# Introduction to the Finite Element Method (FEM)

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Introduction to the Finite Element Method (FEM) ecture First and Second Order One Dimensional Shape Functions Dr. J. Dean

2 Discretisation Consider the temperature distribution along the one-dimensional fin in Fig.1. Figure 1: Depiction of a piecewise approximation to a continuous function A one-dimensional continuous temperature distribution with an infinite number of unknowns is shown in (a). The fin is discretised in (b) i.e. divided into 4 subdomains (or elements). The nodes are numbered consecutively from left to right, as are the elements. The elements are first order elements; the interpolation scheme between the nodes is therefore linear. Note that there are only 5 nodes for this system, since the internal nodes are shared between the elements. Since we are only solving for temperature, there are only 5 degrees of freedom in this model of the continuous system. It should be clear that a better approximation for T(x) would be obtained if the number of elements was increased (i.e. if the element lengths were reduced). It is also apparent that the nodes should be placed closer together in regions where the temperature (or any other unknown solution) changes rapidly. It is useful also to place a node wherever a step change in temperature is expected and where a numerical value of the temperature is needed. It is good practice to continue to increase the number of nodes until a converged solution is reached. In (c), the fin has been divided into two subdomains elements 1 and. However, in this instance we have chosen to use a second order (quadratic) element. These elements contain midside nodes as shown, and the interpolation between the nodes is quadratic which permits a much closer approximation to the real system. For this model system there are still just 5 degrees of freedom. However, the analysis takes longer for (c) than it does for (b) because the quadratic interpolation (which calculates the temperature at locations between the nodes) is more demanding than the corresponding linear case. 1 (There is often a trade-off between a high number of first order elements requiring little computation and a smaller number of second order elements requiring more heavy computation to be made, which affects both the analysis time and the solution accuracy. The choice depends to a large extent on the problem being solved.) 1 For a system with only 5 degrees of freedom the difference would be imperceptible using modern desktop machines. This would not be the case for large systems containing hundreds of thousands of degrees of freedom. 1

3 Shape Functions We can use (for instance) the direct stiffness method to compute degrees of freedom at the element nodes. However, we are also interested in the value of the solution at positions inside the element. To calculate values at positions other than the nodes we interpolate between the nodes using shape functions. A one-dimensional element with length is shown in Fig.. It has two nodes, one at each end, denoted i and j, and known nodal temperatures T i and T j. We can deduce automatically that the element is first order (linear) since it contains no midside nodes. Figure : One dimensional linear element with temperature degrees of freedom We need to derive a function to compute values of the temperature at locations between the nodes. This interpolation function is called the shape function. We demonstrate its derivation for a 1-dimensional linear element here. Note that, for linear elements, the polynomial inerpolation function is first order. If the element was second order, the polynomial function would be second order (quadratic), and so on. Since the element is first order, the temperature varies linearly between the nodes and the equation for T is: T(x) = a + bx (1)

4 We can therefore write: T i = a + bx i () T j = a + bx j (3) which are simultaneous. To determine the coefficients a and b: T i a X i = b (4) T j a X j = b (5) T i a X i = T j a X j (6) (T i a)x j = T j a X i (7) T i X j ax j = T j X i ax i (8) T i X j T j X i = a X j X i (9) T i X j T j X i X j X i = a (10) T i X j T j X i = a (11) and T i bx i = a (1) T j bx j = a (13) T i bx i = T j bx j (14) bx j bx i = T j T i (15) b X j X i = T j T i (16) b = T j T i X j X i b = T j T i (17) (18) Substitution of Eqns.11 and 18 into Eqn.1 yields: T(x) = T ix j T j X i + T j T i x (19) T(x) = T ix j T j X i + T jx T i x (0) 3

5 T(x) = T ix j T jx i + T jx T ix (1) T(x) = T ix j T ix + T jx T jx i () T(x) = T i X j x + T j x X i (3) It should be clear from Eqn.3 that the nodal temperature values are multiplied by linear functions of x the shape functions. The functions are denoted by S with a subscript to indicate the node with which a specific shape function is associated. In the case presented: S i = X j x (4) Draw linear shape functions S j = x X i (5) And Eqn.3 becomes T(x) = S i T i + S j T j (6) In matrix form T x e = [S i S j ] T i T j (7) For the case shown in Fig.3, calculate Tat x = 3.3 T(x = 3.3) = T i X j x + T j x X i T(x = 3.3) = T(x = 3.3) = 50.6 Figure 3: 1-dimensional linear element with known nodal temperatures and positions 4

6 From inspection of Eqn.6 we can deduce that each shape function has a value of 1 at its own node and a value of zero at the other nodes. The sum of the shape functions sums to one. The shape functions are also first order, just as the original polynomial was. The shape functions would have been quadratic if the original polynomial has been quadratic. A continuous, piecewise smooth equation for the one dimensional fin first shown in Fig.1 can be constructed by connecting the linear element equations. We know that the temperature at any point in any element can be found from the nodal temperatures T i and the shape functions, S i. For the following system: T x e = S i T i + S j T j X i x X j (8) Node Element # i j T 1 x = S 1 1 T 1 + S 1 T T x = S T + S 3 T 3 T 3 x = S 3 3 T 3 + S 3 4 T 4 T 4 x = S 4 4 T 4 + S 4 5 T 5 S 1 1 = X x X X 1 S 1 = x X 1 X X 1 X 1 x X (9) S = X 3 x X 3 X S 3 = x X X 3 X X x X 3 (30) S 3 3 = X 4 x X 4 X 3 S 4 3 = x X 3 X 4 X 3 X 3 x X 4 (31) S 4 4 = X 5 x X 5 X 4 S 4 3 = x X 4 X 5 X 4 X 4 x X 5 (3) S 1 S, S 3 S 3 3 and S 4 3 S 4 4 The temperature gradient through the an individual element dte dx derivative of Eqn.33 can be found from the T e x = T i X j x + T j x X i (33) T x e = X jt i T x e = xt j xt i xt i + xt j + X jt i + X it j X it j (34) (35) T x e = x T j T i + 1 X jt i X i T j (36) 5

7 such that dt x e = T j T i dx (37) We can check that our shape functions are correct by knowing that the shape function derivatives sum to zero. 1-Dimensional Quadratic Elements A one-dimensional quadratic element is shown in Fig.4. We can deduce immediately that the element order is greater than one because the interpolation between the nodes in nonlinear. We can determine from inspection that the element is quadratic (second order) because there s a midside node. We know therefore that the function approximating the solution is a second order polynomial: T x e = a + bx + cx (38) x The shape functions S i can be determined by solving Eqn.38 using known T i at known X i to give: T x e = S i T i + S j T j + S k T k j (39) S i = (x X k) x X j (40) S j = 4 (x X i)(x X k ) (41) S k = (x X i) x X j (4) 6

8 Using the quadratic shape functions for a single element (Eqns.40-4), we can assemble a corresponding set of equations for a larger system: Nodes Element # i j k 1 i j k Nodes Element # i j k k m n T x 1 = S i 1 S k 1 T i S 1 j T k Tj T k T x = [S k S m S n ] T m Tn T x 1 = S i 1 T i + S j 1 T j + S k 1 T k (43) S i 1 = (x X k) x X j S j 1 = 4 (x X i)(x X k ) S k 1 = (x X i) x X j T x = S k T k + S m T m + S n T n (44) S k = (x X n)(x X m ) S m = 4 (x X k)(x X n ) S n = (x X k)(x X m ) 7

9 The element shape functions are stored within the element in commercial FE codes. The positions X i are generated (and stored) when the mesh is created. Once the nodal degrees of freedom are known, the solution at any point between the nodes can be calculated using the (stored) element shape functions and the (known) nodal positions. The nodal degrees of freedom are calculated using a method such as that shown previously in lecture 1. For the cases presented above, simple 1-dimensional elements were most appropriate, but for many practical applications we may encounter more complex - and 3-dimensional geometry. A suitable set of element types are shown in Fig.4 encompassing a range of geometry and element order. A meshed geometry is shown in Fig.5. Figure 4: Example element types Figure 5: An exhaust manifold discretised (meshed) using 3-dimensional, quadratic brick elements, enabling the geometrical curvature to be captured 8

10 Coordinate Systems Solution fields (such as displacement fields in solid mechanics) often require integrals of the following kind to be evaluated: X j X j S i (x)s j (x)dx or S X i i (x)dx X i (45) Note that the superscript in Eqn.45 denotes a power now (and not an element label). These integrals can be simplified to make the integration procedures more efficient by deriving new shape functions defined relative to a local (element level) coordinate system. For instance, we have so far derived our one-dimensional linear shape functions using a global coordinate system with its origin at O Fig.6. These shape functions were: S i (x) = X j x (46) S j (x) = x X i (47) x s Figure 6: Representation of a single linear element in global, local and natural frames of reference We wish now to derive the shape functions in the local coordinate reference frame with the origin at X i - shown also in Fig.6. In the new reference frame the position along the element given previously by x is now given by X i + s. We can rewrite the shape functions in Eqns.46 and 47 using the new reference frame by substitution such that: 9

11 S i (s) = X j (X i +s) S j (s) = (X i+s) X i = 1 s = s (48) (49) and X j X j S i (x)s j (x)dx or S X i i (x)dx X i (50) becomes S i (s)s j (s)ds 0 or S i (s)ds 0 (51) where the limits of integration are easier to deal with. However, they can be simplified further by using a non-dimensional natural coordinate system. A natural coordinate system is shown in Fig.6. A coordinate at the centre of the element is specified. From this reference frame, the shape functions are: S i (ξ) = 1 (1 ξ) (5) S j (ξ) = 1 (1 + ξ) (53) To find these, we substitute x in Eqns.46 and 47 with x = X i + + ξ and note that = S i (ξ) = X j x S j (ξ) = x X i = X j X i + +ξ = X i+ +ξ X i = X j X i ξ = 1 ξ = 1 (1 ξ) (54) = X i + + ξ X i = 1 + ξ = 1 (1 + ξ) (55) Recall that the shape functions are equal to one at their respective nodes and 0 at the other nodes i.e: S i ( 1) = 1 (1 ξ) = 1 S i (1) = 1 (1 ξ) = 0 S j ( 1) = 1 (1 ξ) = 1 S j (1) = 1 (1 ξ) = 0 The integrals become: 1 1 S i (ξ)s j (ξ)dξ or 1 1 S i (ξ)dξ (56) 10

12 where the limits of integration are -1 and +1. Integrals of this type can be solved very efficiently using, for instance, the Gauss-egendre method. 11

### The Basics of FEA Procedure

CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring

### Stress Recovery 28 1

. 8 Stress Recovery 8 Chapter 8: STRESS RECOVERY 8 TABLE OF CONTENTS Page 8.. Introduction 8 8.. Calculation of Element Strains and Stresses 8 8.. Direct Stress Evaluation at Nodes 8 8.. Extrapolation

### Finite Elements for 2 D Problems

Finite Elements for 2 D Problems General Formula for the Stiffness Matrix Displacements (u, v) in a plane element are interpolated from nodal displacements (ui, vi) using shape functions Ni as follows,

### To add fractions we rewrite the fractions with a common denominator then add the numerators. = +

Partial Fractions Adding fractions To add fractions we rewrite the fractions with a common denominator then add the numerators. Example Find the sum of 3 x 5 The common denominator of 3 and x 5 is 3 x

### SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

### CAE -Finite Element Method

16.810 Engineering Design and Rapid Prototyping Lecture 3b CAE -Finite Element Method Instructor(s) Prof. Olivier de Weck January 16, 2007 Numerical Methods Finite Element Method Boundary Element Method

### Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver

Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point

### Partial Fractions. (x 1)(x 2 + 1)

Partial Fractions Adding rational functions involves finding a common denominator, rewriting each fraction so that it has that denominator, then adding. For example, 3x x 1 3x(x 1) (x + 1)(x 1) + 1(x +

### the points are called control points approximating curve

Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.

### Back to Elements - Tetrahedra vs. Hexahedra

Back to Elements - Tetrahedra vs. Hexahedra Erke Wang, Thomas Nelson, Rainer Rauch CAD-FEM GmbH, Munich, Germany Abstract This paper presents some analytical results and some test results for different

### FEA Good Modeling Practices Issues and examples. Finite element model of a dual pinion gear.

FEA Good Modeling Practices Issues and examples Finite element model of a dual pinion gear. Finite Element Analysis (FEA) Good modeling and analysis procedures FEA is a powerful analysis tool, but use

### November 16, 2015. Interpolation, Extrapolation & Polynomial Approximation

Interpolation, Extrapolation & Polynomial Approximation November 16, 2015 Introduction In many cases we know the values of a function f (x) at a set of points x 1, x 2,..., x N, but we don t have the analytic

### Finite Element Method

16.810 (16.682) Engineering Design and Rapid Prototyping Finite Element Method Instructor(s) Prof. Olivier de Weck deweck@mit.edu Dr. Il Yong Kim kiy@mit.edu January 12, 2004 Plan for Today FEM Lecture

### Solving Quadratic Equations by Factoring

4.7 Solving Quadratic Equations by Factoring 4.7 OBJECTIVE 1. Solve quadratic equations by factoring The factoring techniques you have learned provide us with tools for solving equations that can be written

### Sample Problems. Practice Problems

Lecture Notes Partial Fractions page Sample Problems Compute each of the following integrals.. x dx. x + x (x + ) (x ) (x ) dx 8. x x dx... x (x + ) (x + ) dx x + x x dx x + x x + 6x x dx + x 6. 7. x (x

### CAE -Finite Element Method

16.810 Engineering Design and Rapid Prototyping CAE -Finite Element Method Instructor(s) Prof. Olivier de Weck January 11, 2005 Plan for Today Hand Calculations Aero Æ Structures FEM Lecture (ca. 45 min)

### The elements used in commercial codes can be classified in two basic categories:

CHAPTER 3 Truss Element 3.1 Introduction The single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for

### Finite Element Formulation for Plates - Handout 3 -

Finite Element Formulation for Plates - Handout 3 - Dr Fehmi Cirak (fc286@) Completed Version Definitions A plate is a three dimensional solid body with one of the plate dimensions much smaller than the

### 19.6. Finding a Particular Integral. Introduction. Prerequisites. Learning Outcomes. Learning Style

Finding a Particular Integral 19.6 Introduction We stated in Block 19.5 that the general solution of an inhomogeneous equation is the sum of the complementary function and a particular integral. We have

### a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)

ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x

### 1.7. Partial Fractions. 1.7.1. Rational Functions and Partial Fractions. A rational function is a quotient of two polynomials: R(x) = P (x) Q(x).

.7. PRTIL FRCTIONS 3.7. Partial Fractions.7.. Rational Functions and Partial Fractions. rational function is a quotient of two polynomials: R(x) = P (x) Q(x). Here we discuss how to integrate rational

### COMPUTATIONAL ENGINEERING OF FINITE ELEMENT MODELLING FOR AUTOMOTIVE APPLICATION USING ABAQUS

International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 7, Issue 2, March-April 2016, pp. 30 52, Article ID: IJARET_07_02_004 Available online at http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=7&itype=2

### AN INTRODUCTION TO THE FINITE ELEMENT METHOD FOR YOUNG ENGINEERS

AN INTRODUCTION TO THE FINITE ELEMENT METHOD FOR YOUNG ENGINEERS By: Eduardo DeSantiago, PhD, PE, SE Table of Contents SECTION I INTRODUCTION... 2 SECTION II 1-D EXAMPLE... 2 SECTION III DISCUSSION...

### INTRODUCTION TO THE FINITE ELEMENT METHOD

INTRODUCTION TO THE FINITE ELEMENT METHOD G. P. Nikishkov 2004 Lecture Notes. University of Aizu, Aizu-Wakamatsu 965-8580, Japan niki@u-aizu.ac.jp 2 Updated 2004-01-19 Contents 1 Introduction 5 1.1 What

### Equations, Inequalities & Partial Fractions

Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

### 1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style

Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with

Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

### Factoring and Applications

Factoring and Applications What is a factor? The Greatest Common Factor (GCF) To factor a number means to write it as a product (multiplication). Therefore, in the problem 48 3, 4 and 8 are called the

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### 3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes

Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same

### Applications to Data Smoothing and Image Processing I

Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is

### Question 2: How do you solve a matrix equation using the matrix inverse?

Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients

### Integration of a fin experiment into the undergraduate heat transfer laboratory

Integration of a fin experiment into the undergraduate heat transfer laboratory H. I. Abu-Mulaweh Mechanical Engineering Department, Purdue University at Fort Wayne, Fort Wayne, IN 46805, USA E-mail: mulaweh@engr.ipfw.edu

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### 1 Lecture: Integration of rational functions by decomposition

Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.

### Integrals of Rational Functions

Integrals of Rational Functions Scott R. Fulton Overview A rational function has the form where p and q are polynomials. For example, r(x) = p(x) q(x) f(x) = x2 3 x 4 + 3, g(t) = t6 + 4t 2 3, 7t 5 + 3t

### is identically equal to x 2 +3x +2

Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3

### Factoring Trinomials: The ac Method

6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### Functions and Equations

Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c

### 3. INNER PRODUCT SPACES

. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.

### SOLVING POLYNOMIAL EQUATIONS

C SOLVING POLYNOMIAL EQUATIONS We will assume in this appendix that you know how to divide polynomials using long division and synthetic division. If you need to review those techniques, refer to an algebra

### Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

### Figure 2.1: Center of mass of four points.

Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would

### Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10

Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction

### LS.6 Solution Matrices

LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

### Systems of Linear Equations

Chapter 1 Systems of Linear Equations 1.1 Intro. to systems of linear equations Homework: [Textbook, Ex. 13, 15, 41, 47, 49, 51, 65, 73; page 11-]. Main points in this section: 1. Definition of Linear

### 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field

3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important

### Problem set #5 EE 221, 09/26/ /03/2002 1

Chapter 3, Problem 42. Problem set #5 EE 221, 09/26/2002 10/03/2002 1 In the circuit of Fig. 3.75, choose v 1 to obtain a current i x of 2 A. Chapter 3, Solution 42. We first simplify as shown, making

### Module 1 : Conduction. Lecture 5 : 1D conduction example problems. 2D conduction

Module 1 : Conduction Lecture 5 : 1D conduction example problems. 2D conduction Objectives In this class: An example of optimization for insulation thickness is solved. The 1D conduction is considered

### An Overview of the Finite Element Analysis

CHAPTER 1 An Overview of the Finite Element Analysis 1.1 Introduction Finite element analysis (FEA) involves solution of engineering problems using computers. Engineering structures that have complex geometry

### LINEAR EQUATIONS IN TWO VARIABLES

66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that

### Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

### 07-Nodal Analysis Text: ECEGR 210 Electric Circuits I

07Nodal Analysis Text: 3.1 3.4 ECEGR 210 Electric Circuits I Overview Introduction Nodal Analysis Nodal Analysis with Voltage Sources Dr. Louie 2 Basic Circuit Laws Ohm s Law Introduction Kirchhoff s Voltage

### INTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y).

INTERPOLATION Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). As an example, consider defining and x 0 =0, x 1 = π 4, x

### Solving Systems of Linear Equations

LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

### a) x 2 8x = 25 x 2 8x + 16 = (x 4) 2 = 41 x = 4 ± 41 x + 1 = ± 6 e) x 2 = 5 c) 2x 2 + 2x 7 = 0 2x 2 + 2x = 7 x 2 + x = 7 2

Solving Quadratic Equations By Square Root Method Solving Quadratic Equations By Completing The Square Consider the equation x = a, which we now solve: x = a x a = 0 (x a)(x + a) = 0 x a = 0 x + a = 0

### Trend and Seasonal Components

Chapter 2 Trend and Seasonal Components If the plot of a TS reveals an increase of the seasonal and noise fluctuations with the level of the process then some transformation may be necessary before doing

### Determinants can be used to solve a linear system of equations using Cramer s Rule.

2.6.2 Cramer s Rule Determinants can be used to solve a linear system of equations using Cramer s Rule. Cramer s Rule for Two Equations in Two Variables Given the system This system has the unique solution

CHAPTER Roots of quadratic equations Learning objectives After studying this chapter, you should: know the relationships between the sum and product of the roots of a quadratic equation and the coefficients

### College Algebra - MAT 161 Page: 1 Copyright 2009 Killoran

College Algebra - MAT 6 Page: Copyright 2009 Killoran Zeros and Roots of Polynomial Functions Finding a Root (zero or x-intercept) of a polynomial is identical to the process of factoring a polynomial.

### Modelling unilateral frictionless contact using. the null-space method and cubic B-Spline. interpolation

Modelling unilateral frictionless contact using the null-space method and cubic B-Spline interpolation J. Muñoz Dep. Applied Mathematics III, LaCàN Univ. Politècnica de Catalunya (UPC), Spain Abstract

### Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.

This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra

### Partial Fractions Examples

Partial Fractions Examples Partial fractions is the name given to a technique of integration that may be used to integrate any ratio of polynomials. A ratio of polynomials is called a rational function.

### Section 6.1 Factoring Expressions

Section 6.1 Factoring Expressions The first method we will discuss, in solving polynomial equations, is the method of FACTORING. Before we jump into this process, you need to have some concept of what

### (Refer Slide Time: 1:42)

Introduction to Computer Graphics Dr. Prem Kalra Department of Computer Science and Engineering Indian Institute of Technology, Delhi Lecture - 10 Curves So today we are going to have a new topic. So far

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### Finite cloud method: a true meshless technique based on a xed reproducing kernel approximation

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2001; 50:2373 2410 Finite cloud method: a true meshless technique based on a xed reproducing kernel approximation N.

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### LEAST SQUARES APPROXIMATION

LEAST SQUARES APPROXIMATION Another approach to approximating a function f(x) on an interval a x b is to seek an approximation p(x) with a small average error over the interval of approximation. A convenient

### Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder).

Math 50, Chapter 8 (Page 1 of 20) 8.1 Common Factors Definitions 1. A factor of integer is an integer that will divide the given integer evenly (with no remainder). Find all the factors of a. 44 b. 32

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method.

A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

### FINITE ELEMENT : MATRIX FORMULATION. Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 7633

FINITE ELEMENT : MATRIX FORMULATION Georges Cailletaud Ecole des Mines de Paris, Centre des Matériaux UMR CNRS 76 FINITE ELEMENT : MATRIX FORMULATION Discrete vs continuous Element type Polynomial approximation

### Piecewise Cubic Splines

280 CHAP. 5 CURVE FITTING Piecewise Cubic Splines The fitting of a polynomial curve to a set of data points has applications in CAD (computer-assisted design), CAM (computer-assisted manufacturing), and

### Mth 95 Module 2 Spring 2014

Mth 95 Module Spring 014 Section 5.3 Polynomials and Polynomial Functions Vocabulary of Polynomials A term is a number, a variable, or a product of numbers and variables raised to powers. Terms in an expression

### Numerical Methods Lecture 5 - Curve Fitting Techniques

Numerical Methods Lecture 5 - Curve Fitting Techniques Topics motivation interpolation linear regression higher order polynomial form exponential form Curve fitting - motivation For root finding, we used

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

### Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results

Wentzville School District Algebra 1: Unit 8 Stage 1 Desired Results Unit Title: Quadratic Expressions & Equations Course: Algebra I Unit 8 - Quadratic Expressions & Equations Brief Summary of Unit: At

### Solutions for 2015 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama

Solutions for 0 Algebra II with Trigonometry Exam Written by Ashley Johnson and Miranda Bowie, University of North Alabama. For f(x = 6x 4(x +, find f(. Solution: The notation f( tells us to evaluate the

### Factoring Polynomials and Solving Quadratic Equations

Factoring Polynomials and Solving Quadratic Equations Math Tutorial Lab Special Topic Factoring Factoring Binomials Remember that a binomial is just a polynomial with two terms. Some examples include 2x+3

### NSM100 Introduction to Algebra Chapter 5 Notes Factoring

Section 5.1 Greatest Common Factor (GCF) and Factoring by Grouping Greatest Common Factor for a polynomial is the largest monomial that divides (is a factor of) each term of the polynomial. GCF is the

### FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS

FUNDAMENTAL FINITE ELEMENT ANALYSIS AND APPLICATIONS With Mathematica and MATLAB Computations M. ASGHAR BHATTI WILEY JOHN WILEY & SONS, INC. CONTENTS OF THE BOOK WEB SITE PREFACE xi xiii 1 FINITE ELEMENT

### Finite Element Formulation for Beams - Handout 2 -

Finite Element Formulation for Beams - Handout 2 - Dr Fehmi Cirak (fc286@) Completed Version Review of Euler-Bernoulli Beam Physical beam model midline Beam domain in three-dimensions Midline, also called

### Linear Equations ! 25 30 35\$ & " 350 150% & " 11,750 12,750 13,750% MATHEMATICS LEARNING SERVICE Centre for Learning and Professional Development

MathsTrack (NOTE Feb 2013: This is the old version of MathsTrack. New books will be created during 2013 and 2014) Topic 4 Module 9 Introduction Systems of to Matrices Linear Equations Income = Tickets!

### Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:

Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than

### 8. Linear least-squares

8. Linear least-squares EE13 (Fall 211-12) definition examples and applications solution of a least-squares problem, normal equations 8-1 Definition overdetermined linear equations if b range(a), cannot

### Solving Quadratic Equations by Completing the Square

9. Solving Quadratic Equations by Completing the Square 9. OBJECTIVES 1. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a geometric application

FACTORING QUADRATIC EQUATIONS Summary 1. Difference of squares... 1 2. Mise en évidence simple... 2 3. compounded factorization... 3 4. Exercises... 7 The goal of this section is to summarize the methods

### Høgskolen i Narvik Sivilingeniørutdanningen

Høgskolen i Narvik Sivilingeniørutdanningen Eksamen i Faget STE67 ELEMENTMETODEN Klasse: 4.ID Dato: 4.2.23 Tid: Kl. 9. 2. Tillatte hjelpemidler under eksamen: Kalkulator. Bok Numerical solution of partial

### Sample Problems. Practice Problems

Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### BEGINNING ALGEBRA ACKNOWLEDMENTS

BEGINNING ALGEBRA The Nursing Department of Labouré College requested the Department of Academic Planning and Support Services to help with mathematics preparatory materials for its Bachelor of Science

### Programming the Finite Element Method

Programming the Finite Element Method FOURTH EDITION I. M. Smith University of Manchester, UK D. V. Griffiths Colorado School of Mines, USA John Wiley & Sons, Ltd Contents Preface Acknowledgement xv xvii

### 2013 MBA Jump Start Program

2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of