Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T15:52:45.276Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  01 May 2013

Y. M. Cheng ,
D. Z. Li ,
N. Li ,
Y. Y Lee  and
S. K. Au
Y. M. Cheng*
Affiliation:
Department of Civil and Environmental Engineering, Hong Kong Polytechnic University, Hong Kong
D. Z. Li
Affiliation:
Department of Civil and Environmental Engineering, Hong Kong Polytechnic University, Hong Kong
N. Li
Affiliation:
Department of Civil and Environmental Engineering, Hong Kong Polytechnic University, Hong Kong
Y. Y Lee
Affiliation:
Department of Civil and Architectural Engineering, City University of Hong Kong
S. K. Au
Affiliation:
Wong and Cheng, Hong Kong
*
*Corresponding author ([email protected])
Get access

Abstract

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.

Keywords

Differential equationUltimate limit stateOptimization methodVariational principle
Type
Articles
Information
Journal of Mechanics , Volume 29 , Issue 3 , September 2013 , pp. 507 - 516
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Sokolovskii, V. V., Statics of Granular Media, Pergamon Press (1965).Google Scholar
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. 628–637 (2011).Google Scholar
3.Denn, M. M., Optimization by Variational Methods, Hills Publishing (1969).Google Scholar
4.Abramson, L. W., Lee, T. S., Sharma, S. and Boyce, G. M., Slope Stability and Stabilization Methods, 2nd Edition, John Wiley, USA (2002).Google Scholar
5.Morgentern, N. R. and Price, V. E., “The Analysis of Stability of General Slip Surface,” Geotechnique, 15, pp 79–93 (1965).CrossRefGoogle Scholar
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. 207–212 (1957).Google Scholar
7.Janbu, N., Slope Stability Computations in Embankment-Dam Engineering, Wiley, New York (1973).Google Scholar
8.Chen, W. F., Limit Analysis and Soil Plasticity, Elsevier (1975).Google Scholar
9.Prandtl, L., “Uber Die Eindringungs-Festigkeit (Harte) Plastischer Baustoffe Und Die Festigkeit Von Schneiden,” Zeitschrift Fur Angewandte Mathematik Und Mechanik, 1, pp. 15–20 (1921).Google Scholar
10.Hill, R., Mathematical Theory of Plasticity, Oxford University Press (1998).Google Scholar
11.Cheng, Y. M. and Au, S. K., “Bearing Capacity Problem by Slip Line Method,” Canadian Geotechnical Journal, 42, pp. 1232–1241 (2005).Google Scholar
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. 1–15 (2007).CrossRefGoogle Scholar
13.Soubra, A. H., “Static and Seismic Passive Earth Pressure Coefficients on Rigid Retaining Structures,” Canadian Geotechnical Journal, 37, pp. 463–478 (2000).Google Scholar
14.Cheng, Y. M., “Seismic Lateral Earth Pressure Coefficients by Slip Line Method,” Computers and Geotechnics, 30, pp. 661–670 (2003).Google Scholar
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).Google Scholar
16.Martin, C. M., “User Guide for ABC – Analysis of Bearing Capacity (v1.0),” Department of Engineering Science, University of Oxford (2004).Google Scholar
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. 1103–1113 (2010).Google Scholar
18.Lawrence, C. T. and Tits, A. L., “A Computationally Efficient Feasible Sequential Quadratic Programming Algorithm,” Society for Industrial and Applied Mathematics, 11, pp. 1092–1118 (2001).Google Scholar
19.Graham, J., Andrew, M. and Shields, D. H., “Stress Characteristics for Shallow Footing in Cohesionless Slopes,” Canadian Geotechnical Journal, 25, pp. 238–249 (1988).Google Scholar
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. 715–720 (1977).Google Scholar
21.Baker, R. and Garber, M., “Theoretical Analysis of the Stability of Slopes,” Geotechnique, 28, pp. 395–341 (1978).Google Scholar
22.Baker, R., “Determination of the Critical Slip Surface in Slope Stability Computations,” International Journal of Numerical and Analytical Methods in Geomechanics, 4, pp. 333–359 (1980).Google Scholar
23.Baker, R., “Tensile Strength, Tension Cracks, and Stability of Slopes,” Soils and Foundation, 21, pp. 1–17 (1981).Google Scholar
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. 3717–3735 (2003).Google Scholar
25.Baker, R., “Variational Slope Stability Analysis of Materials with Non-Linear Failure Criterion,” Electronic Journal of Geotechnical Engineering, 10, Bundle A (2005).Google Scholar
26.Chen, Z. and Morgenstern, N. R., “Extensions to Generalized Method of Slices for Stability Analysis,” Canadian Geotechnical Journal, 20, pp. 104–109 (1983).Google Scholar
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. 137–150, (2007).Google Scholar
28.Cheng, Y. M., Lansivaara, T. and Siu, J., “Impact of Convergence on Slope Stability Analysis and Design,” Computers and Geotechnics, 35, pp. 105–115 (2008).Google Scholar
29.Spencer, E., “A Method of Analysis of the Stability of Embankments Assuming Parallel Inter-Slice Forces,” Geotechnique, 17, pp. 11–26 (1967).Google Scholar
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. 187–195 (2005).Google Scholar