4Course texts and references Course text (for HW problems and reading assignments):Title: A First Course in the Finite Element MethodAuthor: Daryl LoganEdition: FifthPublisher: Cengage LearningISBN:Relevant reference:Finite Element Procedures, K. J. Bathe, Prentice HallA First Course in Finite Elements, J. Fish and T. BelytschkoLecture notes posted on the course website
5Course grades Grades will be based on: Home works (15 %). Mini project (10 %) to be handed in by 17th October.Major course project (25 %)to be handed in by December 5th (by noon)Two in-class exams (2x25%) on 10th October, 5th December1) Form (mini and major) project groups of two by 12th September.2) All write ups that you present MUST containyour name and RIN3) There will be reading quizzes (announced AS WELL AS unannounced) on a regular basis and points from these quizzes will be added on to the homework4) Project grades will be allotted to the group and NOT individually.
6Collaboration / academic integrity Students are encouraged to collaborate in the solution of HW problems, but submit independent solutions that are NOT copies of each other. Funny solutions (that appear similar/same) will be given zero credit.Softwares may be used to verify the HW solutions. But submission of software solution will result in zero credit.2. Groups of 2 for the projects(no two projects to be the same/similar)A single grade will be assigned to the group and notto the individuals.
7Homeworks (15%)1. Be as detailed and explicit as possible. For full credit Do NOT omit steps.2. Only neatly written homeworks will be graded3. Late homeworks will NOT be accepted.4. Two lowest grades will be dropped (except HW #1).5. Solutions will be posted on the course website
8Mini project (10%) Download the miniproject from the course website. You will need to learn the commercial software package ABAQUS (tutorial available at course website + in-class tutorial).Submit a project report by 17th October
9Mini project (10%)..contd. Project report: Must be typed (Text font Times 11pt with single spacing)Must be no more than 5 pages (with figures).Must include the following sections:Problem statementAnalysis (with appropriate figures)Results and discussionAttach an appendix containing the program printouts for the different cases you run.
10Major project (25 %) In this project you will be required to Choose an engineering systemDevelop a simple mathematical model for the systemDevelop the finite element modelSolve the problem using ABAQUS or any other FEM code (you may write one you if you choose to)Discuss whether the mathematical model you chose gives you physically meaningful results. If not, revise your model and perform analysis to improve your results.
11Major project (25 %)..contd. Logistics:Submit 1-page project proposal latest by 26th September (in class). The earlier the better. Projects will go on a first come first served basis.Proceed to work on the project ONLY after it is approved by the course instructor.Submit a one-page progress report on November 4th (this will count as 10% of your project grade)Submit a project report (typed) by noon of 5th December to the instructor.
12Major project (25 %)..contd. Project report:Must be professional (Text font Times 11pt with single spacing)Must include the following sections:IntroductionProblem statementAnalysisResults and DiscussionsYou must also burn the MS Word version of your project report on a CD with your group number, names, and project title written on it and submit it together with the project report.
13Major project (25 %)..contd. Project examples:(two sample project reports from previous year are provided)1. Analysis of a rocker arm2. Analysis of a bicycle crank-pedal assembly3. Design and analysis of a "portable stair climber"4. Analysis of a gear train5.Gear tooth stress in a wind- up clock6. Analysis of a gear box assembly7. Analysis of an artificial knee8. Forces acting on the elbow joint9. Analysis of a soft tissue tumor system10. Finite element analysis of a skateboard truck
14Major project (25 %)..contd. Project grade will depend onOriginality of the ideaTechniques usedCritical discussion
16Finite Element Analysis Cantilever platein plane strainuniform loadingFixed boundaryProblem: Obtain thestresses/strains in theplateApproximate methodGeometric modelNodeElementMeshDiscretizationNodeElementFinite elementmodel
17Course content “Direct Stiffness” approach for springs Bar elements and truss analysisIntroduction to boundary value problems: strong form, principle of minimum potential energy and principle of virtual work.Displacement-based finite element formulation in 1D: formation of stiffness matrix and load vector, numerical integration.Displacement-based finite element formulation in 2D: formation of stiffness matrix and load vector for CST and quadrilateral elements.Discussion on issues in practical FEM modelingConvergence of finite element resultsHigher order elementsIsoparametric formulationNumerical integration in 2DSolution of linear algebraic equations
18For next classPlease read Appendix A of Logan for reading quiz next class (10 pts on Hw 1)
20What is a matrix?A rectangular array of numbers (we will concentrate onreal numbers). A nxm matrix has ‘n’ rows and ‘m’columnsFirst rowSecond rowThird rowFirst columnSecond columnThird columnFourth columnRow numberColumn number
21What is a vector? A vector is an array of ‘n’ numbers A row vector of length ‘n’ is a 1xn matrixA column vector of length ‘m’ is a mx1 matrix
22Special matrices Zero matrix: A matrix all of whose entries are zero Identity matrix: A square matrix which has ‘1’ s on the diagonal and zeros everywhere else.
23Matrix operations Equality of matrices If A and B are two matrices of the same size,then they are “equal” if each and every entry of one matrix equals the corresponding entry of the other.
24Addition of two matrices Matrix operationsAddition of two matricesIf A and B are two matrices of the same size,then the sum of the matrices is a matrix C=A+B whoseentries are the sums of the corresponding entries of A and B
25Addition of of matrices Matrix operationsPropertiesProperties of matrix addition:Matrix addition is commutative (order of addition does not matter)Matrix addition is associativeAddition of the zero matrix
26Multiplication by a scalar Matrix operationsMultiplication by a scalarIf A is a matrix and c is a scalar, then the product cA is a matrix whose entries are obtained by multiplying each of the entries of A by c
27Multiplication by a scalar Matrix operationsSpecial caseIf A is a matrix and c =-1 is a scalar, then the product(-1)A =-A is a matrix whose entries are obtained by multiplying each of the entries of A by -1
28Matrix operationsSubtractionIf A and B are two square matrices of the same size, then A-B is defined as the sum A+(-1)B
29TransposeSpecial operationsIf A is a mxn matrix, then the transpose of A is the nxm matrix whose first column is the first row of A, whose second column is the second column of A and so on.
30TransposeSpecial operationsIf A is a square matrix (mxm), it is called symmetric if
31Scalar (dot) product of two vectors Matrix operationsScalar (dot) product of two vectorsIf a and b are two vectors of the same sizeThe scalar (dot) product of a and b is a scalar obtained by adding the products of corresponding entries of the two vectors
32Matrix multiplication Matrix operationsMatrix multiplicationFor a product to be defined, the number of columns of A must be equal to the number of rows of B.A B = ABm x r r x n m x ninsideoutside
33Matrix multiplication Matrix operationsMatrix multiplicationIf A is a mxr matrix and B is a rxn matrix, then the product C=AB is a mxn matrix whose entries are obtained as follows. The entry corresponding to row ‘i’ and column ‘j’ of C is the dot product of the vectors formed by the row ‘i’ of A and column ‘j’ of B
34Multiplication of matrices Matrix operationsPropertiesProperties of matrix multiplication:Matrix multiplication is noncommutative (order of addition does matter)It may be that the product AB exists but BA does not (e.g. in the previous example C=AB is a 3x2 matrix, but BA does not exist)Even if the product exists, the products AB and BA are not generally the same
35Multiplication of matrices Matrix operationsProperties2. Matrix multiplication is associative3. Distributive law4. Multiplication by identity matrix5. Multiplication by zero matrix6.
36Miscellaneous properties Matrix operationsIf A , B and C are square matrices of the same size, and then does not necessarily mean thatdoes not necessarily imply that either A or B is zero
37DefinitionInverse of a matrixIf A is any square matrix and B is another square matrix satisfying the conditionsThenThe matrix A is called invertible, andthe matrix B is the inverse of A and is denoted as A-1.The inverse of a matrix is unique
38Inverse of a matrix The inverse of a matrix is unique UniquenessInverse of a matrixThe inverse of a matrix is uniqueAssume that B and C both are inverses of AHence a matrix cannot have two or more inverses.
39Some propertiesInverse of a matrixProperty 1: If A is any invertible square matrix the inverse of its inverse is the matrix A itselfProperty 2: If A is any invertible square matrix and k is any scalar then
40PropertiesInverse of a matrixProperty 3: If A and B are invertible square matrices then
41What is a determinant? The determinant of a square matrix is a number obtained in a specific manner from the matrix.For a 1x1 matrix:For a 2x2 matrix:Product along red arrow minus product along blue arrow
42Example 1 Consider the matrix Notice (1) A matrix is an array of numbers(2) A matrix is enclosed by square bracketsNotice (1) The determinant of a matrix is a number(2) The symbol for the determinant of a matrix is a pair of parallel linesComputation of larger matrices is more difficult
43Duplicate column method for 3x3 matrix For ONLY a 3x3 matrix write down the first two columns after the third columnSum of products along red arrowminus sum of products along blue arrowThis technique works only for 3x3 matrices
44Example 32 3 -8 8 Sum of red terms = 0 + 32 + 3 = 35 323-88Sum of red terms = = 35Sum of blue terms = 0 – = 0Determinant of matrix A= det(A) = 35 – 0 = 35
45Finding determinant using inspection Special case. If two rows or two columns are proportional (i.e. multiples of each other), then the determinant of the matrix is zerobecause rows 1 and 3 are proportional to each otherIf the determinant of a matrix is zero, it is called a singular matrix
46Cofactor method If A is a square matrix What is a cofactor? The minor, Mij, of entry aij is the determinant of the submatrix that remains after the ith row and jth column are deleted from A.The cofactor of entry aij is Cij=(-1)(i+j) Mij
47What is a cofactor? Sign of cofactor Find the minor and cofactor of a33MinorCofactor
48Cofactor method of obtaining the determinant of a matrix The determinant of a n x n matrix A can be computed by multiplying ALL the entries in ANY row (or column) by their cofactors and adding the resulting products. That is, for each andCofactor expansion along the jth columnCofactor expansion along the ith row
50det(A)=a13C13 +a23C23+a33C33 Example : evaluate det(A)= 5 1 0 3 -1 -3 1 03 -1-32By a cofactor along the third columndet(A)=a13C13 +a23C23+a33C33det(A)=-3* (-1)4+2*(-1)5+2*(-1)6= det(A)= -3(-1-0)+2(-1)5(-1-15)+2(0-5)=25
51Quadratic form The scalar Is known as a quadratic form If U>0: Matrix k is known as positive definiteIf U≥0: Matrix k is known as positive semidefinite
55OutlineRole of FEM simulation in Engineering DesignCourse Philosophy
56Role of simulation in design: Boeing 777 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 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.Source: Boeing Web site (http://www.boeing.com/companyoffices/gallery/images/commercial/).
57Another success ..in failure: Airbus A380 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.
58Drag Force Analysis of Aircraft QuestionWhat is the drag force distribution on the aircraft?SolveNavier-Stokes Partial Differential Equations.Recent DevelopmentsMultigrid Methods for Unstructured Grids
59Before the 1989 Loma Prieta earthquake San Francisco Oakland Bay BridgeBefore the 1989 Loma Prieta earthquake
60San Francisco Oakland Bay Bridge After the earthquake
61A finite element model to analyze the bridge under seismic loads San Francisco Oakland Bay BridgeIn 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.A finite element model to analyze the bridge under seismic loadsCourtesy: ADINA R&D
62Crush Analysis of Ford Windstar QuestionWhat is the load-deformation relation?SolvePartial Differential Equations of Continuum MechanicsRecent DevelopmentsMeshless Methods, Iterative methods, Automatic Error ControlCrush 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.
63Engine Thermal Analysis Picture fromQuestionWhat is the temperature distribution in the engine block?SolvePoisson Partial Differential Equation.Recent DevelopmentsFast Integral Equation Solvers, Monte-Carlo Methods
64Electromagnetic Analysis of Packages Thanks to CoventorThis 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.SolveMaxwell’s Partial Differential EquationsRecent DevelopmentsFast Solvers for Integral Formulations
65Micromachine Device Performance Analysis FromEquationsElastomechanics, Electrostatics, Stokes Flow.Recent DevelopmentsFast Integral Equation Solvers, Matrix-Implicit Multi-level Newton Methods for coupled domain problems.
68Governed by differential equations General scenario..Engineering designPhysical ProblemQuestion regarding the problem...how large are the deformations?...how much is the heat transfer?Mathematical modelGoverned by differential equationsAssumptions regardingGeometryKinematicsMaterial lawLoadingBoundary conditionsEtc.
69Engineering design Example: A bracket Physical problem Questions: We consider here a simple example of a bracket supporting a vertical load. We need to choose a mathematical model. The choice of this model clearly depends on what phenomena are to be predicted and on the geometry, material properties, loading and support conditions of the bracket.We notice thatThe bracket has been fastened to a “very thick steel column”. The term “very thick” is relative to the thickness “t” and height “h” of the bracket. We translate this statement into the assumption that that the bracket is fastened to a (practically) rigid column.We also assume that the load is applied very slowly. The condition of time “very slowly” is relative to the largest natural period of the bracket: i.e., the time span over which the load W is increased from 0 to its full value is much longer than the fundamental period of the bracket. We translate this statement into meaning that we require static analysis (as opposed to a dynamic analysis).Now we are ready to develop a mathematical model of the bracket-depending on the phenomena to be predicted. Let us assume that we want to answer the following questions:What is the bending moment at section AA?What is the deflection at the pin?Questions:What is the bending moment at section AA?What is the deflection at the pin?Finite Element Procedures, K J Bathe
70Engineering design Example: A bracket Mathematical model 1: beam Moment at section AAFirst we develop a beam mathematical model including shear deformations.We have assumed linear elastic infinitesimal deformation conditions. Hence the load must not be so large as to cause yielding of the material and/or large displacements.Let us now ask whether the mathematical model is reliable and effective.Deflection at loadHow reliable is this model?How effective is this model?
71Engineering design Example: A bracket Mathematical model 2: plane stressDifficult to solve by hand!To measure the reliability and effectiveness of the simple beam model we consider a slightly more complicated mathematical model. This is a linear elastic two-dimensional plane-stress model. This mathematical model represents the geometry of the bracket more accurately than the beam model and assumes a two-dimensional stress situation. However, note that we have not considered the actual bolt fastening and contact conditions between the steel column and the bracket, nor have we modeled the pin carrying the load onto the bracket.
72Engineering design ..General scenario.. Physical Problem Mathematical modelGoverned by differential equationsThe plane stress mathematical model is too complicated to solve by hand. We therefore solve it using a numerical technique- the finite element method.Numerical modele.g., finite element model
73Engineering design PREPROCESSING 1. Create a geometric model ..General scenario..Engineering designFinite element analysisPREPROCESSING1. Create a geometric model2. Develop the finite element modelHow do we perform FEM modeling?We first generate a solid model of the bracket using a commercial package such as Solidworks/ ProE. The next step is to develop the “finite element” model.Solid modelFinite element model
74Engineering design FEM analysis scheme ..General scenario..Engineering designFinite element analysisFEM analysis schemeStep 1: Divide the problem domain into non overlapping regions (“elements”) connected to each other through special points (“nodes”)ElementNodeThe FEM analysis scheme can be broken down into 3 steps (we will use these 3 steps throughout the course).Step 1: We divide the solid model up into non-overlapping regions called “elements”. The elements are connected to each other through special points called “nodes”. The solution, in this case the displacement field, is approximated using piece-wise polynomials in each element. This process is called “mesh generation”. This way we discretize a continuous problem, having infinite degrees of freedom to a discrete problem having a finite number of degrees of freedom.Finite element model
75Engineering design FEM analysis scheme ..General scenario..Engineering designFinite element analysisFEM analysis schemeStep 2: Describe the behavior of each elementStep 3: Describe the behavior of the entire body by putting together the behavior of each of the elements (this is a process known as “assembly”)Step 2: We then describe the behavior of each element in terms of its nodesStep 3: We then describe the behavior of the entire structure by putting together the behavior of of each of these elements. In this process we are able to compute the displacements at the nodal points.Are we done?
76Engineering design POSTPROCESSING Compute moment at section AA ..General scenario..Engineering designFinite element analysisPOSTPROCESSINGCompute moment at section AATo compute the moment, we have to do a bit of extra work and this is known as “postprocessing”.
77Engineering design Preprocessing Analysis Postprocessing ..General scenario..Engineering designFinite element analysisPreprocessingStep 1AnalysisStep 2Step 3Remember that a complete FEM analysis consists ofPreprocessingAnalysisPostprocessingThe Analysis phase alone has 3 steps (which we will be most interested in the rest of this course)Postprocessing
78Engineering design Example: A bracket Mathematical model 2: plane stressFEM solution to mathematical model 2 (plane stress)Moment at section AADeflection at loadConclusion: With respect to the questions we posed, the beam model is reliable if the required bending moment is to be predicted within 1% and the deflection is to be predicted within 20%. The beam model is also highly effective since it can be solved easily (by hand).The beam mathematical model completely neglects the stress increase due to the fillets. Hence a plane stress solution including the fillets is necessary.What if we asked: what is the maximum stress in the bracket?would the beam model be of any use?
79Example: A bracketEngineering designSummaryThe selection of the mathematical model depends on the response to be predicted.The most effective mathematical model is the one that delivers the answers to the questions in reliable manner with least effort.The numerical solution is only as accurate as the mathematical model.
80Modeling a physical problem Example: A bracket...General scenarioModeling a physical problemPhysical ProblemChange physical problemMathematical ModelImprove mathematical modelNumerical modelDoes answermake sense?No!Refine analysisYES!Design improvementsStructural optimizationHappy
81Verification and validation Example: A bracketVerification and validationModeling a physical problemPhysical ProblemValidationMathematical ModelVerificationNumerical model
82Critical assessment of the FEM Reliability:For a well-posed mathematical problem the numerical technique should always, for a reasonable discretization, give a reasonable solution which must converge to the accurate solution as the discretization is refined.e.g., use of reduced integration in FEM results in an unreliable analysis procedure.Robustness:The performance of the numerical method should not be unduly sensitive to the material data, the boundary conditions, and the loading conditions used.e.g., displacement based formulation for incompressible problems in elasticityEfficiency: