MECH4103 Finite Element Method Emneplan

This course covers the fundamentals of the Finite Element method and moves on to include advanced topics on the subject. It focuses on displacement-based isoparametric formulation of elements for an arbitrary discretized geometries in n-dimensional space. The course encompasses enough material for analysts and designers but also allows those keen on conducting research in the field to become aware of the methods and obstacles. As a numerical method, it may only be understood when it is used, therefore both Python coding and commercial software (ABAQUS) are treated as tools and several assignments, an individual project, and a group project are defined to ensure the knowledge learnt may be put into practice.

After completing the course, the student is expected to have achieved the following learning outcomes defined in terms of knowledge, skills and general competence:

Knowledge:

The student has:

• knowledge to establish equation of motion for SDOF and MDOF systems
• knowledge to model structural damping
• knowledge to compute the key-concepts related to structural dynamics, such as natural frequencies, mode shapes, damping and vibration characteristics of structures.
• in-dept knowledge about the assumptions and limitations of the structural dynamics theory.

Skills:

The student

• is capable of recognizing physical phenomenon in the context of structural vibration
• can formulate the equation of motion for dynamics analysis of structures
• can establish necessary matrices for the equation of motion, stiffness, mass and damping matrix
• can calculate response from harmonic and transient loads
• can use computer programming tools (e.g. Matlab) to perform modelling and dynamic analysis of simple structural systems
• can conduct dynamic analysis using commercially available software.

General Competence:

The student is able to:

• design structures with the consideration of structural dynamics.
• solve engineering problems in the context of structural dynamics
• assess the need for dynamic analysis in structural design
• characterize the dynamic properties of a structure such as natural periods and mode shapes.

Knowledge

The candidate

• can explain when and why finite element analyses are required
• can describe the finite element discretization of continuum mechanics equations
• can form mass and stiffness matrices and analyze simple structures using matrix analyses
• can describe Neumann, Dirichlet, and Robin boundary conditions in finite element analyses
• can explain different types of nonlinearities and implicit and explicit dynamic analyses
• can describe the principal idea of the isoparametric finite element formulation.

Skills

The candidate

• can form mass and stiffness matrices based on consistent isoparametric formulation and analyze simple structures using matrix analyses

• can calculate the entries of dense and sparse finite element matrices for continuum, beam, and shell elements
• can study solid mechanics problems, such as statics, implicit and explicit dynamics, and heat transfer, using FEM
• can use FEM to calculate eigenvalues and vibrational modes of a dynamic system
• can analyze the buckling behavior of simple structures and calculate the critical buckling load using the linear perturbation method
• can calculate and evaluate the post-buckling path for a structure under loading using the Riks arc-length method
• can apply relevant methods for solving problems, including ABAQUS and Python coding
• can apply ABAQUS to set up models and run simulations on complex systems.

General competence

The candidate:

• can transfer a practical engineering problem into a FEM problem and assess the numerical results by comparing them with analytical solutions or experimental results
• can communicate numerical results through a report, using accurate and appropriate terminology of FEM
• can contribute to sustainability by allowing for reduction in consumption of volume of materials during manufacturing, also through efficient problem solving and saving on electricity use and quick improved design through virtual testing
• can contribute to innovation in FEM through modeling discontinuities in structures, such as void, crack, and material interfaces, by enriching the approximation space using extended FEM
• can conduct a project in line with the instructions provided and within the bounds of ethical conduct.

Physical classroom lectures, individual exercises, and tutorials. Problem solving sessions with guided questions ranked from simple to difficult. Peer-learning though group formation and allowing students to learn from each other while doing the project related tasks.

The following coursework requirements must have been approved for the student to take the exam:

Four individual assignments (3-5 pages each) comprising deriving the governing equations for a system and solving small systems of equations for low degree-of-freedom prototypes using Python codes. The assignments must include an explanation of the problem and the solution procedure. The solutions must be analyzed and correlated with analytical results when possible.

The exam consists of two parts: One individual and one group project.

Part one: Individual project report, 20-30-page, which counts 40 % of the final grade.

Individual project includes developing a Python code for a medium sized structural system and solving for the response.

Part two: Group project report, 3-5 students per group, 30-40 page, which counts 60 % of the final grade. The project includes simulating a sophisticated system subjected to static, dynamic, or thermal loads using the commercial software ABAQUS and interpreting the results. Part one and two are complementary in the sense that in part one, students code themselves to understand the underlaying code, while in part two they use sophisticated software, ABAQUS to solve the given problem.

New/postponed exam:

In the event of a postponed examination in this course the exam may be held as an oral exam. Oral exams cannot be appealed.

Lecture notes, textbooks, opensource codes, available Python codes, YouTube online lectures

Graded scale A-F

None