Skip to main content Accessibility help

Solution of some Engineering Partial Differential Equations Governed by the Minimal of a Functional by Global Optimization Method

  • Y. M. Cheng (a1), D. Z. Li (a1), N. Li (a1), Y. Y Lee (a2) and S. K. Au (a3)...


Many engineering problems are governed by partial differential equations which can be solved by analytical as well as numerical methods, and examples include the plasticity problem of a geotechnical system, seepage problem and elasticity problem. Although the governing differential equations can be solved by either iterative finite difference method or finite element, there are however limitations to these methods in some special cases which will be discussed in the present paper. The solutions of these governing differential equations can all be viewed as the stationary value of a functional. Using an approximate solution as the initial solution, the stationary value of the functional can be obtained easily by modern global optimization method. Through the comparisons between analytical solutions and fine mesh finite element analysis, the use of global optimization method will be demonstrated to be equivalent to the solutions of the governing partial differential equations. The use of global optimization method can be an alternative to the finite difference/ finite element method in solving an engineering problem, and it is particularly attractive when an approximate solution is available or can be estimated easily.


Corresponding author

*Corresponding author (


Hide All
1.Sokolovskii, V. V., Statics of Granular Media, Pergamon Press (1965).
2.Cheng, Y. M., Li, D. Z., Li, L., Sun, Y. J., Baker, R. and Yang, Y., “Limit Equilibrium Method Based on Approximate Lower Bound Method with Variable Factor of Safety that Can Consider Residual Strength,” Computers and Geotechnic, 38, pp. 628637 (2011).
3.Denn, M. M., Optimization by Variational Methods, Hills Publishing (1969).
4.Abramson, L. W., Lee, T. S., Sharma, S. and Boyce, G. M., Slope Stability and Stabilization Methods, 2nd Edition, John Wiley, USA (2002).
5.Morgentern, N. R. and Price, V. E., “The Analysis of Stability of General Slip Surface,” Geotechnique, 15, pp 7993 (1965).
6.Janbu, N., “Earth Pressures and Bearing Capacity Calculations by Generalized Procedure of Slices,” Proceedings of the 4th International Conference Soil Mechanics Foundation Engineering, 2, pp. 207212 (1957).
7.Janbu, N., Slope Stability Computations in Embankment-Dam Engineering, Wiley, New York (1973).
8.Chen, W. F., Limit Analysis and Soil Plasticity, Elsevier (1975).
9.Prandtl, L., “Uber Die Eindringungs-Festigkeit (Harte) Plastischer Baustoffe Und Die Festigkeit Von Schneiden,” Zeitschrift Fur Angewandte Mathematik Und Mechanik, 1, pp. 1520 (1921).
10.Hill, R., Mathematical Theory of Plasticity, Oxford University Press (1998).
11.Cheng, Y. M. and Au, S. K., “Bearing Capacity Problem by Slip Line Method,” Canadian Geotechnical Journal, 42, pp. 12321241 (2005).
12.Cheng, Y. M., Hu, Y. Y. and Wei, W. B., “General Axisymmetric Active Earth Pressure by Method of Characteristic — Theory and Numerical Formulation,” Journal of Geomechanics, ASCE, 7, pp. 115 (2007).
13.Soubra, A. H., “Static and Seismic Passive Earth Pressure Coefficients on Rigid Retaining Structures,” Canadian Geotechnical Journal, 37, pp. 463478 (2000).
14.Cheng, Y. M., “Seismic Lateral Earth Pressure Coefficients by Slip Line Method,” Computers and Geotechnics, 30, pp. 661670 (2003).
15.Booker, J. R. and Zheng, X., Application of the Theory of Classical Plasticity to the Analysis of the Stress Distribution in Wedges of a Perfectly Frictional Material, Modelling in Geomechanics, John Wiley, New York (2000).
16.Martin, C. M., “User Guide for ABC – Analysis of Bearing Capacity (v1.0),” Department of Engineering Science, University of Oxford (2004).
17.Cheng, Y. M., Zhao, Z. H. and Sun, Y. J., “Determination of the Bounds to the Factor of Safety and the Evaluation of f (x) in Slope Stability Analysis by Lower Bound Method,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 136, pp. 11031113 (2010).
18.Lawrence, C. T. and Tits, A. L., “A Computationally Efficient Feasible Sequential Quadratic Programming Algorithm,” Society for Industrial and Applied Mathematics, 11, pp. 10921118 (2001).
19.Graham, J., Andrew, M. and Shields, D. H., “Stress Characteristics for Shallow Footing in Cohesionless Slopes,” Canadian Geotechnical Journal, 25, pp. 238249 (1988).
20.Shields, D. H., Scott, J. D., Bauer, G. E., Deschenes, J. H. and Barsvary, A. K., “Bearing Capacity of Foundations Near Slopes,” Proceedings of the 9th International Conference on Soil Mechanics and Foundations Engineering, Tokyo, 2, pp. 715720 (1977).
21.Baker, R. and Garber, M., “Theoretical Analysis of the Stability of Slopes,” Geotechnique, 28, pp. 395–341 (1978).
22.Baker, R., “Determination of the Critical Slip Surface in Slope Stability Computations,” International Journal of Numerical and Analytical Methods in Geomechanics, 4, pp. 333359 (1980).
23.Baker, R., “Tensile Strength, Tension Cracks, and Stability of Slopes,” Soils and Foundation, 21, pp. 117 (1981).
24.Baker, R., “Sufficient Conditions for Existence of Physically Significant Solutions in Limiting Equilibrium Slope Stability Analysis,” International Journal of Solids and Structures, 40, pp. 37173735 (2003).
25.Baker, R., “Variational Slope Stability Analysis of Materials with Non-Linear Failure Criterion,” Electronic Journal of Geotechnical Engineering, 10, Bundle A (2005).
26.Chen, Z. and Morgenstern, N. R., “Extensions to Generalized Method of Slices for Stability Analysis,” Canadian Geotechnical Journal, 20, pp. 104109 (1983).
27.Cheng, Y. M., Lansivaara, T. and Wei, W. B., “Two-dimensional Slope Stability Analysis by Limit Equilibrium and Strength Reduction Methods,” Computers and Geotechnics, 34, pp. 137150, (2007).
28.Cheng, Y. M., Lansivaara, T. and Siu, J., “Impact of Convergence on Slope Stability Analysis and Design,” Computers and Geotechnics, 35, pp. 105115 (2008).
29.Spencer, E., “A Method of Analysis of the Stability of Embankments Assuming Parallel Inter-Slice Forces,” Geotechnique, 17, pp. 1126 (1967).
30.Wu, L.Y. and Tsai, Y. F., “Variational Stability Analysis of Cohesive Slope by Applying Boundary integral Equation Method,” Journal of Mechanics, 21, pp. 187195 (2005).



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed