Finite element methods

Transcription

1 Finite element methods Period 2, 2013/2014 Department of Information Technology Uppsala University Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 1

2 Short Bio Patrick Henning, grew up in Germany Educated in Freiburg and Münster Working on Numerical Analysis PostDoc at Uppsala University since 13 I started doing research on FEM in 05. It is still the main focus in my research. Office: 2446 Phone: patrick.henning@it.uu.se Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 2

3 About the course Finite element methods, 5hp Homepage: Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 3

4 Registration If you you want to pass the course, you must be registered! Registration is possible at Studentportalen. Web-registration is possible until November 11, at the latest. Note that registration is a two step procedure consisting of applying and, when you get accepted, a web-registration. Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 4

5 Outline Goals Literature Schedule Assignments Examination Finite element method Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 5

6 Learning goals To pass the course the student shall be able to: formulate and solve (with a computer) second order elliptic boundary value problems in 1d (one spatial dimension) using the finite element method. formulate and solve (with a computer) second order elliptic boundary value problems in 2d (two spatial dimensions) with Dirichlet, Neumann, and Robin boundary conditions, using the finite element method. derive a priori and a posteriori error bounds for elliptic equations in one and two spatial dimensions, and be able to use these error bounds to construct adaptive algorithms for local mesh refinement. solve parabolic and hyperbolic partial differential equations using the finite element method in space and finite differences in time, and to compare different time stepping algorithms and choose appropriate algorithms for the problem at hand. use finite element software to solve more complicated problems, such as coupled systems of equations. evaluate different techniques for solving problems and be able to motivate when to use existing software and when to write new code. Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 6

7 Literature Book: The Finite Element Method: Theory, Implementation, and Practice by M.G. Larson and F. Bengzon, Texts in Computational Science and Engineering, Springer, Compact, covers the course very nicely Was used several times before at Uppsala and at Umeå Several times revised (so few misprints/mistakes) Answers to exercises are published online Available in the student book stores Other relevant books include: Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method, Studentlitteratur, Eriksson, K., Estep, D., Hansbo, P., and Johnson, C. Computational Differential Equations, Studentlitteratur, Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 7

8 (Preliminiary) Schedule Week 44: 30/10 lecture 1 Introduction + (LB) Chap.1, part 1 31/10 lecture 2 (LB) Chap.1, part 2 Week 45: 04/11 exercise 1 (LB) Chap.1 problems 05/11 lecture 3 (LB) Chap.2, part 1 07/11 lecture 4 (LB) Chap.2, part 2 Week 46: 11/11 exercise 2 (LB) Chap.2 problems 12/11 lecture 5 (LB) Chap.3, part 1 14/11 computer lab 1 Finite element method in 1D Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 8

9 (Preliminiary) Schedule Week 47: 19/11 lecture 6 (LB) Chap.3, part 2 20/11 exercise 3 (LB) Chap.3 problems 21/11 lecture 7 (LB) Chap.4, part 1 Week 48: 25/11 exercise 4 (LB) Chap.4 problems 26/11 lecture 8 (LB) Chap.4, part 2 27/11 computer lab 2 Finite element method in 2D 28/11 lecture 9 (LB) Chap.5, part 1 Week 49: 02/02 lecture 10 (LB) Chap.5, part 2 03/12 exercise 5 (LB) Chap.5 problems 05/12 laboration 3 Comsol multiphysics 06/12 lecture 11 discussion of problems from old exams Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 9

10 (Preliminiary) Schedule Week 50: 10/12 exercise 6 Questions Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 10

11 Examination Assignments There will be two assignments. Each group can consist of one or two students. Deadline is 06/12, and final deadline (with corrections if needed) is 18/12. Written exam Exam date 18/12. Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 11

12 The finite element method: History First approach 40 (Courant) Developed in 50 and 60 (Engineers, structural mechanics) Mathematics foundation 70 (Strang and Fix) Leading researcher Babuška 60-present Swedish key contributors Thomée (Chalmers) and Johnson (KTH) Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 12

13 The finite element method: Applications Structural mechanics Electro magnetics Fluid mechanics Geophysics Biophysics General relativity Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 13

14 The finite element method: Research Mathematics Mathematical modeling (equation, data, uncertainties) Functional analysis (existence, uniqueness, regularity) Discretization (elements, disc. func. spaces, time stepping) Error analysis (bounds, estimates) Adaptivity (refinement criteria) Implementation (cad, meshing, refining, linear algebra, visualization) Parallel programming (efficiency, data structure) Computation Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 14

15 Words of wisdom Far better an approximate answer to the right question, which is often vague, than the exact answer to the wrong question, which can always be made precise. - John Tukey Computers are incredibly fast, accurate and stupid. Human beings are incredibly slow, inaccurate and brilliant. Together they are powerful beyond imagination. - Albert Einstein Finite element methods, Uppsala University, Sweden, 30th October 2013 p. 15