# Introduction to Finite Elements And the Big Picture

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### Introduction

Application of physical principles, such as mass balance, energy conservation, and equilibrium, naturally leads many engineering analysis situations into differential equations. The exact solutions for differential equations are not applicable to many practical problems because either their governing differential equations does not have an exact solution or they involve complex geometries.

The finite element method is one of the numerical methods for obtaining approximate solution of ordinary and partial differential equations. It is especially powerful when dealing with boundary conditions defined over complex geometries that are common in practical applications.

### Concepts and Formulas

#### Complete solution procedure:

The approximate solution of any problem governed by a differential equation can be obtained by using the element equations by following the usual finite element steps:

1. Development of element equations
2. Discretization of solution domain into a finite element mesh
3. Assembly of element equations
4. Introduction of boundary conditions
5. Solution for nodal unknowns
6. Computation of solution and related quantities over each element

#### Advantages of using Finite Element Method:

The subdivision of a whole domain into simpler parts has several advantages:

• Accurate representation of complex geometry; while most analytical solutions are restricted to simplified geometries.
• Inclusion of dissimilar material properties; while most analytical methods are applicable to homogeneous materials.
• Easy representation of the total solution
• Capture of local effects.

#### Structural element types:

1. One-dimensional (1D)
1. Spring (rod) elements
2. Beam elements 2. Two-dimensional (2D)
1. Membrane : in-plane loads only
2. Shell : in-plane and bending moments
3. Shear panel : shear loads only 3. Three-dimensional (3D)
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