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عدد المساهمات : 18994
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تاريخ التسجيل : 01/07/2009
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 |  موضوع: كتاب Partial Differential Equations and the Finite Element Method الأحد 03 يونيو 2012, 4:40 am | |
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Partial Differential Equations and the Finite Element Method
P Solin
ويتناول الموضوعات الأتية :
Partial Differential Equations
Selected general properties
Classification and examples
Hadamard’s well-posedness
Exercises
General existence and uniqueness results
Second-order elliptic problems
Weak formulation of a model problem
Bilinear forms, energy norm, and energetic inner product
The Lax-Milgram lemma
Unique solvability of the model problem
Nonhomogeneous Dirichlet boundary conditions
Neumann boundary conditions
Newton (Robin) boundary conditions
Combining essential and natural boundary conditions
Energy of elliptic problems
Maximum principles and well-posedness
Exercises
Second-order parabolic problems
Initial and boundary conditions
Weak formulation
Exercises
Existence and uniqueness of solution
Second-order hyperbolic problems
Initial and boundary conditions
The wave equation
I Exercises
Weak formulation and unique solvability
First-order hyperbolic problems
Conservation laws
S Characteristics
S Riemann problem
S Exercises
Exact solution to linear first-order systems
Nonlinear flux and shock formation
Continuous Elements for D Problems
The general framework
I Exercises
The Galerkin method
Orthogonality of error and CCa’s lemma
Convergence of the Cialerkin method
Ritz method for symmetric problems
Lowest-order elements
Model problem
Piecewise-affine basis functions
Element-by-element assembling procedure
Refinement and convergence
Exercises
Finite-dimensional subspace V,, C v
The system of linear algebraic equations
Higher-order numerical quadrature
Gaussian quadrature rules
Selected quadrature constants
Adaptive quadrature
Exercises
Higher-order elements
Motivation problem
Affine concept: reference domain and reference maps
Transformation of weak forms to the reference domain
Higher-order Lagrange nodal shape functions
Chebyshev and Gauss-Lobatto nodal points
Higher-order Lobatto hierarchic shape functions
Constructing basis of the space Vh,p
Data structures
Assembling algorithm
Exercises
The sparse stiffness matrix
Condition number
Conditioning of shape functions
Exercises
Compressed sparse row (CSR) data format
Stiffness matrix for the Lobatto shape functions
Implementing nonhomogeneous boundary conditions
Dirichlet boundary conditions
Exercises
Combination of essential and natural conditions
Interpolation on finite elements
The Hilbert space setting
Best interpolant
Projection-based interpolant
Nodal interpolant
Exercises
General Concept of Nodal Elements
The nodal finite element
Unisolvency and nodal basis
Checking unisolvency
Example: lowest-order Q' - and PI-elements
Q-element
P-element
Invertibility of the quadrilateral reference map z~
Interpolation on nodal elements
Local nodal interpolant
Global interpolant and conformity
Conformity to the Sobolev space H'
Equivalence of nodal elements
Exercises
Continuous Elements for D Problems
Lowest-order elements
Approximations and variational crimes
Connectivity arrays
Assembling algorithm for Q'/P'-elements
Lagrange interpolation on Q'/P'-meshes
Exercises
Higher-order numerical quadrature in D
Gaussian quadrature on quads
Gaussian quadrature on triangles
Product Gauss-Lobatto points
Lagrange-Gauss-Lobatto Qp,'-elements
The Fekete points
Lagrange-Fekete PP-elements
Data structures
Connectivity arrays
Assembling algorithm for QPIPp-elements
Lagrange interpolation on Qp/Pp-meshes
Exercises
Model problem and its weak formulation
Basis of the space Vh,p
Transformation of weak forms to the reference domain
Simplified evaluation of stiffness integrals
Higher-order nodal elements
Lagrange interpolation and the Lebesgue constant
Basis of the space v,Tl
Transient Problems and ODE Solvers
Method of lines
Model problem
Weak formulation
The ODE system
Construction of the initial vector
Autonomous systems and phase flow
One-step methods, consistency and convergence
Explicit and implicit Euler methods
Selected time integration schemes
Stiffness
Explicit higher-order RK schemes
General (implicit) RK schemes
Embedded RK methods and adaptivity
Introduction to stability
Autonomization of RK methods
A-stability and L-stability
Collocation methods
Solution of nonlinear systems
Stability of linear autonomous systems
Stability functions and stability domains
Stability functions for general RK methods
Maximum consistency order of IRK methods
Higher-order IRK methods
Gauss and Radau IRK methods
Exercises
Beam and Plate Bending Problems
Bending of elastic beams
I Euler-Bernoulli model
Boundary conditions
Weak formulation
Lowest-order Hermite elements in D
Model problem
Cubic Hermite elements
Higher-order Hermite elements in D
Nodal higher-order elements
Hierarchic higher-order elements
Conditioning of shape functions
Basis of the space Vh,p
Transformation of weak forms to the reference domain
Connectivity arrays
Assembling algorithm
Interpolation on Hermite elements
Lowest-order elements
Higher-order Hermite-Fekete elements
Design of basis functions
Reissner-Mindlin (thick) plate model
Kirchhoff (thin) plate model
Boundary conditions
Existence and uniqueness of solution
Hermite elements in D
Global nodal interpolant and conformity
Bending of elastic plates
Weak formulation and unique solvability
BabuSka’s paradox of thin plates
xii CONTENTS
Discretization by H-conforming elements
Local interpolant, conformity
Transformation to reference domains
Design of basis functions
Higher-order nodal Argyris-Fekete elements
Lowest-order (quintic) Argyris element, unisolvency
Nodal shape functions on the reference domain
Exercises
Equations of Electrornagnetics
Electromagnetic field and its basic characteristics
Integration along smooth curves
Conductors and dielectrics
Magnetic materials
Conditions on interfaces
Scalar electric potential
Scalar magnetic potential
Other wave equations
Equations for the field vectors
Interface and boundary conditions
Time-harmonic Maxwell’s equations
Helmholtz equation
Normalization
Model problem
Weak formulation
Maxwell’s equations in integral form
Maxwell’s equations in differential form
Constitutive relations and the equation of continuity
Media and their characteristics
Potentials
Vector potential and gauge transformations
Potential formulation of Maxwell’s equations
Equation for the electric field
Equation for the magnetic field
Time-harmonic Maxwell’s equations
Existence and uniqueness of solution
Conformity requirements of the space H(cur)
Lowest-order (Whitney) edge elements
Higher-order edge elements of NCdClec
Transformation of weak forms to the reference domain
Interpolation on edge elements
Edge elements
Exercises
Conformity of edge elements to the space H(cur)
Appendix A: Basics of Functional Analysis
A Linear spaces
A Exercises
Real and complex linear space
Checking whether a set is a linear space
Intersection and union of subspaces
Linear combination and linear span
Sum and direct sum of subspaces
Linear independence, basis, and dimension
Linear operator, null space, range
Composed operators and change of basis
Determinants, eigenvalues, and eigenvectors
Hermitian, symmetric, and diagonalizable matrices
Linear forms, dual space, and dual basis
A Normed spaces
A Norm and seminorm
A Convergence and limit
A Open and closed sets
A Continuity of operators
A Equivalence of norms
A Banach spaces
A Banach fixed point theorem
A Lebesgue integral and LP-spaces
A Basic inequalities in LP-spaces
A Exercises
Operator norm and C(U, V ) as a normed space
Density of smooth functions in LP-spaces
A Inner product spaces
A Inner product
A Hilbert spaces
A Generalized angle and orthogonality
A Generalized Fourier series
A Projections and orthogonal projections
A Representation of linear forms (Riesz)
A Compactness, compact operators, and the Fredholm alternative
A Weak convergence
A Exercises
A Sobolev spaces
A Domain boundary and its regularity
xiv CONTENTS
Distributions and weak derivatives
Spaces Wklp and Hk
Discontinuity of HI-functions in R", d
PoincarC-Friedrichs' inequality
Embeddings of Sobolev spaces
Traces of W"p-functions
Generalized integration by parts formulae
Exercises
Appendix B: Software and Examples
B Sparse Matrix Solvers
B The sMatrix utility
B An example application
B Interfacing with PETSc
B Interfacing with Trilinos
B Interfacing with UMFPACK
The High-Performance Modular Finite Element System HERMES
B Modular structure of HERMES
B The elliptic module
B The Maxwell's module
B Example : Insulator problem
B Example : Sphere-cone problem
B Example : Diffraction problem
Example : L-shape domain problem
Example : Electrostatic micromotor problem
References
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Admin
مدير المنتدى
عدد المساهمات : 18994
التقييم : 35488
تاريخ التسجيل : 01/07/2009
الدولة : مصر
العمل : مدير منتدى هندسة الإنتاج والتصميم الميكانيكى
| | موضوع: رد: كتاب Partial Differential Equations and the Finite Element Method الأربعاء 07 نوفمبر 2012, 5:00 pm | |
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