Nonlinear Finite Element Methods
The lectures and exercises of the course "Nonlinear Finite Element Methods" will be conducted in presence starting from 2nd of April 2025 (cf. details in schedule). Additionally, webinars and screen casts are provided as supplementary material. Please register to gain access.
Contents
The course focuses on the application of the Finite Element Method (FEM) to physically nonlinear boundary value problems. It includes as well introductory lectures related to thermo-mechanical coupling. Furthermore, the application of FEM in case of dynamical problems is discussed.
- Introduction and fundamentals
- Spatial discretization
- Galerkin's method
- Elements
- Numerical Integration (Quadrature)
- Nonlinear elastic problems
- Newton-Raphson method in FEM
- Irreversible Material
- Runge-Kutta methods
- History-dependent material behavior
- Dynamic problems:
- Newmark method
- Explicit FEM
- FEM for large deformations
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FEM for thermomechanical problems
Requirements
- Engineering Mechanics
- Fundamental knowledge of FEM for linear-elastic problems (e. g. courses "Introduction to the Finite Element Method", "Numerische Methoden der Mechanik", "Einführung in die Methode der finiten Elemente", MIT course, lectures 1 to 7 or a more mathematical approach to FEM by W.Bangerth from Texas A&M / Colorado State University, lectures 3.9, 3.91, 3.92, 3.93, 3.95, 3.98)
- Besides the list of references on Nonlinear FEM, the book Enhanced Introduction to Finite Elements for Engineers can serve as a useful reference for this course.
Schedule for summer term 2025
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Lectures Location: WEI-1051 |
Exercises (Room FOR-024tr) Please download exercise sheet and ensure that you have Python or MatLab (cf. Wiki) installed on your computer. |
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| cw | Date | Time | Date | Time | Type¹ | ||
| 14 | April 2 |
14:30- 16:00 |
Initial lecture |
April 2 |
16:15- 17:45 |
Coding |
Ex 1.1-1.2: Implementations of Algorithms in Python or MatLab |
| 15 | April 9 | Ex 2.1: Linear FEM for rods; consultation on self-assessment |
April 9 |
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| 16 | April 16 |
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Lecture 2: Weak form and Galerkin's method |
April 16 |
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Coding |
Ex 2.2: Implementation of linear FEM for rods |
| 17 | April 23 |
Lecture 3: Galerkin's method / Isoparametric Elements |
April 23 |
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Paper & Pencil |
Ex 3.1, 3.2: Weak form and Galerkin's method Requires: Lecture 2 |
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| 18 | April 30 | Lecture 3: Isoparametric Elements (continued) | April 30 |
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Paper & Pencil |
Ex 3.3: Weak form and Galerkin's method (continued) Ex 4.3, 4.4, 4.2: Shape functions Requires: Lecture 3 |
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| 19 | May 7 |
Lecture 4: Quadrature |
May 7 | Coding |
Ex 5.3 (5.1) & 5.2: Gauss quadrature Requires: Lecture 4 |
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| 20 |
May 14 |
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Lecture 5: Nonlinear elastic problems and Newton-Raphson method |
May 14 |
Paper & Pencil |
Ex 6.1-6.2: Newton-Raphson method Requires: Lecture 5 |
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| 21 | May 21 |
Lecture 6: Irreversible material and Runge-Kutta methods |
May 21 |
Coding |
Ex 6.3: Newton-Raphson method Requires: Lecture 5 |
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| 22 | May 28 | Lecture 7: Irreversible material and Algorithmically Consistent Tangent Stiffness | May 28 | Paper & Pencil |
Ex 7.1: Time integration Requires: Lectures 6 and 7 |
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| 23 | June 4 | Lecture 8: Elastic-plastic material | June 4 | Coding |
Ex 7.2: Time integration Requires: Lectures 6 and 7 |
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| 24 | June 11 |
Lecture 9: Dynamic problems and Newmark method |
June 11 |
Paper & Pencil |
Introduction to assignment |
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| 25 | June 18 |
Lecture 10: Explicit Dynamic FEM |
June 18 | Coding |
Ex 7.3: Dynamic problems Requires: Lectures 9 and 10 |
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| 26 | June 25 | Lecture 11: Large deformations | June 25 | Paper & Pencil | Consultation on Assignment | ||
| 27 | July 2 | Lecture 12: Thermomechanical Problems | July 2 | Coding |
Exercise FE code Abaqus Please install Abaqus Student Edition in advance at your computer.
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| 28 | July 9 | t.b.a. | July 9 | Paper & Pencil |
Ex 8.1: Large deformations Requires: Lectures 11 |
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Examination
Written examination
The examination will be conducted in presence according to the details listed in the examination plan. Please follow the link to check for recent updates of the examination plan.
The examination consists of two parts. A question part, which has to be done without any auxilary material, and a part "tasks" for which two hand-written sheets of paper (DIN A4, double-sided, no complete exercise solutions) , a mathematical formulary and a pocket calculator (without algebra system) may be used . Usage of any other electronic devices during examination is not allowed.
Successful completion of the assignment is mandatory for taking the examination (Prüfungsvorleistung - PVL).
Assignment
The task for the assignment will be available by the mid of the summer semester.