Lecture note in computer science

Then the finite element cretization of these operators form the given matrix and its preconditioner, that dis-is, if the given elliptic boundary value problem Lu = f is suitably discretiz

Lecture Notes in Computer Science 4818

Commenced Publication in 1973

Founding and Former Series Editors:

Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen

Ivan Lirkov Svetozar Margenov

Jerzy Wa´sniewski (Eds.)

Large-Scale

Scientific Computing

6th International Conference, LSSC 2007 Sozopol, Bulgaria, June 5-9, 2007

Revised Papers

1 3

Volume Editors

Ivan Lirkov

Bulgarian Academy of Sciences

Institute for Parallel Processing

1113 Sofia, Bulgaria

E-mail: ivan@parallel.bas.bg

Svetozar Margenov

Bulgarian Academy of Sciences

Institute for Parallel Processing

1113 Sofia, Bulgaria

E-mail: margenov@parallel.bas.bg

Jerzy Wa´sniewski

Technical University of Denmark

Department of Informatics and Mathematical Modelling

2800 Kongens Lyngby, Denmark

E-mail: jw@imm.dtu.dk

Library of Congress Control Number: 2008923854

CR Subject Classification (1998): G.1, D.1, D.4, F.2, I.6, J.2, J.6

LNCS Sublibrary: SL 1 – Theoretical Computer Science and General Issues

ISSN 0302-9743

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ISBN-13 978-3-540-78825-6 Springer Berlin Heidelberg New York

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Springer is a part of Springer Science+Business Media

The 6th International Conference on Large-Scale Scientific Computations(LSSC 2007) was held in Sozopol, Bulgaria, June 5–9, 2007 The conference wasorganized by the Institute for Parallel Processing at the Bulgarian Academy ofSciences in cooperation with SIAM (Society for Industrial and Applied Math-ematics) Partial support was also provided from project BIS-21++ funded bythe European Commission in FP6 INCO via grant 016639/2005

The conference was devoted to the 60th anniversary of Richard E Ewing.Professor Ewing was awarded the medal of the Bulgarian Academy of Sciences forhis contributions to the Bulgarian mathematical community and to the Academy

of Sciences His career spanned 33 years, primarily in academia, but also includedindustry Since 1992 he worked at Texas A&M University being Dean of Scienceand Vice President of Research, as well as director of the Institute for ScientificComputation (ISC), which he founded in 1992 Professor Ewing is internation-ally well known with his contributions in applied mathematics, mathematicalmodeling, and large-scale scientific computations He inspired a generation ofresearchers with creative enthusiasm for doing science on scientific computa-tions The preparatory work on this volume was almost done when the sad newscame to us: Richard E Ewing passed away on December 5, 2007 of an apparentheart attack while driving home from the office

Plenary Invited Speakers and Lectures:

– O Axelsson, Mesh-Independent Superlinear PCG Rates for Elliptic

Problems

– R Ewing, Mathematical Modeling and Scientific Computation in Energy

and Environmental Applications

– L Gr¨une, Numerical Optimization-Based Stabilization: From cobi-Bellman PDEs to Receding Horizon Control

Hamilton-Ja-– M Gunzburger, Bridging Methods for Coupling Atomistic and Continuum

Models

– B Philippe, Domain Decomposition and Convergence of GMRES

– P Vassilevski, Exact de Rham Sequences of Finite Element Spaces on

Agglomerated Elements

– Z Zlatev, Parallelization of Data Assimilation Modules

The success of the conference and the present volume in particular are theoutcome of the joint efforts of many colleagues from various institutions andorganizations First, thanks to all the members of the Scientific Committee fortheir valuable contribution forming the scientific face of the conference, as well asfor their help in reviewing contributed papers We especially thank the organizers

of the special sessions We are also grateful to the staff involved in the localorganization

VI Preface

Traditionally, the purpose of the conference is to bring together scientistsworking with large-scale computational models of environmental and industrialproblems and specialists in the field of numerical methods and algorithms formodern high-speed computers The key lectures reviewed some of the advancedachievements in the field of numerical methods and their efficient applications.The conference lectures were presented by the university researchers and practi-cal industry engineers including applied mathematicians, numerical analysts andcomputer experts The general theme for LSSC 2007 was “Large-Scale ScientificComputing” with a particular focus on the organized special sessions

Special Sessions and Organizers:

– Robust Multilevel and Hierarchical Preconditioning Methods — J Kraus,

S Margenov, M Neytcheva

– Domain Decomposition Methods — U Langer

– Monte Carlo: Tools, Applications, Distributed Computing — I Dimov,

H Kosina, M Nedjalkov

– Operator Splittings, Their Application and Realization — I Farago – Large-Scale Computations in Coupled Engineering Phenomena with Multi-

ple Scales — R Ewing, O Iliev, R Lazarov

– Advances in Optimization, Control and Reduced Order Modeling —

P Bochev, M Gunzburger

– Control Systems — M Krastanov, V Veliov

– Environmental Modelling — A Ebel, K Georgiev, Z Zlatev

– Computational Grid and Large-Scale Problems — T Gurov, A Karaivanova,

The 7th International Conference LSSC 2009 will be organized in June 2009

Svetozar MargenovJerzy Wa´sniewski

Table of Contents

I Plenary and Invited Papers

Mesh Independent Convergence Rates Via Differential Operator

Pairs 3

Owe Axelsson and J´ anos Kar´ atson

Bridging Methods for Coupling Atomistic and Continuum Models 16

Santiago Badia, Pavel Bochev, Max Gunzburger,

Richard Lehoucq, and Michael Parks

Parallelization of Advection-Diffusion-Chemistry Modules 28

Istv´ an Farag´ o, Krassimir Georgiev, and Zahari Zlatev

Comments on the GMRES Convergence for Preconditioned Systems 40

Nabil Gmati and Bernard Philippe

Optimization Based Stabilization of Nonlinear Control Systems 52

Peter Arbenz and Cyril Flaig

Application of Hierarchical Decomposition: Preconditioners and Error

Estimates for Conforming and Nonconforming FEM 78

Radim Blaheta

Multilevel Preconditioning of Rotated Trilinear Non-conforming Finite

Element Problems 86

Ivan Georgiev, Johannes Kraus, and Svetozar Margenov

A Fixed-Grid Finite Element Algebraic Multigrid Approach

for Interface Shape Optimization Governed by 2-Dimensional

Magnetostatics 96

Dalibor Luk´ aˇ s and Johannes Kraus

VIII Table of Contents

The Effect of a Minimum Angle Condition on the Preconditioning of

the Pivot Block Arising from 2-Level-Splittings of Crouzeix-Raviart

Manuel Aldegunde, Antonio J Garc´ıa-Loureiro, and Karol Kalna

Numerical Study of Algebraic Problems Using Stochastic Arithmetic 123

Ren´ e Alt, Jean-Luc Lamotte, and Svetoslav Markov

Monte Carlo Simulation of GaN Diode Including Intercarrier

Interactions 131

A Ashok, D Vasileska, O Hartin, and S.M Goodnick

Wigner Ensemble Monte Carlo: Challenges of 2D Nano-Device

Simulation 139

M Nedjalkov, H Kosina, and D Vasileska

Monte Carlo Simulation for Reliability Centered Maintenance

Management 148

Cornel Resteanu, Ion Vaduva, and Marin Andreica

Monte Carlo Algorithm for Mobility Calculations in Thin Body Field

Effect Transistors: Role of Degeneracy and Intersubband Scattering 157

V Sverdlov, E Ungersboeck, and H Kosina

IV Operator Splittings, Their Application and Realization

A Parallel Combustion Solver within an Operator Splitting Context for

Engine Simulations on Grids 167

Laura Antonelli, Pasqua D’Ambra, Francesco Gregoretti,

Gennaro Oliva, and Paola Belardini

Identifying the Stationary Viscous Flows Around a Circular Cylinder

at High Reynolds Numbers 175

Christo I Christov, Rossitza S Marinova, and Tchavdar T Marinov

On the Richardson Extrapolation as Applied to the Sequential Splitting

Method 184

Istv´ an Farag´ o and ´ Agnes Havasi

Table of Contents IX

A Penalty-Projection Method Using Staggered Grids for Incompressible

Flows 192

C F´ evri` ere, Ph Angot, and P Poullet

Qualitatively Correct Discretizations in an Air Pollution Model 201

K Georgiev and M Mincsovics

Limit Cycles and Bifurcations in a Biological Clock Model 209

Coupled Engineering Problems

Parallel Implementation of LQG Balanced Truncation for Large-Scale

Systems 227

Jose M Bad´ıa, Peter Benner, Rafael Mayo,

Enrique S Quintana-Ort´ı, Gregorio Quintana-Ort´ı, and

Alfredo Rem´ on

Finite Element Solution of Optimal Control Problems Arising in

Semiconductor Modeling 235

Pavel Bochev and Denis Ridzal

Orthogonality Measures and Applications in Systems Theory in One

and More Variables 243

Adhemar Bultheel, Annie Cuyt, and Brigitte Verdonk

DNS and LES of Scalar Transport in a Turbulent Plane Channel Flow

at Low Reynolds Number 251

Jordan A Denev, Jochen Fr¨ ohlich, Henning Bockhorn,

Florian Schwertfirm, and Michael Manhart

Adaptive Path Following Primal Dual Interior Point Methods for Shape

Optimization of Linear and Nonlinear Stokes Flow Problems 259

Ronald H.W Hoppe, Christopher Linsenmann, and Harbir Antil

Analytical Effective Coefficient and First-Order Approximation to

Linear Darcy’s Law through Block Inclusions 267

Rosangela F Sviercoski and Bryan J Travis

X Table of Contents

Optimal Control for Lotka-Volterra Systems with a Hunter

Population 277

Narcisa Apreutesei and Gabriel Dimitriu

Modeling Supply Shocks in Optimal Control Models of Illicit Drug

Ion Chryssoverghi and Juergen Geiser

Approximation of the Solution Set of Impulsive Systems 309

Tzanko Donchev

Lipschitz Stability of Broken Extremals in Bang-Bang Control

Problems 317

Ursula Felgenhauer

On State Estimation Approaches for Uncertain Dynamical Systems

with Quadratic Nonlinearity: Theory and Computer Simulations 326

Tatiana F Filippova and Elena V Berezina

Using the Escalator Boxcar Train to Determine the Optimal

Management of a Size-Distributed Forest When Carbon Sequestration

Is Taken into Account 334

Renan Goetz, Natali Hritonenko, Angels Xabadia, and Yuri Yatsenko

On Optimal Redistributive Capital Income Taxation 342

Mikhail I Krastanov and Rossen Rozenov

Numerical Methods for Robust Control 350

P.Hr Petkov, A.S Yonchev, N.D Christov, and M.M Konstantinov Runge-Kutta Schemes in Control Constrained Optimal Control 358

Nedka V Pulova

Optimal Control of a Class of Size-Structured Systems 366

Oana Carmen Tarniceriu and Vladimir M Veliov

Table of Contents XI

VII Environmental Modelling

Modelling Evaluation of Emission Scenario Impact in Northern Italy 377

Claudio Carnevale, Giovanna Finzi, Enrico Pisoni, and

Comparative Study with Data Assimilation Experiments Using Proper

Orthogonal Decomposition Method 393

Gabriel Dimitriu and Narcisa Apreutesei

Effective Indices for Emissions from Road Transport 401

Kostadin G Ganev, Dimiter E Syrakov, and Zahari Zlatev

On the Numerical Solution of the Heat Transfer Equation in the

Process of Freeze Drying 410

K Georgiev, N Kosturski, and S Margenov

Results Obtained with a Semi-lagrangian Mass-Integrating Transport

Algorithm by Using the GME Grid 417

Wolfgang Joppich and Sabine Pott

The Evaluation of the Thermal Behaviour of an Underground

Repository of the Spent Nuclear Fuel 425

Roman Kohut, Jiˇ r´ı Star´ y, and Alexej Kolcun

Study of the Pollution Exchange between Romania, Bulgaria, and

Advances on Real-Time Air Quality Forecasting Systems for Industrial

Plants and Urban Areas by Using the MM5-CMAQ-EMIMO 450

Roberto San Jos´ e, Juan L P´ erez, Jos´ e L Morant, and

Rosa M Gonz´ alez

VIII Computational Grid and Large-Scale Problems

Ultra-fast Semiconductor Carrier Transport Simulation on the Grid 461

Emanouil Atanassov, Todor Gurov, and Aneta Karaivanova

XII Table of Contents

Simple Grid Access for Parameter Study Applications 470

P´ eter D´ ob´ e, Rich´ ard K´ apolnai, and Imre Szeber´ enyi

A Report on the Effect of Heterogeneity of the Grid Environment on a

Grid Job 476

Ioannis Kouvakis and Fotis Georgatos

Agents as Resource Brokers in Grids — Forming Agent Teams 484

Wojciech Kuranowski, Marcin Paprzycki, Maria Ganzha,

Maciej Gawinecki, Ivan Lirkov, and Svetozar Margenov

Parallel Dictionary Compression Using Grid Technologies 492

D´ enes N´ emeth

A Gradient Hybrid Parallel Algorithm to One-Parameter Nonlinear

Boundary Value Problems 500

D´ aniel Pasztuhov and J´ anos T¨ or¨ ok

Quantum Random Bit Generator Service for Monte Carlo and Other

Stochastic Simulations 508

Radomir Stevanovi´ c, Goran Topi´ c, Karolj Skala,

Mario Stipˇ cevi´ c, and Branka Medved Rogina

A Hierarchical Approach in Distributed Evolutionary Algorithms for

Multiobjective Optimization 516

Daniela Zaharie, Dana Petcu, and Silviu Panica

IX Application of Metaheuristics to Large-Scale Problems

Optimal Wireless Sensor Network Layout with Metaheuristics: Solving

a Large Scale Instance 527

Enrique Alba and Guillermo Molina

Semi-dynamic Demand in a Non-permutation Flowshop with

Constrained Resequencing Buffers 536

Gerrit F¨ arber, Said Salhi, and Anna M Coves Moreno

Probabilistic Model of Ant Colony Optimization for Multiple Knapsack

Problem 545

Stefka Fidanova

An Ant-Based Model for Multiple Sequence Alignment 553

Fr´ ed´ eric Guinand and Yoann Pign´ e

An Algorithm for the Frequency Assignment Problem in the Case of

DVB-T Allotments 561

D.A Kateros, P.G Georgallis, C.I Katsigiannis,

G.N Prezerakos, and I.S Venieris

Table of Contents XIII

Optimizing the Broadcast in MANETs Using a Team of Evolutionary

Algorithms 569

Coromoto Le´ on, Gara Miranda, and Carlos Segura

Ant Colony Models for a Virtual Educational Environment Based on a

Multi-Agent System 577

Ioana Moisil, Iulian Pah, Dana Simian, and Corina Simian

Simulated Annealing Optimization of Multi-element Synthetic Aperture

Imaging Systems 585

Milen Nikolov and Vera Behar

Adaptive Heuristic Applied to Large Constraint Optimisation

Problem 593

Kalin Penev

Parameter Estimation of a Monod-Type Model Based on Genetic

Algorithms and Sensitivity Analysis 601

Olympia Roeva

Analysis of Distributed Genetic Algorithms for Solving a Strip Packing

Problem 609

Carolina Salto, Enrique Alba, and Juan M Molina

Computer Mediated Communication and Collaboration in a Virtual

Learning Environment Based on a Multi-agent System with Wasp-Like

Behavior 618

Dana Simian, Corina Simian, Ioana Moisil, and Iulian Pah

Design of 2-D Approximately Zero-Phase Separable IIR Filters Using

Genetic Algorithms 626

F Wysocka-Schillak

X Contributed Talks

Optimal Order Finite Element Method for a Coupled Eigenvalue

Problem on Overlapping Domains 637

A.B Andreev and M.R Racheva

Superconvergent Finite Element Postprocessing for Eigenvalue

Problems with Nonlocal Boundary Conditions 645

A.B Andreev and M.R Racheva

Uniform Convergence of Finite-Difference Schemes for

Reaction-Diffusion Interface Problems 654

Ivanka T Angelova and Lubin G Vulkov

XIV Table of Contents

Immersed Interface Difference Schemes for a Parabolic-Elliptic Interface

Problem 661

Ilia A Brayanov, Juri D Kandilarov, and Miglena N Koleva

Surface Reconstruction and Lagrange Basis Polynomials 670

Irina Georgieva and Rumen Uluchev

A Second-Order Cartesian Grid Finite Volume Technique for Elliptic

Interface Problems 679

Juri D Kandilarov, Miglena N Koleva, and Lubin G Vulkov

MIC(0) DD Preconditioning of FEM Elasticity Systems on

Unstructured Tetrahedral Grids 688

Nikola Kosturski

Parallelizations of the Error Correcting Code Problem 696

C Le´ on, S Mart´ın, G Miranda, C Rodr´ıguez, and J Rodr´ıguez

Benchmarking Performance Analysis of Parallel Solver for 3D Elasticity

Problems 705

Ivan Lirkov, Yavor Vutov, Marcin Paprzycki, and Maria Ganzha

Re-engineering Technology and Software Tools for Distributed

Computations Using Local Area Network 713

A.P Sapozhnikov, A.A Sapozhnikov, and T.F Sapozhnikova

On Single Precision Preconditioners for Krylov Subspace Iterative

Methods 721

Hiroto Tadano and Tetsuya Sakurai

A Parallel Algorithm for Multiple-Precision Division by a

Single-Precision Integer 729

Daisuke Takahashi

Improving Triangular Preconditioner Updates for Nonsymmetric Linear

Systems 737

Jurjen Duintjer Tebbens and Miroslav T˚ uma

Parallel DD-MIC(0) Preconditioning of Nonconforming Rotated

Trilinear FEM Elasticity Systems 745

Yavor Vutov

Author Index 753

Mesh Independent Convergence Rates Via

Differential Operator Pairs

Owe Axelsson1 and J´anos Kar´atson2

1

Department of Information Technology, Uppsala University,

Sweden & Institute of Geonics AS CR, Ostrava, Czech Republic

2

Department of Applied Analysis, ELTE University, Budapest, Hungary

Abstract In solving large linear systems arising from the discretization

of elliptic problems by iteration, it is essential to use efficient tioners The preconditioners should result in a mesh independent linear

precondi-or, possibly even superlinear, convergence rate It is shown that a generalway to construct such preconditioners is via equivalent pairs or compact-equivalent pairs of elliptic operators

1 Introduction

Preconditioning is an essential part of iterative solution methods, such as jugate gradient methods For (symmetric or unsymmetric) elliptic problems, aprimary goal is then to achieve a mesh independent convergence rate, whichcan enable the solution of extremely large scale problems An efficient way toconstruct such a preconditioner is to base it on an, in some way, simplified differ-ential operator The given and the preconditioning operators should then form

con-an equivalent pair, based on some inner product Then the finite element cretization of these operators form the given matrix and its preconditioner, that

dis-is, if the given elliptic boundary value problem

Lu = f

is suitably discretized to an algebraic system L h u h = f h, then another, equivalent

operator S considerably simpler than L, is discretized in the same FEM subspace

to form a preconditioner S h, and the system which is actually solved is

indepen-to the above, and indepen-to illustrate its applications for some important classes ofelliptic problems Mesh independence and equivalent operator pairs have beenrigorously dealt with previously in [12,15], while superlinear rate of convergenceand compact-equivalent pairs have been treated in [6,8] (see also the references

I Lirkov, S Margenov, and J Wa´ sniewski (Eds.): LSSC 2007, LNCS 4818, pp 3–15, 2008 c

 Springer-Verlag Berlin Heidelberg 2008

4 O Axelsson and J Kar´atson

therein) Since in general the problems dealt with will be nonsymmetric, we firstrecall some basic results on generalized conjugate gradient methods, which will

be used here Equivalent and compact-equivalent pairs of operators are then cussed Then some applications are shown, including a superlinear convergenceresult for problems with variable diffusion coefficients

dis-2 Conjugate Gradient Algorithms and Their Rate of Convergence

Let us consider a linear system

with a given nonsingular matrix A ∈ R n ×n , f ∈ R n and solution u Letting ., 

be a given inner product on Rn and denoting by A ∗ the adjoint of A w.r.t this

inner product, in what follows we assume that

A + A ∗ > 0,

i.e., A is positive definite w.r.t .,  We define the following quantities, to be

used frequently in the study of convergence:

λ0:= λ0(A) := inf {Ax, x : x = 1} > 0, Λ := Λ(A) := A, (2)where. denotes the norm induced by the inner product ., .

2.1 Self-adjoint Problems: The Standard CG Method

If A is self-adjoint, then the standard CG method reads as follows [3]: let u0∈ R n

be arbitrary, d0:=−r0; for given u k and d k , with residuals r k := Au k − b, we

further we user k , d k  = −r k 2for α k , i.e, α k=r k 2/ Ad k , d k  In the study

of convergence, one considers the error vector e k = u − u k and is generallyinterested in its energy norm

e k  A=Ae k , e k  1/2 (5)Now we briefly summarize the minimax property of the CG method and twoconvergence estimates, based on [3] We first note that the construction of the

algorithm implies e k = P k (A)e0 with some P k ∈ π1, where π1 denotes the set

Mesh Independent Convergence Rates Via Differential Operator Pairs 5

of polynomials of degree k, normalized at the origin Moreover, we have the

which is a basis for the convergence estimates of the CG method

Using elementary estimates via Chebyshev polynomials, we obtain from (7)the linear convergence estimate

To show a superlinear convergence rate, another useful estimate is derived if

we consider the decomposition



in (7), where λ j := λ j (A) are ordered

according to|λ1− 1| ≥ |λ2− 1| ≥ Then a calculation [3] yields

Here by assumption |λ1(E) | ≥ |λ2(E) | ≥ If these eigenvalues accumulate

in zero then the convergence factor is less than 1 for k sufficiently large and

moreover, the upper bound decreases, i.e we obtain a superlinear convergencerate

2.2 Nonsymmetric Systems

For nonsymmetric matrices A, several CG algorithms exist (see e.g [1,3,11]).

First we discuss the approach that generalizes the minimization property (6) for

nonsymmetric A and avoids the use of the normal equation, see (18) below.

A general form of the algorithm, which uses least-square residual minimization,

is the generalized conjugate gradient–least square method (GCG-LS method)[2,3] Its full version uses all previous search directions when updating the new

approximation, whose construction also involves an integer t ∈ N, further, we

let t k = min{k, t} (k ≥ 0) Then the algorithm is as follows: let u0 ∈ R n be

arbitrary, d0:= Au0− b; for given u k and d k , with r k := Au k − b, we let

6 O Axelsson and J Kar´atson

where β k (k) −j =−Ar k+1 , Ad k −j /Ad k −j 2 (j = 0, , s k)

and the numbers α (k) k −j (j = 0, , k) are the solution of

There exist various truncated versions of the GCG-LS method that use only a

bounded number of search directions, such as GCG-LS(k), Orthomin(k), and GCR(k) (see e.g [3,11]) Of special interest is the GCG-LS(0) method, which

requires only a single, namely the current search direction such that (11) isreplaced by



u k+1 = u k + α k d k , where α k =−r k , Ad k /Ad k 2;

d k+1 = r k+1 + β k d k , where β k=−Ar k+1 , Ad k /Ad k 2. (12)

Proposition 1 (see, e.g., [2]) If there exist constants c1, c2 ∈ R such that

A ∗ = c1A + c2I , then the truncated GCG-LS(0) method (12) coincides with the

E has imaginary eigenvalues, one can easily verify as in [7] that 1 − (λ0/Λ)2=

E2/(1+ E2) Hence (14) yields that the GCG-LS(0) algorithm (12) convergesas

Mesh Independent Convergence Rates Via Differential Operator Pairs 7

On the other hand, if A is normal and we have the decomposition (9), then

the residual errors satisfy a similar estimate to (10) obtained in the symmetriccase, see [3]:

Another common way to solve (1) with nonsymmetric A is the CGN method,

where we consider the normal equation

and apply the symmetric CG algorithm (3) for the latter [13] In order to preserve

the notation r k for the residual Au k − b, we replace r k in (3) by s k and let

r k = A −∗ s k , i.e., we have s k = A ∗ r k Further, A and b are replaced by A ∗ A

and A ∗ b, respectively From this we obtain the following algorithmic form: let

u0∈ R n be arbitrary, r0:= Au0− b, s0 := d0 := A ∗ r0; for given d k , u k , r k and

0 and A ∗ A = I +(C ∗ +C +C ∗ C), the analogue of the superlinear

estimate (10) for equation A ∗ Au = A ∗ b becomes

8 O Axelsson and J Kar´atson

3 Equivalent Operators and Linear Convergence

We now give a comprehensive presentation of the equivalence property betweenpairs of operators, followed by a basic example for elliptic operators First a briefoutline of some theory from [12] is given

Let B : W → V and A : W → V be linear operators between the Hilbert spaces W and V Let B and A be invertible and let D := D(A) ∩D(B) be dense, where D(A) denotes the domain of an operator A The operator A is said to be equivalent in V -norm to B on D if there exist constants K ≥ k > 0 such that

k ≤ Au V

The condition number of AB −1 in V is then bounded by K/k Similarly, the W

-norm equivalence of B −1 and A −1 implies this bound for B −1 A If A h and B hare

finite element approximations (orthogonal projections) of A and B, respectively, then the families (A h ) and (B h ) are V -norm uniformly equivalent with the same bounds as A and B.

In practice for elliptic operators, it is convenient to use H1-norm equivalence,

since this avoids unrealistic regularity requirements (such as u ∈ H2(Ω)) We

then use the weak form satisfying

A w u, v  H1

D =Au, v L2 (u, v ∈ D(A)), (23)

where H1

D (Ω) is defined in (26) The fundamental result on H1-norm equivalence

in [15] reads as follows: if A and B are invertible uniformly elliptic operators, then

A −1

w and B −1

w are H1-norm equivalent if and only if A and B have homogeneous

Dirichlet boundary conditions on the same portion of the boundary

In what follows, we use a simpler Hilbert space setting of equivalent operatorsfrom [8] that suffices to treat most practical problems We recall that for a

symmetric coercive operator, the energy space H S is the completion of D(S)

under the inner productu, v S=Su, v, and the coercivity of S implies H S ⊂

H The corresponding S-norm is denoted by u S, and the space of bounded

linear operators on H S by B(H S)

Definition 1. Let S be a linear symmetric coercive operator in H A linear operator L in H is said to be S-bounded and S-coercive, and we write L ∈

BC S (H), if the following properties hold:

(i) D(L) ⊂ H S and D(L) is dense in H S in the S-norm;

(ii) there exists M > 0 such that

|Lu, v| ≤ Mu S v S (u, v ∈ D(L));

(iii) there exists m > 0 such that

Lu, u ≥ mu2 (u ∈ D(L)).

Mesh Independent Convergence Rates Via Differential Operator Pairs 9

The weak form of such operators L is defined analogously to (23), and produces

a variationally defined symmetrically preconditioned operator:

Definition 2 For any L ∈ BC S (H), let L S ∈ B(H S) be defined by

L S u, v  S =Lu, v (u, v ∈ D(L)).

Remark 1 (i) Owing to Riesz representation theorem the above definition makes sense (ii) L S is coercive on H S (iii) If R(L) ⊂ R(S) (where R( ) denotes the range), then L S

D(L) = S −1 L.

The above setting leads to a special case of equivalent operators:

Proposition 2 [9] Let N and L be S-bounded and S-coercive operators for the

same S Then

(a) N S and L S are H S -norm equivalent,

(b) N −1

S and L −1

S are H S -norm equivalent.

Definition 3. For given L ∈ BC S (H), we call u ∈ H S the weak solution of

equation Lu = g if L S u, v  S = g, v (v ∈ H S ) (Note that if u ∈ D(L) then u is a strong solution.)

Example. A basic example of equivalent elliptic operators in the S-bounded and S-coercive setting is as follows Let us define the operator

Lu ≡ −div (A ∇u) + b · ∇u + cu for u |ΓD = 0,

(i) Ω ⊂ R d is a bounded piecewise C1 domain; Γ D , Γ N are disjoint open

measurable subsets of ∂Ω such that ∂Ω = Γ D ∪ Γ N;

(ii) A ∈ C1(Ω, R d ×d ) and for all x ∈ Ω the matrix A(x) is symmetric; b ∈

(iv) either Γ D = ∅, or ˆc or ˆα has a positive lower bound.

Let S be a symmetric elliptic operator on the same domain Ω:

Su ≡ −div (G ∇u) + σu for u |ΓD = 0, ∂ν ∂u G + βu |ΓN = 0, (25)

with analogous assumptions on G, σ, β Let

10 O Axelsson and J Kar´atson

Proposition 3 [9] The operator L is S-bounded and S-coercive in L2(Ω).

The major results in this section are mesh independent convergence boundscorresponding to some preconditioning concepts Let us return to a general

Hilbert space H To solve Lu = g, we use a Galerkin discretization in V h =

span {ϕ1, , ϕ n } ⊂ H S , where ϕ i are linearly independent Let

Lh:=



L S ϕ i , ϕ j  S

n i,j=1

and, for the discrete solution, solve

with bh={g, ϕ j } n

j=1 Since L ∈ BC S (H), the symmetric part of L his positivedefinite

First, let L be symmetric itself Then its S-coercivity and S-boundedness

turns into the spectral equivalence relation

h Lhis self-adjoint w.r.t the inner productc, dSh := Shc· d.

Proposition 4 (see, e.g., [10]) For any subspace V h ⊂ H S ,

κ(S −1

h Lh)≤ M

independently of V h

Consider now nonsymmetric problems with symmetric equivalent

precondi-tioners With Sh from (29) as preconditioner, we use the bounds (2) for theGCG-LS and CGN methods:

λ0= λ0(S−1

h Lh) := inf{L hc· c : S hc· c = 1}, Λ = Λ(S −1

h Lh) :=S −1

h Lh Sh These bounds can be estimated using the S-coercivity and S-boundedness

m u2

S ≤ L S u, u  S , |L S u, v  S | ≤ Mu S v S (u, v ∈ H S ). (32)

Mesh Independent Convergence Rates Via Differential Operator Pairs 11

Proposition 5 [9] For any subspace V h ⊂ H S ,

Definition 4 Let L and N be S-bounded and S-coercive operators in H We

call L and N compact-equivalent in H S if

for some constant μ > 0 and compact operator Q S ∈ B(H S)

Remark 2 If R(L) ⊂ R(N), then compact-equivalence of L and N means that

N −1 L is a compact perturbation E of constant times the identity in the space

H , i.e., N −1 L = μI + E.

12 O Axelsson and J Kar´atson

One can characterize compact-equivalence for elliptic operators Let us take twooperators as in (24):

L1u ≡ −div (A1∇u) + b1· ∇u + c1u for u |ΓD = 0,

Now we discuss preconditioned CG methods and corresponding mesh

indepen-dent superlinear convergence rates Let us consider an operator equation Lu = g

in a Hilbert space H for some S-bounded and S-coercive operator L, and its

Galerkin discretization as in (27) Let us first introduce the stiffness matrix Sh

as in (29) as preconditioner

Proposition 7 [8] If L and S are compact-equivalent with μ = 1, then the

CGN algorithm (19) for system (30) yields

where ε k → 0 is a sequence independent of V h

A similar result holds for the GCG-LS method, provided however that Q S is a

normal compact operator in H S and the matrix S−1

h Qh is Sh-normal [6] Theseproperties hold, in particular, for symmetric part preconditioning The sequence

ε k contains similar expressions of eigenvalues as (17) or (21) related to Q S, which

we omit for brevity

For elliptic operators, we can derive a corresponding result Let L be the elliptic operator in (24) and S be the symmetric operator in (25) If the principal parts of L and S coincide, i.e., A = G, then L and S are compact-equivalent by Proposition 6, and we have μ = 1 Hence Proposition 7 yields a mesh independent

superlinear convergence rate Further, by [8], an explicit order of magnitude in

which ε k → 0 can be determined in some cases Namely, when the asymptotics for symmetric eigenvalue problems Su = μu, u |ΓD = 0, r∂u

Mesh Independent Convergence Rates Via Differential Operator Pairs 13

5 Applications of Symmetric Equivalent Preconditioners

We consider now symmetric preconditioning for elliptic systems defined on a

i,j=1 satisfies the coercivity property pointwise in Ω,

λ min (V + V T)−max divb i ≥ 0, pointwise in Ω, where λ mindenotes the smallesteigenvalue

Then system (40) has a unique solution u ∈ H1

D (Ω) l

As preconditioning operator we use the l-tuple S = (S1, · · · , S l) of independent

operators, S i u i ≡ −div(A i ∇u i ) + h i u, where u i = 0 on ∂Ω D , ∂u i

∂ν A

+ β i u i= 0

on ∂Ω N and β i ≥ 0, i = 1, 2, · · · l.

Now we choose a FEM subspace V h ⊂ H1

D (Ω) l and look for the solution u hof

the corresponding system L h c = b using a preconditioner S h being the stiffness

chemi-We have shown that a superlinear convergence takes place for operator pairs(i.e., the given and its preconditioner) which are compact-equivalent The maintheorem states that the principal, i.e., the dominating (second order) parts of theoperators must be identical, apart from a constant factor This seems to exclude

an application for variable coefficient problems, where for reasons of efficiency wechoose a preconditioner which has constant, or piecewise constant coefficients, as-suming we want to use a simple operator such as the Laplacian as preconditioner.However, we show now how to apply some method of scaling or transformation

to reduce the problem to one with constant coefficients in the dominating part

We use then first a direct transformation of the equation Let

14 O Axelsson and J Kar´atson

A case of special importance occurs when a is written in the form a = e −φ , φ ∈

C1(Ω) and b = 0 Then −∇a = e −φ ∇φ = a∇φ and (41) takes the form

1

a Lu = −Δu + ∇φ∇u + e φ cu = e φ g.

This is a convection-diffusion equation with a so called potential vector field,

v =∇φ Such problems occur frequently in practice, e.g in modeling of

semi-conductors

When the coefficient a varies much over the domain Ω one can apply

transfor-mations of both the equation and the variable, to reduce variations of gradients(∇u) of O(max(a)/ min(a)) to O(max √ a/ min( √

a)) Let then u = a 1/2 v and assume that a ∈ C2(Ω) Then a computation shows that

2b· ∇u/a2+ a −1/2 Δ(a −1/2) andg = a −1/2 g.

Remark 3 It is seen that when b = 0 both the untransformed (41) and

trans-formed (42) operators are selfadjoint

The relation N v ≡ a −1/2 Lu shows that

Nv, v L2(Ω =a −1/2 Lu, a 1/2 u  L2(Ω=Lu, u L2(Ω

holds for all u ∈ D(L) The positivity of the coefficient a shows hence that

u H1 and v H1 are equivalent, and N inherits the H1-coercivity of L, i.e.,

the relationLu, u L2(Ω ≥ mu2

Mesh Independent Convergence Rates Via Differential Operator Pairs 15

5 Axelsson, O., Kar´atson, J.: Conditioning analysis of separate displacement conditioners for some nonlinear elasticity systems Math Comput Simul 64(6),649–668 (2004)

pre-6 Axelsson, O., Kar´atson, J.: Superlinearly convergent CG methods via equivalentpreconditioning for nonsymmetric elliptic operators Numer Math 99(2), 197–223(2004)

7 Axelsson, O., Kar´atson J.: Symmetric part preconditioning of the CGM for Stokestype saddle-point systems Numer Funct Anal Optim (to appear)

8 Axelsson, O., Kar´atson J.: Mesh independent superlinear PCG rates via equivalent operators SIAM J Numer Anal (to appear)

compact-9 Axelsson, O., Kar´atson J.: Equivalent operator preconditioning for linear ellipticproblems (in preparation)

10 D’yakonov, E.G.: The construction of iterative methods based on the use of trally equivalent operators USSR Comput Math and Math Phys 6, 14–46 (1965)

spec-11 Eisenstat, S.C., Elman, H.C., Schultz., M.H.: Variational iterative methods fornonsymmetric systems of linear equations SIAM J Numer Anal 20(2), 345–357(1983)

12 Faber, V., Manteuffel, T., Parter, S.V.: On the theory of equivalent operatorsand application to the numerical solution of uniformly elliptic partial differentialequations Adv in Appl Math 11, 109–163 (1990)

13 Hestenes, M.R., Stiefel, E.: Methods of conjugate gradients for solving linear tems J Res Nat Bur Standards, Sect B 49(6), 409–436 (1952)

sys-14 Kar´atson J., Kurics T.: Superlinearly convergent PCG algorithms for some symmetric elliptic systems J Comp Appl Math (to appear)

non-15 Manteuffel, T., Parter, S.V.: Preconditioning and boundary conditions SIAM J.Numer Anal 27(3), 656–694 (1990)

16 Zlatev, Z.: Computer treatment of large air pollution models Kluwer AcademicPublishers, Dordrecht (1995)

Bridging Methods for Coupling Atomistic and

Continuum Models

Santiago Badia1,2, Pavel Bochev1, Max Gunzburger3, Richard Lehoucq1,

and Michael Parks1

1

Sandia National Laboratories, Computational Mathematics and Algorithms,

P.O Box 5800, MS 1320, Albuquerque NM 87185, USA

Abstract We review some recent developments in the coupling of

atomistic and continuum models based on the blending of the two models

in a bridge region connecting the other two regions in which the modelsare separately applied We define four such models and subject them topatch and consistency tests We also discuss important implementationissues such as: the enforcement of displacement continuity constraints inthe bridge region; and how one defines, in two and three dimensions, theblending function that is a basic ingredient in the methods

Keywords: Atomistic to continuum coupling, blended coupling,

molecular statics

1 Coupling Atomistic and Continuum Models

For us, continuum models are PDE models that are derived by invoking a

(phys-ical) continuum hypothesis In most situations, these models are local in nature,

e.g., forces at any point and time depend only on the state at that point istic models are discrete models In particular, we consider molecular staticsmodels; these are particle models in which the position of the particles are de-termined through the minimization of an energy, or, equivalently, by Newton’s

Atom-laws expressing force balances These models are, in general, nonlocal in nature,

e.g., particles other than its nearest neighbors exert a force on a particle.There are two types of situations in which the coupling of atomistic and

continuum models arise In the concurrent domain setting, the atomistic model

is used to determine information, e.g., parameters such as diffusion coefficients,viscosities, conductivities, equations of state, etc., or stress fields, etc., that areneeded by the continuum model Both models are assumed to hold over thesame domain Typically, these parameters are determined by taking statisticalaverages of the atomistic solution at points in the domain and, in this setting,

I Lirkov, S Margenov, and J Wa´ sniewski (Eds.): LSSC 2007, LNCS 4818, pp 16–27, 2008 c

 Springer-Verlag Berlin Heidelberg 2008

Bridging Methods for Coupling Atomistic and Continuum Models 17

In the domain decomposition setting (which is the one we consider in this

pa-per), the atomistic and continuum models are applied in different subdomains.The atomistic model is valid everywhere but is computationally expensive touse everywhere So, it is applied only in regions where “singularities” occur, e.g.,cracks, dislocations, plastic behavior, etc., and a continuum model is applied inregions where, e.g., ordinary elastic behavior occurs There remains the ques-tion of how one couples the atomistic to the continuum model; there are two

approaches to effect this coupling For non-overlapping coupling, the atomistic

and continuum models are posed on disjoint domains that share a common

inter-face For overlapping coupling, the regions in which the atomistic and continuum model are applied are connected by a bridge region in which both models are

applied See the sketches in Fig 1

Atomistic-to-continuum (AtC) coupling is distinct from most

continuum-to-continuum couplings due to the non-local nature of atomistic models Although

the are no “active” particles in the region in which only the continuum model isapplied, in a setting in which particles interact nonlocally, the forces exerted bythe missing particles on the active particles are not accounted for; this discrep-

ancy gives rise to what is known as ghost force phenomena.

In this paper, we consider AtC coupling methods that use overlapping regionsbecause, in that case, it is easier to mitigate the ghost force effect Note thatone should not simply superimpose the two models in the bridge region since

this leads to a non-physical “doubling” of the energy in Ω b Instead, the two

models must be properly blended in this region Such models are considered in

[1,2,3,4,6]; here, we review the results of [1,2,6]

We assume that the atomistic model is valid in the atomistic and bridge gions, Ω a and Ω b, respectively; see Fig 1 The continuum model is valid in the

re-continuum region Ω c and the bridge region Ω b but is not valid in the

atom-istic region Ω a We want to “seamlessly” blend the two models together using

the bridge region Ω according to the following principles: the atomistic model

x in the continuum region.

“dominates” the continuum model near the interface surface between the istic and bridge regions and the continuum model “dominates” the atomisticmodel near the interface surface between the continuum and bridge regions

atom-In the atomistic region Ω a , we assume that the force on the particle α located

at the position xα is due to externally applied force f e;α and the forces exerted

by other particles f α,β within the ballB α={x ∈ Ω : |x − x α | ≤ δ} for some

given δ See the sketch in Fig 2.

The inter-particle forces are determined from a potential function, e.g., if xα

and xβ denote the positions of the particles α and β, then f α,β =−∇Φ|x α −x β |,

where Φ( ·) is a prescribed potential function Instead of using the particle

posi-tions xα, one instead often uses the displacements uαfrom a reference ration

configu-Let N α = {β | x β ∈ B α , β  = α }, i.e., N α is the set of the indices of theparticles1 located withinB α, other than the particle located at xα itself Then,

for any particle α, force equilibrium gives

fα+ fe;α = 0,

where fα=

β ∈Nαfα,β We assume that in Ω a there are two kinds of particles:particles whose positions are specified in advance and particles whose positionsare determined by the force balance equations The set of indices of the secondkind of particles is denoted by N a It is convenient to recast the force balanceequation for the remaining particles in an equivalent variational form

In the continuum region Ω c, the Cauchy hypothesis implies that the forces

acting on any continuum volume ω enclosing the point x are given by the ternally applied volumetric force f e and the force exerted by the surrounding

ex-1

Note that for some α, the set N α may include the indices of some particles whosepositions are specified

Bridging Methods for Coupling Atomistic and Continuum Models 19

material f c =−γ σ · n dγ, where γ denotes the boundary of ω and σ denotes

the stress tensor See the sketch in Fig 2 Note that−σ · n is the stress force

acting on a point on γ.

We assume that σ(x) = σ

x, ∇u(x) and is possibly nonlinear in both its

arguments Here, u(x) denotes the continuous displacement at the point x For

Then, since ω is arbitrary, we conclude that at any point x in the continuum

region, we have the force balance

where we have the strain tensor ε(v) = 12(∇v + ∇v T) and the homogeneous

displacement test space H1(Ω c)

2.1 Blended Models in the Bridge Region

We introduce the blending functions θ a (x) and θ c (x) satisfying θ a + θ c = 1 in

Ω with 0 ≤ θ a , θ c ≤ 1, θ c = 1 in Ω c and θ a = 1 in Ω a Let θ α = θ a(xα) and

We introduce four ways to blend the atomistic and continuum models LetN b

denote the set of indices of the particles in Ω b whose positions are not fixed bythe boundary conditions

Methods I and II were introduced in [1,6] while Method III and IV were

intro-duced in [2] An important observation is that in the bridge region Ω b, near the

continuum region Ω c , we have that θ a is small so that θ α,β and θ α are small

as well Thus, blended models of the type discussed here automatically mitigateany ghost force effects, i.e., any ghost force will be multiplied by a small quantity

such as θ α,β or θ α

2.2 Displacement Matching Conditions in the Bridge Region

In order to complete the definition of the blended model, one must impose

con-straints that tie the atomistic displacements uα and the continuum

displace-ments u(x) in the bridge region Ω b These take the form of

Cuα , u(x)

= 0 for α ∈ N b and x∈ Ω b

for some specified constraint operatorC(·, ·).

One could slave all the atomistic displacements in the bridge region to thecontinuum displacements, i.e., set

uα= u(xα) ∀ α ∈ N b

We refer to such constraints as strong constraints Alternatively, the atomistic

and continuum displacements can be matched in an average sense to define

loose constraints For example, one can define a triangulation T H ={Δ t } T b

t=1of

the bridge region Ω b; this triangulation need not be the same as that used toeffect a finite element discretization of the continuum model LetN t =∅ denote indices of the particles in Δ t One can then match the atomistic and continuum

displacements in an average sense over each triangle Δ t:

Once a set of constraints has been chosen, one also has to choose a means for

enforcing them One possibility is to enforce them weakly through the use of the

Bridging Methods for Coupling Atomistic and Continuum Models 21

Lagrange multiplier rule In this case, the test functions vα and v(x) and trial functions uα and u(x) in the variational formulations are not constrained; one

ends up with saddle-point type discrete systems

A second possibility is to enforce the constraints strongly, i.e., require that

all candidate atomistic and continuum displacements satisfy the constraints In

this case, the test functions vαand v(x) in the variational formulations should

be similarly constrained One ends up with simpler discrete systems of the form



A a,θa,θc ·atomistic unknowns

+ A c,θc,θa ·continuum unknowns

= RHS.

Note that the atomistic and continuum variables are now tightly coupled Thesecond approach involves fewer degrees of freedom and results in better behaveddiscrete systems but may be more cumbersome to apply in some settings

2.3 Consistency and Patch Tests

To define an AtC coupled problem, one must specify the following data sets:

x∈∂(Ωc∪Ωb) (continuum displacements on the boundary)

We subject the AtC blending methods we have defined to two tests whosepassage is crucial to their mathematical and physical well posedness To this end,

we define two types of test problems The set{F, P, B, D} defines a consistency

test problem if the pure atomistic solution u αand the pure continuum solution

u(x) are such that the constraint equations, i.e, C

uα , u(x)

= 0, are satisfied

on Ω Further, a consistency test problem defines a patch test problem if the pure

continuum solution u(x) is such that ε(u) = constant, i.e., it is a solution with

constant strain

If we assume that {F, P, B, D} defines a patch test problem with atomistic

solution uα and continuum solution u(x), then, an AtC coupling method passes

the patch test if {u α , u(x) } satisfies the AtC model equations Similarly, an AtC coupling method passes the consistency test if {u α , u(x) } satisfies the AtC model

equations for any consistency test problem Note that passing the consistencytest implies passage of the patch test, but not conversely

22 S Badia et al.

Our analyses of the four blending methods (see [2]) have shown that Methods

I and IV are not consistent and do not pass patch test problems; Method III

is consistent and thus also passes any patch test problem; and Method II isconditionally consistent: it is consistent if, for a pair of atomistic and continuum

solutions uαand u, respectively

and passes patch tests if this condition is met for patch test solutions

From these results, we can forget about Methods I and IV and it seems thatMethod III is better than Method II The first conclusion is valid but there areadditional considerations that enter into the relative merits of Methods II andIII Most notably, Method II is the only one of the four blended models thatsatisfies2Newton’s third law In addition, the violation of patch and consistencytests for Method II is tolerable, i.e., the error introduced can be made smaller byproper choices for the model parameters, e.g., in a 1D setting, we have shown (see[1]) that the patch test error is proportional to L s2

b h , where s = particle lattice spacing (a material property), h = finite element grid size, and L b = width of

the bridge region Ω b While we cannot control the size of s, it is clear that in

realistic models this parameter is small Also, the patch test error for Method II

can be made smaller by making L b larger (widening the bridge region) and/or

making h larger (having more particles in each finite element).

2.4 Fully Discrete Systems in Higher Dimensions

We now discuss how the fully discrete system can be defined in 2D; we onlyconsider Method II for which we have

Bridging Methods for Coupling Atomistic and Continuum Models 23

To discretize the continuum contributions to the blended model, we use a

finite element method Let W h ⊂ H1

0(Ω b ∪ Ω c) be a nodal finite element spaceand let{w h



j ∈ {1, , J} | x j ∈ Ω b



denote the set of indices of the nodes in Ω c and Ω b, respectively We use

contin-uous, piecewise linear finite element spaces with respect to partition of Ω b ∪ Ω c

into a set of T triangles T h ={Δ t } T

t=1; higher-order finite element spaces canalso be used

j(xΔt ;q)

t ∈T h j

Due to the way we are handling the constraints, we have that the atomistic test

and trial functions in the bridge region Ω b are slaved to the continuum test and

trial functions, i.e., uα= uh(xα) and vα= wh

j(xα)

For j ∈ S b, let N j ={α | x α ∈ supp(w h

j)} denote the set of particle indices

such that the particles are located within the support of the finite element basis

function wh

j Then, in the bridge region Ω b, we have that

2.5 Choosing the Blending Function in 2D and 3D

For the blending function θ c(x) for x ∈ Ω b , we of course have that θ a(x) =

1− θ c (x), θ a (x) = 1, θ c (x) = 0 in Ω a , θ a (x) = 0 and θ c (x) = 1 in Ω c

In many practical settings, the domain Ω b is a rectangle in 2D or is a

rect-angular parallelepiped in 3D In such cases, one may simply choose θ c(x) in

the bridge region to be the tensor product of global 1D polynomials ing the atomistic and continuum regions across the bridge region One couldchoose linear polynomials in each direction such that their values are zero at thebridge/atomistic region interface and one at the bridge/continuum region inter-face If one wishes to have a smoother transition from the atomistic to the bridge

connect-to the continuum regions, one can choose cubic polynomials in each directionsuch that they have zero value and zero derivative at the bridge/atomisitic re-gion interface and value one and zero derivative at the bridge/continuum regioninterface

For the general case in 2D, we triangulate the bridge region Ω b into the set

of triangles having vertices{x b;i } l

i=1 In practice, this triangulation is the same

as that used for the finite element approximation of the continuum model in thebridge region but, in general, it may be different For the two triangulations to

be the same, we must have that the finite element triangulation is conformingwith the interfaces between the bridge region and the atomistic and continuumregions, i.e., those interfaces have to be made up of edges of triangles of thefinite element triangulation The simplest blending function is then determined

by setting θ c (x) = ξ h (x), where ξ h(x) is a continuous, piecewise linear function

with respect to this triangulation The nodal values of ξ h(x) are chosen as follows.

Set ξ h(xb;i) = 0 at all nodes xb;i ∈ Ω a ∩ Ω b, i.e., on the interface between

the atomistic and bridge regions Then, set ξ h(xb;i) = 1 at all nodes xb;i ∈

Ω ∩ Ω , i.e., on the interface between the continuum and bridge regions For

Bridging Methods for Coupling Atomistic and Continuum Models 25

the remaining nodes xb;i ∈ Ω b , there are several ways to choose the values of ξ h.One way is to choose them according to the relative distances to the interfaces A

more convenient way is to let ξ hbe a finite element approximation, with respect

to the grid, of the solution of Laplace’s equation in Ω bthat satisfies the specified

values at the interfaces Once ξ h (x) is chosen, we set θ a(x) = 1− ξ h(x) for all

x∈ Ω b and choose θ α = θ a(xα) = 1− ξ h(xα) so that

This recipe can be extended to 3D settings

One may want θ c(x) to have a smoother transition from the atomistic to the

bridge to the continuum regions To this end, one can choose ξ h(x) to not only be

continuous, but to be continuously differentiable in the bridge region and acrossthe interfaces Note that in 2D, this requires the use of the fifth-degree piecewisepolynomial Argyris element or the cubic Clough-Tocher macro-element; see [5].Such elements present difficulties in a finite element approximation setting, butare less problematical in an interpolatory setting

3 Simple Computational Examples in 1D

For 0 < a < c < 1, let Ω = (0, 1), Ω a = (0, a), Ω b = (a, c), and Ω c = (c, 1).

In Ω c ∪ Ω b = [a, 1], we construct the uniform partition x j = a + (j − 1)h for j = 1, , J having grid size h We then choose W h to be the continuous,piecewise linear finite element space with respect to this partition Without loss

of generality, we define the bridge region Ω busing the finite element grid, i.e., we

assume that there are finite element nodes at x = a and x = c; this arrangement

leads to a more convenient implementation of blending methods in 2D and 3D

In Ω a ∪ Ω b = [0, c], we have a uniform particle lattice with lattice spacing s given by x α = (α − 1)s, α = 1, , N Note that the lattice spacing s is a fixed material property so that there is no notion of s → 0 One would think that one can still let h → 0; however, it makes no sense to have h < s.

We consider the atomistic model to be a one-dimensional linear mass-spring

system with two-nearest neighbor interactions and with elastic moduli K a1 and

K a2for the nearest-neighbor and second nearest-neighbor interactions; only thetwo particles to the immediate left and right of a particle exert a force on thatparticle The continuum model is one-dimensional linear elasticity with elastic

modulus K c We set K a1 = 50, K a2 = 25, K c = K a1 + 2K a2 = 100 A unit

point force is applied at the finite element node at the end point x = 1 and the displacement of the particle located at the end point x = 0 is set to zero.

Using either the atomistic or finite element models, the resulting solutions are

ones having uniform strain 0.01; thus, we want a blended model solution to also

recover the uniform strain solution

We choose h = 1.5s and s = 1/30 so that we have a = 0.3, c = 0.6, 20

particles, and 16 finite element nodes; there are no particles located at either

x = a or x = c, the end points of the bridge region Ω For the right-most

26 S Badia et al.

Fig 3 Strain for Method II (left) and Method III(right)

particle x20< c, we have that θ a (x20)= 0 To avoid the ghost forces associatedwith the missing bond to the right of the 20th particle, a 21st particle is added

to the right of x = c Since x21∈ Ω c , we have that θ a (x21) = 0 so that we neednot be concerned with its missing bond to the right; this is a way that blendingmethods mitigate the ghost force effect We see from Fig 3 that Method IIIpasses the patch test but Method II does not However, the degree of failurefor Method II is “small.” From Fig 4, we see that Method I fails the patchtest; the figure for that method is for the even simpler case of nearest-neighborinteractions Similarly, Method IV fails the patch test

In Fig 5, we compare the blended atomistic solutions in the bridge region,obtained using Method III with both strong and loose constraints, with thatobtained using the fully atomistic solution We consider a problem with a uniformload and zero displacements at the two ends; we only consider nearest-neighborinteractions The loose constraint allows the atomistic solution to be free toreproduce the curvature of the fully atomistic solution, leading to better results.The strong constraint is too restrictive, forcing the atomistic solution to followthe finite element solution; it results in a substantial reduction in the accuracy

in the bridge region

Fig 4 Strain for Method I

0.35 0.4 0.45 0.5 0.55 0.6 0.65 1.12

1.14 1.16 1.18 1.2 1.22 1.24 1.26 x 10

−3

undeformed position

Fig 5 Fully atomistic solution (dashed line);

loose constraints (dash-dotted line); andstrong constraints (solid line)

Bridging Methods for Coupling Atomistic and Continuum Models 27

Acknowledgments

The authors acknowledge the contributions to part of this work by Mohan hally, Catalin Picu, and Mark Shephard and especially Jacob Fish of the Rens-selaer Polytechnic Institute SB was supported by the European Communitythrough the Marie Curie contract NanoSim (MOIF-CT-2006-039522) For PB,

Nugge-RL, and MP, Sandia is a multiprogram laboratory operated by Sandia poration, a Lockheed Martin Company, for the United States Department ofEnergy’s National Nuclear Security Administration under Contract DE-AC04-94-AL85000 MG was supported in part by the Department of Energy grantnumber DE-FG02-05ER25698

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6 Fish, J., et al.: Concurrent AtC Coupling Based on a Blend of the Continuum Stressand the Atomistic Force Comp Meth Appl Mech Engrg (to appear)

Institute for Parallel Processing, Bulgarian Academy of Sciences

Acad G Bonchev, Bl 25A, 1113 Sofia, Bulgaria

georgiev@parallel.bas.bg

3National Environmental Research Institute, Aarhus University

Frederiksborgvej 399, P.O Box 358, DK-4000 Roskilde, Denmark

zz@dmu.dk

Abstract An advection-diffusion-chemistry module of a large-scale air

pollution model is split into two parts: (a) advection-diffusion part and(b) chemistry part A simple sequential splitting is used This meansthat at each time-step first the advection-diffusion part is treated andafter that the chemical part is handled A discretization technique based

on central differences followed by Crank-Nicolson time-stepping is used

in the advection-diffusion part The non-linear chemical reactions aretreated by the robust Backward Euler Formula The performance of thecombined numerical method (splitting procedure + numerical algorithmsused in the advection-diffusion part and in the chemical part) is studied inconnection with six test-problems We are interested in both the accuracy

of the results and the efficiency of the parallel computations

1 Statement of the Problem

Large-scale air pollution models are usually described mathematically by systems

of PDEs (partial differential equations):

(ii) the wind velocities u, v and w, (iii) the diffusion coefficients K x , K y , and K z,

(iv) the emission sources E s , (v) the deposition coefficients κ 1s and κ 2s, and (vi)

the non-linear terms Q s (t, c1, c2, , c q) describing the chemical reactions More

I Lirkov, S Margenov, and J Wa´ sniewski (Eds.): LSSC 2007, LNCS 4818, pp 28–39, 2008 c

 Springer-Verlag Berlin Heidelberg 2008

Parallelization of Advection-Diffusion-Chemistry Modules 29

details about large-scale air pollution models can be found in Zlatev [13] andZlatev and Dimov [14] as well as in the references given in these two monographs

In order to check better the accuracy of the numerical methods used, thefollowing simplifications in (1) were made: (i) the three-dimensional model wasreduced to a two-dimensional model, (ii) the deposition terms were removed, (iii)

a constant horizontal diffusion was introduced, and (iv) a special wind velocitywind was designed The simplified model is given below:

∂ξ2 +∂

2c ∗ s

ξ ∈ [0, 2], η ∈ [0, 2], τ ∈ [0, M(2π)], M ≥ 1. (3)The system of PDEs (2) must be considered together with some initial andboundary conditions It will be assumed here that some appropriate initialand boundary conditions are given Some further discussion related to the initialand boundary conditions will be presented in the remaining part of this paper

The replacement of the general wind velocity terms u = u(t, x, y) and v = v(t, x, y) from (1) with the special expressions η − 1 and 1 − ξ in (2) defines

a rotational wind velocity field (i.e., the trajectories of the wind are concentriccircles with centres in the mid of the space domain and particles are rotatedwith a constant angular velocity)

If only the first two terms in the right-hand-side of (2) are kept, i.e., if pureadvection is considered, then the classical rotation test will be obtained This testhas been introduced in 1968 simultaneously by Crowley and Molenkampf ([3] and

[10]) In this case, the centre of the domain is in the point (ξ1, η1) = (1.0, 1.0).

High concentrations, which are forming a cone (see the upper left-hand-side plot

in Fig 1) are located in a circle with centre at (ξ0, η0) = (0.5, 1.0) and with radius r = 0.25 If ˜ x = 

(ξ − ξ0)2+ (η − η0)2, then the initial values for the

original Crowley-Molenkampf test are given by c ∗

s (ξ, η, 0) = 100(1 −˜x/r) for r < ˜x and c ∗

s (ξ, η, 0) = 0 for r ≥ ˜x It can be proved that c ∗

s (ξ, η, k 2π) = c ∗

s (ξ, η, 0) for k = 1, 2, , M , i.e., the solution is a periodic function with period 2 π It

can also be proved that the cone defined as above will accomplish a full rotation

around the centre (ξ1, η1) of the domain when the integration is carried out

from τ = k 2 π to τ = (k + 1) 2 π, where k = 0, 1, , M − 1 (which explains why

Crowley-Molenkampf test is often called the rotation test)

The advection process is dominating over the diffusion process, which is fined mathematically by the next term in the right-hand-side of (2) This means

de-in practice that the constant K is very small, which is a very typical situation

in air pollution modelling