MANE 4240 & CIVL 4240 Introduction to Finite Elements Prof. Suvranu De Introduction Info Course Instructor: Professor Suvranu De email: [email protected] JEC room: 2049 Tel: 6351 Office hours: T/F 2:00 pm-3:00 pm Course website: http://www.rpi.edu/~des/IFEA2019Fall.html Info Practicum Instructor:
Professor Jeff Morris email: [email protected] JEC room: JEC 7030 Tel: X2613 Office hours: http://homepages.rpi.edu/~morrij5/Office_schedule.pdf Info TA: Jitesh Rane Email: [email protected] Office: CII 7219 Office hours: M: 4:00-5:00 pm R: 1:00-2:00 pm Course texts and references Course text (for HW problems): Title: A First Course in the Finite Element Method Author: Daryl Logan
Edition: Sixth Publisher: Cengage Learning ISBN: 0-534-55298-6 Relevant reference: Finite Element Procedures, K. J. Bathe, Prentice Hall A First Course in Finite Elements, J. Fish and T. Belytschko Lecture notes posted on the course website Course grades Grades will be based on: 1. Home works (15 %). 2. Practicum exercises (10 %) to be handed in within a week of assignment. 3. Course project (25 %)to be handed in by December 10th (by noon) 4. Two in-class quizzes (2x25%) on 18th October, 10th December 1) All write ups that you present MUST contain your name and RIN
2) 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 homework Collaboration / academic integrity 1. 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 not to the individuals. Homeworks (15%) 1. Be as detailed and explicit as possible. For full
credit Do NOT omit steps. 2. Only neatly written homeworks will be graded 3. 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 Practicum (10%) 1. Five classes designated as Practicum. 2. You will need to download and install NX on your laptops and bring them to class on these days. 3. At the end of each practicum, you will be assigned a single problem (worth 2 points). 4. You will need to hand in the solution to the TA within a week of the assignment. 5. No late submissions will be entertained. Course Project (25 %) In this project you will be required to
choose an engineering system develop a mathematical model for the system develop the finite element model solve the problem using commercial software present a convergence plot and discuss whether the mathematical model you chose gives you physically meaningful results. refine the model if necessary. Course project (25 %)..contd. Logistics: Form groups of 2 and email the TA by 24th September. Submit 1-page project proposal latest by 8th October (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
5th (this will count as 10% of your project grade) Submit a project report (hard copy) by noon of 10th December to the instructor. Major project (25 %)..contd. Project report: 1. Must be professional (Text font Times 11pt with single spacing) 2. Must include the following sections: Introduction Problem statement Analysis Results and Discussions Major project (25 %)..contd. Project examples: (two sample project reports from previous year are provided) 1. Analysis of a rocker arm
2. Analysis of a bicycle crank-pedal assembly 3. Design and analysis of a "portable stair climber" 4. Analysis of a gear train 5.Gear tooth stress in a wind- up clock 6. Analysis of a gear box assembly 7. Analysis of an artificial knee 8. Forces acting on the elbow joint 9. Analysis of a soft tissue tumor system 10. Finite element analysis of a skateboard truck Major project (25 %)..contd. Project grade will depend on 1.Originality of the idea 2.Techniques used 3.Critical discussion Fixed boundary
uniform loading Finite element Cantilever plate model in plane strain Approximate method Geometric model Element Node Element Mesh Discretization Node Problem: Obtain the stresses/strains in the plate
Course content 1. Direct Stiffness approach for springs 2. Bar elements and truss analysis 3. Introduction to boundary value problems: strong form, principle of minimum potential energy and principle of virtual work. 4. Displacement-based finite element formulation in 1D: formation of stiffness matrix and load vector, numerical integration. 5. Displacement-based finite element formulation in 2D: formation of stiffness matrix and load vector for CST and quadrilateral elements. 6. Discussion on issues in practical FEM modeling 7. Convergence of finite element results 8. Higher order elements 9. Isoparametric formulation 10. Numerical integration in 2D 11. Solution of linear algebraic equations For next class Please read Appendix A of Logan for reading quiz next class (10 pts on Hw 1)
Linear Algebra Recap (at the IEA level) What is a matrix? A rectangular array of numbers (we will concentrate on real numbers). A nxm matrix has n rows and m columns M 11 M 12 M 13 M 14 First row M 3x4 M 21 M 22 M 23 M 24 Second row M 31 M 32 M 33
M 34 First Second Third Fourth column column column column M 12 Row number Column number Third row What is a vector? A vector is an array of n numbers A row vector of length n is a 1xn matrix a 1 a
2 a 3 a 4 A column vector of length m is a mx1 matrix a1 a 2 a3
Special matrices Zero matrix: A matrix all of whose entries are zero 0 3x 4 0 0 0 0 0 0 0 0 0 0
0 0 Identity matrix: A square matrix which has 1 s on the diagonal and zeros everywhere else. I3x 3 1 0 0 0 1 0 0 0 1 Matrix 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. 1 2 4 A 3 0 7 9 1 5 a 1, a b c B d e f g h i b 2, c 4, A B d 3,
e 0, f 7, g 9, h 1, i 5. Matrix operations Addition of two matrices If A and B are two matrices of the same size, then the sum of the matrices is a matrix C=A+B whose entries are the sums of the corresponding entries of A and B
1 2 4 1 3 10 A 3 0 7 B 3 1 0 9 1 5 1 0 6 0 5 14 C A B 6 1 7 10 1 11 Matrix operations Addition of of matrices Properties Properties of matrix addition: 1. Matrix addition is commutative (order of
addition does not matter) A B B A 2. Matrix addition is associative A B C A B C 3. Addition of the zero matrix A 0 0 A A Matrix operations Multiplication by a scalar If 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
1 A 3 9 3 cA 9 27 2 4 0 7 c 3 1 5 6 12 0 21 3 15
Matrix operations Multiplication by a scalar Special case If 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 1 2 4 A 3 0 7 9 1 5 1 cA -A 3 9
c 1 2 0 1 4 7 5 Matrix operations Subtraction
If A and B are two square matrices of the same size, then A-B is defined as the sum A+(-1)B 1 2 4 1 3 10 A 3 0 7 B 3 1 0 9 1 5 1 0 6 2 1 6 C A B 0 1 7 8 1 1 Note that A - A 0 and 0 - A -A
Transpose Special operations If 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. 1 A 3 9 2 0 1
4 7 5 A T 1 2 4 3 9 0 1 7 5
Special operations Transpose If A is a square matrix (mxm), it is called symmetric if A A T Matrix operations Scalar (dot) product of two vectors If a and b are two vectors of the same size
a1 b1 a a2 ; b b2 a 3 b 3 The scalar (dot) product of a and b is a scalar obtained by adding the products of corresponding entries of the two vectors T a b a 1b 1 a 2 b 2 a 3b 3 Matrix operations Matrix multiplication
For a product to be defined, the number of columns of A must be equal to the number of rows of B. A mxr inside outside B rxn = AB mxn Matrix operations
Matrix multiplication If 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 1 2 4 1 3 A 3x3 3 0 7 B3x2 3 1 9 1 5 1 0 C 3x2 3 AB 10 7
5 1 9 notice 2 4 28 T 1 3 3 1 Matrix operations Multiplication of matrices Properties
Properties of matrix multiplication: 1. Matrix multiplication is noncommutative (order of addition does matter) A B B A in g e n e ra l 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 Matrix operations Multiplication of matrices Properties
2. Matrix multiplication is associative A B C A B C 3. Distributive law A B C A B A C B C A B A C A 4. Multiplication by identity matrix A I A ; IA A 5. Multiplication by zero matrix A 0 0 ; 0 A 0 T T T 6. AB B A
Matrix operations Miscellaneous properties 1. If A , B and C are square matrices of the same size, and A 0 then A B A C does not necessarily mean that B C 2. A B 0 does not necessarily imply that either A or B is zero Inverse of a matrix Definition
If A is any square matrix and B is another square matrix satisfying the conditions AB BA I Then (a)The matrix A is called invertible, and (b) the matrix B is the inverse of A and is denoted as A-1. The inverse of a matrix is unique Inverse of a matrix Uniqueness The inverse of a matrix is unique Assume that B and C both are inverses of A
AB BA I AC CA I (BA)C IC C B(AC) BI B B C Hence a matrix cannot have two or more inverses. Some properties Inverse of a matrix Property 1: If A is any invertible square matrix the inverse of its inverse is the matrix A itself -1 1
A A Property 2: If A is any invertible square matrix and k is any scalar then k A 1 1 -1 A k Properties Inverse of a matrix
Property 3: If A and B are invertible square matrices then 1 1 -1 (AB) AB 1 A B I B A Premultiplying both sides by A-1 -1
A (AB) AB A AB AB -1 B AB 1 1 1 A 1 A 1 A 1 Premultiplying both sides by B -1
AB 1 B 1A 1 What is a determinant? The determinant of a square matrix is a number obtained in a specific manner from the matrix. For a 1x1 matrix: A a 1 1 ; d e t ( A ) a 1 1 For a 2x2 matrix: a11 A a 21 a12 ; det( A ) a 1 1a 2 2 a 1 2 a 2 1 a 22
Product along red arrow minus product along blue arrow Example 1 Consider the matrix 1 3 A 5 7 Notice (1) A matrix is an array of numbers (2) A matrix is enclosed by square brackets d e t( A ) 1
3 5 7 1 7 3 5 8 Notice (1) The determinant of a matrix is a number (2) The symbol for the determinant of a matrix is a pair of parallel lines Computation of larger matrices is more difficult Duplicate column method for 3x3 matrix For ONLY a 3x3 matrix write down the first two columns after the third column a 11 a 12 A a 21 a 22 a 31 a 32
a 13 a 23 a 33 a11 a12 a 21 a 22 a 31 a 32 a13 a11 a12 a 23 a 21 a 22 a 33 a 31 a 32 Sum of products along red arrow minus sum of products along blue arrow d e t( A ) a 1 1a 2 2 a 3 3 a 1 2 a 2 3 a 3 1 a 1 3a 2 1a 3 2 a 1 3 a 2 2 a 3 1 a 1 1a 2 3 a 3 2 a 1 2 a 2 1a 3 3
This technique works only for 3x3 matrices Example 2 4 -3 A 1 0 4 2 - 1 2 2 1 2 0 -8 4 3 2
0 4 1 1 2 2 4 0 1 0 32 8 3 Sum of red terms = 0 + 32 + 3 = 35 Sum of blue terms = 0 8 + 8 = 0
Determinant of matrix A= det(A) = 35 0 = 35 Finding 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 zero 2 7 8 3 2 4 0 2 7 8 because rows 1 and 3 are proportional to each other If the determinant of a matrix is zero, it is called a
singular matrix Cofactor method What is a cofactor? If A is a square matrix a 11 a 12 A a 21 a 22 a 31 a 32 a 13 a 23 a 33 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
M 12 a 21 a 23 a 21 a 23 a 2 1a 3 3 a 2 3 a 3 1 C 1 2 M 1 2 a 31 a 33 a 31 a 33 What is a cofactor? Sign of cofactor -
Find the minor and cofactor of a33 2 4 -3 A 1 0 4 2 - 1 2 Minor
M 33 2 1 4 2 0 4 1 4 0 (3 3) Cofactor C ( 1 ) M 33 M 33 4 33
Cofactor 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 1 i n and 1 j n Cofactor expansion along the jth column d e t ( A ) a 1 jC 1j a 2 jC 2j a n jC
nj Cofactor expansion along the ith row d e t ( A ) a i1C i1 a i2 C i2 a in C in Example: evaluate det(A) for: A=
The scalar T U d k d d v ector k squa re m atrix Is known as a quadratic form If U>0: Matrix k is known as positive definite If U0: Matrix k is known as positive semidefinite Quadratic form Let Then d1 k11
d k d 2 k 21 k12 k 22 Symmetric matrix k11 k12 d 1 U d k d d 1 d 2 k12 k 22 d 2 k11d 1 k12 d 2 d 1 d 2
k12 d 1 k 22 d 2 d 1 ( k11d 1 k12 d 2 ) d 2 ( k12 d 1 k 22 d 2 ) T 2 k11d 1 2k12 d 1 d 2 k 22 d 2 2 Differentiation of quadratic form Differentiate U wrt d1 U 2 k11d 1 2 k12 d 2 d 1 Differentiate U wrt d2
U 2 k12 d 1 2 k 22 d 2 d 2 Differentiation of quadratic form Hence U k11 U d 1 2 d U k12 d 2 2 k d k12 d 1
k 22 d 2 Outline Role of FEM simulation in Engineering Design Course Philosophy Role of simulation in design: Boeing 777 Source: Boeing Web site (http://www.boeing.com/companyoffices/gallery/images/commercial/). Another success ..in failure: Airbus A380 http://www.airbus.com/en/aircraftfamilies/a380/
Drag Force Analysis of Aircraft Question What is the drag force distribution on the aircraft? Solve Navier-Stokes Partial Differential Equations. Recent Developments Multigrid Methods for Unstructured Grids San Francisco Oakland Bay Bridge Before the 1989 Loma Prieta earthquake San Francisco Oakland Bay Bridge After the earthquake San Francisco Oakland Bay Bridge
A finite element model to analyze the bridge under seismic loads Courtesy: ADINA R&D Crush Analysis of Ford Windstar Question What is the load-deformation relation? Solve Partial Differential Equations of Continuum Mechanics Recent Developments Meshless Methods, Iterative methods, Automatic Error Control
Engine Thermal Analysis Picture from http://www.adina.com Question What is the temperature distribution in the engine block? Solve Poisson Partial Differential Equation. Recent Developments Fast Integral Equation Solvers, Monte-Carlo Methods Electromagnetic Analysis of Packages
Thanks to Coventor http://www.cov entor.com Solve Maxwells Partial Differential Equations Recent Developments Fast Solvers for Integral Formulations Micromachine Device Performance Analysis From www.memscap.com Equations Elastomechanics, Electrostatics, Stokes Flow.
Recent Developments Fast Integral Equation Solvers, Matrix-Implicit Multi-level Newton Methods for coupled domain problems. Radiation Therapy of Lung Cancer http://www.simulia.com/academics/research_lung.html Virtual Surgery General scenario.. Engineering design Physical Problem Question regarding the problem ...how large are the deformations? ...how much is the heat transfer? Mathematical model
Governed by differential equations Assumptions regarding Geometry Kinematics Material law Loading Boundary conditions Etc. Example: A bracket Engineering design Physical problem Questions: 1. What is the bending moment at section AA?
2. What is the deflection at the pin? Finite Element Procedures, K J Bathe Example: A bracket Engineering design Moment at section AA Deflection at load Mathematical model 1: beam M W L 27,500 N cm at load W How reliable is this model? How effective is this model?
1 W (L rN )3 W (L rN ) 5 3 EI AG 6 0.053 cm Example: A bracket Engineering design Mathematical model 2: plane stress Difficult to solve by hand!
..General scenario.. Engineering design Physical Problem Mathematical model Governed by differential equations Numerical model e.g., finite element model ..General scenario.. Engineering design Finite element analysis
PREPROCESSING 1. Create a geometric model 2. Develop the finite element model Solid model Finite element model ..General scenario.. Engineering design Finite element analysis FEM analysis scheme Step 1: Divide the problem domain into non overlapping regions (elements) connected to each other through special points (nodes) Element
Node Finite element model ..General scenario.. Engineering design Finite element analysis FEM analysis scheme Step 2: Describe the behavior of each element Step 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) ..General scenario.. Engineering design
POSTPROCESSING Compute moment at section AA Finite element analysis ..General scenario.. Engineering design Finite element analysis Preprocessing Step 1 Analysis Step 2 Step 3
Postprocessing Example: A bracket Engineering design Mathematical model 2: plane stress FEM solution to mathematical model 2 (plane stress) Moment at section AA M 2 7 ,5 0 0 N c m Deflection at load a t lo a d W 0 . 0 6 4 c m Conclusion: 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). What if we asked: what is the maximum stress in the bracket? would the beam model be of any use? Example: A bracket Engineering design Summary 1. The selection of the mathematical model depends on the response to be predicted. 2. The most effective mathematical model is the one that delivers the answers to the questions in reliable manner with least effort. 3. The numerical solution is only as
accurate as the mathematical model. Example: ...GeneralAscenario bracket Modeling a physical problem Change physical problem Physical Problem Mathematical Model Improve mathematical
model Numerical model Does answer make sense? YES! Happy No! Refine analysis Design improvements Structural optimization Modeling a physical problem Verification
Example:and A bracket validation Physical Problem Validation Mathematical Model Verification Numerical model Critical 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 elasticity Efficiency:
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