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![16.810 (16.682) 8
Elastic deformation
Thermal behavior
etc.
Governing
Equation: ( ) 0L fφ + =
Boundary
Conditions: ( ) 0B gφ + = [ ]{ } { }=K u F
A set of simultaneous
algebraic equationsFEM
Approximate!
Fundamental Concepts (2)
Example: Vertical machining center
Geometry is
very complex!
You know all the equations, but
you cannot solve it by hand](https://image.slidesharecdn.com/16-171001173033/85/Finite-Element-Method-8-320.jpg)
![16.810 (16.682) 9
[ ]{ } { }=K u F 1
{ } [ ] { }−
=u K F
Property
Behavior
Action
Elastic
Thermal
Fluid
Electrostatic
Behavior { }uProperty [ ]K Action { }F
stiffness displacement force
conductivity temperature heat source
viscosity velocity body force
dialectri permittivity electric potential charge
Unknown
Fundamental Concepts (3)](https://image.slidesharecdn.com/16-171001173033/85/Finite-Element-Method-9-320.jpg)
![16.810 (16.682) 11
Obtain the algebraic equations for each element (this is easy!)
Put all the element equations together
[ ]{ } { }=K u F
[ ]{ } { }E E E
=K u F [ ]{ } { }E E E
=K u F [ ]{ } { }E E E
=K u F
[ ]{ } { }E E E
=K u F [ ]{ } { }E E E
=K u F [ ]{ } { }E E E
=K u F
[ ]{ } { }E E E
=K u F [ ]{ } { }E E E
=K u F [ ]{ } { }E E E
=K u F
Fundamental Concepts (5)](https://image.slidesharecdn.com/16-171001173033/85/Finite-Element-Method-11-320.jpg)
![16.810 (16.682) 12
[ ]{ } { }=K u F 1
{ } [ ] { }−
=u K F
Solve the equations, obtaining unknown variabless at nodes.
Fundamental Concepts (6)](https://image.slidesharecdn.com/16-171001173033/85/Finite-Element-Method-12-320.jpg)
![16.810 (16.682) 13
Concepts - Summary
- FEM uses the concept of piecewise polynomial interpolation.
- By connecting elements together, the field quantity becomes interpolated
over the entire structure in piecewise fashion.
- A set of simultaneous algebraic equations at nodes.
[ ]{ } { }=K u F
Property
Behavior
Action
K: Stiffness matrix
x: Displacement
F: Load
Kx F=
K
F
x](https://image.slidesharecdn.com/16-171001173033/85/Finite-Element-Method-13-320.jpg)
![16.810 (16.682) 14
Brief History
- The term finite element was first coined by clough in 1960. In the early
1960s, engineers used the method for approximate solutions of problems
in stress analysis, fluid flow, heat transfer, and other areas.
- The first book on the FEM by Zienkiewicz and Chung was published in
1967.
- In the late 1960s and early 1970s, the FEM was applied to a wide variety
of engineering problems.
- Most commercial FEM software packages originated in the 1970s.
(Abaqus, Adina, Ansys, etc.)
- Klaus-Jurgen Bathe in ME at MIT
Reference [2]](https://image.slidesharecdn.com/16-171001173033/85/Finite-Element-Method-14-320.jpg)
![16.810 (16.682) 18
[1] Select analysis type - Structural Static Analysis
- Modal Analysis
- Transient Dynamic Analysis
- Buckling Analysis
- Contact
- Steady-state Thermal Analysis
- Transient Thermal Analysis
[2] Select element type 2-D
3-D
Linear
Quadratic Beam
Truss
Shell
Solid
Plate
[3] Material properties , , , ,E ν ρ α
Preprocess (1)](https://image.slidesharecdn.com/16-171001173033/85/Finite-Element-Method-18-320.jpg)
![16.810 (16.682) 19
Preprocess (2)
[4] Make nodes
[5] Build elements by assigning
connectivity
[6] Apply boundary conditions
and loads](https://image.slidesharecdn.com/16-171001173033/85/Finite-Element-Method-19-320.jpg)
![16.810 (16.682) 20
Process and Postprocess
- Solve the boundary value problem
[7] Process
- See the results
[8] Postprocess
Displacement
Stress
Strain
Natural frequency
Temperature
Time history](https://image.slidesharecdn.com/16-171001173033/85/Finite-Element-Method-20-320.jpg)
![16.810 (16.682) 23
- The computer carries only a finite number of digits.
- Numerical Difficulties
e.g.) Very large stiffness difference
e.g.) 2 1.41421356, 3.14159265π= =
1 2 2, 0k k k ≈
1 2 2 2 2
2
[( ) ]
0
P P
k k k u P u
k
+ − = ⇒ = ≈
Errors Inherent in Computing](https://image.slidesharecdn.com/16-171001173033/85/Finite-Element-Method-23-320.jpg)
Uploaded byTahir Khan
Finite Element Method
This document provides an overview of a course on engineering design and rapid prototyping. It discusses the finite element method (FEM) which will be covered in class. FEM involves cutting a structure into small elements and connecting them at nodes to form algebraic equations that can be solved numerically. This allows for approximate solutions to complex problems. The document outlines the typical FEM procedure of preprocessing, analysis, and postprocessing using software. It also discusses sources of errors in the FEM approach and mistakes users may make.
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Finite Element Method
- 1. 16.810 (16.682)16.810 (16.682) EngineeringDesign and Rapid PrototypingEngineering Design and Rapid Prototyping Instructor(s) Finite Element Method January 12, 2004 Prof. Olivier de Weck Dr. Il Yong Kim deweck@mit.edu kiy@mit.edu
- 2. 16.810 (16.682) 2 Planfor Today FEM Lecture (ca. 50 min) FEM fundamental concepts, analysis procedure Errors, Mistakes, and Accuracy Cosmos Introduction (ca. 30 min) Follow along step-by-step Conduct FEA of your part (ca. 90 min) Work in teams of two First conduct an analysis of your CAD design You are free to make modifications to your original model
- 3. 16.810 (16.682) 3 CourseConcept today
- 4. 16.810 (16.682) 4 CourseFlow Diagram CAD/CAM/CAE Intro FEM/Solid Mechanics Overview Manufacturing Training Structural Test “Training” Design Optimization Hand sketching CAD design FEM analysis Produce Part 1 Test Produce Part 2 Optimization Problem statement Final Review Test Learning/Review Deliverables Design Sketch v1 Analysis output v1 Part v1 Experiment data v1 Design/Analysis output v2 Part v2 Experiment data v2 Drawing v1 Design Intro due today today Wednesday
- 5. 16.810 (16.682) 5 NumericalMethod Finite Element Method Boundary Element Method Finite Difference Method Finite Volume Method Meshless Method
- 6. 16.810 (16.682) 6 Whatis the FEM? Description - FEM cuts a structure into several elements (pieces of the structure). - Then reconnects elements at “nodes” as if nodes were pins or drops of glue that hold elements together. - This process results in a set of simultaneous algebraic equations. FEM: Method for numerical solution of field problems. Number of degrees-of-freedom (DOF) Continuum: Infinite FEM: Finite (This is the origin of the name, Finite Element Method)
- 7. 16.810 (16.682) 7 FundamentalConcepts (1) Elastic problems Thermal problems Fluid flow Electrostatics etc. Many engineering phenomena can be expressed by “governing equations” and “boundary conditions” Governing Equation (Differential equation) ( ) 0L fφ + = Boundary Conditions ( ) 0B gφ + =
- 8. 16.810 (16.682) 8 Elasticdeformation Thermal behavior etc. Governing Equation: ( ) 0L fφ + = Boundary Conditions: ( ) 0B gφ + = [ ]{ } { }=K u F A set of simultaneous algebraic equationsFEM Approximate! Fundamental Concepts (2) Example: Vertical machining center Geometry is very complex! You know all the equations, but you cannot solve it by hand
- 9. 16.810 (16.682) 9 []{ } { }=K u F 1 { } [ ] { }− =u K F Property Behavior Action Elastic Thermal Fluid Electrostatic Behavior { }uProperty [ ]K Action { }F stiffness displacement force conductivity temperature heat source viscosity velocity body force dialectri permittivity electric potential charge Unknown Fundamental Concepts (3)
- 10. 16.810 (16.682) 10 Itis very difficult to make the algebraic equations for the entire domain Divide the domain into a number of small, simple elements Adjacent elements share the DOF at connecting nodes Fundamental Concepts (4) Finite element: Small piece of structure A field quantity is interpolated by a polynomial over an element
- 11. 16.810 (16.682) 11 Obtainthe algebraic equations for each element (this is easy!) Put all the element equations together [ ]{ } { }=K u F [ ]{ } { }E E E =K u F [ ]{ } { }E E E =K u F [ ]{ } { }E E E =K u F [ ]{ } { }E E E =K u F [ ]{ } { }E E E =K u F [ ]{ } { }E E E =K u F [ ]{ } { }E E E =K u F [ ]{ } { }E E E =K u F [ ]{ } { }E E E =K u F Fundamental Concepts (5)
- 12. 16.810 (16.682) 12 []{ } { }=K u F 1 { } [ ] { }− =u K F Solve the equations, obtaining unknown variabless at nodes. Fundamental Concepts (6)
- 13. 16.810 (16.682) 13 Concepts- Summary - FEM uses the concept of piecewise polynomial interpolation. - By connecting elements together, the field quantity becomes interpolated over the entire structure in piecewise fashion. - A set of simultaneous algebraic equations at nodes. [ ]{ } { }=K u F Property Behavior Action K: Stiffness matrix x: Displacement F: Load Kx F= K F x
- 14. 16.810 (16.682) 14 BriefHistory - The term finite element was first coined by clough in 1960. In the early 1960s, engineers used the method for approximate solutions of problems in stress analysis, fluid flow, heat transfer, and other areas. - The first book on the FEM by Zienkiewicz and Chung was published in 1967. - In the late 1960s and early 1970s, the FEM was applied to a wide variety of engineering problems. - Most commercial FEM software packages originated in the 1970s. (Abaqus, Adina, Ansys, etc.) - Klaus-Jurgen Bathe in ME at MIT Reference [2]
- 15. 16.810 (16.682) 15 Canreadily handle very complex geometry: - The heart and power of the FEM Can handle a wide variety of engineering problems - Solid mechanics - Dynamics - Heat problems - Fluids - Electrostatic problems Can handle complex restraints - Indeterminate structures can be solved. Can handle complex loading - Nodal load (point loads) - Element load (pressure, thermal, inertial forces) - Time or frequency dependent loading Advantages of the FEM
- 16. 16.810 (16.682) 16 Disadvantagesof the FEM A general closed-form solution, which would permit one to examine system response to changes in various parameters, is not produced. The FEM obtains only "approximate" solutions. The FEM has "inherent" errors. Mistakes by users can be fatal.
- 17. 16.810 (16.682) 17 TypicalFEA Procedure by Commercial Software Preprocess Process Postprocess User User Computer Build a FE model Conduct numerical analysis See results
- 18. 16.810 (16.682) 18 [1]Select analysis type - Structural Static Analysis - Modal Analysis - Transient Dynamic Analysis - Buckling Analysis - Contact - Steady-state Thermal Analysis - Transient Thermal Analysis [2] Select element type 2-D 3-D Linear Quadratic Beam Truss Shell Solid Plate [3] Material properties , , , ,E ν ρ α Preprocess (1)
- 19. 16.810 (16.682) 19 Preprocess(2) [4] Make nodes [5] Build elements by assigning connectivity [6] Apply boundary conditions and loads
- 20. 16.810 (16.682) 20 Processand Postprocess - Solve the boundary value problem [7] Process - See the results [8] Postprocess Displacement Stress Strain Natural frequency Temperature Time history
- 21. 16.810 (16.682) 21 Responsibilityof the user Results obtained from ten reputable FEM codes and by users regarded as expert.* BC: Hinged supports Load: Pressure pulse * R. D. Cook, Finite Element Modeling for Stress Analysis, John Wiley & Sons, 1995 Fancy, colorful contours can be produced by any model, good or bad!! Displacement(mm) Time (ms) 1 ms pressure pulse 200 mm Unknown: Lateral mid point displacement in the time domain
- 22. 16.810 (16.682) 22 ErrorsInherent in FEM Formulation Quadratic element Cubic element - Field quantity is assumed to be a polynomial over an element. (which is not true) True deformation - Geometry is simplified. Domain Approximated domain FEM Linear element FEM - Use very simple integration techniques (Gauss Quadrature) x f(x) 1-1 1 1 1 1 Area: ( ) 3 3 f x dx f f − ≈ + − ∫
- 23. 16.810 (16.682) 23 -The computer carries only a finite number of digits. - Numerical Difficulties e.g.) Very large stiffness difference e.g.) 2 1.41421356, 3.14159265π= = 1 2 2, 0k k k ≈ 1 2 2 2 2 2 [( ) ] 0 P P k k k u P u k + − = ⇒ = ≈ Errors Inherent in Computing
- 24. 16.810 (16.682) 24 Mistakesby Users - Elements are of the wrong type e.g) Shell elements are used where solid elements are needed - Distorted elements - Supports are insufficient to prevent all rigid-body motions - Inconsistent units (e.g. E=200 GPa, Force = 100 lbs) - Too large stiffness differences Numerical difficulties
- 25. 16.810 (16.682) 25 Planfor Today FEM Lecture (ca. 50 min) FEM fundamental concepts, analysis procedure Errors, Mistakes, and Accuracy Cosmos Introduction (ca. 30 min) Follow along step-by-step Conduct FEA of your part (ca. 90 min) Work in teams of two First conduct an analysis of your CAD design You are free to make modifications to your original model
- 26. 16.810 (16.682) 26 References GlaucioH. Paulino, Introduction to FEM (History, Advantages and Disadvantages), http://cee.ce.uiuc.edu/paulino Robert Cook et al., Concepts and Applications of Finite Element Analysis, John Wiley & Sons, 1989 Robert Cook, Finite Element Modeling For Stress Analysis, John Wiley & Sons, 1995 Introduction to Finite Element Method, http://210.17.155.47 (in Korean) J. Tinsley Oden et al., Finite Elements – An Introduction, Prentice Hall, 1981