The Finite Element Analysis (FEA) is a numerical method for solving problems of engineering and mathematical physics. Useful for problems with complicated. Finite Element. Procedures. Klaus-Jurgen Bathe. Professor of Mechanical Engineering. Massachusetts Institute of Technology. Second Edition. (). Engineering Design and Rapid Prototyping. Instructor(s). Finite Element Method. January 12, Prof. Olivier de Weck. Dr. Il Yong Kim.

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Lecture Notes: The Finite Element Method. Aurélien Larcher, Niyazi Cem De˜ girmenci. Fall Contents. 1 Weak formulation of Partial Differential Equations. transformed into a discretized finite element problem with unknown nodal values. For a linear problem a system of linear algebraic equations should be solved. eral computer programs for finite element analysis of structural and non-structural The analysis was done using the finite element method by K. Morgan.

Skip to main content. Log In Sign Up. Finite Element Methods for Engineering Sciences. Duy Hao Dang. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, , in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. Cover design: For all those years of sacrifice and collusion. Preface This book is dedicated to the use of the finite elements method for the approximation of equations having partial derivatives. Numerical Analysis has undergone a spectacular development during these past forty years.

To evaluate the extent to which performance measures are affected by initial model input, we tested the sensitivity of bite force, strain energy, and stress to changes in seven parameters that are required in testing craniodental function with FEA.

Results showed that unilateral bite force outputs are least affected by the relative ratios of the balancing and working muscles, but only ratios above 0. The constraints modeled at the bite point had the greatest effect on bite force output, but the most appropriate constraint may depend on the study question.

Strain energy is least affected by variation in bite point constraint, but larger variations in strain energy values are observed in models with different number of tetrahedral elements, masticatory muscle ratios and muscle subgroups present, and number of material properties.

These findings indicate that performance measures are differentially affected by variation in initial model parameters.

In the absence of validated input values, FE models can nevertheless provide robust comparisons if these parameters are standardized within a given study to minimize variation that arise during the model-building process. Sensitivity tests incorporated into the study design not only aid in the interpretation of simulation results, but can also provide additional insights on form and function.

Introduction Finite element analysis FEA , the discretization of structures and approximation of their mechanical behavior the response of structure to load , has traditionally been an analytical technique in the engineering disciplines as an important component of the development process to improve design.

More recently, however, its use in functional studies of biological structures has become more common [1] — [4]. FEA has been applied in vertebrate biomechanics research across diverse taxonomic groups, including crocodiles [5] , non-avian dinosaurs [6] — [11] , birds [12] , lizards [13] , [14] , fishes [15] , and a variety of mammals [16] — [26]. FEA complements in vivo experimental studies by allowing simulations using user-defined input assumptions regarding the study system, which could otherwise be impossible to implement.

Currently, most studies of this type address the mechanical behavior of the craniodental system. Given the diverse functional questions that could be examined using the FE approach, the current available data from FE publications are largely incomparable across studies precisely because of the comparative nature of current applications.

Even within narrow clades of closely related genera and species, lack of absolute values from FE results means published stress and strain values cannot be used to evaluate relative performance of models across different studies e. In many studies, different approaches in how muscles and constraints are modeled also make comparisons difficult.

Furthermore, the diversity of taxonomic groups that can potentially be studied using this technique, accompanied by the different software programs and protocols used by researchers in FE model construction, further complicates any attempts at the synthesis of current FE knowledge across vertebrate groups.

The current diversity of input assumptions in FE models used in comparative biology suggests a need to quantify the sensitivity of performance measures to those parameters, in order to build a general context for comparing results within and across different studies.

Several previous studies have addressed the choice of model parameters and their implications for comparing FE analytical results to those obtained from in vivo experiments for masticatory muscle forces [27] , bite forces [28] , and elastic bone properties [29]. Chapter 1, theorem 4 , being given that u belongs to H 2 0, 1 and v to H01 0, 1. The reciprocal is then established. The integration formula is used by parts in the reverse order to the one used to obtain the variational formulation.

This results in: The density theorem 2 is then used as follows, rewriting equ. It is then deduced that: This implies that the space is isomorphic to RN. Moreover, the approximation expressions given by 3. The generic equation of system 3. Thus, the basis functions concerned are: The integrals bearing on those derivatives can then be calculated either exactly or by using the trapezium quadrature formula being exact for constants func- tions.

This substitution leads to an exact correspondence between the scheme with finite differences and the nodal equation 3. It is obvious that the scheme with finite differences 3. This is the reason why the scheme having finite differences 3. Thus, let u be a function of the real variable defined on [0, 1] and has values in R.

The interest is on the solution to the continuous problem CP defined by: Show that the continuous problem CP can be written in a variational formulation VP like the following: What is its order of precision? Remember that the trapezium quadrature is written as: As already mentioned in the presentation of the Dirichlet problem see para- graph [3.

The differential equation of the continuous problem CP is multiplied by v then integrated over the interval [0, 1]. As a result and by considering the above two boundary conditions, the following formulation is obtained: Find u belonging to V being the solution to: It is then observed that this variational formulation is strictly analogous to the one obtained within the framework of the Dirichlet problem — see paragraph [3. That is the reason why, if the functional analysis presented in paragraph [3.

To make that happen, choosing one norm to be defined on the functional space H 1 0, 1 represents one of the key points in the application of the Lax-Milgram Theorem. Then the choice is to measure the dimension of functions v belonging to H 1 0, 1 by the natural norm defined by: The following is a point-by-point summary of this verification: The application of the Lax-Milgram theorem thus implies the existence of one and only one function u belonging to H 1 0, 1 , the solution to the variational prob- lem VP defined by 3.

In fact, as mentioned above, achieving complementary regularity results for weak solutions to the variational problem VP may require sufficiently sophisticated mathematical tools. So, let u be an element of H 1 0, 1 , solution to the variational problem VP , and the following is obtained: To make that happen, the lemma 2 would be used. Then, lemma 1 may be applied to the variational equation 3. Finally, it is inferred that u belongs to H 2 0, 1.

The direct way is simple since if u is a solution to the continuous problem CP searched for in H 2 0, 1 , then u is a weak solution to the variational problem VP. To do so, it is only necessary to revert back to the process that enabled the estab- lishment of the variational formulation VP and to note that the latter makes sense, in particular by using the integration-by-parts formula, see Chap.

The Mathematical Theory of Finite Element Methods

The reciprocal is now calculated. In the variational problem VP , let u be a so- lution belonging to H 2 0, 1 — any solution u to the variational problem henceforth belongs to H 2 0, 1 according to the previous question. The integration-by-parts formula is then used in the inverse direction to the one that enabled the variational formulation to be obtained and this gives: The essential points treated in the rest of the demonstration is thus pointed out: The definition 3.

Moreover, the expressions supplied by the formula 3. In other words, the nodal equation 3. Therefore, the finite differences scheme, with application of a second order dis- cretisation to the differential equation of the continuous problem CP , precisely corresponds to the nodal equation 3.

The calculation of the coefficient is performed in a way analogous to that pre- sented for the calculation of the coefficient Aii in the answer to the question 7. The working out of the coefficient A00 is therefore performed as follows: Therefore, the following is obtained: Let u be a function of a real variable, defined from values [0, 1] in R.

The considered continuous problem CP is defined by: Show that the continuous problem CP can be expressed as a variational formulation VP in the form: In fact, this may be confirmed by re-writing the Fourier condition in the form: This explains why this homogenous Dirichlet condition is imposed on test func- tions v whose solution u constitutes a specific case.

This method guarantees that the full memory of the information contained in the continuous problem CP is maintained in the future variational formulation.

These two boundary conditions, the first one bearing on u via its relationship with 3. Find u belonging to V , solution to: Meanwhile, formulation 3. Actually, this variational formulation is structurally analogous to the one ob- tained within the framework of the Dirichlet problem — see 3. Thus, referring to the functional analysis previously shown in paragraph [3. This is achieved by again choosing the H 1 0, 1 norm see paragraph 3.

The H 1 0, 1 natural norm, previously defined, is as follows: H 1 denotes the inner product of H 1 from which norm 3. Moreover, for any function v belonging to H 1 0, 1 , the result is: The same scheme of analysis previously presented for the bilinear form a. In fact, the ellipticity inequality is immediate if it is reckoned that: Let any K be a closed subset strictly included in ]0, 1[, then: 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? 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.

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. 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!

Practitioner’s guide to finite element modelling of reinforced concrete structures (PDF)

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.

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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.

The nodal equation 6. To ascertain that, it is only necessary to replace the finite differences 6. Moreover, expansion 6. The approximate variational equation 6. From then on, the analysis of the coefficients to be considered in equation 6. Thus, the coefficients of Ai j and Bi j matrices to be retained for evaluation are: The coefficients Ci jk to be evaluated are thus: Thus, the following is obtained: The structural observations, presented in the estimation of coefficients Ci jk of ques- tion 5 are as valid as in the present case.

Now, discretisation by finite differences of the Neumann condition of the continuous problem CP defined by 6. Nevertheless, such an expansion was inevitable in order to obtain an approxi- mation by finite differences that is of the same order as the one established for the approximation of the viscous Burgers equation 6. In order to maintain expansion 6. The first time with a multiple weighting coefficient h, and the second time ac- cording to a weighting in h2.

Insofar an approximation of the second order after a division by h is desired, the first order partial derivative is replaced by its value 6. Equations 6.

Given that the mesh in space 6. In other words, in this particular case of a uniform mesh in space, the global approximation system 6. From then on, it is justified to consider the stability of such a numerical scheme according to usual methods applied to evolution equations and solved by finite dif- ferences. Euvrard, [4] and the approximation of system 6. More precisely, the interest is on the solutions to the following continuous prob- lem CP: It is reminded that Sobolev space H 2 0, 1 is defined by: Show that the continuous problem CP may be written in a varia- tional formulation VP as: To achieve this, a regular mesh of [0, 1] interval with a constant step h is introduced, such that: It is pointed out that the composed trapezium quadrature formula is written as: In fact, using the Cauchy-Schwartz inequality, the following is obtained: The integro- differential equation of continuous problem CP is multiplied by v and the obtained equation is integrated between 0 and 1.

The variational problem VP is thus written as: Dirichlet [3. As for the integral bearing on the non-linear term, it is only necessary to note that: Thus, functional space V that allows giving a sense to variational formula- tion VP is defined by: Moreover, the expressions given by 6. By identification, expressions 6. The generic equation of system 6. That is why equation VP! Then, what may be done is either evaluating every integral of equation 6. To achieve this, it is advisable to write the integro-differential equation of contin- uous problem CP at point xi , then to proceed to the approximation of the second derivative of u on one hand and the approximation of the integral of solution u on the interval [0, 1] on the other hand.

This operation enables one to eliminate the residue of the second order O h2 in equation 6.


Moreover, finite differences scheme 6. In fact, although it may be possible in some cases to establish that solution u possesses more regularity than it seems to have, the majority of cases need to be explained because the regularity of solution u depends on one hand on the regu- larity of the second member and on the regularity inherent to the structure of the differential operator on the other hand.

Thus, having assumed that there is certainly as much regularity as possible, an approximation of the differential operator may be then proposed by an algebraic procedure that would provide an approximation of a certain standard. This standard known as the order of the finite differences scheme in the jargon of numerical analysis finally enables the measurement and appreciation of the per- formances of various finite differences schemes having equal data and sufficiency of regularity of solutions to differential equations.

The 5 coefficients of this nodal equation are evaluated in the same way as for the calculations presented in the previous question. Then, after the use of composed trapezium quadrature formula, finally gives: In other words, the interest is on the scalar function u of variable x, being solution to the continuous problem CP: Show that the continuous problem CP may be written in a varia- tional formulation VP to be given later. To achieve this, a regular mesh of the [0, 1] interval with a constant step h is introduced, such that: Remember that the trapezium quadrature formula is written as: What is the order of approximation of the scheme thus obtained?

The characterization of V will be worked out, a posteriori, once the variational formulation is formally established. The Riccati differential equation of continuous problem CP 6. A variational formulation VP may thus be written as: That is why variational equation 6.

The regularity of functions v of V is now dealt with in order to establish sufficient conditions for the existence of the integrals of equation 6. The first integral of equation 6. The following is thus stated: This non-linearity does pause a problem, because resolution methods of the Gauss, Jacobi, Gauss-Seidel types, successive relaxation methods or simple or con- jugated gradient methods may not be directly operational in such a case. Then the Newton type of methods should be resorted to, e.