text: Analyze and interpret the data. - Steps and Process, Working Scholars® Bringing Tuition-Free College to the Community. Proposal of a... See full answer below. 5. Our experts can answer your tough homework and study questions. Sometimes the steps are combined or added to, but the five are the basic structure for any endeavor to answer a question objectively. Identification of a problem and its possible outcome 2. Develop a hypothesis IV. check. Step 2: Formulate a hypothesis. Share the results with other scientists. Answered. What is the correct order of the steps in the scientific method. Share the results with other scientists V. Design and perform an experiment to test the hypothesis a) I – II – III – IV – V b) III – I – V – II – IV c) I – III – V – II – IV d) V – IV– III – II – I 35. Dependent Variable Definition and Examples, Ph.D., Biomedical Sciences, University of Tennessee at Knoxville, B.A., Physics and Mathematics, Hastings College. 2. What is the correct sequence of the cell cycle? Scientific Method Examples. There are very many examples of the use of the scientific method throughout history because it is the basis for all scientific experiments. It involves making observations, formulating a hypothesis, and conducting scientific experiments. The basic steps in the scientific method are: observation, form hypothesis, design an experiment, collect data In an experimental procedure what is manipulated by the investigator to determine whether it influences the phenomenon of interest is the The number of steps can vary from one description to another (which mainly happens when data and analysis are separated into separate steps), however, this is a fairly standard list of the six scientific method steps that you are expected to know for any science class: Sometimes the scientific method is taught with seven steps instead of six. What are the steps of the scientific method? All rights reserved. The steps that make up the scientific method generally fall into three phases: observation, experimentation and conclusion. The key difference between the scientific method and other ways of acquiring knowledge are forming a hypothesis and then testing it with an experiment. I. It is used in many fields of science, including social and physical. Formal observations are a type of brainstorming that can help you find an idea and form a hypothesis. ThoughtCo, Aug. 27, 2020, thoughtco.com/steps-of-the-scientific-method-p2-606045. (5) Draw conclusions. D. It can be cyclic. What Are the Elements of a Good Hypothesis? It follows one prescribed sequence of steps. What is the correct sequence of steps in the scientific method? \frac{x + 1}{2} - \frac{x +... Why did New Honda's campaign become popular? The scientific method is a method of investigation involving experimentation and observation to acquire new knowledge, solve problems, and answer questions. A scientist learns about the … Analyze and interpret the data III. - Definition, Method & Steps, Common Laboratory Equipment: Types & Uses, Introduction to the Compound Microscope: Parts & Uses, Grounded Theory Design: Definition, Advantages & Disadvantages, What is Scientific Investigation? State the problem II. I- Experimentation II- Formulation of a hypothesis III- Observation IV- Defining a problem V- Formulation a theory a. I, II, III, IV, V b. IV, III, II, I, V c. III, IV, II, I, V d. V, IV, III, II, I 78. The scientific method is a standardized process used to generate scientific evidence. True or false? It begins and ends with an experiment. The scientific method is an ongoing process that repeats itself. The scientific method begins when the scientist / researcher asks a question about something he has observed: How, what, when, who, what, why or where? What is the correct order of the steps in the scientific method. The sequence of steps in the scientific method occurs as followed: 1. Understanding Simple vs Controlled Experiments. (4) Analyze the results. It is involved in modifying, sorting, and packaging macromolecules. Analyze the data. State the problem II. In order to do this, psychologists utilize the scientific method to conduct psychological research. The key steps in the scientific method include the following: Step 1: Make observations. What is the correct sequence of steps in the scientific method? The first four steps in scientific research all fall into the observation phase and include initial observations, asking questions, gathering information and forming hypotheses. Why do owls sleep during the day? heart outlined. Which statement is true about the scientific method? The Scientific Method starts with aquestion, and background research is conducted to try to answer that question. Why do Peacocks dance before rain? What is the reason for Okazaki fragments being formed during DNA replication? C. It is used only in Earth and space science. She has taught science courses at the high school, college, and graduate levels. The scientists in these fields ask different questions and perform different tests. IV. The starting point of most new research is to formulate a general question about an area of research and begin the process of defining it.This initial question can be very broad, as the later research, observation and narrowing down will hone it into a testable hypothesis.For example, a broad question might ask 'whether fish stocks in the North Atlantic are declining or not', based upon general observations about smaller yields of fi… Include colors, timing, sounds, temperatures, changes, behavior, and anything that strikes you as interesting or significant. Helmenstine, Anne Marie, Ph.D. "Six Steps of the Scientific Method." answer! Identification of a problem and its possible outcome. I. There are three types of variables: Dr. Helmenstine holds a Ph.D. in biomedical sciences and is a science writer, educator, and consultant. The scientific method is a set of principles and procedures that are used by researchers to develop questions, collect data, and reach conclusions. The scientific method is not precisely defined, but almost all scientists agree that it involves iterations of these four steps: characterizations (observations and definitions), hypotheses, predictions, and experiments to test all of the previous parts of the sequence. What is the correct sequence of the scientific method? a. In this model, the first step of the scientific method is to make observations. A. The method steps include, observation, research, creating an educated guess, or hypothesis, testing that hypothesis, analyzing the test results, and recording the results as true or untrue. Ask a question, make a hypothesis, test the hypothesis, draw. What Is an Experiment? conclusions, and analyze results, B. 1. Develop a hypothesis. Share the results with other scientists Answer: 1) make an observation that describe the problems 2) create a hypothesis 3) test the hypothesis and draw conclusions and refine the hypothesis. State a problem II. The correct order of the steps in the scientific method are: (1) Ask a question. Scientific investigators to answer specific questions about the natural world earn Transferable Credit & Get your,. Include the following: step 1: ask a question, draw conclusions, communicate results a problem answer question. Method are: ( 1 ) ask a question, draw, 2021 ) 17th century, analyze results... For any endeavor to answer a question, make a hypothesis, test hypothesis... 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text:
# Z transform piecewise function

I have this piecewise function:

$$x(n)= \left\{ \begin{array}{lcc} 1 & 0 \leq n \leq m \\ \\ 0, &\mbox{ for the rest} \\ \\ \end{array} \right.$$

How do I calculate the $z$-transform?

$X(z) = \sum_{n=-\infty}^{\infty}x[n]z^{-n} = \sum_{n=0}^{m}z^{-n} = \frac{1-z^{-(m+1)}}{1-z^{-1}} = \frac{1}{z^{m}}\frac{z^{m+1}-1}{z-1}$, with $z \neq 0$.

format: plain


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# Lagrange Interpolation Polynomial – C PROGRAM

Given a set of $n+1$ data-points $(x_i,y_i)$, the Lagrange Interpolating Polynomial is a polynomial of degree $\leq n$, such that it passes through all the given $n+1$ data-points.

It is given as,

$\boxed{Pn(x) = \sum^n_{i=0} L_i(x)y_i}$

where $(x_0,y_0), (x_1,y_1),.....(x_n,y_n)$ are the $n+1$ data-points.
and,
$\boxed{L_i(x)=\prod^n_{j=0;j \neq i}\frac{x-x_j}{x_i-x_j}}$

### CODE:

/**********************************
******LAGRANGE INTERPOLATION*******
**********************************/
#include<stdio.h>

/*Function to evaluate Li(x)*/

double Li(int i, int n, double x[n+1], double X){
int j;
double prod=1;
for(j=0;j<=n;j++){
if(j!=i)
prod=prod*(X-x[j])/(x[i]-x[j]);
}
return prod;
}

/*Function to evaluate Pn(x) where Pn is the Lagrange interpolating polynomial of degree n*/

double Pn(int n, double x[n+1], double y[n+1], double X){
double sum=0;
int i;
for(i=0;i<=n;i++){
sum=sum+Li(i,n,x,X)*y[i];
}
return sum;
}
main(){
int i,n;  //n is the degree
printf("Enter the number of data-points:\n");
scanf("%d",&n);  //no. of data-points is n+1
n=n-1;
//Arrays to store the (n+1) x and y data-points of size n+1
double x[n+1];
double y[n+1];
printf("Enter the x data-points:\n");
for(i=0;i<n+1;i++){
scanf("%lf",&x[i]);
}

printf("Enter the y data-points:\n");
for(i=0;i<n+1;i++){
scanf("%lf",&y[i]);
}

double X;  //value of x for which interpolated value is required
printf("Enter the value of x for which you want the interpolated value of y(x):\n");
scanf("%lf",&X);
printf("The interpolated value is %lf",Pn(n,x,y,X));
}

f(x)=1/x

f(x)=x^2

### References and Resources:

[wpedon id="7041" align="center"]

## 3 thoughts on “Lagrange Interpolation Polynomial – C PROGRAM”

1. Hello just check the function LI(……) , as per your code i was getting an error while using double x[n+1] , i tried and changed each parameter x[n+1] with x[] and same with y[n+1] with y[] , just check the below code 🙂

8:33: error: use of parameter outside function body before ‘]’ token
8:34: error: expected ‘)’ before ‘,’ token
8:35: error: expected unqualified-id before ‘double’

solution:

/**********************************
******LAGRANGE INTERPOLATION*******
**********************************/
#include

/*Function to evaluate Li(x)*/

double Li(int i,int n,double x[],double X){
int j;
double prod=1;
for(j=0;j

1. Thank you very much for your valuable correction to the above program. I can successfully run a program as per your correction suggestion.


text:
Review of Dual Pairs

# Review of Dual Pairs

• On the Dual Pairs of Vector Spaces page we defined a Dual Pair to be a triple $(E, F, \langle \cdot, \cdot \rangle)$ where $E$ and $F$ are vector spaces and $\langle \cdot, \cdot \rangle : E \times F \to \mathbf{F}$ is a bilinear form with the property that for each nonzero $x \in E$ there exists a $y \in F$ such that $\langle x, y \rangle \neq 0$ and with the property that for each nonzero $y \in F$ there exists a $x \in E$ such that $\langle x, y \rangle \neq 0$.
• We then observed some common examples of dual pairs:
 Examples of Dual Pairs Example 1: If $E$ is a vector space then $(E, E^*)$ is a dual pair with the duality pairing given for all $x \in E$ and for all $f \in E^*$ by $\langle x, f \rangle := f(x)$. Example 2: If $E$ is a Hausdorff locally convex topological vector space then $(E, E')$ is a dual pair with the duality pairing given for all $x \in E$ and for all $f \in E'$ by $\langle x, f \rangle := f(x)$. Example 3: If $(E, F, \langle \cdot, \cdot \rangle)$ is a dual pair then $(F, E, [\cdot, \cdot])$ with the duality pairing given for all $x \in E$ and for all $y \in F$ by $[y, x] := \langle x, y \rangle$.
• On The Weak Topology on E Determined by F page, for a dual pair $(E, F)$, we defined the Weak Topology on $E$ Determined by $F$ denoted by $\sigma(E, F)$, to be the coarsest topology determined by the collection of seminorms $Q := \{ p_y : y \in F \}$ where for each $y \in F$ we defined $p_y : E \to \mathbf{F}$ by:
(1)
\begin{align} \quad p_y(x) := |\langle x, y \rangle| \end{align}
• We also noted the following properties of the weak topology on $E$ determined by $F$, which are summarized below:
 Properties of the Weak Topology on $E$ Determined by $F$ (1) $\sigma(E, F)$ is the coarsest topology for which each of the seminorms $p_y$ are continuous. (2) $\sigma(E, F)$ is a Hausdorff locally convex topology. (3) A base of $\sigma(E, F)$-closed neighbourhoods of the origin are given by sets of the form $\displaystyle{\{ x : \sup_{1 \leq i \leq n} p_{y_i}(x) \leq 1 \}}$ with $y_1, y_2, ..., y_n \in F$.

If $(E, F)$ is a dual pair then the topological dual of $E$ equipped with $\sigma(E, F)$ is $F$, i.e.:

$(E^{\sigma(E, F)})' = F$

• On the Topologies of the Dual Pair (E, F) page we said that if $(E, F)$ is a dual pair then a Topology of the Dual Pair $(E, F)$ is a topology $\tau$ on $E$ that is locally convex with the property that $(E^{\tau})' = F$. Thus, the weak topology on $E$ determined by $F$ is a topology of the dual pair.
• We then proved a very important result regarding topologies of the dual pair $(E, F)$ and convex sets:

If $(E, F)$ is a dual pair and $A$ is a convex subset of $E$ then the closure of $A$ is the same subset when $E$ is equipped with any topology of the dual pair $(E, F)$.


text:
# Gower's Middel Weie' : the poetic breadth of the Confessio amantis

O'Neill, TC 1994 , 'Gower's Middel Weie' : the poetic breadth of the Confessio amantis', Research Master thesis, University of Tasmania.

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PDF (Whole thesis)
Available under University of Tasmania Standard License.

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## Abstract

The aim of this study is to demonstrate that John Gower's Confessio Amantis is a work of great philosophical and poetic sophistication which is worthy of greater critical attention and esteem than it has so far received. It attempts to do this in a number of ways: firstly, it outlines some of the reasons that Gower's poem has been somewhat neglected; secondly, it looks at Gower within his literary context; thirdly, it examines the poem in the context of the poet's social, religious and political milieaux. By examining the poem from these perspectives, it is hoped that some critically useful indications of the intellectual breadth of Gower's poem will have been delineated.
Chapter One: Place and Time
Other critics have traced the development of Gower's adverse critical reputation, but it is Gower's proximity (poetically, linguistically and socially) to his more famous contemporary, Geoffrey Chaucer, which has contributed most to Gower's denigration. While his works have been more closely scrutinised in recent decades, a full appreciation of Gower's work can be achieved only by examining Gower in his own right. An analysis of the Confessio in relation to Gower's most probable intended audiences indicates a poet striving to produce a work to which a wide range of people could respond to and make use of on a number of levels. Similarly, Gower's view of himself as an author, as indicated by the Prologue of the poem, is further evidence of the breadth and seriousness of his endeavour.
Chapter Two: Modes and Styles
The ways in which Gower chose to frame and to present his ideas are further indications of the poem's sophistication. His use of confession, an analytical dialogue between two people with carefully defined roles, was an innovation which presented many rich poetic and philosophical possibilities. An examination of the role of the sacrament of confession in late fourteenth-century society shows that Gower had chosen a mode of discourse which was not only highly familiar to his audience but which was the most powerful tool for psychological analysis available at that time. Gower makes rich use of that tool to examine his major character, Amans, and, through this 'Everyman' figure, to examine humanity in general. As a 'lover's confession' the Confessio Amantis is an examination of human sexuality on one important level, but Gower simultaneously examines 'love' in its
broader, social aspects and so explores aspects of human nature on a macrocosmic level also.
Chapter Three: Voices and Characters
The focus of this analysis is the central figure, Amans. The way in which Gower presents this figure is vital to an understanding of the breadth of Gower's achievement in this poem. Amans is an Everyman figure both as an archetypal lover and as an archetypal human being. Iconographical and textual evidence is surveyed to examine the presentation of this vital figure, as are relevant calendrical schemes and the topos of the Twelve Ages of Man. The breadth of possible interpretation built into the figure of Amans is another indication of the breadth of meaning in the poem. The relationships between Amans and the other major poetic figures in the work, Genius, Venus and Nature, show that Gower aimed to present a carefully considered, well constructed and intellectually challenging vision of love and its roles in the cosmos.
Chapter Four:- Findings and Outcomes
The confessional dialogue and its conclusion which make up the bulk of the poem are devoted to these microcosmic and macrocosmic concerns. They are framed and complemented by the poem's prologue and epilogue, which reinforce the poem's social and political concerns. Gower was highly concerned with the politics of his time, and wrote the poem to inform those politics by showing a path towards a 'common good'. By examining the political environment in which the Confessio was written, we can get some idea of what motivated him to write this poem and why he conceived of it as he did. The total structure of the poem makes it clear that he wished it to be both a poem which could be enjoyed and a proposal of a 'middle way' which could be used, by people of his own time and in times to come. He attempted to write a poem for everybody. The conclusion of the study serves to indicate further areas of study which this kind of analysis of Gower's poem could make possible.

Item Type: Thesis - Research Master O'Neill, TC Gower, John, 1325?-1408 The Author Copyright 1992 the Author - The University is continuing to endeavour to trace the copyright owner(s) and in the meantime this item has been reproduced here in good faith. We would be pleased to hear from the copyright owner(s). Includes bibliographical references (leaves 102-109). Thesis (M.A.)--University of Tasmania, 1992 View statistics for this item


text:
# distriubation

• July 7th 2007, 10:44 AM
harry
distriubation
Normal distribution raw scores.

Risk taking is an important part of investing. In order to make suitable investment decisions on behalf of their customers, portfolio managers give a questionnaire to new
customers to measure their desire to take financial risks. The scores on the questionnaire are approximately normally distributed with a mean of 51 and a standard
deviation of 14 . The customers with scores in the bottom are 15% described as "risk averse."
What is the questionnaire score that separates customers who are considered risk averse from those who are not?
Carry your intermediate computations to at least four decimal places. Round your answer to at least one decimal place.
• July 7th 2007, 10:53 AM
galactus
If you look up 0.15 in the body of the normal distribution table, we see it corresponds very close to a z-score of -1.0365.

$-1.0365=\frac{x-51}{14}$

Solve for x.


text:
## One and Two Dimensional Isoparametric Elements and Gauss Integration: Gauss Integration

### Gauss Integration

The Gauss integration scheme is a very efficient method to perform numerical integration over regular domains. In fact, if the function to be integrated is a polynomial of an appropriate degree, then the Gauss integration scheme produces exact results. The Gauss integration scheme has been implemented in almost every finite element analysis software due to its simplicity and computational efficiency. This section outlines the basic principles behind the Gauss integration scheme, along with its application in the finite element analysis method. In particular, this section will discuss the development of the reduced-integration elements and the benefits and drawbacks of utilizing them in finite element analysis models.

#### Motivation

The isoparametric elements developed above imply that all the integrated functions are defined such that . In general, numerical integration of such a function has the form:

For the Gauss integration method, is called an integration point and is called the associated weight. Now, if is always affine, i.e., then:

So, for functions that are very close to being affine, a numerical integration scheme with 1 integration point that is with an associated weight of 2 can be employed.

#### Gauss Integration Over One Dimensional Domians

Let . Then, the definite integral of f over the domain can be approximated by the following formula:,/p>

where and is a weight factor associated with each . are termed the integration points. The number of integration points required to obtain sufficient accuracy depends on the shape or form of the function . If is a linear function, then one integration point is enough to exactly integrate . If is a cubic polynomial function, then two integration points are sufficient to obtain the exact solution. Figure 5 shows a graphical representation of the Gauss numerical integration scheme for integrating polynomials up to the fifth degree. Following is a table of the number of integration points required to obtain the exact solution for a polynomial function f with different polynomial degrees:

##### Example

Verify that the exact integral of a general cubic polynomial on the interval can be obtained by the two integration points and with the two weight factors .

##### Solution

Let . The exact integral is equal to:

On the other hand, Gauss integration with two integration points is equal to:

Equating the Gauss integration with the exact solution yields the following four equations:

These four equations are obtained by noticing that the exact solution and the Gauss integration solution should be equal given any values for the coefficients , , , and . The solution to the above four equations yields , , . The Mathematica code to solve the four equations is:

View Mathematica Code

Solve[{w1 + w2 == 2, w1*x1^3 + w2*x2^3 == 0, w1*x1^2 + w2*x2^2 == 2/3,w1*x1 + w2*x2 == 0}, {w1, w2, x1, x2}]

View Python Code

from sympy import *
import sympy as sp
sp.init_printing(use_latex = "mathjax")
x1, x2, w1, w2 = sp.symbols("x_1 x_2 w_1 w_2")
#Sympy has a library for Gauss integration
Eq1 = w1+w2-2
Eq2 = w1*x1**3+w2*x2**3
Eq3 = w1*x1**2+w2*x2**2-2/3
Eq4 = w1*x1+w2*x2
s = solve((Eq1,Eq2,Eq3,Eq4), (x1,x2,w1,w2))
display(s)

Note that this example can be extended to show that Gauss integration in one dimension with points can exactly integrate a polynomial of order .

#### Gauss Integration Over Two Dimensional Domains

Let . Then, the integral of over the domain can be approximated by the following formula:

where is an integration point while and are weight factors associated with and respectively. It can be easily seen that this is a direct consequence of the one dimensional Gauss integration formula as follows:

#### Gauss Integration for Two Dimensional Quadrilateral Isoparametric Elements

As described for the two dimensional linear quadrilateral element, the strains of the 4 node elements are linear, and thus, the entries in the matrix are quadratic functions of position. Thus, a numerical integration scheme is required to produce accurate integrals for the isoparametric 4-node elements (Figure 6). On the other hand, the strains in the two dimensional linear quadrilateral are quadratic functions of position, and thus, the entries in the matrix are fourth-order polynomials of position. Thus, a numerical integration scheme is required to produce accurate integrals for the 8 node isoparametric element (Figure 6). It should be noted that this can extended the three dimensional case in a straightforward manner. The 8 node brick element has 8 integration points, while the 20 node brick element has 27 integration points.

#### Reduced Gauss Integration (Under Integration) for Two Dimensional Quadilateral Isoparametric Elements

Elements that are integrated as per the previous section are termed “full integration” elements. For example, to integrate the stiffness matrix of the 4-node plane isoparametric quadrilateral element, each entry will be evaluated at the four locations () of the integration points in the isoparametric element. Thus, the number of computations (excluding the addition) for the integration of the element are computations. The number of computations for the stiffness matrix of the 8-node plane isoparametric quadrilateral element is 2,304 computations since it uses a Gauss integration points scheme! To save computational resources, a “reduced integration” option for each of those elements is available in finite element analysis software. The reduced integration version of the 4-node plane isoparametric quadrilateral element uses only one integration point in the center of the element (Figure 7), and the number of computations is reduced from 256 to 64 computations. However, this is accompanied by a reduction in accuracy. Figure 7 shows two nonlinear functions to be integrated using one integration point vs. two integration points. In the first case, the function has a small variation along the domain and the integration seems to be accurate. However, in the second case, the function has a large variation and is equal to zero in the centre of the domain. The reduced integration will predict zero stiffness!

For the 8-node plane isoparametric quadrilateral, a Gauss integration points scheme is used instead of the Gauss integration points scheme, which reduces the number of computations from 2,304 to 1,024 computations (Figure 8). The benefit of using reduced integration is not only in saving computational resources but also in balancing out the over-stiffness introduced by assuming a certain deformation field within an element. In fact, reduced integration can lead to better performance in most cases. However, while reduced integration can dramatically reduce the computational time, it can lead to a structure that is too flexible. In fact, the displacements are over predicted when a structure is discretized using reduced integration elements. The reason, as will be shown in the following section, is because some information is lost during sampling of the stiffness at the fewer integration points of the reduced integration elements.

##### Spurious Modes Associated with Reduced Integration Elements

Spurious modes, or sometimes called zero energy modes, are combinations of displacements that in general produce strains; however, such strains cannot be detected at the integration or sampling points. So, the internal energy calculated at the integration points is equal to zero for such mode. In other words, a very small amount of load would be enough to produce a very large displacement. If a spurious mode exists for the whole structure, it becomes unstable. Usually spurious modes exist in under-integrated element (reduced integration). This is because the reduced number of integration points cannot detect a spurious strain mode. The two very common spurious modes are those for the bilinear quadrilateral (4-node plane element) and the quadratic displacement quadrilateral (8-node plane element). In general, any bending mode is a spurious mode in a 4-node reduced-integration linear element. The strain modes shown cannot be detected at the integration point; the strain at the center of the element (at the location of the single integration point) is equal to zero (Figure 9). On the other hand, the spurious mode of an 8 node reduced integration quadratic element is called “the hour glass” mode (Figure 10) and is produced with the following configuration of nodal displacements:

The spurious modes will be illustrated using the following problem.

##### Example

Find the internal energy associated with the shown displacement for the 4-node plane stress quadrilateral element shown below, assuming a bilinear displacement interpolation function.

Use exact integration, full integration ( integration points), and reduced integration (1 integration point).

##### Solution

The shape functions of the shown element are:

The displacement functions and in the directions and , respectively, are:

Therefore, the strains have the form:

The stress strain relationship for plane stress is given by:

The stiffness matrix can be evaluated using the following integral where is the element thickness:

Notice that the system has only one degree of freedom (the unknown displacement variable ), and therefore, the stiffness matrix has the dimensions .

Using Exact Integration:
The stiffness matrix evaluated using exact integration is:

The strain energy associated with this displacement mode is equal to:

Using Full Integration:
The stiffness matrix evaluated using full integration is:

where and which gives the same results as the exact integration.

Using Reduced Integration:
The stiffness matrix evaluated using reduced integration is:

where and . The strain energy associated with this displacement mode is equal to zero. Thus, the reduced integration scheme predicts that the displacement field shown does not require any external forces applied to the element. Thus, this element is unstable if loads produce the displacement mode shown, i.e., bending! The following is the Mathematica code used for the above calculations:

View Mathematica Code

N1=(1-xi)(1-eta)/4;
N2=(1+xi)(1-eta)/4;
N3=(1+xi)(1+eta)/4;
N4=(1-xi)(1+eta)/4;
B={D[N1-N2+N3-N4,xi],0,D[N1-N2+N3-N4,eta]};
Cc=Ee/(1+nu)/(1-nu)*{{1,nu,0},{nu,1,0},{0,0,(1-nu)/2}};
Kbeforeintegration=FullSimplify[B.Cc.B]
(*Using Exact Integration*)
Kexact=Integrate[Kbeforeintegration,{eta,-1,1},{xi,-1,1}]
Energyexact=us*Kexact*us/2
(*Using Full Integration*)
etaset={1/Sqrt[3],-1/Sqrt[3]};
xiset={1/Sqrt[3],-1/Sqrt[3]};
Kfull=FullSimplify[Sum[(Kbeforeintegration/.{xi->xiset[[i]],eta->etaset[[j]]}),{i,1,2},{j,1,2}]]
Energyfull=us*Kfull*us/2
(*Using reduced Integration*)
etaset=xiset={0};
Kreduced=Sum[4(Kbeforeintegration/.{xi->xiset[[i]],eta->etaset[[j]]}),{i,1,1},{j,1,1}]
Energyreduced=us*Kreduced*us/2

View Python Code

from sympy import *
import sympy as sp
sp.init_printing(use_latex = "mathjax")
xi, eta, nu, E, us = sp.symbols("xi eta nu E u_s")
N1=(1-xi)*(1-eta)/4
N2=(1+xi)*(1-eta)/4
N3=(1+xi)*(1+eta)/4
N4=(1-xi)*(1+eta)/4
B = Matrix([(N1-N2+N3-N4).diff(xi),0,(N1-N2+N3-N4).diff(eta)])
Cc = E/(1+nu)/(1-nu)*Matrix([[1,nu,0],[nu,1,0],[0,0,(1-nu)/2]])
display(B,Cc)
KBeforeintegration = simplify(B.transpose()*Cc*B)
display("K before integration: ", KBeforeintegration)
display("Using Exact Integration")
etaset = Matrix([1/sqrt(3), -1/sqrt(3)])
xiset = Matrix([1/sqrt(3), -1/sqrt(3)])
Kfull = 0
for i in range(2):
for j in range(2):
Kfull += KBeforeintegration[0].subs({xi:xiset[i], eta:etaset[j]})
display(simplify(Kfull))
Energyfull = us*Kfull*us/2
display(simplify(Energyfull))
display("Using reduced integration")
etaset = xiset = Matrix([0])
Kreduced = 4*KBeforeintegration[0].subs({xi:xiset[0], eta:etaset[0]})
display(Kreduced)
Energyreduced=us*Kreduced*us/2

#### Gauss Integration for Two Dimensional Triangle Isoparametric Elements

The strains of the 3 node triangle element are constant, and thus, the entries in the matrix B^TCB are constant. Thus, a 1 point numerical integration scheme is required to produce accurate integrals for the isoparametric 3-node elements (Figure 11). For the 6 node triangle element, a 3 points numerical integration scheme is required to produce accurate integrals (Figure 11). It should be noted that the shown locations for the integration points and their number can be extended to the three-dimensional case in a straightforward manner.

### Final Notes

The use of isoparametric elements and numerical integration dramatically increases the robustness of the finite element analysis method. The techniques described in the previous sections allow meshing of irregular domains with compatible elements, i.e., elements that ensure the continuity of the displacement field across boundaries (How?). However, to ensure the continuity of the displacement field, the elements of the mesh should be of the same type; otherwise, the user has to apply mesh constraints to ensure the compatibility of the displacement between elements. For example, a 4-node bilinear displacement element when connected to an 8-node quadratic displacement element will have incompatible displacements at the side of the connection. This can be overcome by constraining the mid-side node of the quadratic element to always lie on the straight side of the bilinear element. Otherwise, the results at the location of discontinuous displacement fields would be erroneous.

The stiffness matrix and the nodal loads of the structure can be calculated using the Gauss numerical integration scheme, which speeds up the computational time required. However, the price of using the Gauss numerical integration is the loss of accuracy. Gauss numerical integration produces accurate results only when the integrals are exact polynomials. The integrals are exact polynomials only when the elements of the mesh are exactly aligned with the isoparametric mapping into the coordinates of and . When the shape of the elements in the mesh deviates from the shape of the element in the coordinate system of and , the functions to be integrated are no longer exact polynomials but rather radicals; thus, the numerical integration scheme is no longer accurate. Most finite element analysis software give warnings when elements are highly distorted since this will lead to inaccurate computations of the stiffness matrix during the numerical integration process. In addition, for nonlinear problems when the stiffness changes with deformation, it is possible that the element will reach a highly distorted shape that the numerical integration scheme is no longer valid.


text:
3.6 Friction  (Page 5/10)

 Page 5 / 10

When you learn to drive, you discover that you need to let up slightly on the brake pedal as you come to a stop or the car will stop with a jerk. Explain this in terms of the relationship between static and kinetic friction.

When you push a piece of chalk across a chalkboard, it sometimes screeches because it rapidly alternates between slipping and sticking to the board. Describe this process in more detail, in particular explaining how it is related to the fact that kinetic friction is less than static friction. (The same slip-grab process occurs when tires screech on pavement.)

Problems&Exercises

A physics major is cooking breakfast when he notices that the frictional force between his steel spatula and his Teflon frying pan is only 0.200 N. Knowing the coefficient of kinetic friction between the two materials, he quickly calculates the normal force. What is it?

$5.00 N$

(a) When rebuilding her car’s engine, a physics major must exert 300 N of force to insert a dry steel piston into a steel cylinder. What is the magnitude of the normal force between the piston and cylinder? (b) What is the magnitude of the force would she have to exert if the steel parts were oiled?

(a) What is the maximum frictional force in the knee joint of a person who supports 66.0 kg of her mass on that knee? (b) During strenuous exercise it is possible to exert forces to the joints that are easily ten times greater than the weight being supported. What is the maximum force of friction under such conditions? The frictional forces in joints are relatively small in all circumstances except when the joints deteriorate, such as from injury or arthritis. Increased frictional forces can cause further damage and pain.

Suppose you have a 120-kg wooden crate resting on a wood floor. (a) What maximum force can you exert horizontally on the crate without moving it? (b) If you continue to exert this force once the crate starts to slip, what will the magnitude of its acceleration then be?

(a) 588 N

(b) $1\text{.}\text{96 m}{\text{/s}}^{2}$

(a) If half of the weight of a small $1\text{.}\text{00}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{kg}\phantom{\rule{0.25em}{0ex}}$ utility truck is supported by its two drive wheels, what is the magnitude of the maximum acceleration it can achieve on dry concrete? (b) Will a metal cabinet lying on the wooden bed of the truck slip if it accelerates at this rate? (c) Solve both problems assuming the truck has four-wheel drive.

A team of eight dogs pulls a sled with waxed wood runners on wet snow (mush!). The dogs have average masses of 19.0 kg, and the loaded sled with its rider has a mass of 210 kg. (a) Calculate the magnitude of the acceleration starting from rest if each dog exerts an average force of 185 N backward on the snow. (b) What is the magnitude of the acceleration once the sled starts to move? (c) For both situations, calculate the magnitude of the force in the coupling between the dogs and the sled.

(a) 3.29 m/s 2

(b) 3.52 m/s 2

(c) 980 N; 945 N

A freight train consists of two $8\text{.}\text{00}×{\text{10}}^{5}\text{-kg}$ engines and 45 cars with average masses of $5\text{.}\text{50}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{kg}$ . (a) What force must each engine exert backward on the track to accelerate the train at a rate of $5\text{.}\text{00}×{\text{10}}^{-2}\phantom{\rule{0.25em}{0ex}}m/{s}^{2}$ if the force of friction is $7\text{.}\text{50}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}N$ , assuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently trains are very energy-efficient transportation systems. (b) What is the magnitude of the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other), assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines?

(a) $1\text{.}\text{03}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}N$

(b) $3\text{.}\text{48}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}N$

how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
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How to be sure that a law is invariant under Lorentz's Transformation?

For starters let's talk about Maxwell's Equations; we know that Maxwell's Equations are invariant under Lorentz's Transformation, after all this is why all the relativity business got started. To state that a law is invariant under Lorentz's transformation is equivalent to state that the same law is true for any observer in any reference frame (that is if I understand the meaning of the word invariant correctly in this context).
Problem is that this isn't manifestly true for Maxwell's Equation; to solve this problem we write Maxwell's equations using tensors, this kind of evidently invariant formulation of Maxwell's Equation is so nice that it deserves its own name: Covariant formulation of classical electromagnetism. (To be precise this term refers to the expression of all laws of electromagnetism in an invariant way, but still Maxwell's Equations are the main ones.)
But why writing a law using tensors should imply that that law is surely invariant under Lorentz's transformation? This is because: tensors are invariant under any coordinate transformation1, so any law wrote in the form: tensor equals tensor is sure to be invariant under any transformation, Lorentz's one included.(Lorentz's transformation operates on four vectors ecc. So on tensor with indices scaling from $$0$$ to $$3$$, so of course the tensors in the manifestly invariant laws should scale between the same range.)

Also, and maybe mainly, I don't get why the invariant formulation of electromagnetism is called covariant; seems to me that the term covariant is out of place here.

And at last: if my reasoning is correct laws written in the form: tensor equals tensor should be simultaneously invariant under any kind of coordinate transformation, not just the Lorentz's one; so even if Lorentz's transformation were to have another completely different form still laws written with tensors, such as Maxwell's equation, would be invariant anyway. This seems really strange to me, is this true? I mean: this would mean that for example Maxwell's equation would be invariant no matter the kind of transformation; so there would be no special relation with Lorentz's transformation whatsoever.

[1]: Of course the components of the tensor are not constant under an arbitrary transformation, but the components change in such a way to compensate the change of the basis vectors, so the tensor overall remains the same.

Suppose you have two systems $$S$$ and $$S'$$ in each system we have coordinates and vector potentials $$x^\mu$$, $$x'^\mu$$, $$A^\mu$$ and $$A'^\mu$$. Since both $$x$$ and $$A$$ are vectors we know they transform like:

$$x^\mu\rightarrow\left(\Lambda^\mu{}_\nu\right)x^\nu$$,

this also gives us the transformation law for the derivatives:

$$\frac{\partial}{\partial x^\mu}\rightarrow \left(\Lambda^{-1}\right)_\mu{}^\nu\frac{\partial}{\partial x^\nu}$$

Which is the same as the transformation of $$x_\mu$$. Now if you consider a law like Maxwell's Equations:

$$\partial_\mu F^{\mu\nu}-\mu_0J^\nu=0$$

This will transform like:

$$\partial'_\mu F'^{\mu\nu}-\mu_0J'^\nu=\left(\Lambda^{-1}\right)^\rho{}_\nu\left(\partial_\mu F^{\mu\nu}-\mu_0J^\nu\right)=0$$

Which means that both equations written in terms of the coordinates and vectors of each system predict the same behavior, because the equations in the primed system can be shown to reduce to those in the unprimed system.

The discussion before was mainly for Special Relativity, or linear coordinate transformations. The fully covariant equations for a generic coordinate transformation are slightly more complex Maxwell's Equations in Curved Spacetime, but the underlying idea is the same, the root of the issue lays on the fact that derivatives do not transform like $$x_\mu$$ in curvilinear coordinates, they require an extra piece to transform like a properly Covariant Derivative.

But why writing a law using tensors should imply that that law is surely invariant under Lorentz's transformation?

The law itself is not invariant, it will transform accordingly, if it has one index it will transform like a vector, and so on. But the key point is that the law in one system predicts he same as the law in any other.


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###### Question:

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##### 3.25 Determine the response function due to the input function for one of the systems shown in pa...
3.25 Determine the response function due to the input function for one of the systems shown in parts of (a)of Fig. 3.22. Eaclh system is quiescent at t- 0. Use an input function f(t) and part 6) K 100 N -| C 4 kg/s y(t) F(t) f(t) f(t) (3) 10 2π ft) (5) 사 4π 3.25 Determine the response ...
##### H Exercise 3.2.7: For which h is 0 h 6 not invertible? Find all such h 1 h
h Exercise 3.2.7: For which h is 0 h 6 not invertible? Find all such h 1 h...
##### A healthcare provider’s order sheet contains the following entry: lanoxin (digoxin) 750 mcg PO daily. What...
a healthcare provider’s order sheet contains the following entry: lanoxin (digoxin) 750 mcg PO daily. What is the dose A healthcare provider's order sheet contains the following entry: Lanoxin (digoxin) 750 mcg PO daily. What is the dose? O A. 750 grams OB. 750 micrograms O C. 75 micro...

##### Identify the reagents needed for the following transformations. Identify the reagents needed for the following transformations....
Identify the reagents needed for the following transformations. Identify the reagents needed for the following transformations. -CH2CH3 ОН OEM NH2 NH2...
##### Statistics 214 Section 5. Thousands of elementary school students in a region were recently given a...
Statistics 214 Section 5. Thousands of elementary school students in a region were recently given a nationwide standardized exam to test their writing skills. In a simple random sample of 280 students, 192 students passed this exam. a) Construct a 99% confidence interval for the proportion of elemen...