ACME2014

Proceedings of the 22nd

UK National

Conference of the Association for

Computational Mechanics in Engineering

2nd

- 4th

April 2014

College of Engineering, Mathematics and Physical Sciences,

University of Exeter, UK

Edited by :

Akbar Javadi

Mohammed S. Hussain

First published 2014 by

The University of Exeter

Electronic Version Only

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library

ISBN 978-0-902746-30-5

Copyright © 2014 the University of Exeter. All rights reserved.

No parts of this publication may be produced, stored in a retrieval system, or

transmitted in any form or by any means, electronic, mechanical, photocopying,

recording or otherwise, without the prior written permission of the University of

Exeter, Exeter, UK.

ACME2014

Proceedings of the 22nd

UK National

Conference of the Association for

Computational Mechanics in Engineering

2nd

- 4th April 2014

College of Engineering, Mathematics and Physical Sciences,

University of Exeter, UK

FULL PAPERS

Edited by:

Akbar Javadi

Mohammed S. Hussain

ISBN 978-0-902746-30-5

University of Exeter, UK

PREFACE

The UK Association for Computational Mechanics in Engineering (ACME) was founded with the

aim of promoting research in computational mechanics and various engineering applications and

establishing formal links and with similar organisations in Europe and the International

Association of Computational Mechanics (IACM). One of the principal activities of ACME

involves the organisation of the annual conference. The first such conference took place at the

University College of Wales Swansea in 1993. Since then, the conferences have provided a forum

for reviewing research activities in many areas of mechanics, with an emphasis on

interdisciplinary aspects. The conferences have proved to be particularly useful events for

bringing together researchers from different disciplines, and especially for providing young

researchers with opportunities to present their work.

These conference proceedings contain more than 80 four-page papers presented at the 22nd

ACME Conference that was held in the Department of Engineering at the University of Exeter

from 3rd to 4th April 2014, following the 3rd ACME School on 2nd April, where four lectures

were delivered on the topics of Isogeometric Analysis and Automatic Meshing by renowned

academics.

On behalf of the local organising committee, I would like to thank many people who have

contributed to this conference, especially, all the authors who meticulously prepared their papers

and presented their original research. I would like to express my gratitude to the three invited

keynote speakers, Professor Hywel Thomas, Mr Adrian Gaylard and Professor Bassam Izzuddin

for their thought provoking lectures and seminal scholarly contributions. I would also like to

thank the speakers of the ACME School, Professor Sven Klinkel, Dr Ido Akkerman, Dr Robert

Simpson and Professor Philippe Young, for their excellent lectures on the very important topics

covered in the School.

I would like to acknowledge the support received from the College of Engineering, Mathematics

and Physical Sciences of the University of Exeter in the organisation of this conference. In

particular, I would like to thank Ms Karen Pope, Ms Denise Watts and Mr Chris Snow for

providing assistance and advise on financial issues and IT services. Finally, I would like to thank

the local organising committee members, with special thanks to Mr Mohammed S. Hussain for

his dedication and hard work in the day-to-day organisation of ACME2014 conference.

Akbar Javadi

Vice-Chairman of ACME 2014

INVITED PLENARY LECTURES

Prof. Hywel Thomas

Prof. Hywel Thomas is Pro Vice-Chancellor, Engagement and International,

Professor of Civil Engineering, and Founder Director of the Geoenvironmental

Research Centre (GRC) at Cardiff University. Prof. Thomas will give a lecture

on Geoenergy Research Applications.

Mr. Adrian Gaylard

Mr. Adrian Gaylard is Jaguar Land Rover's Aerodynamics Technical

Specialist. He is the Chair of the Aerodynamics Activity at the SAE and a

member of the European Car Aerodynamics Research Association. Adrian will

speak on the Application of Unsteady Aerodynamic Simulation in Automotive Engineering.

Prof. Bassam Izzuddin

Prof. Bassam Izzuddin is Head of the Computational Structural Mechanics

(CSM) Group at Imperial College London. He will give a lecture on Partitioned Modelling for Nonlinear Analysis of Large-Scale Structural Systems using High Performance Computing, presenting applications to realistic structures subject to extreme loading.

CONTENTS

NUMERICAL METHODS

A COUPLED PARTITION OF UNITY FEM-COLLOCATION BEM FOR ACOUSTIC WAVE

SCATTERING IN HETEROGENEOUS MEDIA IN TWO DIMENSIONS

Ganesh Diwan, Jon Trevelyan and Graham Coates

1

AN ENRICHED QUADTREE/OCTREE IMPLICIT BOUNDARY FINITE ELEMENT

METHOD FOR THE SIMULATION OF INCOMPRESSIBLE HYPERELASTIC MATERIALS

Jack S. Hale, Pierre Kerfriden, Juan J. Ródenas García and Stéphane P. A. Bordas

5

APPLICATION OF A DENSITY-NUMBER (D-N) FRACTAL MODELLING TO CORRELATE

MINERALISED ZONES IN KAHANG CU-MO PORPHYRY DEPOSIT

Amir Bijan Yasrebi, Andrew Wetherelt, Patrick Foster, John Coggan and Peyman Afzal

9

A Q-REFINEMENT PROCEDURE FOR RADIATIVE HEAT TRANSFER IN GLASS

M Shadi Mohamed, Mohammed Seaid, Jon Trevelyan and Omar Laghrouche 13

A REVIEW OF THE MATERIAL POINT METHOD AND ITS LINKS TO OTHER

COMPUTATIONAL METHODS

Tim Charlton, Will Coombs and Charles Augarde

17

A THREE-DIMENSIONAL IMPLEMENTATION OF THE BOUNDARY ELEMENT AND

LEVEL SET BASED STRUCTURAL OPTIMISATION

Baseer Ullah, Jon Trevelyan and Ioannis Ivrissimtzis

21

ENERGY-MOMENTUM METHOD E FOR BEAM DYNAMICS

Tien Long Nguyen, Carlo Sansour and Mohammed Hjiaj 25

MESHFREE VOLUME-AVERAGED NODAL PRESSURE METHODS FOR

INCOMPRESSIBLE ELASTICITY

Jack S. Hale, Alejandro Ortiz and Christian J. Cyron

29

NONLINEAR ANALYSIS OF MASONRY ARCHES AND BRIDGES USING MESOSCALE

PARTITIONED MODELLING

Yanyang Zhang, Lorenzo Macorini and Bassam Izzuddin

33

NUMERICAL METHODS FOR SUBSURFACE RESERVOIR SIMULATION: BOUNDARY

ALIGNED GRID GENERATION AND FLUX APPROXIMATION SCHEMES

Shahid Manzoor, Michael Edwards, Ali Dogru and Tareq Al-Shaalan

37

SUMMATION RULES FOR HIGHER ORDER QUASI-CONTINUUM METHODS

Claire Heaney, Lars Beex and Pierre Kerfriden 41

3D LEAP-FROG SCHEME FOR ELECTROMAGNETIC SCATTERING BY DIELECTRIC

MEDIA USING UNSTRUCTURED MESHES

Alex Gansen, Mohamed El Hachemi, Salim Belouettar , Oubay Hassan and Kenneth Morgan

45

WAVE MODELING USING ENRICHED FINITE ELEMENTS

Konstantinos Christodoulou, Omar Laghrouche and Shadi Mohamed 49

GEOMECHANICS

A COUPLED HYDRO-MECHANICAL ANALYSIS OF SLOPE INSTABILITY PROCESSES

IN SAN LEO (RN, ITALY)

Long Nguyen-Tuan, Margherita C. Spreafico, Maria Datcheva, Lisa Borgatti and Tom Schanz

53

AN ANISOTROPIC CONSTITUTIVE MODEL FOR CYCLIC LOADING OF SOFT CLAYS

Mohammad Rezania, Hesam Dejaloud, Mohaddeseh Mousavi Nezhad and Yaser Jafarian 57

A NUMERICAL INVESTIGATION INTO THE EFFECT OF PARTICLE FORM ON THE

STRENGTH OF GRANULAR MATERIALS

Matthew Potticary, Antonis Zervos and John Harkness

61

COUPLED DISCRETE ELEMENT AND LATTICE BOLTZMANN SCHEME FOR

SIMULATING SOIL EROSION

Min Wang, Y.T. Feng and Gyan Pande

65

DEFORMATION AND SAFETY CONTROL OF BURIED PIPELINES CROSSED BY VERY

LARGE DIAMETER SLURRY SHIELD TUNNELLING

Jing Ni, You-Liang Chen and Hui-Zi Zhao

69

DIGITAL DATA ACQUISITION FOR EFFECTIVE SLOPE MANAGEMENT

Matthew Eyre, Denise Pascoe, Patrick Foster and John Coggan 73

EFFECT OF PERMEABILITY VARIATION WITH DEPTH ON SOIL RESISTANCE TO

CAISSON PENETRATION

Moura Mehravar, Ouahid Harireche and Amir.M Alani

77

FINITE ELEMENT STUDY OF THE BEARING CAPACITY OF SHALLOW FOUNDATIONS

WITH NON-ASSOCIATED FLOW RULE

Madani Hamlaoui, Abdelbacet Oueslati and Gery De Saxcé

81

NUMERICAL MODELLING OF THE IMPACTS OF SEA LEVEL RISE ON SEAWATER

INTRUSION IN UNCONFINED COASTAL AQUIFERS

Mohammed Hussain, Akbar Javadi and Victor Robin

85

RELIABILITY ASSESSMENT OF COHESIVE VERTICAL CUT STABILITY USING

RANDOM FINITE ELEMENT METHOD

Ali Johari, Amir Gholampour and Zahra Abedini

89

SOIL LIQUEFACTION RELIABILITY ANALYSIS UNDER MONOTONIC LOADING USING

RANDOM FINITE ELEMENT METHOD

Ali Johari, Jaber Rezvani Pour and A.Reza Fazeli

93

3D-FEM NUMERICAL SIMULATION OF GROUND SURFACE DISPLACEMENT DUE TO

SHORT-DISTANCE MULTI-LINE OVERLAPPED SHIELD TUNNELLING

Lei Li and Mengxi Zhang

97

FRACTURE, FAILURE MECHANICS

A DIRECT METHOD TO EVALUATE STRESS INTENSITY FACTORS USING ENRICHED

BEM

Ibrahim Alatawi and Jon Trevelyan

101

A MULTI-SCALE STOCHASTIC FRACTURE MODELLING CALIBRATION USING

MONTE CARLO SIMULATIONS

Razvan Sencu, Zhenjun Yang and Yong Wang

105

COMPUTATIONAL MODELLING OF HYDRAULIC FRACTURE

Guoqiang Xue, Lukasz Kaczmarczyk and Chris Pearce 109

EXTRACTING STRESS INTENSITY FACTORS OF 3-D CRACKS USING DISPLACEMENT

CORRELATION METHOD APPLIED ON AN UNSTRUCTURED MESH

Morteza Nejati, Adriana Paluszny and Robert W. Zimmerman

113

FAILURE ANALYSIS OF 3D GRAPHITE BRICKS IN AN AGR CORE

Costy Kodsi, Łukasz Kaczmarczyk and Chris Pearce 117

GLOBAL ENERGY MINIMIZATION FOR MULTI-CRACK GROWTH IN LINEAR ELASTIC

FRACTURE USING THE EXTENDED FINITE ELEMENT METHOD

Danas Sutula, Stephane P. A. Bordas, Pierre Kerfriden and Alexandre Barthelemy

121

MESO-SCALE FRACTURE MODELLING OF CONCRETE WITH RANDOM AGGREGATES

AND PORES

Xiaofeng Wang, Zhenjun Yang and John R. Yates

125

MIXED-DIMENSIONAL MULTI-POINT FLUX APPROXIMATION FOR DISCRETE

FRACTURE-MATRIX SIMULATION

Raheel Ahmed, Michael Edwards and Mayur Pal

129

NUMERICAL MODELLING OF BRITTLE FRACTURE IN GEOLOGICAL MATERIALS

John Coggan, Doug Stead and Fuqiang Gao 133

THREE–DIMENSIONAL BRITTLE FRACTURE: CONFIGURATIONAL-FORCE-DRIVEN

CRACK PROPAGATION

Lukasz Kaczmarczyk and Chris Pearce

137

3D MESO-SCALE IMAGE-BASED FRACTURE MODELLING OF CONCRETE USING

COHESIVE ELEMENTS

Wenyuan Ren, Zhenjun Yang and Rajneesh Sharma

140

USING SEMI-FUZZY RMR TO DETERMINE THE REQUIRED SUPPORT SYSTEM FOR

GHESHLAGH COAL MINE

Mahdi Mahdizadeh, Navid Hosseini, Payman Afzal, Daryoush Kaveh Ahangaran and Amir Bijan

Yasrebi

144

STRUCTURES

AN EQUILIBRIUM FINITE ELEMENT MODEL FOR SHELLS WITH LARGE

DISPLACEMENTS

Edward Maunder and Bassam Izzuddin

148

ASSESSING THE REMAINING SAFE LIFE OF CONCRETE SEWER PIPES USING

STOCHASTIC FINITE ELEMENT METHOD

Asaad Faramarzi and Amir M. Alani

152

CO-ROTATIONAL FORMULATION FOR SANDWICH PLATES AND SHELLS

Yating Liang and Bassam A. Izzuddin 156

DELAY OPTIMIZATION IN REAL TIME DYNAMIC SUBSTRUCTURING TESTS ON A

CABLE-DECK SYSTEM

Maria Rosaria Marsico and David Wagg

160

DYNAMIC MODELING AND ANALYSIS FOR FLEXIBLE SPACE WEB

Qingbin Zhang, Zhiwei Feng and Tao Yang 164

KRIGING MODELS FOR COMPENSATING THERMAL RESPONSE IN MEASUREMENTS

FROM BRIDGE MONITORING

Rolands Kromanis and Prakash Kripakaran

169

PERFORMANCES OF RUBBER BEARINGS BY VARYING THE RUBBERS PROPERTIES

Maria Rosaria Marsico and Harry Norman 173

MATERIALS, ELASTICITY

A COMPUTATIONAL FRAMEWORK FOR POLYCONVEX LARGE STRAIN

ELECTROMECHANICS

Rogelio Ortigosa, Antonio J. Gil and Javier Bonet

177

ANALYSIS OF THERMAL STRESS DISTRIBUTION IN BRAKE DISC DURING SINGLE

BRAKING UNDER NON-AXISYMMETRIC LOAD

Adam Adamowicz

181

BREAKING PERIODICITY IN MATERIAL SIMULATIONS

Jan Novak, Martin Doskar and Martin Horak 185

COMPUTATIONAL MODELLING OF BRAIDED FIBRE EMBEDDED IN CONCRETE

Michael Cortis, Lukasz Kaczmarczyk and Chris Pearce 189

CONSTITUTIVE MODELING FOR CYTOSKELETAL CONTRACTILITY OF SMOOTH

MUSCLE CELLS

Tao Liu

193

FINITE STRAIN MODELLING FOR THE CURING PROCESS IN MAGNETO-

VISCOELASTICITY

Mokarram Hossain, Prashant Saxena and Paul Steinmann

197

HIGH TEMPERATURE PROTECTIVE COATINGS FOR AVIATION GAS TURBINES

Arif J. Pashayev, Adalat S. Samedov, Tural B. Usubaliyev and Rzagulu Agaverdiyev 201

HOMOGENEOUS RESPONSE OF RANDOM STRUCTURED MATERIAL CONCRETE

WITH REALISTIC MICROSTRUCTURE

Rajneesh Sharma, Wenyuan Ren and Zhenjun Yang

206

MIXED FINITE ELEMENT FORMULATIONS FOR STRAIN GRADIENT ELASTICITY IN

FENICS

Veena Phunpeng

210

MULTISCALE MODELLING OF THE TEXTILE COMPOSITE MATERIALS

Zahur Ullah, Łukasz Kaczmarczyk, Michael Cortis and Chris Pearce 214

VISCOELASTIC MODELLING OF WOOD-WATER INTERACTIONS USING FINITE

DIFFERENCE MODELLING

Euan Richardson and Karin de Borst

218

FLUID MECHANIC, COMPUTATIONAL FLUID DYNAMIC

AN HP-FEM FRAMEWORK FOR THE SIMULATION OF COUPLED MAGNETO-FLUID

EFFECTS

Darong Jin, Paul D. Ledger and Antonio J. Gil

222

AN IMMERSED STRUCTURAL POTENTIAL METHOD FRAMEWORK FOR

INCOMPRESSIBLE FLEXIBLE/RIGID/MULTI-PHASE FLOW INTERACTION

Liang Yang, Aurelio Arranz Carreño, Antonio Gil and Javier Bonet

226

COMPUTATIONAL MODELLING OF AN ESTUARY IN THE FRAME OF THE

OPTIMISATION OF TIDAL FARMS

Miriam Garcia, Gavin Tabor and Slobodan Djordjevic

230

FLUID-STRUCTURE INTERACTION SIMULATION OF PARACHUTE FINITE MASS

INFLATION

Xinglong Gao, Qingbin Zhang and Qiangang Tang

234

HIGHER RESOLUTION CELL-CENTRED FINITE-VOLUME SCHEMES ON

UNSTRUCTURED GRIDS FOR MULTI-PHASE FLOW IN A POROUS MEDIUM

Yawei Xie, Michael G. Edwards and Mayur Pal

240

MODELLING OF MICRO-SCALE DROPLETS SUBJECT TO SURFACE ACOUSTIC

WAVES

Ross Mackenzie, Lukasz Kaczmarczyk and Chris Pearce

244

MULTI-DIMENSIONAL MODELS OF THE INTERIOR OF STARS

Jane Pratt, Maxime Viallet, Isabelle Baraffe, Chris Geroux, Tom Goffrey, Mikhail Popov, Doris

Folini and Rolf Walder

248

PRESSURE GAUGE PLACEMENT DESIGN FOR WATER NETWORK HYDRAULIC

CALIBRATION

Xiongfei Xie, Mahmoud Nachabe and Bo Zeng

252

ARTIFICIAL INTELLIGENCE, DATA MINING

ANALYSIS OF THE DYNAMIC RESPONSE OF A LARGE DEEP KARST AQUIFER BY

DATA-DRIVEN MODELLING

Angelo Doglioni and Vincenzo Simeone

256

IDENTIFICATION OF MORPHOLOGICAL ANOMALIES BY 2D DISCRETE WAVELET

TRANSFORM

Angelo Doglioni and Vincenzo Simeone

260

LATERAL LOAD BEARING CAPACITY MODEL FOR PILES IN COHESIVE SOILS

Alireza Ahangar-Asr, Akbar Javadi, Ali Johari and Youliang Chen 264

MODELLING SWELLING INDUCED LATERAL PRESSURE TRANSMISSION CONTROL

ON RETAINING STRUCTURES IN EXPANSIVE SOILS

A.Ahangar-Asr, L. Fonseca1, R Carvalheira, I. Silva, S. Penha, L. Braga,A. A. Javadi and , A.

Johari

268

OPTIMISATION

LEVEL SET OPTIMISATION METHOD FOR CONTINUOUS FIBRES PATHS

Christopher Brampton and H Alicia Kim 272

OPTIMISATION OF A VENTURI MIXER DESIGN TO PROMOTE EFFICIENT

COMBUSTION IN A PRE-MIX BURNER

Jennifer Thompson, Oubay Hassan and Sam Rolland

276

RESEARCH ON POTABLE WATER NETWORK AUTOMATIC FLUSH DEVICE

OPERATION OPTIMIZATION

Xiongfei Xie, Mahmoud Nachabe and Bo Zeng

280

WEIGHT OPTIMIZATION OF LAMINATED COMPOSITE STRUCTURES USING GENETIC

ALGORITHMS

Nasser Taghizadieh and Mokhtar Jalilian

284

ISOGEOMETRIC ANALYSIS

A HYBRID IGAFEM/IGABEM FORMULATION FOR TWO-DIMENSIONAL STATIONARY

MAGNETIC AND MAGNETO-MECHANICAL FIELD PROBLEMS

Stefan May, Markus Kästner, Sebastian Müller and Volker Ulbricht

288

AN ISOGEOMETRIC MODEL FOR EARTHQUAKE FAULT RUPTURE WITH

WEAKENING FRICTION

Julien Vignollet, Stefan May and René de Borst

292

MULTIPHYSICS SIMULATIONS OF SMART STRUCTURES USING ISOGEOMETRIC

FINITE ELEMENTS

Christian Willberg and Sascha Duczek

296

3D EXTENDED ISOGEOMETRIC BOUNDARY ELEMENT METHOD (XIBEM) FOR

ACOUSTIC WAVE SCATTERING

Michael Peake, Jon Trevelyan and Graham Coates

300

THERMAL ANALYSIS

COMPUTATIONAL THERMAL CONDUCTIVITY IN POROUS MATERIALS: NUMERICAL

AND STATISTICAL APPROACHES

Ahmed El Moumen, Abdellatif Imad and Toufik Kanit

304

STOCHASTIC THERMOMECHANICAL ANALYSIS OF NUCLEAR GRAPHITE USING

PARAFEM

Jose David Arregui-Mena, Lee Margetts, Louise Lever, Graham Hall and Paul Mummery

308

TEMPERATURE DATA ANALYSIS FROM FIELD SCALE THERMAL MONITORING

Majid Sedighi, Daniel P. Bennett, Shakil A. Masum, Hywel R. Thomas, Erik Johansson and

Topias Siren/

312

CONTACT MECHANICS

FINITE ELEMENT BASED CONTACT SOLUTION FOR TRACTIVE ROLLING CONTACT

OF GRADED COTING

Reza Yazdanparast, Abbas Moradi and Mohsen Safajoy

316

NON-LINEAR MODELS OF FRICTIONAL HEATING DURING BRAKING

Piotr Grzes 320

QUASI-STATIC ANALYSIS OF SELF-CLEANING SURFACE MECHANISMS

Muhammad Osman and Roger A. Sauer 324

SPACE-TIME DISCETIZETION OF FRICTIONAL ROLLING CONTACT IN DEFORMABLE

BODIES

Reza Yazdanparast, Abbas Moradi and Mohsen Safajuy

328

Proceedings of the 22nd UK Conference of theAssociation for Computational Mechanics in Engineering

2 - 4 April 2014, University of Exeter, Exeter

A coupled Partition of Unity FEM-collocation BEM for acoustic wavescattering in heterogeneous media in two dimensions

Ganesh C. Diwan*1, Jon Trevelyan1 and Graham Coates1

1School of Engineering and Computing Sciences, Durham University, Durham, DH1 3LE

*g.c.diwan@durham.ac.uk

ABSTRACT

The Finite Element Method (FEM) is known for its versatility to handle complex geometries and het-erogeneities in the medium properties, while the Boundary Element Method (BEM) is known for itsability of exact treatment of waves propagating to infinity for wave problems. In this paper, we pro-pose a coupled formulation of partition of unity FEM (PUFEM) and partition of unity BEM (PUBEM)to solve acoustic scattering problems in a heterogeneous medium. Numerical examples are presentedto establish the convergence of the proposed method. Two of the well known non-reflecting boundaryconditions for wave problems are implemented and their performance is compared against the coupledPUFEM-PUBEM results.

Key Words: Wave scattering; Partition of unity method; Non-reflecting boundary condition; Boundaryintegral equation; Heterogeneous media

1. Introduction

Modelling of waves in a heterogeneous medium can become a difficult task for the element based meth-ods with polynomial basis as i) a heterogeneous medium requires a high resolution of mesh when thedegree of heterogeneity is high and ii) high frequency wave problems become increasingly costly duethe requirement of at least ten nodes per wavelength. Plane wave based methods have been found tooffer a clear advantage over polynomial based methods when modelling wave problems in the mediumfrequency range, see [1][2]. Cases where the wavenumber is not constant in the computational domainare of practical interest. PU based methods have been used in such cases, see [3][4]. It is known that theSommerfeld radiation condition is satisfied exactly for BEM (at least for the continuous equations) thusthe waves propagating to infinity are modelled exactly. FEM (or PUFEM) on the other hand needs the socalled non-reflecting boundary conditions (NRBCs) which are approximate and for an improved accu-racy the non-reflecting boundary needs to be kept as far as possible from the scatterer to avoid spuriousreflections. The heterogeneity, if limited to a relatively small part of the domain, can be modelled usingPUFEM. The rest of the domain, if homogeneous and unbounded, can be modelled using PUBEM. Itis therefore natural to couple PUFEM and PUBEM for modelling exterior acoustic problems involvingwave scattering in heterogeneous medium which is the focus of this paper. The rest of the paper is or-ganised as follows. Section 2 provides an overview of the governing PDE for wave scattering, its weakform for PUFEM and the boundary integral equation. Section 3 gives the partition of unity concept andcoupling of PUFEM and PUBEM through matrix equations. This is followed by numerical examples(Section 4) and conclusions (Section 5).

2. Weak form and boundary integral equation

We consider the case of a time-harmonic (e−iωt) acoustic wave scattering by a rigid sound hard obstacleΩs bounded by Γs in R2 (Fig. 1). We seek the total acoustic pressure p exterior to Γs that admits thefollowing BVP:

1

2.3 Time-stepping algorithms

in the section 2.2 we have drived the spatially discretized form of the problem while the time dimension kept continuously at all time. So at following via the timestepping procedures the total time discretization will be accomplished. So we divide the time interval T into subintervals with non-

overlapping and equal lengths as following: 1

10[ , ]

L

k kk

T t t

(18)

Where 0 0, .Lt t T and

1kk t and 1 .k kt t t for simplicity one can may use the of these

notations ( ), ( ), ( )k k k

k k ku U t U t a U t . At this process the solution satisfying equation

(16) only in a finite number of time-steps defined by tk. Thus in order to find the other solutions the totally discretized problem defines as follows:

1 1 1

1 1 1 1 1

0 0

0 0

0,1,..., 1 , ,

,

k k k

n t

k k k T k T k

n n t t

k L find a such that t T

Ma K u L B B

u U U

(19)

For the solution of this problem the direct integration algorithms have been proposed which may be

derived from Taylor’s expansion of the displacement U and the velocity U at kt With respect to

time.

References

[1] G. Hu, P.Wriggers. on adaptive finite element method of steady-state rolling contact for hyperelastic in finite deformations. Comput. Methods Appl. Mech. Engrg, 191,1333-1348, 2002.

[2] K.P. Singh, B. Poul. Numerical solution of non-Hertzian elastic contact problems. J. Appl. Mech,24,484-

490, 1974.

[3] J.J. Kaller. Three-dimensional elastic bodies in rolling contract. Kluwer Academic Pablishers,Dordrecht,

1990.

[4] J.M. Bass. Three-dimensional finite deformation rolling contact of a hyperelastic cylinder: formulation of

problem and computational results. Comput struct, 26,991-1004, 1987.

[5] J.T. Oden, T.L. Lin. On the general rolling contact problem for finite deformations of a viscoelastic cylinder, Comput. MethodsAppl. Mech. Engrg, 57,297–376, 1986.

[6] L.O. Faria, J.M. Bass, J.T. Oden, E.B. Becker. A three-dimensional rolling contact model for a reinforced rubber tire.Tire Sci.Technol, 17 (1), 217–233, 1989.

[7] R. Kennedy, J. Padovan. Finite element analysis of steady and transiently moving/rolling nonlinear viscoelastic structure–II. Shell and three-dimensional simulations, Comp. Struct, 27 (2) , 259–273, 1987.

[8] U. Nackenhorst.The ALE-formulation of bodies in rolling contact Theoretical foundations and finite element approach. Comput. Methods Appl. Mech. Engrg, 193,4299-4322, 2004.

331

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