Computational methods in structural analysis are of prime importance in industry as tools to assess the performance of structures in the field of aerospace, mechanical, civil, and biomedical engineering. To design structures safely, engineers need to be able to predict their performance limits which often entails answering questions such as: when would damage appear? How much plastic deformation would there be? Where and how far would the crack grow? What are the structural responses after bucking? These questions go beyond the scope of linear elastic theories and linear modeling methods.
This course builds on the first course linear modeling and takes one step further. It delivers the skillsets in non-linear structural modeling & analysis using the non-linear Finite Element Method (FEM). The weekly lectures and practical sessions impart both theoretical knowledge on non-linear FEM and practical experience with the modeling pipeline of commercial FE software. There will be optional online hangout sessions for the discussion of theories, exercises, assignments, etc. Professionals in many engineering sectors working on material & structural simulations will benefit from adding non-linear FEM to their skills array. By the end of this course, you will be able to:
- understand and explain the theories of non-linear FEM, and use them to perform analytical work;
- apply non-linear FEM to solve practical non-linear structural problems
- identify and employ efficient modelling techniques
The previous course, linear modeling, is a prerequisite. Working professionals who are already experienced with structural/stress analysis may choose to enroll directly in this course.
Non-Linear modeling is an advanced course as a follow up to linear modeling. It is based on the extension of the Finite Element (FE) method to the non-linear domain. Applications include geometric, material and contact nonlinearities. The course uses a free FE package (for students) in weekly practical sessions, where you will model sample problems to gain hands-on experience. Students are expected to be well versed with FE software usage if they have not followed the linear modeling course. Please note that this is not a software training course. The software is treated as an application platform to try out the nonlinear FE theories.
Multiple assignments are provided, with both theoretical and practical questions. The assessment of this course will be based on the assignment reports.
Week 0: Self-study/Revision. Recap of linear finite element method formulation and usage.
Week 1: An introduction to the types of nonlinearities; general form of a nonlinear finite element equation; demonstrate some common nonlinear solution methods; convergence criteria and common convergence issues.
Practical: Familiarize with the non-linear analysis module of ABAQUS; create a simple model with 1D and 2D elements; carry out a non-linear analysis using two Newton methods.
Week 2: The limitations of Newton type methods; arc-length method; start the theory of geometrical nonlinear FEM (fundamental concepts of continuum mechanics).
Practical: carry out a non-linear analysis using Newton and Arc-length method
Week 3:Finish geometrical nonlinear FEM: finite element discretization of the weak form and derive the expressions of internal & external force vectors and tangent stiffness matrix; examples for truss, beam, and plate elements; buckling & post-buckling analysis.
Practical: Buckling & post-buckling analysis
Week 4: Contact modeling: finite element weak form; penalty and Lagrange multiplier methods; some practical issues; exercise on a contact problem.
Practical: Try out different contact algorithms on a problem with analytical stress solution
Week 5: Plasticity: yield condition, flow rule, hardening, loading/unloading conditions, Prager's consistency condition, tangent modulus, return mapping algorithm.
Practical: Model plastic deformation in metal
Week 6: Fracture and damage: energy concepts of fracture mechanics (Griffith & Irwin) and the cohesive zone theory; the formulations of cohesive element and smeared crack/continuum damage model; viscous regularization.
Practical: Model a debonding problem using cohesive zone approaches
Week 7: some advanced methods for modeling crack propagations, we will discuss remeshing, nonlocal damage model, XFEM, Phantom Node Method and Floating Node Method.
Practical: Model a crack propagation problem with XFEM
If you successfully complete your online course you will be awarded with a TU Delft certificate.
This certificate will state that you were registered as a non-degree-seeking student at TU Delft and successfully completed the course. The certificate will also indicate the number of ECTS credits this course is equal to (3 ECTS) when this course is taken as part of a degree program at the university.
If you decide that you would like to apply to the full Master's program in Aerospace Engineering, you will need to go through the admission process as a regular MSc student. If you are admitted, you can then request an exemption for this course that you completed as a non-degree-seeking student. The Board of Examiners will evaluate your request and will decide whether or not you are exempted.
General admission to this course
Required prior knowledge
- A relevant BEng or BSc degree in a subject closely related to the content of the course or specialized program in question, such as aerospace engineering, aeronautical engineering, mechanical engineering, civil engineering or (applied) physics.
- If you do not meet these requirements because you do not have a relevant Bachelor's degree but you have a Bachelor's degree from a reputable institution and you think you have sufficient knowledge and experience to complete the course, you are welcome to apply, stating your motivation and reasons for admission. The faculty of aerospace engineering will decide whether you will be admitted based on the information you have provided. Appeal against this decision is not possible.
Expected prior knowledge
In addition to the entry requirements mentioned above, prior knowledge of the topic is necessary in order to complete this course. For admission purposes, TU Delft will not ask you for proof of this prior knowledge, but it is your responsibility to ensure that you have the sufficient knowledge, obtained through relevant work experience or prior education.
To view the essential background knowledge, please check your knowledge against the learning objectives of these comparable TU Delft courses:
- Basic Structural Mechanics
- Structural Analysis & Buckling
- Differential Equations and Linear Algebra
- Linear Modeling including relevant linear FEM experience
Expected Level of English
English is the language of instruction for this online course. If your working language is not English or you have not participated in an educational program in English in the past, please ensure that your level of proficiency is sufficient to follow the course. TU Delft recommends an English level equivalent to one of the following certificates (given as an indication only; the actual certificates are not required for the admission process):
- TOEFL score 90+ (this is an internet-based test)
- IELTS (academic version) overall Band score of at least 6.5
- University of Cambridge: "Certificate of Proficiency in English" or "Certificate in Advanced English"
In order to complete your admission process you will be asked to upload the following documents:
- a CV which describes your educational and professional background (in English)
- a copy of your passport or ID card
- a copy of relevant transcripts and diplomas
If you have any questions about this course or the TU Delft online learning environment, please visit our Help & Support page.