UID:
almahu_9949602158002882
Format:
1 online resource (152 pages)
Edition:
1st ed.
ISBN:
9783319524627
Series Statement:
Simula SpringerBriefs on Computing Series ; v.3
Note:
Intro -- Foreword -- Contents -- Preface -- 1 Preliminaries -- 1.1 The FEniCS Project -- 1.2 What you will learn -- 1.3 Working with this tutorial -- 1.4 Obtaining the software -- 1.4.1 Installation using Docker containers -- 1.4.2 Installation using Ubuntu packages -- 1.4.3 Testing your installation -- 1.5 Obtaining the tutorial examples -- 1.6 Background knowledge -- 1.6.1 Programming in Python -- 1.6.2 The finite element method -- 2 Fundamentals: Solving the Poisson equation -- 2.1 Mathematical problem formulation -- 2.1.1 Finite element variational formulation -- 2.1.2 Abstract finite element variational formulation -- 2.1.3 Choosing a test problem -- 2.2 FEniCS implementation -- 2.2.1 The complete program -- 2.2.2 Running the program -- 2.3 Dissection of the program -- 2.3.1 The important first line -- 2.3.2 Generating simple meshes -- 2.3.3 Defining the finite element function space -- 2.3.4 Defining the trial and test functions -- 2.3.5 Defining the boundary conditions -- 2.3.6 Defining the source term -- 2.3.7 Defining the variational problem -- 2.3.8 Forming and solving the linear system -- 2.3.9 Plotting the solution using the plot command -- 2.3.10 Plotting the solution using ParaView -- 2.3.11 Computing the error -- 2.3.12 Examining degrees of freedom and vertex values -- 2.4 Deflection of a membrane -- 2.4.1 Scaling the equation -- 2.4.2 Defining the mesh -- 2.4.3 Defining the load -- 2.4.4 Defining the variational problem -- 2.4.5 Plotting the solution -- 2.4.6 Making curve plots through the domain -- 3 A Gallery of finite element solvers -- 3.1 The heat equation -- 3.1.1 PDE problem -- 3.1.2 Variational formulation -- 3.1.3 FEniCS implementation -- 3.2 A nonlinear Poisson equation -- 3.2.1 PDE problem -- 3.2.2 Variational formulation -- 3.2.3 FEniCS implementation -- 3.3 The equations of linear elasticity -- 3.3.1 PDE problem.
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3.3.2 Variational formulation -- 3.3.3 FEniCS implementation -- 3.4 The Navier-Stokes equations -- 3.4.1 PDE problem -- 3.4.2 Variational formulation -- 3.4.3 FEniCS implementation -- 3.5 A system of advection-diffusion-reaction equations -- 3.5.1 PDE problem -- 3.5.2 Variational formulation -- 3.5.3 FEniCS implementation -- 4 Subdomains and boundary conditions -- 4.1 Combining Dirichlet and Neumann conditions -- 4.1.1 PDE problem -- 4.1.2 Variational formulation -- 4.1.3 FEniCS implementation -- 4.2 Setting multiple Dirichlet conditions -- 4.3 Defining subdomains for different materials -- 4.3.1 Using expressions to define subdomains -- 4.3.2 Using mesh functions to define subdomains -- 4.3.3 Using C++ code snippets to define subdomains -- 4.4 Setting multiple Dirichlet, Neumann, and Robin conditions -- 4.4.1 Three types of boundary conditions -- 4.4.2 PDE problem -- 4.4.3 Variational formulation -- 4.4.4 FEniCS implementation -- 4.4.5 Test problem -- 4.4.6 Debugging boundary conditions -- 4.5 Generating meshes with subdomains -- 4.5.1 PDE problem -- 4.5.2 Variational formulation -- 4.5.3 FEniCS implementation -- 5 Extensions: Improving the Poisson solver -- 5.1 Refactoring the Poisson solver -- 5.1.1 A more general solver function -- 5.1.2 Writing the solver as a Python module -- 5.1.3 Verification and unit tests -- 5.1.4 Parameterizing the number of space dimensions -- 5.2 Working with linear solvers -- 5.2.1 Choosing a linear solver and preconditioner -- 5.2.2 Choosing a linear algebra backend -- 5.2.3 Setting solver parameters -- 5.2.4 An extended solver function -- 5.2.5 A remark regarding unit tests -- 5.2.6 List of linear solver methods and preconditioners -- 5.3 High-level and low-level solver interfaces -- 5.3.1 Linear variational problem and solver objects -- 5.3.2 Explicit assembly and solve -- 5.3.3 Examining matrix and vector values.
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5.4 Degrees of freedom and function evaluation -- 5.4.1 Examining the degrees of freedom -- 5.4.2 Setting the degrees of freedom -- 5.4.3 Function evaluation -- 5.5 Postprocessing computations -- 5.5.1 Test problem -- 5.5.2 Flux computations -- 5.5.3 Computing functionals -- 5.5.4 Computing convergence rates -- 5.5.5 Taking advantage of structured mesh data -- 5.6 Taking the next step -- References -- Index.
Additional Edition:
Print version: Langtangen, Hans Petter Solving PDEs in Python Cham : Springer International Publishing AG,c2017 ISBN 9783319524610
Language:
English
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