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 |  موضوع: كتاب Finite Element Methods in Civil and Mechanical Engineering الأربعاء 10 فبراير 2021, 8:47 pm | |
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أحضرت لكم كتاب
Finite Element Methods in Civil and Mechanical Engineering
A Mathematical Introduction
Dr Arzhang Angoshtari
Ali Gerami Matin
و المحتوى كما يلي :
Contents
Preface .xi
Chapter 1 Overview .1
Chapter 2 Mathematical Preliminaries .5
2.1 Real Numbers .5
2.2 Functions .5
2.3 Linear Spaces, Linear Mappings, and Bilinear Forms .6
2.4 Linear Independence, Hamel Bases, and Dimension .9
2.5 The Matrix Representation of Linear Mappings and
Bilinear Forms 10
2.6 Normed Linear Spaces 12
2.7 Functionals and Dual Spaces 13
2.8 Green’s Formulas 14
Exercises 15
Comments and References .17
Chapter 3 Finite Element Interpolation 19
3.1 1D Finite Element Interpolation .19
3.1.1 The Global Level .19
3.1.2 The Local Level .24
3.2 Finite Elements .26
3.2.1 Simplicial Lagrange Finite Elements of Type (k) 26
3.2.2 Simplicial Hermite Finite Elements of Type (3) 30
3.2.3 The Raviart-Thomas Finite Element 31
3.2.4 The Nedelec Finite Element .33
3.3 Meshes 34
3.4 Finite Element Spaces and Interpolations .37
3.4.1 H1-Conformal Finite Element Spaces .38
3.4.1.1 Lagrange Elements 39
3.4.1.2 Hermite Elements 43
3.4.2 H(div)-Conformal Finite Element Spaces .43
3.4.3 H(curl)-Conformal Finite Element Spaces 45
3.4.4 Affne Families of Finite Elements 45
3.5 Convergence of Interpolations 46
Exercises 50
Computer Exercises .51
viiviii Contents
Comments and References .52
Chapter 4 Conforming Finite Element Methods for PDEs 55
4.1 Second-Order Elliptic PDEs .55
4.2 Weak Formulations of Elliptic PDEs 56
4.2.1 Dirichlet Boundary Condition 57
4.2.2 Neumann Boundary Condition 58
4.2.3 Robin Boundary Condition 59
4.3 Well-posedness of Weak Formulations .60
4.4 Variational Structure .61
4.5 The Galerkin Method and Finite Element Methods .62
4.5.1 The Stiffness Matrix 63
4.5.2 Well-posedness of Coercive Discrete Problems 64
4.5.3 Convergence of Finite Element Solutions 64
4.6 Implementation: The Poisson Equation 66
4.6.1 Dirichlet Boundary Condition 66
4.6.2 Mixed Dirichlet-Neumann Boundary Condition .70
4.6.3 Robin Boundary Condition 74
4.7 Time-Dependent Problems: Parabolic Problems 76
4.7.1 Finite Element Approximations using the Method
of Lines 78
4.7.2 Temporal Discretization .79
4.7.3 Implementation: A Diffusion Problem 79
4.8 Mixed Finite Element Methods 82
4.8.1 Mixed Formulations .83
4.8.2 Mixed Methods and inf-sup Conditions 84
4.8.3 Implementation 85
Exercises 90
Computer Exercises .92
Comments and References .93
Chapter 5 Applications .97
5.1 Elastic Bars .97
5.2 Euler-Bernoulli Beams .100
5.3 Elastic Membranes 104
5.4 The Wave Equation .105
5.5 Heat Transfer in a Turbine Blade 107
5.6 Seepage in Embankment .112
5.7 Soil Consolidation 117
5.8 The Stokes Equation for Incompressible Fluids .122
5.9 Linearized Elasticity .126
5.10 Linearized Elastodynamics: The Hamburg Wheel-Track
Test 129
5.11 Nonlinear Elasticity 134Contents ix
Exercises 143
Computer Exercises .145
Appendix A FEniCS Installation .147
Appendix B Introduction to Python .149
B.1 Running Python Programs 149
B.2 Lists .150
B.3 Branching and Loops 151
B.4 Functions .152
B.5 Classes and Objects 152
B.6 Reading and Writing Files 154
B.7 Numerical Python Arrays .155
B.8 Plotting with Matplotlib 157
References
Index
abstract problem, 60
coercivity, 61, 64
ellipticity, 61
variational structure, 61
well-posedness, 60, 64
advection-diffusion equation, 56, 108
affne family of fnite elements, 46
affne mapping, 8
affne-equivalent, 45
approximability property, 65
approximation space, 19
Babuska-Brezzi condition, ˇ 124
barycenter, 27
barycentric coordinates, 27
basis (Hamel), 9
bilinear form, 9
boundary condition, 56
Dirichlet, 57
essential, 58
mixed Dirichlet-Neumann, 59
natural, 59
Neumann, 58
Robin, 59
boundary value problem, 56
bounded set, 5
from above, 5
from below, 5
Box, 36
Cea’s lemma, ´ 65
CG, 30
conformal space, 38
conforming method, 62
consolidation problem, 117
conventional fnite element diagram, 28
convergence, 46, 64
convergence rate, 23, 47, 65
optimal, 24, 47, 66
curl, 8
degree of freedom
global, 19, 38
local, 25, 26
diameter of element, 35
diffusion problem, 79
dimension, 9
DirichletBC, 68, 73, 87
divergence div, 7
dolfin, 22
ds, 68, 73
dual basis, 14
dual space, 13
duality brackets h·, ·i, 14
dx, 68
elastic membrane, 56, 104
elasticity
linearized, 56, 126, 130
nonlinear, 134
elliptic PDE, 55
error, 46, 64
errornorm, 23
Euler method, 79
explicit, 79
implicit, 79
existence of solution, 56
Expression, 22
FEniCS, 21, 66
installation, 147
fenics, 22
fnite element, 26
assembly process, 25, 37
Crouzeix-Raviart, 50
Hermit, 30
Hermite n-simplex of type (3), 30
Lagrange, 25, 27
Ned ´ ´ elec, 33
node, 27, 30
Raviart-Thomas, 32
161162 INDEX
simplex of type (k), 29
space, 37
fnite element method, 62
conforming, 62
mixed, 84, 123
fnite-dimensional, 9
FiniteElement, 22, 41
full rank, 10, 85
function, 5
continuous Cm(Ω), Cm(Ω), 6
bijective, 6
domain, 5
extension, 5
invertible, 6
one-to-one, injective, 6
onto, surjective, 6
range, 6
restriction, 5
functional, 13
FunctionSpace, 22, 41, 87
Gaussian elimination, 69
Gelerkin method, 62
generate_mesh, 36
grad, 68
gradient ∇, 14
Green’s formulas, 14
h-type approach, 50
hat function, 20
heat transfer equation, 56, 77, 82
hyperbolic PDE, 92
inf-sup condition, 85
infmum, 5
infnite-dimensional, 9
initial condition, 77, 92
initial-boundary value problem, 77, 79,
92, 105
inner, 68
inner product, 14
interpolant, 26
global, 38
Lagrange, 19, 27, 41
local, 24
Ned ´ elec, ´ 45
Raviart-Thomas, 44
interpolate, 23, 41
interpolation operator, 21
global Ih, 38
local IK, 26
Ned ´ elec ´ I
K
N
, IhN, 34, 45
Raviart-Thomas IKRT, IhRT, 32, 44
simplex of type (k) IKk , Ihk, 29, 40
isoparametric family of fnite elements,
46
iterative method, 69
kernel, 8
Kronecker delta δi j, 14
Lagrange, 22, 25, 30
Laplacian Δ, 15, 122
Lax-Milgram lemma, 61
Lebesgue space L2(Ω), 7
linear combination, 9
linear mapping, 8
linear space, 6
linear subspace, 8
linearly independent, 9
locally supported, 38
lower bound, 5
mass matrix, 78
maximum, 5
Measure, 73
mesh, 19, 34
affne, 35
cell, 34
element, 19, 34
geometrically conformal, 35
number of edges ne, 35
number of elements nel, 35
number of faces nf, 35
number of vertices nv, 35
number of vertices on boundary nv ∂ ,
85
vertex, 19
MeshFunction, 72
method of lines, 78n
INDEX 163
minimum, 5
mixed formulation, 83, 102, 123
MixedElement, 87
mshr, 36
N1curl, 34
nodal basis, 27
non-conforming method, 62
norm, 12
H1, k·k1,2, 12
L2, k·k2, 12
k·kc, 13
k·kd, 13
normal derivative ∂ , 15
normed linear space, 12
null space, 8
Numerical Python, 155
NumPy, 155
on_boundary, 68
p-type approach, 50
parabolic PDE, 77
partly Sobolev class, 7
H(curl;Ω), 8
[H(curl;Ω)]n, 135
H(div;Ω), 7
[H(div;Ω)]n, 135
Point, 36
Poisson’s equation, 56, 66, 85, 104
Polygon, 36
polynomial space
NE, 33
Pk(Rn), 7
Qk(Rn), 16
RT, 31
positive defnite matrix, 64
preconditioned Krylov solver, 69
project, 90
projection, 83, 89
Python programming language, 149
rank-nullity theorem, 10
real numbers, 5
Rectangle, 36
reference element, 35
reference fnite element, 45
regularity of solution, 56
Ritz method, 62
RT, 33
saddle-point variational structure, 84,
123
seepage problem, 112
set
closure, 6
open, 6
shape function
global, 21, 38
local, 25, 26
shape-regular, 46
simplex, 27
center of gravity, 27
edge, 27
face, 27
vertex, 27
size_t, 72
Sobolev space, 7
Hm(Ω), 7
[H1(Ω)]n, 7
1H0
(Ω), 8
[H01(Ω)]n, 122
H
D
1(Ω), 59
[HD1(Ω)]n, 127
solution space, 57, 62
solve, 69
span of a set, 9
sparse LU decomposition, 69
sparse matrix, 63
Sphere, 36
split(), 88
stiffness matrix, 63
time-dependent, 78
Stokes equation for fuids, 144
strong solution, 57
SubDomain, 72
supremum, 5
Taylor-Hood element, 124
test function, 57, 62164 INDEX
test problem, 66
test space, 57, 62
TestFunction, 68
TestFunctions, 87
the global degrees of freedom, 38
three-point centered-difference formula,
92
implicit, 106, 130
time step, 79
trial space, 57, 62
TrialFunction, 68
TrialFunctions, 87
triangulation, 35
UFL, 68
uniqueness of solution, 56
unit n-simplex, 27, 51
unit ball, 6
unit cube, 36
unit sphere, 6
unit square, 15, 36
UnitCubeMesh, 36
UnitIntervalMesh, 22
UnitSquareMesh, 36
upper bound, 5
variational crimes, 94
variational structure, 61
vector space, 6
VectorFunctionSpace, 42
vectorization, 156
wave equation, 93, 105
weak formulation, 56, 57
weak solution, 56, 57
well-posedness, 60, 64
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