Finite Element AnalysisIntroduction

Finite Element MethodFEM is a numerical method for solving a system of governing equations over the domain of a continuous physical system, which is discretized into simple geometric shapes called finite element.

FEA IntroductionNumerical method used for solving problems that cannot be solved analytically (e.g., due to complicated geometry, different materials)Well suited to computers

The finite element method is a computational scheme to solve field problems in engineering and science. The technique has very wide application, and has been used on problems involving stress analysis, fluid mechanics, heat transfer, diffusion, vibrations, electrical and magnetic fields, etc. The fundamental concept involves dividing the body under study into a finite number of pieces (subdomains) called elements (see Figure). Particular assumptions are then made on the variation of the unknown dependent variable(s) across each element using so-called interpolation or approximation functions. This approximated variation is quantified in terms of solution values at special element locations called nodes. Through this discretization process, the method sets up an algebraic system of equations for unknown nodal values which approximate the continuous solution. Because element size, shape and approximating scheme can be varied to suit the problem, the method can accurately simulate solutions to problems of complex geometry and loading and thus this technique has become a very useful and practical tool.

Chronicle of Finite Element Method

Chronicle of Finite Element Method

Applications of Finite Element Method

Structural ProblemNon-structural ProblemStress Analysis - truss & frame analysis - stress concentrated problemBuckling problemVibration AnalysisImpact ProblemHeat TransferFluid MechanicsElectric or Magnetic Potential

Approximate method Geometric model Node Element Mesh Discretization

Finite Element Method PhasesPreprocessingGeometryModeling analysis typeMeshMaterial propertiesBoundary conditionsSolutionSolve linear or nonlinear algebraic equations simultaneously to obtain nodal results (displacements, temperatures)PostprocessingObtain other results (stresses, heat fluxes)

FEA Discretization Process - MeshingContinuous elastic structure (geometric continuum) divided into small (but finite), well-defined substructures, called elementsElements are connected together at nodes; nodes have degrees of freedomDiscretization process known as meshing

Spring AnalogyElements modeled as linear springs

Matrix FormulationLocal elastic behavior of each element defined in matrix form in terms of loading, displacement, and stiffnessStiffness determined by geometry and material properties (AE/l)

SolutionMatrix operations used to determine unknown dofs (e.g., nodal displacements)Run time proportional to # of nodes/elementsError messagesBad elementsInsufficient disk space, RAMInsufficiently constrained

PostprocessingDisplacements used to derive strains and stresses

FEA PrerequisitesFirst Principles (Newtons Laws)Body under external loadingArea Moments of InertiaStress and StrainPrincipal stressesStress states: bending, shear, torsion, pressure, contact, thermal expansionStress concentration factorsMaterial PropertiesFailure ModesDynamic Analysis

Advantages of Finite Element Analysis- Models Bodies of Complex Shape- Can Handle General Loading/Boundary Conditions- Models Bodies Composed of Composite and Multiphase Materials- Model is Easily Refined for Improved Accuracy by Varying Element Size and Type (Approximation Scheme)- Time Dependent and Dynamic Effects Can Be Included- Can Handle a Variety Nonlinear Effects Including Material Behavior, Large Deformations, Boundary Conditions, Etc.

Basic Concept of the Finite Element MethodAny continuous solution field such as stress, displacement, temperature, pressure, etc. can be approximated by a discrete model composed of a set of piecewise continuous functions defined over a finite number of subdomains.One-Dimensional Temperature Distribution

Two-Dimensional Discretizationu(x,y)Approximate Piecewise Linear Representation

Common Types of ElementsOne-Dimensional Elements Line Rods, Beams, Trusses, FramesTwo-Dimensional Elements Triangular, Quadrilateral Plates, Shells, 2-D ContinuaThree-Dimensional Elements Tetrahedral, Rectangular Prism (Brick) 3-D Continua

Discretization ExamplesOne-Dimensional Frame ElementsTwo-Dimensional Triangular ElementsThree-Dimensional Brick Elements

Mesh for the design of scaled model of aircraft for dynamic analysis

Mesh for a boom showing the stress distribution (Picture used by courtesy of EDS PLM Solutions)

Mesh of a hinge joint

Applications

Role of simulation in design: Boeing 777Source: Boeing Web site (http://www.boeing.com/companyoffices/gallery/images/commercial/).

Another success ..in failure: Airbus A380http://www.airbus.com/en/aircraftfamilies/a380/

Drag Force Analysis of AircraftQuestionWhat is the drag force distribution on the aircraft?SolveNavier-Stokes Partial Differential Equations.Recent DevelopmentsMultigrid Methods for Unstructured Grids

San Francisco Oakland Bay BridgeBefore the 1989 Loma Prieta earthquake

San Francisco Oakland Bay BridgeAfter the earthquake

San Francisco Oakland Bay BridgeA finite element model to analyze the bridge under seismic loadsCourtesy: ADINA R&D

Crush Analysis of Ford WindstarQuestionWhat is the load-deformation relation?SolvePartial Differential Equations of Continuum MechanicsRecent DevelopmentsMeshless Methods, Iterative methods, Automatic Error Control

Engine Thermal AnalysisPicture fromhttp://www.adina.com

QuestionWhat is the temperature distribution in the engine block?SolvePoisson Partial Differential Equation.Recent DevelopmentsFast Integral Equation Solvers, Monte-Carlo Methods

Electromagnetic Analysis of PackagesSolveMaxwells Partial Differential EquationsRecent DevelopmentsFast Solvers for Integral FormulationsThanks to Coventorhttp://www.coventor.com

Micromachine Device Performance AnalysisFrom www.memscap.comEquationsElastomechanics, Electrostatics, Stokes Flow.Recent DevelopmentsFast Integral Equation Solvers, Matrix-Implicit Multi-level Newton Methods for coupled domain problems.

Radiation Therapy of Lung Cancerhttp://www.simulia.com/academics/research_lung.html

Analytical -> exact solutionNumerical -> approximate solutionMostly from solid mechanics*The Boeing 777 is the first jetliner to be 100 percent digitally designed using three-dimensional solids technology. Throughout the design process, the airplane was "preassembled" on the computer, eliminating the need for a costly, full-scale mock-up. The 230 000 kg plane is the biggest twin-engine aircraft ever to fly-it can carry 375 passengers 7400 km-and from its first service flight in June 1995, has been certified for extended-range twin-engine operations.Boeing invested more than $4 billion (and insiders say much more) in CAD infrastructure for the design of the Boeing 777 and reaped huge benefits from design automation. The more than 3 million parts were represented in an integrated database that allowed designers to do a complete 3D virtual mock-up of the vehicle.Boeing based its CAD system on CATIA (short for Computer-aided Three-dimensional Interactive Application) and ELFINI (Finite Element Analysis System), both developed by Dassault Systemes of France (Dassault systems acquired ABAQUS in 2005 and ABAQUS+CATIA is known as SIMULIA) and licensed in the United States through IBM. Designers also used EPIC (Electronic Preassembly Integration on CATIA) and other digital preassembly applications developed by Boeing. Much of the same technology was used on the B-2 program. To design the 777, Boeing organized its workers into 238 cross-functional "design build teams" responsible for specific products. The teams used 2200 terminals and the computer-aided three dimensional interactive application (CATIA) system to produce a "paperless" design that allowed engineers to simulate assembly of the 777. The system worked so well that only a nose mockup (to check critical wiring) was built before assembly of the first flight vehicle which was only 0.03 mm out of alignment when the port wing was attached. Boeing also included customers and operators, down to line mechanics, to help tell them how to design the plane.

*On Tuesday, Feb. 14, 2006, a resounding crack echoed through the Airbus static test facility in Toulouse, France. The wing of Airbus' mammoth A380 had snapped between the inboard and outboard engines at an applied pressure approximately 1.45 times limit load about 3.3 percent short of the planned ultimate load.Just before the snap, an extreme wingtip deflection of 24.3 feet had been recorded.Limit load is the highest aerodynamic load expected during an aircraft's lifetime of normal service often 30 years or more. Ultimate load includes a built-in and required safety factor: 1.5 times limit load. Just to be sure.Furthermore, we are talking about the huge, double-decked A380, a brand-new aircraft and the world's largest commercial jetliner, scheduled to enter service at the end of the year with Singapore Airlines. It will seat 555 passengers in three classes far more in an all-economy configuration. It lists for about $295 million per aircraft, of which 159 have been ordered by 16 airlines around the world.The failure occurred last Tuesday between 1.45 and 1.5 times the limit load at a point between the inboard and outboard engines, says Airbus executive vice president engineering Alain Garcia. This is within 3% of the 1.5 target, which shows the accuracy of the FEM. But is this sufficient?The European Aviation Safety Agency (EASA) says that the maximum loading conditions are defined in the A380 certification basis. The aircraft structure is analysed and tested to demonstrate that the structure can withstand the maximum loads, including a factor of safety of 1.5. This process is ongoing and will be completed before type certification. ABAQUS v6.6 was used to accurately model the failed wing section including every rivet and bolt as well as their interactions. Nonlinear analysis was carried out and finally the problem was fixed and A380 received certification from EASA and FAA on 12 Dec 2006.****In response to the extensive damage caused by the 1989 Loma Prieta earthquake, the California Department of Transportation (Caltrans) commissioned a seismic-retrofit project for six of the major toll bridges in California, including the famous San Francisco Oakland Bay Bridge. The finite element software company ADINA (located in Watertown, MA) was selected to be used for the seismic analysis of these bridges to determine what retrofitting or new sections of bridges should be constructed.

ADINA was used for the linear and highly nonlinear analyses (including large displacements, plasticity, and contact) of the six major toll bridges in California after the 1989 Loma Prieta earthquake. Here, the model of a section of the old Bay Bridge subjected to an earthquake load is shown. Subsequent to this analysis, ADINA was used by the California Department of Transportation and its contractors to design a brand new bridge section spanning from Oakland to Treasure Island. *Crush analysis of automobiles is the simulation of a slow, dynamic (virtually static) process. In general, only codes based on explicit dynamic analysis procedures have been employed to analyze a crush response. These analyses are computationally expensive, require tuning of the model, and do not accurately represent the actual physics of the crush process. ADINA can be employed to perform crush analyses using the implicit dynamic (practically static) analysis procedures, thereby accurately representing the actual physical process. The solution of the Ford Windstar shown here was solved using ADINA on a Silicon Graphics O2000 machine in about 24 hours of computing time. **This is an example of electromagnetic analysis of packages for electronic chips. The problem is that if one of the pins is switching (i.e. conducting) then current may be induced in the neighboring pins due to electromagnetic induction. The purpose of the simulation is to detect such a glitch or false signal.**