16.810 (16.682) 16.810 (16.682) Engineering Design and Rapid PrototypingEngineering Design and Rapid PrototypingInstructor(s)Finite Element MethodJanuary 12, 2004Prof. Olivier de Weck Dr. Il Yong [email protected] [email protected] (16.682)2Plan 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 model16.810 (16.682)3Course Concepttoday16.810 (16.682)4Course Flow DiagramCAD/CAM/CAE IntroFEM/Solid Mechanics OverviewManufacturing TrainingStructural Test “Training”Design OptimizationHand sketchingCAD designFEM analysisProduce Part 1TestProduce Part 2OptimizationProblem statementFinal ReviewTestLearning/Review DeliverablesDesign Sketch v1Analysis output v1Part v1Experiment data v1Design/Analysis output v2Part v2Experiment data v2Drawing v1Design Introdue todaytodayWednesday16.810 (16.682)5Numerical MethodFinite Element MethodBoundary Element MethodFinite Difference MethodFinite Volume MethodMeshless Method16.810 (16.682)6What 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 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: InfiniteFEM: Finite(This is the origin of the name, Finite Element Method)16.810 (16.682)7Fundamental Concepts (1)Elastic problemsThermal problemsFluid flowElectrostaticsetc.Many engineering phenomena can be expressed by “governing equations” and “boundary conditions”Governing Equation(Differential equation)() 0Lfφ+=Boundary Conditions() 0Bgφ+=16.810 (16.682)8Elastic deformationThermal behavioretc.Governing Equation:() 0Lfφ+=Boundary Conditions:() 0Bgφ+=[]{}{}=Ku FA set of simultaneous algebraic equationsFEMApproximate!Fundamental Concepts (2)Example: Vertical machining centerGeometry is very complex!You know all the equations, but you cannot solve it by hand16.810 (16.682)9[]{}{}=Ku F1{} [ ] {}−=uKFPropertyBehaviorActionElasticThermalFluidElectrostaticBehavior{}uProperty[]KAction{}Fstiffnessdisplacementforceconductivity temperature heat sourceviscosityvelocitybody forcedialectri permittivity electric potentialchargeUnknownFundamental Concepts (3)16.810 (16.682)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: Small piece of structureA field quantity is interpolated by a polynomial over an element16.810 (16.682)11Obtain the algebraic equations for each element (this is easy!)Put all the element equations together[]{}{}=Ku F[]{}{}EEE=Ku F []{}{}EEE=Ku F []{}{}EEE=Ku F[]{}{}EEE=Ku F []{}{}EEE=Ku F []{}{}EEE=Ku F[]{}{}EEE=Ku F[]{}{}EEE=Ku F[]{}{}EEE=Ku FFundamental Concepts (5)16.810 (16.682)12[]{}{}=Ku F1{} [ ] {}−=uKFSolve the equations, obtaining unknown variabless at nodes.Fundamental Concepts (6)16.810 (16.682)13Concepts - 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.[]{}{}=Ku FPropertyBehaviorActionK: Stiffness matrixx: DisplacementF: LoadKxF=KFx16.810 (16.682)14Brief 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 MITReference [2]16.810 (16.682)15Can 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 solved.Can handle complex loading- Nodal load (point loads)- Element load (pressure, thermal, inertial forces)- Time or frequency dependent loadingAdvantages of the FEM16.810 (16.682)16Disadvantages of the FEMA 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.16.810 (16.682)17Typical FEA Procedure by Commercial SoftwarePreprocessProcessPostprocessUserUserComputerBuild a FE modelConduct numerical analysisSee results16.810 (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 type2-D3-DLinearQuadraticBeamTrussShellSolidPlate[3] Material properties,,,,Eνρα"Preprocess (1)16.810 (16.682)19Preprocess (2)[4] Make nodes[5] Build elements by assigning connectivity[6] Apply boundary conditionsand loads16.810 (16.682)20Process and Postprocess- Solve the boundary value problem[7] Process- See the results[8] PostprocessDisplacementStressStrainNatural frequencyTemperatureTime history16.810 (16.682)21Responsibility of the userResults obtained from ten reputable FEM codes and by users regarded as expert.*BC: Hinged supportsLoad: Pressure pulse* R. D. Cook, Finite Element Modeling for Stress Analysis, John Wiley & Sons, 1995Fancy, 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 domain16.810 (16.682)22Errors Inherent in FEM FormulationQuadratic element Cubic element- Field quantity is assumed to be a polynomial over an element. (which is not true)True deformation- Geometry is simplified.DomainApproximated domainFEMLinear elementFEM- Use very simple integration techniques (Gauss Quadrature) xf(x)1-11111A rea: ( )33fxdx f