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Analysis of traffic flow by the finite element method. (English) Zbl 0583.90028

A finite element methodology is developed for the numerical solution of traffic flow problems encountered in arterial streets. The simple continuum traffic flow model consisting of the equation of continuity and an equilibrium flow-density relationship is adopted. A Galerkin type finite element method is used to formulate the problem in discrete form and the solution is obtained by a step-by-step time integration in conjunction with the Newton-Raphson method. The proposed finite element methodology, which is of the shock capturing type, is applied to flow traffic problems. Two numerical examples illustrate the method and demonstrate its advantages over other analytical or numerical techniques.

MSC:

90B10 Deterministic network models in operations research
65K05 Numerical mathematical programming methods
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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[1] Lighthill, M. J.; Whitham, G. B., On kinematic waves II. A theory of traffic flow on long crowded roads, (Proc. Roy. Soc., Ser., A229 (1955)), 317-345 · Zbl 0064.20906
[2] Stephanopoulos, G.; Michalopoulos, P. G.; Stephanopoulos, G., Modelling and analysis of traffic queue dynamics at signalized in tersections, Trans. Res., 13A, 295-307 (1979)
[3] Michaelopoulos, P. G.; Stephanopoulos, G.; Pisharody, V. B., Modelling of traffic flow at signalized links, Transp. Sci., 14, 9-41 (1980)
[4] Michalopoulos, P. G.; Pisharody, V. B., Platoon dynamics at signal controlled arterials, Transp. Sci., 14, 365-396 (1980)
[5] Michalopoulos, P. G.; Pisharody, V. B., Derivation of delays based on improved microscopic traffic models, Transp. Res., 15B, 299-317 (1981)
[6] Michalopoulos, P. G.; Beskos, D. E.; Lin, J., Analysis of interrupted traffic flow by finite difference methods, Transp. Res., 18B, 4/5, 409-421 (1984)
[7] Michalopoulos, P. G.; Beskos, D. E.; Yamauchi, Y., Multilane traffic flow dynamics: some macroscopic considerations, Transp. Res., 18B, 4/5, 377-395 (1984)
[8] Zienkiewicz, O. C., The finite element method in engineering science (1977), McGraw-Hill: McGraw-Hill New York · Zbl 0435.73072
[9] Chung, T. J., Finite element analysis in fluid dynamics (1978), McGraw-Hill: McGraw-Hill New York · Zbl 0432.76003
[10] Wellford, L. C.; Oden, J. T., Discontinuous finite-element approximations for the analysis of shock waves in nonlinearly elastic solids, J. Comput. Phys., 19, 179-210 (1975) · Zbl 0328.73034
[11] Hafez, M.; South, J.; Murman, E., Artificial compressibility methods for numerical solution of transonic full potential equation, (Proc. 11th AIAA Fluid and Plasma Dynamics Conf.. Proc. 11th AIAA Fluid and Plasma Dynamics Conf., Seattle (July, 1978)) · Zbl 0409.76013
[12] Jepps, S. A., A finite element method for computing transonic potential flow, (Rizzi, A.; Viviand, H., Numerical methods for the computation of inviscid transonic flows with shock waves (1981), F. Vieweg & Sohn: F. Vieweg & Sohn Braunschweig), 91-99
[13] Greeshields, B. D., A study of traffic capacity, (Proc. Hwy. Res. Board, 14 (1934)), 448-477
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