Classical and isogeometric finite element method

Pod redakcją:Maciej Paszynski

Autorzy/Autorki:Maciej Paszynski

Afiliacja autorów:AGH University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications

Wydawca:Akademia Górniczo-Hutnicza im. St. Staszica w Krakowie

Data publikacji:2021

Recenzja: prof. dr hab. inż. Maciej Pietrzyk, dr hab. inż. Krzysztof Banaś

ISBN:978-83-952566-9-1

Preface

Introduction to the finite element method handbook

1. Basics of finite element method

Exemplary problem of two-dimensional bitmap projection

Approximation with B-spline basis functions

Derivation of the system of linear equations

Generation of the system of linear equations with the analytical method

The solution of the system of linear equations

Interpretation of the solution

MATLAB implementation of the bitmap projection problem

2. Discretization with different basis functions

Linear basis functions

Higher-order Ck basis functions in 1D

Lagrange polynomials

Refined isogeometric analysis in 1D

Two-dimensional generalization of basis functions with tensor products

Three-dimensional generalization of basis functions with tensor products

Non-regular meshes

MATLAB implementation of the generation of basis functions based on the knot vector

3. Weak formulations for different problems and methods

Heat transfer with isogeometric finite element method

Heat transfer with classical finite element method

Advection-diffusion equations

Stokes problem

Linear elasticity

Weak formulations and numerical integration

MATLAB implementation of the heat transfer problem

MATLAB implementation of the advection-diffusion with Galerkin method

4. Solvers of linear equations generated from finite element method

Gaussian elimination algorithm

Gaussian elimination with a pivoting algorithm

LU factorization algorithm

Algorithm of LU factorization with pivoting

Frontal solver algorithm

Multi-frontal solver algorithm

Direction-splitting algorithm

Pre-conditioner

Iterative solver algorithm

Selection of the solver based on problem kind

MATLAB implementation of the alternating-direction solver for the bitmap projection problem

5. Stabilization methods

Stabilization of finite element method

Stabilization of the advection-diffusion equations with the residual minimization method

Stabilization of the advection-diffusion equations with the Streamline Upwind Petrov-Galerkin (SUPG)

Stabilization of the Stokes problem with the residual minimization method

Stabilization of the Stokes problem with the Discontinuous Galerkin method (DG)

MATLAB implementation of the advection-diffusion problem with SUPG

MATLAB implementation of the advection-diffusion problem with residual minimization method

6. Computational grids

Generation and adaptation of computational grids

Algorithm of adaptation of triangular and tetrahedral elements

Algorithm of h-adaptation

Algorithm of p-adaptation

Algorithm of hp-adaptation

Approximation on the mesh with hanging nodes

Isogeometric analysis on adaptive grids

MATLAB implementation of the adaptive algorithm for the bitmap projection problem

7. Finite element method for non-stationary and non-linear problems

Non-stationary problems

Non-stationary problems as a generalization of a sequence of the isogeometric L2 projections

Examples of differential operators and right-hand-sides for selected problems

Explicit method

Implicit method

Crank-Nicolson scheme

Alpha scheme

Example of the non-stationary linear problem: propagation of pollution

MATLAB implementation of the alpha scheme for the heat transfer problem

Example of the non-stationary non-linear problem: non-linear flow in heterogeneous media

8. Mathematical foundations of the finite element method

Functional spaces

One-dimensional finite element

The one-dimensional strong and weak formulation for the elliptic problem

One-dimensional finite element method

Example of a one-dimensional finite element method

Two-dimensional finite element

The two-dimensional strong and weak formulation for the elliptic problem

Two-dimensional finite element method

Adaptive algorithm

The convergence of the finite element method

Isogeometric finite element method

The future of the finite element method

Appendices

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