Determination of relaxation and retardation spectrum by inverse functional filtering

The article is devoted for the determination of the relaxation and retardation spectrum (RRS) from monotonic time- and frequency-domain material functions by the inverse functional filters executing discrete convolution algorithms for geometrically spaced data. It is shown that the problem of RRS determination from a wide variety of material functions leads to the three inverse filtering tasks on a logarithmic time or frequency scale with the three specific frequency responses concerning: (i) the time-domain functions, (ii) the real parts and (iii) the imaginary parts of the frequency-domain functions, and three algorithms (having the versions with even and odd number of coefficients) are to be applied to: (i) time-domain compliance and modulus functions, (ii) their derivatives, and (iii) frequency-domain functions. It is demonstrated that ill-posedness of an inverse filter manifests as large sampling-rate-dependent noise amplification coefficients. A novel regularization strategy allowing to ensure the desired noise immunity is proposed based on choosing sampling rate for geometrically spaced data. The performance of the algorithms is investigated. Optimal sampling rates are disclosed for specific material functions. The frequency range of 2–3 decades is established to be optimal for the recovery of a single RRS point estimate ensuring maximum accuracy with reasonable noise immunity. Practical algorithms are proposed for recovering RRS from the real and imaginary parts of frequency-domain functions. Some known non-parametric methods are compared with the suggested functional filters.     

Resin cure monitoring system based on nondestructive dielectric spectrometry

An on-line indirect cure monitoring prototype system is described based on dielectric spectrometric measurements (frequency dependences of the dielectric modulus and/or dielectric permittivity and/or dielectric loss factor and/or relaxation spectrum). A key block of the system is the Dielectric Spectrometer. Model DS 7056-C interfaced with a standard PC. A variation of the dielectric frequency spectra during the cure process for two epoxy systems based on araldite resins LY556 and LY561-1 are considered as an illustration of capabilities of the described hardware and software.Institute of Polymer Mechanics, Riga, Latvia LV-1006. Published in Mekhanika Kompozitnykh Materialov, Vol. 32. No. 3, pp. 401–409, May–June, 1996.     

Determination of the relaxation and retardation spectra – a view from the up-to-date signal processing perspective

The problem of determination of relaxation and retardation spectra (RRS) is considered from the viewpoint of up-to-date signal processing. It is shown that the recovery of RRS represents the Mellin deconvolution problem, which transforms into the Fourier deconvolution problem for data on a logarithmic time or frequency scale, where it can also be treated as the inverse filtering problem. On this basis, discrete deconvolution (inverse) filters operating with geometrically sampled data are proposed to use as RRS estimators. Appropriate frequency responses and algorithms are derived for estimating RRS from eight different material functions. The noise amplification coefficient is suggested to use as a measure for quantifying the degree of ill-posedness and illconditioness of the RRS recovery problem and algorithms. A methodology is developed for designing RRS estimators with a desired noise amplification, producing maximum accurate spectra for available limited input data. Practical algorithms for determining RRS are proposed, and their performance is studied. The algorithms suggested are compared with the so-called moving-average formulae. It is demonstrated that the minimum frequency range for recovering the point estimate of a relaxation spectrum depends on the allowable noise amplification (the degree of ill-conditioness) and is in no way limited by 1.36 decades, as it is stated by the sampling localization theorem.     

Signal Processing in Relaxation Experiments

Signal processing problems arising in the study of the linearly viscoelastic behavior of polymers and composites are considered. It is shown that the great amount of data conversions is associated with integral transforms using kernels which depend on the ratio or product of arguments for monotonic long-time-interval and wide-frequency-band functions (signals). A unified method of carrying out these integral transforms is developed by combining a logarithmic transformation of the signal time scale with digital filtering. For integral transforms leading to ill-conditioned inverse problems, a method of regularization is proposed based on choosing a sampling rate which ensures an acceptable error variance of the output signal. The specific features of the functional filters used for performing the functional (integral) transforms are discussed. Examples of performing the Heaviside-Carson sine transform and an inherently ill-conditioned problem of inverting the integral transform for determining the relaxation spectrum are represented by digital functional filters.     

Structure evaluation of materials by electrical methods

A theoretical and practical framework of different problems has been discussed to explain advantages and limitations of techniques based on electrical sensing methods which are consistent to implement structure evaluation of materials. The term structure evaluation is generally used here to mean extracting the quantitative information of interest of a material or an object to be examined from a physical measurement, or more typically from a set of physical measurements, that themselves may be only indirectly related to the quantitative information desired. The paper emphasizes the point that the structure evaluation is a typical inverse problem and as such is made more difficult, since inverse problems are characteristically illconditioned, that is, small errors in the measurement typically lead to large errors in the solution. Use of physically motivated a priori information to diminishing such ill-conditioning is discussed. Features of updated electrical sensing methods, particularly those used for nondestructive testing (NDT) are considered. Potential advantages of the electrical methods compared to other NDT techniques are that they offer inexpensive, safe, nonharmful, and fast testing. Some novel achievements including excitation with a number of complex electric field excitation patterns on the object together with computer-assisted information acquisition have considerably extended both the potential and the scope of the electrical methods which will still be recognized. In order to present an up-to-date perspective, some conceptual and methodologic aspects have been considered for two structure evaluation problems, i.e., for generating a) algorithms for determining structural parameters of materials, such as density, moisture content, polymerization degree, etc., from dielectric spectra; b) tomographic cross-sectional maps of electrical parameters of the object to be examined.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October, 1995.Institute of Polymer Mechanics, Riga, Latvia. Published in Mekhanika Kompozitnykh Materialov, No. 1, pp. 124–134, January–February, 1996.     

Structural evaluation of materials by artificial neural networks

It is shown that due to the complexity of interaction of the excitation field with a material in determining its physical characteristics, as well as sophisticated correlation relationships between the physical characteristics and structure of a real material, in many cases, relization of the structural evaluation of materials on the basis of idealized mathematical models of the underlying physical processes is of limited use. As an alternative, it is proposed to use an artificial neural network for the extraction of quantitative information of interest from measurements of the physical characteristics. A general overview of artificial neural networks is given. A methodology of the use of a multilayer perceptron for determining various structural parameters from the dielectric spectra is described. As an example, determination of the moisture content and density of sawdust from the dielectric modulusis considered by the neural-network technique. The noise performance of the neural network is analyzed by applying an additive and multiplicative noise to the input data.Institute of Polymer Mechanics, University of Latvia, Riga, LV-1006 Latvia. Published in Mekhanika Kompozitnykh Materialov, Vol. 35, No. 1, pp. 109–124, January–February, 1999.     

A Study on the Information Capability of Dielectric Spectra in Monitoring Nonisothermal Cure Processes

The monitoring of nonisothermal cure processes by means of nondestructive dielectric spectrometry is considered for two epoxy systems with araldite resins. It is demonstrated that the dielectric characteristics at different frequencies have different sensitivities to the recognition of cure processes. An attempt to evaluate the cure process from dielectric measurements by artificial neural networks is made. It is shown that a properly trained artificial neural network allows one to detect all stages of the cure process to a high accuracy.