Simple Formula Research Articles

About Filtration in a Geophysical Bridge with a Seepage Site

Based on the representation of a modular elliptic function in the form of a combination of simple algebraic formulas that conformally map the area of the velocity hodograph (curved triangle) onto a half-plane, a direct definition of filtration rates in a geophysical bridge is given. For the first time, a family of isotope lines of equal filtration rates was constructed for the inner area of the bridge in the absence of water in the downstream. For special cases, the values of the proposed formulas almost completely coincide with the hydromechanical solution of Masket M. (for 4 cases) and the numerical calculations of Khairullin Z.E. (for 2 cases). The well-known Nelson-Skornyakov F.B. formula for the output filtration rates through the lower face of the jumper, adopted by analogy with the flow outlet from the base of the flatbed into the horizontal drainage, gives an underestimation of the results by up to 45%, and therefore cannot be recommended for practical use.

Simple Scaling as a Tool to Help Assess the Closure-Free da/dN Versus ΔKeff Curve in a Range of Materials

Recent studies have proposed a simple formula, which is based on Elber’s original approach to account for R-ratio effects, for determining the crack closure-free ΔKeff versus da/dN curve from the measured R-ratio-dependent ΔK versus da/dN curves. This approach, which is termed “Simple Scaling,” has been shown to collapse the various R-ratio-dependent curves onto a single curve. Indeed, this approach has been verified for a number of tests on metals, polymers, and a medium-entropy alloy. However, it has not yet been used to help assess/determine the closure-free ΔKeff versus da/dN curve. The current paper addresses this shortcoming and illustrates how to use this methodology to assess the ΔKeff versus da/dN curves given in the open literature for tests on a number of steels, aluminum alloys, STOA Ti-6Al-4V, a magnesium alloy, and Rene 95. As such, it would appear to be a useful tool for assessing fatigue crack growth.

Efficiency Ranking of Photovoltaic Microinverters and Energy Yield Estimations for Photovoltaic Balcony Power Plants

The market for microinverters is growing, especially in Europe. Driven by rising electricity prices and an easing in legislation since 2024, the number of mini-photovoltaic energy systems (mini-PVs) being installed is increasing substantially. Indoor and outdoor studies of microinverters have been carried out at Paderborn University since 2014. In the indoor lab, conversion efficiencies as a function of load have been measured with high accuracy and ranked according to Euro and CEC weightings; the latest rankings from 2024 are included in this paper. In the outdoor lab, energy yields have been measured using identical and calibrated crystalline silicon PV modules; until 2020, measurements were carried out using 215 Wp modules. Because of increasing PV module power ratings, 360 Wp modules were used from 2020 until 2024. In 2024, the test modules were upgraded to 410 Wp modules, taking into account the increase from 600 W to 800 W of inverter power limits, which is suitable for simplified operation permission (“plug-in”) in many European countries within a homogenised legislation area for such mini-photovoltaic energy systems or “balcony power plants”. This legislation for simplified operation also covers overpowered mini-plants, although the maximum AC output remains limited to 800 W. Presently, yield assessments are being carried out in the outdoor lab, which will take at least a year to be valid and comparable. Kits consisting of PV modules, inverters, and mounting systems are also being evaluated. Yield rankings sometimes differ from efficiency rankings due to the use of different MPPT algorithms with different MPP approach speeds and accuracies. To accelerate yield assessment, we developed a novel, simple formula to determine energy yield for any module and inverter configuration, including overpowered systems. This is a linear approach, determined by just two coefficients, a and b, which are given for several inverters. To reduce costs, inverters will be integrated into the module frame or the module terminal box in the future.

Analysing Knowledge Gain in Women Farmers: Insights from Tuber Crop Cultivation Training

Aims: This study aimed to assess the knowledge gain of women farmers participating in a training program on tuber crop cultivation in Thrissur district, Kerala. The training focused on improving their understanding of key agricultural practices related to crops such as cassava, elephant foot yam, coleus, and sweet potatoes. Study Design: Ex post facto design. Place and Duration of Study: The training was organized and conducted by the Center for Gender Studies in Agriculture and Farm Entrepreneurship Development, College of Agriculture, Vellanikkara, Thrissur, Kerala on March 2024. Methodology: A total of 50 women farmers were randomly selected from various panchayats in the Thrissur district to attend the training. A pre- and post-training evaluation was conducted using a structured questionnaire comprising 15 questions, covering topics such as variety selection, soil fertility management, pest and disease control, and post-harvest value addition, and so on. Using grapesAgri1 in R Studio 4.3.1 software, the data were analysed using statistical methods like frequency, percentage, t-test and coefficient of correlation. Results: Pre- and post-training knowledge scores were calculated and analysed using simple percentage formulas to assess knowledge gain. The study revealed a significant improvement in participants' knowledge post-training. The overall average pre-training knowledge score was 9.67 out of 50 (19.34%), while the average post-training score increased to 41.53 out of 50 (83.06%). T-test was also employed to compare the mean of pre-test and post-test evaluation. The percentage of knowledge gain in the pre-training and post-training evaluation were correlated with the independent variables such as age, education, family size, category and information source utilization. The results of correlation revealed that the independent variable such as age (X1), Education (X2), Category (X3) and farm size (X4) are negatively correlated and information source (X5) was positively correlated with the dependent variable. Conclusion: The training program effectively enhanced the knowledge of women farmers in key aspects of tuber crop cultivation, contributing to improved agricultural practices and the potential for increased productivity.

Flexural performance of BFRP grid–ECC in strengthening damaged RC beams

An innovative strengthening method combining basalt fiber reinforced polymer (BFRP) grid and engineered cementitious composites (ECC) is proposed to address the shortcomings in the durability of bonded connections with FRP materials and concrete materials. In this method, the BFRP grid wraps around the original damaged reinforced concrete (RC) structure, and the outer layer is cast with ECC, which mainly acts as a binder to enhance the bonding effect. In order to verify the effectiveness of the proposed method, 10 specimens were prepared and their flexural characteristics and load carrying capacity were investigated. Some factors that may affect the effectiveness of strengthening, such as the degree of initial damage (expressed in terms of different initial damage loads), the thickness of the BFRP grid, and the method of reinforcement, were considered in the experiments. The experimental results show that the proposed strengthening method can significantly improve the bending capacity of the damaged RC beams (7%~43%) without significantly enlarging the cross-section and self-weight. In addition, during the loading tests, localized peeling between the BFRP grid–ECC layer and the RC beams occurs at the time of damage, meanwhile the BFRP grid breaks. This suggests that it is feasible to enhance the bond between BFRP grid and RC by using ECC layer instead of other materials as the bonding agents. The proposed strengthening method has a promising application in strengthening old/residual bridges. In addition, a simple formula for predicting the flexural load capacity of the strengthened beams is derived. The ratio of the proposed formula to test value is between 0.853 and 1.304. The flexural load capacity predicted by the proposed formula is in better agreement with the experimental results.

Toward Implementation of the Pressure-regulated, Feedback-modulated Model of Star Formation in Cosmological Simulations: Methods and Application to TNG

Traditional star formation subgrid models implemented in cosmological galaxy formation simulations, such as that of V. Springel & L. Hernquist (hereafter SH03), employ adjustable parameters to satisfy constraints measured in the local Universe. In recent years, however, theory and spatially resolved simulations of the turbulent, multiphase, star-forming interstellar medium (ISM) have begun to produce new first-principles models, which when fully developed can replace traditional subgrid prescriptions. This approach has advantages of being physically motivated and predictive rather than empirically tuned, and allowing for varying environmental conditions rather than being tied to local-Universe conditions. As a prototype of this new approach, by combining calibrations from the TIGRESS numerical framework with the pressure-regulated feedback-modulated (PRFM) theory, simple formulae can be obtained for both the gas depletion time and an effective equation of state. Considering galaxies in TNG50, we compare the “native” simulation outputs with postprocessed predictions from PRFM. At TNG50 resolution, the total midplane pressure is nearly equal to the total ISM weight, indicating that galaxies in TNG50 are close to satisfying vertical equilibrium. The measured gas scale height is also close to theoretical equilibrium predictions. The slopes of the effective equations of states are similar, but with effective velocity dispersion normalization from SH03 slightly larger than that from current TIGRESS simulations. Because of this and the decrease in PRFM feedback yield at high pressure, the PRFM model predicts shorter gas depletion times than the SH03 model at high densities and redshift. Our results represent a first step toward implementing new, numerically calibrated subgrid algorithms in cosmological galaxy formation simulations.

Formula Design of Hyperbranched Polyarylether‐Based Low Dielectric Photosensitive Resin and Study of UV‐Cured Films Properties

ABSTRACTFor large‐area electronic and high shape complexity structure applications, various solution‐processable materials are chosen as dielectric layer due to their simple molding process via spin‐coating, casting, or 3D printing at room temperature. Herein, the acrylic‐ended fluorinated hyperbranched polyaryletherketone (hb‐P6FAEK‐Ace) was used as the prepolymer and then, the composition of photosensitive resin formula was regulated by adjusting the addition ratio of hb‐P6FAEK‐Ace and diluents (acrylimorpholine [ACMO] 1,6‐Hexanediol diacrylate [HDDA]). These photosensitive resin solutions showed the relative low viscosities ranged ranging from 18 to 253 mPa S at 100°C which was in favor of the future 3D printing method and printing accuracy. Then, the obtained UV‐cured films show the maximum tensile strength up to 71.9 MPa and superior thermal properties with the Tg of 157°C–177°C and T5% of 314°C–351°C. Moreover, they also exhibited the good dielectric performance with dielectric constants of 2.43–2.20 and dielectric loss of 0.013–0.014 at 1 MHz–1 GHz. These abovementioned satisfactory performance could be ascribed to the aromatic skeleton of hb‐P6FAEK‐Ace, introduced polarizability‐CF3 groups and the inner cavities for the hyperbranched structures, also the precise control of formula composition. Additionally, the curing degree, shrinkage of solid, and moisture properties were also researched thoroughly. This work provides a simple formula design though to construct the high performance and low dielectric photosensitive resin for the requirements of microelectronic applications.

Estimation of the friction coefficient by identifying the evolution of rough surface topography

The complexity and randomness of surface roughness make the origin of friction always puzzling. Is there a potential correlation mechanism between the evolution of friction force and surface topography, and is it possible to estimate the friction coefficient from the topography evolution? Here, we explore the evolution of surface topography in metallic materials during friction by defining the characteristic orientation of the surface and analyzing its changes over time, which closely mirrors the behavior of friction forces. Then, we propose a simple relationship that relates the characteristic orientation and friction force via the friction coefficient. Therefore, the friction coefficient can be evaluated based on topography evolution without relying on any theoretical modeling assumptions. These results reveal the mechanism between complex surface topography and friction force through a simple formula and provide a method to estimate the friction coefficient. This method has promising applications when the friction coefficient is difficult to measure, such as in large-scale landslides and geologic faults.

Quantitative analysis of the impact of room temperature stability on cryostat performance

The constant temperature in laboratories is crucial for precise low-temperature measurements. Currently, there are no reports on the specific temperature needs for such experiments or quantitative analyses of how room temperature fluctuations affect cryostat control. This paper proposes a method to quantitatively analyze these fluctuations and their impact on cryostat temperature control. Taking the cryostat using for 2 K-5 K thermodynamic temperature measurement as the research object, and conducts model simulation and experimental validation. The research results indicate the primary influence on the sphere’s temperature is the time-dependent fluctuation of the room temperature. As temperature fluctuations propagate from the cryostat’s outer casing to the sphere, heat conduction is the dominant factor. The temperature fluctuations inside the system caused by environmental temperature fluctuations can be quantitatively expressed by a simple linear formula, with the coefficient being the fluctuation attenuation rate, which is 6.590 × 10-4 in this system. The deviation between experimental results and simulation results is within 5 %. To keep the sphere’s ambient temperature influence below 0.077mK, or 20 % of cryocooler-induced fluctuations, the room temperature must be controlled within 0.12 °C. In addition, the fluctuation period of room temperature is controlled below 1 h, the 0th flange is thickened, the rod is lengthened, and the material of the rod is changed to G10 above 4.2 K and stainless steel below 4.2 K can also effectively attenuate the influence on system temperature control.

A novel formula to assess the QT interval in ventricular paced rhythm

Abstract Background QT interval measurement in the presence of right ventricular pacing (RVP) represents a clinical challenge due to altered ventricular activation resulting in a prolonged QT-interval. The aim of this study is to compare the QT interval in patients before and after ventricular pacing to develop a new formula for QT interval assessment in patients with RVP. Methods and results All patients presenting to our pacemaker clinic with single- or dual chamber device were included in this prospective observational study. Inclusion criteria were no permanent right ventricular stimulation, an intrinsic QRS interval of <120 ms, and ventricular rate >35 bpm and <150 bpm. Three 12-lead electrocardiograms were recorded for each patient: during intrinsic rhythm and 2 following an atrial paced and a ventricular paced QRS at the same rate. The intrinsic QRS and QT time/JT time (QTi/JTi) were measured during intrinsic and subsequent RVP. To obtain the formula, all measurements were performed on 100 patients (mean age 74 ± 11.3, male 33%) (RV apical lead 30% and non-apical lead 70%). Based on our measurements, a new formula QT (non-paced) = 139.23 + 0.55 xQT (paced) was derived. A population of 482 individuals (mean age 75.5±13.2 years (SD), 64% male) with permanent RV pacing were used as a validation group. The intrinsic QRS and QT time were measured during SR and subsequent RVP. To assess the relationship between the predicted QT with RVP and the measured QT without RVP, linear correlations analyses were performed. Our formula had a strong correlation with the irrespective QT without RVP (r=0.50) Conclusion We proposed a new simple formula to accurately estimate QT during RVP. This formula can be easily utilized in clinical practice. Figure 1: MAD characteristic value via plots stands for Median Absolute Difference (i.e. by how many ms the prediction is estimated on average). Figure 2: "Forest Plot" illustrates the correlation coefficients in the form of a forest plot. The dot r represents r and the bars the 95% confidence intervalsMAD characteristic value via plots standForest Plot" illustrates the correlation

Bremsstrahlung radiation power in fusion plasmas revisited: towards accurate analytical fitting

Abstract In fusion plasmas, where electron temperatures $T_e$ range from keV to hundreds of keV, Bremsstrahlung radiation constitutes a significant energy loss mechanism. While various thermal average fitting formulas exist in the literature, their accuracy is limited, particularly for $T_e \leq 20$ keV with error $>10\%$. Additionally, non-relativistic fitting formulas become invalid for $T_e \geq 50$ keV. The accurate calculation of Bremsstrahlung radiation is important for fusion gain study, especially of advanced fuels fusion. In this work, we develop new but still simple fitting formulas that are valid for electron temperatures ranging from small ($\lesssim0.1$ keV) to extreme relativistic range, with errors of less than 1\% even for high atomic number ions (e.g., $Z=30$), which could be sufficient for fusion plasma applications. Both electron-ion (e-i) and electron-electron (e-e) Bremsstrahlung radiations are considered. Both polynomial fitting with truncated ($t=k_BT_e/m_ec^2\leq10$) and one-fit-all formulas are provided. Additionally, we offer fitting formulas for e-i and e-e specifically for electron energies, which could prove useful in non-Maxwellian Bremsstrahlung radiation studies.

The role of losses in determining hyperbolic material figures of merit

Uniaxial materials have achieved new prominence in photonics because they can have hyperbolic spectral regions with metallic (ε<0) and dielectric (ε>0) permittivities along different crystal axes. In the lossless case, this results in an open hyperboloid dispersion relation, allowing materials to support highly confined modes with extremely large wavevectors. However, even small losses change the character of the hyperbolic dispersion from open hyperboloids to closed surfaces with finite maximum k, significantly limiting the extent to which highly-confined modes can be achieved. Here, we derive a simple analytic formula for the dispersion relation in the presence of loss and show that for some typical materials the maximum wavevector in hyperbolic materials is roughly ten times the free-space. The scaling of the maximum wavevector is derived, and it is shown that there is a universal scaling relation between the propagation length and the wavelength, which implies that the shortest wavelengths in any hyperbolic material are strongly attenuated.

Fast and Accurate Numerical Integration of the Langevin Equation with Multiplicative Gaussian White Noise.

A univariate stochastic system driven by multiplicative Gaussian white noise is considered. The standard method for simulating its Langevin equation of motion involves incrementing the system's state variable by a biased Gaussian random number at each time step. It is shown that the efficiency of such simulations can be significantly enhanced by incorporating the skewness of the distribution of the updated state variable. A new algorithm based on this principle is introduced, and its superior performance is demonstrated using a model of free diffusion of a Brownian particle with a friction coefficient that decreases exponentially with the kinetic energy. The proposed simulation technique proves to be accurate over time steps that are an order of magnitude longer than those required by standard algorithms. The model used to test the new numerical technique is known to exhibit a transition from normal diffusion to superdiffusion as the environmental temperature rises above a certain critical value. A simple empirical formula for the time-dependent diffusion coefficient, which covers both diffusion regimes, is introduced, and its accuracy is confirmed through comparison with the simulation results.

Flexural behaviour and shear capacity of RC flat plate sections during fire exposure

PurposeFire safety is a pivotal requirement in building codes. Prescribed design criteria have been the norm to achieve it, which imposes limitations on engineers, including the inability to accommodate new solutions/materials. The shift towards performance-based design offers the potential to address shortcomings of the prescribed design. However, this shift also significantly increases the workload on structural engineers without a corresponding increase in their engineering fees. Simplified design tools are needed to assist engineers in this transition.Design/methodology/approachThe paper is divided into sections investigating equivalent standard fire duration, thermal deformations, flexural behaviour and shear capacity of flat slabs when exposed to fire. The first section conducts a parametric study correlating equivalent and realistic fire durations using the average internal temperature profile (AITP) method, resulting in statistical equations estimating equivalent fire duration. The second section evaluates thermal deformations and flexural behaviour through a parametric study considering various parameters. This section results in statistical equations estimating thermal deformations and flexural behaviour of flat slab sections during fire exposure. The final section focuses on shear capacity, developing simplified heat transfer formulas and statistical equations predicting compression zone depth reduction. The section presents methodologies predicting flat slab sections' one-way and two-way shear capacities during fire exposure.FindingsStructural engineers can use the proposed methods for daily design work without applying complex heat transfer calculations. When the equivalent standard fire duration is utilized, a flat slab’s thermal deformations, flexural behaviour and shear capacity under an actual fire condition can be calculated. As such, the methods would be highly beneficial in assessing the structural integrity of a building during an active fire incident.Originality/valueThe paper provides engineers with the tools required to evaluate the safety of flat slab sections during fire exposure. The methodologies presented in the paper enable engineers to use performance-based design for slab sections by (1) converting any real fire scenario to a standard fire with an equivalent duration, (2) assessing their thermal behaviour, (3) evaluating their flexural behaviour and (4) evaluating their flexural and shear capacities. The paper concludes with a case study example demonstrating the detailed application of the developed methods.

Efficient Computation of Overlap Reduction Functions for Pulsar Timing Arrays.

Pulsar timing arrays seek and study gravitational waves (GWs) through the angular two-point correlation function of timing residuals they induce in pulsars. The two-point correlation function induced by the standard transverse-traceless GWs is the famous Hellings-Downs curve, a function only of the angle between the two pulsars. Additional polarization modes (vector or scalar) that may arise in alternative-gravity theories have different angular correlation functions. Furthermore, anisotropy, linear, or circular polarization in the stochastic GW background gives rise to additional structure in the two-point correlation function that cannot be written simply in terms of the angular separation of the two pulsars. In this Letter, we provide a simple formula for the most general two-point correlation function-or overlap reduction function (ORF)-for a gravitational-wave background with an arbitrary polarization state, possibly containing anisotropies in its intensity and polarization (linear and/or circular). We provide specific expressions for the ORFs sourced by the general-relativistic transverse-traceless GW modes as well as vector (or spin-1) modes that may arise in alternative-gravity theories.

Thermal-deformation behaviors of the primary sealants in double, triple, and multi glazed insulating glass units

Polyisobutylene (PIB), commonly used as the primary sealant of double, triple, and multi glazed insulating glass units (IGUs), provides the key moisture barrier function and determines the expected lifespan of the IGUs and even the entire glass curtain wall systems (GCWSs). Slipping and debonding of the PIB, caused by temperature changes, have resulted in numerous instances of premature failure of building envelopes. However, available research is inadequate in accurately evaluating the service environment of IGUs and thermal-deformation behaviors of this energy-saving building material. This paper presented a simplified method for analyzing the thermal environment of IGUs, considering outdoor air temperature, solar radiation, wind speed, and angle to the horizontal. A numerical modeling method was proposed and validated with the heat transfer tests. Three finite element (FE) models were built and utilized for the precise analysis of temperature-induced deformations of double, tripled, and quadruple glazed IGUs. Simplified calculation formulas for the maximum thermal deformation of PIB in the X and Y directions under different temperature conditions were obtained, taking a new concept “cavity-to-pane ratio” into consideration. Finally, the relationship between the PIB’s temperature-induced deformation and its probability was proposed. The results show that the IGUs in Beijing suffer more than 22 % of their time in harsh service conditions where the difference between indoor and outdoor temperatures exceeds 20 °C. For DIGUs, the maximum temperature-induced deformations of the PIB in X (along the short side of the panel) and Y (along the long side of the panel) directions are 0.142 and 0.214 mm, respectively, corresponding to shear strains of 28.4 % and 42.8 % for the PIB with a thickness of only 0.5 mm. For TIGUs, the maximum deformations increase to 0.159 and 0.239 mm, corresponding to shear strains of 31.8 % and 47.8 %. For QIGUs and the other multi-glazed IGUs, these values can increase to 36.8 % and 55.4 % or more. The methodology proposed in this paper aims to lay the groundwork for further addressing premature failure issues and optimizing edge bond constructions of multi-glazed IGUs.

Розрахунок несної здатності аркової дорожньої споруди з гофрованого металу аналітичним способом

Introduction. One of the possible options for a simple, practical design of an arched road structure (small bridge) made of corrugated metal is considered, which can be considered as one that is more accurate and labor-intensive for the design of such a structure under the road pressure, as the “structure – soil” system for finite element method. Problem Statement. Regular normative and methodological documents for the design and maintenance of road structures (culverts, small bridges, tunnels) made of corrugated metal provide a simple formula for the structure of the structure and deformation of the body for such structures round pipes. The world is constantly streaming a wider range of rational and fairly simple types of road structures. Purpose. To show one of the possible ways of calculating an arched road structure made of corrugated metal on the basis of generally accepted initial provisions and a simple, understandable calculation scheme is the goal and task of this work.

Acoustic performances of earthen walls, review of modeling approaches and tools

Earth construction benefits of a renewed interest particularly for environmental reasons. However, due to its mainly artisanal construction process, acoustic characterization data of earthen walls are quite rare which can limit their use for projects involving acoustic requirements. Moreover, the great diversity in materials as well as in related construction techniques could result in a large spread in terms of acoustical performances. In this context, acoustic characterizations and modelings are necessary to provide a better understanding on earth materials behaviors. In this study, the results from different modeling approaches are compared to few earth walls sound transmission loss data publicly available. The modeling tools including simplified formulas or more elaborated softwares present different levels of complexities and assumptions sometimes highlighting the specificities of this material. Depending on the construction techniques used (earth blocks, light earth, earth panels etc ...), different assumptions can be appropriate. Extrapolations of performances on several walls for both sound transmission loss and sound absorption provide information regarding tendencies and potentially achievable performances for future earth construction projects. Nevertheless, the comparison between the modeling approaches allows gaining insight on the limits of some approaches and the necessary critical analysis before a possible generalization to other earth walls.

Accretion tori around rotating neutron stars II. Oscillations and precessions

The four characteristic oscillation frequencies of accretion flows (in addition to the Keplerian orbital frequency) are often discussed in the context of the time variability of black hole and neutron star (NS) low-mass X-ray binaries (LMXBs). These four frequencies are the frequencies of the axisymmetric radial and vertical epicyclic oscillations, and the frequencies of non-axisymmetric oscillations corresponding to the periastron (radial) and Lense-Thirring (vertical) precessions. In this context, we investigated the effect of the quadrupole moment of a slowly rotating NS and provide complete formulae for calculating these oscillation and precession frequencies, as well as convenient approximations. Simple formulae corresponding to the geodesic limit of a slender torus (and test-particle motion) and the limit of a marginally overflowing torus (a torus exhibiting a critical cusp) are presented, and more general approximate formulae are included to allow calculations for arbitrarily thick tori. We provide the Wolfram Mathematica code used for our calculations together with the C++ and PYTHON codes for calculating the frequencies. Our formulae can be used for various calculations regarding the astrophysical signatures of the NS super-dense matter equation of state. For instance, we demonstrate that even for a given fixed number of free parameters, a model that accounts for fluid flow precession matches the frequencies of twin-peak quasiperiodic oscillations observed in NS LMXBs better than a model that uses geodesic precession.

Direct Identification of the Continuous Relaxation Time and Frequency Spectra of Viscoelastic Materials

Relaxation time and frequency spectra are not directly available by measurement. To determine them, an ill-posed inverse problem must be solved based on relaxation stress or oscillatory shear relaxation data. Therefore, the quality of spectra models has only been assessed indirectly by examining the fit of the experiment data to the relaxation modulus or dynamic moduli models. As the measures of data fitting, the mean sum of the moduli square errors were usually used, the minimization of which was an essential step of the identification algorithms. The aim of this paper was to determine a relaxation spectrum model that best approximates the real unknown spectrum in a direct manner. It was assumed that discrete-time noise-corrupted measurements of a relaxation modulus obtained in the stress relaxation experiment are available for identification. A modified relaxation frequency spectrum was defined as a quotient of the real relaxation spectrum and relaxation frequency and expanded into a series of linearly independent exponential functions that are known to constitute a basis of the space of square-integrable functions. The spectrum model, given by a finite series of these basis functions, was assumed. An integral-square error between the real unknown modified spectrum and the spectrum model was taken as a measure of the model quality. This index was proved to be expressed in terms of the measurable relaxation modulus at uniquely defined sampling instants. Next, an empirical identification index was introduced in which the values of the real relaxation modulus are replaced by their noisy measurements. The identification consists of determining the spectrum model that minimizes this empirical index. Tikhonov regularization was applied to guarantee model smoothness and noise robustness. A simple analytical formula was derived to calculate the optimal model parameters and expressed in terms of the singular value decomposition. A complete identification algorithm was developed. The analysis of the model smoothness and model accuracy for noisy measurements was carried out. The equivalence of the direct identification of the relaxation frequency and time spectra has been demonstrated when the time spectrum is modeled by a series of functions given by the product of the relaxation frequency and its exponential function. The direct identification concept can be applied to both viscoelastic fluids and solids; however, some limitations to its applicability have been pointed out. Numerical studies have shown that the proposed identification algorithm can be successfully used to identify Gaussian-like and Kohlrausch–Williams–Watt relaxation spectra. The applicability of this approach to determining other commonly used classes of relaxation spectra was also examined.