A model for a network of conveyor belts with discontinuous speed and capacity

Transcription

1 A model for a network of conveyor belts with discontinuous speed and capacity Adriano FESTA Seminario di Modellistica differenziale Numerica work in collaboration with M. Pfirsching, S. Goettlich

2 Introduction

3 Motivations Conveyor Belts - Simple scenario - Transport equation Micro-meso-macro models - Simple interaction rules Networks and discontinuities - Complex behaviors Basic issue for network control, output tracking, etc. Figure: Inside a brewery 1 a pressureless-accumulation device. 1 Image courtesy of Sidel Blowing & Services SAS.

4 Basic model A production system is described by a conservation law with discontinuous flux function and constant speed a > 0. More precisely for a closed set Ω R and calling ρ : Ω [0, T ] [0, ] the product density, the evolution of the system is { t ρ(x, t) + x (ah( ρ(x, t))ρ(x, t)) = 0, ρ(x, 0) = ρ 0 (x), where H is the Heaviside function and the initial data ρ 0 is a BV function s.t. ρ 0 (x).

5 On a single arc: On a single arc the equation simply reduces to an advection equation and therefore the solution is simply ρ(x, t) = ρ 0 (x at). The presence of discontinuities in the flux function, anyway, poses some problems in the case of ρ 0 (x) = for some x Ω.

6 On a single arc: On a single arc the equation simply reduces to an advection equation and therefore the solution is simply ρ(x, t) = ρ 0 (x at). The presence of discontinuities in the flux function, anyway, poses some problems in the case of ρ 0 (x) = for some x Ω.

7 Weak solutions: Obtained using some Friedrichs mollifiers (classic). We denote by f the following multivalued function f (ρ) = aρ if ρ, f ( ) = [0, a ]. Definition A function ρ L (R [0, T ]) is called weak solution to the Cauchy problem (4) if there exists a function v L (R [0, T ]) such that v(x, t) f (ρ) a.e. and T 0 Ω ρ φ T t dxdt + v φ 0 Ω x dxdt + ρ 0 (x)φ(x, 0)dx = 0 Ω for each φ C 1 c (R [0, T ]) (where φ C 1 c means φ C 1 with compact support).

8 Entropy Weak Solutions: Denote by H the following multivalued function H(ρ) = H(ρ) if ρ 0, H(0) = [0, 1]. Definition A weak solution ρ of the Cauchy problem (4) is called an entropy weak solution if, for each entropy η C 1 (R), η convex, there exists a function w L (R [0, T ]) such that w(x, t) H(ρ(x, t)) a.e. and t η(ρ) + x F (ρ) η ( ) w x 0, where F (ρ) = a ρ 0 η (s)h( s)ds.

9 One arc case We remark that that the solution ρ(x, t) = ρ 0 (x at). is a weak entropy solution. This can be shown by choosing v(x, t) = aρ f (ρ), for ρ [0, ] and (Kruskov Entropy functions) η(ρ) = ρ k, for ρ [0, ], w(x, y) 1 for any constant k R. We observe that Entropy Definition reduces to the advection equation with constant speed a andthe function above provides the solution to the problem for every initial condition ρ 0 (x).

10 Analytic discussion of the Network case

11 Extension to networks: We now extend the model to a network Γ = (V, E) with V the set of vertices and E V V the set of arcs. For a fixed node v V, the sets δv, δ v + denote the ingoing and outgoing arcs, respectively. We consider the following network problem: t ρ(x, t) + x (f i(ρ(x, t))) = 0, x Ω i, t [0, T ] ρ(x, 0) = ρ 0 i (x), x Ω i f i (ρ(v, t)) = f i (ρ(v, t)), v V, where i δ v i δ + v f i (ρ(x, t)) = a i H( i ρ(x, t))ρ(x, t), x Ω i, t [0, T ] is the flux function on arc i E and a i R + the transport velocity.

12 One-to-one junction We consider the problem on Ω = Ω 1 {0} Ω 2 = (, 0) {0} (0, ), where the intersection is located at x = 0. The model equations are given by (10), where the junction condition is specified as f 1 (ρ(0, t)) = f 2 (ρ(0, t)) with flux function (10) on arc i = 1, 2.

13 The solution can be easily computed if no congestion occurs during the transportation, i.e. a 1 ρ 0 1(x) a 2 2 x Ω 1. For x Ω 1, the characteristics of the problem are simply the straight lines y(t) = x a 1 t, which lead to the solution Analogously, we find ρ(x, t) = ρ 0 1(x a 1 t) for (x, t) : x < 0. ρ(x, t) = ρ 0 2(x a 2 t) for { (x, t) Ω 2 (0, T ] x > a 2 t } which corresponds to the density initially placed on Ω 2. The solution for (x, t) in the case 0 < x < a 2 t is found by f 1 (ρ(x +, t)) = f 2 (ρ(x, t)) where x ± denotes lim h 0 x ± h. This leads to ρ(x, t) = a 1 a 2 ρ(x +, t) at the interface x = 0.

14 t x = a 2 t + c x = a 1 t + c 0 Figure: Characteristics in the non-congested case The solution is then described by ρ 0 1 (x a 1t), )) (x, t) Ω 1 (0, T ] a ρ(x, t) = 1 a 2 ρ 0 1 ( a 1 (t x, (x, t) Ω a2 2 (0, T ] : 0 < x a 2 t ρ 0 2 (x a 2t), (x, t) Ω 2 (0, T ] : x > a 2 t. x

15 If a 1 ρ 0 1 (x) a 2 2 is not satisfied, congestion arises and the problem is more complex. Let t 0 denote the first time of congestion: { t 0 = inf t 0 such that ρ 0 1( a 1 t) > a } 2 2. a 1 We track the interface describing the congested area (Λ). The interface is a time-dependent function g(t). We can track g(t) by integrating the difference between the fluxes entering and exiting the region Λ as well as the current density. t t 0 0 ( a1 ρ 0 1( a 1 s) a 2 ) 2 ds = g(t) ( 1 ρ 0 1(y a 1 t))dy.

16 Rearranging the terms, we can describe the congested region Λ as { } Λ := (x, t) Ω 1 [t 0, t0 E ], such that g(t) x 0 with interface g defined as g : [0, ) R 0, where t is mapped to the solution of x + (t t 0 ) ρmax 2 1 a 2 1 if this is negative, and to zero otherwise. The final time of congestion is t E = min {t t 0 such that g(t) = 0}. 1 a 1 t 0 x a 1 t ρ 0 1(s)ds = 0,

17 t t E 0 x = a 2 t + c x = a 1 t + c g(t) Λ 0 Trajectories of the problem with congestion (a 1 > a 2 ). Inside Λ, the transport velocity ā is t 0 2 ā = a 2 1. x

18 Ze derive a general solution discussing the Riemann problem on [0, T ) R with the initial data { ρl, x < 0, ρ(x, 0) = ρ r, x 0. 1 ρ 2 ρ l ρ r Shock, s l < 0 x = 0 Shock, s r > 0 Shock, s = 0 x

19 We allow waves with negative velocity and we distinguish: a) f 1 f2 max : no congestion arises standard advection. b) f 1 > f2 max : a congestion arises the solution consists of three shock waves starting at x = 0: i) At x = 0, a shock with velocity s = 0 arises, where the solution ρ jumps from 1 to 2. For the special case 1 = 2, there is no jump in the solution at x = 0. ii) Left-going shock wave, where the density jumps from ρ l to 1 with negative velocity s l < 0 (Rankine-Hugoniot c.): s l = āρmax 1 f 1 (ρ l ) = f max 2 f 1 (ρ l ) 1 ρ l < 0. 1 ρ l This shock wave describes the left boundary of the congested area Λ, meaning that it follows the function g(t). iii) Right-going shock wave, where the density jumps from 2 to ρ r with positive velocity s r = a 2 > 0 (Rankine-Hugoniot c.): s r = f 2(ρ r ) f 2 ( 2 ) ρ r 2 = a2ρ r a 2 ρmax 2 ρ r 2 = a 2 > 0.

20 In the general case with non-constant initial data, we obtain the following solution: ρ 0 1 (x a 1t), (x, t) Ω 1 \ Λ (0, T ] 1, (x, t) Λ (0, T ] 2, (x, t) Ω 2 (0, T ] ρ(x, t) = )) x a 2 t, g(t x a 2 ) 0 a 1 a 2 ρ 0 1 ( a 1 (t x, (x, t) Ω a2 2 (0, T ] x a 2 t, g(t x a 2 ) = 0 ρ 0 2 (x a 2t), (x, t) Ω 2 (0, T ] : x > a 2 t. If g(t) = 0 for all t [0, T ], we have Λ = and recover the previously described solution. Contrary to traffic flow models, rarefaction waves do not appear in the conveyor belt problem.

21 One-to-two junction We denote by i = 1 the incoming and by i = 2, 3 the outgoing arcs. We consider two kind of intersections: a) if the intersection is passive, we impose a fixed rate between the outgoing fluxes. b) if the intersection is active the ratio changes in order to reduce congestion. a) f 1 b) f 1 D f 2 f 3 f 2 f 3

22 We consider the problem on Ω = Ω 1 e 1 {0} Ω 2 e 1 Ω 3 e 2 = (, 0) e 1 {0} (0, ) e 1 (0, ) e 2, where (e 1, e 2 ) is the standard base of R 2, and we identify the element x Ω 1 with the vector (x, 0) T, and analogously for arcs i = 2, 3. The equations that we consider are as before, where the junction condition is specified as f 1 (ρ(0, t)) = f 2 (ρ(0, t)) + f 3 (ρ(0, t)), equipped with the discontinuous flux function on arc i = 1, 2, 3.

23 Case 1: passive junction. An intersection device D keeps the ratio of the two outgoing fluxes constant, i.e., for a fixed distribution parameter µ [0, 1], it holds f 2 = µf 1, f 3 = (1 µ)f 1, even if only one outgoing arc is congested. No congestion arises iff { 1 ρ 0 a 2 1(x) min µ a 1 2, } 1 a 3 3, x Ω 1. 1 µ a 1 We have the actual density on arc 2 and 3 as { } { } µ a 3 1 µ ρ 2 = min 3, a 2 2, ρ 3 = min 2, 3. 1 µ a 2 µ a 3

24 The first time of congestion t 0 can be determined as { { 1 t 0 = inf t 0 such that ρ 0 a 2 1( a 1 t) > min µ a 1 We can introduce the interface g(t). The interface is g : [0, ) R 0 mapped to the solution of 2, }} 1 a µ a 1 g(t) + (t t 0 ) ρmax 2 1 a a 1 t 0 g(t) a 1 t ρ 0 1(s)ds = 0, if this is negative, and to zero otherwise. The region of congestion Λ on Ω 1 is given by { } Λ := (x, t) Ω 1 [t 0, t0 E ], such that g(t) x 0, analogously to the 1-to-1 case.

25 The general solution ρ on Ω (passive junction) is given by ρ 0 1 (x a 1t), (x, t) Ω 1 \ Λ (0, T ] 1, (x, t) Λ (0, T ] ρ i, (x, t) Ω i (0, T ] : x a i t, g(t x a ρ(x, t) = i ) 0, i = 2, 3 α i a 1 a i ρ 1 0 ( a 1 (t x ai )), (x, t) Ω i (0, T ] : x a i t, g(t x a i ) = 0, i = 2, 3 ρ 0 i (x a it), (x, t) Ω i (0, T ] : x > a i t, i = 2, 3, where α 2 = µ and α 3 = 1 µ for a compact notation.

26 Case 2: active junction. If congestion arises, the device adapts the fluxes to ensure the maximal total outgoing flux, i.e., f 2 = f2 max and f 3 = f3 max. We fix a parameter µ [0, 1]. We have (no congestion) µf 1 = f 2 and (1 µ)f 1 = f 3. To get a unique solution also in the congested case, we define β i with the following properties: a) The flux conservation f 1 = f 2 + f 3 at the coupling is satisfied. b) The parameters β i are equal to α i, i = 2, 3 if no congestion. c) If arc i is congested, i.e., ρ 0 1 (x a 1t) > a i α i a 1 i parameter β i changes to β i = a i a 1 d) If the value β i = a i i a 2 2 +a 3 3 i ρ 0 1 ( a 1t). at time t the is reached the congestion starts.

27 No congestion arises as long as the following inequality holds true: ρ 0 1(x) a 2 a a 3 a 1 3, x Ω 1. If not, the time t 0 (first time of congestion) is { t 0 = inf t 0 such that ρ 0 1( a 1 t) > a a } 3 3. a 1 a 1 The interface g is the solution of ( ρ max 2 g(t) + (t t 0 ) 1 a 2 + ρmax 3 1 a 3 ) 1 1 a 1 t 0 g(t) a 1 t ρ 0 1(s)ds = 0, if this solution is negative, and to zero otherwise. The region of congestion Λ on Ω 1 is { } Λ := (x, t) Ω 1 [t 0, t0 E ], such that g(t) x 0.

28 We define for i = 2, 3 and α 2 = µ, α 3 = 1 µ, { { β i (t) := min max α i, a i } i a i } i a 1 ρ 0 1 (x a, 1t) a a 3 3 The general solution ρ on Ω (active junction) is ρ 0 1 (x a 1t), (x, t) Ω \ Λ (0, T ] 1, (x, t) Λ (0, T ] i, (x, t) Ω i (0, T ] x a i t, g(t x a ρ(x, t) = i ) 0 )) β i (t x a 1 ) a 1 a i ρ 1 0 ( a 1 (t x, (x, t) Ω ai i (0, T ] x a i t, g(t x a i ) = 0 with i = 2, 3. ρ 0 i (x a it), (x, t) Ω i (0, T ] x > a i t

29 Two-to-one junction We know from traffic flow that in the free flow regime no additional information is needed. Conversely, in the congested case, we need a priority rule between the two incoming arcs, i.e., how to use released capacities of the outgoing arc. We denote by i = 1, 2 the incoming and by i = 3 the outgoing arcs. We consider the problem on Ω = Ω 1 e 1 Ω 2 e 2 {0} Ω 3 e 1 = (, 0) e 1 (, 0) e 2 {0} (0, ) e 1 with the same interpretation as before, where the system is given by conservation laws as before, the coupling condition reads as f 1 (ρ(0, t)) + f 2 (ρ(0, t)) = f 3 (ρ(0, t))..

30 The solution can be directly computed if it holds a 1 ρ 0 1( a 1 t) + a 2 ρ 0 2( a 2 t) a 3 3, i.e. no congestion arises. Then, the solution is given by ρ 0 i (x a it), (x, t) Ω i (0, T ], i = 1, 2 2 )) a ρ(x, t) = i a 3 ρ 0 i ( a i (t x, (x, t) Ω a3 3 (0, T ] : x a 3 t i=1 ρ 0 3 (x a 3t), (x, t) Ω 3 (0, T ] : x > a 3 t and the incoming mass can be totally absorbed by the outgoing arc. This is independent of the capacity of the incoming arcs, and no further priority rule is needed to obtain a unique solution.

31 In the congested case, to obtain a unique solution, we impose a priority rule. We want to use the whole capacity of the outgoing arc i = 3, which implies f 3 = f3 max = a 3 3. We set the merging parameter q [0, 1] such that This leads to f 1 = qf max 3 and f 2 = (1 q)f max 3. f 2 = 1 q q f 1, describing a ratio of the actual fluxes on the corresponding arcs.

32 a) f max 2 f 2 f 1 + f 2 = f max 3 Θ P f max 1 f 2 = q 1 q f 1 f2 max f 1 b) f 2 f 1 + f 2 = f max 3 Figure: Choice of the merging parameter q Θ f 2 = q Q f max 1 P 1 q f 1 f 1 The admissible region for the fluxes is Θ = {(f 1, f 2 ) : 0 f 1 f max 1, 0 f 2 f max 2, 0 f 1 + f 2 f max 3 } and shaded gray in the Figure above.

33 Merging parameters: a) The intersection point P between the maximal outgoing flux (the line f 1 + f 2 = f3 max ) and the priority ratio is inside the admissible set Θ. In this case, we keep q (0, 1) fix. b) The intersection point P is outside Θ. We choose the closest point Q inside Θ on the line f 1 + f 2 = f3 max, which guarantees maximal throughput, i.e., f 1 = f1 max. The merging parameter changes to q = f1 max /f3 max. Then, the resulting fluxes are f 1 = f1 max and f 2 = f3 max f1 max.

34 Λ i is the congested region on arc i = 1, 2, g i (t) its interface ) solution of g i (t) + (t t 0 ) a 3 1 a i t 0 ρ 0 i (s)ds = 0 ( qi 3 i i g i (t) a i t and q i {q, 1 q} the corresponding merging parameters. The solution is then (for i = 1, 2) ρ 0 i (x a it), (x, t) Ω i \ Λ i (0, T ], i, (x, t) Λ i (0, T ], 3, (x, t) { Ω 3 (0, T ] ( ) } x a 3 t, max g i t x ρ(x, t) = i=1,2 a 3 0 )) a i a 3 ρ 0 i ( a i (t x, (x, t) Ω a3 3 (0, T ] i=1,2 { ( ) } x a 3 t, max g i t x i=1,2 a 3 = 0 ρ 0 3 (x a 3t), (x, t) Ω 3 (0, T ] : x > a 3 t.

35 Numerical Tools

36 Numerical approximation We discretize Ω i = (ā i, b i ) X i = (x i,0 = ā i, x i,1,..., x i,mi = b i ) with constant discretization x = x i,j x i,j 1, j = 1,..., m i. The spatial grid cells are defined as C i,j = (x i,j 1/2, x i,j+1/2 ) R. We discretize [0, T ] with {t n = n t, n = 0, 1,..., T / t}, where t [0, T ]. We define the piecewise constant approximation of the solution ρ as ρ n i,j ρ(x i,j, t n ) with ρ n constant in each grid cell C i,j. The scheme to update the approximation in each time step is for all i E, j 2,..., m i 1 given by { ( ρ n+1 i,j = ρ n i,j x t h ( ρ n i,j, i,j+1) ρn h ( ρ n i,j 1, ρ n i,j) ), ρ 0 i,j = ρ0 (x i,j ). For j = 1 and all i E.

37 The inflow and outflow fluxes must be adapted at the junction a) For an one-to-one junction, i.e. δ v = δ + v = 1, we set h n,out i = h n,in î = h ( ρ n i,m i, ρ ṋ i,1) for i δ v, î δ + v. b) For an one-to-two junction, i.e. 1 = δ v δ + v = 2, we choose in the non-congested case for i δ v, î δ + v h n,out i = f ( ρ n i,m i ) for i δ v, and h n,in î b.i) In the congested case: on a passive junction { h n,in 2 = min f2 max, h n,out i = î δ + v µ 1 µ f 3 max h n,in î for i δ v, }, h n,in 3 = min = αîf i ( ρ n i,m i ). { f3 max, 1 µ } µ f 2 max.

38 b.ii) On an active junction h n,out i = î δ + v f max î for i δ v, and h n,in î = f max î for î δ + v. c) For a two-to-one junction, i.e. 2 = δv δ v + = 1, in the non-congested case = h n,out i = f i ( ρ n i,m i ) for i δ v, and h n,in î i δ v f i ( ρ n i,m i ) for î δ + v. In the congested case, we apply the merging parameter q i and set h n,out i = q i f max î for i δ v, î δ + v, and h n,in î = f max î for î δ + v.

39 To define h( ρ n i,j, ρn i,j+1 ) we look for a function h satisfying m (ũ) ũ h(0, 0) = h( i, i ) = 0, h(ũ, u) 0 u h(ũ, u) m +(u), with continuous function m : R R and m = min(m, 0), m + = max(m, 0).

40 Numerical Mollifiers Since f i is discontinuous in ρ, we need a suitable regularization. We define a Friedrichs mollifier ϕ C0 (R) ϕ( y) = ϕ(y), ϕ(y)dy = 1. In our case, we use the mollifier ϕ(y) := max(0, 1 y ) and define ϕ ξ (y) := 2 ξ ϕ( 2y ξ ) for a small parameter ξ > 0, which implies that ϕ ξ (y) has compact support in [ ξ 2, ξ 2 ]. For each arc i E, we introduce the following smooth regularization of the flux function f ξ,i (ρ) := a i ρ ( 1 ρ i R ( ϕ ξ y i ξ ) ) dy 2,

41 f ξ,i (ρ) ā ρ a i i 0 i ρ i ρ + ξ The function coincides with the original flux function in x [0, i ] but there is a continuously differentiable connection to the value f ξ,i ( i + ξ) = 0.

42 We choose the numerical flux function h as Godunov flux min f ξ,i(z), if ρ n h( ρ n i,j, ρ n z [ ρ i,j+1) = n i,j, ρn i,j+1 ] i,j ρn i,j+1 max f ξ,i(z), if ρ n z [ ρ n i,j+1, ρn i,j ] i,j ρn i,j+1. The scheme is stable, if the following CFL condition is fulfilled. x t max v V δ v m L (0, +ξ) In particular, if max δ v 2 for a fixed v V, the CFL condition reduces to t x ξ 4.

43 Tests

44 We compute the numerical solution based on scheme described before. The space step size is x = , the time step size t = 10 5 and the smoothing parameter is fixed to ξ = We choose = 1. One-to-one junction The linear network is given by Ω 1 = ( π, 0) and Ω 2 = (0, π), i.e. the coupling is at x = 0. We fix the initial solution ρ 0 on Ω 1 Ω 2 as ( ρ 0 (x) = exp 3 (x + 3 ) ) 2 5 π.

45 t = 0 t = 1 t = 1.4 t = 1.8 t = 2.2 t = 2.6 Figure: Test 1: non-congested case with a 1 = 1 and a 2 = 2

46 One-to-one congested case t = 0 t = 0.5 t = 0.8 t = 1.1 t = 1.4 t = 1.7 Figure: Test 2: congested case with a 1 = 2 and a 2 = 1

47 Numerical solution x t error Analytical solution ξ error Table: space-time diagram for the congested case (above) and decreasing step sizes (left), decreasing smoothing parameter ξ (right)

48 One-to-two passive junction with µ = 0.5 t = 0 : t = 0.5 : t = 1.0 :

49 One-to-two active junction with µ = 0.5 t = 0 : t = 0.5 : t = 1.0 :

50 Two-to-one junction with q = 0.3 t = 0 : t = 2 : t = 4 :

51 Further steps: Control problems related to this model 1 Tracking of a production output (K. Lux SCICOM) 2 Optimization of the network Adding noise and stochastic disturbance 1 Appearance of viscosity terms 2 Evolution prediction in complex networks

52 D. Armbruster, S. Göttlich, and M. Herty, A scalar conservation law with discontinuous flux for supply chains with finite buffers, SIAM J. Appl. Math., 71 (2011), pp A. Festa, M. Pfirsching, and S. Göttlich, A model for a network of conveyor belts with discontinuous speed and capacity, ArXiv (2018). M. Garavello, K. Han, and B. Piccoli, Models for vehicular traffic on networks, vol. 9, American Institute of Mathematical Sciences (AIMS), Springfield, MO, S. Göttlich, A. Klar, and P. Schindler, Discontinuous conservation laws for production networks with finite buffers, SIAM J. Appl. Math., 73 (2013), pp M. Herty, C. Joerres, and B. Piccoli, Existence of solution to supply chain models based on pde with discontinuous flux function, J. Math. Anal. Appl., 401 (2013), pp Thank you!