Lect20
•Download as PPTX, PDF•
0 likes•93 views
1) The document derives the shape functions for a two-noded bar element in finite element analysis. 2) It expresses the displacement function in terms of the nodal displacements using the coordinates of the two nodes. 3) The shape functions are obtained by equating the displacement function to the product of the shape function matrix and the nodal displacement vector.
Report
Share
Report
Share
![Shape functions for two nodded bar element
in (x, y) coordinates
Let consider two nodded bar element of length L.
x1 and x2 are the Cartesian coordinates of nodes 1 and 2.
u1 and u2 are the displacements of nodes 1 and 2.
u is the displacement of any point in x-direction.
Total DOF = 02 (one at each node)
I) Displacement function
1 2
u x
In matrix form
1
2
1
u x
u P
--------------- (1)
where, [P] = Parametric matrix](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Recommended
Lect19
This document discusses shape functions in finite element analysis. It explains that shape functions approximate the variation of displacement within an element since the true variation is unknown. Shape functions, also called interpolation functions, are used to replace difficult analytical solutions with easier mathematical functions. Important properties of shape functions are that the magnitude at each node is unity, the number of functions equals the number of nodes, and the sum of functions is always unity. Common methods for deriving shape functions include using polynomials in Cartesian and natural coordinate systems and Lagrange interpolation in natural coordinates.
Lect15
1) The document discusses natural coordinates for a 1D bar element in finite element analysis.
2) It defines the natural coordinates (L1, L2) of any point P on the bar element in terms of the Cartesian coordinates (x1, x2) of the two nodes and the length L of the element.
3) The natural coordinates at node 1 are derived to be (1, 0) and at node 2 to be (0, 1).
Lect16
The document discusses the natural coordinates of a 1D bar element in finite element analysis. It defines the natural coordinate ξ as the ratio of the distance from any point P to the origin, over the maximum distance from the origin to node 2. This yields expressions for ξ at each node. At node 1, ξ = 0, and at node 2, ξ = 1. Therefore, the natural coordinates of a 1D bar element are ξ, which ranges from 0 at node 1 to 1 at node 2.
Lect17
The document discusses natural coordinates, also called area coordinates, for a three-node triangular element in finite element analysis. It defines the natural coordinates (L1, L2, L3) of any point P within the triangle in terms of the Cartesian coordinates of the three nodes and the fractions of the total triangle area (A1, A2, A3) associated with each node. The natural coordinates represent the point P shifted to each node and satisfy the properties L1 + L2 + L3 = 1 and L1, L2, L3 ≥ 0.
Lect21
1. The document derives the shape functions for a three-noded constant strain triangular (CST) element.
2. It presents the displacement function in terms of nodal displacements using the element connectivity matrix.
3. The shape functions are obtained by relating the displacement function to the nodal displacement terms, resulting in a set of three linear shape functions that describe the displacement across the element in terms of the displacements at its nodes.
Lect13
This document discusses displacement functions or polynomial functions for various finite elements. It provides examples of the displacement functions for:
1) A two-noded bar element with two degrees of freedom.
2) A three-noded bar element with three degrees of freedom.
3) A two-noded beam element with four degrees of freedom.
4) A constant strain triangular element with six degrees of freedom.
It explains that the displacement functions are selected from Pascal's triangle and include terms for x- and y-coordinates for two-dimensional elements.
Fem class notes
This document contains lecture notes on the finite element method from Dr. Atteshamuddin S. Sayyad. It covers the fundamentals of elasticity theory including assumptions, basic terms like stresses and strains, equilibrium equations, and strain-displacement relationships. The notes provide definitions and equations for normal and shear stresses and strains, the state of stress and strain at a point, and developing the governing equations of equilibrium in three dimensions from an infinitesimal element.
Matrix Methods of Structural Analysis
This document contains lecture notes on matrix methods of structural analysis. It discusses three methods for solving systems of linear equations that arise in structural analysis problems - Gauss elimination method, Gauss-Jordan method, and Gauss-Seidel iterative method. For each method, an example problem is provided and solved step-by-step to illustrate the application of the method.
Recommended
Lect19
This document discusses shape functions in finite element analysis. It explains that shape functions approximate the variation of displacement within an element since the true variation is unknown. Shape functions, also called interpolation functions, are used to replace difficult analytical solutions with easier mathematical functions. Important properties of shape functions are that the magnitude at each node is unity, the number of functions equals the number of nodes, and the sum of functions is always unity. Common methods for deriving shape functions include using polynomials in Cartesian and natural coordinate systems and Lagrange interpolation in natural coordinates.
Lect15
1) The document discusses natural coordinates for a 1D bar element in finite element analysis.
2) It defines the natural coordinates (L1, L2) of any point P on the bar element in terms of the Cartesian coordinates (x1, x2) of the two nodes and the length L of the element.
3) The natural coordinates at node 1 are derived to be (1, 0) and at node 2 to be (0, 1).
Lect16
The document discusses the natural coordinates of a 1D bar element in finite element analysis. It defines the natural coordinate ξ as the ratio of the distance from any point P to the origin, over the maximum distance from the origin to node 2. This yields expressions for ξ at each node. At node 1, ξ = 0, and at node 2, ξ = 1. Therefore, the natural coordinates of a 1D bar element are ξ, which ranges from 0 at node 1 to 1 at node 2.
Lect17
The document discusses natural coordinates, also called area coordinates, for a three-node triangular element in finite element analysis. It defines the natural coordinates (L1, L2, L3) of any point P within the triangle in terms of the Cartesian coordinates of the three nodes and the fractions of the total triangle area (A1, A2, A3) associated with each node. The natural coordinates represent the point P shifted to each node and satisfy the properties L1 + L2 + L3 = 1 and L1, L2, L3 ≥ 0.
Lect21
1. The document derives the shape functions for a three-noded constant strain triangular (CST) element.
2. It presents the displacement function in terms of nodal displacements using the element connectivity matrix.
3. The shape functions are obtained by relating the displacement function to the nodal displacement terms, resulting in a set of three linear shape functions that describe the displacement across the element in terms of the displacements at its nodes.
Lect13
This document discusses displacement functions or polynomial functions for various finite elements. It provides examples of the displacement functions for:
1) A two-noded bar element with two degrees of freedom.
2) A three-noded bar element with three degrees of freedom.
3) A two-noded beam element with four degrees of freedom.
4) A constant strain triangular element with six degrees of freedom.
It explains that the displacement functions are selected from Pascal's triangle and include terms for x- and y-coordinates for two-dimensional elements.
Fem class notes
This document contains lecture notes on the finite element method from Dr. Atteshamuddin S. Sayyad. It covers the fundamentals of elasticity theory including assumptions, basic terms like stresses and strains, equilibrium equations, and strain-displacement relationships. The notes provide definitions and equations for normal and shear stresses and strains, the state of stress and strain at a point, and developing the governing equations of equilibrium in three dimensions from an infinitesimal element.
Matrix Methods of Structural Analysis
This document contains lecture notes on matrix methods of structural analysis. It discusses three methods for solving systems of linear equations that arise in structural analysis problems - Gauss elimination method, Gauss-Jordan method, and Gauss-Seidel iterative method. For each method, an example problem is provided and solved step-by-step to illustrate the application of the method.
APLICACIONES DE ESPACIO VECTORIALES
The document discusses applications of vector spaces and subspaces in electronics and automation engineering. It provides examples of how vector spaces are used in areas like RGB color coordinates, polarization of electromagnetic waves, RGB to YUV conversion, and Lorentz force. Vector spaces and linear transformations allow complex processes and algorithms to be developed for machines to help people. Understanding vector spaces is fundamental for electronics and automation careers.
Lab no.07
This document describes analysis of linear time-invariant (LTI) systems through impulse response and convolution. It defines LTI systems and explains that their behavior is characterized by impulse response. The input and output of an LTI system are related by convolution. Impulse response is the output of an LTI system when the input is a unit impulse. Given impulse response, the output can be found for any input using convolution. Difference equations are also used to represent LTI systems. The document provides examples of finding impulse responses and performing convolution in MATLAB. It discusses properties of convolution like associativity and commutativity. Tasks are given on finding impulse responses and convolving various signals.
Lect18
This document introduces 3D elements that can model structural components in three dimensions. It discusses two families of 3D elements: tetrahedrons generated from triangular elements, and parallelepipeds generated from rectangular elements. Common 3D solid elements for finite element analysis include the 4-noded tetrahedron and 8-noded hexahedron/brick. The tetrahedron has 12 degrees of freedom while the hexahedron has 24 degrees of freedom. Displacement functions are provided for both elements. 3D elements can be used for fully 3D analysis and modeling axisymmetric solids.
Convolution&Correlation
Convolution and correlation are similar mathematical operations used to extract information from images. Convolution operates on two functions to produce a third, and is equivalent to multiplication in the frequency domain. It can be linear or circular. Linear convolution of signals x(n) and h(n) produces output y(n)=x(n)*h(n). Circular convolution uses the maximum length of the signals. Correlation provides a measure of similarity between two functions and can be auto correlation, comparing a function to its shifted self, or cross correlation, comparing two different functions. Both are used in applications like image processing, signal processing, and more.
Finite element analysis of space truss by abaqus
As a part of seminar of FEM we presented ppt on 3D Tues Analysis by ABAQUS Software & Checked by manual calculation
Lab no.08
This document describes a lab experiment on complex exponentials and phasors. It introduces complex numbers and defines common operations like obtaining the real and imaginary parts. It then presents tasks involving calculating values for complex exponential signals and adding phasors. The document was designed by Dawar Awan for a class at CECOS College of Engineering and IT from March to July 2012.
G031201040042
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
1 d analysis
The document discusses concepts related to 1D finite element analysis including shape functions, natural coordinates, strain-displacement matrices, and stiffness matrices. It covers defining linear shape functions and interpolation functions within an element using natural coordinates. It also derives the strain-displacement matrix and expresses the normal strain relation in matrix form using this. Finally, it discusses properties of the global stiffness matrix, including it being symmetric, singular, and banded for 1D structural problems.
Higher order elements
The document describes the formulation of a Linear-Strain Triangular (LST) finite element. The LST element has 6 nodes, 12 degrees of freedom, and a quadratic displacement function, offering advantages over the Constant Strain Triangular (CST) element. The procedure to derive the LST element stiffness equations is identical to that used for the CST element. Key steps include discretizing the element, selecting displacement functions, defining strain-displacement relationships, and deriving the element stiffness matrix using the total potential energy approach.
Matlab cheatsheet
This document provides a cheat sheet for common Matlab commands and functions. It summarizes basics like saving/loading variables, constructing matrices, defining variables, arithmetic operations, solving equations, plotting, and transposes/dot products. Key commands include save, load, rand, zeros, ones, eye, diag, sin, plot, eig, and dot for constructing matrices, numerical operations, solving equations, plotting, and linear algebra.
Gaussian Elimination Method
Gaussian elimination is a method for solving systems of linear equations. It involves converting the augmented matrix into an upper triangular matrix using elementary row operations. There are three types of Gaussian elimination: simple elimination without pivoting, partial pivoting, and total pivoting. Partial pivoting interchanges rows to choose larger pivots, while total pivoting searches the whole matrix for the largest number to use as the pivot. Pivoting strategies help prevent zero pivots and reduce round-off errors.
Finite Element Analysis of Truss Structures
The document discusses the finite element method (FEM) for analyzing truss structures. It begins with objectives of becoming familiar with FEM concepts for truss elements like stiffness matrices and assembling the global stiffness matrix. It then covers derivation of the element stiffness matrix in local coordinates, transforming it to global coordinates, and assembling the global stiffness matrix of the overall structure from the element matrices. Strain and stress calculations are also briefly discussed. Finally, an example problem is presented to demonstrate the FEM process for a simple truss structure.
solution for 2D truss1
This document describes using the finite element method to analyze stresses in a truss structure. It defines the truss geometry, elements, and nodes. Stiffness matrices are developed for each element and combined into a global stiffness matrix. Boundary conditions are applied and the system of equations is solved to determine displacements. Stresses are then calculated for each element using the displacements. Finally, reactions are computed at fixed supports.
Introduction to finite element method(fem)
it tells u what is finite element method and it's use in solving typical 1D, 2D rods along with some soved problems .
Bresenham's line drawing algorithm
Bresenham's line algorithm is an efficient method for drawing lines on a digital display. It works by calculating the next pixel coordinate along the line using integer math only. This avoids complex floating point calculations. It starts at the initial coordinate and iteratively calculates the next x,y coordinate using integer addition and comparisons until it reaches the final endpoint.
Direct Methods to Solve Linear Equations Systems
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
Directs Methods
Gaussian elimination and Gauss-Jordan elimination are methods for solving systems of linear equations by reducing the coefficients matrix to row-echelon form. Gauss-Jordan elimination transforms the matrix into an identity matrix by eliminating each variable in turn from all equations. Gauss-Jordan with pivoting chooses rows strategically to minimize rounding errors during calculations.
Laplaces transformer
This document summarizes a lecture on Laplace transforms given by Prof. Mihir Suthar to engineering students in Sem 3 of the 2015-2016 academic year. It covers key concepts in Laplace transforms including the unit step function, second shifting theorem, Dirac delta function, differentiation and integration of transforms, and convolution and integral equations. Example problems were provided to demonstrate how to apply these concepts to specific functions.
Week 15 state space rep may 25 2016 final
This document provides an overview of state space analysis for dynamic systems. It introduces state space equations and models using matrix representations. Key matrices include the A matrix for the state, B for input, C for output, and D for direct transmission. The document discusses converting transfer functions to state space models and provides a mass-damper-spring example. It also outlines related terminology and using Matlab functions to create state space models from matrices.
Introduction to FEM
The document provides an introduction to the finite element method (FEM) for numerical analysis of engineering problems. It discusses how FEM can be used to solve complex problems by dividing them into smaller, simpler elements. FEM allows the use of computers to solve problems that cannot be solved through analytical methods. It also describes the different types of FEM formulations including implicit, which tries to obtain structural equilibrium at each time step for faster solutions, and explicit, which does not require iterations but has longer calculation times. Finally, the document gives examples of how FEM can be applied to problems in solid mechanics, fluid mechanics, and thermodynamics.
Shape1 d
The document discusses the benefits of meditation for reducing stress and anxiety. Regular meditation practice can help calm the mind and body by lowering heart rate and blood pressure. Making meditation a part of a daily routine, even if just 10-15 minutes per day, can have mental and physical health benefits over time by helping people feel more relaxed and better able to handle life's stresses.
Constant strain triangular
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
More Related Content
What's hot
APLICACIONES DE ESPACIO VECTORIALES
The document discusses applications of vector spaces and subspaces in electronics and automation engineering. It provides examples of how vector spaces are used in areas like RGB color coordinates, polarization of electromagnetic waves, RGB to YUV conversion, and Lorentz force. Vector spaces and linear transformations allow complex processes and algorithms to be developed for machines to help people. Understanding vector spaces is fundamental for electronics and automation careers.
Lab no.07
This document describes analysis of linear time-invariant (LTI) systems through impulse response and convolution. It defines LTI systems and explains that their behavior is characterized by impulse response. The input and output of an LTI system are related by convolution. Impulse response is the output of an LTI system when the input is a unit impulse. Given impulse response, the output can be found for any input using convolution. Difference equations are also used to represent LTI systems. The document provides examples of finding impulse responses and performing convolution in MATLAB. It discusses properties of convolution like associativity and commutativity. Tasks are given on finding impulse responses and convolving various signals.
Lect18
This document introduces 3D elements that can model structural components in three dimensions. It discusses two families of 3D elements: tetrahedrons generated from triangular elements, and parallelepipeds generated from rectangular elements. Common 3D solid elements for finite element analysis include the 4-noded tetrahedron and 8-noded hexahedron/brick. The tetrahedron has 12 degrees of freedom while the hexahedron has 24 degrees of freedom. Displacement functions are provided for both elements. 3D elements can be used for fully 3D analysis and modeling axisymmetric solids.
Convolution&Correlation
Convolution and correlation are similar mathematical operations used to extract information from images. Convolution operates on two functions to produce a third, and is equivalent to multiplication in the frequency domain. It can be linear or circular. Linear convolution of signals x(n) and h(n) produces output y(n)=x(n)*h(n). Circular convolution uses the maximum length of the signals. Correlation provides a measure of similarity between two functions and can be auto correlation, comparing a function to its shifted self, or cross correlation, comparing two different functions. Both are used in applications like image processing, signal processing, and more.
Finite element analysis of space truss by abaqus
As a part of seminar of FEM we presented ppt on 3D Tues Analysis by ABAQUS Software & Checked by manual calculation
Lab no.08
This document describes a lab experiment on complex exponentials and phasors. It introduces complex numbers and defines common operations like obtaining the real and imaginary parts. It then presents tasks involving calculating values for complex exponential signals and adding phasors. The document was designed by Dawar Awan for a class at CECOS College of Engineering and IT from March to July 2012.
G031201040042
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
1 d analysis
The document discusses concepts related to 1D finite element analysis including shape functions, natural coordinates, strain-displacement matrices, and stiffness matrices. It covers defining linear shape functions and interpolation functions within an element using natural coordinates. It also derives the strain-displacement matrix and expresses the normal strain relation in matrix form using this. Finally, it discusses properties of the global stiffness matrix, including it being symmetric, singular, and banded for 1D structural problems.
Higher order elements
The document describes the formulation of a Linear-Strain Triangular (LST) finite element. The LST element has 6 nodes, 12 degrees of freedom, and a quadratic displacement function, offering advantages over the Constant Strain Triangular (CST) element. The procedure to derive the LST element stiffness equations is identical to that used for the CST element. Key steps include discretizing the element, selecting displacement functions, defining strain-displacement relationships, and deriving the element stiffness matrix using the total potential energy approach.
Matlab cheatsheet
This document provides a cheat sheet for common Matlab commands and functions. It summarizes basics like saving/loading variables, constructing matrices, defining variables, arithmetic operations, solving equations, plotting, and transposes/dot products. Key commands include save, load, rand, zeros, ones, eye, diag, sin, plot, eig, and dot for constructing matrices, numerical operations, solving equations, plotting, and linear algebra.
Gaussian Elimination Method
Gaussian elimination is a method for solving systems of linear equations. It involves converting the augmented matrix into an upper triangular matrix using elementary row operations. There are three types of Gaussian elimination: simple elimination without pivoting, partial pivoting, and total pivoting. Partial pivoting interchanges rows to choose larger pivots, while total pivoting searches the whole matrix for the largest number to use as the pivot. Pivoting strategies help prevent zero pivots and reduce round-off errors.
Finite Element Analysis of Truss Structures
The document discusses the finite element method (FEM) for analyzing truss structures. It begins with objectives of becoming familiar with FEM concepts for truss elements like stiffness matrices and assembling the global stiffness matrix. It then covers derivation of the element stiffness matrix in local coordinates, transforming it to global coordinates, and assembling the global stiffness matrix of the overall structure from the element matrices. Strain and stress calculations are also briefly discussed. Finally, an example problem is presented to demonstrate the FEM process for a simple truss structure.
solution for 2D truss1
This document describes using the finite element method to analyze stresses in a truss structure. It defines the truss geometry, elements, and nodes. Stiffness matrices are developed for each element and combined into a global stiffness matrix. Boundary conditions are applied and the system of equations is solved to determine displacements. Stresses are then calculated for each element using the displacements. Finally, reactions are computed at fixed supports.
Introduction to finite element method(fem)
it tells u what is finite element method and it's use in solving typical 1D, 2D rods along with some soved problems .
Bresenham's line drawing algorithm
Bresenham's line algorithm is an efficient method for drawing lines on a digital display. It works by calculating the next pixel coordinate along the line using integer math only. This avoids complex floating point calculations. It starts at the initial coordinate and iteratively calculates the next x,y coordinate using integer addition and comparisons until it reaches the final endpoint.
Direct Methods to Solve Linear Equations Systems
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
Directs Methods
Gaussian elimination and Gauss-Jordan elimination are methods for solving systems of linear equations by reducing the coefficients matrix to row-echelon form. Gauss-Jordan elimination transforms the matrix into an identity matrix by eliminating each variable in turn from all equations. Gauss-Jordan with pivoting chooses rows strategically to minimize rounding errors during calculations.
Laplaces transformer
This document summarizes a lecture on Laplace transforms given by Prof. Mihir Suthar to engineering students in Sem 3 of the 2015-2016 academic year. It covers key concepts in Laplace transforms including the unit step function, second shifting theorem, Dirac delta function, differentiation and integration of transforms, and convolution and integral equations. Example problems were provided to demonstrate how to apply these concepts to specific functions.
Week 15 state space rep may 25 2016 final
This document provides an overview of state space analysis for dynamic systems. It introduces state space equations and models using matrix representations. Key matrices include the A matrix for the state, B for input, C for output, and D for direct transmission. The document discusses converting transfer functions to state space models and provides a mass-damper-spring example. It also outlines related terminology and using Matlab functions to create state space models from matrices.
Introduction to FEM
The document provides an introduction to the finite element method (FEM) for numerical analysis of engineering problems. It discusses how FEM can be used to solve complex problems by dividing them into smaller, simpler elements. FEM allows the use of computers to solve problems that cannot be solved through analytical methods. It also describes the different types of FEM formulations including implicit, which tries to obtain structural equilibrium at each time step for faster solutions, and explicit, which does not require iterations but has longer calculation times. Finally, the document gives examples of how FEM can be applied to problems in solid mechanics, fluid mechanics, and thermodynamics.
What's hot (20)
Similar to Lect20
Shape1 d
The document discusses the benefits of meditation for reducing stress and anxiety. Regular meditation practice can help calm the mind and body by lowering heart rate and blood pressure. Making meditation a part of a daily routine, even if just 10-15 minutes per day, can have mental and physical health benefits over time by helping people feel more relaxed and better able to handle life's stresses.
Constant strain triangular
constant strain triangular which is used in analysis of triangular in finite element method with the help of shape function and natural coordinate system.
D1_YT_English_VEctor_Calculus_Gradient_&_Directional_Derivative.pptx
This document provides an overview of vector calculus concepts including gradient, directional derivative, dot product, and cross product. It includes examples of calculating directional derivatives of multivariate functions at given points in specified directions. It also provides practice problems testing the concepts covered, such as finding the gradient of a function, or the directional derivative of a function at a point along a given direction.
Mixed Spectra for Stable Signals from Discrete Observations
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the
modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral
measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to
estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we
propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial
kernel to build a periodogram which we then smooth by two spectral windows taking into account the
width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing
often encountered in the case of estimation from discrete observations of a continuous time process.
Mixed Spectra for Stable Signals from Discrete Observations
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the
modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral
measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to
estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we
propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial
kernel to build a periodogram which we then smooth by two spectral windows taking into account the
width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing
often encountered in the case of estimation from discrete observations of a continuous time process.
MIXED SPECTRA FOR STABLE SIGNALS FROM DISCRETE OBSERVATIONS
This paper proposes a method to estimate the spectral density of a continuous-time stable alpha symmetric process from discrete observations of the process. Specifically, it considers when the spectral measurement is a mixture of a continuous component and discrete jumps. It samples the process at periodic times to create a periodogram, which is shown to be an asymptotically unbiased but inconsistent estimator. The periodogram is then smoothed using two spectral windows to account for the bandwidth of the spectral density, providing a consistent estimator of the spectral density at the jump points.
Mixed Spectra for Stable Signals from Discrete Observations
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the
modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral
measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to
estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we
propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial
kernel to build a periodogram which we then smooth by two spectral windows taking into account the
width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing
often encountered in the case of estimation from discrete observations of a continuous time process.
Mixed Spectra for Stable Signals from Discrete Observations
This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.
Additional Mathematics form 4 (formula)
This document provides a summary of various math formulae for Form 4 students in Malaysia, including:
1. Functions, quadratic equations, and quadratic functions
2. Simultaneous equations, indices and logarithms, and coordinate geometry
3. Statistics, circular measures, and differentiation
It lists common formulae for topics like the quadratic formula, completing the square, differentiation rules, and measures of central tendency and dispersion. The document is intended as a study guide for students to review essential formulae.
Spm add-maths-formula-list-form4-091022090639-phpapp01
This document provides a summary of various math formulae for Form 4 students in Malaysia, including:
1. Functions, quadratic equations, and quadratic functions
2. Simultaneous equations, indices and logarithms, and coordinate geometry
3. Statistics, circular measures, and differentiation
It lists common formulae for topics like the quadratic formula, completing the square, differentiation rules, and measures of central tendency and dispersion. The document is intended as a study guide for students to review essential formulae.
Spm add-maths-formula-list-form4-091022090639-phpapp01
This document provides a summary of various math formulae for Form 4 students in Malaysia, including:
1. Functions, quadratic equations, and quadratic functions
2. Simultaneous equations, indices and logarithms, and coordinate geometry
3. Statistics, circular measures, and differentiation
It lists common formulae for topics like the quadratic formula, completing the square, differentiation rules, and measures of central tendency and dispersion. The document is intended as a study guide for students to review essential formulae.
Diffusion kernels on SNP data embedded in a non-Euclidean metric
This document discusses using diffusion kernels on single nucleotide polymorphism (SNP) data embedded in non-Euclidean metric spaces. It defines kernels as weighting functions that provide a similarity metric based on a chosen distance metric, such as Euclidean or Manhattan distance. The document explores using graphs to represent SNP data, with nodes for each genotype and edges connecting similar genotypes. It describes constructing diffusion kernels on these graphs by taking Cartesian graph products of one-dimensional genotype graphs. This allows modeling SNP data as grids where similarity decreases as the number of differing loci increases.
Isoparametric Elements 4.pdf
This document discusses isoparametric finite elements. It begins by defining shape functions and how they relate node displacements to the displacement field within an element. Isoparametric elements are introduced, which use the same shape functions for geometric representation and displacement approximation. A 1D bar element demonstration shows how shape functions define element geometry and displacements. Higher-order elements, including triangular elements, are also discussed. Different types of shape functions like Lagrange, serendipity, and Hermitian polynomials are covered.
Scalar and vector differentiation
This document discusses a course on electromagnetic theory taught by Arpan Deyasi. It covers topics related to vector differentiation, including the vector differential operator in Cartesian, cylindrical and spherical coordinates. It also covers differentiation of scalar functions, including calculating gradients, directional derivatives and finding normals to surfaces. Finally, it discusses differentiation of vector functions, specifically divergence, which represents the volume density of the net outward flux from a vector field.
1526 exploiting symmetries
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Solution set 3
The document discusses quantum mechanical concepts including:
1) The time derivative of the momentum expectation value satisfies an equation involving the potential gradient.
2) For an infinite potential well, the kinetic energy expectation value is proportional to n^2/a^2 and the potential energy expectation value vanishes.
3) Eigenfunctions of an eigenvalue problem under certain boundary conditions correspond to positive eigenvalues that are sums of squares of integer multiples of pi.
Em10 fl
The document summarizes key concepts about functions and lines. It defines a function as a rule that assigns unique output values to inputs. Functions can be represented graphically by plotting the points (x, f(x)). Lines on a plane can be defined by two points and have a slope that represents the rise over run. The slope formula is used to find the equation of a line in y=mx+b form, where m is the slope and b is the y-intercept. Perpendicular and parallel lines have specific relationships between their slopes.
dalrymple_slides.ppt
This document discusses various methods for modeling shallow water flows and waves using numerical techniques. It covers topics like wave theories, wave modeling approaches, meshfree Lagrangian methods, smoothed particle hydrodynamics (SPH), and the use of graphics processing units (GPUs) for real-time simulations. SPH is presented as a meshfree Lagrangian technique for modeling wave breaking processes. The document outlines the governing SPH equations, kernel approximations, time stepping approaches, and submodels for viscosity and turbulence. Validation examples are shown comparing SPH simulations to experimental data.
Deblurring in ct
Presentation about basics of image deblurring, two popular approaches, and two computed tomography applications.
Similar to Lect20 (20)
D1_YT_English_VEctor_Calculus_Gradient_&_Directional_Derivative.pptx
D1_YT_English_VEctor_Calculus_Gradient_&_Directional_Derivative.pptx
Mixed Spectra for Stable Signals from Discrete Observations
Mixed Spectra for Stable Signals from Discrete Observations
Mixed Spectra for Stable Signals from Discrete Observations
Mixed Spectra for Stable Signals from Discrete Observations
MIXED SPECTRA FOR STABLE SIGNALS FROM DISCRETE OBSERVATIONS
MIXED SPECTRA FOR STABLE SIGNALS FROM DISCRETE OBSERVATIONS
Mixed Spectra for Stable Signals from Discrete Observations
Mixed Spectra for Stable Signals from Discrete Observations
Mixed Spectra for Stable Signals from Discrete Observations
Mixed Spectra for Stable Signals from Discrete Observations
Spm add-maths-formula-list-form4-091022090639-phpapp01
Spm add-maths-formula-list-form4-091022090639-phpapp01
Spm add-maths-formula-list-form4-091022090639-phpapp01
Spm add-maths-formula-list-form4-091022090639-phpapp01
Diffusion kernels on SNP data embedded in a non-Euclidean metric
Diffusion kernels on SNP data embedded in a non-Euclidean metric
shape functions of 1D and 2 D rectangular elements.pptx
shape functions of 1D and 2 D rectangular elements.pptx
More from DrASSayyad
Theory of Plates and Shells
This document provides an overview of basic equations for the theory of plates and shells. It discusses the state of stress and strain at a point, including defining the six independent stress and strain components. It presents the relationships between strain and displacement, and discusses the equilibrium equations relating stress and body forces. Finally, it provides the equations for both Cartesian and cylindrical coordinate systems. The key concepts covered are the fundamental equations that form the basis of plate and shell theory.
Lect14
This document compares the constant strain triangular (CST) element and the linear strain triangular (LST) element in finite element analysis. It states that the CST element has constant strains inside each element but strains are not continuous across boundaries. In contrast, the LST element has linear strains inside each element and is preferred for stress analysis due to its higher accuracy. The document recommends using LST elements over multiple CST elements for better representation of stresses and displacements.
Lect12
The document discusses geometric isotropy or invariance in finite element modeling, which means the displacement shapes of an element do not change with a change in the local coordinate system. This can be achieved in 2D modeling by selecting terms on either side of the Pascal's triangle that correspond to the same order. In 3D modeling, geometric invariance requires selecting terms based on the 3D Pascal's triangle.
Lect11
The document discusses three criteria for the convergence of displacement/shape functions in the finite element method:
1) Displacement functions must be continuous and compatible within elements, with no discontinuities between elements.
2) Displacement functions must be capable of representing constant strain states within elements.
3) Displacement functions must be capable of representing rigid body displacements, with elements experiencing no strain and zero nodal forces under rigid body motion.
Lect10
The document discusses the different types of finite elements used in civil engineering. It outlines three main types: 1) 1D elements which are used to analyze trusses, beams, pipes and tubes; 2) 2D elements which are used for plane stress, plane strain and plate bending as well as shell structures; and 3) 3D elements which are used for three-dimensional analysis and axisymmetric solids.
Lect09
1. The document outlines the step-by-step procedure for finite element analysis, which includes: selecting suitable field variables and elements, discretizing the continuum, selecting interpolation functions, finding element properties, assembling global properties, imposing boundary conditions, and making additional calculations.
2. Key steps include discretizing the continuum by dividing it into sub-domains called elements, expressing unknown field variables within each element using interpolation functions, assembling element properties like stiffness matrices, and assembling global properties from the element properties to obtain system equations.
3. The procedure allows reducing infinite unknowns to a finite number and solving the system of equations imposed with boundary conditions to find nodal values and make additional calculations for required outputs.
Lect08
The document discusses element aspect ratios in finite element analysis for civil engineering structures. It defines aspect ratio as the ratio of the largest to smallest size of an element. For analysis accuracy, the aspect ratio should be as close to unity as possible. It provides an example of correctly sizing square elements of equal size compared to incorrectly using long rectangular elements.
Lect07
This document discusses effective node numbering schemes for finite element analysis. It explains that effective node numbering should minimize the bandwidth of the stiffness matrix. The bandwidth is calculated as B = (1+D)f, where D is the maximum difference in node numbers between any two connected nodes, and f is the degrees of freedom per node. Several examples are provided to demonstrate calculating the bandwidth for different node numbering approaches applied to structures like rectangular slabs and trusses. Effective node numbering reduces computation time by producing a banded, sparse stiffness matrix.
Lect06
Nodes are points where basic unknowns are determined in finite element analysis and where elements connect. There are two main types of nodes: external nodes located at element edges or surfaces and shared by multiple elements, and internal nodes located inside a single element. External nodes are further divided into primary nodes at corners or ends, and secondary nodes along edges but not at corners. Internal nodes are specific to each individual element.
Lect05
Three coordinate systems are used in finite element analysis: the local coordinate system defines each element's properties, the global coordinate system defines points in the entire structure, and the natural coordinate system specifies points within an element using dimensionless numbers between -1 and 1 with the origin at the element's center.
Lect04
The document discusses applications of the finite element method (FEM) in various engineering fields. It lists applications of FEM in civil engineering such as analyzing structures like bars, trusses, beams and frames; earthquake analysis; and bending, buckling and vibration analysis of beams and plates. It also lists applications of FEM in mechanical engineering, aerospace engineering, medical engineering, and electrical engineering.
Lect03
The document discusses the advantages and disadvantages of the finite element method (FEM) in civil engineering. It states that FEM can effectively handle problems with irregular shapes, boundary conditions, and loading conditions. It can also solve problems involving anisotropic and non-linear material and geometric properties that classical methods cannot. However, FEM provides approximate solutions rather than exact ones, requires skill in element selection and modeling, and demands significant computer resources.
Lect02
Discretization is the process of dividing a continuous structure into smaller, finite pieces or elements to simulate the structure on a computer for finite element analysis. The structure is divided into a number of elements that are connected at nodes. The number, shape, size and configuration of elements must closely simulate the real structure, and the discretization must allow results to converge to the true solution. For example, a rectangular slab can be discretized by dividing it into n subdomains, with each subdomain being an element connected to others at nodes to form a finite element mesh.
Lect01
The document discusses the finite element method (FEM), a numerical technique used to solve complex engineering problems. FEM was originally developed for aerospace engineering but is now widely used in civil, mechanical, and electrical engineering. It works by dividing a structure into finite elements and obtaining an approximate solution by analyzing the properties of individual elements and their interaction. The key principles of FEM include using computational methods to solve boundary value problems representing physical structures, with field variables governed by differential equations and boundary conditions specified on the field boundaries.
Recently uploaded
The Python for beginners. This is an advance computer language.
Python language is very important language at this time. we can easily understand this language by these notes.
Modelagem de um CSTR com reação endotermica.pdf
Modelagem em função de transferencia. CSTR não-linear.
Iron and Steel Technology Roadmap - Towards more sustainable steelmaking.pdf
Iron and Steel Technology towards Sustainable Steelmaking
ACEP Magazine edition 4th launched on 05.06.2024
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
Unit-III-ELECTROCHEMICAL STORAGE DEVICES.ppt
Batteries -Introduction – Types of Batteries – discharging and charging of battery - characteristics of battery –battery rating- various tests on battery- – Primary battery: silver button cell- Secondary battery :Ni-Cd battery-modern battery: lithium ion battery-maintenance of batteries-choices of batteries for electric vehicle applications.
Fuel Cells: Introduction- importance and classification of fuel cells - description, principle, components, applications of fuel cells: H2-O2 fuel cell, alkaline fuel cell, molten carbonate fuel cell and direct methanol fuel cells.
学校原版美国波士顿大学毕业证学历学位证书原版一模一样
原版一模一样【微信：741003700 】【美国波士顿大学毕业证学历学位证书】【微信：741003700 】学位证，留信认证（真实可查，永久存档）offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原海外各大学 Bachelor Diploma degree, Master Degree Diploma
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
Advanced control scheme of doubly fed induction generator for wind turbine us...
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
Understanding Inductive Bias in Machine Learning
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
New techniques for characterising damage in rock slopes.pdf
rock mass characterization and New techniques for characterising damage in rock slopes
2008 BUILDING CONSTRUCTION Illustrated - Ching Chapter 02 The Building.pdf
2008 BUILDING CONSTRUCTION Illustrated - Ching Chapter 02 The Building
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECT
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
Casting-Defect-inSlab continuous casting.pdf
Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting. Casting-Defect-inSlab continuous casting
Harnessing WebAssembly for Real-time Stateless Streaming Pipelines
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
Recycled Concrete Aggregate in Construction Part III
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Recently uploaded (20)
ML Based Model for NIDS MSc Updated Presentation.v2.pptx
ML Based Model for NIDS MSc Updated Presentation.v2.pptx
The Python for beginners. This is an advance computer language.
The Python for beginners. This is an advance computer language.
Iron and Steel Technology Roadmap - Towards more sustainable steelmaking.pdf
Iron and Steel Technology Roadmap - Towards more sustainable steelmaking.pdf
Advanced control scheme of doubly fed induction generator for wind turbine us...
Advanced control scheme of doubly fed induction generator for wind turbine us...
Manufacturing Process of molasses based distillery ppt.pptx
Manufacturing Process of molasses based distillery ppt.pptx
New techniques for characterising damage in rock slopes.pdf
New techniques for characterising damage in rock slopes.pdf
2008 BUILDING CONSTRUCTION Illustrated - Ching Chapter 02 The Building.pdf
2008 BUILDING CONSTRUCTION Illustrated - Ching Chapter 02 The Building.pdf
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECT
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECT
Harnessing WebAssembly for Real-time Stateless Streaming Pipelines
Harnessing WebAssembly for Real-time Stateless Streaming Pipelines
Recycled Concrete Aggregate in Construction Part III
Recycled Concrete Aggregate in Construction Part III
Lect20
- 1. Dr. A. S. Sayyad Professor & Head Department of Structural Engineering Sanjivani College of Engineering, Kopargaon 423603. (An Autonomous Institute, Affiliated to Savitribai Phule Pune University, Pune) Finite Element Method In Civil Engineering Shape functions for two nodded bar element in (x, y) coordinates 2 1 1 2 and x x x x N N L L
- 2. Shape functions for two nodded bar element in (x, y) coordinates Let consider two nodded bar element of length L. x1 and x2 are the Cartesian coordinates of nodes 1 and 2. u1 and u2 are the displacements of nodes 1 and 2. u is the displacement of any point in x-direction. Total DOF = 02 (one at each node) I) Displacement function 1 2 u x In matrix form 1 2 1 u x u P --------------- (1) where, [P] = Parametric matrix
- 3. II) Displacement function in-terms of nodal displacements Express displacement function in terms of nodal displacements using the coordinates of nodes x1 and x2. 1 1 1 2 2 2 1 1 u x u x e x A where, [A] = Connectivity matrix Obtained from Eq. (2) and put into the Eq. (1), we get ------------------- (2) 1 e u P A x e u N x where [N] = Shape functions 1 N P A
- 4. III) Shape functions 1 1 1 2 1 1 1 N P A x N x x Inverse is obtained by using method of adjoin 1 2 1 1 1 1 1 2 x x N P A x 1 2 2 1 1 2 N x x N x x 2 1 1 2 and x x x x N N L L
- 5. Sum of shape functions is always unity Validation of properties of shape functions 2 1 2 1 1 2 1 x x x x x x L N N L L L L At node 1, x=x1 2 1 1 1 1 2 1 and 0 x x L x x N N L L L At node 2, x=x2 2 2 2 1 1 2 0 and 1 x x x x L N N L L L Value of shape function at each node is equal to unity