# Nonlinear Finite Element Method (MFGE576) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Nonlinear Finite Element Method MFGE576 3 0 0 3 5
Pre-requisite Course(s)
MFGE 505
Course Language English N/A Ph.D. Face To Face Lecture, Drill and Practice, Problem Solving. Asst. Prof. Dr. Celalettin Karadoğan The objective of this course is to introduce basic topics in nonlinear finite element analysis of metal forming operations. Sources of nonlinearities will be covered. Solution methods of nonlinear equation systems will be introduced. Based on these preliminary information one dimensional nonlinear problems will be used to deepen knowledge on the nonlinearities and their nature. Further lectures will cover two and three dimensional rigid plastic and large strain elasto-plastic behavior of metals and the necessary finite element concepts for the solution of metal forming processes. The students who succeeded in this course; An ability to identify and solve metal forming problems using finite element method. An ability to solve nonlinear transient problems in the form of nonlinear equation systems. Understand the basic assumptions and equations of nonlinear finite element method. Attain necessary knowledge on plasticity theory of materials and rigid as well as elasto-plastic material formulations. Understand the issues such as contact, friction, remeshing, implicit formulations and explicit methods. Review of the linear FE-concepts, solution of nonlinear equations, one-dimensional nonlinear problems, two/there-dimensional rigid-plastic finite element solution, two/three-dimensional large-strain elasto-plastic FE-solutions.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Chapter 1: Introduction Linearity Assumption in Mechanics, Sources of Nonlinearity, Examples of Nonlinear Problems in Solid Mechanics
2 Chapter 2: Review of Linear FEM-Concepts Common Procedure of FEA, Direct Approach (Example: Truss Solution), Types of Elements, Variational Approach, Example: Tapered Slab
3 Chapter 3: Solution of Nonlinear Equations Incremental Solution Methods (Euler Method, Self-Correcting Euler Method), Iterative Solution Methods (Direct Iteration Method, Full Newton-Raphson Method)
4 Chapter 3: Solution of Nonlinear Equations Iterative Solution Methods (Modified Newton-Raphson Method, Quasi-Newton Methods), Numerical Errors (Condition Number , Ill-Conditioned Set of Equations)
5 Chapter 4: One-Dimensional Nonlinear Problems Material Nonlinearities: Small-Strain Elasto-Plasticity (Fundamentals , Finite Element Discretization, Incremental Newton-Raphson Solution, Initial Stiffness Solution)
6 Chapter 4: One-Dimensional Nonlinear Problems Geometric Nonlinearities: Small-Strain Large-Displacements (Introduction, A Finite Strain Measure, Finite Element Discretization by Energy Method, An Example: Spring-Truss System)
7 Chapter 5: Two/Three-Dimensional Rigid-Plastic Finite Element Solution One-Dimensional Observations on Theory of Plasticity (Idealized Observations, Idealized Stress-Strain Models, Microstructural Mechanisms of Plastic Deformation)
8 Chapter 5: Two/Three-Dimensional Rigid-Plastic Finite Element Solution General Potential Theory of Plasticity (The Yield Condition, The Flow Rule-Drucker's Postulate, Work-Hardening Assumption, Extremum Principles of Plasticity)
9 Chapter 5: Two/Three-Dimensional Rigid-Plastic Finite Element Solution Finite Element Solution: Problem Description, Finite Element Discretization
10 Chapter 5: Two/Three-Dimensional Rigid-Plastic Finite Element Solution Finite Element Solution: Solution Procedure (Direct Iteration Solution, Newton-Raphson Solution, Element Selection and Integration Orders, Modelling Friction)
11 Chapter 5: Two/Three-Dimensional Rigid-Plastic Finite Element Solution Finite Element Solution: Solution Procedure (Treatment of Rigid Regions, Contact-Algorithms, Remeshing-Algorithms, Application Codes)
12 Chapter 6: Two/Three-Dimensional Large-Strain Elasto-Plastic FE-Solutions Static Implicit Methods: Governing Variational Statement
13 Chapter 6: Two/Three-Dimensional Large-Strain Elasto-Plastic FE-Solutions Static Implicit Methods: Governing Variational Statement (Objective Stress Increment, Finite Strain Increment, Time Integration of the Constitutive Equation), Finite Element Equations
14 Chapter 6: Two/Three-Dimensional Large-Strain Elasto-Plastic FE-Solutions Dynamic Explicit Methods (Mass-Spring-Damper System, Finite Element Equation of Motion, Computational Issues, Dynamic Relaxation)
15 Final Examination Period
16 Final Examination Period

### Sources

Course Book 1. Cook, R. D.; Malkus, D. S.; Plesha, M. E.: Concepts and Applications of Finite Element Anlaysis, New York: John Wiley & Sons, 1989 2. Malvern, L. E.: Introduction to Mechanics of a Continuous Media, Englewood Cliffs/New Jersey: Prentice-Hall, 1969 3. Kobayashi, S.; Oh, S.; Altan, T.: Metal Forming and the Finite-Element Method; New York: Oxford University Press, 1989. 4. Lubliner, J.: Plasticity Theory, New York: Macmillan, 1990

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 6 30
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 1 30
Final Exam/Final Jury 1 40
Toplam 8 100
 Percentage of Semester Work 60 40 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Gains the ability to understand and apply knowledge in the fields of mathematics, science and basic sciences at the level of expertise.
2 Gains the ability to access wide and deep knowledge in the field of Engineering by doing scientific research with current techniques and methods, evaluate, interpret and implement the gained knowledge.
3 Being aware of the latest developments his/her field of study, defines problems, formulates and develops new and/or original ideas and methods in solutions.
4 Designs and applies theoretical, experimental, and model-based research, analyzes and interprets the results obtained at the level of expertise.
5 Gains the ability to use the applications, techniques, modern tools and equipment in his/her field of study at the level of expertise.
6 Designs, executes and finalizes an original work process independently.
7 Can work in interdisciplinary and interdisciplinary teams, lead teams, use the information of different disciplines together and develop solution approaches.
8 Pays regard to scientific, social and ethical values in all professional activities and acquires responsibility consciousness at the level of expertise.
9 Contributes to the literature by communicating the processes and results of his/her academic studies in written form or orally in national and international academic environments, communicates effectively with communities and scientific staff working in the field of specialization.
10 Gains the skill of lifelong learning at the level of expertise.
11 Communicates verbally and in written form using a foreign language at least at the European Language Portfolio B2 General Level.
12 Recognizes the social, environmental, health, safety, legal aspects of engineering applications, as well as project management and business life practices, being aware of the limitations they place on engineering applications.

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application 16 2 32
Special Course Internship
Field Work
Study Hours Out of Class 16 6 96
Presentation/Seminar Prepration
Project
Report
Homework Assignments 6 6 36
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury
Prepration of Final Exams/Final Jury 1 15 15