The objective of this course is to provide to the students thorough theoretical and practical basis for the Finite Element Method (FEM). It will be shown how to apply this method in order to find an approximate solution to the boundary value problems in elasticity. The solution will be fully developed for displacement-based finite elements. Recovery of strains and stresses will be further demonstrated for single field approximations (just the displacement field) but also two field and three field independent approximations. These strategies will be demonstrated for finding the approximate solution of coupled problems, with reference to thermo-mechanical coupling problems. The problem of shear locking for quasi- incompressible solids will be treated in details with the employment of selective integration techniques and incompatible modes elements. The implementation of constitutive models into existing FEM codes will be exemplified on commercial software ABAQUS and open source software CODE_ASTER.

Upon completing the course the students will be able to apply linear theory of finite element method in order to: -write codes for building the stiffness matrix for structural and continuum finite elements; -perform finite element analysis within commercial software ABAQUS; -perform analysis within open source software package CODE_ASTER; -write their own subroutines for the implementation of constitutive models within the existing software.

Introduction to numerical modeling. The application of virtual work principle for the formulation of stiffness matrix. Full and selective numerical integration. Coupled problems and solution strategy for two field and three field independent approximations. Constitutive modeling and its numerical implementation within FEM codes. The solution of dynamic problems. Implicit and explicit integration scheme. The stability of the solution.

Writing codes in FORTRAN, C++ or Python, for the assembling of stiffness matrix for structural finite elements (trusses and beams) and continuum elements in 2D and 3D. Performing quasi-static and dynamic analysis in commercial software ABAQUS and open source software CODE_ASTER. Examples of enlarging the capabilities of existing software by the implementation of user sub-routines for the implementation of material constitutive modeling.

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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: 0
Review and grading of lab reports: 0
Review and grading of seminar papers: 10
Review and grading of the project: 0
Test: 0
Test: 5
Final exam: 0

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.; An introduction to Nonlinear Finite Element Analysis. J.N Reddy, Oxford University Press, 2015.; Computational methods in plasticity: Theory and applications. EA de Souza Neto, D. Peric and DRJ Owen, Wiley, 2008.