Non linear Finite Element Methods

The objective of this course is to provide a comprehensive introduction to the methods and theory of nonlinear finite element analysis. The focus is given to the formulation and solution of the discrete equation for various classes of problems that are of principal interest in applications in solid mechanics and structural mechanics. In the introductory part of the course the discretization by finite elements of continua in one-dimension and in multi-dimension is presented. Discrete equations are developed for Lagrangian meshes, and different strategies for the solution of nonlinear problems are discussed. In problems of large displacements, the differences between total and updated Lagrangian formulations are demonstrated. Further on, the material nonlinearity is covered by introducing the formulation of constitutive equations for plasticity and behavior in large deformation regime. The course has an engineering style rather than a mathematical, although it includes analyses of the stability of numerical methods, as the objective is to teach methods of finite element analysis and the properties of the solution.

After fulfilling this course the students will be able to: -Understand and successfully use strategies for solving various nonlinear problems by methods that are implemented in most modern commercial FEM software. -Write their own code for iterative solution of nonlinear FEM problems in FORTRAN or MATLAB surrounding. -Will gain full understanding of implementation of nonlinear constitutive models into the algorithms for numerical solutions of boundary value problems. -Will be capable to develop their own sub-routines written in FORTRAN for specific material constitutive models which can be used within commercial FEM software.

Theoretical concepts of nonlinear methods and their implementation will be presented within theoretical lectures. Since a fundamental understanding of the equations requires substantial familiarity with continuum mechanics, the lectures will summarize the continuum mechanics which is pertinent to the topics taught in the course. Strategy solutions for given problems are fully derived on one dimensional elements and the concept is then extended to the multi dimensional elements.

Each topic covered is thoroughly demonstrated by numerical examples. Practical part of the course includes implementation of discussed strategies into fully working computer programs in MATLAB or FOTRAN surrounding. The use of discussed methods is evidenced and exemplified in commercial FEM software ABAQUS. Students will get familiar with using of such software for performing advanced nonlinear structural analysis.

Knowledge of basic FEM concepts and basic knowledge of structural analysis are required.

1.Non-linear finite element analysis of solids and structures – Volume 1. M.A. Crisfield 2.Introduction to computational plasticity. Fionn Dunne and Nik Petrinic. 3.An introduction to Nonlinear Finite Element Analysis. J.N Reddy 4.Computational methods in plasticity: Theory and applications. EA de Souza Neto, D. Peric and DRJ Owen.

Total assigned hours: 65

New material: 30
Elaboration and examples (recapitulation): 20

Auditory exercises: 0
Laboratory exercises: 0
Calculation tasks: 0
Seminar paper: 0
Project: 0
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 4
Review and grading of lab reports: 0
Review and grading of seminar papers: 5
Review and grading of the project: 5
Test: 0
Test: 0
Final exam: 1

Activity during lectures: 0
Test/test: 0
Laboratory practice: 5
Calculation tasks: 5
Seminar paper: 40
Project: 0
Final exam: 50
Requirement for taking the exam (required number of points): 30

Non-linear finite element analysis of solids and structures – Volume 1. M.A. Crisfield, Wiley, 2012.; Introduction to computational plasticity. Fionn Dunne and Nik Petrinic, Oxford University Press, 2005.; Computational methods in plasticity: Theory and applications. EA de Souza Neto, D. Peric and DRJ Owen, Wiley, 2008; Constitutive modeling of engineering materials: Theory, numerical implementation and characterization; V.Buljak and G. Ranzzi, Academic Press, 2021.