Introduction to Finite Element Method - iut.ac.ir

1 Introduction to Finite Element MethodByS. Ziaei-RadWhere the Course FitsThe field of Mechanics can be subdivided into 3 major areas:MechanicsTheoreticalAppliedComputa tionalComputational MechanicsNano and MicromechanicsContinuum MechanicsComputationalMechanicsBranches of Computational Mechanics can be distinguishedAccording to the physical focus of attentionSolid & StructuresFluidsMultiphysicsComputationa l Solid and Structural MechanicsA convenient subdivision of problems in ComputationalSolid and Structural Mechanics (CSM) isComputationalSolid and StructuralMechanics (CSM)StaticsDynamicsLinearNon-LinearCSM Linear StaticsFor the numerical simulation on the computer we must nowchose a spatial discretization Method :CSM Linear StaticsFinite Element MethodFinite Difference MethodBoundary Element MethodFinite Volume MethodSpectral MethodMesh-Free MethodCSM Linear Statics by FEMH aving selected the FEM for discretization, we must nextpick a formulation and a solution Method :Formulation of FEM ModelSolution of FEM Model Direct MethodVariational MethodWeighted ResidualsStiffnessFlexibilityMixedFormul ation of FEM Model1- The Direct Method - Limited to very simple Element - It worth studying because it enhances the The Variational Method - Applicable to problems that can be stated by certainintegral Weighted Residual methods - Applicable to problems for which differential equationsare known but no variational statement is ConceptsThe Finite Element Method (FEM), or Finite Element analysis(FEA), is based on the idea of building a complicated object wisimple blocks, or, dividing a complicated object into small andmanageable pieces.

2 Application of this simple idea can be founeverywhere in everyday life as well as in : Lego (kids play) Buildings Approximation of the area of a circle:Basic Concepts Archimedes' problem (circa 250 ): rectification of thecircle as limit of inscribed regular polygonsArea of one triangle:Area of the circle:where N = total number of triangles ( elements ).)2/sin)(2/cos(iiiRRS Computing "by Archimedes FEM" Why Finite Element Method ? Design analysis: hand calculations, experiments, andcomputer simulations FEM/FEA is the most widely applied computer simulation Method in engineering Closely integrated with CAD/CAM applications ..Applications of FEM in Engineering Mechanical/Aerospace/Civil/Automobile Engineering Structure analysis (static/dynamic, linear/nonlinear) Thermal/fluid flows Electromagnetics Geomechanics Biomechanics Fluid/solid Interactions Fluid/thermal/solid Interactions ..A Brief History of the FEM 1941 ----- Hrennikoff Used 1D Element (bars and beams) for the solution of stresscontinuous solids.

3 1943 ----- Courant(Variational methods )First to propose the FEM as we know today, he used principof stationary potential energy. 1956 ----- Turner, Clough, Martin and Topp(Stiffness)Stiffness equations in matrix format and solved equations wdigital computers. (100 DOFs) 1960 ----- Clough( 2D Finite Element , plane problems)Triangular plane stress Element to model skin of a delta winA Brief History of the FEM 1961 ----- Martin (3D tetrahedral elements ) 1962 ---- Callagher, Padlog and Bijlaard (3D elements ) 1963, 1964 ----- Melosh and Argyris (3D elements ) 1965 ---- Clough and Rashid , Wilson (Axisymmetric solid) 1970s -----Applications on mainframe computers 1980s -----Microcomputers, pre- and postprocessors 1990s -----Analysis of large structural systemsA Brief History of the FEMFE DOFs1950s ----- 100 DOFs1960s ----- 1000 DOFs1980s ----- 10000 DOFs1990s ----- 100000 DOFs2000s -----500000-Several millions DOFsPapers Published in FEM1961 ----- 10 1966 ----- 1341971 ----- 8441976 ----- 70001986 ----- in Structural AnalysisProcedures: Divide structure into pieces ( elements with nodes).

4 Describe the behavior of the physical quantities on eachelement,. Connect (assemble) the elements at the nodes to form anapproximate system of equations for the whole structure. Solve the system of equations involving unknownquantities at the nodes ( , displacements). Calculate desired quantities ( , strains and stresses) atComputer Implementations Preprocessing (build FE model, loads and constraints) FEA solver (assemble and solve the system of equations) Postprocessing (sort and display the results)Available Commercial FEM Software Packages ANSYS (General purpose, PC and workstations) NISA (PC and workstation) SDRC/I-DEAS (Complete CAD/CAM/CAE package) NASTRAN (General purpose FEA on mainframes) ABAQUS (Nonlinear and dynamic analyses) COSMOS (General purpose FEA) ALGOR (PC and workstations) PATRAN (Pre/Post Processor) HyperMesh (Pre/Post Processor) LsDyna, Dyna-2D, Dyna-3D (Crash/impact analysis) Pro-Mechanica (PTC Company) ..Available Commercial FEM Software Packages1969 ---Pedro Marcal taught at Brown University for a time butHe set up a firm to market the first nonlinear commercial FEprogram called ---John Swanson was developing a NFE program atWestinghouse for Nuclear applications.

5 He left Westinghouseto market program ---David Hibbit who worked for Marcal until 1972 andThen co-founded HKS which markets ABAQUS. The program was the first to introduce gateways for researchers to add elements and material launched his program after completing his PhDUnder supervision of Wilson at MIT. ADINA was an outgrowthOf Commercial FEM Software Packages1975 ---A milestone in the advancement of explicit FE wasJohn Halliquist s work at Lawrence Livermore. He releasedhis code called DYNA in success was the development of contact-impact interfaceswith Dave Benson and the resulting codes DYNA-2D and DYNA code first commercialized by French firm ESI in 1980sand called PAMCRASH with many routines from Halliquist left Livemore and started his own firm todistribute LSDYNA, a commercial version of and Disadvantages of general-purpose programsAdvantages1- The input is well The are large systems and can solve many types of problemsof large or small Many programs have the ability for adding new modules fornew kinds of problems or new technology with minimum Many of them can run on Many of them have become very attractive in price and cansolve a wide range of programs are less efficient than The initial cost of developing general-purpose programs is The user has little access to the logic of the of This FEM Course Understand the fundamental ideas of the FEM Know the behavior and usage of each type of elementscovered in this course Be able to prepare a suitable FE model for given problems Can interpret and evaluate the quality of the results (knowthe physics of the problems) Be aware of the limitations of the FEM (don t misuse theFEM - a numerical tool)

6 Course Coverage Finite Element Discretization Concepts Formulation of Finite elements Computer Implementation of FEMWhat do we need?1- ANSYS2- MATLABE xamples:Boot SealBoot seals are used to protect steering mechanisms in automobiles. These flexible components must accommodate the motions associated with angulation of the steering mechanism. Some regions of the boot seal are always in contact with an internal metal shaft, while other areas come into contact with the metal shaft during the angulation. In addition, the boot seal may also come into contact with itself, both internally and externally. The contacting regions affect the performance and longevity of the seal. Boot SealDeformed configuration at 20 degrees rotation of of maximum principal stress in Manifold AssemblyThe assembly considered (Fig. 1) consists of a four-tube exhaust manifold fastened to a partial section of an engine head by seven bolts acting on three flanges. The analysis consists of three steps. First, prescribed bolt loads fasten the manifold to the head.

7 Then, the assembly is heated to a steady- state thermal operating condition, shown in Fig. 2. Finally, the assembly is cooled to a uniformambient temperature. The variation of the bolt loads is monitored as the boltsrespond to the thermal loading of the assembly. Fig. 1 Exhaust manifold assemblyExhaust Manifold AssemblyThe base of the engine head is constrained vertically. Furthermore, it is assumed that the bolts are threaded tightly into the head so that the bottoms of the bolt shanks share nodes with the surrounding head elements and consequently are constrained vertically. The bolts and engine head are modeled as elastic materials; the manifold is modeled as an elastic-plastic, temperature-dependent material. Fig. 2 Steady-state temperature MeshingGears of various types are commonly used in modern machinery. Historically, gear design has been based largely on textbook formulas, extensive testing, and previous design experience. This application brief describes the simulation of gear meshing to predict gear tooth stresses and overall gear performance during operation.

8 Contours of maximum principal stressRail CrushCrash simulations are performed on entire vehicle models, but the design of individual components often requires their study on a stand-alone basis. This application brief describes a rail crush calculation. The rectangular, box-section rail has an initial velocity of 160 km/h and impacts a rigid wall. Because of symmetry only half of the rail needs to be modeled. The rail is made of an elastic-plastic, material. Its initial geometry is designed to induce a collapse mechanism that will maximize energy absorption. The shell elements account for Finite membrane strain, which is required for accurate simulation of this crushing processRail CrushThe program accounts for self contact throughout the simulation, including the effects of changing shell thickness, as points come into contact and surfaces slide along one another. Intermediate deformed dissipation history in Analysis of a Jack-Up PlatformMobile jack-ups play an important role in the initial development of shallow-water oil reserves.

9 They must be designed to withstand severe and random ocean wave, wind, and current loading caused by storm conditions. Figure 1: Elevated jack-up platformDynamic Analysis of a Jack-Up PlatformThe next phase of the investigation involves a geometrically nonlinear, transient dynamic simulation of the jack-up subjected to prescribed wave and current loadings. Gravity, buoyancy, fluid inertial, drag, and structural and hydrodynamic damping effects should all be time history of the wave trace. Partial time history of hull sidesway. Thermal Fatigue of a Surface Mount AssemblyLow-cycle fatigue is a common failure mechanism in solder joints of surface mount assemblies in the electronic packaging industry. Cyclic thermal loading combined with differences in thermal expansion properties for the various components of the assembly lead to stress reversals and the accumulation of inelastic strain in the joints. Predictions of fatigue life in solder joints require a thorough understanding of the deformation and failure mechanisms of the solder alloy and an accurate calculation of the stresses and strains in the joint.

10 Thermal Fatigue of a Surface Mount AssemblyThe analysis consists of a single superelement generation step and three cycles (12 steps) of thermal loading. The automatic time stepping scheme uses a combination of implicit and explicit time integration techniques to maximize solution efficiency for problems involving creep behavior. Deformation of corner legs at the end of the first holding C 0 C Equivalent creep strain distribution in the solder joint after three thermal cycles Thermal Fatigue of a Surface Mount AssemblyThe corresponding Mises stress history for point A is plotted in Figure 1 (right). The second and third cycles appear to be the same because the initial stress state conditions of the second and subsequent cycles are similar. The initial dip and subsequent peak in stress during the heating stage of the second and third cycles are due to the combination of the initial stress state and the competing effects of creep relaxation and CTE mismatch between the PCB and chip.