2008_Application of Optimization and Other Evolutionary Techniques in Geotechnical Engineering

8/3/2019 2008_Application of Optimization and Other Evolutionary Techniques in Geotechnical Engineering

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The 12th

International Conference ofInternational Association for Computer Methods and Advances in Geomechanics (IACMAG)1-6 October, 2008Goa, India

Applicationof Optimizationandother Evolutionary Techniques in

Geotechnical Engineering

Prabir K. BasudharDepartment of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur - 208016

Keywords: Optimization, Evolutionary Techniques, Artificial Neural Networks, Nonlinear Programming

ABSTRACT: The paper describes methods that are available for optimal design and analysis of geotechnicalengineering problems. An overview of the different application areas has been presented giving a brief account ofthe available literature on the subject highlighting the studies undertaken by the author and his students. Twotypical example problems have been presented to demonstrate successful application of non-linear programmingtechniques in solving such problems. Finally, general observation has been made with regard to the scope anddevelopment of these methods.

1. Introduction

1.1 Geotechnical Problems and Optimization

Most problems in soil and foundation engineering involve analysis for stability. Thus, foundations, retaining wallsand slopes are designed for safety against failure. In analyzing these problems one obtains, either the maximumload the structure can transmit to the soil or the minimum of the ratio of the stabilizing and activating forces andmoments. In many design problems, it is necessary to not only check the safety against shear failure but alsoensure that the structure is safe from settlement consideration. Another aspect is the economy; it is theresponsibility of the engineer in charge to provide the most economic and efficient design. Thus, almost all practicalproblems lead to either maximization or minimization of a function subjected to certain behavior or side constraintsor both. The problems are of such magnitude and complexity as to require the most systematic and rationalapproach of solution.

The differential calculus provides classical methods for determining maxima and minima of functions of realvariables. These methods are applicable to decision problems in which the objective function is at least twicedifferentiable with in the feasible region, and the constraints are either nonexistent or consists of equations only. Ifthere are no constraints, optimization consists nothing more than the direct differentiation of the objective function.This leads to a set of simultaneous equations that are in general nonlinear. Solution of such a system iscomplicated and analytically involved. When equality constraints are present, constraint equations may be solvedfirst for pertinent variables, which are substituted into the objective function before the differentiation is performed,or use Lagrange multipliers in the optimization scheme. When the objective function and the constraints arenonlinear, the above approach will generally involve some nonlinear equations.Many contemporary problems of design and analysis in geotechnical engineering involve not only equalityconstraints but also inequalities. Mathematical problems that arose in tackling these problems stretched the limits ofconventional analyses, and required new methods for their successful treatment. The classical techniques ofcalculus and calculus of variation are occasionally valuable in these new areas but are limited in their range andversatility. The recognition of this fact led to the development of a number of novel mathematical programmingtechniques concerned with the solution of optimization problems. It involves finding a set of decision variables thatoptimize the objective function and satisfy a set of predefined design restrictions. The advent of digital computer

and their use in analysis and design problems led to a remarkable and increased use of automated design andanalysis requiring the development of more and more sophisticated algorithms to minimize the wastage ofcomputer time. As with the use of any analytical or numerical techniques with in the context of complex problemsolving, the focus for discussion falls not only on the various techniques available for the analysis but also on the artof how such mathematical procedures are applied. Large-scale systems may pose considerable problems in termsof the number of decision variables and objectives. These issues must be acknowledged and addressed in astraightforward manner with proper attention paid to the particular important aspects of the problem at hand. Noone procedure or series of procedures would be the panacea that can solve all problems to the last details.Optimization and other evolutionary methods are useful methods for design and analysis, but their successfulapplication depends largely on how it is used and how the results are interpreted. However, it is to be noted thateven though optimization is an elegant tool, its use is not as much as it is expected. This is because thegeotechnical engineering designers are in general conservative considering the complexity of soil behavior.

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Nevertheless, there is a scope and need to adapt to the optimization techniques as a philosophy of design.Optimization is essentially the art, science and mathematics of choosing the best solution out of severalalternatives. With in the limitation of space and brevity it is not possible to discuss elaborately all the methods thatare available in this context.

With the above in view, an attempt has been made in this paper to present an overview of different approaches likeoptimization, artificial neural network and other evolutionary techniques that are available and then briefly describethe scope of these techniques to analyze different problems in Geotechnical engineering leaving out considerabledetails.

1.2 Mathematical Programming Problem

Optimization is the act of obtaining the best result under given circumstances satisfying all limitation and restrictionplaced on it. In the analysis or design of any engineering system, the ultimate goal is to minimize the effort requiredor maximize the desired benefit. In such situation, this can be viewed as a function of certain decision variables,and the optimization can be considered as the process of finding the conditions that gives the maximum orminimum value of the function.All analysis and design problems of optimization can be stated in a standard form as follows:

Findm

D such that MinimumDfm )( (1)

Such that,

max

( ) 0, 1,2,...., ( )

( ) 0, ( 1),( 2),...., ( )

( ) ( ) ( )

j m

j m

mim m

g D j m a

l D j m m p b

D D D c

= = + +

(2)

)(),(),(, DlDgDfD jj are the design or decision vector, objective function, inequality constraints and equality

constraints respectively.maxmin )(,)( DD are the minimum and maximum bound on the design vector.

A maximization problem can be converted to one of minimization by multiplying the function by -1. The constraintsrepresented by the equations 2(c) can also be written in the form given by the equations 2(a). Therefore, theproblem is one of optimization with equality and inequality constraints.The number of elements n of the design vector and p the sum of the equality and inequality constraints areunrelated. Thus p could be greater than, equal or less than n in a given problem. If there are no constraints, theproblem is one of unconstrained minimization. The type of mathematical programming problems that are of anyconcern to geotechnical engineering can be classified depending on the nature of the objective function and

constraints, as linear, nonlinear programming problems. Another class of optimization problems frequentlyencountered in geotechnical engineering belongs to dynamic programming developed by Bellman (1967) thatdecomposes a multistage decision problem as a sequence of single stage decision problems, which are solvedsuccessively.

1.3 Optimization techniques

In most of these methods, optimum solution is approached starting from an initial guess. Then using some pre-specified rule determined based on localized information, the search direction is chosen and one-dimensionalminimization is carried out to find the best solution along that direction. Starting from the solution thus found, thefinal optimal value is obtained through an iterative process.Optimization methods can be classified in to two groups depending on how the search direction is found: Direct andgradient-based techniques (Fox, 1971; Rao, 1984). Direct methods use only the information regarding the objectivefunction and the constraints unlike the gradient-based techniques that use the first and/or the second derivative ofthe objective function and the constraint functions as well. If the gradients can be explicitly found, the gradient-

based techniques generally would converge faster than the direct methods. However, these are not efficient if thefunctions are not easily differentiable or are discontinuous. In such situations, direct methods work better. However,all these techniques have some associated difficulties. These are:1. The obtained optimal solution is starting point dependent.2. Most of the algorithms converge to the local minimum.3. The efficiencies of the algorithms are generally problem dependent. An algorithm that is found to be efficient for

a problem may not be as efficient in tackling other problems.4. The algorithms are not suitable for discrete variables and highly nonlinear problems with too many constraints.Out of the various direct and indirect methods of solving constrained optimization problems, an indirect method thathas been used by many investigators is the penalty function method; here the constrained problem is converted toan unconstrained one developing a composite function by appending the constraints with the objective functionterm. Thereafter, sequential unconstrained minimization of the composite function is carried out.

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Wide use of the method is not because of its simplicity but due to the availability in literature of several efficientalgorithms for unconstrained minimization. Penalty function methods are of two types namely exterior and interiorpenalty function methods. In exterior penalty function method, the iterative process can be started from an initialinfeasible design vector in contrast to the interior penalty function method that requires an initial feasible design

vector to start the solution. Designers generally prefer the interior penalty function method as it always results in asafe design. In exterior penalty function method, the optimal solution is approached from an infeasible region.Therefore, the final optimal solution may not satisfy strictly the design constraints. In many practical problems withtoo many constraints, it is sometime impossible to find an initial feasible design vector to start with. Under suchcircumstances, the problem may be solved either by using exterior penalty function method or obtaining an initialfeasible design vector by picking up the violated constraints and satisfying each of them one by one using interiorpenalty function method itself following (Fox;1971). This method of finding an acceptable design is very timeconsuming. Even when an initial feasible solution is available, due to the long step nature of the unconstrainedoptimization algorithms, the path may be diverted into the infeasible region. In such situations, the function isgenerally set to an arbitrarily high value and the minimization procedure is left to correct the situation on its own.The composite function that is developed is of the form:

1

( , ) ( ) [ ( )]n

k k j j

j

D r f D r G g D=

= + (3)

Where,j

G is a function of the constraintsj

g . kr is a positive constant known as the penalty parameter or the

response factor. The second term in the right hand side of the equation 3 is called the penalty term. If theunconstrained minimization of the composite function is repeated for a sequence of the penalty parameter, thesolution may be brought to converge to that of the original problem stated in equation 1. This is why the penaltyfunction methods are also known as Sequential Unconstrained Minimization Technique (SUMT).As already stated out of the exterior and interior penalty function methods, interior penalty function method ispreferred as it always results in a safe design strictly satisfying all the constraints. In the absence of any equalityconstraints, the most commonly used form of the composite function as suggested by Fiacco and McCormick (Fox,1971) is:

=

=m

J j

kkDg

rDfrD1 )(

1)(),( (4)

where )(Df is minimized over all D satisfying .....,2,1,0)( mjDg j = If the penalty parameter kr is

positive, it adds a positive penalty to the objective function )(Df as all ,the terms in the sum is negative at theminterior points. If the design vector is near any boundary of a constraint surface, then the function

)(Dg j representing the particular constraint tends to zero and the penalty term tend to infinity. Using a reduction

factor c, the penalty parameterk

r is made successively smaller in order to obtain the constraint minimum of

)(Df .Thus,kk

crr =+1 .When both equality and inequality constraints are present, Fiacco and McCormick (Fox, 1971) reported somesuccess with the formulation:

2

1/ 21 1

1 1( , ) ( ) ( )

( )

pm

k k j

j j mj

D r F D r l Dg D r

= = +

= + (5)

The function ),( krD is minimized for a decreasing sequence of the penalty parameter kr . As kr is made small

the second term allows the minimum to approach the constraints from the inside and the third term successivelyforces the satisfaction of the equality constraints. It has been already pointed out that use of interior penalty functionneeds an initial feasible design vector for starting the solution. In a complex analysis and design problem, it is notalways possible to find initially feasible design vector. A procedure to tackle such situation has been brieflymentioned earlier. However, an alternative approach (extended penalty function method) proposed by Kavlie (1971)has been proved to be very useful. In this approach, the constrained problem is transformed to an unconstrainedone as follows:

=

+=p

j

jkkDgGrDfrD

1

)]([)(),( (6)

Where, the function G is chosen as follows:

)(

1)]([

DgDgG

j

j = for )(Dg j (7a)

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And.2

[2 ( )]( )

j

j

g DG D

= for >)(Dg j (7b)

where,tk

r = andt

is a constant defining the transition between the two types of penalty terms.

The progress in optimization in the last 50 years is enormous. The credit for this goes to the strong foundation laidby mathematical giants like Cauchy, Euler, Gauss, Lagrange, Newton and others. Many innovative and powerfulalgorithms like method of conjugate direction (Powells method), conjugate gradient method (Fletcher and Reevesmethod) based on Gram-Schmidt orthogonalization and Variable metric methods (Davidon-Fletcher-Powell method)have been developed for unconstrained minimization (Fox, 1971; Rao, 1984).

2. Recent Progress in Optimization Methods: Evolutionary Algorithms

Most advances in the development and application of classical optimization methods has taken place over the last50 years and is still going on strong. However, since the eighties, unorthodox, intriguing yet less mathematicalprocedures have been suggested. Development and application of such non-traditional procedures also known asevolutionary algorithms (EA) are gaining ground in different disciplines.EA mimics the natural evolutionary principles on randomly picked solutions from the search space and iterativelyprogress towards the optimal solution. In the search space both good and bad solutions of a given problem co-existand EA uses the natures ruthless selective advantage to fittest individuals and creation of new and fit individualsusing re-combinative and mutative genetic processing. Thus, EA avoids the bad solutions in the search space,taking clue from the good ones and reaches eventually close to the best possible solution. Out of several EAs somebased on principles of recombination, some emphasizing mutation operation and some using a niching strategywhile some other using mating restriction strategy, Genetic Algorithm (GA) is most popular. The methods can beobtained from Deb (1995). GA generally begins its search from a random population of solutions. If the solutiondoes not satisfy the required acceptance criterion, three different operators- reproduction, crossover, and mutationare applied to update the population of solutions. Single iteration of the three operators is known as generation inGAs. As the representation of a solution in a GA is similar to that of natural chromosome and GA operators aresimilar to genetic operators, such procedures are called genetic algorithms.

3. Artificial Neural Network (ANN)

Artificial neural networks (ANNs) process information the way biological nervous system, such as brain, for optimaldecision making in contrast to the GAs that use an analogy of chromosome encoding and natural selection toevolve a good optimized solution. Learning capability of neural networks is useful in approximating any continuous

function to a desired accuracy by an approximating network architecture establishing a relationship betweencauses and effects. Information is passed from one neuron to all other neurons connected to it. Inmathematical model, neurons are represented as processing elements or nodes, input paths are defined asinterconnections, the process combining the signals and generating the output neurons is modeled through atransfer function, synaptic strength of each connection is represented as weight and the change in synaptic strengthis defined as learning. Each input is multiplied by a corresponding weight analogous to synaptic strength, and allthe weighted inputs are then summed to determine the activation of the neuron. Figure 1a shows a mathematicalmodel that implements this idea. Despite the diversity of network paradigms, nearly all are based on this

configuration. Here a set of inputs labelednxxx ,....,, 21 (called input vectors) is applied to the artificial neuron.

Each input (signal) is multiplied by an associated weightn

www ,...,2,1 (called weight vector) before it is applied to

the summation bloc ( ). This weighted sum of the inputs is then transmitted to another neuron via a transferfunction or activation function. A typical transfer function is a sigmoid transfer function which is an S-shaped,continuous and nonlinear curve. The sigmoid transfer function modulates the weighted sum of the inputs so that the

output approaches unity when the input gets larger than the threshold value ( i ) of that particular neuron andapproaches zero when the input gets smaller.There are different types of ANN configurations. One of the most popular is the Feed Forward Neural Network(FFNN) consisting of input layer, hidden layer and output layer (Figure 1b). The hidden layer enables thesenetworks to represent and compute complicated association between inputs and outputs. FFNNs have beenapplied successfully to model complex physical phenomenon by training them in a supervised manner with apopular and efficient algorithm known as back propagation algorithm (BPA). During training and testing severalissues like number of hidden layers and nodes in them, initial network weights, learning rate etc need to be carefullyconsidered.

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Figure 1a: Typical Neural Network architecture Figure 1b: Artificial model of a neuron and sigmoid function

4. Applications in Geotechnical Engineering: An overview

Even though optimization is an elegant mathematical tool and had been used quite early by the researchers in

other disciplines like aerospace and structural engineering, geotechnical engineers took notice of the techniquesquite belatedly. Since then due to the efforts of various investigators, a vast amount of literature is available on thesubject. It is not possible to cover all these literatures in a single paper. Therefore, many works have been omitteddue to lack of space and brevity. However, an effort has been made here to provide a cursory glance of the samecovering the development.Serious attempts to apply these techniques were made only in the early seventies. Calculus of variation has beenused (Ramamurthy et al., 1977, Baker and Garber, 1977) in locating the critical slip surface and the correspondingminimum factor of safety for homogeneous slopes. However, these methods were not very useful in analyzing non-homogeneous slopes and zoned dams. Even these attempts were hindered due to lack of computational facilitieswith severe constraints in terms of memory and speed of computation. In spite of all these odds, more and moreresearchers took up challenging problems cutting across different braches of engineering and science.Theoretically the task of optimization for a given problem is to find the most or the best suitable solution. Thus inmathematical optimization a great deal of effort is spent on trying to describe the properties of such an idealsolution. Researchers from Geotechnical Engineering also took initiative in this direction. Realizing the potential ofthese methods in geotechnical engineering, they started developing programs to solve nonlinear optimizationproblems subjected to both equality and inequality constraints in order to analyze stability problems like earth

pressure, bearing capacity and slope stability problems, as standard all purpose optimization programs were notfreely available at that time. Earlier either graphical, classical methods based on calculus or very primitive numericaltechniques like grid search techniques were used for solving such problems. Later on many advanced searchtechniques like methods of conjugate directions, conjugate gradient and variable metric methods for unconstrainedminimization were developed. From the initial early attempts to date, the art of application of these techniques havebeen extended to different areas such as slope stability, bearing capacity, earth pressure, parameter estimation,optimal design of foundations. From the initial hesitation of using anything other than the classical approach oftackling such problems now a days the potential for using these non-classical approaches is well recognized andapart from non-linear and dynamic programming, application of genetic and other evolutionary algorithms andartificial neural network are gaining grounds in geotechnical engineering.Using the fundamental ideas from geometry and calculus to reach an optimum in an iterative manner, many

efficient algorithms have been developed. Researches in these areas are continued strongly for over fifty years;adoption of these techniques is being pursued vigorously not only in geotechnical engineering but in other branchesof engineering as well. Starting from the middle of eighties unorthodox and less mathematical yet intriguingoptimization procedures have been suggested to find approximate solution that is as similar to the ideal solution as

possible. Though ideal solution is desirable, the restriction of the availability of computing power and time, veryoften the practicing engineers are happy with the approximate solutions. Out of all nontraditional approaches,Evolutionary Algorithm (EA) is used extensively. EA mimics the natural evolutionary principles on randomly pickedsolutions from the search space of the problem and iteratively progresses towards the optimum point.The available literatures on the subject undertaken and their scope are so vast that it is extremely difficult to do justice and treat all the topics with equal emphasis and consideration. Here more emphasis is given to theapplication of non-linear programming techniques to such problems.

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In the early phase because of the limitation of memory and speed of computation, very simple problems used to betaken up. Little and Price (1958) presented a computerized approach of slope stability analysis using Bishopsadaptation of Swedish method. With an elementary beginning in 1960 (Horn, 1960) application of optimizationtechniques in solving geotechnical engineering problems made a modest progress through out the seventies.

Lysmer (1970) solved bearing capacity and earth pressure problems by converting the nonlinear programmingproblem to a linear one by approximating the nonlinear constraints by a series of straight lines and applied linearprogramming techniques. Potchman and Kolesnichenko (1972) used dynamic programming to bearing capacityproblems. Krugman and Krizek (1973), Chen (1975) solved unconstrained slope stability problems with only 2 to 3variables and used pattern search technique for direction finding as proposed by Hookes and Jeeves (1961) andPowells algorithm (1964) respectively. Basudhar (1976), Basudhar et al (1978, 1979 a, b) used nonlinearprogramming techniques to bearing capacity, slope stability and passive earth pressure problems.Use of such techniques in geotechnical engineering increased significantly there after especially in analyzing

stability problems (Lysmer, 1970; Baker,1980; Basudhar et al.1980, 1981; Martin, 1982; Munro, 1982; Arai andTagyo, 1965 a, b; Nguyen, 1984; Yudhbir et al., 1987; De Natale and Gillet, 1988; Sloan, 1988, 1989; Bhowmik andBasudhar, 1989; Sing and Basudhar,1992a,1992b,1993a,1993b, 1995; Basudhar and Singh, 1994; Sabahit et al,1994, 1996, 1997). Efforts have also been made to compare the efficiency of these techniques (De Natale andGillett, 1988).Though the power of linear (Lysmer, 1970; Sloan, 1988, 1989) and nonlinear optimization (Basudhar, 1976)techniques was well recognized, by the late nineties their inadequacy in tackling problems under special situationswas being increasingly appreciated. Thus, need for applying new evolutionary methods, such as, simulated

annealing and genetic algorithm was felt. These methods are used in other branches of engineering and science. Incivil engineering problems application of ANN in water resource engineering (Mayer and Dandy, 2000) isphenomenal and there are various applications of GA in transportation engineering and structural engineeringproblems as well. In the early nineties geotechnical engineers ventured out and looked for application of newtechniques like artificial neural networks in modeling complex geotechnical engineering problems like liquefactionanalysis (Goh, 1994), soil constitutive relations (Ghaboussi et al.1998), pile foundation analysis (Goh, 1995a,1995b), parameter estimation from pile load test data (Puttaraju,1999). Even though ANN is being used in variousgeotechnical engineering problems since the last decade, its application is mostly confined to soil liquefaction andrelated studies, pile foundation and soil constitutive relations with limited application in prediction of soil properties(Shahin et al. 2001; Shahin, 2002, 2004). Recently Das (2005) undertook the following studies under thesupervision of the author:i. Rock failure criteria: Model Parameter Estimation using various Statistical and Optimization Algorithms e.g.

GA and presented a Comparative Study.ii. Application of Artificial neural Networks to some Geotechnical Engineering Problems like prediction of

conductivity of clay liners, prediction of co-efficient of lateral earth pressure at rest, prediction of residualfriction angle of clays, prediction of lateral load capacity of piles.

iii. Characterization of Alluvial soil site utilizing unsupervised learning and fuzzy clustering for soil stratificationwith piezocone data.

The importance and potential of ANN and GA in solving geotechnical problems have been acknowledged by havingan exclusive session in the conference on Numerical Models in Geomechanics-NUMOG X. These are on:i. Constitutive modeling (Drakos et al., 2007; Javedi et al., 2007; Pedroso and Farius, 2007)ii. Site investigation (Shin et al., 2007; Samui and Sitaram, 2007)iii. Determination of Embankment Safety (Koelewijn, 2007)iv. Soil Parameter identification (Levasseur et al., 2007)

5. Typical Example Problems: Nonlinear Programming

Two typical example problems (the first one is an analysis problem and the second one is a design problem) arepresented here in brief. The details are available in literature (Basudhar and Singh, 1994; Basudhar, Lakshman andDey, 2006).

5.1 Uplift Capacity of Anchors ( Basudhar and Singh, 1994)

This is a problem involving the application of non-linear programming in conjunction with finite element technique toestimate the optimal lower bound solution of stability problems in geotechnical engineering; it is presented verybriefly as follows.

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In geotechnical engineering, very often the stability of foundation structures and slopes needs to be evaluated.There are many reliable methods based on limit equilibrium or finite elements for the same. However, the abovemethods give upper bound solutions. However, lower bound solution is more important from engineering viewpointas it satisfies equilibrium condition at all points without violating the yield condition constraints at any point with in

the soil medium and thus results in a safe design. Literature on lower bound formulation of stability problems is veryscanty because it is very difficult to construct statically admissible stress field. Extending the Lysmer method, ageneralized procedure based on finite elements and non-linear programming (for isolating the optimal solution) wasdeveloped at I.I.T. Kanpur for finding lower bound solutions of stability problems. The general steps of the methodare:i) The soil medium under consideration is divided into a mesh of finite number of triangular elements. The

triangular mesh is defined by nodal points, co-ordinates and indexing systems for nodal points, elements andelement sides. of the nodal points

ii) The material properties and the body forces of each element are prescribed.iii) Equations that satisfy element equilibrium are generated satisfying the equilibrium equations of continuum

mechanics.iv) Conditions of continuity at the element interfaces are satisfied.v) Boundary conditions are specified.vi) Yield conditions for each element are formulated.vii) The objective function is identified.viii) Optimal solution is isolated by formulating the problem as one of non-linear programming.

From the interface equilibrium and boundary conditions, a set of linear equations in terms of the stress variables areobtained. The numbers of such equations are less than the number of unknowns. With the help of these equations,some of the principal unknowns are expressed in terms of the remaining variables. Thus, we can reduce the totalnumber of principal unknowns.Figures 2 and 3 show typical mesh pattern for shallow and deep horizontal and vertical anchors whose break outcapacity is to be determined.

Fig. 2: Mesh patterns shallow and deep horizontal Fig 3: Forces and nodal stresses

anchors

Extended penalty function method capable of handling infeasible decision vector is used to isolate the optimalbreakout factors. Figure.4 shows excellent convergence of the objective function with decreasing value of penaltyparameter and increasing values of number of function evaluation.

Fig.2 Typical forces on an element (left) and continuity ofnodal stresses (right).

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Fig. 4: Convergence studies . Fig. 5: Comparison of results: Horizontal anchors

A comparative study of the predictions using the present approach and experiments conducted by variousinvestigators is presented in Figures 5 and 6 for horizontal and vertical anchors respectively to demonstrate itssuccessful application.

5.2 Optimum Design of Cantilever Retaining walls (Basudhar et al., 2006)

Fig.7 shows the three dimensional view of a cantilever retaining wall. Given the loading, geotechnical and structuralproperties with freely draining horizontal backfill, the problem is to determine the minimum dimensions of the wallsatisfying all the design restrictions. The cost of the structure is directly proportional to the volume of differentquantities like volume of excavation, filling, concrete and steel. Thus, the cost function developed by summing up allthese values is optimized.

Fig. 6: Comparative study of results Fig. 7 Cantilever retaining wall with design variables

The angle of internal friction of the backfill is equal to 30, the unit weight of backfill soil is equal to 18 kN/m3. The

backfill is assumed to have a horizontal top surface with good drainage facility and it is filled with coarse-grainedsoil.

The design variables that control the cost of the retaining wall are base width )( 1d , width of the toe )( 2d , vertical

stem thickness )( 3d , base thickness )( 4d , minimum depth of embedment )( 5d , diameter of reinforcing rod

)( 6d and stem top width )( 7d . They are collected in a design vectorT

D , where,

TTdddddddD ),,,,,,( 7654321=

The total cost of the raft foundation is considered as the objective function for the analysis to be carried out.

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fillfillsteelsteelconcconcexcexc

fillsteelconcexc

VRVRVRVR

CCCCF

+++=

+++=(8)

where, F=Total cost of the raft, =fillsteelconcexec CCCC &,, Cost of excavation, concrete, steel, and filling of theremaining excavated earth volume, =fillsteelconcexec RRRR &,, Rates of respective items &

=fillsteelconcexec VVVV &,, Volumes of excavation, concrete, steel and filling respectively which depends upon the

design variables.

The geotechnical aspects that control the structural design and safety of the structure are stability and settlementconsiderations. The constraints that are imposed on the design, arising out of these considerations, can becategorized as behavior constraints. Such considerations also arise from structural design considerations. Someside constraints arise out of geometric restrictions imposed on the design from code provisions and practice.Behavior constraints such as the maximum base pressure (which occurs at the toe of the wall) must not exceed theallowable bearing capacity of the soil. In addition, the base pressure should not be tensile in nature as soil is weakin tension, i.e. the pressure developed should be compressive throughout the length of the base. proper factor ofsafety should be imposed against sliding and overturning of the wall; the settlements should be within specifiedlimits and that the maximum overturning moment should be less than the resisting moment. Figure 8 shows theefficacy of the implemented numerical scheme in isolating the optimal solution; it is observed that beyond 4iterations convergence is achieved. Figure 9 shows the variation of the objective and composite function withpenalty parameter indicating that for decreasing sequence of the penalty parameter the solution converges to theminimum.

Fig. 8 Variation of composite and objective function Fig. 9 Variation of composite and objective function withwith number of iterations penalty parameter

6. RESEARCH AT I.I.T.KANPUR

The studies using these techniques that have been carried out by the author and his students, as a part of Ph. Dand M. Tech theses over the years are listed here. It provides an idea about the type of application areas inGeotechnical engineering. Some of the resulting publications are given in the reference even though they have notbeen cited in the main text. The theses that are listed below are not included in the reference list as their source isobvious.

6.1 Ph.D Thesis

1. Some application of mathematical programming problems in geotechnical engineering, 1976.P.K.Basudhar

2. Sequential Unconstraint minimization technique in slope stability analysis, 1990. G.Bhattacharyya3. Lower-bound solutions of some stability problems in geotechnical engineering, 1992. D.N.Singh4. Stability analysis of soil reinforcement problems: A nonlinear programming approach, 1994, N.Sabahit5. Sequential unconstrained minimization technique to the optimum design of slopes with or without nails,

1998, C.R.Patra.6. Parametric estimation from pile load tests: Using optimization and Neural network, 1999, Puttaraju7. Application of Genetic Algorithm and Artificial Neural Network to Some Geotechnical Engineering

Problems, 2005, Sarat Kr. Das.

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6.2 M.Tech Thesis

1. Settlement controlled optimum design of shallow footings,1984, R.Madan Mohan2. Sequential unconstrained minimization in slope stability analysis, 1984., S.K.Bhowmik3. Optimization techniques in stability analysis of zoned dams, 1986., N.Satyambabu4. Parametric studies in slope stability using SUMSTAB package, 1986., R.K.Dhawan 5. Parametric estimation from pile load test data: An inverse formulation, 1987. , S.K.Garg6. Optimum design of shallow circular footing, 1988., A.S.Mandal7. Reanalysis of two failed embankments built on soft clays using SSOPT, PCSTABLE5 and SUMSTAB II:

A comparative study, 1993., S.K.Prasad8. A comparative study of lower bound bearing capacity solutions, 1993. , S. Srivastava9. Lower bound bearing capacity of surface strip footings on reinforced foundation beds, 1995.,

P.V.Ramakrishna10. Some studies towards generalization of mesh pattern in lower bound solutions of bearing capacity

problems, 1996., S.Roy11. Three dimensional stability analysis of nailed soil slopes, 1997., M.R.Lakshminarayana12. Optimal cost design of rigid raft foundation, 1998., Anuradha Das13. Optimal cost design and drafting of cantilever and counter fort retaining walls, 1998., B.Lakshman

7. GENERAL COMMENTS AND OBSERVATIONS

With a modest beginning in the early sixties, application of Optimization techniques and Artificial Neural Network intackling geotechnical engineering problems has increased phenomenally especially with the advent of high speeddigital computers and increase in their memory. With the availability of excellent personal computers, more andmore researchers are now applying these techniques in efficient and economic design of foundations and earthstructures. In the initial phase non-linear problems were used to be converted to linear problems by treating thenonlinear equations to be piecewise linear so that the problems could be solved by linear programming. However,in practical problems most of the design constraints are in general in the inequality form. Therefore many slackvariables had to be introduced to convert those to equalities and there by increase the number of variables needingmore memory in the computer Treating very large number of linear equations is generally difficult and lot of effortswere therefore given to solve such problems developing new algorithms. With the development of algorithms totackle nonlinear equations in an optimization frame, efforts were initiated to solve these problems by using non-linear programming techniques. These efforts has been found to be fruitful in solving many geotechnicalengineering problems like optimal design of retaining structures, and shallow foundations, automated slope stability

analysis with and without reinforcement, reinforced earth walls, lower bound limit analysis of anchors, shallowfootings and parameter estimation etc. Adoption of extended penalty function method and carrying out sequentialunconstrained minimization of the composite function is very handy in isolating the optimal solution especially whenthe initial design vector chosen to start the solution is infeasible. If the objective function and the constraints are notexplicitly available and one cannot adopt direct differentiation of the functions, it is better to use non-gradienttechniques like method of conjugate direction (Powells technique) for finding the search direction and use quadraticsearch for linear minimization. However, one needs to start from different starting point to ensure global optimalityof the solution. The method has been found to be very effective in handling both equality and inequality constrainsand infeasible initial design points. Dynamic programming has been successfully applied to solve bearing capacityand especially the slope stability problems. However, this approach did not gain popularity as its applicabilitybecomes increasingly difficult with the increase in the number of variables. Most of the developments in thisdirection took place in the seventies and eighties. There after efforts were made to use evolutionary techniques likeGenetic algorithm to analyze slope stability problems and ensure global optimality of the solution. It has beenreported that GA gives results better or comparable to those obtained from other NLP approaches. It should benoted that the cross over and mutation operation plays an important role in the solution and it may be extremelydifficult to implement the same to cases where there is more number of variables. GA has been introduced to

geotechnical engineering in the early nineties and applied broadly in solving two types of problems namelyparameter estimation and un-reinforced and reinforced slope stability analysis. Side by side, efforts were made toapply artificial neural network in solving problems with noisy data and for cases where analytical solution is notavailable (impossible). It has been applied to various types of geotechnical engineering problem with success.Modeling with ANN has an edge over other types of system modeling.

Thus, it is observed that that these techniques play a great role in finding solutions to problems, which was notthought of earlier.Though considerable progress has been made in applying these techniques to different geotechnical engineeringproblems, there exists considerable scope to apply all these techniques for economic & better design and modelingof physical phenomenon.

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