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Finite Element Method - Massachusetts Institute of …

( ) ( ) Engineering Design and Rapid PrototypingEngineering Design and Rapid PrototypingInstructor(s) Finite Element MethodJanuary 12, 2004 Prof. Olivier de WeckDr. Il Yong ( )2 Plan for 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 ( )3 Course ( )4 Course Flow DiagramCAD/CAM/CAE IntroFEM/Solid Mechanics OverviewManufacturing TrainingStructural Test Training Design OptimizationHand sketchingCAD designFEM analysisProduce Part 1 TestProduce Part 2 OptimizationProblem statementFinal ReviewTestLearning/ReviewDeliverablesDes ign Sketch v1 Analysis output v1 Part v1 Experiment data v1 Design/Analysis output v2 Part v2 Experiment data v2 Drawing v1 Design Introdue ( )5 Numerical MethodFinite Elem

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.

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Transcription of Finite Element Method - Massachusetts Institute of …

1 ( ) ( ) Engineering Design and Rapid PrototypingEngineering Design and Rapid PrototypingInstructor(s) Finite Element MethodJanuary 12, 2004 Prof. Olivier de WeckDr. Il Yong ( )2 Plan for 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 ( )3 Course ( )4 Course Flow DiagramCAD/CAM/CAE IntroFEM/Solid Mechanics OverviewManufacturing TrainingStructural Test Training Design OptimizationHand sketchingCAD designFEM analysisProduce Part 1 TestProduce Part 2 OptimizationProblem statementFinal ReviewTestLearning/ReviewDeliverablesDes ign Sketch v1 Analysis output v1 Part v1 Experiment data v1 Design/Analysis output v2 Part v2 Experiment data v2 Drawing v1 Design Introdue ( )

2 5 Numerical MethodFinite Element MethodBoundary Element MethodFinite Difference MethodFinite Volume MethodMeshless ( )6 What is 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 process results in a set of simultaneous algebraic : Method for numerical solution of field of degrees-of-freedom (DOF)Continuum:InfiniteFEM: Finite (This is the origin of the name, Finite Element Method ) ( )7 Fundamental Concepts (1)Elastic problemsThermal problemsFluid engineering phenomena can be expressed by governing equations and boundary conditions Governing Equation(Differential equation)()0Lf +=Boundary Conditions() 0Bg += ( )8 Elastic deformationThermal Equation:()0Lf +=Boundary Conditions.

3 () 0Bg +=[]{}{}=KuFA set of simultaneous algebraic equationsFEMA pproximate!Fundamental Concepts (2)Example: Vertical machining centerGeometry is very complex!You know all the equations, but you cannot solve it by ( )9[]{}{}=KuF1{} [ ]{} =uKFPropertyBehaviorActionElasticThermal FluidElectrostaticBehavior{}uProperty[]K Action{}Fstiffnessdisplacementforcecondu ctivitytemperatureheat sourceviscosityvelocitybody forcedialectri permittivityelectric potentialchargeUnknownFundamental Concepts (3) ( )10It is very difficult to make the algebraic equations for the entire domain Divide the domain into a number of small, simple elementsAdjacent elements share the DOF at connecting nodesFundamental Concepts (4) Finite Element .

4 Small piece of structureA field quantity is interpolated by a polynomial over an ( )11 Obtain the algebraic equations for each Element (this is easy!)Put all the Element equations together[]{}{}=KuF[]{}{}EEE=KuF[]{}{}EEE =KuF[]{}{}EEE=KuF[]{}{}EEE=KuF[]{}{}EEE= KuF[]{}{}EEE=KuF[]{}{}EEE=KuF[]{}{}EEE=K uF[]{}{}EEE=KuFFundamental Concepts (5) ( )12[]{}{}=KuF1{} [ ]{} =uKFSolve the equations, obtaining unknown variabless at Concepts (6) ( )13 Concepts - Summary- FEM uses the concept of piecewise polynomial connecting elements together, the field quantity becomes interpolated over the entire structure in piecewise set of simultaneous algebraic equations at nodes.

5 []{}{}=KuFPropertyBehaviorActionK: Stiffness matrixx: DisplacementF: LoadKxF= ( )14 Brief History- The term Finite elementwas 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 The first book on the FEM by Zienkiewiczand Chung was published in In the late 1960s and early 1970s, the FEM was applied to a wide variety of engineering commercial FEM software packagesoriginated in the 1970s.(Abaqus, Adina, Ansys, etc.)-Klaus-Jurgen Bathe in ME at MITR eference [2] ( )15 Can readily handle very complex geometry:- The heart and power of the FEMCan handle a wide variety of engineering problems- Solid mechanics- Dynamics- Heat problems- Fluids- Electrostatic problemsCan handle complex restraints- Indeterminate structures can be handle complex loading- Nodal load (point loads)- Element load (pressure, thermal, inertial forces)- Time or frequency dependent loadingAdvantages of the ( )16 Disadvantages of the FEMA general closed-form solution, which would permit one to examine system response to changes in various parameters, is not FEM obtains only "approximate" FEM has "inherent" by userscan be ( )

6 17 Typical FEA Procedure by Commercial SoftwarePreprocessProcessPostprocessUser UserComputerBuild a FE modelConduct numerical analysisSee ( )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 type2-D3-DLinearQuadraticBeamTrussShellS olidPlate[3] Material properties,,,,E "Preprocess (1) ( )19 Preprocess (2)[4] Make nodes[5] Build elements by assigning connectivity[6] Apply boundary conditionsand ( )20 Process and Postprocess- Solve the boundary value problem[7] Process- See the results[8] PostprocessDisplacementStressStrainNatur al frequencyTemperatureTime ( )21 Responsibility of the userResults obtained from ten reputable FEM codes and by users regarded as expert.

7 *BC:Hinged supportsLoad: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 pulse200 mmUnknown:Lateral mid point displacement in the time ( )22 Errors Inherent in FEM FormulationQuadratic elementCubic Element -Field quantity is assumed to be a polynomialover an Element . (which is not true)True deformation-Geometry is domainFEML inear elementFEM-Use very simple integrationtechniques (Gauss Quadrature) xf(x)1-11111 Area:( )33fxdxff + ( )23-The computer carries only a Finite number of ) Very large stiffness ) 2 , ==12 2,0kkk 12 2222[() ]0 PPkkkuPuk+ = = Errors Inherent in ( )24 Mistakes by Users- elements are of the wrong ) Shell elements are used where solid elements are needed- Distorted elements - Supports are insufficient to prevent all rigid-body motions- Inconsistent units ( E=200 GPa, Force = 100 lbs)- Too large stiffness differences Numerical ( )25 Plan for Today FEM Lecture (ca.

8 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 ( )26 ReferencesGlaucio H. Paulino, Introduction to FEM (History, Advantages and Disadvantages), 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 , (in Korean)J. Tinsley Oden et al., Finite elements An Introduction, Prentice Hall, 1981


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