NUOVO METODO DI INTEGRAZIONE NUMERICA DELL'ING. LAMBERTO BERTOLI - ADVANCED QUADRATURE RULES FOR FINITE ELEMENTS

Ingegneria Strutturale

· ADVANCED QUADRATURE RULES FOR FINITE ELEMENTS

· INTRODUCTION

· OUTLINE OF THE PROBLEM

· ANALYTICAL DEVELOPEMENTS

· CONCLUSIONS

I fondamenti scientifici del metodo proposto dall’Ing. Lamberto Bertoli, i cui contenuti fondamentali sono discussi in questo articolo, sono stati presentati dai docenti Carmelo Majorana, Stefano Odorizzi e Renato Vitaliani dell’Università di Padova al “Fourth seminar about method and variational methods” tenutosi a Plzen (Praga) tra il 12 e il 15 maggio 1981.

Successivamente la trattazione fu pubblicata in versione ridotta sulla rivista “Advanced in Engineering Software” (C. Majorana, S. Odorizzi, R. Vitaliani, “Shortened quadrature rules for finite elements”, Advanced in Engineering Software, 1982, Vol. 4, N.2) e, come selected paper, nel volume “Software in Engineering Problems”.

Le soluzioni del problema dell’integrazione numerica, a partire dai casi più semplici, oggetto della tesi di laurea, fino ai più sofisticati, che sono stati affrontati nelle ricerche degli ultimi anni, sono state pubblicate dall’Ing. Bertoli nel testo:

“Quadratura diretta degli integrali multipli”, 2006, Ed. Libreria Internazionale Cortina, Padova.

In 1982 a new method for the quadrature of complete polynomials of 2, 3 and 4 dimensions was published in the paper: “Advances in Engineering Software”. This method permit a 25 % reduction of time in confront of Gauss-Legendre’s one and a par precision two variables, a 50 % for three variables, and 77 % for four variables adaptables to the resolution of dynamic problems.

The complexity of the calculation due to the bad conditioning of the nonlinear systems which govern the new numerical integration limited the study of the problem of the complete polynomials up the seventh degree. Here in the same procedure to obtain a numerical reduction of double integrals formed by polynomials of the nineth and the eleventh degree is presented. This method, unlike the one presented by Gauss, deals directly with the solution of the multi variable integration allowing us to obtain the solution with a significant saving of time of calculation in comparison with other procedures, thanks to the reduction of the number of sampling points.

Moreover, the availability of an alternative algorithm allows us to control those solutions obtained by other procedures and surpass possible singularities due to the transformations operated inside the elements. The case of the reduction of the interpoling polynomial of the nineth degree was solved using inside the normalized element, while when treating the eleventh level the solution includes 4 external points. This circumstance does not limit the applicability of finite elements to the method because, on the contrary, it garantees a major continuity among these interpoling functions which operate within contigous elements.

This paper also contains the tables which give you the normalized coordinates and weighting factors of the sampling points on the quadrature orders.

INTRODUCTION

A lot of formulations in the method of the finite elements require the calculation of multiple integrals. Such operations noticeably fall on the complete time of computing and are generally conducted through numerical schemes.

A first procedure consists in an aprioristic selection of a given number of sampling points which is determined by the grade fixed by the interpoling polynomial. The unknown of the problem are therefore only represented by the weighting coefficients of every point. More frequently, both the weighting and the sampling point coefficients are treated as unknown, so to reduce the number of points necessary to solve the problem. While the Newton Cotes scheme belongs to the first category, the Gauss scheme belongs to the second. The unknown factors of the sampling points are worked out by means of orthogonal functions, like for example the Legendre polynomials, the Hermite ones, the Laguerre ones or the Chebyshev ones. The Gauss has been formulated to obtain numerically simple integrals and it has been extended to the multiple integrals thus increasing exponentially the number of sampling points.

The indirect procedure, if applicable to a number of coordinates includes the control of the integration, and does not however make its approach as quicq when the variables are multiple, and for this reason an alternative scheme is given for the case when there are two variables. This procedure belongs to the class of problems where the position of the points and the weighting factors are considered as the unknown of the problem. Therefore it is a method similar to the Gauss one, of which its essential aspects are presented, even though it is more specific because it directly deals with the study of the bi variable polynomial functions. Even this method allows the elimination of odd grade functions, thanks to the symmetry of the position of the points with regard to the couple of Cartesian axis. They are not therefore considered in this paper, because their contribution is anyhow of no value for the calculation of the integral, as their effect consist uniquely in giving a mayor grade of freedom to the value of the polynomial functions which, in this case, are able to express in each point of the element a value of the integrand function which is completely independent of that of any point.

OUTLINE OF THE PROBLEM

Taken a continuous function defined in an n-dimensional space, its integral in a multi interval (-1,1) can be numerically obtained as:

(1)

where the coordinates and the weighting coefficients have to meet a sufficient degree of precision to satisfy the relation:

(2)

with : (p + q + … + r ) = (0, 1,…,m-1)

As to the field of problems regarded herein, E is equal to zero, as the integrand function is supposed in the form of polynomials, whose maximum degree is equal to (m-1). The relation (1), in explicit terms, given rise to a non linear system of equations, the relative solution of which can also not be unique.

In fact the uniqueness of the solution depends on a number of adeguate conditions which can be fixed on the coordinates of the points and also on their weighting coefficient. The Gauss Legendre method makes use of conditions which consider the positions of the points and thus proceding the roots of the Legendre polynomials are yelded.

The Gauss method reduces the number of integration points fixing that any polynomial of degree (m-1) passing through the integration points has zero integral. The terms of odd degree are eliminated giving a symmetrical position to the points x, which regard to the origin of the axis, and also an equal weighting coefficient, this because the numerical integral of any arithmetical function is made zero. This way the Gauss method reduces the number of integration points per polynomial function which regards only one unknown x, giving the roots of linear combinations of orthogonal polynomials of the known families.

The aim of the present paper is to generalize the basic principles of the Gauss method for functions of more variables, obtaining an appraisable reduction in the number of sampling points compared to the one offered by the direct application of the same method on more unknown.

ANALYTICAL DEVELOPEMENTS

DOUBLE INTEGRATION WITH m = 9

The case of the problem of the integration of a function on (x, y) of m = 9 degree is treated. It can be observed that these polynomials accord with the functions derived from finite elements of the serendipity type.

Integration scheme with 21 points

A complete polynomial of the nineth degree on (x, y), once eliminated the odd terms leads to:

The sampling points and their weighting coefficients have to be chosen in order to satisfy the relation:

As each polynomial function is made up of the sum of the various terms, the integral of the function corresponds to the sum of the integrals of the term wich form it. Consequently, to find a solution to the problem, it is necessary to fix the points so that each of its components is able to satisfy, when considered separate from the others, one single equation indipendant from the ones obtained by other terms. Disposing symmetrically the points with regard to the bisecting lines of the coordinates planes, the number of solving equations is further reduced because the terms x^p y^q are completely equivalent to the terms x^q y^p as the contribution offered by the couple of symmetrical points with regard to the bisector is the same in both cases. Arranging one point on the bisector of each quadrant and disposing a couple of symmetrical points with regard to the bisector in each quadrant, we obtain a reduction of the system of non linear equations leading to:

The system offers infinite, true and positive solutions, of which one of them corresponds to the following scheme:

Example with function test

The precision of the numerical resolution is of 99,997 %

Integration scheme with 20 points

The above given scheme is not the quickest solution to the problem. The same equations can be solved by posing the condition that the weighting coefficient of the point situated on the origin of the axis were zero. Thus doing the quadrature has been solved by following a procedure which allows the lowest numerical integration of 20 points whose weighting coefficients are all positive.

Example with function test

The precision of the numerical integration is of 99,997 %

DOUBLE INTEGRALS WITH m = 11

The complete 11° degree polynomial shows the following equal terms:

The sampling points have to be chosen so that the equality can be satisfied for any value of the terms of the coefficients of the polynomial:

The problem can be solved by arranging on each quadrant 2 points on the bisector and a couple of points symmetrically arranged which regard to the same. Once again the research for the numerical quadrature of any polynomial of inferior o equal degree to 11 yields to the solution of a system of non linear equations:

28 points form the scheme which allows to obtain the same accuracy as the one offered by the 36 sampling points of the Gauss method. The solution of the system is given by the following values:

Example with function test:

The precision of the function test is of 99,999584 %.

CONCLUSIONS

An effective procedure is proposed in this paper for the reduction of double integrals. The method is based on the same principles of the numerical integration that in 1977 allowed to reduce the problem of the numerical quadrature of multiple integrals of 2, 3 and 4 dimensions up to the polynomials of 7° degree. The proposed method allows to achieve the same degree of accuracy as using the Gauss-Legendre method, but with a remarkable 20 % reduction in the computing time for the integration of polynomials of bi variables of 9 degree and 22,2 % reduction for polynomials of 11° degree.

The systems of equations which form the essence of the problem are strongly and badly conditioned and such difficulties had not allowed to develop the method further in past times. This was due to the fact that the calculation techniques used in that period were not able to improve the level of accuracy achieved.

Also the numerical methods of the reduction of non linear systems, like the Newton- Raphson method, do not allow to obtain satisfying results as they require to start the iteration by points very near to the zeros of the polynomial.

However, the development of modern computer science technologies has allowed to resume the approach started in the 70s with the mathematical reduction of the problem to those cases which are relatively simpler.

The auctor of this article solved a lot of others cases which are published in the text: “Quadratura diretta degli integrali multipli” edited by Libreria Internazionale Cortina of Padua (Italy) with ISBN 88-7784-263-6 and it can be found in the same bookshop in via Marzolo in Padua.

In this text are published all the solutions of the integration of two variables functions from 2^nd degree polynomials to 15^th degree, of which is given the correct solution.

There are also the quadratures of more variables polynomials and the exact integration of one nine degree polynomial in three variables with a temporal reduction of the 50% in confront of Gauss- Legendre’s one.

It is also been solved the quadrature of the four variables functions until the correct integration of 7^th degree. It is possible in this way the resolution of dynamic problems with a reduction of time of the 77 %.

In the end there are polynomials with 5, 6 and 7 variables for the correct quadrature of polynomials until the 5^th degree. These cases solved specialistic problems like the explosion of supernovas, that need 7^th dimensions’ elements.

The auctor of the herein article is going to carry on this research although the complexity of calculation exponentially grows as the number of variables and of the degree of the polynomial increase for the worsening of the degree of the bad conditioning of the system of equations which govern the reduction of the problem. Anyway, the advantages which this method implies compared to other techniques of calculation are thought such to justify the efforts made to follow it.