Non-linear approximation method by an approach in stages

Computational Mechanics 32 (2003) 177–184 Ó Springer-Verlag 2003 DOI 10.1007/s00466-003-0473-9

Non-linear approximation method by an approach in stages T. Beda, Y. Chevalier

177 Abstract This work presents a new methodology of approximation of functions. The identification is done graphically, stage-by-stage, taking into account the form of the generating functions. The above method is developed in this article within the scope of an efficient tool for modeling physical systems and as a reliable curve-fitting method that can use any basic function. Thus, it represents a non-linear identification procedure. On the one hand, simulated cases that do not represent any physical model are studied, and on the other hand, real cases that model physical systems as rubber behavior law are illustrated. The Ogden hyperelastic and the extended Gent–Thomas models for rubber materials are shown to give an accurate prediction of Treloar’s classical data with the help of mechanical parameters evaluated by the stage approach. Critical and comparative analyses of the stage approach are presented with the usual non-linear iterative procedures such as the non-linear least squares. Keywords Approximation, Nonlinearity, Basic functions, Behavior, Identification, Stage-by-stage

1 Introduction In numerical analysis, discrete values of a function are usually approximated using polynomials, exponentials or trigonometric functions. The methodology consists of making a choice of a limited sort (type, kind) of basic functions: monomial, exponential, orthogonal polynomials: Legendre, Tchebychev, Hermite, Laguerre, etc... The selection of basic functions is based on ones judgment and experience. Problems arise when the model to approximate is generated by a series of basic functions that are not used by the above classical methods. Such situations usually occur if one investigates a material experimentally: the physical law has to be approximated by a mathematical model, i.e. basic functions, chosen to reflect the true physical behavior. Consequently, numerical analysts do Received: 1 November 2002 / Accepted: 24 June 2003

not have the possibility to choose for the approximation the usual well-known basic functions cited above. That sort of problem happens frequently when discrete values of an experimental function of a material are given. Ordinarily, the approximating function is generated by the usual basic functions cited above. This work presents an approach that can be used to approximate a function by considering any basic function. The method restores the exact form of any given mathematical model by a fit of discrete functions.

2 Approximation methodology 2.1 Usual methods The usual methodology of approximation is based on given basic functions uj ðxÞ for which one evaluates the weighting coefficients [1–5]. Exponents of the basic functions are pre-established and are integer numbers in general. The approximating function takes the form of a combination of basic functions: M X fa ðxÞ ¼ cj uj ðxÞ ð1Þ j¼0

All usual numerical methods are based on this process i.e. orthogonal polynomials. For these different approximation or interpolation methods, the basic functions are:

uj ðxÞ ¼ xj

(polynomials)

ð2Þ

uj ðxÞ ¼ ejx

(exponentials)

ð3Þ

uj ðxÞ ¼ cos jx or sin jx (trigonometric) i 1 dj h 2 j (Legendre) ðx
1Þ uj ðxÞ ¼ j 2 j! dxj dj
j
x  (Laguerre) xe dxj j h i 2 d
x2 (Hermite) e uj ðxÞ ¼ ð
1Þj ex dxj uj ðxÞ ¼ cosðj arcos xÞ (Tchebychev) uj ðxÞ ¼ ex

ð4Þ ð5Þ ð6Þ ð7Þ ð8Þ

T. Beda (&) ENS Polytechnic, University of Yaounde I, BP 8390, Yaounde, Cameroon E-mail: [email protected]

uj ðxÞ ¼

Y. Chevalier LISMMA (ISMCM), 3 rue F. Hainaut, 93407 Saint-Ouen, Paris, France

and also for other interpolations [3] such as: Newton, Neville–Aitken, Cubic Spline, etc...

N Y ðx
xk Þ
xk Þ ðx j6¼k¼1 j

(Lagrange)

ð9Þ

178

One sees that all these numerical calculus methods pre-establish their own basic functions uj ðxÞ. While in modeling, the basic functions are given by the mathematical model reflecting the real physical behavior. Thus, the usual basic functions (i.e. usual methods) become unsuitable for the approximation. They are adequate when the real function to approximate is generated by classical basic functions. If not, they result in curve-fitting parameters only. So, they are efficient and excellent tools of approximation when one wants just a record of data, but not always adapted to approximate physical laws such as: behavior laws, transfer functions etc... In many cases, they fail in resolution and estimation of parameters. The method by approach in stages presented in this work is an attempt to resolve such cases.

determines the elementary weighted basic function of the maximum ‘‘order’’ i.e. aN uN ðxÞ. To evaluate the order N, one plots the curve ðuk ðxÞ; fex ðxÞÞ step-by-step. From the previous analysis, the rationale for doing it is that if the absolute value of the plotted curve is: – convex increasing, it means that the order k is inferior to the real one i.e. N. One has to raise the order of the basic function, that is to say one takes the following elementary basic function ukþ1 ðxÞ for the new plot, – concave, one has to reduce the order of the basic function by taking uk
1 ðxÞ for the new plot, – convex decreasing, orders of uk ðxÞ and fex ðxÞ are of opposite sign. One has to plot and study the curve ðu 1ðxÞ ; fex ðxÞÞ; k

– straight line (segment), uk ðxÞ is a partial solution and 2.2 the slope p of the segment corresponds to the relative Stage approach weighting coefficient. So, the order N of fex ðxÞ is equal The stage approach is based on a methodology completely to the solution obtained and the coefficient aN to the different from the classical one: the elementary basic slope p. functions generating the model are determined stepby-step, while the usual methodology does it globally at once. It affects the approximation of any arbitrary model 2.2.2 generated by any series of arbitrary basic functions uk ðxÞ. Basic functions Suppose one wishes to approximate the function fex ðxÞ The choice of basic functions must be done taking evenin the interval ½a; b by yðxÞ. Let uk ðxÞ be the series of basic tually into account the analytical form of the model functions generating yðxÞ, then: studied. If one makes a bad choice of basic functions, the error of approximation by stage approach is not optimal. N X For example, by stage approach method, an exponential y¼ ak uk ðxÞ ð10Þ function can also be approximated by polynomials and k¼0 vice versa. This is a wrong way that engenders errors. The where ak is the weighting coefficient. choice of basic functions of approximation remains funLet us assume that the weighted basic functions ak uk ðxÞ damental when one has to fit experimental functions or are arranged in ‘‘increasing order’’. That is to say, by acquisitions. considering the equivalent function aN uN ðxÞ for x 2 ½a; b, one has: N X

ak uk ðxÞ ffi aN uN ðxÞ

ð11Þ

k¼0

so,

y ffi aN uN ðxÞ

ð12aÞ

Taking X ¼ uN ðxÞ, relation (12a) gives:

y ffi aN X

2.3 Generalized stage approach When the first stage basic solution (elementary component) i.e. aN uN ðxÞ is determined, one defines a new function to approximate for the second stage by the relation: f1 ðxÞ ¼ fex ðxÞ
aN uN ðxÞ

ð13Þ

ð12bÞ

Then one re-begins the study by considering the function f1 ðxÞ. One plots the curve ðuk ðxÞ; f1 ðxÞÞ. By analogy to the study of the a-power function If the new curve is linear (a segment), the second stage a X ; X > 0 and a a real number; one finds that if the curve solution is obtained. If not, one takes the relative coeffiðX; jyjÞ i.e. ðX; jfex ðxÞjÞ; jyj being the absolute value of y, is: cient ak to be null. – convex increasing, one has jyj > X for jyj > 1 and In general, for the stage k, the function fk ðxÞ to jyj < X for 0 < X < 1, approximate is defined by the recurrence formula: – concave, jyj < X for jyj > 1 and jyj > X for 0 < X < 1, – convex decreasing, powers of jyj and X are of opposite fk ðxÞ ¼ fk
1 ðxÞ
aN
kþ1 uN
kþ1 ðxÞ k
1 sign, X ð14Þ aN
j uN
j ðxÞ k ¼ 1; N fk ðxÞ ¼ fex ðxÞ
– linear, jyj is proportional to X. j¼0

2.2.1 Stage of an identification The stage approach consists of determining the basic functions solutions stage-by-stage. At first stage, one

with f0 ðxÞ ¼ fex ðxÞ. The last partial solution is obtained when one gets a stage function fk ðxÞ being constant. The global solution is the sum of partial stage solutions.

– for the third stage, the solution is
3x2 :

3 Application of stage approach method

f3 ðxÞ ¼ f2 ðxÞ þ 3x2

ð19Þ

3.1 – for the fourth stage, the solution is 2x: Linear stage approach f4 ðxÞ ¼ f3 ðxÞ
2x ð20Þ In linear approximation, unknown parameters are linearly combined i.e. polynomial coefficients for example. – for the fifth stage, the solution is the constant function equal to five. 3.1.1 The approximating function PðxÞ is given by the sum of all Complete polynomial basis stage monomial solutions: Let us consider the function f ðxÞ to approximate, defined PðxÞ ¼ f ðxÞ
fN ðxÞ by a complete base: ð21Þ 4 3 2 f ðxÞ ¼ 5 þ 2x
3x2 þ x3 þ 2x4 for x 2 ½0; 10 ð16Þ PðxÞ ¼ 2x þ x
3x þ 2x þ 5 Figure 1.1 shows that the curve ðx3 ; f ðxÞÞ is convex, so the order of f ðxÞ is greater than three. The monomial x3 is not a solution at first stage, but can be one at inferior stages. Figure 1.3 shows that the curve ðx5 ; f ðxÞÞ is concave: the order of f ðxÞ is less than five. The monomial x5 is not a solution. Figure 1.2 shows that the curve ðx4 ; f ðxÞÞ is straight. One concludes that the monomial x4 is the sought-after partial solution and is also the ‘‘maximum order’’ monomial of the function f ðxÞ. The slope of the segment (calculated for extreme values of x 2 ½a bÞ is equal to two. So, the first stage solution is the monomial 2x4 . One defines a new function to approximate by:

3.1.2 Incomplete polynomial basis The Fig. 1.4, 1.5 and 1.6 show the first stage identification of the function of incomplete base gðxÞ: gðxÞ ¼ x2 þ 15x3
10x7

for x 2 ½0; 10

ð22Þ

After having obtained the first stage solution, if the curve ðxk ; f ðxÞÞ is not linear on the inferior stage, it means that xk is not a solution of the problem. So, one has the relative weighting coefficient equal to zero. For the resolution of gðxÞ, one deduces that the coefficients of x6 ; x5 ; x4 , and x are null. The same process can be adopted for exponential generating basic functions [7].

Remark By linear least squares, one obtains exact solutions if one takes by a mere chance the order of approximation n superior to the real order, which is N ¼ 7 for the And the same reasoning is applied to f1 ðxÞ. The study of studied example. For n  7, one has: Pn ðxÞ ¼ gðxÞ. the curve ðxk ; f1 ðxÞÞ; k  3, leads to the second stage When one takes the order of approximation inferior to solution equal to x3 . So, one has: the real order, the solution obtained is not the exact one; f2 ðxÞ ¼ f1 ðxÞ
x3 ð18Þ for example, let us take the order n ¼ 6, the program

f1 ðxÞ ¼ f ðxÞ
2x4

ð17Þ

Fig. 1. Stage of a monomial evaluation

179

‘‘polyfit’’ of MATLAB [6], gives as a solution the polyno- 3.2.1 mial P6 ðxÞ, which is different from the real function gðxÞ: Correct basic functions Consider the function FðeÞ: 5 5 5 2

P6 ðxÞ ¼
3:2434  10 þ6:8755  10 x
5:5320  10 x þ2:2243  105 x3
0:4905  105 x4 þ0:0602  105 x5
0:0039  105 x6 ð23Þ

180

  1 a b FðeÞ ¼ a e2
pffiffi þ e ðe þ e2 Þb

ð25Þ

Let us extrapolate with the help of the computed P6 ðxÞ by taking x beyond the given values, that is to say, x 62 ½0; 10. For x > 10 or x < 0, the function P6 ðxÞ diverges from the where a; b; a and b are unknown parameters. The relative basic functions are in two different forms solution gðxÞ, so it has poor extrapolation properties. up and wp , defined by:

3.1.3   1 ap 2 Influence of weighting coefficient up ðeÞ ¼ e
pffiffi Consider a case where the value of a weighting coefficient e is relatively high, for example, let us take: hðxÞ ¼ 104 x þ x3 for x 2 ½0; 10 ð24Þ and The curve ðx3 ; hðxÞÞ of the Fig. 1.7 suggests a solution xk with k < 3. Figure 1.9 shows that the first stage solution is equal to the monomial 104 x. Here, the equivalent function of hðxÞ in the study domain [0, 10] is the monomial 104 x instead of x3 , contrary to the previous view. So, a2 u2 ðxÞ ¼ 104 x, which corresponds to the first stage solution, is on ‘‘higher order’’ in this case. The monomial x3 remains the second stage solution: a1 u1 ðxÞ ¼ x3 . Thus, the weighting coefficient value influences the arrangement by ‘‘equivalent function’’ of the weighted basic functions, consequently modifies succession of obtaining partial solutions.

wp ðeÞ ¼ ðe þ e2 Þbp

ð26aÞ

ð26bÞ

Let us simulate by taking a ¼ 3; b ¼
0:7; a ¼ 4 and b ¼
10; Fig. 2a. Figure 3 shows that, at first stage, the pffiffi exploitation of the curves ðup ðeÞ; FðeÞÞ gives ðe2
1= eÞ3 as partial solution. The corresponding calculated slope is equal to four. At second stage, the function to approximate becomes, Fig. 2b:

  1 3 2 F1 ðeÞ ¼ FðeÞ
4 e
pffiffi e

ð27Þ 3.2 Non-linear stage approach Figure 4 shows that, at second stage, the exploitation of In non-linear approximation, unknown parameters are non-linearly combined i.e. both coefficients and exponents different curves ðwp ðeÞ; FðeÞÞ gives as elementary solution 0:7 of polynomials. the function
10=ðe þ e2 Þ :

Fig. 2. Stages of an approximation. a First stage FðeÞ, b Second stage F1 ðeÞ

181

Fig. 3. Identification at first stage

Fig. 4. Identification at second stage

  1 3 10 2 PðeÞ ¼ 4 e
pffiffi
e ðe þ e2 Þ0:7

One defines a new stage function F2 ðeÞ:

10 ðe þ e2 Þ0:7   1 3 10 F2 ðeÞ ¼ FðeÞ
4 e2
pffiffi þ e ðe þ e2 Þ0:7 F2 ðeÞ ¼ F1 ðeÞ þ

ð29Þ

ð28Þ 3.2.2

Incorrect basic functions F2 ðeÞ is a null function. So, one deduces the approximating function PðeÞ by the Suppose that one chooses basic functions that do not correspond to real models. relation:

For example let us approximate the exponential func- where lp and ap are real numbers characteristic of matetion ex for x 2 ½1 10 by a monomial. Using stage method, rials. Let us represent the nominal stress by the function one obtains: f : f  rnom . The elementary basic functions generating x
6 9:7 e ffi 4:41  10 x ð30Þ the function f are: 2   ap Let us approximate the monomial function x by stage ap
1
2
1 ð36Þ u ðxÞ ¼ k
k p approach. When taking one exponential term, one obtains:

x2 ffi 23:5e0:17x
27:8546640060 182

ð31Þ

Relation (35) is an N-term model with 2N unknown parameters. One realizes that one makes more errors for a wrong One sees that no classical methods of approximation choice of basic functions. For these two examples, a good use this kind of basic functions, relations (34) or (36). choice of basic functions gives exact solutions. Thus, there is no possibility to characterize the behavior of a material given by relations (32) or (35) using the classical 4 methods, from experimental acquisitions. The stage Mechanical results approach seems adequate in these cases that impose conditions on the form of the generating functions and are 4.1 non-linear identification. Rubber materials behavior: theoretical models In experimental investigations for materials characterization, usually one has to identify mechanical parameters by 4.2 Parameters’ identification fitting data with the help of a theoretical model. In this paragraph, one presents a practical application of We consider incompressible constrained isotropic the stage method in real cases and by comparison to least Green elastic materials for our study. squares, results fitted discrete experimental tensile curve of Treloar rubber [10, 11]. 4.1.1

Extended Gent–Thomas model In simple tension, the nominal stress is given by [8, 9]:   1 k rnom ¼ 2 k
2 K1 þ K2 k 1 þ 2k3  a
1 # 2 2 þK3 k þ
3 ð32Þ k

4.2.1 Extended Gent–Thomas model For extended Gent–Thomas model, the number of basic functions is pre-established and equal to three.

(a) Non-linear stage approach Taking into consideration the model given by the relation where k is the stretch, Kj and a are real numbers charac- (33), the stage approach method permits us to evaluate as teristic of materials; with a > 2. mechanical parameters of Treloar rubber material from a The relation (32) can be rewritten as reduced stress n: fit of data, the values given in Table 1.

 a
1 At first stage, the study of the curve ðk2 þ 2=k
3Þb ; n k 2 n ¼ K1 þ K2 þ K3 k2 þ
3 ð33Þ permits us to get b ¼ 1:745 and K3 ¼ 2:2  10
4 .

k 1 þ 2k3 At second stage, one studies the curve k=ð1 þ 2k3 Þ; n1 The reduced stress n is generated by the following three where: basic functions:  1:745

u0 ðkÞ ¼ 1

ð34aÞ

2 n1 ¼ n
2:2  10
4 k2 þ
3 k

One obtains a segment (linear curve) of slope K2 ¼ 0:3 and intercept K1 ¼ 0:127.

u1 ðkÞ ¼

k 1 þ 2k3

ð34bÞ

u2 ðkÞ ¼

 b 2 k2 þ
3 k

ð34cÞ

with b ¼ a
1 and b > 1. The relation (33) is a three-term model with four unknown parameters.

4.1.2 Ogden model In simple tension, the behavior law is given by [10]:   N X 1 ap
1 ð35Þ rnom ¼ lp k
ap k 2 þ1 p¼1

ð37Þ

(b) Non-linear least squares Considering the three-term model of the relation (33), by a fit of Treloar data in a non-linear least squares sense, we obtain the computed parameters listed in Table 1. Figure 5a shows curves resulting from stage approach and that from least squares compared to experimental Table 1. Identified parameters of extended Gent–Thomas model Parameters

K1

K2

K3

Stage approach Least squares Unit

0.127 0.1288 MPa

0.3 0.3116 MPa

2:2  10
4 2.745 10
4 2.9379 MPa /

a

data. Both methods are shown to give a good accurate prediction of experimental acquisitions.

(b) Non-linear least squares For least squares, one must fix a priori the number of generating basic functions, and the chosen number influences the improvement on the quality of the results: for 4.2.2 example a two-term of relation (35) fails to fit Treloar data Ogden model while a three-term found to capture quite well the data of For Ogden model, the number of basic functions is not any mode of deformation [10]. pre-established. Table 2 summarizes non-linear least squares results obtained by Ogden using a three-term with six unknown (a) Non-linear stage approach parameters. The stage approach is an identification procedure that Figure 5b shows theoretical curves of Ogden model finds out and optimizes directly the number of basic functions generating the theoretical model i.e. number of obtained by stage approach and by least squares compared model terms.

to Treloar data. The theoretical predicted tensile curves of At first stage, one studies the curve ka
1
1=ka=2þ1 ; rnom both methods take a good shape of discrete experimental curves. that leads to a1 ¼ 29:5 and l1 ¼ 3  10
25 . At second stage, one studies the curve a
1

1 k
ka=2þ1 ; rnom1 where: 4.3
25

rnom1 ¼ rnom
3  10

 k28:5


1



Comparative and critical analysis

ð38Þ

k15:75 The second stage solutions evaluated are a2 ¼ 5 and l2 ¼ 0:013.  At third stage, the  curve to study is a
1 1 k
ka=2þ1 ; rnom2 where:   1 4 ð39Þ rnom2 ¼ rnom1
0:013 k
3:5 k After progressively studying all stages, the mechanical parameters evaluated for the material are given in Table 2.

4.3.1 Usual methods The use of least squares requires the user to provide the number of generating basic functions, which can be the order of approximation for linear least squares or the choice of an appropriate effective model for non-linear least squares. All usual non-linear methods i.e. Newton–Raphson, Gauss–Newton, Marquardt, etc... being iterative procedures [12, 13], optimal use of non-linear least squares requires good judgment and experience to establish good starting values and to avoid local minima for stationary

Fig. 5. Stage approach and least squares results compared to Treloar data. a Extended Gent–Thomas model, b Ogden model

Table 2. Computed parameters of Ogden model

parameters

l1

l2

l3

l4

Stage approach Least squares Unit

6.276 6.2995 MPa

0.013 0.0127 MPa

)0.11 )0.1001 MPa

3  10
25 1.3 / 1.3 MPa /

a1

a2

a3

a4

5 5 /

)2 )2 /

29.7 / /

183

solution. The convergence of these procedures depends on many factors: regularity of the Jacobian matrix, choice of the line search, etc... When all requirements are under control, the least squares method is a great and a very efficient tool of approximation and is simple enough to program.

184

4.3.2 Stage approach The main advantage of stage approach over other techniques is its capacity to find out and optimize the number of generating basic functions, thus to determine the optimal order of linear or non-linear approximation. Consequently, the order of approximation and the number of terms defining a model are not left to user appreciation. It’s a method that converges to the same solution for any starting order of basic functions. It can assume over long ranges with good extrapolation properties. Difficulties arise when by equivalence of function one cannot bring out clearly a distinct, separate basic function at a step of identification. Apparently, it’s a method less simple to program than the least squares. 5 Conclusion The generalized stage approach treated in this paper is a method with original methodology of approximation of functions. The identification is done stage-by-stage, which is a phase of elementary generating function evaluation. This process permits one to optimize the approximation procedure and therefore reduces the difficulties of resolution. The stage approach unlike the well-known methods, takes into account any analytical form of a model and permits only the basic functions that generate the model to emerge, thus, the choice of basic functions remains very important for the best resolution. It also enables one to palliate difficulties inherent in the complexity to identify simultaneously exponent parameters and weighting coefficients, which is a delicate problem in numerical calculus i.e. non-linear identification procedure. One of the high qualities of the stage approach lies in the ability to find out the exact order of real function. This

possibility is not given by classical methods, for which the numerical analyst fixes by himself a priori, from empirical considerations, the order of approximation in the linear case or the number of generating basic functions in nonlinear case.

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