Numerical Continuum Mechanics

✍ Scribed by Vladimir N. Kukudzhanov; Alexei Zhurov

• Publisher: De Gruyter
• Year: 2012
• Tongue: English
• Leaves: 447
• Series: De Gruyter Studies in Mathematical Physics; 15
• Category: Library

✦ Synopsis

This work focuses on computational methods in continuum thermomechanics. The text is based on the author's lectures, which ensures a didactical and coherent buildup. The main emphasis is put on the presentation of ideas and qualitative considerations, illustrated by specific examples and applications. Conditions and explanations that are essential for the practical application of methods are discussed thoroughly.

✦ Table of Contents

Preface
I Basic equations of continuum mechanics
1 Basic equations of continuous media
1.1 Methods of describing motion of continuous media
1.1.1 Coordinate systems and methods of describing motion of continuous media
1.1.2 Eulerian description
1.1.3 Lagrangian description
1.1.4 Differentiation of bases
1.1.5 Description of deformations and rates of deformation of a continuous medium
1.2 Conservation laws. Integral and differential forms
1.2.1 Integral form of conservation laws
1.2.2 Differential form of conservation laws
1.2.3 Conservation laws at solution discontinuities
1.2.4 Conclusions
1.3 Thermodynamics
1.3.1 First law of thermodynamics
1.3.2 Second law of thermodynamics
1.3.3 Conclusions
1.4 Constitutive equations
1.4.1 General form of constitutive equations. Internal variables
1.4.2 Equations of viscous compressible heat-conducting gases
1.4.3 Thermoelastic isotropic media
1.4.4 Combined media
1.4.5 Rigid-plastic media with translationally isotropic hardening
1.4.6 Elastoplastic model
1.5 Theory of plastic flow. Theory of internal variables
1.5.1 Statement of the problem. Equations of an elastoplastic medium
1.5.2 Equations of an elastoviscoplastic medium
1.6 Experimental determination of constitutive relations under dynamic loading
1.6.1 Experimental results and experimentally obtained constitutive equations
1.6.2 Substantiation of elastoviscoplastic equations on the basis of dislocation theory
1.7 Principle of virtual displacements. Weak solutions to equations of motion
1.7.1 Principles of virtual displacements and velocities
1.7.2 Weak formulation of the problem of continuum mechanics
1.8 Variational principles of continuum mechanics
1.8.1 Lagrange’s variational principle
1.8.2 Hamilton’s variational principle
1.8.3 Castigliano’s variational principle
1.8.4 General variational principle for solving continuum mechanics problems
1.8.5 Estimation of solution error
1.9 Kinematics of continuous media. Finite deformations
1.9.1 Description of the motion of solids at large deformations
1.9.2 Motion: deformation and rotation
1.9.3 Strain measure. Green-Lagrange and Euler-Almansi strain tensors
1.9.4 Deformation of area and volume elements
1.9.5 Transformations: initial, reference, and intermediate configurations
1.9.6 Differentiation of tensors. Rate of deformation measures
1.10 Stress measures
1.10.1 Current configuration. Cauchy stress tensor
1.10.2 Current and initial configurations. The first and second Piola-Kirchhoff stress tensors
1.10.3 Measures of the rate of change of stress tensors
1.11 Variational principles for finite deformations
1.11.1 Principle of virtual work
1.11.2 Statement of the principle in increments
1.12 Constitutive equations of plasticity under finite deformations
1.12.1 Multiplicative decomposition. Deformation gradients
1.12.2 Material description
1.12.3 Spatial description
1.12.4 Elastic isotropic body
1.12.5 Hyperelastoplastic medium
1.12.6 The von Mises yield criterion
II Theory of finite-difference schemes
2 The basics of the theory of finite-difference schemes
2.1 Finite-difference approximations for differential operators
2.1.1 Finite-difference approximation
2.1.2 Estimation of approximation error
2.1.3 Richardson’s extrapolation formula
2.2 Stability and convergence of finite difference equations
2.2.1 Stability
2.2.2 Lax convergence theorem
2.2.3 Example of an unstable finite difference scheme
2.3 Numerical integration of the Cauchy problem for systems of equations
2.3.1 Euler schemes
2.3.2 Adams-Bashforth scheme
2.3.3 Construction of higher-order schemes by series expansion
2.3.4 Runge-Kutta schemes
2.4 Cauchy problem for stiff systems of ordinary differential equations
2.4.1 Stiff systems of ordinary differential equations
2.4.2 Numerical solution
2.4.3 Stability analysis
2.4.4 Singularly perturbed systems
2.4.5 Extension of a rod made of a nonlinear viscoplastic material
2.5 Finite difference schemes for one-dimensional partial differential equations
2.5.1 Solution of the wave equation in displacements. The cross scheme
2.5.2 Solution of the wave equation as a system of first-order equations (acoustics equations)
2.5.3 The leapfrog scheme
2.5.4 The Lax-Friedrichs scheme
2.5.5 The Lax-Wendroff Scheme
2.5.6 Scheme viscosity
2.5.7 Solution of the wave equation. Implicit scheme
2.5.8 Solution of the wave equation. Comparison of explicit and implicit schemes. Boundary points
2.5.9 Heat equation
2.5.10 Unsteady thermal conduction. Explicit scheme (forward Euler scheme)
2.5.11 Unsteady thermal conduction. Implicit scheme (backward Euler scheme)
2.5.12 Unsteady thermal conduction. Crank-Nicolson scheme
2.5.13 Unsteady thermal conduction. Allen-Cheng explicit scheme
2.5.14 Unsteady thermal conduction. Du Fort-Frankel explicit scheme
2.5.15 Initial-boundary value problem of unsteady thermal conduction. Approximation of boundary conditions involving derivatives
2.6 Stability analysis for finite difference schemes
2.6.1 Stability of a two-layer finite difference scheme
2.6.2 The von Neumann stability condition
2.6.3 Stability of the wave equation
2.6.4 Stability of the wave equation as a system of first-order equations. The Courant stability condition
2.6.5 Stability of schemes for the heat equation
2.6.6 The principle of frozen coefficients
2.6.7 Stability in solving boundary value problems
2.6.8 Step size selection in an implicit scheme in solving the heat equation
2.6.9 Step size selection in solving the wave equation
2.7 Exercises
3 Methods for solving systems of algebraic equations
3.1 Matrix norm and condition number of matrix
3.1.1 Relative error of solution for perturbed right-hand sides. The condition number of a matrix
3.1.2 Relative error of solution for perturbed coefficient matrix
3.1.3 Example
3.1.4 Regularization of an ill-conditioned system of equations
3.2 Direct methods for linear system of equations
3.2.1 Gaussian elimination method. Matrix factorization
3.2.2 Gaussian elimination with partial pivoting
3.2.3 Cholesky decomposition. The square root method
3.3 Iterative methods for linear system of equations
3.3.1 Single-step iterative processes
3.3.2 Seidel and Jacobi iterative processes
3.3.3 The stabilization method
3.3.4 Optimization of the rate of convergence of a steady-state process
3.3.5 Optimization of unsteady processes
3.4 Methods for solving nonlinear equations
3.4.1 Nonlinear equations and iterative methods
3.4.2 Contractive mappings. The fixed point theorem
3.4.3 Method of simple iterations. Sufficient convergence condition
3.5 Nonlinear equations: Newton’s method and its modifications
3.5.1 Newton’s method
3.5.2 Modified Newton-Raphson method
3.5.3 The secant method
3.5.4 Two-stage iterative methods
3.5.5 Nonstationary Newton method. Optimal step selection
3.6 Methods of minimization of functions (descent methods)
3.6.1 The coordinate descent method
3.6.2 The steepest descent method
3.6.3 The conjugate gradient method
3.6.4 An iterative method using spectral-equivalent operators or reconditioning
3.7 Exercises
4 Methods for solving boundary value problems for systems of equations
4.1 Numerical solution of two-point boundary value problems
4.1.1 Stiff two-point boundary value problem
4.1.2 Method of initial parameters
4.2 General boundary value problem for systems of linear equations
4.3 General boundary value problem for systems of nonlinear equations
4.3.1 Shooting method
4.3.2 Quasi-linearization method
4.4 Solution of boundary value problems by the sweep method
4.4.1 Differential sweep
4.4.2 Solution of finite difference equation by the sweep method
4.4.3 Sweep method for the heat equation
4.5 Solution of boundary value problems for elliptic equations
4.5.1 Poisson’s equation
4.5.2 Maximum principle for second-order finite difference equations
4.5.3 Stability of a finite difference scheme for Poisson’s equation
4.5.4 Diagonal domination
4.5.5 Solution of Poisson’s equation by the matrix sweep method
4.5.6 Fourier’s method of separation of variables
4.6 Stiff boundary value problems
4.6.1 Stiff systems of differential equations
4.6.2 Generalized method of initial parameters
4.6.3 Orthogonal sweep
4.7 Exercises
III Finite-difference methods for solving nonlinear evolution equations of continuum mechanics
5 Wave propagation problems
5.1 Linear vibrations of elastic beams
5.1.1 Longitudinal vibrations
5.1.2 Explicit scheme. Sufficient stability conditions
5.1.3 Longitudinal vibrations. Implicit scheme
5.1.4 Transverse vibrations
5.1.5 Transverse vibrations. Explicit scheme
5.1.6 Transverse vibrations. Implicit scheme
5.1.7 Coupled longitudinal and transverse vibrations
5.1.8 Transverse bending of a plate with shear and rotational inertia
5.1.9 Conclusion
5.2 Solution of nonlinear wave propagation problems
5.2.1 Hyperbolic system of equations and characteristics
5.2.2 Finite difference approximation along characteristics. The direct and semi-inverse methods
5.2.3 Inverse method. The Courant-Isaacson-Rees grid-characteristic scheme
5.2.4 Wave propagation in a nonlinear elastic beam
5.2.5 Wave propagation in an elastoviscoplastic beam
5.2.6 Discontinuous solutions. Constant coefficient equation
5.2.7 Discontinuous solutions of a nonlinear equation
5.2.8 Stability of difference characteristic equations
5.2.9 Characteristic and grid-characteristic schemes for solving stiff problems
5.2.10 Stability of characteristic and grid-characteristic schemes for stiff problems
5.2.11 Characteristic schemes of higher orders of accuracy
5.3 Two- and three-dimensional characteristic schemes and their application
5.3.1 Spatial characteristics
5.3.2 Basic equations of elastoviscoplastic media
5.3.3 Spatial three-dimensional characteristics for semi-linear system
5.3.4 Characteristic equations. Spatial problem
5.3.5 Axisymmetric problem
5.3.6 Difference equations. Axisymmetric problem
5.3.7 A brief overview of the results. Further development and generalization of the method of spatial characteristics and its application to the solution of dynamic problems
5.4 Coupled thermomechanics problems
5.5 Differential approximation for difference equations
5.5.1 Hyperbolic and parabolic forms of differential approximation
5.5.2 Example
5.5.3 Stability
5.5.4 Analysis of dissipative and dispersive properties
5.5.5 Example
5.5.6 Analysis of properties of finite difference schemes for discontinuous solutions
5.5.7 Smoothing of non-physical perturbations in a calculation on a real grid
5.6 Exercises
6 Finite-difference splitting method for solving dynamic problems
6.1 General scheme of the splitting method
6.1.1 Explicit splitting scheme
6.1.2 Implicit splitting scheme
6.1.3 Stability
6.2 Splitting of 2D/3D equations into 1D equations (splitting along directions)
6.2.1 Splitting along directions of initial-boundary value problems for the heat equation
6.2.2 Splitting schemes for the wave equation
6.3 Splitting of constitutive equations for complex rheological models into simple ones. A splitting scheme for a viscous fluid
6.3.1 Divergence form of equations
6.3.2 Non-divergence form of equations
6.3.3 One-dimensional equations. Ideal gas
6.3.4 Implementation of the scheme
6.4 Splitting scheme for elastoviscoplastic dynamic problems
6.4.1 Constitutive equations of elastoplastic media
6.4.2 Some approaches to solving elastoplastic equations
6.4.3 Splitting of the constitutive equations
6.4.4 The theory of von Mises type flows. Isotropic hardening
6.4.5 Drucker-Prager plasticity theory
6.4.6 Elastoviscoplastic media
6.5 Splitting schemes for points on the axis of revolution
6.5.1 Calculation of boundary points
6.5.2 Calculation of axial points
6.6 Integration of elastoviscoplastic flow equations by variation inequality
6.6.1 Variation inequality
6.6.2 Dissipative schemes
6.7 Exercises
7 Solution of elastoplastic dynamic and quasistatic problems with finite deformations
7.1 Conservative approximations on curvilinear Lagrangian meshes
7.1.1 Formulas for natural approximation of spatial derivatives
7.1.2 Approximation of a Lagrangian mesh
7.1.3 Conservative finite difference schemes
7.2 Finite elastoplastic deformations
7.2.1 Conservative schemes in one-dimensional case
7.2.2 A conservative two-dimensional scheme for an elastoplastic medium
7.2.3 Splitting of the equations of a hypoelastic material
7.3 Propagation of coupled thermomechanical perturbations in gases
7.3.1 Basic equations
7.3.2 Conservative finite difference scheme
7.3.3 Non-divergence form of the energy equation. A completely conservative scheme
7.4 The PIC method and its modifications for solid mechanics problems
7.4.1 Disadvantages of Lagrangian and Eulerian meshes
7.4.2 The particle-in-cell (PIC) method
7.4.3 The method of coarse particles
7.4.4 Limitations of the PIC method and its modifications
7.4.5 The combined flux and particle-in-cell (FPIC) method
7.4.6 The method of markers and fluxes
7.5 Application of PIC-type methods to solving elastoviscoplastic problems
7.5.1 Hypoelastic medium
7.5.2 Hypoelastoplastic medium
7.5.3 Splitting for a hyperelastoplastic medium
7.6 Optimization of moving one-dimensional meshes
7.6.1 Optimal mesh for a given function
7.6.2 Optimal mesh for solving an initial-boundary value problem
7.6.3 Mesh optimization in several parameters
7.6.4 Heat propagation from a combustion source
7.7 Adaptive 2D/3D meshes for finite deformation problems
7.7.1 Methods for reorganization of a Lagrangian mesh
7.7.2 Description of motion in an arbitrary moving coordinate system
7.7.3 Adaptive meshes
7.8 Unsteady elastoviscoplastic problems on moving adaptive meshes
7.8.1 Algorithms for constructing moving meshes
7.8.2 Selection of a finite difference scheme
7.8.3 A hybrid scheme of variable order of approximation at internal nodes
7.8.4 A grid-characteristic scheme at boundary nodes
7.8.5 Calculation of contact boundaries
7.8.6 Calculation of damage kinetics
7.8.7 Numerical results for some applied problems with finite elastoviscoplastic strains
7.9 Exercises
8 Modeling of damage and fracture of inelastic materials and structures
8.1 Concept of damage and the construction of models of damaged media
8.1.1 Concept of continuum fracture and damage
8.1.2 Construction of damage models
8.1.3 Constitutive equations of the GTN model
8.2 Generalized micromechanical multiscale damage model
8.2.1 Micromechanical model. The stage of plastic flow and hardening
8.2.2 Stage of void nucleation
8.2.3 Stage of the appearance of voids and damage
8.2.4 Relationship between micro and macro parameters
8.2.5 Macromodel
8.2.6 Tension of a thin rod with a constant strain rate
8.2.7 Conclusion
8.3 Numerical modeling of damaged elastoplastic materials
8.3.1 Regularization of equations for elastoplastic materials at softening
8.3.2 Solution of damage problems
8.3.3 Inverse Euler method
8.3.4 Solution of a boundary value problem. Computation of the Jacobian
8.3.5 Splitting method
8.3.6 Integration of the constitutive relations of the GTN model
8.3.7 Uniaxial tension. Computational results
8.3.8 Bending of a plate
8.3.9 Comparison with experiment
8.3.10 Modeling quasi-brittle fracture with damage
8.4 Extension of damage theory to the case of an arbitrary stress-strain state
8.4.1 Well-posedness of the problem
8.4.2 Limitations of the GTN model
8.4.3 Associated viscoplastic law
8.4.4 Constitutive relations in the absence of porosity (k < 0.4, f = 0, σr = 0)
8.4.5 Fracture model. Fracture criteria
8.5 Numerical modeling of cutting of elastoviscoplastic materials
8.5.1 Introduction
8.5.2 Statement of the problem
8.6 Conclusions. General remarks on elastoplastic equations
8.6.1 Formulations of systems of equations for elastoplastic media
8.6.2 A hardening elastoplastic medium
8.6.3 Ideal elastoplastic media: a degenerate case
8.6.4 Difficulties in solving mixed elliptic-hyperbolic problems
8.6.5 Regularization of an elastoplastic model
8.6.6 Elastoplastic shock waves
Bibliography
Index