Cyclic Loading Using Intelligent Finite Element Method

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Analysis of Behaviour of Soils Under CyclicLoading Using EPR-based Finite Element

Method

ARTICLE in FINITE ELEMENTS IN ANALYSIS AND DESIGN · OCTOBER 2012

Impact Factor: 2.02 · DOI: 10.1016/j.finel.2012.04.005

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3 AUTHORS:

Akbar Javadi

University of Exeter

102 PUBLICATIONS 696 CITATIONS

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Asaad Faramarzi

University of Birmingham


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Alireza Ahangar-Asr

University of Salford


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Available from: Asaad Faramarzi

Retrieved on: 31 March 2016

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Analysis of behaviour of soils under cyclic loading using EPR-based finite

element method

Akbar A. Javadi a,n, Asaad Faramarzi b,1, Alireza Ahangar-Asr a,2

a Computational Geomechanics Group, College of Engineering Mathematics and Physical Sciences, University of Exeter, Harrison Building, North Park Road, Exeter EX4 4QF, UK b Department of Civil Engineering, School of Engineering, University of Greenwich, Central Avenue, Chatham Maritime, Kent ME4 4TB, UK

a r t i c l e i n f o

Article history:

Received 9 April 2011Received in revised form

17 April 2012

Accepted 18 April 2012Available online 16 May 2012

Keywords:

Finite element

Evolutionary computation

Material modelling

Cyclic loading

EPR

a b s t r a c t

In this paper, a new approach is presented for modelling of behaviour of soils in finite element analysis

under cyclic loading. This involves development of a unified approach to modelling of complexmaterials using evolutionary polynomial regression (EPR) and its implementation in the finite element

method. EPR is a data mining technique that generates a clear and structured representation of

the system being studied. The main advantage of an EPR-based constitutive model (EPRCM) over

conventional models is that it provides the optimum structure and parameters of the material model

directly from raw experimental (or field) data. The development and validation of the method will be

presented followed by the application to study of behaviour of soils under cyclic loading. The results of

the analyses will be compared with those obtained from standard finite element analysis using

conventional constitutive models. It will be shown that the EPR-based models offer an effective

and unified approach to modelling of materials with complex behaviour in finite element analysis of

boundary value problems.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Finite element method has, in recent decades, been widely

used as a powerful tool in the analysis of engineering problems. In

this numerical analysis, the behaviour of the actual material is

approximated with that of an idealised material that deforms

in accordance with some constitutive relationships. Therefore, the

choice of an appropriate constitutive model that adequately

describes the behaviour of the material plays an important role

in the accuracy and reliability of the numerical predictions. During

the past few decades several constitutive models have been

developed for various materials including soils. Among these

models there are simple elastic models [1], plastic models (e.g.,

[2]), models based on critical state theory [3], and single or double

hardening models [4,5], etc. Most of these models involve deter-mination of material parameters, many of which have little or no

physical meaning [6]. In conventional constitutive material mod-

elling, an appropriate mathematical model is initially selected and

the parameters of the model (material parameters) are identified

from appropriate physical tests on representative samples to

capture the material behaviour. When these constitutive models

are used in finite element analysis, the accuracy with which theselected material model represents the various aspects of the

actual material behaviour and also the accuracy of the identified

material parameters affect the accuracy of the finite element

predictions.

In the past few years, the use of artificial neural networks

(ANN) has been introduced as an alternative approach to con-

stitutive modelling of materials. The application of ANN for

modelling of the behaviour of concrete was first proposed by

Ghaboussi et al. 1991 [7]. Ghaboussi and Sidarta [8] presented an

improved technique of ANN approximation for learning the

mechanical behaviour of drained and undrained sand. Ghaboussi

et al. 1998 and Sidarta and Ghaboussi [9,10] presented a new

autoprogressive approach to train ANN constitutive model

(autoprogressive ANN). In this approach initially, a finite elementmodel of an available experimental test (with the measured

boundary forces and displacements) is created using a pre-trained

ANN model as the constitutive material model. Then the mea-

sured forces and displacements data are applied incrementally to

the FE model and through the increments the ANN model is

updated with more data. The data for training ANN come from the

stresses and strains at gauss points of all elements. In this method

the main idea is that the FE model of the experimental tests

usually contain a large number of stresses and strains with a wide

range of different values that can be used for training of the ANN

model. Hashash et al. [11] continued and extended the autopro-

gressive training methodology in a new framework, self learning

Contents lists available at SciVerse ScienceDirect

journal homepage: www .elsevier.com/locate/finel

Finite Elements in Analysis and Design

0168-874X/$ - see front matter & 2012 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.finel.2012.04.005

n Corresponding author. Tel.: þ44 1392 723640; fax: þ44 1392 217965.

E-mail addresses: [email protected] (A.A. Javadi),

[email protected] (A. Faramarzi), [email protected] (A. Ahangar-Asr).1 Tel.: þ44 1634 883126.2 Tel.: þ44 1392 723909.

Finite Elements in Analysis and Design 58 (2012) 53–65

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simulation (SelfSim), to extract soil constitutive behaviour from a

sequence of construction stages of a braced excavation using

measurements of lateral wall deflection and surface settlements.

The role of autoprogressive and self-learning simulation was also

studied by other researchers [12–19]. These works indicated that

neural network based constitutive models can capture nonlinear

material behaviour with a high accuracy. The developed ANN

models are versatile and have the capacity to continuously learn

as additional material response data become available.On the subject of studying material behaviour under cyclic and

hysteric loading using ANNs, a few works have been reported so

far. Furukawa and Hoffman [20] proposed an approach to mate-

rial modelling using neural networks, which can describe mono-

tonic and cyclic plastic deformation and its implementation in a

FEA system. They developed two neural networks, each of which

was used separately to represent the back stress and the drag

stress. After training and validation stages of neural networks

(NNs), the neural network constitutive models (NNCMs) were

implemented in FEA to update the stiffness matrix ( D). In this

approach D is made of an elastic matrix De, and a plastic matrix

Dp. In the proposed approach the elastic matrix was derived from

Young’s modulus and Poisson’s ratio and only the plastic matrix

was updated using the developed ANNs.

Tsai and Hashash [21] showed the application of SelfSim

method in dynamic soil behaviour. They described the imple-

mentation of SelfSim to integrate data from field measurements

and numerical simulations of seismic site response to obtain the

underlying cyclic soil response. They applied the SelfSim to study

1D seismic site response.

Yun et al. [22] introduced an approach for ANN-based model-

ling of the cyclic behaviour of materials. They focused on the issue

in the hysteric behaviour of material where one strain value may

correspond to multiple stresses and this can be a major reason

that stops NNs from learning hysteretic and cyclic behaviour. To

overcome this issue, they introduced two new internal variables

in addition to the other ordinary inputs of ANN-based constitutive

material models to help the learning of the hysteretic and cyclic

behaviour of materials. The ANN models trained in this way were

implemented in a FE model to analyse a boundary value problem

(steel beam–column connection) under cyclic loading.

Yun et al. [23] extended the ANN-based cyclic material model

developed by [22] to beam–column connections by adding the

mechanical and design parameters as inputs of the ANN model.

Moreover Yun et al. [24] used self-learning simulation to char-

acterize cyclic behaviour of beam–column connections in steel

frames.

Ghaboussi et al. [25] developed a hybrid modelling framework

to analyse engineering systems. The hybrid method combines the

mathematical models of engineered systems (derived based on

physics and mechanical laws) with artificial neural network

models using autoprogressive and self-learning simulation. They

applied this hybrid method to model and analyse a steel jointunder cyclic load.

The implementation of the ANN based constitutive models in FE

codes has been the interest of many researchers. Shin and Pande

[26] used a self learning code to identify elastic constants for

orthotropic materials from a structural test. Hashash et al. [27]

described some of the issues related to the numerical implementa-

tion of NNCM in finite element analysis and derived a closed-form

solution for material stiffness matrix for the neural network-based

constitutive model. Jung and Ghaboussi 2006 [28] presented a rate

dependant ANN material model for creep behaviour of concrete

and its implementation in finite element software (ABAQUS),

through its user material subroutine (UMAT). Kessler et al. [29]

demonstrated the implementation of an ANN material model in

ABAQUS, through user subroutine VUMAT. Haj-Ali and Kim [30]

presented a neural network based constitutive model for fibre

reinforced polymeric (FRP) composites. The developed ANN model

was implemented in ABAQUS user material subroutine to analyse

a notched composite plate with an open hole. Yun et al. [22,23]

also implemented the developed ANN models for materials under

cyclic load in ABAQUS.

The authors have also carried out extensive research on applica-

tion of neural networks in constitutive modelling of complex

materials in general and soils in particular. They have developedan intelligent finite element method (NeuroFE code) based on the

incorporation of a back propagation neural network (BPNN) in finite

element analysis (e.g., [31,32]). The intelligent finite element model

has been applied to a wide range of boundary value problems

including cyclic loading and has shown that ANNs can be very

efficient in learning and generalising the constitutive behaviour of

complex materials [33].

In this paper a fundamentally different approach is presented

for constitutive modelling using Evolutionary Polynomial Regres-

sion (EPR). In the proposed EPR approach the optimum structure

for the material constitutive model representation and its para-

meters are determined directly from raw data. Furthermore, it

provides a transparent and structured representation of the

constitutive relationships that can be readily incorporated in a

finite element code. Javadi and Rezania [34] and Javadi et al. [35]

presented the application of the EPR-based constitutive models in

material modelling under monotonic loading conditions. How-

ever, this paper focuses on the application of the EPR-based

constitutive models, to the simulation of behaviour of soils under

cyclic loading and its integration in a FE model. The development

and validation of the EPRCM and its integration in FEA are

presented and the efficiency of the methodology is examined by

application to the complex problem of cyclic loading of soils. It is

shown that the proposed methodology can simulate the real

behaviour of complex materials under cyclic loading with very

high accuracy. The main advantages of using an EPR approach are

highlighted.

In what follows, the main principles of EPR will be outlined. The

application of EPR in modelling of nonlinear constitutive relation-

ships and the implementation of developed EPRCMs in FE analysis

will be illustrated with two examples. An EPR will be trained with

data from results of a synthetic triaxial cyclic loading tests. The

trained EPR will then be incorporated into a finite element model

which will in turn be used to analyse the behaviour of the soil

under cyclic loading. The training and generalisation capabilities of

the EPR in extending the learning to cases of multiple, variable and

irregular cycles will be investigated.

2. Evolutionary polynomial regression

Evolutionary polynomial regression (EPR) is a data-driven

method based on evolutionary computing, aimed to search forpolynomial structures representing a system. A general EPR

expression can be presented as [36]

y ¼Xn j ¼ 1

F ð X , f ð X Þ,a jÞþa0 ð1Þ

where y is the estimated vector of output of the process; a j is a

constant; F is a function constructed by the process; X is the

matrix of input variables; f is a function defined by the user;

n is the number of terms of the target expression. The general

functional structure represented by F is constructed from ele-

mentary functions by EPR using a genetic algorithm (GA) strategy.

The GA is employed to select the useful input vectors from X to be

combined. The building blocks (elements) of the structure of F can

be defined by the user based on understanding of the physical

A.A. Javadi et al. / Finite Elements in Analysis and Design 58 (2012) 53 –6554

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processes. While the selection of feasible structures to be com-

bined is done through an evolutionary process the parameters a jare estimated by the least square method.

EPR is a technique for data-driven modelling. In this technique,

the combination of the genetic algorithm to find feasible struc-

tures and the least square method to find the appropriate

constants for those structures implies some advantages. In parti-

cular, the GA allows a global exploration of the error surface

relevant to specifically defined objective functions. By using suchobjective functions some criteria can be set in order to (i) avoid

overfitting of models, (ii) push the models towards simpler

structures, and (iii) avoid unnecessary terms representative of

the noise in data [36]. Selecting an appropriate objective function,

assuming pre-selected elements in Eq. (1) based on engineering

judgment, and working with dimensional information enable

refinement of final models. Application and capability of EPR in

modelling and analysing different civil and geotechnical engineer-

ing problems have been investigated by the authors, [37–39].

Detailed explanation of the method is out of the scope of this

paper and can be found in [36,40].

3. Application of EPR for modelling of material behaviour

In modelling using EPR, the raw experimental or in-situ test

data are directly used for training the EPR model. In this approach,

there are no mathematical models to select and as the EPR learns

the constitutive relationships directly from the raw data it is the

shortest route from experimental research to numerical model-

ling. In this approach there are no material parameters to be

identified and as more data become available, the material model

can be improved by re-training of the EPR using the additional

data. Furthermore, the incorporation of an EPR in a finite element

procedure avoids the need for complex yield/failure functions,

flow rules, etc. An EPR equation can be incorporated in a finite

element code/procedure in the same way as a conventional

constitutive model. It can be incorporated either as incremental

or total stress–strain strategy [35]. In this study the incrementalstrategy has been successfully implemented in the EPR-based

finite element model.

3.1. Input and output parameters

The choice of input and output quantities is determined by both

the source of the data and the way the trained EPR model is to be

used. A typical scheme to train most of the neural network based

material models includes an input set providing the network with

the information relating to the current state units (e.g., current

stresses and current strains) and then a forward pass through the

neural network yields the prediction of the next expected state of

stress and/or strain relevant to an input strain or stress increment [9].

The same idea has been utilised in this work. Thus depending on theproblem and the available data, typically the mean stress p0i,

deviatoric stress qi, volumetric strain eiv and shear strain eiq are used

as the input parameters representing the current state of stress and

strain in a load increment i, and the devatoric stress qiþ1and/or

volumetric strain eiþ1v corresponding to the input incrementaldeviatoric strain Deiq are used as the output parameters.

The database is divided into two separate sets. One set is used

for training to obtain the EPR model and the other one is used for

validation to appraise the applicability of the trained model.

3.2. EPR procedure and developed EPRCMs

Before starting the training procedure, a number of constraints

can be implemented to control the evolutionary process in terms

of length of equations, type of functions used, number of terms,

range of exponents, number of generations etc. Therefore, there is

a potential to achieve different models for a particular problem

which enables the user to gain additional information for differ-

ent scenarios [40]. Applying the EPR procedure, the evolutionary

constitutive material modelling starts from a constant mean of

output values. By increasing the number of evolutions it gradually

picks up the different participating parameters in order to form

equations representing the constitutive relationship. Each pro-posed model is trained using the training data and tested using

the testing data. The level of accuracy at each stage is evaluated

based on the coefficient of determination (COD), i.e., the fitness

function as

COD¼ 1

PN ðY aY pÞ

2PN ðY að1=N Þ

PN Y aÞ

2 ð2Þ

where Y a is the actual target value; Y p is the EPR predicted value;

N is the number of data points on which the COD is computed. If

the model fitness is not acceptable or the other termination

criteria (in terms of maximum number of generations and max-

imum number of terms) are not satisfied, the current model

should go through another evolution in order to obtain a new

model [40,34].

4. Incorporating of EPRCM in FEA

The developed EPRCMs are implemented in the widely used

general-purpose finite element code ABAQUS through its user

defined material module (UMAT). UMAT updates the stresses and

provide the material Jacobian matrix for every increment in every

integration point [41]. The manner, in which EPRCM is incorpo-

rated in UMAT is shown in Fig. 1. In the developed methodology,

the EPRCM replaces the role of a conventional constitutive model.

The source of knowledge for EPR is a set of raw experimental (or

in situ) data representing the mechanical response of the material

to applied load. When EPR is used for constitutive description, the

physical nature of the input–output data for the EPR is deter-mined by the measured quantities, e.g., stresses, strains, etc.

The constitutive relationships are generally given in the

following form [42]

Dr¼DDe ð3Þ

where D is material stiffness matrix, also known as the Jacobian.

For an isotropic and elastic material, matrix D is given in terms of

Young’s modulus, E , and Poisson’s ratio, n.Therefore the function of an EPR-based constitutive model in a

FE model (at every element’s integration point) is as follows:

(i) For the iþ1th load increment, the input pattern for the

EPRCM contains (1) the values of ( p0i,

qi,

eiv,

eiq) which have

already been calculated in the previous load increment and

(2) an arbitrary value of Deiq. The new value of qiþ1 and e iþ1v

are then calculated for the next step.

(ii) For each load increment the material Young’s modulus, E EPR ,

and the Poisson’s ratio, nEPR can be calculated from therelationship between the relevant stresses and strains. For

example for axisymetric condition:

E EPR ¼ Dqi

Dei1ð4Þ

nEPR ¼ 1DeivDei1

!=2 ð5Þ

where e1 is the axial strain.

A.A. Javadi et al. / Finite Elements in Analysis and Design 58 (2012) 53–65 55

https://www.researchgate.net/publication/235694738_A_symbolic_data-driven_technique_based_on_evolutionary_polynomial_regression?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/235694738_A_symbolic_data-driven_technique_based_on_evolutionary_polynomial_regression?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/233807667_An_evolutionary_based_approach_for_assessment_of_earthquake-induced_soil_liquefaction_and_lateral_displacement?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/233807667_An_evolutionary_based_approach_for_assessment_of_earthquake-induced_soil_liquefaction_and_lateral_displacement?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/233807667_An_evolutionary_based_approach_for_assessment_of_earthquake-induced_soil_liquefaction_and_lateral_displacement?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/233807667_An_evolutionary_based_approach_for_assessment_of_earthquake-induced_soil_liquefaction_and_lateral_displacement?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/235694738_A_symbolic_data-driven_technique_based_on_evolutionary_polynomial_regression?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/235694738_A_symbolic_data-driven_technique_based_on_evolutionary_polynomial_regression?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/235694738_A_symbolic_data-driven_technique_based_on_evolutionary_polynomial_regression?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/235694738_A_symbolic_data-driven_technique_based_on_evolutionary_polynomial_regression?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/260125327_An_Artificial_Intelligence_Based_Finite_Element_Method?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/260125327_An_Artificial_Intelligence_Based_Finite_Element_Method?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/227815375_Autoprogressive_training_of_neural_network_constitutive_models?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/227815375_Autoprogressive_training_of_neural_network_constitutive_models?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/229888248_An_investigation_on_stream_temperature_analysis_based_on_evolutionary_computing?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/229888248_An_investigation_on_stream_temperature_analysis_based_on_evolutionary_computing?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/229888248_An_investigation_on_stream_temperature_analysis_based_on_evolutionary_computing?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/229888248_An_investigation_on_stream_temperature_analysis_based_on_evolutionary_computing?el=1_x_8&enrichId=rgreq-4f930a6b-c500-42da-b5fc-0551da8f478e&enrichSource=Y292ZXJQYWdlOzI0MDAzMjc4MjtBUzo5NzUyMDMyMzY2MTgyOEAxNDAwMjYyMDY1MjE2https://www.researchgate.net/publication/229888248_An_investigat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5/14

Therefore with D being recalculated in the FE procedure, the

element stiffness matrix for every single element is updated at

each load increment as

Z O

BT DBdO ð6Þ

where B is the strain matrix and O is the elemental area.

Consequently the global stiffness matrix for a particular problem

is refined for each load increment. The whole procedure ensures

that the constitutive model follows the actual behaviour of

the material, both at the element level and at the global level.

This is primarily because EPRCM avoids the errors associated

with idealisation of the material behaviour and captures and

accurately represents (as will be shown in the examples) the

constitutive behaviour of the material directly from data.

5. Numerical examples

To illustrate the developed computational methodology,

three numerical examples of application of the developed

EPR-based finite element method to engineering problems are

presented. In the first example, the application of the methodol-

ogy to a simple case of linear elastic material behaviour is

examined. In the second example, the method is applied to a

boundary value problem involving the analysis of stresses and

strains around a tunnel considering nonlinear and elasto-plastic

Start

Input Data (Geometry,

Applied Load, Initial and

Boundary Condition)

Increase the Applied Load

Incrementally

Convergence

I t er a t i onL o o p

Output Result

L o a d I n c r em en t

L o o p

STOP

Start

Input Data (Experimental

Data, Physical Insight)

Genetic

Algorithm

Mathematical

Structure

Least Square

Symbolic

Function

Fitness

Check based on fitness criteria

and or generation number

YESNO

NO

YES

Whole load

applied?

YES

NO

EPR

Constitutive

equation

EPR FEA

Current state of

stresses and strains

1- Next state of

Stresses

2- Jacobian Matrix

EPRCM(s)

Solve the Main

Equation

UMAT

Fig. 1. The incorporation of EPR-based material model in ABAQUS finite element software.



6/14

material behaviour. In the third example, the application of

the method to the analysis of the behaviour of soil under cyclic

loading is presented.

5.1. Example 1

This example involves a thick circular cylinder conforming to

plane strain conditions. Fig. 2 shows the geometric dimensions

and the element discretization employed in the solution where 12

eight-node isoparametric elements have been used. The cylinder

is made of linear elastic material with a Young’s modulus of

E ¼2.1105 N/mm2 and a Poisson’s ratio of 0.3 [42]. This example

was deliberately kept simple in order to verify the computationalmethodology by comparing the results with those of a linear

elastic finite element model. The compressibility of the material

is assumed to be negligible and hence the EPR model for

volumetric strain is not considered in this example. The loading

case considered involves an internal pressure of 8.0104 kPa as

shown in Fig. 2.

Fig. 3a shows a linear elastic stress–strain relationship with a

gradient of 2.1105 MPa. The slope of this line represents the

elastic modulus, E , for the material. The data from this figure were

used to train the EPR model in order to capture the linear stress–

strain relationship for the material.

After training, the selected EPR model to represent the stress–

strain:

qiþ1 ¼1:5 108

Deq3:4662 106

ffiffiffiffiqi

q þ2:4231 1011Deq

þ2:4231 1011eq0:00504eq

ffiffiffiffiqi

q þ0:01197 ð7Þ

Fig. 3(b) shows the stress–strain relationship predicted by the

EPR model, together with the original one. It is seen that after

training, the EPR model has successfully captured the stress–

strain relationship with a precise accuracy.

100 mm

200 mm

P

Fig. 2. FE Mesh in symmetric quadrant of a thick cylinder.

0

50

100

150

200

250

0

Strain

S t r e s s ( M P a )

0

50

100

150

200

250

0 0.001

Strain

S t r e s s ( M P a )

0.0002 0.0004 0.0006 0.0008 0.001

0.0002 0.0004 0.0006 0.0008

Fig. 3. (a) Linear stress–strain relationship used for training and (b) the results of

EPR predictions for stress–strain values.

0

40

80

120

160

200

0

radius (mm)

R a d i a l S t r e s s ( M P

a )

Standard FEM

IFEM (EPRCM)

0

0.02

0.04

0.06

0.08

radius (mm)

R a d i a l D i s p l a c e m e n t ( m m )

Standard FEM

IFE (EPRCM)

40 80 120 160 200

0 40 80 120 160 200

Fig. 4. Comparison of the results of the EPR-FEM and standard FEM solution in

terms of (a) radial stress and (b) radial displacement.


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7/14

The EPR-based finite element model incorporating the trained

EPR model was used to analyse the behaviour of the cylinder

under applied internal pressure. The results are compared with

those obtained using a standard linear elastic finite element

method. Fig. 4 shows the radial displacements and radial stresses

along a radius of the cylinder, predicted by the two methods.

Comparison of the results shows that the results obtained using

the EPR-based FEM are in excellent agreement with those

attained from the standard finite element analysis. This showsthe potential of the developed EPR-based finite element method

in deriving constitutive relationships from raw data using EPR

and using these relationships to solve boundary value problems.

5.2. Example 2

This example involves the analysis of deformations around a

tunnel subjected to excavation and gravity loadings. The geometry of

the tunnel and the finite element mesh are shown in Fig. 5. The finite

element mesh includes 142 eight-node isoparametric elements and

451 nodes. The depth of the tunnel crown from the ground surface is

12 m. The analysis is done in two steps. The first step includes a

geostatic analysis where all the elements are subjected to gravity

loading. In the second step 46 elements representing the tunnelelements, are removed to simulate the excavation process.

The results from a series of drained triaxial tests from

literature [43], containing information on both shear and volu-

metric behaviour, were used for the training of the EPR based

constitutive model with an incremental stress–strain (tangential

stiffness) strategy. It was assumed that the soil tested was

representative of the soil material around the tunnel. The test

data were arranged as shown in Table 1 and used to train an

EPRCM to model the stress–strain relationship for the soil.

The results from five tests conducted at confining pressures

of 50, 100, 300, 400, 600 kPa were used for training of the

EPR models while those for the sixth and seventh tests at the

confining pressure of 200, 500 kPa were used for validation of the trained EPR models. At the end of the training and testing

procedure, the selected best EPR models representing the beha-

viour of the soil are:

qiþ1 ¼7:3384 107 p03ev

qi

5:077eivqiDeq

þ0:416 p0eq

qi

0:47537eqþ0:90403qi0:09154qiDeq

þ0:00011719ðqiÞ2þ0:06081 p0þ0:12364 p0Deq

0:0025518 p0eqDeq0:00006795 p02þ4:4456 ð8aÞ

eiþ1v ¼ 1:0005eivþ0:0010163q

iDeqþ0:000016185q

ieqDeq

0:001178 p0

Deqþ0:0017108 ð8bÞFig. 5. Geometry of the tunnel and the FE mesh.

Table 1

Input and output parameters used for training the

EPR constitutive model for the tunnel example.

Input parameters Output parameter

p0i , qi , eiv , eiq , De

iq q

iþ1

p0i , qi , eiv , eiq , De

iq e

iþ1v

0

100

200

300

400

500

600

700

800

D e v i a t o r i c S t r e s s ( k P a )

Deviatoric Strain %

0

100

200

300

400

500

600

700

0


Deviatoric Strain %

5 10 15 20 25 30

0 5 10 15 20 25 30

Fig. 6. (a) Results of training of the ANN and (b) stress–strain relationship

predicted by the trained EPR.


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Figs. 6(a) and 7(a) shows the stress–strain curves predicted by

Eqs. (8a) and (8b) against those expected and used as training

data. It is clearly seen that, the EPR was able to capture the

constitutive (nonlinear) stress–strain relationship for the soil with

very good accuracy. The generalisation capability of the EPRCMs

is shown in Fig. 6(b) and Fig. 7(b). The data from the test

conducted at the confining pressures of 200 and 500 kPa (which

did not form a part of the training data) were used to test the

trained EPRCMs. The predicted output values of the EPR modelsare compared with the experimentally measured values in

Fig. 6(b) and Fig. 7(b). It is seen that the generalisation capability

of the trained EPRCM is excellent.

In addition, the EPR models (Eqs. (8a) and (8b)) are used to

predict the entire stress paths, incrementally, point by point, in

the q : eq and ev : eq spaces. This is used to evaluate the capabilityof the incremental EPR models to predict the behaviour of the soil

during the entire stress paths. Fig. 8 illustrates the procedure

followed for updating of the input parameters and building the

entire stress path for a shearing stage of a triaxial test. At the start

of the shearing stage in a conventional triaxial experiment, the

values of all parameters are known. For example in a test on a

sample of a saturated soil, the values of effective mean stress, p0i,

deviator stress qi

, deviatoric strain ei

q and volumetric strain, ei

vareknown from values of applied cell pressure, pore water pressure

and volume change at the end of the previous stage (e.g., at the start

of shearing stage eiq ¼ 0, eiv ¼ 0 and q

i ¼ 0). Then, for a given

increment of deviatoric strain, Deq, the values of qiþ1 and eiþ1v arecalculated from the EPR models (Eqs. (8a) and (8b), respectively). For

the next increment, the values of p 0i, e iq, eiv q

i and are updated as

qi ¼ qiþ1 ð9Þ

eiv ¼ eiþ1v ð10Þ

p0i ¼ p0iþ qiþ1qi

3

ð11Þ

eiq ¼ eiqþDeq ð12Þ

and the next points on the curves are predicted using the EPR

models. The incremental procedure is continued until all the points

on the curves are predicted. Fig. 9(a) and (b) show the comparison

between two complete curves predicted using the EPR models

following the above incremental procedure and the experimental

results. The predicted results are in good agreement with the

experimental results and the facts that (i) the entire curves have

been predicted point by point; (ii) the errors of prediction of the

individual points are accumulated in this process, and still the EPR

-7

-6

-5

-4

-3

-2

-1

0

1

2

V o l u m e t r i c

S t r a i n %

Deviatoric Strain %

-6

-5

-4

-3

-2

-1

00

V o l u m e t r i c S t r a i n %

Deviatoric Strain %

EPR Prediction (200 kPa)

EPR Prediction (500 kPa)

Actual Data (200 kPa)

Actual Data (500 kPa)

5 10 15 20 25 30

0 5 10 15 20 25 30

Fig. 7. (a) Results of training of the EPR and (b) volumetric strain predicted by the

trained EPR.

Fig. 8. procedure followed for updating of the input parameters and building the

entire stress path for a shearing stage of a triaxial test.

0

100

200

300

400

500

600

700

800

0


Deviatoric Strain (%)

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

0

V o l u m e t r i c S t r a i n ( % )

Deviatoric Strain (%)

5 10 15 20 25 30

5 10 15 20 25 30

Fig. 9. Comparison of EPR incremental simulation with the actual data

(a) simulation of deviatoric stress and (b) simulation of volumetric strain.


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models are able to predict the complete stress paths with a good

degree of accuracy are testaments to the robustness of the

developed EPR framework for modelling of soils.

These figures show that the EPR has been able to capture the

general trend of the nonlinear relationship of stress and strain

with a good accuracy. It also shows that the EPR model was

trained sufficiently to adequately model the stress–strain beha-

viour of the soil. The trained EPRCMs were incorporated in the

EPR-based finite element (EPR-FEM) using UMAT in ABAQUS. TheFE incorporating the EPR models was then used to simulate the

behaviour of the tunnel under gravity and excavation loadings.

For the conventional finite element analyses, the results of the

triaxial tests were used to derive the material parameters for the

Modified Cam Clay (MCC) and Mohr–Coulomb (MC) models for

the soil (see Table 2).

Fig. 10 shows the comparison between the displacements in the

tunnel predicted using standard finite element analyses using the

MCC and MC models as well as the EPR-based finite element

method where the raw data from the triaxial tests were directly

used in deriving the EPR-based constitutive model. Deformations of

tunnel is magnified by a factor of 5. The patterns of deformation are

similar in all three analyses. Despite the relatively small difference

between the results from the different analyses, it can be argued

that the EPR-based FE results are more reliable, as this method used

the original raw experimental data to learn the constitutive

relationships for the material and it did not assume a priori any

particular constitutive relationships, yield conditions, etc.

From the results obtained, it is shown that the developed

intelligent finite element method is also capable of capturing

more complex constitutive relationships of materials and can

offer very realistic prediction of the behaviour of structures.

5.3. Example 3: behaviour of soil under cyclic loading

In this example, the behaviour of a soil is studied in triaxial

tests under cyclic axial loading. The test data for this example

were generated by numerical simulation of triaxial experiments.

In general, generating data by numerical simulation has advan-

tages including: (i) it is more economic (ii) it is far less time

demanding, (iii) it can simulate loading paths and test conditions

that cannot be easily achieved in physical testing due to physical

constraints of the testing equipment. The data for training and

validation of the EPR were created by finite element simulation of

triaxial cyclic loading tests at constant cell pressures using the

Modified Cam Clay model. The material parameters assumed for

the soil are:

l¼0.174 (slope of the virgin consolidation line),

k¼0.026 (slope of the unloading/reloading lines in the v:Lnp0

space),

M ¼1 (slope of the critical state line in the q: p0 space),

P 1¼100.0 kPa (isotropic preconsolidation pressure),

Table 2

Material parameters for Modified Cam Clay and Mohr–Coulomb models.

C 0 (kPa) f0 ðdeg:Þ M k e0 n g (kN/m3) l

11.7 21 0.8 0.00715 0.921 0.3 17 0.091

Fig. 10. Comparison of the results of the intelligent FEA and conventional FE

analyses using Mohr–Coulomb and Duncan–Chang models.

0

50

100

150

200

250

0.00

Strain

D e v i a t o r S t r e s s ( k P

a )

100 kPa

150 kPa

200 kPa250 kPa

300 kPa

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Axial Strain

V o l u m e t r i c S t r a i n

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

Fig. 11. Typical cyclic loading test data used for training and validation of EPR.

(a) Shear stress and (b) volumetric strain.


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The generated data were used to train and test EPRCMs. The

EPRCMs were then incorporated in the intelligent EPR-based

finite element model to represent the soil behaviour under cyclic

loading. The results of the EPR-based finite element analyses were

compared with those attained using conventional finite element

method. The performance of the model was evaluated for two

separate cases of loading where the soil was subjected to:

1- Multiple and regular cycles of loading and unloading; and2- Multiple and irregular cycles of loading and unloading.

The data generated by numerical simulation of the cyclic

loading tests at confining pressures of 100, 150, 200, 250 and

300 kPa are shown in Fig. 11(a) and Fig. 11(b). In order to

introduce a level of noise that inevitably exists in real triaxial

test data, numerical simulation for each confining pressure was

repeated by changing the total number of load increments in

the simulation and the obtained data were combined and used

in training of the EPR models. Fig. 12 shows typical results of

the tests conducted at confining pressure of 150 kPa with four

different load increments.

The data from the tests at confining pressures of 100, 150, 200and 300 kPa were used for the training of the two EPR models.

The first model was developed to predict the deviator stress qiþ1

and the second one to predict the volumetric strain eiþ1v . Thetrained EPR models were validated using the data from the test at

the confining pressure of 250 kPa. The input and output para-

meters used for training of the EPR models are presented in

Table 3. In the table p0 is net mean stress, q is deviator stress, ev isvolumetric strain and e1 is axial strain. The indices i and iþ1represent the state of stress or strain at current (incremental)

step, and the next step, respectively.

The selected EPR models for q and ev are:

qiþ1 ¼0:0084ðqiÞ3

p02eiv

2667:247ðqiÞ2ðDe1Þ2

p0e1

0:060714ðqiÞ3De1 p0eiv

þ1:8866evDe1

qi

1:9676qiDe1eiv

þ888:4ðqiÞ2ðDe1Þ

3

eiv

þ104:4964 p0De11:4 105 p02þ0:018826 p02qieivDe1

þ1:0525qi0:71525 ð13aÞ

eiþ1v ¼ 0:02369qiDe1

p0e1

0:4217qiDe1 p0

þ9:3 106e1

qi

þ0:45727De1þ0:99eivþ0:000041535 ð13bÞ

Fig. 13(a) and (b) show the results predicted by Eqs. (13a)

and (13b), respectively, for the training data set together with the

actual training data. Fig. 13(a) shows the results of the devitoric

stress model for different confining pressures. Typical results of

the volumetric strain model (Eq. (13b)) at confining pressure of

100 kPa are presented in Fig. 13(b) together with the actual data.

It is seen from the figures that the EPR models were capable of

learning, with very good accuracy, the constitutive relationship of

the soil under cyclic loading paths. The trained EPRCMs were

validated using a data set corresponding to the confining pressureof 250 kPa. The results of the validation tests are shown in

Fig. 14(a) and (b). It is shown that the trained EPR models were

able to generalise the training to loading cases that were not

introduced to the EPR during training. Moreover the incremental

prediction capability (described in Example 2) of the developed

0

10

20

30

40

50

0.000

Strain

D e v i a t o r S t r e s s ( k P a )

4 increments

8 increments20 increments40 increments

0.002 0.004 0.006 0.008 0.010 0.012 0.014

Fig. 12. Typical cyclic loading test results with different load increments at

confining pressure of 150 kPa.

Table 3

Input and output parameters used for training and

testing of the EPR models for shear stress and

volumetric strain cyclic models.

Input parameters Output parameter

p0i, qi , eiv, ei1 , De

i1 q

iþ1

p0i, qi , eiv, ei1 , De

i1 e

iþ1v

0

50

100

150

200

250

300

D e v i a t o r i c S

t r e s s ( k P a )

Axial Strain

Actual Data (100 kPa) Actual Data (150 kPa)

Actual Data (200 kPa) Actual Data (300 kPa)

EPR prediction (100 kPa) EPR prediction (150 kPa)

EPR prediction (200 kPa) EPR prediction (300 kPa)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.00

V o l u m e t r i c S t r a i n

Axial Strain

Actual Dat

Cyclic Loading Using Intelligent Finite Element Method

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