2. Introduction to FEA&

General Steps of FEA2.1. Definitions2.2. Typical Steps In F.E. Analysis2.3. Modeling Requirements for FE

What is Finite Element? The Finite Element Method A CAE technique in which a model of physical

configuration is developed. It permits computer modeling prior to prototype building.

2.1. Definitions

Finite Element Analysis A group of numerical methods for approximating the

solution of governing equations of any continuous system.

Example of problems that can be treated by FE:

• Structural Analysis• Heat Transfer• Fluid Flow• Mass Transport• Electromagnetic Potential• Acoustic• Bioengineering

2.1. Definitions

The primary commercial FE codes NASTRAN for aircraft industry ANSYS for nuclear industry ABAQUS MARC SAP ADINA MIT PATRAN

2.1. Definitions

Steps 1 - 5 are typically performed in sequence using Computer Aided Engineering tools.

The flow chart of the process using CAE tools is:

2.2. Typical Steps in FE

2.2. Typical Steps in FE

Pre-Processor

Solver

Post-Processor

5 steps involved in the procedure

1. Computer modeling, mesh generation

2. Definition of materials properties.

3. Assemble of elements

4. Boundary conditions and loads defined

5. Solution using the required solver and display results/data

1. Divide / discretize the structure or

continuum into finite elements.

This is typically done using mesh

generation program, called pre-processor.

2.2. Typical Steps in FE

2. Formulate the properties of each element.

Ex.: Nodal loads associated with all elements,

deformation states that are allowed.

2.2. Typical Steps in FE

3. Assemble elements to obtain FEA model

2.2. Typical Steps in FE

4. Specify the load and boundary conditions.

Constraints, force, known temperatures, etc.

5. Solve simultaneous linear algebraic equations

to obtain the solutions.

2.2. Typical Steps in FE

1. Model geometry

2. Material Properties

3. Meshing (s)

4. Load Cases

5. Boundary conditions

2.3. Modeling Requirements

simplify from actual dimensions

Is it necessary to model all the details of the components?

The problem can be reduced to part-modeling via symmetry?

1. Model Geometry

2.3. Modeling Requirements

2. Material Properties Standard or based on test data

Can we use standard data for the selected materials? Elastic modulus, poisson ratio, thermal conductivity,

electromagnetic permeability, etc. If it is not standard materials, do we need to confirm the

properties first through testing? Composite materials, new types of alloys, honeycomb

structure, etc.

2.3. Modeling Requirements

3. Meshing practical considerations in the meshing can lead to better accuracy of results and efficient computation.

• Aspect ratio• Element shape• Use of symmetry• Mesh refinement

2.3. Modeling Requirements

2-D meshing

3-D meshing

3. Meshing (examples)

2.3. Modeling Requirements

3. Meshing (Practical Considerations)

* Aspect Ratio is defined as the ratio of the longest dimension

to the shortest dimension of a quadrilateral element. as the aspect ratio increases, the inaccuracy

of the solution increases.

Large aspect ratio moderate aspect ratio good aspect ratio

2.3. Modeling Requirements

Aspect Ratio, (AR) = longest dimension/shortest dimension

exact solution

Per

cen

t o

f ac

cura

cy in

d

isp

lace

men

t

FEA results

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

AR

2.3. Modeling Requirements3. Meshing (Practical Considerations)

* Aspect Ratio

* Element shape An element yields best results if its shape is

compact and regular.

• Elements with poor shapes tend to yield poor results.

• in general try to: 1. Maintain aspect ratio as low as possible (closest to 1) 2. Maintain the corner angles of quadrilateral near 90°.

2.3. Modeling Requirements3. Meshing (Practical Considerations)

Very large and very small corner angles

Triangular quadrilateral With Large and small angles

Large aspect ratio

Examples of elements with poor shape

2.3. Modeling Requirements3. Meshing (Practical Considerations)

* Element shape

2.3. Modeling Requirements3. Meshing (Practical Considerations)

* Element shape

Use of Symmetry

The use of symmetry allows us to consider a reduced problem instead of the actual problem.

Then we can either:Model the problem with less number of elements.Use a finer meshing with less labor and computational cost.

2.3. Modeling Requirements3. Meshing (Practical Considerations)

Example on application of symmetry

F-F

Dog bone specimen

2.3. Modeling Requirements

Use of Symmetry

3. Meshing (Practical Considerations)

Modeling half of the flow over a circular pipe

CFD of half car

2.3. Modeling Requirements

Use of Symmetry

3. Meshing (Practical Considerations)

Breaking up the load

2.3. Modeling Requirements

Use of Symmetry

Not only the geometry, the forces as well

3. Meshing (Practical Considerations)

2.3. Modeling Requirements3. Meshing (Practical Considerations)•Mesh refinement

Use a relatively fine discretization in regions where you expect a high gradient of strains and/or stresses.

Regions to watch out for high stress gradients are:• Near entrant corners or sharply curved edges.• In the vicinity of concentrated (point) loads, concentrated reactions, cracks and cutouts.

In the interior of structures with abrupt changes in thickness, material properties or cross sectional areas.

2.3. Modeling Requirements3. Meshing (Practical Considerations)

•Mesh refinement

Examples.

2.3. Modeling Requirements3. Meshing (Practical Considerations)•Mesh refinement

Examples

Refine mesh use near internal hole and sharp angle

FEA model of welding joints

4. Load Cases

• Is it point load or distributed load?• Is the force applied to the whole body ? (Inertia,

gravity)• What is the estimated magnitude of forces (and

direction)

100 N

point load distributed load, Snow on a surface

2.3. Modeling Requirements

2.3. Modeling Requirements4. Load Cases

In practical structural problems, distributed loads are more common than concentrated (point) loads. Distributed loads may be of surface or volume type.

Distributed surface loads are associated with actions such as wind or water pressure, snow weight on roofs, lift in airplanes, live loads on bridges, and the like. They are measured in force per unit area.

Volume loads (called body forces in continuum mechanics) are associated with own weight (gravity), inertial, centrifugal, thermal, pre-stress or electromagnetic effects. They are measured in force per unit volume.

2.3. Modeling Requirements4. Load Cases

Examples

Pressure Vessel (Surface Load) Snow on the roof (Surface Load)

Structure deformation due to gravity (Volume load)

5. Boundary conditions

• Support locations and point of contacts.

• Types of support. • Fully constraints or free to

translate/rotate in certain direction?

• Friction?

• Temperatures distribution at the boundaries?

• Flow parameters at inlets and outlets.

2.3. Modeling Requirements

2.3. Modeling Requirements

Numerical Method?

The finite element method is a numerical method for solving problems of engineering and mathematical physics. In FEA, the continuum is divided into finite number of elements

and the governing equations are represented in matrix form. Method for solutions developed to solve complex mathematical

problems:• Runge-Kutta, Gauss-Seidel, Galerkin, Rayleigh, Ritz, Forward

Difference, etc.

1. Physical problem

2. Global Stiffness Matrix

3. Governing Equations

In obtaining the approximate solution, the continuum is discretized into finite elements.

Useful for problems with complicated geometries, loadings, and material properties where analytical solutions can not be obtained.

Approximation?

Finite element analysis is broadly defined as a group of numerical methods for approximating the governing equations of any continuous system. For a regular types bodies/surfaces (constant cross section,

cylinder, square, etc) , it might be possible to find closed-loop analytical solution.

For irregular types bodies/surfaces, the boundaries are irregular and the analytical solution might not exist.

Discretize?

In obtaining the approximate solution, the continuum is discretized into finite elements. The structure/parts/components are divided into

finite number of elements. The selection of elements types are based on

many factors – geometry, processing power, types of loadings, etc.

1. Actual geometry & loading 2. Discretization (Meshing) 3. Solution (Von Mises Stress)

Discretize?

The elements are interconnected at points common to two or more elements (nodes or nodal points) and/or boundary lines and/or surfaces.

The transfer of load (force, displacement, heat flux, etc) between elements occurred at the common nodes between elements.

Elements

Node

Discretize?

The transfer of load (force, displacement, heat flux, etc) between elements occurred at the common nodes between elements.

Primary Assumptions in FEA

Typical Steps in FEA

Matrix Operation Review

Vectors & Matrix

Examples

3 x 1: vector 4 x 4: matrix

0486

3612

0486

6901

]K[

2.3

2

1

}u{

The elements of a matrix are defined by their row and their column position:

Note, the 1st subscript is the row position and the 2nd subscript is the column position.

Therefore, is the element in the ith row and the jth column.

2221

1211

kk

kk]k[

Matrix Definition

ijk

If the matrix elements are defined as:

B1,1=1, B1,2=3, B2,1=4, B2,2=5

The matrix B is:

54

31]B[

Element Definition

Matrices can be multiplied by another matrix, but only if the left-hand matrix has the same number of columns as the right hand matrix has rows.

A*B=C

625

341A

109

1811

127

B

136111

7478C

Matrix Multiplication

The product of a Matrix, A, and it’s inverse, A-1 is the identity matrix, I. Only square matrices can be inverted.

Not all square matrices are invertible. A matrix has an inverse if and

only if it is nonsingular (its determinant is nonzero)

212

5

2

31A

32

54A

10

01* 1AA

10

01I

Identity Matrix

Announcement

Lecture & Lab

Please check lists of lecture group posted at Block 18, 3rd Floor.

Starting this week attendance will be recorded and you have to attend your assigned lecture session.