CAE -Finite Element Method16.81016.810Engineering Design and Rapid PrototypingEngineering Design and Rapid PrototypingInstructor(s)CAE -Finite Element MethodJanuary 16, 2007Prof. Olivier de WeckLecture 3b16.810 (16.682)2Numerical MethodsFinite Element MethodBoundary Element MethodFinite Difference MethodFinite Volume MethodMeshless Method16.810 (16.682)3What 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)4Fundamental 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)5Elastic deformationThermal behavioretc.Governing Equation:() 0Lfφ+=() 0BgBoundary Conditions:φ+=[]{}{}=Ku FA set of simultaneous algebraic equationsFEMApproximate!Example: Vertical machining centerGeometry is very complex!Fundamental Concepts (2)You know all the equations, but you cannot solve it by hand16.810 (16.682)6[]{}{}=Ku F1{} [ ]{}−=uKFPropertyBehaviorActionElastThermalFluidElectrostaticicBehavior{}uProperty[]KAction{}Fstiffnessdisplacementforceconductivity temperature heat sourceviscosityvelocitybody forDielectric permittivity electric potentialcechargeUnknownFundamental Concepts (3)16.810 (16.682)7It is very difficult to solve 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)8Obtain 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)9[]{}{}=Ku F1{} [ ]{}−=uKFSolve the equations, obtaining unknown variables at nodes.Fundamental Concepts (6)16.810 (16.682)10Concepts - 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)11Brief 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)12Can 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 loads - distributed (pressure, thermal, inertial forces)- Time or frequency dependent loadingAdvantages of the FEM16.810 (16.682)13Disadvantages 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 remain undetected.16.810 (16.682)14Typical FEA Procedure by Commercial SoftwarePreprocessProcessPostprocessUserUserComputerBuild a FE modelConduct numerical analysisSee results16.810 (16.682)15[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)16Preprocess (2)[4] Make nodes[5] Build elements by assigning connectivity[6] Apply boundary conditionsand loads16.810 (16.682)17Process and Postprocess- Solve the boundary value problem[7] Process- See the results[8] PostprocessDisplacementStressStrainNatural frequencyTemperatureTime history16.810 (16.682)18Responsibility 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)19Errors 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)1-11111Area: ( )33fxdx f f−⎛⎞ ⎛ ⎞≈+−⎜⎟ ⎜ ⎟⎝⎠ ⎝ ⎠∫16.810 (16.682)202-D vs. 3-DIn reality, everything is 3-D. But some problems can be simplified to 2-D (in structures, plane stress and plane strain).Plane StressPlane Strainthickness 0≈thickness≈∞sheetdam3-D2-Dzz0zσ=0zε=16.810 (16.682)21Truss vs. BeamTrussBeamOnly supports axial loadsSupports axial loads and bending loads16.810 (16.682)22- The computer carries only a finite number of digits.- Numerical Difficultiese.g.) Very large stiffness differencee.g.) 2 1.41421356, 3.14159265π==122,0kkk>> ≈12 22 22[( ) ]0PPkk ku P uk+− =⇒=≈Errors Inherent in Computing16.810 (16.682)23Mistakes by Users- Elements are of the wrong typee.g) Shell elements are used where solid elements are needed- Distorted elements- Supports are insufficient to prevent all rigid-body motions- Inconsistent units