text: argument,

text: we

text: can

text: prove

text: g(t,x,y)

text: β

text: max(

text: x

text: y

text: R

text: for

text: any

text: t

text: and

text: Therefore,

text: assumption

text: of

text: the

text: Theorem

text: is

text: satisﬁed

text: when

text: α,β

text: with

text: k

text: and

text: r

text: }In

text: our

text: case,

text: sup

text: t

text: α2

text: and

text: sup

text: t

text: β

text: The

text: inequality

text: appearing

text: in

text: of

text: Theorem

text: has

text: the

text: expression

text: α

text: β

text: For

text: α

text: and

text: β

text: we

text: have

text: This

text: inequality

text: is

text: satisﬁed

text: for

text: Moreover,

text: Therefore,

text: Problem

text: has

text: at

text: least

text: one

text: solution

text: x(t)

text: for

text: α

text: and

text: β

text: with

text: x

text: An

text: interesting

text: question

text: is

text: the

text: non-oscillatory

text: character

text: of

text: the

text: solutions

text: of

text: Problem

text: i.e.,

text: that

text: the

text: solutions

text: of

text: Problem

text: have

text: a

text: constant

text: sign.

text: Notice

text: that

text: if

text: f(t,x,y)

text: and

text: g(t,x,y)

text: have

text: the

text: same

text: sign

text: (this

text: means

text: that

text: f(t,x,y)

text: and

text: g(t,x,y)

text: or

text: R

text: f(t,x,y)

text: and

text: g(t,x,y)

text: for

text: any

text: t

text: and

text: x,y

text: and

text: assumptions

text: of

text: Theorem

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 are satisﬁed, then the solutions x(t) of Problem (1) are nonnegative due to the fact that these solutions satisfy the integral (cid:90)equation t f(t,x(t),x(ϕ(t))) g(s,x(s),x(ρ(s))) ≤ ≤ x(t) = − − ds, 0 t 1. Γ(α) (t s)1 α 0 In connection with the above-mentioned question, we have the following result. Proposition 1. Under assumptions of Theorem 3 and suppose that g(t,x,y) has (cid:54) ∈ ∈ R constant sign and g(t,x,y) = 0 for t [0,1] and x,y , we have that the solution x(t) (cid:54) of Problem (1) obtained in Theorem 3 satisﬁes that x(t) = 0 for 0 < t < 1. ∗ ∈ ∗ Proof: Suppose the contrary case. Then we can ﬁnd t (0,1) with x(t ) = 0. Since x(t) satisﬁes the last integral equation, we(cid:90)have ∗ ∗ ∗ ∗ t ∗ f(t ,x(t ),x(ϕ(t ))) g(s,x(s),x(ρ(s))) 0 = x(t ) = ∗ − − ds. Γ(α) (t s)1 α 0 (cid:54) ∈ ∈ R Taking into account that(cid:90)f(t,x,y) = 0 for any t [0,1] and x,y we infer that ∗ t ∗ − − (t s)1 αg(s,x(s),x(ρ(s)))ds = 0. 0 ∗ − − ∈ ∗ Since g(t,x,y) has constant sign and (t s)1 α > 0 for s [0,t ) it follows that ∈ ∗ g(s,x(s),x(ρ(s))) = 0 a.e.s [0,t ). (cid:54) ∈ ×R×R This contradicts the fact that g(t,x,y) = 0 for any (t,x,y) [0,1] . Therefore, (cid:54) ∈ x(t) = 0 for t (0,1). As an application of Bolzano’s theorem, we have the following corollary. Corollary 1. Under assumptions of Proposition 2, the solution x(t) of Problem ∈ ∈ (1) obtained in Theorem 3 satisﬁes that x(t) > 0 for t (0,1) or x(t) < 0 for t (0,1).  (cid:20) (cid:21) On the other hand, if we perturb the data function in Problem (1) of the following way Dα x(t) = g(t,x(t),x(ρ(t)))+η(t), 0 < t < 1 0+ f(t,x(t),x(ϕ(t))) x(0) = 0, where η ∈ C[0,1], α ∈ (0,1),ϕ,ρ ∈ C[0,1], f ∈ C([0,1]×R×R,R\{0}), and g ∈ ×R×R R C([0,1] , ). In this case, assumptions (H1), (H2) and (H3) of Theorem 3 if f(t,x,y) and g(t,x,y) satisfy (H3) of Theorem 3 and, only, we would have to check assumption (H4). This fact makes that Theorem 2 is applicable to a great number of cases. In the sequel, we compare our results with ones appearing in the literature. In [13], the authors studied the fractional hybrid differential equation 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: are

text: satisﬁed,

text: then

text: the

text: solutions

text: x(t)

text: of

text: Problem

text: are

text: nonnegative

text: due

text: to

text: the

text: fact

text: that

text: these

text: solutions

text: satisfy

text: the

text: integral

text: t

text: f(t,x(t),x(ϕ(t)))

text: g(s,x(s),x(ρ(s)))

text: x(t)

text: ds,

text: t

text: Γ(α)

text: (t

text: α

text: In

text: connection

text: with

text: the

text: above-mentioned

text: question,

text: we

text: have

text: the

text: following

text: result.

text: Proposition

text: Under

text: assumptions

text: of

text: Theorem

text: and

text: suppose

text: that

text: g(t,x,y)

text: has

text: R

text: constant

text: sign

text: and

text: g(t,x,y)

text: for

text: t

text: and

text: x,y

text: we

text: have

text: that

text: the

text: solution

text: x(t)

text: of

text: Problem

text: obtained

text: in

text: Theorem

text: satisﬁes

text: that

text: x(t)

text: for

text: t

text: Proof:

text: Suppose

text: the

text: contrary

text: case.

text: Then

text: we

text: can

text: ﬁnd

text: t

text: with

text: x(t

text: Since

text: x(t)

text: satisﬁes

text: the

text: last

text: integral

text: equation,

text: t

text: f(t

text: ,x(t

text: ),x(ϕ(t

text: g(s,x(s),x(ρ(s)))

text: x(t

text: ds.

text: Γ(α)

text: (t

text: α

text: R

text: Taking

text: into

text: account

text: for

text: any

text: t

text: and

text: x,y

text: we

text: infer

text: that

text: t

text: (t

text: αg(s,x(s),x(ρ(s)))ds

text: Since

text: g(t,x,y)

text: has

text: constant

text: sign

text: and

text: (t

text: α

text: for

text: s

text: it

text: follows

text: that

text: g(s,x(s),x(ρ(s)))

text: a.e.s

text: ×R×R

text: This

text: contradicts

text: the

text: fact

text: that

text: g(t,x,y)

text: for

text: any

text: (t,x,y)

text: Therefore,

text: x(t)

text: for

text: t

text: As

text: an

text: application

text: of

text: Bolzano’s

text: theorem,

text: we

text: have

text: the

text: following

text: corollary.

text: Corollary

text: Under

text: assumptions

text: of

text: Proposition

text: the

text: solution

text: x(t)

text: of

text: Problem

text: obtained

text: in

text: Theorem

text: satisﬁes

text: that

text: x(t)

text: for

text: t

text: or

text: x(t)

text: for

text: t

text: On

text: the

text: other

text: hand,

text: if

text: we

text: perturb

text: the

text: data

text: function

text: in

text: Problem

text: of

text: the

text: following

text: way

text: Dα

text: x(t)

text: g(t,x(t),x(ρ(t)))+η(t),

text: t

text: f(t,x(t),x(ϕ(t)))

text: where

text: η

text: α

text: f

text: and

text: g

text: ×R×R

text: R

text: In

text: this

text: case,

text: assumptions

text: and

text: of

text: Theorem

text: if

text: f(t,x,y)

text: and

text: g(t,x,y)

text: satisfy

text: of

text: Theorem

text: and,

text: only,

text: we

text: would

text: have

text: to

text: check

text: assumption

text: This

text: fact

text: makes

text: that

text: Theorem

text: is

text: applicable

text: to

text: a

text: great

text: number

text: of

text: cases.

text: In

text: the

text: sequel,

text: we

text: compare

text: our

text: results

text: with

text: ones

text: appearing

text: in

text: the

text: literature.

text: In

text: the

text: authors

text: studied

text: the

text: fractional

text: hybrid

text: differential

text: equation

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 (cid:90) Particularly, 1 ∗ ∗ 0 = u(t ) = G(T , s)f(s, u(s))ds. 0 Since that G and f are non-negative functions, we infer that ∗ G(t , s)f(s, u(s)) = 0 a.e. (s) (2) ∞ On the other hand, as lim f(t, 0) = for given M > 0 there exists δ > 0 such → ∈ t 0+ ≥ that for s (0, δ) f(s, 0) > M. Since tσf(t, y) is increasing and u(t) 0, ≥ ≥ ∈ sσf(s, u(s)) sσf(s, 0) sσM for s (0, δ) ≥ ∈ (cid:54) and, therefore, f(s, u(s)) M for s (0, δ) and f(s, u(s)) = 0 a.e. (s). But this ∗ is a contradiction since G(t , s) is a function of rational type in the variable s and, ∗ (cid:54) ∈ consequently, G(t , s) = 0 a.e. (s). Therefore, u(t) > 0 for t (0, 1). This ﬁnishes the proof. 4. Acknowledgement The present work is partially supported by the project MTM-2013-44357-P. References [1] J.Caballero, J.Harjani, K.Sadarangani. Positive solutions for a class of singular frac- tional boundary value problems. Computers and Mathematics with Applications. 62(2011), pp.1325-1332. [2] S.Stanˆek. The existence of positive solutions of singular fractional boundary value problems. Computers and Mathematics with Applications. 62(2011), pp.1379-1388. [3] Z.Bai, W.Sun. Existence and multiplicity of positive solutions for singular fractional bound- ary problems. Computers and Mathematics with Applications. 63(2012), pp.1369-1381. [4] J.Xu, Z.Wei, W.Dong. Uniqueness of positive solutions for a class of fractional boundary value problems. Applied Mathematics Letters. 25(2012), pp.590-593. [5] S.Zhang. Positive solutions to singular boundary value problem for nonlinear fractional diﬀerential equation. Computers and Mathematics with Applications. 59(2010), pp.1300- 1309. [6] X.Xu, D.Jiang, C.Yuan. Multiple positive solutions for the boundary value problem of a nonlinear fractional diﬀerential equation. Nonlinear Analysis 71 (2009), pp. 4676-4688. [7] D.Wardowski. Fixed points of a new type of contractive mappings in complete metric spaces. Fixed point theorey and applications (2012), 2012:14.

text: Particularly,

text: u(t

text: G(T

text: s)f(s,

text: u(s))ds.

text: Since

text: that

text: G

text: and

text: f

text: are

text: non-negative

text: functions,

text: we

text: infer

text: that

text: G(t

text: s)f(s,

text: u(s))

text: a.e.

text: (s)

text: On

text: the

text: other

text: hand,

text: as

text: lim

text: f(t,

text: for

text: given

text: M

text: there

text: exists

text: δ

text: such

text: t

text: that

text: for

text: s

text: δ)

text: f(s,

text: M.

text: Since

text: tσf(t,

text: y)

text: is

text: increasing

text: and

text: u(t)

text: sσf(s,

text: u(s))

text: sσf(s,

text: sσM

text: for

text: s

text: δ)

text: and,

text: therefore,

text: f(s,

text: u(s))

text: M

text: for

text: s

text: δ)

text: and

text: f(s,

text: u(s))

text: a.e.

text: (s).

text: But

text: this

text: is

text: a

text: contradiction

text: since

text: G(t

text: s)

text: is

text: a

text: function

text: of

text: rational

text: type

text: in

text: the

text: variable

text: s

text: and,

text: consequently,

text: G(t

text: s)

text: a.e.

text: (s).

text: Therefore,

text: u(t)

text: for

text: t

text: This

text: ﬁnishes

text: the

text: proof.

text: Acknowledgement

text: The

text: present

text: work

text: is

text: partially

text: supported

text: by

text: the

text: project

text: References

text: J.Caballero,

text: J.Harjani,

text: K.Sadarangani.

text: Positive

text: solutions

text: for

text: a

text: class

text: of

text: singular

text: frac-

text: tional

text: boundary

text: value

text: problems.

text: Computers

text: and

text: Mathematics

text: with

text: Applications.

text: S.Stanˆek.

text: The

text: existence

text: of

text: positive

text: solutions

text: of

text: singular

text: fractional

text: boundary

text: value

text: problems.

text: Computers

text: and

text: Mathematics

text: with

text: Applications.

text: Z.Bai,

text: W.Sun.

text: Existence

text: and

text: multiplicity

text: of

text: positive

text: solutions

text: for

text: singular

text: fractional

text: bound-

text: ary

text: problems.

text: Computers

text: and

text: Mathematics

text: with

text: Applications.

text: J.Xu,

text: Z.Wei,

text: W.Dong.

text: Uniqueness

text: of

text: positive

text: solutions

text: for

text: a

text: class

text: of

text: fractional

text: boundary

text: value

text: problems.

text: Applied

text: Mathematics

text: Letters.

text: S.Zhang.

text: Positive

text: solutions

text: to

text: singular

text: boundary

text: value

text: problem

text: for

text: nonlinear

text: fractional

text: diﬀerential

text: equation.

text: Computers

text: and

text: Mathematics

text: with

text: Applications.

text: X.Xu,

text: D.Jiang,

text: C.Yuan.

text: Multiple

text: positive

text: solutions

text: for

text: the

text: boundary

text: value

text: problem

text: of

text: a

text: nonlinear

text: fractional

text: diﬀerential

text: equation.

text: Nonlinear

text: Analysis

text: pp.

text: D.Wardowski.

text: Fixed

text: points

text: of

text: a

text: new

text: type

text: of

text: contractive

text: mappings

text: in

text: complete

text: metric

text: spaces.

text: Fixed

text: point

text: theorey

text: and

text: applications


text: Quantile-based optimization under uncertainties using adaptive Kriging surrogate models M. Moustapha1,2,3, B. Sudret1, J.-M. Bourinet2, and B. Guillaume3 1Chair of Risk, Safety and Uncertainty Quantiﬁcation, ETH Zurich, Stefano-Franscini-Platz 5, 8093 Zurich, Switzerland 2Institut Pascal, Sigma Clermont, CNRS UMR 6602, Aubi`ere, France 3PSA Group, Centre Technique de V´elizy, V´elizy-Villacoublay, France Abstract Uncertainties are inherent to real-world systems. Taking them into account is crucial in industrial design problems and this might be achieved through reliability-based design optimiza- tion (RBDO) techniques. In this paper, we propose a quantile-based approach to solve RBDO problems. We ﬁrst transform the safety constraints usually formulated as admissible probabili- ties of failure into constraints on quantiles of the performance criteria. In this formulation, the quantile level controls the degree of conservatism of the design. Starting with the premise that industrial applications often involve high-ﬁdelity and time-consuming computational models, the proposed approach makes use of Kriging surrogate models (a.k.a. Gaussian process modeling). Thanks to the Kriging variance (a measure of the local accuracy of the surrogate), we derive a procedure with two stages of enrichment of the design of computer experiments (DoE) used to construct the surrogate model. The ﬁrst stage globally reduces the Kriging epistemic uncertainty and adds points in the vicinity of the limit-state surfaces describing the system performance to be attained. The second stage locally checks, and if necessary, improves the accuracy of the quantiles estimated along the optimization iterations. Applications to three analytical exam- ples and to the optimal design of a car body subsystem (minimal mass under mechanical safety constraints) show the accuracy and the remarkable eﬃciency brought by the proposed procedure. Keywords: Quantile-based design optimization – RBDO – Kriging – Adaptive design of experiments 1

text: Quantile-based

label: title

text: optimization

text: under

text: uncertainties

text: using

text: adaptive

text: Kriging

text: surrogate

text: models

text: M.

label: author

text: B.

text: J.-M.

text: and

text: B.

text: of

text: Risk,

text: Safety

text: and

text: Uncertainty

text: Quantiﬁcation,

text: ETH

text: Zurich,

text: Stefano-Franscini-Platz

text: Zurich,

text: Switzerland

text: Pascal,

text: Sigma

text: Clermont,

text: CNRS

text: UMR

text: Aubi`ere,

text: France

text: Group,

text: Centre

text: Technique

text: de

text: V´elizy,

text: V´elizy-Villacoublay,

text: France

text: Abstract

text: Uncertainties

label: abstract

text: are

text: inherent

text: to

text: real-world

text: systems.

text: Taking

text: them

text: into

text: account

text: is

text: crucial

text: in

text: industrial

text: design

text: problems

text: and

text: this

text: might

text: be

text: achieved

text: through

text: reliability-based

text: design

text: optimiza-

text: tion

text: (RBDO)

text: techniques.

text: In

text: this

text: paper,

text: we

text: propose

text: a

text: quantile-based

text: approach

text: to

text: solve

text: RBDO

text: problems.

text: We

text: ﬁrst

text: transform

text: the

text: safety

text: constraints

text: usually

text: formulated

text: as

text: admissible

text: probabili-

text: ties

text: of

text: failure

text: into

text: constraints

text: on

text: quantiles

text: of

text: the

text: performance

text: criteria.

text: In

text: this

text: formulation,

text: the

text: quantile

text: level

text: controls

text: the

text: degree

text: of

text: conservatism

text: of

text: the

text: design.

text: Starting

text: with

text: the

text: premise

text: that

text: industrial

text: applications

text: often

text: involve

text: high-ﬁdelity

text: and

text: time-consuming

text: computational

text: models,

text: the

text: proposed

text: approach

text: makes

text: use

text: of

text: Kriging

text: surrogate

text: models

text: (a.k.a.

text: Gaussian

text: process

text: modeling).

text: Thanks

text: to

text: the

text: Kriging

text: variance

text: (a

text: measure

text: of

text: the

text: local

text: accuracy

text: of

text: the

text: surrogate),

text: we

text: derive

text: a

text: procedure

text: with

text: two

text: stages

text: of

text: enrichment

text: of

text: the

text: design

text: of

text: computer

text: experiments

text: (DoE)

text: used

text: to

text: construct

text: the

text: surrogate

text: model.

text: The

text: ﬁrst

text: stage

text: globally

text: reduces

text: the

text: Kriging

text: epistemic

text: uncertainty

text: and

text: adds

text: points

text: in

text: the

text: vicinity

text: of

text: the

text: limit-state

text: surfaces

text: describing

text: the

text: system

text: performance

text: to

text: be

text: attained.

text: The

text: second

text: stage

text: locally

text: checks,

text: and

text: if

text: necessary,

text: improves

text: the

text: accuracy

text: of

text: the

text: quantiles

text: estimated

text: along

text: the

text: optimization

text: iterations.

text: Applications

text: to

text: three

text: analytical

text: exam-

text: ples

text: and

text: to

text: the

text: optimal

text: design

text: of

text: a

text: car

text: body

text: subsystem

text: (minimal

text: mass

text: under

text: mechanical

text: safety

text: constraints)

text: show

text: the

text: accuracy

text: and

text: the

text: remarkable

text: eﬃciency

text: brought

text: by

text: the

text: proposed

text: procedure.

text: Keywords:

text: Quantile-based

text: design

text: optimization

text: RBDO

text: Kriging

text: Adaptive

text: design

text: of

text: experiments


text: 1 Introduction In engineering design, one often seeks to lower the product cost while ensuring its integrity. These are by construction two conﬂicting objectives. Optimization has therefore been used as an automatic procedure to ﬁnd a good trade-oﬀ. The optimal solution usually lies at the boundary of the feasible space. However uncertainties are ubiquitous to engineering systems whether arising from modeling approximations or input parameters inherent variability. They make any optimal design solution likely to depart from its real-world counterpart. Such discrepancy may turn a feasible solution into an unfeasible one. It is therefore of prime importance to account for uncertainties during optimization. This is generally achieved through robust and reliability- based design optimization (respectively RDO and RBDO). In the former, emphasis is put on the cost function. The designer actually searches for a design that is immune to the inputs uncertainties. The cost function is in this case replaced by robustness measures which include worst-case scenarios or moment-based criteria (Trosset, 1997). Beyer and Sendhoﬀ (2007) and Baudoui (2012) give a comprehensive review of such techniques. On the other hand, reliability- based design optimization rather seeks to balance the cost and the safety requirements by moving the solution away from the boundary of the admissible space. The work presented in this paper is concerned with the latter approach. Following the notations in Dubourg et al. (2011), a reliability-based design optimization may be formulated as follows:    ∗ f (d) ≤ 0, {j = 1, . . . , n } , j s d = arg min c (d) subject to: (1) ∈D P d (g (X (d) , Z) ≤ 0) ≤ P¯ , {k = 1, . . . , n } , k fk h where a cost function c is minimized with respect to design variables d. This minimization task is carried out under a set of constraints divided into two groups respectively denoted by soft and hard constraints. The n soft constraints f are simple analytical functions, often bounding s j the design space while the n hard constraints g are actually the system performance functions. h k They rely on the mechanical model M used to predict the structural behavior. In our case, k they result from a ﬁnite element model and may be written as g = g¯ − M , where g¯ is a k k k k threshold not to be exceeded by the structural response which is computed from a simulation model x (cid:55)→ M (x) (usually a time-consuming ﬁnite element model). When safety requirements k are of interest, performance may be measured in terms of a failure probability. To this end, random variables accounting for the uncertainties in the inputs are introduced and denoted respectively by X ∼ f (x) for the design variables and Z ∼ f for the environmental variables. X Z The former notation means that the distribution of X is conditioned on the design parameters. 2

text: Introduction

text: In

text: engineering

text: design,

text: one

text: often

text: seeks

text: to

text: lower

text: the

text: product

text: cost

text: while

text: ensuring

text: its

text: integrity.

text: These

text: are

text: by

text: construction

text: two

text: conﬂicting

text: objectives.

text: Optimization

text: has

text: therefore

text: been

text: used

text: as

text: an

text: automatic

text: procedure

text: to

text: ﬁnd

text: a

text: good

text: trade-oﬀ.

text: The

text: optimal

text: solution

text: usually

text: lies

text: at

text: the

text: boundary

text: of

text: the

text: feasible

text: space.

text: However

text: uncertainties

text: are

text: ubiquitous

text: to

text: engineering

text: systems

text: whether

text: arising

text: from

text: modeling

text: approximations

text: or

text: input

text: parameters

text: inherent

text: variability.

text: They

text: make

text: any

text: optimal

text: design

text: solution

text: likely

text: to

text: depart

text: from

text: its

text: real-world

text: counterpart.

text: Such

text: discrepancy

text: may

text: turn

text: a

text: feasible

text: solution

text: into

text: an

text: unfeasible

text: one.

text: It

text: is

text: therefore

text: of

text: prime

text: importance

text: to

text: account

text: for

text: uncertainties

text: during

text: optimization.

text: This

text: is

text: generally

text: achieved

text: through

text: robust

text: and

text: reliability-

text: based

text: design

text: optimization

text: (respectively

text: RDO

text: and

text: RBDO).

text: In

text: the

text: former,

text: emphasis

text: is

text: put

text: on

text: the

text: cost

text: function.

text: The

text: designer

text: actually

text: searches

text: for

text: a

text: design

text: that

text: is

text: immune

text: to

text: the

text: inputs

text: uncertainties.

text: The

text: cost

text: function

text: is

text: in

text: this

text: case

text: replaced

text: by

text: robustness

text: measures

text: which

text: include

text: worst-case

text: scenarios

text: or

text: moment-based

text: criteria

text: (Trosset,

text: Beyer

text: and

text: Sendhoﬀ

text: and

text: Baudoui

text: give

text: a

text: comprehensive

text: review

text: of

text: such

text: techniques.

text: On

text: the

text: other

text: hand,

text: reliability-

text: based

text: design

text: optimization

text: rather

text: seeks

text: to

text: balance

text: the

text: cost

text: and

text: the

text: safety

text: requirements

text: by

text: moving

text: the

text: solution

text: away

text: from

text: the

text: boundary

text: of

text: the

text: admissible

text: space.

text: The

text: work

text: presented

text: in

text: this

text: paper

text: is

text: concerned

text: with

text: the

text: latter

text: approach.

text: Following

text: the

text: notations

text: in

text: Dubourg

text: et

text: al.

text: a

text: reliability-based

text: design

text: optimization

text: may

text: be

text: formulated

text: as

text: follows:

text: f

text: (d)

text: {j

text: n

text: j

text: s

text: d

text: arg

text: min

text: c

text: (d)

text: subject

text: to:

text: ∈D

text: P

text: d

text: (g

text: (X

text: (d)

text: Z)

text: P¯

text: {k

text: n

text: k

text: fk

text: h

text: where

text: a

text: cost

text: function

text: c

text: is

text: minimized

text: with

text: respect

text: to

text: design

text: variables

text: d.

text: This

text: minimization

text: task

text: is

text: carried

text: out

text: under

text: a

text: set

text: of

text: constraints

text: divided

text: into

text: two

text: groups

text: respectively

text: denoted

text: by

text: soft

text: and

text: hard

text: constraints.

text: The

text: n

text: soft

text: constraints

text: f

text: are

text: simple

text: analytical

text: functions,

text: often

text: bounding

text: s

text: j

text: the

text: design

text: space

text: while

text: the

text: n

text: hard

text: constraints

text: g

text: are

text: actually

text: the

text: system

text: performance

text: functions.

text: h

text: k

text: They

text: rely

text: on

text: the

text: mechanical

text: model

text: M

text: used

text: to

text: predict

text: the

text: structural

text: behavior.

text: In

text: our

text: case,

text: k

text: they

text: result

text: from

text: a

text: ﬁnite

text: element

text: model

text: and

text: may

text: be

text: written

text: as

text: g

text: g¯

text: M

text: where

text: g¯

text: is

text: a

text: k

text: k

text: k

text: k

text: threshold

text: not

text: to

text: be

text: exceeded

text: by

text: the

text: structural

text: response

text: which

text: is

text: computed

text: from

text: a

text: simulation

text: model

text: x

text: M

text: (x)

text: (usually

text: a

text: time-consuming

text: ﬁnite

text: element

text: model).

text: When

text: safety

text: requirements

text: k

text: are

text: of

text: interest,

text: performance

text: may

text: be

text: measured

text: in

text: terms

text: of

text: a

text: failure

text: probability.

text: To

text: this

text: end,

text: random

text: variables

text: accounting

text: for

text: the

text: uncertainties

text: in

text: the

text: inputs

text: are

text: introduced

text: and

text: denoted

text: respectively

text: by

text: X

text: f

text: (x)

text: for

text: the

text: design

text: variables

text: and

text: Z

text: f

text: for

text: the

text: environmental

text: variables.

text: X

text: Z

text: The

text: former

text: notation

text: means

text: that

text: the

text: distribution

text: of

text: X

text: is

text: conditioned

text: on

text: the

text: design

text: parameters.


text: Note that the proposed framework naturally encompasses the two following cases: • When the analysts disregard uncertainties in the design parameters (e.g. , ignore manu- − facturing tolerances), q − and q are simply set to d and d+ respectively. d d+ X • When no environmental variables are considered, the augmented space reduces to . Once the augmented space is deﬁned for a speciﬁc problem, one may build a single global sur- rogate model. This may be achieved by a space-ﬁlling design of experiments, i.e. by using uniformly distributed samples so as to cover the entire space. The built Kriging model may henceforth be used for any reliability analysis during the RBDO. This one-shot approach is theoretically possible but would require the Kriging model to be accurate in the entire space. However, during the optimization only a subset of the space is actually of interest, i.e. regions in the vicinity of the limit-state surface and those where the objective function decreases. These two issues can be dealt with using so-called adaptive design of experiments. We propose in this paper a two-stage enrichment scheme where each stage is geared toward achieving one of the two above goals. 4.2 Adaptive design of experiments: A short literature review Adaptive design of experiments have been developed from the premise that only a limited region of the space is of interest to the designer during an optimization analysis. Thus, instead of densely ﬁlling the space so as to have an evenly accurate model in the entire space, the optimiza- tion starts with a not so accurate model built upon a scarcely sampled design of experiments. Enrichment is then made so as to improve the surrogate accuracy in regions that matter. Jones et al. (1998) proposed the eﬃcient global optimization (EGO) scheme relying on an expected improvement function which focuses on sequentially updating a Kriging model so as to converge to a global minimum. From the same idea, numerous authors have proposed inﬁll sampling criteria to achieve the same goal. As for RBDO, emphasis is rather put on the vicinity of the limit-state surface in order to accurately estimate failure probabilities. A ﬁrst family of inﬁll sampling criteria comes from EGO techniques as they are mere adaptation, e.g. adjusted ex- pected improvement (Schonlau et al., 1998), expected violation (Audet et al., 2000) or expected improvement for contour approximation (Ranjan et al., 2008). On the other hand, Bichon et al. (2008) introduced a so-called eﬃcient global reliability analysis (EGRA) where an expected fea- sibility function is used to improve the surrogate model in the vicinity of the limit-state surface. Similarly, Dubourg (2011) used in his PhD thesis work the margin probability function. In the present work, we will focus on the deviation number developed by (Echard et al., 2011) for their Active Kriging Monte Carlo simulation technique (AK-MCS). In AK-MCS, some candidates to enrichment are considered among a Monte Carlo set. The point that is most likely to improve 13

text: Note

text: that

text: the

text: proposed

text: framework

text: naturally

text: encompasses

text: the

text: two

text: following

text: cases:

label: list

text: When

text: the

text: analysts

text: disregard

text: uncertainties

text: in

text: the

text: design

text: parameters

text: (e.g.

text: ignore

text: manu-

text: facturing

text: tolerances),

text: q

text: and

text: q

text: are

text: simply

text: set

text: to

text: d

text: and

text: d+

text: respectively.

text: d

text: d+

text: X

text: When

text: no

text: environmental

text: variables

text: are

text: considered,

text: the

text: augmented

text: space

text: reduces

text: to

text: Once

text: the

text: augmented

text: space

text: is

text: deﬁned

text: for

text: a

text: speciﬁc

text: problem,

text: one

text: may

text: build

text: a

text: single

text: global

text: sur-

text: rogate

text: model.

text: This

text: may

text: be

text: achieved

text: by

text: a

text: space-ﬁlling

text: design

text: of

text: experiments,

text: i.e.

text: by

text: using

text: uniformly

text: distributed

text: samples

text: so

text: as

text: to

text: cover

text: the

text: entire

text: space.

text: The

text: built

text: Kriging

text: model

text: may

text: henceforth

text: be

text: used

text: for

text: any

text: reliability

text: analysis

text: during

text: the

text: RBDO.

text: This

text: one-shot

text: approach

text: is

text: theoretically

text: possible

text: but

text: would

text: require

text: the

text: Kriging

text: model

text: to

text: be

text: accurate

text: in

text: the

text: entire

text: space.

text: However,

text: during

text: the

text: optimization

text: only

text: a

text: subset

text: of

text: the

text: space

text: is

text: actually

text: of

text: interest,

text: i.e.

text: regions

text: in

text: the

text: vicinity

text: of

text: the

text: limit-state

text: surface

text: and

text: those

text: where

text: the

text: objective

text: function

text: decreases.

text: These

text: two

text: issues

text: can

text: be

text: dealt

text: with

text: using

text: so-called

text: adaptive

text: design

text: of

text: experiments.

text: We

text: propose

text: in

text: this

text: paper

text: a

text: two-stage

text: enrichment

text: scheme

text: where

text: each

text: stage

text: is

text: geared

text: toward

text: achieving

text: one

text: of

text: the

text: two

text: above

text: goals.

text: Adaptive

text: design

text: of

text: experiments:

text: A

text: short

text: literature

text: review

text: Adaptive

text: design

text: of

text: experiments

text: have

text: been

text: developed

text: from

text: the

text: premise

text: that

text: only

text: a

text: limited

text: region

text: of

text: the

text: space

text: is

text: of

text: interest

text: to

text: the

text: designer

text: during

text: an

text: optimization

text: analysis.

text: Thus,

text: instead

text: of

text: densely

text: ﬁlling

text: the

text: space

text: so

text: as

text: to

text: have

text: an

text: evenly

text: accurate

text: model

text: in

text: the

text: entire

text: space,

text: the

text: optimiza-

text: tion

text: starts

text: with

text: a

text: not

text: so

text: accurate

text: model

text: built

text: upon

text: a

text: scarcely

text: sampled

text: design

text: of

text: experiments.

text: Enrichment

text: is

text: then

text: made

text: so

text: as

text: to

text: improve

text: the

text: surrogate

text: accuracy

text: in

text: regions

text: that

text: matter.

text: Jones

text: et

text: al.

text: proposed

text: the

text: eﬃcient

text: global

text: optimization

text: (EGO)

text: scheme

text: relying

text: on

text: an

text: expected

text: improvement

text: function

text: which

text: focuses

text: on

text: sequentially

text: updating

text: a

text: Kriging

text: model

text: so

text: as

text: to

text: converge

text: to

text: a

text: global

text: minimum.

text: From

text: the

text: same

text: idea,

text: numerous

text: authors

text: have

text: proposed

text: inﬁll

text: sampling

text: criteria

text: to

text: achieve

text: the

text: same

text: goal.

text: As

text: for

text: RBDO,

text: emphasis

text: is

text: rather

text: put

text: on

text: the

text: vicinity

text: of

text: the

text: limit-state

text: surface

text: in

text: order

text: to

text: accurately

text: estimate

text: failure

text: probabilities.

text: A

text: ﬁrst

text: family

text: of

text: inﬁll

text: sampling

text: criteria

text: comes

text: from

text: EGO

text: techniques

text: as

text: they

text: are

text: mere

text: adaptation,

text: e.g.

text: adjusted

text: ex-

text: pected

text: improvement

text: (Schonlau

text: et

text: al.,

text: expected

text: violation

text: (Audet

text: et

text: al.,

text: or

text: expected

text: improvement

text: for

text: contour

text: approximation

text: (Ranjan

text: et

text: al.,

text: On

text: the

text: other

text: hand,

text: Bichon

text: et

text: al.

text: introduced

text: a

text: so-called

text: eﬃcient

text: global

text: reliability

text: analysis

text: (EGRA)

text: where

text: an

text: expected

text: fea-

text: sibility

text: function

text: is

text: used

text: to

text: improve

text: the

text: surrogate

text: model

text: in

text: the

text: vicinity

text: of

text: the

text: limit-state

text: surface.

text: Similarly,

text: Dubourg

text: used

text: in

text: his

text: PhD

text: thesis

text: work

text: the

text: margin

text: probability

text: function.

text: In

text: the

text: present

text: work,

text: we

text: will

text: focus

text: on

text: the

text: deviation

text: number

text: developed

text: by

text: (Echard

text: et

text: al.,

text: for

text: their

text: Active

text: Kriging

text: Monte

text: Carlo

text: simulation

text: technique

text: (AK-MCS).

text: In

text: AK-MCS,

text: some

text: candidates

text: to

text: enrichment

text: are

text: considered

text: among

text: a

text: Monte

text: Carlo

text: set.

text: The

text: point

text: that

text: is

text: most

text: likely

text: to

text: improve