On rheology, cure kinetics and chemorheology of gum rubbers

Mitra, Arpita; Leonov, A. I.

doi:10.1134/s0965545x10110052

Cited by 2 publications

(3 citation statements)

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“…All the papers that implemented the PL method [ 4 , 8 , 9 , 10 , 11 , 12 ] considered the condition

for the construction of the Padé approximants, Equation (6), which can result in ill-posedness of the PL method. In effect, as will be demonstrated in this section, after applying

, a system of equations will be extracted from Equation (6) that, by performing analysis and numerical computations later in the paper, will show that these equations are ill-conditioned.…”

Section: The Padé-laplace Theorymentioning

confidence: 99%

“…Theoretically, the results of the computation must be independent of

; however, this is not the case in reality. According to previous papers [ 4 , 8 , 9 , 10 , 11 , 12 ], the problem with some values of

is attributed to the round-off errors. Therefore, it was suggested that

must be chosen in an optimal range.…”

Section: The Padé-laplace Theorymentioning

confidence: 99%

“…The PL method has been used in deconvolution of the relaxation spectra of polymers [ 8 , 9 , 10 , 11 , 12 ], but the points raised by Tang and Norris [ 6 ] regarding the instability of the Longman method and in general the problematic computation of Padé approximants have been, so far, overlooked. Knowing the fact that the computation of Padé approximants is a crucial step in PL numerical procedure, an analysis of the PL method, specifically the computation of Padé approximants and the possibility of ill-posedness caused by this computation, is still lacking.…”

Section: Introductionmentioning

confidence: 99%

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A Critical Evaluation and Modification of the Padé–Laplace Method for Deconvolution of Viscoelastic Spectra

Es‐haghi

Gardner

2021

Molecules

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This paper shows that using the Padé–Laplace (PL) method for deconvolution of multi-exponential functions (stress relaxation of polymers) can produce ill-conditioned systems of equations. Analysis of different sets of generated data points from known multi-exponential functions indicates that by increasing the level of Padé approximants, the condition number of a matrix whose entries are coefficients of a Taylor series in the Laplace space grows rapidly. When higher levels of Padé approximants need to be computed to achieve stable modes for separation of exponentials, the problem of generating matrices with large condition numbers becomes more pronounced. The analysis in this paper discusses the origin of ill-posedness of the PL method and it was shown that ill-posedness may be regularized by reconstructing the system of equations and using singular value decomposition (SVD) for computation of the Padé table. Moreover, it is shown that after regularization, the PL method can deconvolute the exponential decays even when the input parameter of the method is out of its optimal range.

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“…All the papers that implemented the PL method [ 4 , 8 , 9 , 10 , 11 , 12 ] considered the condition

for the construction of the Padé approximants, Equation (6), which can result in ill-posedness of the PL method. In effect, as will be demonstrated in this section, after applying

, a system of equations will be extracted from Equation (6) that, by performing analysis and numerical computations later in the paper, will show that these equations are ill-conditioned.…”

Section: The Padé-laplace Theorymentioning

confidence: 99%

“…Theoretically, the results of the computation must be independent of

; however, this is not the case in reality. According to previous papers [ 4 , 8 , 9 , 10 , 11 , 12 ], the problem with some values of

is attributed to the round-off errors. Therefore, it was suggested that

must be chosen in an optimal range.…”

Section: The Padé-laplace Theorymentioning

confidence: 99%