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E-raamat: Solving PDEs in Python: The FEniCS Tutorial I

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Solving PDEs in Python: The FEniCS Tutorial ISuurem pilt
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This book offers a concise and gentle introduction to finite element programming in Python based on the popular FEniCS software library. Using a series of examples, including the Poisson equation, the equations of linear elasticity, the incompressible Navier-Stokes equations, and systems of nonlinear advection-diffusion-reaction equations, it guides readers through the essential steps to quickly solving a PDE in FEniCS, such as how to define a finite variational problem, how to set boundary conditions, how to solve linear and nonlinear systems, and how to visualize solutions and structure finite element Python programs.This book is open access under a CC BY license.

1 Preliminaries.- 2 Fundamentals: Solving the Poisson Equation.- 3 A Gallery of Finite Element Solvers.- 4 Subdomains and Boundary Conditions.- 5 Extensions: Improving the Poisson Solver.- References.

Arvustused

This book of about 150 pages introduces the FEniCS software toolset in 5 chapters ... . The targeted audience includes applications scientists in the area of PDEs who seek for an easy way to implement their problems, and who are willing to invest the time to learn how to work with FEniCS. (Gudula Rünger, zbMATH 1376.65144, 2018)

Muu info

This is an open access book, the electronic versions are freely accessible online.
Preface 1(2)
1 Preliminaries
3(8)
1.1 The FEniCS Project
3(1)
1.2 What you will learn
4(1)
1.3 Working with this tutorial
4(1)
1.4 Obtaining the software
5(3)
1.4.1 Installation using Docker containers
6(1)
1.4.2 Installation using Ubuntu packages
7(1)
1.4.3 Testing your installation
8(1)
1.5 Obtaining the tutorial examples
8(1)
1.6 Background knowledge
8(3)
1.6.1 Programming in Python
8(1)
1.6.2 The finite element method
9(2)
2 Fundamentals: Solving the Poisson equation
11(26)
2.1 Mathematical problem formulation
11(6)
2.1.1 Finite element variational formulation
12(3)
2.1.2 Abstract finite element variational formulation
15(1)
2.1.3 Choosing a test problem
16(1)
2.2 FEniCS implementation
17(2)
2.2.1 The complete program
17(1)
2.2.2 Running the program
18(1)
2.3 Dissection of the program
19(11)
2.3.1 The important first line
19(1)
2.3.2 Generating simple meshes
20(1)
2.3.3 Defining the finite element function space
20(1)
2.3.4 Defining the trial and test functions
20(1)
2.3.5 Defining the boundary conditions
21(3)
2.3.6 Defining the source term
24(1)
2.3.7 Defining the variational problem
24(1)
2.3.8 Forming and solving the linear system
25(1)
2.3.9 Plotting the solution using the plot command
25(2)
2.3.10 Plotting the solution using ParaView
27(1)
2.3.11 Computing the error
28(1)
2.3.12 Examining degrees of freedom and vertex values
29(1)
2.4 Deflection of a membrane
30(1)
2.4.1 Scaling the equation
31(1)
2.4.2 Defining the mesh
32(1)
2.4.3 Defining the load
32(1)
2.4.4 Defining the variational problem
33(1)
2.4.5 Plotting the solution
33(1)
2.4.6 Making curve plots through the domain
34(3)
3 A Gallery of finite element solvers
37(46)
3.1 The heat equation
37(9)
3.1.1 PDE problem
37(1)
3.1.2 Variational formulation
38(2)
3.1.3 FEniCS implementation
40(6)
3.2 A nonlinear Poisson equation
46(4)
3.2.1 PDE problem
46(1)
3.2.2 Variational formulation
47(1)
3.2.3 FEniCS implementation
47(3)
3.3 The equations of linear elasticity
50(6)
3.3.1 PDE problem
51(1)
3.3.2 Variational formulation
51(1)
3.3.3 FEniCS implementation
52(4)
3.4 The Navier-Stokes equations
56(17)
3.4.1 PDE problem
56(1)
3.4.2 Variational formulation
57(3)
3.4.3 FEniCS implementation
60(13)
3.5 A system of advection-diffusion-reaction equations
73(10)
3.5.1 PDE problem
73(2)
3.5.2 Variational formulation
75(1)
3.5.3 FEniCS implementation
75(8)
4 Subdomains and boundary conditions
83(26)
4.1 Combining Dirichlet and Neumann conditions
83(3)
4.1.1 PDE problem
83(1)
4.1.2 Variational formulation
84(1)
4.1.3 FEniCS implementation
85(1)
4.2 Setting multiple Dirichlet conditions
86(1)
4.3 Defining subdomains for different materials
87(5)
4.3.1 Using expressions to define subdomains
88(1)
4.3.2 Using mesh functions to define subdomains
88(3)
4.3.3 Using C++ code snippets to define subdomains
91(1)
4.4 Setting multiple Dirichlet, Neumann, and Robin conditions
92(7)
4.4.1 Three types of boundary conditions
93(1)
4.4.2 PDE problem
93(1)
4.4.3 Variational formulation
94(1)
4.4.4 FEniCS implementation
95(2)
4.4.5 Test problem
97(1)
4.4.6 Debugging boundary conditions
98(1)
4.5 Generating meshes with subdomains
99(10)
4.5.1 PDE problem
100(2)
4.5.2 Variational formulation
102(1)
4.5.3 FEniCS implementation
102(7)
5 Extensions: Improving the Poisson solver
109(34)
5.1 Refactoring the Poisson solver
109(6)
5.1.1 A more general solver function
110(1)
5.1.2 Writing the solver as a Python module
111(1)
5.1.3 Verification and unit tests
111(3)
5.1.4 Parameterizing the number of space dimensions
114(1)
5.2 Working with linear solvers
115(3)
5.2.1 Choosing a linear solver and preconditioner
115(1)
5.2.2 Choosing a linear algebra backend
115(1)
5.2.3 Setting solver parameters
116(1)
5.2.4 An extended solver function
117(1)
5.2.5 A remark regarding unit tests
117(1)
5.2.6 List of linear solver methods and preconditioners
117(1)
5.3 High-level and low-level solver interfaces
118(5)
5.3.1 Linear variational problem and solver objects
118(1)
5.3.2 Explicit assembly and solve
119(3)
5.3.3 Examining matrix and vector values
122(1)
5.4 Degrees of freedom and function evaluation
123(4)
5.4.1 Examining the degrees of freedom
123(2)
5.4.2 Setting the degrees of freedom
125(1)
5.4.3 Function evaluation
126(1)
5.5 Postprocessing computations
127(14)
5.5.1 Test problem
127(1)
5.5.2 Flux computations
128(2)
5.5.3 Computing functionals
130(2)
5.5.4 Computing convergence rates
132(4)
5.5.5 Taking advantage of structured mesh data
136(5)
5.6 Taking the next step
141(2)
References 143(2)
Index 145
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