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Introduction to Finite Element Method

ByS. Ziaei-Rad

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

Non-Linear

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

Mesh-Free 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 MethodWeighted ResidualsStiffness

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 (kidsplay)BuildingsApproximation of the area of a circle:

Basic ConceptsArchimedes' 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 ----- 500000-Several 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/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 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 co-founded 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 Halliquists work at Lawrence Livermore. He releasedhis code called DYNA in 1976.His success was the development of contact-impact interfaceswith Dave Benson and the resulting codes DYNA-2D and DYNA-3D.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 general-purpose 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- General-purpose programs are less efficient than special-purposeprograms.2- The initial cost of developing general-purpose 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 (dont 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 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. 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 elastic-plastic, temperature-dependent material.

Fig. 2 Steady-state 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