## Course Title : Advanced Finite Element Methods

Code 10G041 Master and Doctor Course 1st term Wed 1st C3-Lecture Room 2 2 No Restriction Lecture and Practice English Kotera and Nishiwaki,

### Course Description

This course presents the basic concept and mathematical theory of the Finite Element Method (FEM), and explains how the FEM is applied in engineering problems. We also address important topics such as the physical meaning of geometrical non-linearity, material non-linearity, and non-linearity of boundary conditions, and we explore numerical methods to deal with these nonlinearities. Also, we guide students in class in the use of software to solve several numerical problems, to develop practical skill in applying the FEM to engineering problems.

Grading is based the quality of two or three reports and the final exam.

### Course Goals

The course goals are for students to understand the mathematical theory of the FEM and the numerical methods for analyzing non-linear problems based on the FEM.

### Course Topics

Theme Class number of times Description
Basic knowledge of the FEM 3 What is the FEM? The history of the FEM, classifications of partial differential equations, linear problems and non-linear problems, mathematical descriptions of structural problems (stress and strain, strong form and weak form, the principle of energy).
Mathematical background of the FEM 2 Variational calculus and the norm space, the convergence of the solutions.
FEM formulations 3 FEM approximations for linear problems, formulations of iso-parametric elements, numerical instability problems such as shear locking, formulations of reduced integration elements, non-conforming elements, the mixed approach, and assumed-stress elements.
Classifications of nonlinearities and their formulations 4 Classifications of nonlinearities and numerical methods to deal with these nonlinearities.
Numerical practice 2 Numerical practice using COMSOL.
Evaluation of student achievements 1

### Textbook(supplemental)

Bath, K.-J., Finite Element Procedures, Prentice Hall
Belytschko, T., Liu, W. K., and Moran, B.., Nonlinear Finite Elements for Continua and Structures, Wiley

Solid Mechanics