^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

^{1}

^{*}

The Asymptotic Numerical Method (ANM) is a family of algorithms for path following problems, where each step is based on the computation of truncated vector series [1]. The Vector Padé approximants were introduced in the ANM to improve the domain of validity of vector series and to reduce the number of steps needed to obtain the entire solution path [1,2]. In this paper and in the framework of the ANM, we define and build a new type of Vector Padé approximant from a truncated vector series by extending the definition of the Padé approximant of a scalar series without any orthonormalization procedure. By this way, we define a new class of Vector Padé approximants which can be used to extend the domain of validity in the ANM algorithms. There is a connection between this type of Vector Padé approximant and Vector Padé type approximant introduced in [3, 4]. We show also that the Vector Padé approximants introduced in the previous works [1,2], are special cases of this class. Applications in 2D nonlinear elasticity are presented.

Many engineering problems can be reduced to solving nonlinear problems depending on a control parameter λ. These problems are written in general form:

where

The Asymptotic Numerical Method (ANM) [1,2] is a family of algorithms for path following problems. The principle is simply to expand the unknown

where

where

The Vector Padé approximants were introduced in the ANM to improve the domain of validity

Many applications in structural mechanics (for instance nonlinear elasticity and contact), [1,2] have established that Vector Padé approximants with a common denominator can reduce the number of poles and permit to obtain more regular solutions. By using this rational representation in a continuation procedure, the number of steps to obtain the entire solution path has been reduced [

The aim of this paper is to discuss some techniques to define new Vector Padé approximants in the framework of the ANM and to show that their utilization can improve clearly the classical Vector Padé representation.

In the second part, we propose a new type of Vector Padé approximant which can be directly defined from the vector series (1) by extending the definition of the Padé approximant of a scalar series [5,8] and without any orthonormalization procedure. By this way, we show that a family of Vector Padé approximants is possible. There is a connection between this type of Vector Padé approximant and Vector Padé type introduced in [3,4]. We show also that the Vector Padé approximant introduced in the previous works [1,2] are special cases of this class.

All the approximants are applied on some examples from nonlinear two-dimensional elasticity which are presented and analyzed in the third part. Among this family of Vector Padé approximant, we show on numerical examples, that there are some approximants which increase the range of validity

In this Section, we will give the definition and the construction of a new type of vector Padé approximants.

A Vector Padé approximant of a vector function

where

The aim of this paper is to define Vector Padé approximant

where

The vector ”polynomial”

In Appendix 1, we show that the

and the

The system (8) can be written in the following matrix form (see Appendix 1)

It may be noted that if the terms of the series in Equations (6) are scalar, we find exactly the system defining the scalar Padé approximant in [

The construction of the new type of Vector Padé approximants requires the solution of the matrix system (10) verified by the matrices

where

rows and

Note that in the scalar case, the matrix

According to the definition of the Vector Padé approximant (4), the construction of this new type of Vector Padé approximant usually requires high computational cost due to the fact that for each value of a, we need to calculate the inverse of the matrix

For example, if we look for matrices,

It corresponds to the Vector Padé approximant that would be built from the scalar Padé approximant corresponding to each component of the vector

Another choice of the form of the matrices

where the polynomial

In Appendix 2, we show that the scalars

We thus find the Vector Padé approximant (13) introduced in the work of the ANM algorithms [1,2] where the coefficients

Therefore, we constructed a new family of Vector Padé approximants given by Equation (12) or (13) without any condition on the coefficients

The representations (2) or (13) permit to compute only a part of the solution path of the nonlinear problem (1). To obtain the entire solution path, Cochelin [

To introduce the vector Padé representation in a continuation algorithm, Elhage et al. [

which gives an evaluation of the radius of validity

The numerical robustness of the approximate solutions obtained by the vector series representation (2) and by the new family of Vector Padé representation (12) is discussed on the basis of tests emanating from plane stress two-dimensional nonlinear elasticity analysis. The studied structure is discretized using a classical CST finite element [

More precisely, we plot the load-displacement curves with three ANM steps. Three calculations are carried out:

• the first calculation will be made using ANM continuation with a series representation (2)• the second calculation will be made using ANM continuation with classical Vector Padé representation (12), the coefficients

The performance of the three calculations are compared in terms of the step lengths of the three ANM continuations, the quality of the solutions is given by the residual curves.

The first numerical example concerns the bending of a plate; see

In

The first calculation, ANM continuation with series representation (2), at orders 10, 15 and 20, shows that the step length increases with the truncation order. Three steps at order 20, allow obtaining the curve until a displacement equal to 62 mm with accuracy of the order of 10^{−5}. This result is classical in the works of ANM algorithm [

The second calculation, ANM continuation with classical Padé representation, at orders 10, 15 and 20, shows

that the step lengths are greater than the first calculation using series representation. Three steps with ANM Padé representation at order 10 allows obtaining the curve until a displacement equal to 73 mm with a good quality as can be seen on the residual curve of

For this second calculation, the results obtained by using the Householder orthonormalisation method are compared with those obtained by using Gram-Schmidt orthonormalization in

The third calculation, ANM continuation with the proposed Vector Padé representation (12) the coefficients being arbitrary, is performed by using the orders 10, 15 and 20. We carried out the calculations by slightly modifying the values of the coefficients

This first test was very successful. Indeed, it shows that an arbitrary choice of the coefficients

For the second numerical experiment, we chose the example of the bending of an elastic arch, see

was discretized using 41 nodes along the radius and five nodes along the width; a total of 320 elements and 410 degrees of freedom is used.

We represent in

The results obtained by respectively the first calculation, ANM continuation by using series representation (2) at orders 10, 15 and 20, the second calculation, ANM continuation by using classical vector Padé approximation at orders 10, 15 and 20, the coefficients

Three steps with the proposed ANM Padé representation at order 20 allows obtaining the curve until a displacement equal to −72 mm with a good quality as can be seen on the residual curve of

This second numerical test confirms that an arbitrary choice of the coefficients

For the third numerical experiment, we consider the traction of an elastic plate; see

We represent in

In this work, we introduced a new way to build directly a new type of Vector Padé approximants from a truncated vector series in the framework of the asymptotic numerical method. We have shown that the vector Padé approximants introduced in references [1,2], are a special case of this class. The proposed Vector Padé approximants can be determined without any orthonormalisation procedure which saves the time computation for problems with a large number of degrees of freedom. In

fact, the orthonormalization procedure is time consuming because of the very large number of scalar products to be evaluated. It remains to explore different choices of this new class of Vector Padé approximants.

To determine the equations satisfied by the matrices

Injecting (6) and (14) into (7) yields:

which can be written as:

where

By identifying the terms corresponding to coefficients

and that the last L equations are

As

which are written in matrix form

where:

We will look in this part to particular solutions

where

If, for any j,

If the system has a solution, then by (9), the

As the matrix

With the aim of building a Vector Padé approximant such that all its components are rational fractions with the same denominator, we denote by

where

Using this vector

These Equation (29) show that the vectors

Using Equation (9), we deduce that

It is obvious that if the matrix

is invertible, then its inverse is of the form

Which is equivalent to

As a result,

By choosing

equality (4) is written

(37)

To express the Vector Padé approximant

and

then, taking into account the equality (29), we obtain

Hence

it is concluded that:

Using this Equation (41) in the expression of the Vector Padé approximant

We show in this section that in the definition of the Vector Padé approximant (37), the scalar

Therefore, it suffices to take

Hence, if one chooses

then we have the equality (12) for all m . Note also that if