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Introduction to Finite Element Method
ByS. ZiaeiRad
Where the Course Fits
The field of Mechanics can be subdivided into 3 major areas:
Mechanics
Theoretical
Applied
Computational
Computational Mechanics
Nano and Micromechanics
Continuum Mechanics
ComputationalMechanics
Branches of Computational Mechanics can be distinguishedAccording to the physical focus of attention
Solid & StructuresFluidsMultiphysics
Computational Solid and Structural Mechanics
A convenient subdivision of problems in ComputationalSolid and Structural Mechanics (CSM) is
ComputationalSolid and StructuralMechanics (CSM)
Statics
Dynamics
Linear
NonLinear
CSM Linear StaticsFor the numerical simulation on the computer we must nowchose a spatial discretization method:
CSM Linear Statics
Finite Element MethodFinite Difference Method
Boundary Element MethodFinite Volume MethodSpectral Method
MeshFree Method
CSM Linear Statics by FEMHaving selected the FEM for discretization, we must nextpick a formulation and a solution method:
Formulation of FEM Model
Solution of FEM Model
Direct MethodVariational Method
Weighted Residuals
Stiffness
Flexibility
Mixed
Formulation of FEM Model1 The Direct Method
 Limited to very simple element It worth studying because it enhances the physicalmeaning.
2 The Variational Method Applicable to problems that can be stated by certainintegral expression.
3 Weighted Residual Methods Applicable to problems for which differential equationsare known but no variational statement is available.
Basic 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. Application of this simple idea can be founeverywhere in everyday life as well as in engineering.
Examples:·Lego (kids’play)·Buildings·Approximation of the area of a circle:
Basic Concepts·Archimedes' problem (circa 250 B.C.): rectification of thecircle as limit of inscribed regular polygons
Area of one triangle:
Area of the circle:
where N = total number of triangles (elements).
)2/sin)(2/cos( iii RRS
Computing "by Archimedes FEM"
Why Finite Element Method?
Design analysis: hand calculations, experiments, andcomputer simulationsFEM/FEA is the most widely applied computer simulation method in engineeringClosely integrated with CAD/CAM applications ……
Applications of FEM in Engineering
Mechanical/Aerospace/Civil/Automobile EngineeringStructure analysis (static/dynamic, linear/nonlinear)Thermal/fluid flowsElectromagneticsGeomechanicsBiomechanicsFluid/solid InteractionsFluid/thermal/solid Interactions ……
A Brief History of the FEM1941  Hrennikoff Used 1D element (bars and beams) for the solution of stresscontinuous solids.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 win
A 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 computers1980s  Microcomputers, pre and postprocessors1990s  Analysis of large structural systems
A Brief History of the FEMFE DOFs
1950s  100 DOFs1960s  1000 DOFs1980s  10000 DOFs1990s  100000 DOFs2000s  500000Several millions DOFs
Papers Published in FEM
1961  10 1966  1341971  8441976  70001986  20000……………..
FEM in Structural AnalysisProcedures:Divide structure into pieces (elements with nodes).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 (e.g., displacements).Calculate desired quantities (e.g., strains and stresses) at
Computer 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/IDEAS (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, Dyna2D, Dyna3D (Crash/impact analysis) ProMechanica (PTC Company)……..
Available Commercial FEM Software Packages
1969  Pedro Marcal taught at Brown University for a time butHe set up a firm to market the first nonlinear commercial FEprogram called MARC.1969  John Swanson was developing a NFE program atWestinghouse for Nuclear applications. He left Westinghouseto market program ANSYS.1972  David Hibbit who worked for Marcal until 1972 andThen cofounded HKS which markets ABAQUS. The program was the first to introduce gateways for researchers to add elements and material models.1970s Bathe launched his program after completing his PhDUnder supervision of Wilson at MIT. ADINA was an outgrowthOf NONSAP.
Available Commercial FEM Software Packages
1975  A milestone in the advancement of explicit FE wasJohn Halliquist’s work at Lawrence Livermore. He releasedhis code called DYNA in 1976.His success was the development of contactimpact interfaceswith Dave Benson and the resulting codes DYNA2D and DYNA3D.The DYNA code first commercialized by French firm ESI in 1980sand called PAMCRASH with many routines from WHAMS.John Halliquist left Livemore and started his own firm todistribute LSDYNA, a commercial version of DYNA.
Advantages and Disadvantages of generalpurpose programs
Advantages1 The input is well organized.2 The are large systems and can solve many types of problemsof large or small size.3 Many programs have the ability for adding new modules fornew kinds of problems or new technology with minimum efforts.4 Many of them can run on PCs.5 Many of them have become very attractive in price and cansolve a wide range of problems.Disadvantages1 Generalpurpose programs are less efficient than specialpurposeprograms.2 The initial cost of developing generalpurpose programs is high.3 The user has little access to the logic of the program.
Objectives of This FEM Course
Understand the fundamental ideas of the FEMKnow the behavior and usage of each type of elementscovered in this courseBe able to prepare a suitable FE model for given problemsCan 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)
Course Coverage Finite Element Discretization
Concepts Formulation of Finite Elements Computer Implementation of FEM
What do we need?1 ANSYS2 MATLAB
Examples: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 Seal
Deformed configuration at 20 degrees rotation of shaft.
Contours of maximum principal stress in boot.
Exhaust Manifold Assembly
The assembly considered (Fig. 1) consists of a fourtube 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. 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 assembly
Exhaust 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 elasticplastic, temperaturedependent material.
Fig. 2 Steadystate temperature distribution.
Gear 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.
Contours of maximum principal stress
Rail CrushCrash simulations are performed on entire vehicle models, but the design of individual components often requires their study on a standalone basis. This application brief describes a rail crush calculation.
The rectangular, boxsection 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 elasticplastic, 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 process
Rail 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 configurations. Plastic dissipation history in rail.
Dynamic Analysis of a JackUp Platform
Mobile jackups play an important role in the initial development of shallowwater oil reserves. They must be designed to withstand severe and random ocean wave, wind, and current loading caused by storm conditions.
Figure 1: Elevated jackup platform
Dynamic Analysis of a JackUp Platform
The next phase of the investigation involves a geometrically nonlinear, transient dynamic simulation of the jackup subjected to prescribed wave and current loadings. Gravity, buoyancy, fluid inertial, drag, and structural and hydrodynamic damping effects should all be modeled.
Partial time history of the wave trace. Partial time history of hull sidesway.
Thermal Fatigue of a Surface Mount Assembly
Lowcycle 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.
Thermal Fatigue of a Surface Mount Assembly
The 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 period.
120 C 0 CEquivalent creep strain
distribution in the solder joint after three thermal cycles
Thermal Fatigue of a Surface Mount Assembly
The 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.
Figure 1: Strain and stress histories at point A.
Hydroforming of a Square Box
Hydroforming of sheet metal components is widely used in several industries. While numerous variations of hydroforming exist, the basic principle remains the same: utilize fluid pressure to form a component.
Hydroforming of a Square Box
A critical parameter in hydroforming is the chamber pressure magnitude, which typically varies as a function of punch displacement. Excessive pressure may lead to tearing of the blank, while insufficient pressure may result in wrinkling.
Hydroforming specifications rely heavily on the intuition and experience of design engineers. Iterative cutandtry development cycles are costly and timeconsuming. As demonstrated by the square box hydroforming problems described here, use of an explicit FE is an accurate and efficient simulation tool that reduces time and costs associated with physical cutandtry methods.
Hydroforming of a Square Box
Initial configuration for trial 1. Initial configuration for trial 2.
Hydroforming of a Square Box
Final configuration of the box. Contours of box wall thickness.
To suppress wrinkling of the box, a rigid draw cap is added to the model, as shown in Figure trial 1. The position of the draw cap (shown in orange) in the actual hydroforming process is depicted in the inset to Figure trial 1. For modeling purposes only the surfaces of the draw cap that contact the blank are required
Continuous Casting
Continuous casting simulation.
In a continuous casting process liquid aluminum (Al0.7% Mg) is passed through a waterchilled mold to initiate the solidification process on the outer skin of the liquid. Water is then sprayed on the top and bottom of the casting to continue the cooling process. The objective of this analysis is to determine the shape and location of the freeze front under steadystate conditions.
Continuous Casting
Steadystate temperature distribution
The upstream boundary condition consists of a fixed inlet temperature, while the downstream condition allows convection but no axial conduction. Heat transfer between the aluminum and the mold and between the aluminum and the water spray is modeled with "surfacebased" thermal interaction. A thermal boundary layer is simulated along this interface, with surface heat transfer coefficients and cooling fluid temperatures specified as functions of axial position.
Continuous Casting
Steadystate temperature distribution
Hip ImplantWhile total hip replacement has become a common surgical practice, there are continuing efforts to optimize further the implant design and to extend the durability and life of the joint. This application brief examines the interaction between the implant and femur resulting from an initial interference fit and subsequent service loads.
Hip ImplantThe analysis is conducted in two steps. In the first step an interference fit between the implant and femur, simulating an implant using pressfit fixation, is resolved.
Creep of a Pipe IntersectionCreep is the permanent elongation of a component under a static load maintained for a period of time. Most metals and their alloys creep only at elevated temperatures, but several materials such as thermoplastics and rubbers do so at room temperature. Designers estimating the service life and structural integrity of components must account for creep effects in their designs.
This model represents the intersection of a pipe with a cylindrical pressure vessel. The system operates at an elevated temperature and carries internal pressure. The calculation consists of two steps. In the first step a static analysis is performed, during which the internal pressure is applied. In a second step a transient analysis is carried out to determine the creep of the pressurized vessel.
Creep of a Pipe Intersection
The onequarter model shown consists of 904 secondorder brick elements. Symmetry boundary conditions are enforced on the appropriate sectioned surfaces (displayed in red). The remaining sectioned surfaces (displayed in blue) are under tensile load and are constrained to remain planar. A single node is constrained in the vertical direction to eliminate rigid body motion.
Creep of a Pipe IntersectionIn this case a power law creep model is used. The automatic time stepping scheme uses a combination of implicit and explicit time integration techniques to maximize solution efficiency for problems involving creep behavior.
Creep of a Pipe IntersectionStress and strain histories at point A.
Circuit Board Drop TestElectronic components frequently have drop test requirements. This application brief describes the simulation of a circuit board dropped onto a flat, rigid floor. The objective of the simulation is to assist in the design of the packaging material (a crushable foam), which is intended to protect the circuit board from damage in such an event.
The analysis, performed with Explicit FE, uses the crushable foam model for the packaging material. This inelastic material model has been implemented for use with foamed, lightweight, energy absorbing materials. Strainratedependent effects, as well as high volumetric compressibility, are included in the model. The packaging is modeled with 1200 brick elements. The circuit board is treated as a rigid body with appropriate mass and rotary inertia: we are interested in the accelerations experienced by the circuit board but not in the details of any deformation in this component. The circuit board and foam assemblage is dropped at an angle onto a flat, rigid surface from a height of 1 meter. The illustrations shown correspond to deformed shapes at separate points in time. An acceleration history of the center of gravity of the circuit board is also included. The simulation shows that part of the board disengages from the packaging material during the impact event.
Circuit Board Drop Test
Three configurations during impact
Acceleration history of the center of gravity of the circuit board (g's).
Rolling of Thick Plates
Hot rolling is a basic manufacturing technique that is used to transform preformed shapes into forms that are suitable for further processing. Important aspects of the manufacturing operation are elongation and spread of the material during the rolling process.
Friction plays a key role in the simulations since it provides the mechanism by which the plate is pulled through a roller. When a point on the surface of the plate has just made contact with a roller, the roller surface is moving faster than the point on the workpiece, and there is relative slip between the two surfaces. As the point on the plate is drawn into the process zone under the roller, it moves faster and, after a certain distance, sticks to the roller. As the point on the workpiece is pushed out of the process zone, it speeds up and moves faster than the roller, causing slip in the opposite direction before separation takes place.
A singleroller operation A refined mesh is used for the single roller isothermal simulation. There are 2944 8node brick elements used for the workpiece and 89 rigid elements used for the roller.
A tworoller operationA coarse mesh is used for the tworoller simulation, which includes the effects of adiabatic heating due to plastic work. There are 80 8node brick elements used for the workpiece and 60 rigid elements used for each of the rollers.
A tworoller operation
Airplane
References1An Introduction to the Finite Element Method J. N. Reddy, McGrawHill, 19932 The Finite Element Method Linear Static and Dynamic Finite Element Analysis
Thomas J. R. Hughes, PrenticeHall, 1987 3Finite Element Modeling for Stress AnalysisR. D. Cook, John Wiley & Sons, 1995
4 Building Better Products with Finite Element Analysis, Vince Adams & Abraham Askenazi, OnWord Press, 1998 5 The Finite Element Method: Volume 1, Basic Formulation
and Linear ProblemsO. C. Zienkiewicz and R. L. Taylor, Fourth Edition, McGraw Hill, 19756 Introduction To Finite Elements in Engineering, T. Chandrupatla and A. D. Belegundu, Prentice Hall, 1997 7A First Course in the Finite Element Method, D. L. Logan, PWSKent, 1986
References8Numerical Methods using MATLAB J. Penny & G. Lindfield, Ellis Horwood Limited, 19959 Programming the Finite Element MethodSmith & Griffiths, John Wiley and Sons, 199210 Finite Element for Analysis and DesignJ. E. Akin, Academic press, 199411 The Finite Element Method using MATLABY. W. Hown, H. Bang, CRC Press, 1996