# FINITE ELEMENT METHOD PDF

Lecture Notes: The Finite Element Method. Aurélien Larcher, Niyazi Cem De˜ girmenci. Fall Contents. 1 Weak formulation of Partial Differential Equations. INTRODUCTION TO THE FINITE. ELEMENT METHOD. G. P. Nikishkov. Lecture Notes. University of Aizu, Aizu-Wakamatsu , Japan. eral computer programs for finite element analysis of structural and non-structural The analysis was done using the finite element method by K. Morgan.

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The Finite Element Analysis (FEA) is a numerical method for solving problems of engineering and mathematical physics. Useful for problems with complicated. Finite Element Analysis as an Integral Part of Computer-Aided Engineering Formulation of the Finite Element Method-Linear Analysis in Solid. OUR BASIC AIM has been to present some of the mathematical as- pects of the finite element method, as well as some applications of the finite element method .

This comprehensive new two-volume work provides the reader with a detailed insight into the use of the finite element method in geotechnical engineering. As specialist knowledge required to perform geotechnical finite element analysis is not normally part of a single engineering degree course, this lucid work will prove invaluable. It brings together essential information presented in a manner understandable to most engineers. Volume 1 presents the theory, assumptions and approximations involved in finite element analysis while Volume 2 concentrates on its practical applications. Back to Book Listing. Buy this book in print.

Back to Book Listing. Buy this book in print. David M. View Chapters. Select All. For selected items: Table of Contents. Geotechnical analysis. Finite element theory for linear materials. Dr Margetts' main areas of expertise are in structural mechanics, geotechnical engineering, high performance computing and tomographic imaging. His main research activities concern investigating how real materials behave both organic and inorganic and how this can be simulated using computers.

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## Introduction to Finite Element cittadelmonte.info | Pham Ngoc Tinh - cittadelmonte.info

First published: Print ISBN: About this book Many students, engineers, scientists and researchers have benefited from the practical, programming-oriented style of the previous editions of Programming the Finite Element Method, learning how to develop computer programs to solve specific engineering problems using the finite element method.

The direct sense is simple as it is the result of the construction of a solution to the variational formulation VP using a given solution to the continuous problem CP. The reciprocal is now considered. The integration-by-parts formula used in the reverse order to the one that yielded the variational formulation VP leads to the following: In that case, the equation 3.

The essential points for the continuation of the demonstration are: Once it is proved that the differential equation of the problem CP is satisfied by the solution to the variational problem VP , the family of equations 3.

In that case, equation 3. Once again, definition 3. Moreover, the expressions given by 3.

That is why the results demonstrated in the Dirichlet problem are directly reused in order to exploit them directly in the Fourier-Dirichlet problem. This discretisation is written as: The result of this substitution is that the finite differences scheme obtained cor- responds exactly to the nodal equation 3.

Moreover, the finite differences scheme 3. This equation is written as: This expansion is written as: Actually, interest is axed on the solutions to the following continuous problem: Show that the continuous problem CP can be expressed as a variational formulation VP under the form of: What is its precision order?

For reminder, the trapezium quadrature formula is expressed as: The corresponding equation of the VP pressed as: The corresponding equation to the VP expressed as: The differential equation of the continuous prob- lem CP is multiplied by v and integrated upon [0, 1] interval: Thus, the result obtained is: The result obtained is that u is solution to the following formal variational for- mulation: In other words, from then on, take the test functions v — and from there, solution u — as belonging to H 1 0, 1.

Considering the periodic boundary conditions 3. To achieve this, the space Hper1 0, 1 is fitted with the H 1 -norm 3. It is established that Cf.

Moreover, since vn converges towards v in H 1 0, 1 , it is inferred that Cf. Therefore, a unique solution exists to the variational formulation VP that be- 1 0, 1. Effectively, let u be a solution to the problem VP and the result is: The rest of the demonstration remains unchanged. In other words, the following expression is finally obtained: Int , for all values of i varying from 1 to N.

VP For the remainder, the same formalism, as considered for of the Dirichlet, Neu- mann and Fourier-Dirichlet problems, is observed. Hence, direct use is made of the results obtained while solving these problems for the calculation of the coefficients of matrix Ai j, as well as from the second mem- ber bi.

In other words, the following formula is obtained: This substitution immediately leads to the nodal equation 3. This is why the generic equation of system 3. By adding the two nodal equs. Thus we obtain: What is the order of its precision?

## Introduction to the Finite Element Method for Structural Analysis

It is pointed out that the trapezium quadrature formula applied to a triangle T , whose vertices are given as A1 , A2 and A3 is written as: Formula 4. This operation yields the equation below: Equation 4.

On sev- eral occasions cf. This is precisely what has been assumed at the beginning of this problem. Concerning the two terms that are under the integral of the quantity a u, v , it is seen that only linear combinations of partial derivatives of the first order appear in the expression of a u, v.

The Cauchy-Schwartz inequality thus enables one to write in a generic manner: In general, more of the a. Theorem 10 has to be applied. In this case, it is necessary to establish that form a. From then on, a V-elliptical form is automatically positive.

In addition, D. Euvrard [4] demonstrates the equivalence between MP and VP u problems under previously inventoried conditions. Thus the formula below is obtained: To achieve this, it is only necessary to replace the equilibrium equation 4. Subsequently, the boundary conditions 4. To achieve this, consider an arbitrary field of vectors v belonging to a functional space Vd that will be specified later.

To be more precise, it is the evaluations of the integrals by triangle that will be directly used in the solution below. This was done by systematic exploitation of the local numeration presented in the statement, cf.

Then the following formula is thus obtained: This choice is guided by two reasons: Since a second order approximation is sought, the developments have to be writ- ten up to the fourth order. Moreover, in order to simplify writings, the notation convention will be as fol- lows: Finally, the fact that partial derivatives, of the order O h2 , have been ignored in all approximations, the finite differences scheme is globally of the second order 2. In fact, the following is obtained: As for the developments shown in the answer to question 12, the following nota- tions will be used in order to simplify writings: The following is thus obtained: The formula is stated as follows: Then make explicit that part of system 4.

Equation associated with a characteristic basis function of a node interior to seg- ment OA. That is the reason why functions v of V are required to meet the following bound- ary conditions: By taking into account the boundary conditions 4. The integral on the left side of equation 4. Likewise, by using once again the Cauchy-Schwartz inequality, the member on the right side of equation 4. In fact, the second partial derivatives of a function whose degree is less or equal to one with respect to the ordered pair x, y being nil, it follows that the Laplacian of such a function is alike!

Thus, the use of such finite elements to numerically solve the problem VP by variational approximation is not really recommended since the right side of 4. In fact, it is only neces- sary to inject of the function change 4. Laplace equation: The other equations forming the rest of problem CP2 are trivial.

In order to obtain a variational formulation VP2 associated with the continuous problem CP2 , multiply each of the partial differential equations of continuous problem CP2 by its corresponding test function and obtain: In fact, concerning the normal differential coefficient of u, given the fact that the values of u do not intervene at all in the integrals of double formulation 4. Moreover, given the bilinearity of form a. To make that happen, it is observed that having considered a regular mesh, cf.

Then the following is obtained: These calculations are conventional and may be consulted, for further details, in the work of D. Euvrard, [4]. The generic equation 4.

In other words, it is a rectangular linear system. Thus, not having as many equa- tions as unknowns, it cannot be solved numerically in an autonomous manner.

But hold on since each thing is performed in its own time. Fig 4. In this case, only the functions of nodes locally numbered 0, 1, 2, 3 and 4 display a potential contribution in the writing of the nodal equation associated with the characteristic function of node 0. In other words, the nodal equation is in this case written as: The same thing is done for the approximation function u. Finally, the nodal equation is written as: It consists in discretising successively two second-order Laplacians and one Neumann condition.

Problems [3. Chapter 5 Finite Elements Applied to Strength of Materials Preamble This chapter is dedicated to the application of the finite elements method within the framework of strengths of materials.

The main objective of this chapter is to clear up, as much as possible, a state of confusion that predominates within the community of graduate students in mechan- ics, physics, and also within certain graduate schools of engineering. Let there not be the slightest ambiguity. The finite elements method though ap- plied to solid mechanics is, and should be standardized for the benefit of students on one hand, and for its own further use in applications that can only benefit from its practical and indisputable performance and flexibility on the other hand.

In order to set up this standardization, the aim of this presentation is to propose and expound, within the framework of the beam theory, a double application of the finite elements method. O; X1 refers to the axis of the beam and the current abscissa is x.

In order to propose the application of this method as a complementary to the one exposed in the previous chapters, this approximation will be worked out by underlining the assembly technique on one hand and by applying it in the particular framework of approximation of the minimization problem MP on the other hand.

The global framework of the approximation is that of the finite elements P1 and a regular mesh of the interval [0, L] having constant step h is considered, such as: Thus, the following is written: Qualitatively explain the assembly technique inferred from it.

Therefore, the evaluation of the virtual work of internal forces can be performed: In the same way, the virtual work of external forces is evaluated: The variational problem EVP is obtained: From the variational formulation 5.

To achieve this, considering that the density f is a given function belonging to L2 0, L , the application of the Cauchy-Schwartz inequality to the integral of equ. It can further be remarked that the functions of H 2 0, L being C1 on [0, L] cf. The continuous problem CP is then solved in two steps: Firstly, the equ. For such functions, equ. The fact that D 0, L is dense in L2 0, L is then used: Thus, when g is fixed, while remaining an undefined value , in L2 0, L , the sequence gn of D 0, L defined by 5.

In fact, it would be convenient for 5. This easily yields the differential equation of the continuous problem CP. To achieve this, the left member of equ. As mentioned earlier, the differential equation of the continuous problem CP is then obtained immediate; it only requires the choice of a function among all the functions g belonging to L2 0, L that satisfies 5.

The variational equality 5. Indeed, it only suffices to revert to formulation 5. Eu- vrard, [4] or P. Raviard, [7]. Indeed, it suffices to state the formula below in the framework of the variational formulation 5. The approximate minimisation problem MP is therefore written as: Therefore, a necessary condition of minimisation of J is written as: These relationships are then expressed as: In addition, there is: Start assembling with the first element [x0 , x1 ].

In other words, in this case, the elementary matrix a 1 is degenerate. In fact, in this particular case, the following is obtained: This contribution is exactly equivalent to b2 as defined in 5. The second element [x1 , x2 ] is now examined.

The final result corresponds to the following matrix A: In other words, the integrands bear on constant functions by mesh. To ensure matters, retrieve expression 5. It is im- mediately obvious that the nodal equation 5. As for the nodal equation 5. From equ. This justifies why u1 is used to refer to the displacement at this end of the beam in relation to the fixed frame. The formulation of minimisation problem 5. The potential energy of minimisation problem, 5.

Solving the linear system 5. It is one of the main differentiation points, in relation to the finite differences method, producing a sequence of approximations at points fixed on a discretisation grid, namely on the nodes of a predefined mesh of the beam. To do this, just revert to the global matrix as well as to the second member 5. Density f2 is given and has all properties of functional regularity so that the integration calculations of the first two questions of the theoretical part may be per- formed.

Thus, let the regular mesh of the interval [0, L], having a constant step h, be such as: Let MP , the approximate minimization problem associated with problem MP be defined by: Similarily, the virtual work of the external forces is obtained from: To achieve this, first of all the variational equation 5.

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The variational problem EVP is then written as: A reasonable functional framework is now defined to give a sense to the writing up of problem 5. Under these conditions, it is easily noticed that the integral equation 5.

## Table of Contents

Then, consider the particular case where equ. The particular formulation of 5. Finally, the following is obtained: At last, having the two differential equations of the continuous problem CP , the equ. Then, each equation is integrated within [0, L] to obtain: Then, an integration by parts of the integral equations 5.

In order to obtain the variational formulation VP defined by 5. Euvrard [4] may be consulted to make a list of the whole properties — so as to obtain, by equivalence, the mini- mization problem MP defined by: To ascertain that, it is only necessary to proceed to the visualization of such functions cf.

That is why the finite sums intervening in equs. Furthermore, the trapezium quadrature formula is used to evaluate the integrals that occur in equs. Hence the equs. Therefore, the common regressive finite difference is considered: To achieve this, it suffices to use the approximations in the equ. To constitute the global matrix of the linear system 5.

The assembly process is initiated by considering the mesh [x0 , x1 ]. This sub-matrix is presented in bold characters in the elementary matrix A 1 , as shown below: Consequently, by maintaining the norm of bold characters for writing down the coefficients of matrix A 2 that would be considered during the assembly process, the matrix in question is written as: To easily visualise the global matrix structure, three levels of analysis are pro- posed: Left upper corner of the stiffness matrix: The density f2 is given and has all the functional regularity properties so as to perform the integration calculations of the first two questions of the theoretical part.

To achieve this, the Hermite finite elements can be applied as follows: Let the regular mesh be at interval [0, L] and of constant step h, such as: For reminder, the Simpson formula is expressed as: In an analogous way, the virtual work of external forces is obtained from: To achieve this, formulation 5. The reader may refer to the elaboration of the density method as applied in the case of the problem of a beam subjected to traction [5.

The following is then easily obtained: In other words, u is the solution of the continuous problem CP defined by 5. Finally, it can be noted that when the distribution of forces f shows more regu- larity, at least continuous along the interval [0, L], the solution u of the continuous problem CP is then the classical solution belonging to C4 ]0, L[.

The fourth orde r differential equation 5. The following formulation is then found: Firstly, it can be observed that subsequent to the double integration by parts, variational equation 5. In order to record this information in the variational formulation VP , it is com- pulsory that the functions v of V area of investigation of solution u satisfy the following boundary conditions: Consequently, equation 5. The variational formulation VP as defined in 5.

Thus, the variational formulation can be written in a generic form: These properties ensure that an equivalent minimisation problem defined by MP exists, cf. Raviart [7] or Euvrard [4]. Furthermore, following the derivation of 5.

It would be observed that such a result is essentially based on properties 5. These four functions are third degree polynomials and must satisfy properties 5.

The following is finally obtained: Then x0 , x1 , x2 and x3 are the four nodes of the mesh resulting from this three-meshed discretisation. In this case, only the node at abscissa x1 makes a contribution to the global ma- trix A.

Elements of the elementary matrix a 1 see definition 5. Thus, after integrating the contribution of the first mesh [x0 , x1 ], the second mem- ber b is written as: Thus, the elementary matrix a 2 , relative to element [x1 , x2 ] is full and is written according to definition 5.

This contribution is shown in bold characters in matrix A, the terms from the elementary matrix a 1 being neutralized in normal font size: In the present case, there is a situation of symmetry in relation to the one pre- sented for mesh [x0 , x1 ]. In fact, in the present case, it is node x3 that is restrained and only the degrees of freedom of the node at abscissa x2 i.

The corresponding analytical solution produced by a computational solver is given by: The variational formulation VP is written according to formula 5. In order! Then, it is only necessary to note that the 2N equations parameterised by i in formu- lation 5.

This distinction then strictly yields the same formulation as that of the system of equations 5. Wherefrom, it is inferred that, in the case of a discretisation with three meshes, the nodal equations of the approximate variational formulation VP! Given the properties of the bilinear form a. In other words, only the formal aspects of the variational formulations and of the numerical application of the finite elements are considered in all that will follow.

Statement 1 Here, the scalar function u of variables x,t is of interest as solution to the fol- lowing partial differential equation: Show that the CP problem can be expressed in the following variational formulation VP: Find u belonging to V solution of: To achieve this, a regular mesh of constant step h is introduced at interval [0, L], such that: What is its order?

It is reminded that the trapezium quadrature formula is expressed as: The corresponding equation to the system DS is then expressed as: DS2 6. The interested reader may consult the work of D. Euvrard [4] for an elemen- tary presentation intended for mechanics or physics graduate students.

The work of Edwige Godlewski and Pierre-Arnaud Raviart provide further in depth studies requiring a good command of the basic techniques in functional analysis [6]. To achieve this, consider test functions v, defined on [0, L] and having real values.

In other words, test functions v are a function of the only space variable x. Then, the equation with partial derivatives of the continuous problem CP de- fined in 6. It would be noticed that the variational formulation in space VP is only formal, insofar as the functional framework V , in which this formulation makes sense, was completely omitted. To ascertain that, it is only necessary to proceed to the visualization of such func- tions.

Then, by introducing notations 6. Fig 6. Thus, the following pairs of indices to be considered are already available: Then, the values of index k are determined for each of these pairs — those likely to produce non-zero terms in the non-linear system DS.

Likewise, the pair i, i requires consideration of the following triplet of indices: All coefficients that can produce non-zero terms in the non-linear system DS are grouped below: Calculation of the three coefficients Bi j may be performed either by the exact method or by approximation via the trapezium quadrature formula, insofar as the latter is exact on the constant functions. Then the following is then obtained: Starting with a qualitative observation: The results obtained from 6.

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