Finite Element Analysis Notes (computer Aided Engineering )

2. Introduction to FEA & General Steps of FEA 2.1. Definitions 2.2. Typical Steps In F.E. Analysis 2.3. Modeling Requirements for FE

2.1. Definitions 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.  Finite Element Analysis A group of numerical methods for approximating the solution of governing equations of any continuous system.

2.1. Definitions 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

2.2. Typical Steps in FE 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 5 steps involved in the procedure 1. Computer modeling, mesh generation

Pre-Processor

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

Solver

Post-Processor

2.2. Typical Steps in FE 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.

2.3. Modeling Requirements 1. Model geometry 2. Material Properties 3. Meshing (s) 4. Load Cases 5. Boundary conditions

2.3. Modeling Requirements 1. Model Geometry 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?

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 3. Meshing (examples) 2-D meshing

3-D meshing

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 3. Meshing (Practical Considerations) * Aspect

Ratio exact solution

FEA results

ca f ot necr e P

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

AR

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

2.3. Modeling Requirements 3. Meshing (Practical Considerations) * 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 Requirements 3. Meshing (Practical Considerations) * Element

shape Examples of elements with poor shape

Very large and very small corner angles Triangular quadrilateral With Large and small angles Large aspect ratio

2.3. Modeling Requirements 3. Meshing (Practical Considerations) * Element

shape

2.3. Modeling Requirements 3. Meshing (Practical Considerations) 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 Requirements 3. Meshing (Practical Considerations) Use of Symmetry

Example on application of symmetry -F

F

Dog bone specimen

2.3. Modeling Requirements 3. Meshing (Practical Considerations) Use of Symmetry

CFD of half car

Modeling half of the flow over a circular pipe

2.3. Modeling Requirements 3. Meshing (Practical Considerations) Use of Symmetry

Breaking up the load Not only the geometry, the forces as well

2.3. Modeling Requirements 3. 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 Requirements 3. Meshing (Practical Considerations) •Mesh refinement Examples.

2.3. Modeling Requirements 3. Meshing (Practical Considerations) •Mesh refinement Examples

FEA model of welding joints

Refine mesh use near internal hole and sharp angle

2.3. Modeling Requirements 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 4. 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 Requirements 4. Load Cases Examples

Pressure Vessel (Surface Load)

Snow on the roof (Surface Load)

Structure deformation due to gravity (Volume load)

2.3. Modeling Requirements 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

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. 2. Global Stiffness Matrix

1. Physical problem 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.

Node Elements

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

1 {u} = − 2   3.2

4 x 4: matrix

1 6 [K] =  2  6

0 8 1 8

9 4 6 4

6 0  3  0

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

 k11 [k] =  k 21

k12  k 22 

Note, the 1st subscript is the row position and the 2nd subscript is the column position. Therefore, k ij is the element in the ith row and the jth column.

Element Definition If the matrix elements are defined as: B1,1=1, B1,2=3, B2,1=4, B2,2=5 The matrix B is:

1 3 [ B] =   4 5

Matrix Multiplication Matrices can be multiplied by another matrix, but only if the lefthand matrix has the same number of columns as the right hand matrix has rows. A*B=C 1 4 3 A=  5 2 6

 7 12 B = 11 18  9 10

 78 74  C=  111 136  

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

4 5 A=  2 3  

1 0 −1 A* A =   0 1  

5 3 − A =2 2 − 1 2    −1

1 0 I =  0 1 

Not all square matrices are invertible. A matrix has an inverse if and only if it is nonsingular (its determinant is nonzero)

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