Abstract
We are interested in an optimal packing density problem for material flows on conveyor belts in two spatial dimensions. The control problem is concerned with the initial configuration of parts on the belt to ensure a high overall flow rate and to further reduce congestion. An adjoint approach is used to compare the optimization results from the microscopic model based on a system of ordinary differential equations with the corresponding macroscopic model relying on a hyperbolic conservation law. Computational results highlight similarities and differences of both optimization models and emphasize the benefits of the macroscopic approach.
References
Aggarwal A, Colombo RM, Goatin P (2015) Nonlocal systems of conservation laws in several space dimensions. SIAM J Numer Anal 53:963–983
Albi G, Bongini M, Cristiani E, Kalise D (2016) Invisible control of self-organizing agents leaving unknown environments. SIAM J Appl. Math 76:1683–1710
Allain P, Courty N, Corpetti T (2014) Optimal crowd editing. Graph Model 76:1–16
Babic M (1997) Average balance equations for granular materials. Int J Eng Sci 35:523–548
Betts JT, Campbell ST (2005) Discretize then optimize. In: Ferguson D, Peters T (eds) Mathematics for industry: challenges and frontiers, SIAM, Philadelphia, pp 140–157
Bressan A (2003) An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend Semin Mat Univ Padova 110:103–117
Che J, Chen L, Göttlich S, Wang J (2016) Existence of a classical solution to complex material flow problems. Math Methods Appl Sci 39:4069–4081
Colombo RM, Garavello M, Lécureux-Mercier M (2012) A class of nonlocal models for pedestrian traffic. Math Models Methods Appl Sci 22:1150023, 34
Colombo RM, Lécureux-Mercier M (2012) Nonlocal crowd dynamics models for several populations. Acta Math Sci Ser B Engl Ed 32:177–196
Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Geotechnique 29:47–65
Etikyala R, Göttlich S, Klar A, Tiwari S (2014) Particle methods for pedestrian flow models: from microscopic to nonlocal continuum models. Math Model Methods Appl Sci 24:2503–2523
Evers JH, Hille SC, Muntean A (2015) Mild solutions to a measure-valued mass evolution problem with flux boundary conditions. J Differ Eqs 259:1068–1097
Göttlich S, Hoher S, Schindler P, Schleper V, Verl A (2014) Modeling, simulation and validation of material flow on conveyor belts. Appl Math Model 38:3295–3313
Göttlich S, Klar A, Tiwari S (2015) Complex material flow problems: a multi-scale model hierarchy and particle methods. J Eng Math 92:15–29
Göttlich S, Schindler P (2015) Discontinuous Galerkin method for material flow problems. Math Probl Eng Article ID 341893
Herty M, Klar A, Singh AK (2007) An ODE traffic network model. J Comput Appl Math 203:419–436
Hifi M, M’hallah R (2009) A literature review on circle and sphere packing problems: models and methodologies. Adv Oper Res 150624:1–22
La Marca M, Armbruster D, Herty M, Ringhofer C (2010) Control of continuum models of production systems. IEEE Trans Autom Control 55:2511–2526
Landry JW, Grest GS, Silbert LE, Plimpton SJ (2003) Confined granular packings: structure, stress, and forces. Phys Rev E 67:041303
LeVeque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge
McNamara A, Treuille A, Popović Z, Stam J (2004) Fluid control using the adjoint method. ACM Trans Graph 23:449–456
Reghelin R, de Arruda LVR (June 2012) An optimization model for microscopic centralized traffic management of intelligent vehicles in a segment of a single lane. In: 2012 IEEE intelligent vehicles symposium, pp 908–913
Reilly J, Samaranayake S, Delle Monache ML, Krichene W, Goatin P, Bayen AM (2015) Adjoint-based optimization on a network of discretized scalar conservation laws with applications to coordinated ramp metering. J Optim Theory Appl 167:733–760
Rosen JB (1960) The gradient projection method for nonlinear programming. I. Linear constraints. J Soc Ind Appl Math 8:181–217
Schlitzer R (2000) Applying the adjoint method for biogeochemical modeling: export of particulate organic matter in the world ocean. In: Kasibhata P (ed) Inverse methods in biogeochemical cycles, vol 114. AGU Monograph, Washington, pp 107–124
Wojtan C, Mucha PJ, Turk G (2006) Keyframe control of complex particle systems using the adjoint method. In: Cani M-P, O’Brien J (eds) ACM SIGGRAPH/Eurographics symposium on computer animation. The Eurographics Association, Boston
Zhu HP, Yu AB (2002) Averaging method of granular materials. Phys Rev E 66:021302
Zhu HP, Yu AB (2005) Micromechanic modeling and analysis of unsteady-state granular flow in a cylindrical hopper. J Eng Math 52:307–320
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This work was financially supported by the German Research Foundation (DFG), Grant OptiFlow (Project-ID GO 1920/3-1).
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Appendices
Appendix 1: Derivatives of macroscopic force terms
We present the technical details used in Sect. 2.1 for the derivation of the adjoint approach. In particular, the computation of the flux derivatives F with respect to the density \(\rho\) are discussed, i.e., \(F^{(1),+},\, F^{(1),-},F^{(2),+},F^{(2),-}\). These derivatives appear in Eqs. (2.5) and (2.6). A smooth approximation of the Heaviside function \(H(\rho )\) is needed. Here, we use
Then the derivative is \({\partial} _{\rho } H_{\text {app}}(\rho ) = \frac{k}{2}\cdot \frac{1}{\mathrm {cosh}^2(k\rho )}\) For simplicity, the approximation \(H_{app}(\rho )\) will be denoted as \(H(\rho )\) in the following.
First, we compute the derivative \({\partial} _{\rho _{i,j}^k} F_{i,j}^{(1),+}\). This computation includes four cases.
Next, we need the derivative \({\partial} _{\rho _{i,j}^k} F_{i,j}^{(1),-}\). The calculation is similar to the one above.
To complete the flux derivatives in the first dimension we additionally have to compute \({\partial} _{\rho _{i,j}^k} F_{i+1,j}^{(1),-}\) and \({\partial} _{\rho _{i,j}^k} F_{i-1,j}^{(1),+}\).
That means it is sufficient to calculate only the two derivatives (A.1) and (A.2) for solving Eq. (2.5) backward in time. Second, the adjoint Eq. (2.6) requires the derivatives of fluxes in the second dimension with respect to \(\tilde{\rho }\). Similarly to the first part, we start with the consideration of \({\partial} _{\tilde{\rho }_{i,j}^k} F^{(2),+}_{i,j}\) and that also includes four cases.
The derivative of the flux from the left \({\partial} _{\tilde{\rho }_{i,j}^k} F_{2,(i,j)}^-\) is stated as
As before, a short calculation yields \({\partial} _{\tilde{\rho }_{i,j}^k} F^{(2),-}_{i,j+1}={\partial} _{\tilde{\rho }_{i,j}^k} F^{(2),+}_{i,j}\) and \({\partial} _{\tilde{\rho }_{i,j}^k} F^{(2),+}_{i,j-1} = {\partial} _{\tilde{\rho }_{i,j}^k} F^{(2),-}_{i,j}.\)
Hence, the second adjoint Eq. (2.6) can be also solved backward in time. However, the derivatives of the non-local operator \(I(\rho )\) with respect to \(\rho\) and \(\tilde{\rho }\) are still needed. We introduce the following notation:
with weights c as defined in Sect. 2.
The discretization of the integrals is done by a two-dimensional trapezoid formula with eight intermediate points between the start and end points of the corresponding interval.
Then, the derivative \({\partial} _{\rho _{i,j}^k} I_1(\rho )({\mathbf x} _{i+\frac{1}{2},j})\) in the first dimension can be computed as:
The same computation holds true for \({\partial} _{\rho _{i,j}^k} I_1(\rho )({\mathbf x} _{i-\frac{1}{2},j})\).
Analogously, the derivatives of the non-local operator with respect to \(\tilde{\rho }\) in the second dimension are:
and \({\partial} _{\tilde{\rho }_{i,j}^k} I_2(\tilde{\rho })({\mathbf x} _{i,j-\frac{1}{2}})\) accordingly.
Appendix 2: Derivatives of microscopic force terms
We present the technical details used in Sect. 3.1 for the derivation of the adjoint approach. In particular, the computation of the force derivatives F with respect to x and v are discussed. These derivatives appear in Eqs. (3.2) and (3.3). The Heaviside function is approximated as before.
Then, \({\partial} _{x_i^{(q)}} F^{(k)}_{j,i}(x,v)\) can be expressed for \(q,k \in {1,2}\) as:
The partial derivative of \(F^{(k)}_{i,j}(x,v)\) with respect to \({\partial} _{v^{(q)}_i}\) is written as follows for \(q,k \in {1,2}\):
Note that the derivatives of interactions with walls can be computed in the same way.
Furthermore, in Eq. (3.3), the derivative of the bottom friction force G with respect to the second velocity component \(v^{(2)}\) is needed:
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Erbrich, M., Göttlich, S. & Pfirsching, M. Optimal packing of material flow on conveyor belts. Optim Eng 19, 71–96 (2018). https://doi.org/10.1007/s11081-017-9362-5
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DOI: https://doi.org/10.1007/s11081-017-9362-5