This course builds on the first course linear modeling and takes one step further. This course delivers the skill set in non-linear structural modeling & analysis in the framework of the finite element method that is required to solve problems from the engineering practice. The weekly lectures and practicals impart both, practical experience with the modeling pipeline of commercial FE software and the theory fundamentals for the numerical approximation of non-linear engineering phenomena.
The majority of professionals in many branches of mechanical engineering will benefit from adding FEM to their skills array. The ability to develop modeling skills under supervision in a non-critical environment means that skills and techniques can be acquired in a logical, progressive manner.
The main topics of this course:
- nonlinear finite element method
- Geometrical nonlinearity (large displacements)
- Material nonlinearity (plasticity, continuum damage, etc.)
- Modeling of contact
- Modeling of cracks
In this course students will gain:
- a strong theoretical understanding of non-linear FEM
- how to apply non-linear FEM to practical engineering problems
- efficient modeling and solution techniques
- understanding for the importance of verification and validation
While the previous course, linear modeling, is the required background, working professionals who are experienced with structural/stress analysis may choose to enroll directly in this course.
Computational methods in structural analysis are of prime importance in industry as tools to assess the efficiency and performance of structures in the field of aerospace, mechanical, civil and biomedical engineering. The reduced performance and destructive failure of structures is mostly due to the effects of physical and geometric non-linearities. The combination of theoretical and practical knowledge in finite element analysis are valuable skills with which to address such problems in industry.
In order to efficiently model a real life engineering problem using finite element analysis and predict its future behavior, an engineer must possess a strong theoretical understanding of non-linearities in the finite element method (FEM) along with the understanding of the importance of verification and validation of such computational models.
On completing this course, you will have the skillset in non-linear (structural) modeling required to solve structural problems for industry with a special emphasis on implementation of physical and geometric non-linearities. The course equips you with a theoretical background of non-linear FEM enabling you to develop finite element (FE) models for practical applications and correctly interpret their results.
The course uses a free FE package (for students) in weekly practical sessions where you will model sample problems and validate your results against simplified analytical models or open literature. These weekly exercises also serve as a continuous assessment of the course.
Multiple assignments are provided, both theoretical and practical. Assignments may be altered to particular needs to develop competencies you are looking for.
There will be optional online hangout sessions for the discussion of theories, exercises, assignments, etc.
The assessment of this course will be based on both the weekly exercises and assignments.
Non-Linear modeling is an advanced course as a follow up to linear modelling. It is based on the extension of formulation of the finite element equilibrium equation to the non-linear domain. Applications include geometric, material and contact-based nonlinearities. Extra knowledge is also shared in damage modelling and fracture. Some programming exercises are part of the assignment. These are modifiable for online students since not all of them are experienced programmers.
Students are expected to be well versed with software usage if they have not followed the linear modelling course. And especially non-linear solution solvers for the software they choose. Please not that this is not a software training course. The software is treated as an application platform to understand the nonlinear problems, solutions and comprehension.
Week 0: Self-study/Revision. Recap of linear finite element method formulation and usage.
Week 1: An introduction to the types of nonlinearities; how to setup a nonlinear problem; Newton types solution methods (Newton-Raphson and Modified Newton-Raphson).
Practical:Get familiarised 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; report and reflect on results in the results sheet.
Week 2: The limitations of Newton type methods; some other solution methods; arc-length method; the need for adding damping in models; convergence criteria.
Practical: carry out a non-linear analysis using Newton and Arc-length method
Week 3: the types of geometric nonlinearities; basic concepts related to continuum mechanics; finite element formulation of a truss element.
Practical: Assignment based on geometric nonlinearity
Week 4: Geometric stiffness matrix derivation for bar, beam and plate element; linear bifurcation buckling; post buckling; some modelling strategies for buckling analysis.
Practical: Carry out a twin analysis of buckling and post buckling.
Week 5: Physical nonlinearity - material behaviour; plasticity in uniaxial stress state; continuum damage; inclusion of material non-linear behaviour in a finite element description; computation procedures for elastic, plastic and yield states.
Practical: Carry out a material and geometric nonlinear analysis.
Week 6: The basic theory behind contact analysis; why contact is a non-linear problem; how contact is searched and implemented in a model; different interfaces and bodies in contact.
Practical: Exercise and assignment based on contact, material and geometric nonlinearities
Week 7: Introduction to Cohesive Zone Theory, cohesive elements, and modeling methods for crack propagations.
Practical: Special cases; Spill overs.
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.
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 (no driver's license)
- 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.