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Optimal packing of material flow on conveyor belts

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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.

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Acknowledgements

This work was financially supported by the German Research Foundation (DFG), Grant OptiFlow (Project-ID GO 1920/3-1).

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Correspondence to Simone Göttlich.

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

$$H(\rho ) \approx H_{{{\text{app}}}} (\rho ) = \frac{1}{2} + \frac{1}{2}{\text{tanh}}(k\rho ) = \frac{1}{{1 + e^{{ - 2k\rho }} }},\quad k \in \mathbb{R}^{ + }$$

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.

$$\begin{aligned} \partial _{{\rho _{{i,j}}^{k} }} F_{{i,j}}^{{(1), + }} &= \partial _{{\rho _{{i,j}}^{k} }} F_{{i,j}}^{{(1)}} + \partial _{{\rho _{{i,j}}^{k} }} G_{{i,j}}^{{(1)}} \\ & = \left\{ {\begin{array}{*{20}l} {H\left( {\rho _{{i,j}}^{k} - \rho _{{{\text{max}}}} } \right)I^{{(1)}} (\rho )\left( {{\mathbf{x}}_{{i + \frac{1}{2},j}} } \right)} \hfill & {} \hfill \\ { +\, \rho _{{i,j}}^{k} H^{\prime } (\rho _{{i,j}}^{k} - \rho _{{{\text{max}}}} )I^{{(1)}} (\rho )\left( {{\mathbf{x}}_{{i + \frac{1}{2},j}} } \right)} \hfill & {} \hfill \\ { +\,\rho _{{i,j}}^{k} H\left( {\rho _{{i,j}}^{k} - \rho _{{{\text{max}}}} } \right)\partial _{{\rho _{{i,j}}^{k} }} I^{{(1)}} (\rho ),} \hfill & {I^{{(1)}} (\rho ) \le \left( {{\mathbf{x}}_{{i + \frac{1}{2},j}} } \right) \ge 0} \hfill \\ {\mkern 1mu} \hfill & {} \hfill \\ {\rho _{{i + 1,j}}^{k} H\left( {\rho _{{i + 1,j}}^{k} - \rho _{{{\text{max}}}} } \right)\partial _{{\rho _{{i,j}}^{k} }} I^{{(1)}} (\rho )\left( {{\mathbf{x}}_{{i + \frac{1}{2},j}} } \right),} \hfill & {I^{{(1)}} (\rho )\left( {{\mathbf{x}}_{{i + \frac{1}{2},j}} } \right) \le 0} \hfill \\ \end{array} } \right.\quad + \left\{ {\begin{array}{*{20}l} {\mkern 1mu} \hfill & {} \hfill \\ {{\mathbf{v}}_{{i + \frac{1}{2},j}}^{{{\text{stat}},(1)}} ,} \hfill & {{\mathbf{v}}_{{i + \frac{1}{2},j}}^{{{\text{stat}},(1)}} \ge 0} \hfill \\ {\mkern 1mu} \hfill & {} \hfill \\ {0,} \hfill & {{\mathbf{v}}_{{i + \frac{1}{2},j}}^{{{\text{stat}},(1)}} \le 0} \hfill \\ \end{array} } \right. \\ \end{aligned}$$
(A.1)

Next, we need the derivative \({\partial} _{\rho _{i,j}^k} F_{i,j}^{(1),-}\). The calculation is similar to the one above.

$$\begin{aligned} \partial _{{\rho _{{i,j}}^{k} }} F_{{i,j}}^{{(1), - }} & = \partial _{{\rho _{{i,j}}^{k} }} F_{{i - 1,j}}^{{(1)}} + \partial _{{\rho _{{i,j}}^{k} }} G_{{i - 1,j}}^{{(1)}} \\ & \quad = \left\{ {\begin{array}{*{20}l} {\rho _{{i - 1,j}}^{k} H\left( {\rho _{{i - 1,j}}^{k} - \rho _{{{\text{max}}}} } \right)\partial _{{\rho _{{i,j}}^{k} }} I^{{(1)}} (\rho )\left( {{\mathbf{x}}_{{i - \frac{1}{2},j}} } \right),} \hfill & {I^{{(1)}} (\rho )\left( {{\mathbf{x}}_{{i - \frac{1}{2},j}} } \right) \ge 0} \hfill \\ {\mkern 1mu} \hfill & {} \hfill \\ {H\left( {\rho _{{i,j}}^{k} - \rho _{{{\text{max}}}} } \right)I^{{(1)}} (\rho )\left( {{\mathbf{x}}_{{i - \frac{1}{2},j}} } \right)} \hfill & {} \hfill \\ { +\, \rho _{{i,j}}^{k} H^{\prime } \left( {\rho _{{i,j}}^{k} - \rho _{{{\text{max}}}} } \right)I^{{(1)}} (\rho )\left( {{\mathbf{x}}_{{i - \frac{1}{2},j}} } \right)} \hfill & {} \hfill \\ {+\, \rho _{{i,j}}^{k} H\left( {\rho _{{i,j}}^{k} - \rho _{{{\text{max}}}} } \right)\partial _{{\rho _{{i,j}}^{k} }} I^{{(1)}} (\rho )\left( {{\mathbf{x}}_{{i - \frac{1}{2},j}} } \right),} \hfill & {I^{{(1)}} (\rho )\left( {{\mathbf{x}}_{{i - \frac{1}{2},j}} } \right) \le 0} \hfill \\ \end{array} } \right.\quad + \left\{ {\begin{array}{*{20}l} {} \hfill & {} \hfill \\ {0,} \hfill & {{\mathbf{v}}_{{i - \frac{1}{2},j}}^{{{\text{stat}},(1)}} \ge 0} \hfill \\ {} \hfill & {} \hfill \\ {{\mathbf{v}}_{{i - \frac{1}{2},j}}^{{{\text{stat}},(1)}} ,} \hfill & {{\mathbf{v}}_{{i - \frac{1}{2},j}}^{{{\text{stat}},(1)}} \le 0} \hfill \\ \end{array} } \right. \\ \end{aligned}$$
(A.2)

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 \({\partial} _{\rho _{i,j}^k} F_{i-1,j}^{(1),+}\).

$$\begin{aligned} \partial _{{\rho _{{i,j}}^{k} }} F_{{i + 1,j}}^{{(1), - }} & = \partial _{{\rho _{{i,j}}^{k} }} \left[ {F^{{(1)}} \left( {\rho ,\rho _{{i,j}}^{k} ,\rho _{{i + 1,j}}^{k} ,{\mathbf{x}}_{{i + \frac{1}{2},j}} } \right) + G^{{(1)}} \left( {\rho _{{i,j}}^{k} ,\rho _{{i,j}}^{k} ,{\mathbf{v}}_{{i + \frac{1}{2},j}}^{{{\text{stat}}}} } \right)} \right] \\ & = \partial _{{\rho _{{i,j}}^{k} }} \left[ {F_{{i,j}}^{{(1)}} + G_{{i,j}}^{{(1)}} } \right] \\ & = \partial _{{\rho _{{i,j}}^{k} }} F_{{i,j}}^{{(1), + }} \\ \partial _{{\rho _{{i,j}}^{k} }} F_{{i - 1,j}}^{{(1), + }} & = \partial _{{\rho _{{i,j}}^{k} }} \left[ {F^{{(1)}} \left( {\rho ,\rho _{{i - 1,j}}^{k} ,\rho _{{i,j}}^{k} ,{\mathbf{x}}_{{i - \frac{1}{2},j}} } \right) + G^{{(1)}} \left( {\rho _{{i - 1,j}}^{k} ,\rho _{{i,j}}^{k} ,{\mathbf{v}}_{{i - \frac{1}{2},j}}^{{{\text{stat}}}} } \right)} \right] \\ & = \partial _{{\rho _{{i,j}}^{k} }} \left[ {F_{{i - 1,j}}^{{(1)}} + G_{{i - 1,j}}^{{(1)}} } \right] \\ & = \partial _{{\rho _{{i,j}}^{k} }} F_{{i,j}}^{{(1), - }} . \\ \end{aligned}$$

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.

$$\begin{aligned} \partial _{{\tilde{\rho }_{{i,j}}^{k} }} F_{{i,j}}^{{(2), + }} & = \partial _{{\tilde{\rho }_{{i,j}}^{k} }} F_{{i,j}}^{{(2)}} + \partial _{{\tilde{\rho }_{{i,j}}^{k} }} G_{{i,j}}^{{(2)}} \\ & = \left\{ {\begin{array}{*{20}l} {H\left( {\tilde{\rho }_{{i,j}}^{k} - \rho _{{{\text{max}}}} } \right)I^{{(2)}} (\tilde{\rho })\left( {{\mathbf{x}}_{{i,j + \frac{1}{2}}} } \right)} \hfill & {} \hfill \\ {+\, \tilde{\rho }_{{i,j}}^{k} H^{\prime } \left( {\tilde{\rho }_{{i,j}}^{k} - \rho _{{{\text{max}}}} } \right)I^{{(2)}} (\tilde{\rho })\left( {{\mathbf{x}}_{{i,j + \frac{1}{2}}} } \right)} \hfill & {} \hfill \\ { +\, \tilde{\rho }_{{i,j}}^{k} H\left( {\tilde{\rho }_{{i,j}}^{k} - \rho _{{{\text{max}}}} } \right)\partial _{{\tilde{\rho }_{{i,j}}^{k} }} I^{{(2)}} (\tilde{\rho })\left( {{\mathbf{x}}_{{i,j + \frac{1}{2}}} } \right),} \hfill & {I^{{(2)}} (\tilde{\rho })\left( {{\mathbf{x}}_{{i,j + \frac{1}{2}}} } \right) \ge 0} \hfill \\ {\mkern 1mu} \hfill & {} \hfill \\ {\tilde{\rho }_{{i,j + 1}}^{k} H\left( {\tilde{\rho }_{{i,j + 1}}^{k} - \rho _{{{\text{max}}}} } \right)\partial _{{\tilde{\rho }_{{i,j}}^{k} }} I^{{(2)}} (\tilde{\rho })\left( {{\mathbf{x}}_{{i,j + \frac{1}{2}}} } \right),} \hfill & {I^{{(2)}} (\tilde{\rho })\left( {{\mathbf{x}}_{{i,j + \frac{1}{2}}} } \right) \le 0} \hfill \\ \end{array} } \right.\quad + \left\{ {\begin{array}{*{20}l} {\mkern 1mu} \hfill & {} \hfill \\ {{\mathbf{v}}_{{i,j + \frac{1}{2}}}^{{{\text{stat}},(2)}} ,} \hfill & {{\mathbf{v}}_{{i,j + \frac{1}{2}}}^{{{\text{stat}},(2)}} \ge 0} \hfill \\ {\mkern 1mu} \hfill & {} \hfill \\ {0,} \hfill & {{\mathbf{v}}_{{i,j + \frac{1}{2}}}^{{{\text{stat}},(2)}} \le 0} \hfill \\ {\mkern 1mu} \hfill & {} \hfill \\ \end{array} } \right. \\ \end{aligned}$$

The derivative of the flux from the left \({\partial} _{\tilde{\rho }_{i,j}^k} F_{2,(i,j)}^-\) is stated as

$$\begin{aligned} \partial _{{\tilde{\rho }_{{i,j}}^{k} }} F_{{i,j}}^{{(2), - }} & = \partial _{{\tilde{\rho }_{{i,j}}^{k} }} F_{{i,j - 1}}^{{(2)}} + \partial _{{\tilde{\rho }_{{i,j}}^{k} }} G_{{i,j - 1}}^{{(2)}} \\ & = \left\{ {\begin{array}{*{20}l} {\tilde{\rho }_{{i,j - 1}}^{k} H\left( {\tilde{\rho }_{{i,j - 1}}^{k} - \rho _{{{\text{max}}}} } \right)\partial _{{\tilde{\rho }_{{i,j}}^{k} }} I^{{(2)}} (\tilde{\rho })\left( {{\mathbf{x}}_{{i,j - \frac{1}{2}}} } \right),} \hfill & {I^{{(2)}} (\tilde{\rho })\left( {{\mathbf{x}}_{{i,j - \frac{1}{2}}} } \right) \ge 0} \hfill \\ {} \hfill & {} \hfill \\ {H\left( {\tilde{\rho }_{{i,j}}^{k} - \rho _{{{\text{max}}}} } \right)I^{{(2)}} (\tilde{\rho })\left( {{\mathbf{x}}_{{i,j - \frac{1}{2}}} } \right)} \hfill & {} \hfill \\ { +\, \tilde{\rho }_{{i,j}}^{k} H^{\prime } \left( {\tilde{\rho }_{{i,j}}^{k} - \rho _{{{\text{max}}}} } \right)I^{{(2)}} (\tilde{\rho })\left( {{\mathbf{x}}_{{i,j - \frac{1}{2}}} } \right)} \hfill & {} \hfill \\ { + \,\tilde{\rho }_{{i,j}}^{k} H\left( {\tilde{\rho }_{{i,j}}^{k} - \rho _{{{\text{max}}}} } \right)\partial _{{\tilde{\rho }_{{i,j}}^{k} }} I^{{(2)}} (\tilde{\rho })\left( {{\mathbf{x}}_{{i,j - \frac{1}{2}}} } \right),} \hfill & {I^{{(2)}} (\tilde{\rho })\left( {{\mathbf{x}}_{{i,j - \frac{1}{2}}} } \right) \le 0} \hfill \\ \end{array} } \right.\quad + \left\{ {\begin{array}{*{20}l} {\mkern 1mu} \hfill & {} \hfill \\ {0,} \hfill & {{\mathbf{v}}_{{i,j - \frac{1}{2}}}^{{{\text{stat}},(2)}} \ge 0} \hfill \\ {\mkern 1mu} \hfill & {} \hfill \\ {{\mathbf{v}}_{{i,j - \frac{1}{2}}}^{{{\text{stat}},(2)}} ,} \hfill & {{\mathbf{v}}_{{i,j - \frac{1}{2}}}^{{{\text{stat}},(2)}} \le 0} \hfill \\ {\mkern 1mu} \hfill & {} \hfill \\ \end{array} } \right. \\ \end{aligned}$$

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:

$$\begin{aligned} D_{{x_{1} }} \rho _{{i,j}} & : = \sum\limits_{{p,q}} {\rho _{{p,q}}^{k} } \cdot c_{{i - p,j - q}}^{1} \\ D_{{x_{2} }} \rho _{{i,j}} & : = \sum\limits_{{p,q}} {\rho _{{p,q}}^{k} } \cdot c_{{i - p,j - q}}^{2} \\ D_{{x_{1} ,x_{2} }} \rho _{{i,j}} & : = D_{{x_{1} }} \rho _{{i,j}} \cdot D_{{x_{2} }} \rho _{{i,j}} \\ {\text{Nor}}_{{i,j}} & : = \sqrt {1 + (D_{{x_{1} }} \rho _{{i,j}} )^{2} + (D_{{x_{2} }} \rho _{{i,j}} )^{2} } \\ \widetilde{{{\text{Nor}}}}_{{i,j}} & : = \sqrt {1 + (D_{{x_{1} }} \tilde{\rho }_{{i,j}} )^{2} + (D_{{x_{2} }} \tilde{\rho }_{{i,j}} )^{2} } \\ \end{aligned}$$

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:

$$\begin{aligned} & \partial _{{\rho _{{i,j}}^{k} }} I_{1} (\rho )\left( {{\mathbf{x}}_{{i + \frac{1}{2},j}} } \right) \\ & = \partial _{{\rho _{{i,j}}^{k} }} \left( { - \epsilon \frac{{\sum\limits_{{p,q}} {\rho _{{p,q}}^{k} } \cdot c_{{i - p,j - q}}^{1} }}{{\sqrt {1 + \left( {\sum\limits_{{p,q}} {\rho _{{p,q}}^{k} } \cdot c_{{i - p,j - q}}^{1} } \right)^{2} + \left( {\sum\limits_{{p,q}} {\rho _{{p,q}}^{k} } \cdot c_{{i - p,j - q}}^{2} } \right)^{2} } }}} \right) \\ & = \partial _{{\rho _{{i,j}}^{k} }} \left( { - \epsilon \frac{{D_{{x_{1} }} \rho _{{i,j}} \left( {{\mathbf{x}}_{{i + \frac{1}{2},j}} } \right)}}{{{\text{Nor}}\left( {{\mathbf{x}}_{{i + \frac{1}{2},j}} } \right)}}} \right) \\ & = - \epsilon \cdot \left( {\frac{{c_{{0,0}}^{1} \cdot {\text{Nor}}_{{i,j}} }}{{({\text{Nor}}_{{i,j}} )^{2} }} - \frac{{(D_{{x_{1} }} \rho _{{i,j}} \cdot (c_{{0,0}}^{1} \cdot D_{{x_{1} }} \rho _{{i,j}} + c_{{0,0}}^{2} \cdot D_{{x_{2} }} \rho _{{i,j}} )({\text{Nor}}_{{i,j}} )^{{ - 1}} }}{{({\text{Nor}}_{{i,j}} )^{2} }}} \right) \\ & = - \epsilon \cdot \left( {\frac{{c_{{0,0}}^{1} }}{{{\text{Nor}}_{{i,j}} }} - \frac{{(D_{{x_{1} }} \rho _{{i,j}} )^{2} \cdot c_{{0,0}}^{1} + c_{{0,0}}^{2} \cdot D_{{x_{1} ,x_{2} }} \rho _{{i,j}} }}{{({\text{Nor}}_{{i,j}} )^{3} }}} \right). \\ \end{aligned}$$

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:

$$\begin{aligned} & \partial _{{\tilde{\rho }_{{i,j}}^{k} }} I_{2} (\tilde{\rho })\left( {{\mathbf{x}}_{{i + \frac{1}{2},j}} } \right) \\ & = \partial _{{\tilde{\rho }_{{i,j}}^{k} }} \left( { - \epsilon \frac{{\sum\limits_{{p,q}} {\tilde{\rho }_{{p,q}}^{k} } \cdot c_{{i - p,j - q}}^{2} }}{{\sqrt {1 + \left( {\sum\limits_{{p,q}} {\tilde{\rho }_{{p,q}}^{k} } \cdot c_{{i - p,j - q}}^{1} } \right)^{2} + \left( {\sum\limits_{{p,q}} {\tilde{\rho }_{{p,q}}^{k} } \cdot c_{{i - p,j - q}}^{2} } \right)^{2} } }}} \right) \\ & = \partial _{{\tilde{\rho }_{{i,j}}^{k} }} \left( { - \epsilon \frac{{D_{{x_{2} }} \tilde{\rho }_{{i,j}} }}{{\widetilde{{{\text{Nor}}}}_{{i,j}} }}} \right) \\ & = - \epsilon \cdot \left( {\frac{{c_{{0,0}}^{2} \cdot {\text{Nor}}_{{i,j}} }}{{(\widetilde{{{\text{Nor}}}}_{{i,j}} )^{2} }} - \frac{{(D_{{x_{2} }} \tilde{\rho }_{{i,j}} \cdot \left( {c_{{0,0}}^{1} \cdot D_{{x_{1} }} \tilde{\rho }_{{i,j}} } \right) + c_{{0,0}}^{2} \cdot D_{{x_{2} }} \tilde{\rho }_{{i,j}} )({\text{Nor}}_{{i,j}} )^{{ - 1}} }}{{(\widetilde{{{\text{Nor}}}}_{{i,j}} )^{2} }}} \right) \\ & = - \epsilon \cdot \left( {\frac{{c_{{0,0}}^{2} \cdot }}{{\widetilde{{{\text{Nor}}}}_{{i,j}} }} - \frac{{(D_{{x_{2} }} \tilde{\rho }_{{i,j}} )^{2} \cdot c_{{0,0}}^{1} + c_{{0,0}}^{2} \cdot D_{{x_{1} ,x_{2} }} \tilde{\rho }_{{i,j}} }}{{(\widetilde{{{\text{Nor}}}}_{{i,j}} )^{3} }}} \right) \\ \end{aligned}$$

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:

$$\begin{aligned} \partial _{{x_{i}^{{(q)}} }} F_{{j,i}}^{{(k)}} (x,v) & = \tilde{H}\left( {2R - ||x_{j} - x_{i} ||} \right) \\ & \quad \times \left( {( - 1) \cdot \left( {k_{n} \frac{{2R - ||x_{j} - x_{i} ||}}{{||x_{j} - x_{i} ||}}} \right) \cdot \delta _{{q,k}} } \right. \\ & \quad + \left( {x_{j}^{{(k)}} - x_{i}^{{(k)}} } \right) \cdot \left( {k_{n} \cdot (x_{j}^{{(q)}} - x_{i}^{{(q)}} ) \cdot \frac{{2R - ||x_{j} - x_{i} ||}}{{||x_{j} - x_{i} ||^{3} }} + \frac{{k_{n} \cdot (x_{j}^{{(q)}} - x_{i}^{{(q)}} )}}{{||x_{j} - x_{i} ||^{2} }}} \right) \\ & \quad + \left( {\left( {\gamma _{n} - \gamma _{t} } \right) \cdot \frac{{\left\langle {x_{j} - x_{i} ,v_{j} - v_{i} } \right\rangle }}{{||x_{j} - x_{i} ||^{2} }}} \right) \cdot \delta _{{q,k}} p \\ & \quad - \left( {x_{j}^{{(k)}} - x_{i}^{{(k)}} } \right) \cdot \left( {\gamma _{n} - \gamma _{t} } \right) \cdot \left( { - \frac{{\left( {v_{j}^{{(q)}} - v_{i}^{{(q)}} } \right)}}{{||x_{j} - x_{i} ||^{2} }} + \frac{{2 \cdot \left( {x_{j}^{{(q)}} - x_{i}^{{(q)}} } \right) \cdot \left\langle {x_{j} - x_{i} ,v_{j} - v_{i} } \right\rangle }}{{||x_{j} - x_{i} ||^{4} }}} \right) \\ & \quad + \tilde{H}^{\prime } \left( {2R - ||x_{j} - x_{i} ||} \right) \cdot \frac{{\left( {x_{j}^{{(k)}} - x_{i}^{{(k)}} } \right)}}{{||x_{j} - x_{i} ||}} \cdot \left( {\left( {x_{j}^{{(q)}} - x_{i}^{{(q)}} } \right) \cdot \left( {k_{n} \frac{{2R - ||x_{j} - x_{i} ||}}{{||x_{j} - x_{i} ||}}} \right)} \right. \\ & \quad - \left( {x_{j}^{{(q)}} - x_{i}^{{(q)}} } \right) \cdot \left( {\left( {\gamma _{n} - \gamma _{t} } \right) \cdot \left. {\frac{{\left\langle {x_{j} - x_{i} ,v_{j} - v_{i} } \right\rangle }}{{||x_{j} - x_{i} ||^{2} }}} \right) - \left( {v_{j}^{{(q)}} - v_{i}^{{(q)}} } \right) \cdot \gamma _{t} } \right). \\ \end{aligned}$$

The partial derivative of \(F^{(k)}_{i,j}(x,v)\) with respect to \({\partial} _{v^{(q)}_i}\) is written as follows for \(q,k \in {1,2}\):

$$\partial _{{v_{i}^{{(q)}} }} F_{{i,j}}^{{(k)}} (x,v) = H\left( {2R - ||x_{j} - x_{i} ||} \right) \cdot \left( {\left( {\gamma _{n} - \gamma _{t} } \right) \cdot \frac{{(x_{j}^{{(q)}} - x_{i}^{{(q)}} ) \cdot (x_{j}^{{(k)}} - x_{i}^{{(k)}} )}}{{||x_{j} - x_{i} ||^{2} }} + \gamma _{t} \cdot \delta _{{q,k}} } \right)$$

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:

$$\partial _{{v_{i}^{{(2)}} }} G^{{(2)}} (v_{i} ) = \left\{ {\begin{array}{*{20}l} {\mu _{b} mg\frac{{||v_{i} - v_{T} || - \left( {v_{i}^{{(2)}} - v_{T}^{{(2)}} } \right)^{2} }}{{||v_{i} - v_{T} ||^{2} }}} \hfill & {{\text{if }}\;\mu _{b} mg < \gamma _{b} ||v_{i} - v_{T} ||} \hfill \\ {\gamma _{b} } \hfill & {{\text{if }}\;\mu _{b} mg > \gamma _{b} ||v_{i} - v_{T} ||.} \hfill \\ \end{array} } \right.$$

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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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