Journal of Applied and Numerical Optimization

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DOI: 10.23952/jano.7.2025.3.03
Volume 7, Issue 3, 1 December 2025, Pages 309-332

 

Abstract. In this paper, topological derivatives are defined and employed for gas transport networks governed by nonlinear hyperbolic systems of PDEs. The concept of topological derivatives of a shape functional is introduced for optimum design and control of gas networks. First, the dynamic model for the network is considered. The cost for the control problem includes the deviations of the pressure at the inflow and outflow nodes. For dynamic control problems of gas networks when the turnpike property occurs, the synthesis of control and optimum design of the network can be simplified. That is, the design of the network can be performed for optimal control of the steady-state network model. The cost of design is defined by the optimal control cost for the steady-state network model. The topological derivative of the design cost, given by the optimal control cost with respect to the nucleation of a small cycle, is determined. Tree-structured networks can be decomposed into single network junctions. The topological derivative of the design cost is systematically evaluated at each junction of the decomposed network. This allows for the identification of internal nodes with negative topological derivatives, where replacing the node with a small cycle leads to an improved design cost. As the set of network junctions is finite, the iterative procedure is convergent. This design procedure is applied to representative examples and it can be generalized to arbitrary network graphs. A key feature of such modeling approach is the availability of exact steady-state solutions, enabling a fully analytical topological analysis of the design cost without numerical approximations.

 

How to Cite this Article:
M. Schuster, J. Sokolowski, The topological derivative method for optimum shape design and control of gas networks, J. Appl. Numer. Optim. 7 (2025), 309-332.