In the framework of limit equilibrium theory, the isoperimetric model of functional extremum regarding the seismic active earth pressure is deduced according to the variational method. On this basis, Lagrange multipliers are introduced to convert the problem of seismic active earth pressure into the problem on the functional extremum of two undetermined function arguments. Based on the necessary conditions required for the existence of functional extremum, the function of the slip surface and the normal stress distribution on the slip surface is obtained, and the functional extremum problem is further converted into a function optimization problem with two undetermined Lagrange multipliers. The calculated results show that the slip surface is a plane and the seismic active earth pressure is minimal when the action point is at the lower limit position. As the action point moves upward, the slip surface becomes a logarithmic spiral and the corresponding value of seismic active earth pressure increases in a nonlinear manner. And the seismic active earth pressure is maximal at the upper limit position. The interval estimation constructed by the minimum and maximum values of seismic active earth pressure can provide a reference for the aseismic design of gravity retaining walls.

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Study on the Seismic Active Earth Pressure by Variational Limit Equilibrium Method

Equivalence between Bearing-Capacity, Lateral Earth-Pressure, and Slope-Stability Problems by Slip-Line and Extremum Limit Equilibrium Methods

The solutions of many engineering problems can be formulated as the optimized results of a functional. While many engineering problems are governed by a continuous convex optimization process, this is not the case for many geotechnical problems. Many geotechnical problems have irregular solution domains, with the objective function being nonconvex and may not be a continuous function. The presence of multiple local minima is common in many geotechnical problems, and the occurrence of local zones where there is rapid changes in the material parameters is not uncommon. The corresponding governing problems are hence usually NP-type nonconvex optimization problem, and by nature, such NP-type problems with the various constraints pose great difficulty in analysis. While the classical heuristic optimization methods may work well for some of these problems, there are also some practical cases where the classical methods may fail to perform satisfactorily. To maintain a balance between the computation time and accuracy, several hybrid metaheuristic algorithms are proposed by the author which can work well for many practical geotechnical problems. In this chapter, the author will illustrate the basic concept of hybrid metaheuristic algorithms and the applications to some difficult geotechnical problems.

Hybrid Metaheuristic Algorithms in Geotechnical Engineering

The paper is concerned with a parallel implementation of the progressive hedging algorithm (PHA) which is applicable for the solution of stochastic optimization problems. We utilized the Message Passing Interface (MPI) and the General Algebraic Modelling System (GAMS) to concurrently solve the scenario-related subproblems in parallel manner. The standalone application combining the PHA, MPI, and GAMS was programmed in C++. The created software was successfully applied to a steel production problem which is considered by means of the two-stage stochastic PDE-constrained program with a random failure. The numerical heat transfer model for the steel production was derived with the use of the control volume method and the phase changes were taken into account with the use of the effective heat capacity. Numerical experiments demonstrate that parallel computing facility has enabled a significant reduction of computational time. The quality of the stochastic solution was evaluated and discussed. The developed system seems computationally effective and sufficiently robust which makes it applicable in other applications as well.

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Two-stage stochastic programming approach to a PDE-constrained steel production problem with the moving interface

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.

S. K. Au

Papers

Solution of some engineering partial differential equations governed by the minimal of a functional by global optimization method