Fluid-Solid Interaction Dynamics: Theory, Variational Principles, Numerical Methods, and Applications

Description

Fluid-Solid Interaction Dynamics: Theory, Variational Principles, Numerical Methods and Applications gives a comprehensive accounting of fluid-solid interaction dynamics, including theory, numerical methods and their solutions for various FSI problems in engineering. The title provides the fundamental theories, methodologies and results developed in the application of FSI dynamics. Four numerical approaches that can be used with almost all integrated FSI systems in engineering are presented. Methods are linked with examples to illustrate results. In addition, numerical results are compared with available experiments or numerical data in order to demonstrate the accuracy of the approaches and their value to engineering applications. The title gives readers the state-of-the-art in theory, variational principles, numerical modeling and applications for fluid-solid interaction dynamics. Readers will be able to independently formulate models to solve their engineering FSI problems using information from this book. Cover Half Title Fluid–Solid Interaction Dynamics Copyright Preface 1 Introduction 1.1 Fluid–solid interaction dynamics and its characteristics 1.2 Fluid–solid interaction problems in engineering 1.3 Solution approaches to fluid–solid interaction problems 1.3.1 Approximate solution with no fluid–solid interaction 1.3.2 Quasicoupling approximation method 1.3.3 Solution of integrated coupling fields 1.4 Approaches to deriving numerical equations 1.4.1 Problem and its governing equations 1.4.2 Analytical solution 1.4.2.1 Natural vibration 1.4.2.2 Orthogonal relationships of natural modes 1.4.2.3 Generalized mass and stiffness 1.4.2.4 General solution of free vibrations 1.4.2.5 General solution of forced vibrations 1.4.3 Variational formulations and Rayleigh–Ritz method 1.4.3.1 Admissible displacement field 1.4.3.2 Variational formulation 1.4.3.3 Rayleigh–Ritz method 1.4.3.4 Example 1.1 1.4.4 Finite element method 1.4.4.1 Mesh 1.4.4.2 Element analysis 1.4.4.3 Element assembling 1.4.4.4 Displacement boundary conditions 1.4.5 Weighted residual methods 1.4.6 Finite difference method 1.5 Short historical review on fluid–solid interaction 1.5.1 Terms of fluid–solid interaction and its subdisciplines in literatures 1.5.2 Historical remarkable events and progress on fluid–solid interaction 1.5.2.1 Aeroelasticity 1.5.2.2 Dam–water interactions 1.5.2.3 Pipe vibrations induced by flowing fluids 1.5.2.4 Ship–water interaction 1.5.2.5 Sloshing 1.5.2.6 Dynamic responses of structures to explosions in water 1.5.3 World-recognized conferences 1.5.3.1 ASME conferences 1.5.3.2 Fluid–structure interaction conferences by Wessex Institute of Technology 1.5.3.3 International Union for Theoretical and Applied Mechanics conferences 1.5.3.4 Hydroelasticity of water–structure interactions 1.5.3.5 Aeroelasticity of air–structure interactions 1.5.3.6 Aeroacoustics and aeroelasticity of turbomachines 1.5.3.7 Computational methods 1.5.3.8 Dynamics, vibrations, and acoustics 1.5.4 Influential review papers 1.5.4.1 Flow-induced vibrations 1.5.4.2 Acoustics 1.5.4.3 Sloshing dynamics 1.5.4.4 Aeroelasticity 1.5.4.5 Hydroelasticity and water–solid interactions 1.5.4.6 Computational methods 1.5.5 Important books on fluid–solid interaction 1.6 Main aim and characteristics of this book 1.7 Suggestions how to choose some contents as lecture notes 2 Cartesian tensor and matrix calculus 2.1 Cartesian tensor 2.1.1 Vector 2.1.2 Summation convention 2.1.3 Kronecker delta 2.1.4 Permutation symbol 2.1.5 e−δ Identity 2.1.6 Differentiation of a function f(x1, x2, x3) 2.1.7 Transformation of coordinates 2.1.8 Tensor 2.1.9 Quotient rule 2.1.10 Index forms of some important variables 2.1.11 Two primary identities 2.2 Matrix calculus 2.2.1 Types of matrix derivatives 2.2.2 Derivatives with vectors 2.2.2.1 Vector by scalar 2.2.2.2 Scalar by vector 2.2.2.3 Vector by vector 2.2.3 Derivatives with matrices 2.2.3.1 Matrix by scalar 2.2.3.2 Scalar by matrix 2.2.3.3 Matrix by matrix 2.2.4 Identities 2.2.4.1 Vector by vector 2.2.4.2 Scalar by vector 2.3 Exercise problems 2.3.1 Problem 1: prove the following formulations 2.3.2 Problem 2: prove the identity of three arbitrary vectors 2.3.3 Problem 3: express the constitutive equation in a tensor form 2.3.4 Problem 4: write the tensor equation in a coordinate (xyz) form 2.3.5 Problem 5: prove the following identities using index notations 2.3.6 Problem 6: prove eijkaiajbk=0 for nonzero vectors a and b 3 Fundamentals of continuum mechanics 3.1 Descriptions of the motion of a continuum 3.1.1 Material frame of reference 3.1.2 Spatial frame of reference 3.1.3 Arbitrary Lagrange Euler frame of reference 3.1.4 Updated Lagrangian system 3.1.5 Updated arbitrary Lagrange Euler system 3.2 Analysis of deformation 3.2.1 Displacement and strain 3.2.1.1 Strain tensors 3.2.1.2 Line element transformation and its stretch ratio Line element transformation Stretch ratio 3.2.1.3 Area element transformation and area ratio Area element transformation Area ratio Unit normal vector of deformed area element 3.2.1.4 Volume element transformation and volume ratio Volume element transformation Volume ratio 3.2.2 Velocity field and rate of deformation of fluids 3.3 Stress tensor 3.3.1 Cauchy’s stress 3.3.2 Piola–Kirchhoff stress 3.4 Constitutive equation 3.4.1 Solids 3.4.2 Fluids 3.5 Laws of conservation 3.5.1 Green theorem 3.5.2 Material derivatives of volume integral with mass density 3.5.2.1 Spatial system 3.5.2.2 Arbitrary Lagrange Euler system 3.5.3 Material derivatives of arbitrary integrands in a spatial system 3.5.4 Material derivatives of arbitrary integrands in the arbitrary Lagrange Euler system 3.5.4.1 The form based on the spatial coordinates 3.5.4.2 The form based on the referential coordinates 3.5.5 General forms of the conservation laws 3.5.5.1 Spatial system Integration form Differential form 3.5.5.2 Lagrange system Integration form Differential form 3.5.5.3 Arbitrary Lagrange Euler system Integration form Differential form 3.5.5.4 Updated arbitrary Lagrange Euler system Integration form Differential form 3.5.5.5 Updated Lagrange system Integration form Differential form 3.5.6 Jump condition and equation 3.5.7 Conservation of mass and the equation of continuity 3.5.7.1 Spatial system Integration form Differential system 3.5.7.2 Lagrange system Integration form Differential form 3.5.7.3 Arbitrary Lagrange Euler system Integration form Differential form 3.5.7.4 Updated arbitrary Lagrange Euler system Integration form Differential form 3.5.7.5 Updated Lagrange system Integration form Differential form 3.5.8 Conservation of momentum and equations of motion 3.5.8.1 Spatial system Integration form Differential form Angular momentum conservation 3.5.8.2 Lagrange system Integration form Differential form 3.5.8.3 Arbitrary Lagrange Euler system Integration form Differential form 3.5.8.4 Updated arbitrary Lagrange Euler system Integration form Differential form 3.5.8.5 Updated Lagrange system Integration form Differential form 3.5.9 Conservation of energy and equation of energy 3.5.9.1 Spatial system Integration form Differential form 3.5.9.2 Lagrange system Integration form Differential form 3.5.9.3 Arbitrary Lagrange Euler system Integration form Differential form 3.5.9.4 Updated arbitrary Lagrange Euler system Integration form Differential form 3.5.9.5 Update Lagrange system Integration form Differential form 3.6 Navier–Stokes equations and boundary conditions 3.6.1 Displacement solution of solid mechanics 3.6.1.1 Governing equations 3.6.1.2 Displacement solution equation 3.6.1.3 Linear waves in solids Dilatational wave in solids Shear wave in solids 3.6.2 Velocity–pressure solution equations of fluid mechanics 3.6.2.1 Governing equations Solid boundary Free surface 3.6.2.2 Navier–Stokes equation 3.6.2.3 Different cases of fluids Stokes fluid Perfect fluids Incompressible fluids 3.6.3 Bernoulli equation and potential flows 3.6.3.1 Bernoulli equation 3.6.3.2 Potential flows 3.6.3.3 Incompressible potential flows 3.6.3.4 Free surface conditions for incompressible potential flows 3.6.4 Linear waves in fluids 3.6.4.1 Wave equations 3.6.4.2 Radiation conditions Sommerfeld condition Radiation condition involving both free surface and pressure waves 4 Variational principles of linear fluid–solid interaction systems 4.1 Short review on historic background 4.2 Fluid–solid interaction problems and interaction conditions 4.2.1 Geometric and dynamic conditions on material interfaces 4.2.1.1 Dynamic condition 4.2.1.2 Geometrical condition 4.2.2 Interactions on nonfloating fluid–solid interaction interface 4.2.2.1 Velocity potential–displacement model 4.2.2.2 Pressure–acceleration model 4.2.3 Interaction on floating fluid–solid interaction interface 4.2.3.1 Velocity potential–displacement model 4.2.3.2 Pressure–acceleration model 4.2.4 Interactions on air–liquid interface 4.2.4.1 Two synthesized formulations Pressure form Velocity–potential form 4.2.4.2 Velocity potential–displacement model Same fluid Free surface of water 4.2.4.3 Pressure–acceleration model Same fluid Free surface of water 4.2.5 Conditions of surface–tension interactions 4.2.5.1 Synthesized equations on surface tension interface Velocity potential form Pressure form Surface membrane acceleration form 4.2.5.2 Surface tension interaction on air–liquid interface 4.2.5.3 Energy integration of surface tension Velocity–potential form Pressure form 4.2.6 Boundary conditions on infinity and moving structures 4.2.6.1 Conditions on infinity boundaries Γ±∞ 4.2.6.2 Boundary condition caused by a moving structure 4.3 A complementary energy model: pressure–acceleration form 4.3.1 Governing equations 4.3.1.1 Solid structure Dynamic equation Strain-displacement Constitutive equation Boundary conditions 4.3.1.2 Fluid subdomains (β=LorA) Dynamic equation Boundary conditions 4.3.1.3 Conditions on interaction interfaces Fluid–solid interaction interface Σ Gas–liquid interaction interfaces Γf(LA) Conditions on infinity boundaries Γ±∞ 4.3.2 Variational formulation 4.4 A potential–energy model: displacement–velocity potential form 4.4.1 Governing equations 4.4.1.1 Solid structure Dynamic equation Strain-displacement Constitutive equation Boundary conditions 4.4.1.2 Fluid subdomains (β=LorA) Dynamic equation Boundary conditions 4.4.1.3 Conditions on interaction interfaces Fluid–solid interaction interface Σ Gas–liquid interaction interfaces Γf(LA) 4.4.1.4 Conditions on infinity boundaries Γ±∞ 4.4.2 Variational formulation 4.5 Mixed energy models: displacement–pressure and acceleration–velocity potential forms 4.5.1 Displacement–pressure form 4.5.2 Acceleration–velocity potential form 4.6 Three field variational formulations 4.6.1 Displacement–pressure–velocity potential form 4.6.2 Displacement–acceleration–pressure form 4.7 Formulations with displacement potential or pressure impulse as a variable 4.7.1 Displacement–potential form 4.7.2 Pressure impulse form 4.8 Variational formulations for dissipative systems 4.8.1 Damping types 4.8.1.1 Viscous damping 4.8.1.2 Structural damping 4.8.1.3 Complex modulus 4.8.1.4 Rayleigh damping 4.8.1.5 Radiation and absorption conditions 4.8.2 Virtual variational formulations 4.8.3 Complex variational formulation 4.9 Variational formulations for pipes conveying fluid 4.9.1 Description and assumptions of the problem 4.9.1.1 Geometrical relationships Rotation angle of beam cross section Horizontal motion of beam section center due to bending Vertical component of fluid speed and pressure 4.9.2 Variational formulation 4.9.2.1 Boundary conditions 4.9.2.2 Kinetic energy density 4.9.2.3 Potential energy density 4.9.2.4 Variational functional of the system 4.9.3 Variational stationary conditions for governing equations 4.9.3.1 Variation of solid functional 4.9.3.2 Variations of the fluid functional 4.9.3.3 Governing equation of the system 4.9.4 Natural vibrations and first approximate frequency 5 Solutions of some linear fluid–solid interaction problems 5.1 One-dimensional problems 5.1.1 Dynamic response of one-dimensional fluid–solid interaction system to a pressure wave 5.1.1.1 Governing equations 5.1.1.2 Solution approach 5.1.1.3 Natural vibration No fluid–solid interactions Fluid–solid interactions 5.1.1.4 Force vibration 5.1.2 A mass-spring system coupled to a one-dimensional infinite fluid domain 5.1.2.1 Governing equations 5.1.2.2 Complex eigenvalues of the problem 5.1.3 Dynamic response of a Sommerfeld system 5.1.4 Natural vibration of an fluid–solid interaction system with free surface wave 5.1.4.1 Governing equations and corresponding functional 5.1.4.2 Approximate equation by the Rayleigh–Ritz method 5.1.4.3 Approximate solutions 5.1.5 Natural vibration of a vertical system with a floating fluid–solid interaction interface 5.1.5.1 Governing equations and corresponding functional 5.1.5.2 Approximate equation by the Rayleigh–Ritz method 5.1.5.3 Approximate solutions 5.2 Two dimensional problems 5.2.1 Sloshing modes of a two dimensional rectangular water container 5.2.1.1 Governing equations 5.2.1.2 Variable separation method for solution 5.2.1.3 Surface tension effect 5.2.2 Radiations of mixed compressive and gravity waves 5.2.2.1 Governing equations 5.2.2.2 Solution Functions Y(y) Orthogonality relationship Functions Xn 5.2.2.3 Discussion 5.2.3 Dam–water pond excited by earthquake 5.2.3.1 Problem and governing equation Water domain and boundaries On the free surface On wet interface On the infinite boundary 5.2.3.2 Solution Function Y(y) Function X(x) Dynamic pressure 5.2.3.3 Discussion 5.2.4 Beam–water interaction 5.2.4.1 Beam equations 5.2.4.2 Solution Function X(x) Function Y(y) No free surface waves Free surface wave considered Eigenvalue equations 5.2.4.3 Discussion Water depth less than the beam height Dry mode functions of beam Two-sided water case Radiation case 5.3 Three-dimensional problems 5.3.1 Vibrations of a floating rigid mass on the water 5.3.1.1 Governing equations Mass motion Water motion Boundary conditions Interaction condition 5.3.1.2 Solution Function Z(z) Function R(r) Velocity potential and pressure 5.3.1.3 Discussion 5.3.2 A water–spherical shell–damping layer interaction system 5.3.2.1 Mathematical model Solids Water Boundary and coupling conditions 5.3.2.2 General solution Power flows 5.3.2.3 Effect on noise reduction Relative pressure amplitude Relative power amplitude Pressure reduction factor Power reduction factor 5.3.2.4 Discussion 6 Preliminaries of waves 6.1 d’Alembert’s solution, dispersive, dissipation 6.1.1 d’Alembert’s solution 6.1.2 Dispersive wave 6.1.3 Dissipation wave 6.2 Nonlinear waves 6.2.1 A nonlinear wave and its characteristic solution 6.2.2 Burgers’ equation 6.2.3 Korteweg–de Vries equation 6.2.4 Generalized nonlinear water wave equation 6.2.5 Analytical solutions of nonlinear wave equations 6.2.5.1 Jacobi elliptic functions 6.2.5.2 Jacobi elliptic function expansion method 6.2.5.3 Solution of Korteweg–de Vries equation 6.2.5.4 Solution of Boussinesq equation 6.3 Linear water waves 6.3.1 Three-dimensional water waves 6.3.1.1 Governing equations 6.3.1.2 Initial conditions 6.3.2 Plane water wave 6.3.2.1 Governing equations 6.3.2.2 Solution by the method of variable separation 6.3.2.3 Deep and shallow water waves 6.3.2.4 Stationary wave or standing wave Wave profile on free surface Velocity field of water waves Orbit of particle motions 6.3.2.5 Traveling waves Wave profile on free surface Wave velocity fields Orbit of particle motion Pressure 6.3.2.6 Group velocity and phase velocity Wave profile on free surface Phase velocity and group velocity 6.3.2.7 Wave energies Kinetic energy Potential energy Mechanical energy 6.3.2.8 Wave energy transmission and wave resistance Wave energy transmission Wave resistance 6.3.3 Approximate theory for long waves 7 Finite element models for linear fluid–structure interaction problems 7.1 Introduction on finite element models for linear fluid–structure interaction 7.2 Displacement–velocity potential finite element model 7.2.1 Description of the problem 7.2.2 Governing equations 7.2.3 Variational formulation 7.2.4 Finite element equations 7.2.5 Natural vibration 7.2.5.1 Quadratic eigenvalue problem Pure imaginary eigenvalues 7.2.5.2 Solution of natural frequencies and modes Linearized method Orthogonality of the eigenvectors with matrices Ψ and Η¯ Orthogonality with matrix H Symmetric formulation 7.2.5.3 Free vibration 7.2.5.4 Dynamic response Mode summation approach in state space Noncoupling mode method 7.2.6 Examples 7.2.6.1 Mode solution of a spring–mass–water interaction system Governing equations Variational formulation Approximate solution by mode reduction of water domain 7.2.6.2 Finite element solution of a one-dimensional fluid–structure problem Governing equations Variational formulation Finite element equation Natural vibration Complex mode method Linearized solution Dynamic response to ground motion Complex response function Mode summation method in state space 7.3 Mixed finite element displacement–pressure model 7.3.1 General description of the problem 7.3.2 Variational formulation 7.3.3 Mixed finite element model 7.3.4 Symmetric matrix equations and approximations 7.3.4.1 Method I: Nonsingular fluid matrix k 7.3.4.2 Method II: Nonsingular solid mass matrix M 7.3.4.3 Method III: Nonsingular fluid matrix m⌢ 7.3.4.4 Method IV: Nonsingular solid stiffness matrix K⌢ 7.3.5 Effect of solid/fluid natural frequencies on fluid–structure interaction process 7.4 Substructure–subdomain approaches 7.4.1 Variational formulations in substructure–subdomain form 7.4.2 Displacement consistency model for solid substructure 7.4.2.1 Sets of mode vectors Consistent displacement modes between interfaces Normalized modes of fixed interface substructure 7.4.2.2 Rule and method for mode reduction Frequency reduction and its influence Reduction depending on vibration shape factor 7.4.2.3 Synthesis of substructure equations 7.4.3 Hybrid displacement model of solid substructure 7.4.3.1 Sets of mode vector Consistent force modes between interfaces Normalized modes for free interface substructure 7.4.3.2 Rule and method of mode reduction 7.4.3.3 Synthesis of substructure equations 7.4.4 Pressure equilibrium model of fluid domain 7.4.4.1 Sets of mode vector Pressure equilibrium modes between interfaces Normalized pressure modes of zero pressure interface subdomain 7.4.4.2 Synthesis of subdomain equations 7.4.5 Mixed model of substructure–subdomain 7.4.5.1 Set of mode vectors 7.4.5.2 Synthesis of substructure–subdomain equations 7.4.6 Special techniques for fluid–structure interaction systems 7.4.6.1 Frequency shift Constant pressure mode with zero frequency Global eigenvalue problem Substructure or subdomain method with no fluid–structure interaction Fluid–structure interaction case 7.4.6.2 Ground motion 7.4.7 A simple example: one-dimensional fluid–structure interaction problem 7.4.7.1 Governing equations 7.4.7.2 Variational formulation 7.4.7.3 Finite element equation 7.4.7.4 Natural vibration 7.4.7.5 Ground vertical motion 7.4.7.6 Pressure excitation on the free surface 7.4.8 Computer code design 7.4.9 Application examples 7.4.9.1 One-dimensional pressure wave in the water 7.4.9.2 Natural vibration and dynamic response of a two-dimensional fluid container excited by a ground motion 7.4.9.3 Dynamic response of a two-dimensional dam-water system subjected to a ground motion 7.4.9.4 Dynamic response of a two-dimensional beam-water system excited by a pressure wave 7.4.9.5 A liquid storage tank excited by earthquake loads 7.4.9.6 Structure–acoustic volume interaction system 7.4.9.7 A wave energy harvest device–water interaction system Governing equations Energy harvest device Electrical equations Mechanical motion equations Fluid domain Variational formulation Finite element equation Energy harvest device design Dry harvest device Dry harvest device affected by water buoyancy Natural vibration of fluid–structure interaction system Dynamic response and energy harvest system 8 Mixed finite element–boundary element model for linear water–structure interactions 8.1 Formulation of the boundary element method 8.1.1 Fundamental solution of Laplace equation 8.1.2 Formulation of boundary integral equation 8.1.2.1 Green’s second identity 8.1.2.2 Green’s third identity 8.1.2.3 Numerical solution 8.1.2.4 Example 8.1: A mass–spring unit landing on a mass floating on the water Initial conditions Equations of motion Boundary element equation for water 8.2 Mixed finite element–boundary element method for very large floating structure subjected to airplane landing impacts 8.2.1 Introduction 8.2.2 General description of the problem 8.2.3 Governing equations 8.2.3.1 Fluid domain 8.2.3.2 Solid substructures 8.2.3.3 Fluid–structure interaction interface conditions 8.2.3.4 Initial conditions 8.2.4 Mixed finite element–boundary element method 8.2.4.1 Finite element model of solid substructures Finite element equation Mode equation 8.2.4.2 Boundary element model of fluid domain 8.2.4.3 Mixed finite element–boundary element equation 8.2.4.4 Numerical solution procedure Characteristics of coupling equation Nonsymmetrical behavior Time-dependent matrices Solution strategies Simultaneous solution Partitioned solution Numerical integration Time integration scheme 8.2.4.5 Example 8.2: A mass–damper–spring unit landing on a floating beam 8.2.4.6 Example 8.3: A landing beam lands on a floating beam Theoretical mode functions of free–free beam Geometrical and physical parameters Nondimensional form of results 8.2.4.7 Example 8.4: A simulation for a car running test 8.3 Finite element–boundary element modeling for dynamic response of structures excited by incident water waves 8.3.1 General description of the problem and governing equations 8.3.1.1 Structure 8.3.1.2 Fluid domain Dynamic equation Boundary conditions 8.3.1.3 Interaction condition 8.3.2 Solution approach 8.3.3 Variational formulation 8.3.4 Mixed finite element–boundary element equations 8.3.4.1 Displacement interpolation of structure 8.3.4.2 Interpolation of fluid velocity potential 8.3.4.3 Derivation of finite element–boundary element equation 8.4 Mirror image method for acoustic radiations of underwater structures 8.4.1 Green functions of Helmholtz equation 8.4.2 Green identity 8.4.3 Acoustic radiation in infinite fluid domain 8.4.3.1 Governing equations of the problem 8.4.3.2 Solution approach Mode summation for solid structure Mode functions of fluid corresponding the Ith solid mode Mode summation of the pressure and fluid–solid interaction equation 8.4.3.3 Example 8.5: Acoustic radiation of an underwater small ball Vibration of elastic ball Pressure mode Fluid–solid interaction radiation equation 8.4.4 Generalized acoustic radiation problems 8.4.4.1 Governing equations 8.4.4.2 Image method Relationships valid for the symmetrical image system Radiation pressure for the quarter water with no free surface wave Radiation pressure for the quarter water with free surface wave 8.4.4.3 Example 8.6: A quarter radiation Case for no free surface wave Case for free surface wave 9 Hydroelasticity theory of ship–water interactions 9.1 Fundamentals for ship–water interactions 9.1.1 Frames of reference 9.1.2 Governing equations 9.1.2.1 Ship structure 9.1.2.2 Water On the far-field boundary Γ±∞ Interaction conditions 9.1.3 Equations for static equilibrium state 9.1.4 Equations for steady motion 9.2 Incident waves 9.2.1 Equation of incident water waves 9.2.2 Linear plane gravity waves 9.2.3 Frequency of wave encounter 9.3 Linear hydroelasticity theory 9.3.1 Linearized governing equations 9.3.1.1 Ship structure 9.3.1.2 Water 9.3.1.3 Interaction interface 9.3.2 Equations in the modal space 9.3.2.1 Mode summation method for ship motions Natural frequencies and modes with orthogonality Mode equation of ship motions 9.3.2.2 Water domain Potential of velocity Bernoulli equation Boundary conditions 9.3.2.3 Interaction interface Kinematic condition Fluid pressure on wet interface 9.3.2.4 Generalized fluid forces 9.3.2.5 Generalized equation of motion Equation of static equilibrium Equation of steady motion Equation of flexible structure subject waves 9.3.3 Numerical solutions 9.3.3.1 Free surface Green functions Equations for free surface Green functions Two-dimensional case Three-dimensional case 9.3.3.2 Green identity and the solution of ship excited by waves Green third identity 9.3.3.3 Solution of the ship–water interaction in hydroelasticity Generalized coordinate equation of structure Matrix equation for the radiated velocity potential Solution process of ship–water interaction 9.3.3.4 Comparison with finite element–boundary element method 9.3.4 Examples 9.3.4.1 Dynamic response of a two-dimensional beam-like ship to an incident wave Radiated potential of beam-like ship Diffracted potential of beam-like ship Dynamic response of interaction system 9.3.4.2 A wave energy harvesting device Electrical equations Mechanical motion equations Fluid domain Solution discussion 10 Variational principles for nonlinear fluid–solid interactions 10.1 A short review on variational principles for nonlinear dynamical systems 10.2 Fundamental variational concepts for nonlinear systems 10.2.1 The motion of a continuum 10.2.2 Translation and transmission velocities of a curved surface 10.2.3 Time derivative of an integral over a moving volume in space 10.2.4 A local variation and a material variation 10.2.5 Local variation of an integral over a moving volume in space 10.3 Governing equations 10.3.1 Solid domain 10.3.1.1 Dynamic equation 10.3.1.2 Strain-displacement and velocity-displacement relations 10.3.1.3 Constitutive equations 10.3.1.4 Boundary conditions 10.3.2 Fluid domain 10.3.2.1 State equation 10.3.2.2 Equation of continuity 10.3.2.3 Conservation of energy 10.3.2.4 Conservation of the identity of particles 10.3.2.5 Dynamic equation 10.3.2.6 Boundary conditions 10.3.3 Fluid–structure interface 10.3.4 Variational conditions at initial time t1 and final time t2 10.4 Variational principles 10.4.1 Fluid motion assumed rotational 10.4.1.1 Compressible fluid with s,i≠0 in the fluid domain Ωf 10.4.1.2 Compressible fluid with s,i≡0 in the fluid domain Ωf 10.4.1.3 Incompressible fluid 10.4.2 Fluid flow assumed irrotational 10.4.2.1 Compressible fluid 10.4.2.2 Incompressible fluid 10.4.3 Discussion 10.4.3.1 Fluid domain Ωf and its boundary Γv excluded 10.4.3.2 Solid domain Ωs and its boundary ST excluded 10.4.3.3 Special cases for fluid–solid interaction 10.4.3.4 Linear models 10.5 Two simple examples of applications 10.5.1 A one-dimensional water–mass–spring interaction problem 10.5.2 A forced one-dimensional gas–mass–spring dynamic interaction problem 10.5.2.1 Dynamic equations derived from the functional 10.5.2.2 Approximate solution 10.6 Variational principles for nonlinear elastic ship–water interactions 10.6.1 Short introduction 10.6.2 Governing equations in the moving reference frame 10.6.2.1 Ship structure 10.6.2.2 Water 10.6.2.3 Water–ship interface 10.6.3 Variational formulations in the moving reference frame 10.6.3.1 Compressible water 10.6.3.2 Incompressible water 10.6.4 Rigid ship dynamics 10.6.4.1 Motion equations for rigid ship 10.6.4.2 Steady-state problem: rigid ship traveling in calm water 10.6.4.3 Sea keeping problem: rigid ship traveling in waves 10.6.5 Offshore and hydroelastic examples 10.6.5.1 Dynamic response of fixed rigid rod to an incident wave 10.6.5.2 A two-dimensional elastic beam traveling in waves 11 Mixed finite element–computational fluid dynamics method for nonlinear fluid–solid interactions 11.1 Updated Lagrangian formulation in finite element methods 11.1.1 Principle of virtual work and dynamic equilibrium equations 11.1.2 Expression of stress virtual work 11.1.3 Total Lagrangian formulation in solid dynamics 11.1.3.1 Dynamic equation 11.1.3.2 Incremental decompositions 11.1.3.3 Dynamic equation with incremental decompositions 11.1.3.4 Linearization of dynamic equation 11.1.3.5 Matrix equation of displacement-based element Dynamic equation with implicit time integration Finite element matrices 11.1.4 Updated Lagrangian formulation in solid dynamics 11.1.4.1 Dynamic equation 11.1.4.2 Incremental decompositions 11.1.4.3 Dynamic equation with incremental decompositions 11.1.4.4 Linearization of dynamic equation 11.1.4.5 Matrix equation of displacement-based element Dynamic equation with implicit time integration Finite element matrices 11.1.5 Solution of nonlinear equations 11.1.5.1 Newton–Raphson iteration 11.1.5.2 Solution of explicit nonlinear dynamics equation 11.1.5.3 Solution of implicit nonlinear dynamics equation 11.1.6 A one-dimensional example 11.1.6.1 Coordinate interpolations 11.1.6.2 Displacement interpolations 11.1.6.3 Total Lagrangian formulations Longitudinal strain Increment strain Strain–displacement matrices Stiffness matrices and nodal point stress vector Mass matrix and external load vector Dynamic Eq. (11.15) Introducing the displacement boundary condition 11.1.6.4 Updated Lagrangian formulations Longitudinal strain Increment strain Strain–displacement matrices Stiffness matrices and nodal point stress vector Mass matrix and external load vector Dynamic equation Introducing the displacement boundary condition 11.1.6.5 Iteration equation 11.2 Updated arbitrary Lagrangian–Eulerian formulations in computational fluid dynamics 11.2.1 History and development of arbitrary Lagrangian–Eulerian description 11.2.2 Basic discretization techniques of computational fluid dynamics 11.2.2.1 Finite difference method Basic concept Finite difference approximations for first-order derivative Finite difference approximations for second-order derivative Difference formulas with multiple points General operator methods for finite difference formulas Higher-order derivatives Implicit finite difference formulas and its general generation Multidimensional finite difference formulas Nonuniform meshes 11.2.2.2 Finite volume method Conservative discretization Fundamental concepts of finite volume method Evaluation formulas for finite volume method 11.2.3 Consistency, stability, and convergence of numerical schemes 11.2.3.1 Consistency 11.2.3.2 Stability 11.2.3.3 Convergence 11.2.3.4 von Neumann method for stability analysis Error equation and its Fourier series Amplification factor Unconditionally unstable example Conditional stability example Unconditional stability example General von Neumann’s formation Spectral analysis of numerical errors 11.2.3.5 Nature of flow equations with its quasi-linear form Two first-order equations in two-dimensional space Stationary shallow-water equations 11.2.4 Updated arbitrary Lagrangian–Eulerian formulations in finite difference method and finite volume method 11.2.4.1 Generalized numerical conservative laws in finite volume method Matrix integration form Discretized matrix form in finite volume method Numerical process 11.2.4.2 Generalized numerical conservative laws in finite difference method Differential matrix equation for finite difference method Numerical matrix equation for finite difference method 11.3 Mixed finite element–computational fluid dynamics solutions for nonlinear fluid–solid interaction problems 11.3.1 Direct or simultaneous integration 11.3.2 Partitioned iteration 11.3.3 A simple example 11.3.3.1 Governing equations of water domain 11.3.3.2 Differential matrix formulations 11.3.3.3 Integrated dynamic equations 11.3.3.4 Numerical equations 11.3.3.5 Mixed finite element–computational fluid dynamics equations 11.4 Numerical examples by the mixed finite element–finite difference method 11.4.1 Description of nonlinear rigid body–water fluid–solid interaction systems 11.4.1.1 Reference frames 11.4.1.2 Governing equations Fluid domain Boundary conditions Solid domain Fluid–structure interaction interface Initial conditions 11.4.2 Arbitrary Lagrangian–Eulerian description of the water motion 11.4.3 Numerical formulations 11.4.3.1 Artificial compressibility 11.4.3.2 Discretization of momentum equation Time derivatives Spatial derivatives 11.4.3.3 Discretization of kinematic condition on free surface Time derivatives Spatial derivatives 11.4.3.4 Dynamic equation of solid motion 11.4.3.5 Numerical solution 11.4.4 A fluid–mass–spring interaction system 11.4.5 A spring-supported fluid–rigid-dam interaction system 11.4.5.1 Free vibration 11.4.5.2 Forced vibration 11.4.6 Two-dimensional rigid body floating in a free-surface fluid 11.4.6.1 Free vibration 11.4.6.2 Forced vibration 11.4.7 Prescribed motion of a rigid cylinder floating in the water 11.4.8 Flows around a bluff body 11.4.8.1 Overset grids 11.4.8.2 The definition of the problem 11.4.8.3 Case 1: Fixed cylinder at γ=0=θ 11.4.8.4 Case 2: Cylinder rotating about its mass center at γ=0 line 11.4.8.5 Case 3: Cylinder with 2 degrees of freedom of motion, γ and θ variables 11.4.8.6 Remark 12 Mixed finite element—smoothed particle methods for nonlinear fluid–solid interactions 12.1 Introduction 12.1.1 Limitations of grid-based methods for violent flows 12.1.2 Key ideas of meshfree particle methods 12.1.3 History and developments with applications 12.1.3.1 Meshfree Galerkin methods 12.1.3.2 Smoothed particle hydrodynamics Application on incompressible fluid Application on multiphase flow Application on solids Application on fluid–solid interaction problems 12.2 Fundamentals of smoothed particle hydrodynamics 12.2.1 Smoothed particle hydrodynamics interpolations 12.2.1.1 Integral representation of a function 12.2.1.2 Integral representation of the derivatives of a function First derivative Second derivative 12.2.1.3 Rules for nth-order accuracy 12.2.1.4 Consistency of kernel approximation 12.2.2 Particle approximation 12.2.2.1 Approximation of a function 12.2.2.2 Techniques in deriving smoothed particle hydrodynamics formulations 12.2.2.3 Consistency of the particle approximation 12.2.3 Construction of smoothing functions 12.2.3.1 Polynomial form Example I: Quadratic smoothing function (κ=1) Example II: Quartic smoothing function (κ=1) 12.2.3.2 Kernel functions Quadratic (Hicks and Liebrock, 2000) Quartic (Lucy, 1977) Johnson’s quadratic (Johnson and Beissel, 1996) Gaussian (Gingold and Monaghan, 1977) Super Gaussian (Monaghan and Poinracic, 1985a) Cubic spline (Monaghan and Poinracic, 1985a) Quartic spline Quintic spline New quartic (Liu et al., 2003) 12.2.3.3 Numerical tested functions and error estimation Integration approximation Particle approximation Error parameter for accuracy estimation 12.2.3.4 Numerical test results Integration approximation Particle approximations 12.2.3.5 Choosing smoothing length and symmetrization of particle interaction 12.2.3.6 Smoothed particle hydrodynamics approximation rules 12.2.4 Smoothed particle hydrodynamics formulation for Navier–Stokes equations 12.2.4.1 Particle approximation of density Summation density Continuity density 12.2.4.2 Particle approximation of momentum 12.2.4.3 Particle approximation of energy 12.2.5 Numerical techniques for fluid flows 12.2.5.1 Artificial viscosity 12.2.5.2 Artificial heat 12.2.5.3 Spurious zero-energy mode 12.2.5.4 Pressure calculation for incompressible fluids Artificial compressibility Projection method Kinematic constraint method 12.2.5.5 Boundary treatment Free surface Fixed solid boundary Enforced essential boundary conditions 12.2.5.6 Time integration 12.2.5.7 Particle interactions 12.2.5.8 Open source smoothed particle hydrodynamics solvers 12.2.6 Improved methods based on smoothed particle hydrodynamics 12.2.6.1 Corrective smoothed particle method 12.2.6.2 Moving particle semiimplicit method Function approximation Modeling of incompressibility Derivative approximation Approximation of Laplacian 12.3 Meshfree Galerkin methods 12.3.1 Moving least square representing kernel interpolant 12.3.1.1 Local standard least square approximation 12.3.1.2 Moving process for global approximation 12.3.1.3 An example 12.3.2 Shepard interpolant 12.3.3 Orthogonal basis for local approximations 12.3.4 Applications of the moving least square reproducing kernels 12.4 Mixed finite element—smoothed particle method for fluid–solid interaction problems 12.4.1 Generalized solution procedure 12.4.2 Modeling of fluid–solid interaction involving large rigid motions with small elastic deformation 12.4.2.1 Governing equations Solid domain Displacement and velocity fields Strain field and strain energy density Kinetic energy density Generalized force Dynamic equation for numerical simulation Fluid domain Navier–Stokes equation Fluid boundary conditions Fluid–solid interaction boundary condition Pressure Neumann condition Laplacian operator compensation near fluid–solid interaction interface Intermediate velocity of fluid–solid interaction boundary 12.4.2.2 Integrated coupling equations Solid domain Fluid domain Fluid–solid interaction interface Initial conditions Solid body Fluid domain 12.4.2.3 Numerical simulation process 12.4.3 Application examples 12.4.3.1 Rigid wedge dropping on the water 12.4.3.2 Flexible wedge dropping on the water 12.4.3.3 Two-dimensional water dam breaking impact on rigid/flexible beams Spring-supported rigid beam Fixed elastic beam 12.4.3.4 Flow-induced vibration of two-dimensional cylinder Motion equation of two-dimensional cylinder Fluid domain 12.4.3.5 Wave/wind energy harvesting systems Fundamental principle for wave/wind energy harvesting device design Interdisciplinary research on fluid–solid interaction and electric–mechanical interaction Resonance Periodical orbit of nonlinear system Aerofoil flutter system Active aerofoil Equations of aerofoil motion Electromagnetic power generation Solution approach Solution results Semiactive aerofoil Equations of aerofoil motions Fluid motion Solution Harvested energy Appendix Numerical methods solving finite element dynamic equations A.1 Introduction A.2 Fundamental iteration formulations A.2.1 Generalized dynamic equation A.2.2 Iteration criterions A.2.2.1 Energy criterion A.2.2.2 Force criterion A.2.2.3 Displacement criterion A.3 Five numerical integration schemes A.3.1 Newmark method A.3.2 Wilson-θ method A.3.3 Hilber-α method A.3.4 Hilber collocation method A.3.5 Central difference method A.4 Stability and accuracy A.4.1 Stability and spectral radius A.4.2 Period elongation and amplitude decay A.5 Implementation A.5.1 Initial calculations A.5.1.1 Newmark method A.5.1.2 Hilber-α method A.5.1.3 Wilson-θ method A.5.1.4 Hilber collocation method A.5.1.5 Central difference method A.5.2 Form effective stiffness matrix A.5.3 Triangular decomposition of matrix Kˆ A.5.4 Calculations at each time step A.6 Time element program A.6.1 Main characteristics A.6.2 Flowchart of the program A.7 Examples A.7.1 Example 1 A.7.2 Example 2 A.7.3 Example 3 A.8 Program of Time Element Methods A.9 User manual A.10 Input and output files for Examples 1 and 2 in Section A.7 A.10.1 Example 1 Bibliography Index Cover 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