## Finite Element Methods

#### Course Information

- Course Instructor VISWANADH CH., IIT HYDERABAD
- Level Undergraduate
- Language English
- Enrolled Colleges 35
- Total Students 1360
- Course Duration 16 Weeks
- Course Start June 28, 2017

**Description**

Finite Element Methods

A finite element method (abbreviated as FEM) is a numerical technique to obtain an approximate solution to a class of problems governed by elliptic partial differential equations. Such problems are called as boundary value problems as they consist of a partial differential equation and the boundary conditions. The finite element method converts the elliptic partial differential equation into a set of algebraic equations which are easy to solve. The initial value problems which consist of a parabolic or hyperbolic differential equation and the initial conditions (besides the boundary conditions) cannot be completely solved by the finite element method. Thus, to solve an initial value problem, one needs both the finite element method as well as the finite difference method where the spatial derivatives are converted into algebraic expressions by FEM and the temporal derivatives are converted into algebraic equations by FDM.

#### **Prerequisites**

Knowledge about basic calculus and differential equations.

**Course Objectives:**

Upon completion of this course, students will be able to do the following:

•To learn basic principles of finite element analysis procedure |

•To learn the theory and characteristics of finite elements that represent engineering structures. |

•To learn and apply finite element solutions to structural, thermal, dynamic problem to develop the knowledge and skills needed to effectively evaluate finite element analyses performed by others |

•Learn to model complex geometry problems and solution techniques. |

**Course Outcomes:**

After completing this course the candidate should be able to:

•Understand the concepts behind variational methods and weighted residual methods in FEM |

•Identify the application and characteristics of FEA elements such as bars, beams, plane and isoparametric elements, and 3-D element. |

• Develop element characteristic equation procedure and generation of global stiffness equation will be applied. |

• Able to apply Suitable boundary conditions to a global structural equation, and reduce it to a solvable form. |

• Able to identify how the finite element method expands beyond the structural domain, for problems involving dynamics, heat transfer, and fluid flow. |

**Syllabus**

### Unit 1

Introduction to finite element method, stress and equilibrium, strain – displacement relations, stress – strain relations, plane stress and plane strain conditions, variational and weighted residual methods, concept of potential energy, one dimensional problems.

### Unit 2

Discretization of domain, element shapes, discretization procedures, assembly of stiffness matrix, band width, node numbering, mesh generation, interpolation functions, local and global coordinates, convergence requirements, treatment of boundary conditions.

### Unit 3

Analysis of Trusses: Finite element modeling, coordinates and shape functions, assembly of global stiffness matrix and load vector, finite elementequations, treatment of boundary conditions, stress, strain and support reaction calculations. Analysis of Beams: Element stiffness matrix for Hermite beam element, derivation of load vector for concentrated and UDL, simple problems on beams.

### Unit 4

Finite element modeling of two dimensional stress analysis with constantstrain triangles and treatment of boundary conditions, formulation of axisymmetric problems.

### Unit 5

Higher order and isoparametric elements: One dimensional quadratic and cubic elements in natural coordinates, two dimensional four noded isoparametric elements and numerical integration.

### Unit 6

Steady state heat transfer analysis : one dimensional analysis of a fin and two dimensional analysis of thin plate, analysis of a uniform shaft subjected to torsion. Dynamic Analysis: Formulation of finite element model, element consistent and lumped mass matrices, evaluation of eigen values and eigen vectors, free vibration analysis.