COMPUTATIONAL MECHANICS RESEARCH TRENDS

COMPUTATIONAL MECHANICS RESEARCH TRENDS

CONTENTS

PREFACE

Chapter 1 A NATURAL NEIGHBOUR METHOD BASED ON FRAEIJS DE VEUBEKE VARIATIONAL PRINCIPLE

Abstract

Introduction

Virtual Work Principle

Approximation of the Displacement Field

Discretized Virtual Work Principle

Linear Elastic Theory

The Fraeijs de Veubeke Functional

The Fraeijs de Veubeke Variational Principle

Domain Decomposition and Discretization

Equations Deduced from the Fraeijs de Veubeke Variational Principle

Matrix Notation

Numerical Integration

Patch Tests

Application to Pure Bending

Application to a Square Membrane with a Circular Hole

Extention to non Linear Materials

Variational Equation `

Domain Decomposition and Discretization

Matrix Notation

Solution of the Matrix Equations

Elasto-plastic Material with von Mises Linear Hardening

Patch Tests

Pure Bending of a Beam

Square Membrane with a Circular Hole

Extention to Linear Fracture Mechanics

Introduction

Domain Decomposition and Discretization

Solution of the Equation System

Patch Tests

Translation Tests

Mode 1 Tests

Mode 2 Tests

Bar with a Single Edge Crack

Conclusions

Annex 1: Construction of the Voronoi Cells

Case of a Convex Domain

Case of a non Convex Domain

Annex.2: Laplace Interpolant

Case of a Point X Inside the Domain

Case of a Point X on the Domain Contour

Annex 3. Particular Case of a Regular Grid of Nodes

Laplace Interpolant

Case 1: X between A and B

Case 2: X between B and C

Case 3: X between C and D

Calculation of

Annex 4. Introduction of the Hypotheses in the FdV Principle

Calculation of1

Calculation of2

Calculation of3

Calculation of4

Calculation of5

Calculation of6

Calculation of 0

Annex 5. Analytical Calculation of V andIH.

Analytical Integration of V over a Triangle

Analytical Integration of IH over a Triangle

References

Chapter 2 NUMERICAL AND THEORETICAL INVESTIGATIONS OF THE TENSILE FAILURE OF SHRUNK CEMENT-BASED COMPOSITES

Abstract

1. Introduction

1.1. Characteristics of Shrunk Concrete

1.2. Algorithm to Produce a Shrunk Specimen

1.3. Lattice-Type Modeling of Concrete

1.4. Paper Structure

2. GB Lattice Model

3. Method to Simulate Mismatch Deformation Due to Matrix Uniform Shrinkage

4. Global Numerical Procedure

4.1. Mohr-Coulomb Criterion

4.2. Event-By-Event Algorithm

5. Theoretical Analyses of Influences of Pre-stressed Field

6. Numerical Examples and Discussions

6.1. Production of Shrunk Specimens

6.2. Tensile Examples on Specimens without the Shrinkage-Induced Stress: Case 1 and Case 2

6.3. Analysis of a Typical Case for Shrunk Specimens: Case 3

6.4. Influence of the Shrinkage Rate: Case 3-5

7. Conclusions

Acknowledgements

References

Chapter 3 RECENT ADVANCES IN THE STATIC ANALYSIS OF STIFFENED PLATES – APPLICATION TO CONCRETE OR TO COMPOSITE STEEL-CONCRETE STRUCTURES

Abstract

Introduction

Statement of the Problem

A. In the Plate (at the Traces of the Two Interface Lines J=1,2 of the I-Th Plate-Beam Interface)

B. In Each (I-Th) Beam (iiiiOxyz System of Axes)

For the Plate

B. For Each (I-Th) Beam

Numerical Solution

A. For the Plate Transverse Deflection pw.

B. For the Plate Inplane Displacement Components pu, pv.

C. For the Beam Transverse Displacements ibw, ibv and for the Angle of Twist ibxθ.

D. For the Axial Deformation ibu

Numerical Examples

Example 1

Example 2.

Conclusion

References

Chapter 4 A SPRING-BASED FINITE ELEMENT MODEL FOR THE PREDICTION OF MECHANICAL PROPERTIES OF CARBON NANOTUBES AND THEIR COMPOSITES

Abstract

1. Introduction

2. CNTs Nanostructure

3. Interatomic Interactions

4. Finite Element Formulation

4.1. Swcnt Modeling

4.2. MWCNT

4.3. Swcnt Reinforced Composite

4.3.1. Swcnt Reinforcement Modeling

4.3.2. Matrix Modeling

4.3.3. Interface Modeling

5. Numerical Results and Discussion

5.1. Static Behavior of SWCNTs

5.2. Static Behavior of Swcnt Reinforced Composites

5.3. Dynamic Behavior of SWCNTs

5.4. Dynamic Behavior of MWCNTs

6. Conclusion

References

Chapter5COMPUTATIONALMECHANICSOFMOLECULARSYSTEMS

Abstract

1.Introduction

2.MolecularPhaseSpaceTrajectoryasaComplexDynamicalSystem

3.TheProblemofSymbolisation

4.HowComputationalMechanicsCanBeUsedtoFindaSuit-ableSymbolisation

4.1.TheDynamicsMakesthePartitionFiner

4.2.ComputationalMechanicsCoarsensthePartition

4.3.ThePartitionGeneratedbyComputationalMechanicsIstheMostIn-formativeOne

4.4.ThreeStagesofSymbolisation

5.Implementation

5.1.MolecularDynamicsSimulation

5.2.Symbolisation

5.3.-machineReconstruction:CSSR

5.4.SurrogateTimeSeries

6.Results

6.1.-MachineGrowswiththeLengthofTimeSeries

6.2.AnalysisoftheCausalStates

6.3.Non-stationaryModelofGrowing-Machine

7.Conclusions

Acknowledgements

References

Chapter 6 MESHLESS APPROACH AND ITS APPLICATION IN ENGINEERING PROBLEMS

Abstract

1. Introduction

2. Potential Problems

2.1. The Analog Equation Method

2.2. RBF Approximation for the Particular Solution pu

2.3. VBCM for the Homogeneous Solution

2.4. The Construction of Solution System

Example 1: Nonlinear Poisson Problems

3. Steady-State Heat Conduction in Inhomogeneous Materials

3.1. Governing Equation for Steady-State Heat Conduction in Isotropic Heterogeneous Media

3.2. Governing Equation for Steady-State Heat Conduction in Anisotropic Media

3.3. Implementation of the Meshless Method

3.4. The Virtual Boundary Collocation Method for the Homogeneous Solution

3.5. The Construction of Solving Equations

3.6. Numerical Assessment

4. Transient Heat Conduction in Functionally Graded Materials

4.1. Basic Formulas of Transient Heat Conduction

4.2. Meshless Formulation

4.3. The Backward Time Stepping Scheme

4.4. Numerical Assessment

5. Thermo-Mechanical Analysis of FGMs

5.1. Governing Equations for FGMs

5.2. Graded Types of FGM

5.3. RBF Approximation

5.4. Method of Fundamental Solutions

3.4. Final Complete Solutions

6. Thin Plate Bending

6.1.Basic Equations of Thin Plate Bending

6.2. Fundamental Solution And Determination of Source Points

6.3. Radial Basis Function (RBF)

3.4. Solution w(x)

Appendix. First and Second Order Differentials of Fundamental Solutions and Approximated Particular Solutions

A1. Fundamental Solutions and Their Derivatives

A2. Approximated Particular Solutions and Their Derivatives

A2.1. Power Spline (PS) Function

A2.2. Thin Plate Spline (TPS) Function

References

Chapter 7 EXPLICIT DYNAMIC FINITE ELEMENT METHOD FOR FRACTURE OF SHELLS

1. Introduction

2. Representation of Fractured Shell Element

2.1. Shell Formulation with Fracture

2.2. Representation of Fractured Shell Elements

2.3. Computation of Element Kinematics

2.3.1. Belytschko Lin Tsay 4 Nodes Element

2.3.2. Discrete Kirchhoff Triangular Shell Element

3. Computation Procedures

3.1. Newmark Scheme

3.2. Computation of Lumped Mass Matrix for Cracked Elements

3.3. Computation of Element Internal Forces

4. Material Model and Modeling of Fracture

4.1. Hardening Plasticity for Quasi-Brittle Material

4.2. Fracture Criterion

4.3. Cohesive Crack Model

5. Numerical Examples

5.1. The Simulations of a Thin Shell Cylinder under Hydrostatic and Impulsive External Pressure

5.2. Tearing of a Plate by Out-of-Plane Loading

6. Conclusion

References

Chapter 8 PROBABILISTIC INTERPRETATIONS OF THE TLM NUMERICAL METHOD

Abstract

Introduction

Theory (Part I)

The Fundamentals of TLM

Lossless TLM Formulations in Two and Three Dimensions

Lossy TLM Formulations in One Dimension

Mesh Scaling

Lossy TLM Formulations in Two and Three Dimensions

Theory (Part II)

Probabilistic Interpretations of TLM (The One-Dimensional Case)

Probabilistic Interpretations in Two and Three Dimensions

Problem with Existing Theory

An Apparent Paradox in the TLM/Random Walk Equivalence

Extension of Theory

A Resolution to the Paradox of a Discrete Random Walker With Negative Probability

A General Transition Probability for Walker Pairs

Discrete Random Walks and Diffraction

Application of Extended Theory

A Particle Approach to Wave Diffraction Phenomena

Conclusion

Appendix I

Generating Function Derivation of Eqn (1) in Main Text

Two-Dimensional Expressions: Eqns (11), (12)

Three-Dimensional Expressions: Eqns (13), (14)

References

Chapter 9 SOME OBSERVATIONS ON ACCELERATED NUMERICAL SCHEMES FOR THE LAPLACE EQUATION

Abstract

Introduction

Theory I

The Reverse Engineering of Numerical Schemes for the Laplace Equation

I Jacobi Scheme

II. Gauss-Seidel Scheme

III. Successive Over-Relaxation

IV. A Fraction of Error at (X) Over Two Previous Time-Steps Added to Gauss-Seidel Scheme

V. A Fraction of the Error at (X) between N-th and (N+1)th Steps Added to Gauss-Seidel Scheme

VI. A Fraction of the Error at (X) iver the Two Previous Steps Is Added to a Jacobi Scheme

VII. A Fraction of the Error at (X) Observed between N-th and (N+1)th Steps Added to a Jacobi Scheme

VIII. A Fraction of the Error Difference over Two Time-Steps Added to a Gauss-Seidel Scheme

IX. A Fraction of the Error Difference over Two Time-Steps Added to a Jacobi Scheme

X. A Fraction of the Mean Error over Two Time-Steps Added to a Gauss-Seidel Scheme

XI. A Fraction of the Mean Error over Two Time-Steps Added to a Jacobi Scheme

XII. Du Fort- Frankel Scheme

XIII. TLM Scheme

Theory II

Matrix Stability Analysis and Optimum Convergence Conditions

Numerical Experiments

Scheme (V)

Scheme (VI)

Scheme (VIII)

Discussion

Conclusion

Appendix I

TLM Schemes for The Solution of the Laplace Equation

Appendix II

References

Chapter10INSEARCHOFIMPROVEMENTSFORTHECOMPUTATIONALSIMULATIONOFINTERNALCOMBUSTIONENGINES

Abstract

1.Introduction

2.GoverningEquationsandNumericalApproximation

2.1.GoverningEquations

2.1.1.TurbulenceModeling

2.1.2.BoundaryConditions

2.1.3.ArbitraryLagrangianEulerianDescriptionofGoverningEquations

2.2.NumericalImplementation

2.2.1.FiniteElementFormulation

2.2.2.TimeDiscretization

2.2.3.DynamicBoundaryConditionsUsingLagrangeMultipliers

3.MeshDynamics

3.1.MeshQuality

3.2.TheMeshDynamicsStrategy

3.2.1.FunctionalDesign

3.2.2.DifferentialPredictor

3.2.3.AvoidingtheRelaxationoftheInitialMesh

3.2.4.Results

3.3.SimultaneousMeshUntanglingandSmoothing

3.3.1.FunctionalRegularization

3.3.2.SolutionStrategy

3.3.3.Results

4.ResolutionofCompressibleFlowsintheLowMachNumberLimit

4.1.ProblemDefinitionandEigenvaluesAnalysis

4.1.1.PreconditioningStrategies

4.2.NumericalImplementation

4.2.1.VariationalFormulation

4.2.2.DynamicBoundaryConditions

4.3.Results

4.3.1.FlowinaLidDrivenCavity

4.3.2.FlowinaChannelwithaMovingIndentation

4.3.3.Opposed-PistonEngine

5.Couplingof1D/multi-DDomainsforCompressibleFlows

5.1.CouplingforImplicitSchemes‘Monolithically’Solved

5.1.1.Couplingof1D/multi-DDomains

5.2.Results

5.2.1.1D/1DCoupling

5.2.2.2D/1DCoupling

5.2.3.3D/1DCoupling

6.NumericalSimulationoftheMRCVCEngine

6.1.OperationandGeometryofMRCVC

6.2.NumericalSimulationofFluidFlowintheMRCVCEngine

6.2.1.ComputationalMeshDynamicProblem

6.2.2.ComputationalFluidDynamicProblem

7.Conclusion

Appendix:PipeJunction0DModel

References

Chapter11MPIANDPETSCFORPYTHON

Abstract

1.Introduction

1.1.ThePythonProgrammingLanguage

1.2.ToolsforScientificComputing

1.2.1.NumericalPython

1.2.2.ScientificToolsforPython

1.2.3.FortrantoPythonInterfaceGenerator

1.2.4.Cython

2.MPIforPython

2.1.AnOverviewofMPI

2.1.1.CommunicationDomainsandProcessGroups

2.1.2.Point-to-PointCommunication

2.1.3.CollectiveCommunication

2.1.4.DynamicProcessManagement

2.1.5.One-SidedOperations

2.1.6.ParallelInput/Output

2.2.RelatedWorkonMPIandPython

2.3.DesignandImplementation

2.3.1.CommunicatingGeneralPythonObjectsandArrayData

2.4.UsingMPIforPython

2.4.1.Communicators

2.4.2.BlockingPoint-to-PointCommunications

2.4.3.NonblockingPoint-to-PointCommunications

2.4.4.CollectiveCommunications

2.4.5.DynamicProcessManagement

2.4.6.One-sidedOperations

2.4.7.ParallelInput/OutputOperations

2.5.EfficiencyTests

3.PETScforPython

3.1.AnOverviewofPETSc

3.1.1.Vectors

3.1.2.Matrices

3.1.3.LinearSolvers

3.1.4.NonlinearSolvers

3.1.5.Time-Steppers

3.2.UsingPETScforPython

3.2.1.WorkingwithVectors

3.2.2.WorkingwithMatrices

3.2.3.UsingLinearSolvers

3.2.4.UsingNonlinearSolvers

3.3.EfficiencyTests

3.3.1.ThePoissonProblem

3.3.2.AMatrix-FreeApproachfortheLinearProblem

3.3.3.MeasuringOverhead

4.Conclusion

References

Chapter12DOMAINDECOMPOSITIONMETHODSINCOMPUTATIONALFLUIDDYNAMICS

Abstract

1.Introduction

2.SchurComplementDomainDecompositionMethod

2.1.TheSteklovOperator

2.2.EigenvaluesofSteklovOperator

3.PreconditionersfortheSchurComplementMatrix

3.1.TheNeumann-NeumannPreconditioner

3.2.TheInterfaceStripPreconditioner(ISP)

4.TheAdvective-DiffusiveCase

5.ImplementationoftheNeumann-NeumannPreconditioner

6.TheInterfaceStripPreconditioner:SolutionoftheStripProblem

6.1.ImplementationDetailsoftheIISDSolver

7.ClassicalOverlappingDomainDecompositionMethod:AlternatingSchwarzMethods

8.NumericalExamplesinSequentialEnvironments

8.1.ThePoisson’sProblem

8.2.TheScalarAdvective-DiffusiveProblem

8.2.1.SUPGVariationalFormulation

8.3.TheHypersonicFlowOveraFlatPlateTest

8.3.1.PhysicalModel

8.3.2.InviscidApproximation

8.3.3.VariationalFormulation

8.3.4.TestandResults

9.NumericalExamplesinParallelEnvironment

9.1.ThePoisson’sProblem

9.2.TheScalarAdvective-DiffusiveProblem

9.3.TheCoupledHydrologicalFlowModel

9.3.1.SubsurfaceFlow

9.3.2.SurfaceFlow[42,21]

9.3.3.BoundaryConditions

9.3.4.Saint-VenantNumericalExample

9.3.5.CoupledSurface-SubsurfaceFlowNumericalTest

9.4.TheStokesFlowinaLongHorizontalChannel

9.4.1.IncompressibleNavier-StokesEquations

9.4.2.TestandResults

9.5.TheViscousIncompressibleNavier-StokesFlowAroundanInfiniteCylinder

9.6.TheFractionalStepScheme.TheLidDrivenCavity

9.6.1.DisaggregatedScheme

9.6.2.TestandResults

9.6.3.SomeCommentsontheScalabilityoftheIISD+ISPPreconditioner

9.7.TheWindFlowArounda3DImmersedBody.TheAHMEDModel

9.7.1.AhmedBody:NumericalResultsforVeryLowReynoldsNumber

9.7.2.AhmedBody:NumericalResultsforHighReynoldsNumber

10.Conclusions

References

Chapter13MESHADAPTATIONALGORITHMBASEDONGRADIENTOFSTRAINENERGYDENSITY

Abstract

1.Introduction

2.Nearest-NodesFiniteElementMethodwithInter-dependentShapeFunctions

3.GradientofStrainEnergyDensityasaGuideforMeshMod-ification

4.MeshModificationOperators

4.1.MeshRefinement

4.2.MeshCoarsening

4.3.MeshSmoothing

5.NumericalExamples

6.ConcludingRemarks

References

INDEX

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