Research Article | Open Access
Grégory Antoni, "A New Iterative Numerical Continuation Technique for Approximating the Solutions of Scalar Nonlinear Equations", International Journal of Engineering Mathematics, vol. 2017, Article ID 2783682, 21 pages, 2017. https://doi.org/10.1155/2017/2783682
A New Iterative Numerical Continuation Technique for Approximating the Solutions of Scalar Nonlinear Equations
The present study concerns the development of a new iterative method applied to a numerical continuation procedure for parameterized scalar nonlinear equations. Combining both a modified Newton’s technique and a stationary-type numerical procedure, the proposed method is able to provide suitable approximate solutions associated with scalar nonlinear equations. A numerical analysis of predictive capabilities of this new iterative algorithm is addressed, assessed, and discussed on some specific examples.
The resolution of scalar nonlinear equations is an issue frequently encountered in many branches of physical sciences such as mechanics [1–5]. Although, in the literature, the most used numerical methods are either the classical Newton’s technique [3, 4, 6] or modified Newton-type procedures [7–10], they suffer from the main disadvantage of being held in check in the presence of critical points . In order to overcome this deficiency, we propose to develop a new iterative algorithm applied to a numerical continuation procedure  for providing the approximate solutions associated with parameterized scalar nonlinear equations. The presented algorithm is based on a modified Newton-type method coupled with a stationary numerical technique. This study is organized in the following manner: (i) in Section 2, the standard numerical continuation procedure is briefly recalled including some classical algorithms; (ii) in Section 3, the new proposed iterative numerical method is presented in detail; (iii) in Section 4, the predictive abilities associated with this new iterative algorithm are tested and evaluated on some examples.
2. Standard Numerical Continuation Methods
2.1. Problem Statement
We consider the parameterized scalar nonlinear equation in the following form:where denotes the real-valued “solution” variable associated with the nonlinear problem under consideration and is the real-valued scalar "parameter" variable. It is important to emphasize the following: (i) the parametrized scalar nonlinear equation (see (1)) may include critical points (see Figure 1); (ii) the couple can depend on another parameter such as (1) reading ; (iii) in the framework of solid mechanics, (1) represents the mechanical equilibrium equation and the “solution” variable and the scalar “parameter” variable denote the displacement and the mechanical load, respectively (i.e., and ). Within this context, the natural parameter is the physical time ; that is, (1) can be written as follows:
2.2. Some Commonly Used Algorithms
2.2.1. Classical Newton and Newton-Type Procedures
By placing in the context of solid mechanics and considering that the physical time (with ) is divided into -subintervals (i.e., with and , with ), we have the displacement and the mechanical load at the incremental time (resp., ) that are written as follows: and (resp., and ).
For solving numerically (2), we consider the discrete-time interval and we perform a Taylor series expansion of the function (representing here the mechanical equilibrium of solid) in the first order at point (with being fixed and constant):where denotes the higher-order terms with the Landau notation associated with the asymptotic behaviour of the function (considering only the variable quantity ), is Euclidean norm associated with the quantity (here, Euclidean distance reduces to the absolute value since that there is only one-variable , i.e., ) and (resp., ) denote the th (resp., th) iterative solution associated with the variable at the incremental time (with ), and is the first-order partial derivative operator associated with the function with respect to (at point which is th iterative displacement of the incremental time ) for fixed and constant mechanical load (at point which is the value of parameter of the incremental time ). It should be underscored that the variable (resp., ) without the exponent or represents a converged (resp., known) quantity at the incremental time (resp., ).
In line with (3), we can define the following:(i)classical Newton’s algorithm (see case (a) of Figure 2; cf. [3, 4, 7]):(ii)Newton-type algorithm (see case (b) of Figure 2; cf. [3, 4, 7]): where denotes a coefficient which checks with , is the sign function (such as when , when , and when ), and is the absolute-value function (such as when and when ). It is worth noting that if then this is the classical Newton’s algorithm and if (with being the constant coefficient which does not depend on the point and checking and ) then this is the modified Newton’s algorithm.
2.2.2. Standard Arclength and Pseudo-Arclength Procedures
Similar to previous approach (see Section 2.2.1), we perform a Taylor series expansion of the function (representing the mechanical equilibrium of solid) in the first order at point :where denotes the higher-order terms with the Landau notation associated with the asymptotic behaviour of the function (considering the variable quantities ) and (resp., is the first-order partial derivative operator associated with the function with respect to (resp., ) at point (resp., ) for fixed and constant displacement (resp., mechanical load ) at point (resp., ) which is th iterative variable of incremental time .
Combined with (5), we introduce another scalar equation (so-called “constraint condition”); that is,(i)the equation is nonlinear type (second order) in the case of an arclength algorithm (see case (c) of Figure 2) (e.g., Crisfield procedure [5, 12, 13]);(ii)the equation is linear type in the case of a pseudo-arclength algorithm (see case (d) of Figure 2) (e.g., Riks [14–17], Ramm [18, 19], or Wempner  procedures).
It may be stressed that there exist many other methods used for numerical continuation procedures; one of them, which is not present here, is called “normal flow algorithm” or “Davidenko’s flow algorithm” (see [21, 22] for more details); the mechanical equilibrium equation of solid associated with the Davidenko’s flow reads , where denotes the perturbation parameter.
3. A New Iterative Numerical Continuation Method
3.1. Proposed Algorithm
In this section, we present a new iterative numerical continuation procedure for approximating the solutions associated with any parameterized scalar nonlinear equations. The proposed iterative algorithm belongs to the family of predictor-corrector methods, and it uses both a modified Newton’s method and a stationary-type numerical technique. The stationary procedure allows reducing the considered scalar nonlinear equation (see (1)) to only one explicit equation such aswhere is the considered variable and is the fixed and constant parameter. It should be stressed that the derivative of the function (see (1)) checks thatwhere denotes the first-order total derivative operator (with .
When considering both a discrete-time interval and an orthonormal basis , the direction vector associated with the tangent straight line at point can be written as follows (with the stationary procedure (7)):withwhere denotes the unit vector of the basis such as () and are the components associated with the vector in the orthonormal basis . It should be noted that, in (11), the term since that is the mechanical equilibrium point at the time .
For crossing more easily some critical points associated with the nonlinear function , we introduce a new director vector associated with the straight line at point at the time such aswithwhere (with ) are the components associated with the vector in the orthonormal basis and and are two parameters ().
For the first iterative step (i.e., ), the iterative point must satisfy the following relation (with , ):where is a parameter ().
For the other iterative steps (i.e., ), we introduce the straight line passing thought the point and with the director vector must satisfy the following relations (see Figure 3):withwhere (with ) are the components associated with the vector in the orthonormal basis .
The th iterative solution is obtained when with ;that is,with
In the same way with (20), the iterative solution can be determined as follows (with ):
3.2. Some Comments
It is important to emphasize the following:(1)For the first iterative step at the first increment (i.e., and ), the choice of sign “” in (17) is arbitrary and depends only on the direction for beginning the numerical continuation procedure. Further in this study, we adopt the sign “+.” Nevertheless, for the other first iterative steps at other increments (i.e., and ), we consider a certain continuity of path with respect to the previous converged increment and we adopt the following rule: (i) the sign “+” when or and ; (ii) the sign “−” when or and .(2)The new proposed iterative algorithm (see Section 3.1) uses the first-order partial derivative operator as classical Newton’s algorithm (see Section 2.2.1), but unlike this latter, the critical points can be passed without relatively strong difficulties. Based on a modified Newton’s procedure, the new algorithm considers iterative steps of predictions and corrections which depend on both the first-order partial derivative operator , , and its inverse , , modulated by two parameters ( and ) allowing to pass through the limit points for the first (see (17) and (18)) and other iterations (see (23) and (24)) during a discrete-time interval.(3)The two values associated with the function (see (15) and (24)), which is a linear combination of the first-order partial derivative operator and its inverse , , , represent the fact that are considered: (i) when , the initial straight line previously converged (i.e., with , ); (ii) when , the current iterative straight line is (, ).(4)The new proposed algorithm is composed of three parameters: , , and . The optimal values of these parameters depend clearly on the nonlinear function under consideration. Therefore, a sensibility analysis must be conducted in order to obtain these optimal values. An alternative approach consists, in the first time, to choose the following values: (i) for given: (i-a) and ; (i-b) and ; and (i-c) and ; (ii) for given: (ii-a) and . On the other hand, in the case where and , the new proposed algorithm reduces to a pseudo-arclength type procedure (see Section 2.2.2).
4. Some Numerical Examples
4.1. Preliminary Remarks
In the current section, we propose to test and evaluate the accuracy, efficiency, and robustness associated with the developed iterative method associated with the numerical continuation procedure in Section 3 on some scalar nonlinear equations. Moreover, all the numerical results of this section have been obtained with MATLAB software (see ).
New Iterative Numerical Continuation Algorithm (See Figure 3)(i)For the iterative solution (), one has the following:(a)For the first iteration (), with(b)For the other iterations (), with (ii)For the iterative solution (),
It is important to emphasize the following:(i)On the one hand, we consider only the case where for the new continuation algorithm (see (26)–(30)) used in this section.(ii)On the other hand, we introduce four types of Convergence Criterion () (with ) in order to stop the iterative process associated with the new proposed algorithm: where represents the maximum number of iterations, and (resp., ) are the tolerance parameters associated with the residue error of the function and approximation error criterion of displacement (resp., mechanical load ). In what follows, we consider the following values for each CC: , , and .
We consider the following scalar nonlinear equations:where , , , , and are five parameters chosen with these values: , , , , and .
4.3. Results and Discussion
4.3.1. Predictive Numerical Capabilities
All the numerical results associated with scalar nonlinear functions to (see Section 4.2) are presented in Figures 4–6. The obtained numerical results show fairly clearly that the new proposed iterative numerical algorithm (see Section 3) is capable of giving convenient approximate solutions associated with parameterized scalar nonlinear equations including critical points. Indeed, in Example 1 , there are four critical points and we can see that the new iterative algorithm is able to pass all these points: (i) the first (representing case (c) of Figure 1); (ii) the second (representing case (b) of Figure 1); (iii) the third (representing case (d) of Figure 1); and (iv) the fourth (representing case (a) of Figure 1). In Example 2 , there is only one critical point (representing case (a) of Figure 1) and the new iterative algorithm passes without excessive difficulty this singularity. In the same way, in Example 3 , there are periodic critical points and we can see that the new algorithm passes again the first (representing case (a) of Figure 1) and the second (representing case (b) of Figure 1) singularity associated with the nonlinear function in the interval (where denotes the period of the sinusoidal function). In the light of all the numerical results obtained in Figures 4, 5, and 6, the new iterative numerical continuation algorithm enables passing through the critical points and predicting the approximate solutions associated with parameterized scalar nonlinear equations.
4.3.2. Sensibility Analysis
In this section, we propose a sensibility analysis for evaluating the influence of different values assigned with the parameters (, , and ) used by the new iterative numerical continuation algorithm (see Section 3). All the numerical results associated with the scalar nonlinear functions , , and (see Section 4.2) with different values of the parameters (, , and ) are presented in Figures 7–30.