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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
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f ac
cura
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d
isp
lace
men
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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
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Starting this week attendance will be recorded and you have to attend your assigned lecture session.