Number Of Fitting Parameters Research Articles

New two-parameter constitutive models for rubber-like materials: Revisiting the relationship between single chain stretch and continuum deformation

The connection between macroscopic deformation and microscopic chain stretch is a key element in constitutive models for rubber-like materials that are based on the statistical mechanics of polymer chains. A new micro-macro chain stretch relation is proposed, using the Irving–Kirkwood–Noll procedure to construct a Cauchy stress tensor from forces along polymer chains. This construction assumes that the deformed polymer network remains approximately isotropic for low to moderate macroscopic stretches, a starting point recently adopted in the literature to propose a non-affine micro-macro chain stretch relation (Amores et al., 2021). Requiring the constructed Cauchy stress to be consistent with the stress tensor derived from the strain energy density results in a new chain stretch relation involving the exponential function. A hybrid chain stretch relation combining the new chain stretch with the well-known affine relation is then proposed to account for the whole range of stretches in experimental datasets. Comparison of the model predictions to experimental data in the literature shows that the two new micro-macro chain stretch relations in this work result in two-parameter constitutive models that outperform those based on existing chain stretches with no increase in the number of fitting parameters used.

Raman spectroscopy analysis of the chemical structure of diamond-like carbon films deposited via high-frequency inclusion high-power impulse magnetron sputtering

Aiming to develop a new high-power impulse magnetron sputtering (HiPIMS) method to further improve the functionality of thin films, we previously designed a high-frequency inclusion HiPIMS (HF-HiPIMS) power supply for obtaining high-performance diamond-like carbon (DLC) films. Herein, the chemical structure of DLC films deposited via HF-HiPIMS was analyzed using Raman spectroscopy. First, the peak positions were fixed, and the number of fitting parameters was reduced by conducting a waveform separation of the initial Raman spectrum using a differential spectrum method. Next, the Raman spectra of the DLC films subjected to five-peak separation and two-peak separation analyses were compared, and the limitations of the two-peak separation analysis are discussed. Furthermore, the relation between the Raman parameters of the five-peak separation analysis and film properties suggested that Raman spectroscopy could be used to estimate the sp3 C–C/(sp3 C–C + sp2 C=C) ratio.

Electron beam impact parameters for the creation of excited species in N2 gas

The number of electron–ion pairs and the distribution of excited species created by the passage of an intense electron beam in a gas are important parameters for many applications. The previously published values for molecular nitrogen rely on a differential ionization cross section that uses a number of fitting parameters and excitation cross sections determined from analytical fitting functions [S. P. Slinker, A. W. Ali, and R. D. Taylor, J. Appl. Phys. 67, 679 (1990)]. Slinker used cross section fits to solve the Boltzmann equation which was then used to compute the important beam-impact parameters. In this work, it is shown that an alternative approach based on the continuous slowing down approximation (CSDA) can be used to compute the energy expended per electron-ion pair and the distribution of excited gas species. This method results in an integral equation that can be solved iteratively and converges rapidly. The binary-encounter Bethe (BEB) differential ionization cross section is used [Y. K. Kim and M. E. Rudd, Phys. Rev. A 50, 3954 (1994); W. Hwang, Y.-K. Kim and M. E. Rudd, J. Chem. Phys. 104, 2956 (1996)]. The BEB model naturally extends to relativistic energies and has no free parameters. This makes the BEB considerably easier to use than previous models based on fitting parameters. The BEB model requires orbital constants obtained from quantum chemistry calculations. To demonstrate the technique, the electron-beam impact parameters are computed for nitrogen gas. The tabulated low-energy excitation cross sections are extended to relativistic energies using Bethe's asymptotic value for the inelastic cross sections [M. Inokuti, Rev. Mod. Phys. 43, 297 (1971)]. It is shown that the results for the energy expended per electron–ion pair as well as the distribution of excited states agree with published experimental values and are similar to previously published theoretical results.

X and Zcs in B+→J/ψϕK+ as s -wave threshold cusps and alternative spin-parity assignments to X(4274) and X(4500)

Recent LHCb amplitude analysis on ${B}^{+}\ensuremath{\rightarrow}J/\ensuremath{\psi}\ensuremath{\phi}{K}^{+}$ suggests the existence of exotic $X$ and ${Z}_{cs}$ hadrons, based on an assumption that Breit-Wigner resonances describe all the peak structures. However, all the peaks and also dips in the spectra are located at relevant meson-meson thresholds where threshold kinematical cusps might cause such structures. This points to the importance of an independent amplitude analysis with due consideration of the kinematical effects, and this is what we do in this work. Our model fits well $J/\ensuremath{\psi}\ensuremath{\phi}$, $J/\ensuremath{\psi}{K}^{+}$, and ${K}^{+}\ensuremath{\phi}$ invariant mass distributions simultaneously, demonstrating that all the $X$, ${Z}_{cs}$, and dip structures can be well described with the ordinary $s$-wave threshold cusps. Spin-parity of the $X(4274)$ and $X(4500)$ structures are respectively ${0}^{\ensuremath{-}}$ and ${1}^{\ensuremath{-}}$ from our model, as opposed to ${1}^{+}$ and ${0}^{+}$ from the LHCb model. With all relevant threshold cusps considered, the number of fitting parameters seems to be significantly reduced. The LHCb data requires ${D}_{s}^{(*)}{\overline{D}}^{*}$ scattering lengths in our model to be consistent with zero, disfavoring ${D}_{s}^{(*)}{\overline{D}}^{*}$ molecule interpretations of ${Z}_{cs}(4000)$ and ${Z}_{cs}(4220)$ and, via the SU(3) relation, being consistent with previous lattice QCD results.

Bias-variance decomposition of overparameterized regression with random linear features.

In classical statistics, the bias-variance trade-off describes how varying a model's complexity (e.g., number of fit parameters) affects its ability to make accurate predictions. According to this trade-off, optimal performance is achieved when a model is expressive enough to capture trends in the data, yet not so complex that it overfits idiosyncratic features of the training data. Recently, it has become clear that this classic understanding of the bias variance must be fundamentally revisited in light of the incredible predictive performance of overparameterized models-models that avoid overfitting even when the number of fit parameters is large enough to perfectly fit the training data. Here, we present results for one of the simplest examples of an overparameterized model: regression with random linear features (i.e., a two-layer neural network with a linear activation function). Using the zero-temperature cavity method, we derive analytic expressions for the training error, test error, bias, and variance. We show that the linear random features model exhibits three phase transitions: two different transitions to an interpolation regime where the training error is zero, along with an additional transition between regimes with large bias and minimal bias. Using random matrix theory, we show how each transition arises due to small nonzero eigenvalues in the Hessian matrix. Finally, we compare and contrast the phase diagram of the random linear features model to the random nonlinear features model and ordinary regression, highlighting the additional phase transitions that result from the use of linear basis functions.

Adaptive fitting of potential energy surfaces of small to medium-sized molecules in sum-of-product form: Application to vibrational spectroscopy.

A semi-automatic sampling and fitting procedure for generating sum-of-product (Born-Oppenheimer) potential energy surfaces based on a high-dimensional model representation is presented. The adaptive sampling procedure and subsequent fitting rely on energies only and can be used for re-fitting existing analytic potential energy surfaces in the sum-of-product form or for direct fits from ab initio computations. The method is tested by fitting ground electronic state potential energy surfaces for small to medium sized semi-rigid molecules, i.e., HFCO, HONO, and HCOOH, based on ab initio computations at the coupled-cluster single double and perturbative triples-F12/cc-pVTZ-F12 or MP2/aug-cc-pVTZ levels of theory. Vibrational eigenstates are computed using block improved relaxation in the Heidelberg multi-configurational time dependent Hartree package and compared to available experimental and theoretical data. The new potential energy surfaces are compared to the best ones currently available for these molecules in terms of accuracy, including resulting vibrational states, required number of sampling points, and number of fitting parameters. The present procedure leads to compact expansions and scales well with the number of dimensions for simple potentials such as single or double wells.

High-Dimensional Learning Under Approximate Sparsity with Applications to Nonsmooth Estimation and Regularized Neural Networks

In “High-Dimensional Learning Under Approximate Sparsity with Applications to Nonsmooth Estimation and Regularized Neural Networks,” Liu, Ye, and Lee study a model fitting problem where there are much fewer data than problem dimensions. Of their particular focus are the scenarios where the commonly imposed sparsity assumption is relaxed, and the usual condition of the restricted strong convexity is absent. The results show that generalization performance can still be ensured in such settings, even if the problem dimensions grow exponentially. The authors further study the sample complexities of high-dimensional nonsmooth estimation and neural networks. Particularly for the latter, it is shown that, with explicit regularization, a neural network is provably generalizable, even if the sample size is only poly-logarithmic in the number of fitting parameters.

A new experimental and theoretical approach for viscosity Iranian heavy crude oils based on tuning friction theory and friction volume theory parameters

In this research work, Friction Theory and Free Volume Theory are applied to live oil characterized based on SARA TEST for viscosity modeling and make a new model in combination with two equation of state (PR and PC-SAFT). Parameters for pseudo-components are obtained by tuning the viscosity at atmospheric pressure and temperatures of 10, 20, and 40 ℃. A new fitting approach is suggested where the number of fitting parameters is 17 and 12 for FT and FVT model, respectively. These parameters are tuned using the Genetic Algorithm and Particle Swarm Optimization and make eight new models. The results show that PC-SAFT provides viscosity predictions for all models with less deviation from experimental viscosity. The FT and FVT models have less error for oils with API > 40 and API < 40, respectively. The PC-SAFT + PSO improves the accuracy in viscosity modeling for both FT and FVT models. PSO can play a significant role even more than PC-SAFT.

Prediction of solidification behavior of biodiesel containing monoacylglycerols above the solubility limit

Monoacylglycerols (MAGs) are typical impurities in biodiesel (fatty acid methyl esters, FAMEs) and are often the cause of solid precipitation because of their high melting points. In this study, the liquidus temperature of biodiesel, below which solidification of biodiesel components can occur, was measured by differential scanning calorimetry or visual observation, and was predicted by thermodynamic models. First, the solubility limit of MAGs, defined as the total MAG content above which MAGs can solidify before FAMEs, was found to be about 0.25 wt% for coconut methyl esters and about 0.5 wt% for palm methyl esters and rapeseed methyl esters. For biodiesel containing MAGs above the solubility limit, the compound formation (CF) model showed good agreement with the experimentally determined liquidus temperatures. This thermodynamic model assumed different types of MAGs solidifying simultaneously while forming molecular compounds. However, within the range of the total MAG content of actual biodiesel (typically less than 0.7 wt%), the number of fitting parameters in the CF model was excessive. This led to the use of a simplified version of the CF model with only one parameter, which still fitted the experimental results well. One parameter value was determined for biodiesel from one feedstock, allowing the liquidus temperature of biodiesel from a known feedstock to be predicted based only on the total MAG content, even for biodiesel containing diacylglycerols in addition to MAGs.

The Delplot kinetic method applied to systems with adsorbates: Hydrodeoxygenation of benzofuran on a bimetallic CoPd phosphide catalyst supported on KUSY

The Delplot method provides a means of analyzing conversion and selectivity data to derive a reaction sequence and has been applied so far to gas-phase or liquid-phase species. This study analyses the applicability of the method for catalytic reactions and considers for the first time adsorbed intermediates. The method is applied to a contact time study of the hydrodeoxygenation (HDO) of benzofuran at 0.5 MPa and 350 °C over a catalyst consisting of bimetallic CoPd phosphide supported on a potassium ion-exchanged ultra-stable Y (KUSY) zeolite. Both simple and detailed reaction networks were derived. The simple networks considered only gas-phase species and were modeled by a first-order sequence of steps, whereas the detailed network took into account adsorbed intermediates and was simulated using a rake mechanism. The networks were compared using F-statistics, which accounted for the differences in the number of fitting parameters. The detailed network gave a better fit to the experimental data because it represented a more realistic description of the transformation. A suggested sequence from the Delplot method was consistent with the simple network but not with the detailed network. The lack of applicability of the Delplot analysis to the network with adsorbates was linked to overly large equilibrium constants, a determination supported by Delplot fitting for a model sequence. This study indicated the inapplicability of the Delplot method in determining reaction sequences that involve adsorbed species with large equilibrium constants for formation.

An ab-initio based semi-empirical thermal conductivity model for multiphase uranium-zirconium alloys

In this work, a three-stage thermal conductivity model for complex metal alloys is developed, based on density functional theory (DFT) calculations, physics rules, and experimental data. The model is broken down into three stages for incorporating incremental complexity—Stage 1: modeling single-phase alloys with fixed compositions over a range of temperatures; Stage 2: modeling single-phase alloys across a range of compositions; and Stage 3: modeling multi-phase mixtures. The model is demonstrated using U-Zr and U-Mo alloys and displays good agreement with experimental data, with a root-mean-square error (RMSE) of about 1.3 W/m-K (~10% error against experimental data) over the typical operating temperature range for all phases of U-Zr. This model delivers the same level of accuracy as existing models for U-Zr and U-Mo, along with the same (or fewer) number of fitting parameters. More importantly, all the fitting parameters have clear physical meaning, providing significant insights into the physics of various electron- and phonon-scattering mechanisms and enabling predictions regarding the thermal conductivities in alloys, with a reduced reliance on experimental data.

A hydrodynamic-plastic formulation for modelling sand using a minimal set of parameters

Conventional plasticity models follow rich mathematical formulations. However, they typically lack a physical basis, so their predictive quality is strongly correlated with the number of fitting parameters used. More recent hydrodynamic principles inject physical clarity by separating scales using distinct thermal and granular temperatures. However, existing hydrodynamic models also require many parameters to reproduce a wide variety of experimental observations. This paper combines the advantages of these two approaches to construct a new physics-based hydrodynamic-plasticity model formulation, which is comprehensively predictive. An illustrative model is developed that relies on only five parameters, all of which have been previously defined and calibrated. Four parameters are easily determined from common geotechnical experiments and address diverse rate-independent phenomena, including pressure and density dependent elasticity, pressure- and density-dependent dilation behaviours, critical state relationships under continuous shearing, liquefying cyclic hystereses, and initial stress-ratio dependent extensional shear response. The one remaining parameter enables the recovery of the rate-dependent rheology of friction and density during steady flow, as well as material response during transient loading. A key feature that further distinguishes the current formulation from previous hydrodynamic ones is that here granular temperature is being related to the measurable kinetic pressure.

Study of the Kinetic Features of Bimodal Gelation in Viscous Flow Fluids

The kinetic features of bimodal gelation in viscous flow fluids, in particular, petroleum products, are analyzed by viscometry. The gelation and densification of petroleum products facilitate the elimination of consequences of their spill. Experimental dependences on the viscosity time of gasoline, kerosene, and winter and summer diesel fuel mixtures with lithium tetralkoxyborate as a gelator are measured. These dependences are found to have a bimodal sigmoidal shape called the dose-response form in scientific publications. A mathematical formula is proposed to describe these bimodal dependences. In accordance with this formula, the gelation process consists of two stages: (1) a reaction of lithium tert-butylate Li–O–C(CH3)3 and trialkoxyborate B(OC8H17)3 with the formation of lithium tetralkoxyborate; (2) a gelation reaction of lithium tetralkoxyborate and hydrocarbons of the mentioned petroleum products. The formula contains a number of fitting parameters, e.g., tinf1 and tinf2 (h) are the onset times of gelation and the formation of a stable gel respectively; h1 and h2 (h–1) are the rate constants of the first and second stages of gelation, and so on. The concentration dependences of gelators on these parameters are determined. Based on these dependences, the conclusions are drawn about the kinetic features of bimodal gelation in viscous flow fluids.

Coarse graining pi pi scattering

We carry out an analysis of pi pi scattering in the IJ=00, 11 and 20 channels in configuration space up to a maximal center-of-mass energy sqrt{s}=1.4 GeV. We separate the interaction into two regions marked by an elementarity radius of the system; namely, a long distance region above which pions can be assumed to interact as elementary particles and a short distance region where many physical effects cannot be disentangled. The long distance interaction is described by chiral dynamics, where a two-pion-exchange potential is identified, computed and compared to lattice calculations. The short distance piece corresponds to a coarse grained description exemplified by a superposition of delta-shell potentials sampling the interaction with the minimal wavelength. We show how the so constructed non-perturbative scattering amplitude complies with the proper analytic structure, allowing for an explicit N/D type decomposition in terms of the corresponding Jost functions and fulfilling dispersion relations without subtractions. We also address renormalization issues in coordinate space and investigate the role of crossing when fitting the scattering amplitudes above and below threshold to Roy-equation results. At higher energies, we show how inelasticities can be described by one single complex and energy dependent parameter. A successful description of the data can be achieved with a minimal number of fitting parameters, suggesting that coarse graining is a viable approach to analyze hadronic processes.

Measuring Activation and Luminescence Time Scales of Upconverting NaYF4:Yb,Er Nanocrystals

Accurate determination of upconversion and luminescence lifetimes requires kinetic modeling of the complete time-resolved response for upconversion luminescence of NaYF4:Yb,Er nanocrystals. Prior investigations typically perform exponential fitting of the tail in the time profile or employ complex systems of differential equations to extract lifetimes. To simplify analysis while fitting the entire time-resolved response, this work introduces a set of simplified models that model the response as a convolution of upconversion and fluorescence processes. Models for two- and three-photon upconversion processes are developed and tested for NaYF4:Yb,Er nanocrystals excited by a 980 nm laser. The results are presented for the transitions 2H11/2, 4S3/2 → 4I15/2 (530 nm, green emission), 4F9/2 → 4I15/2 (650 nm, red emission), and 2H9/2 → 4I15/2 (400 nm, blue emission). Even with the same number of fitting parameters, the two- and three-photon models resulted in better fitting than the simpler models of single and ...

Analysis of flow field design on vanadium redox flow battery performance: Development of 3D computational fluid dynamic model and experimental validation

Homogeneous distribution of the electrolyte over the porous electrode is a critical issue hindering the commercialization of vanadium redox flow batteries, owing to increased overpotential at high current and limited power density of the system. Therefore, an understanding of the physical phenomena regulating mass transport of the electrolyte is crucial to improving system performance. The present work describes the development and experimental validation of a 3D computational fluid dynamic model of a vanadium redox flow battery in a half-cell configuration with an active area of 25 cm2. The model simulates the influence of a single serpentine and an interdigitated flow field. The adoption of the half-cell configuration allows the negative electrode to be considered as a pseudo-reference electrode with zero potential loss, leading to a reduction in computation time and the number of fitting parameters, which can be determined with reduced uncertainty. The developed model includes a traditional fluid dynamic analysis of the electrolyte in the flow field and in the porous electrode, coupled with the electrochemistry of the reactions involved. In both the experiments and the simulations, the single serpentine distributor exhibits better performance and higher pressure drops compared to those of the interdigitated geometry under all the investigated operating conditions. In the analysis of the local reaction rate, both distributors experienced increased reaction rates under the rib, induced by a by-pass flow between adjacent channels. The reaction rate shows a highly heterogeneous distribution in the serpentine geometry, while it is more uniform in the interdigitated configuration.

Thymic involution and rising disease incidence with age

For many cancer types, incidence rises rapidly with age as an apparent power law, supporting the idea that cancer is caused by a gradual accumulation of genetic mutations. Similarly, the incidence of many infectious diseases strongly increases with age. Here, combining data from immunology and epidemiology, we show that many of these dramatic age-related increases in incidence can be modeled based on immune system decline, rather than mutation accumulation. In humans, the thymus atrophies from infancy, resulting in an exponential decline in T cell production with a half-life of ∼16 years, which we use as the basis for a minimal mathematical model of disease incidence. Our model outperforms the power law model with the same number of fitting parameters in describing cancer incidence data across a wide spectrum of different cancers, and provides excellent fits to infectious disease data. This framework provides mechanistic insight into cancer emergence, suggesting that age-related decline in T cell output is a major risk factor.

Quantum enhancement of accuracy and precision in optical interferometry

White-light interferometry is one of today’s most precise tools for determining the properties of optical materials. Its achievable precision and accuracy are typically limited by systematic errors due to a high number of interdependent data-fitting parameters. Here, we introduce spectrally resolved quantum white-light interferometry as a novel tool for optical property measurements, notably, chromatic dispersion in optical fibres. By exploiting both spectral and photon-number correlations of energy-time entangled photon pairs, the number of fitting parameters is significantly reduced, which eliminates systematic errors and leads to an absolute determination of the material parameter. By comparing the quantum method to state-of-the-art approaches, we demonstrate the quantum advantage of 2.4 times better measurement precision, despite requiring 62 times fewer photons. The improved results are due to conceptual advantages enabled by quantum optics, which are likely to define new standards in experimental methods for characterising optical materials.

Unification proposals for fatigue crack propagation laws

PurposeThe purpose of this paper is to propose the new dependences of cycles to failure for a given initial crack length upon the stress amplitude in the linear fracture approach. The anticipated unified propagation function describes the infinitesimal crack-length growths per increasing number of load cycles, supposing that the load ratio remains constant over the load history. Two unification functions with different number of fitting parameters are proposed. On one hand, the closed-form analytical solutions facilitate the universal fitting of the constants of the fatigue law over all stages of fatigue. On the other hand, the closed-form solution eases the application of the fatigue law, because the solution of nonlinear differential equation turns out to be dispensable. The main advantage of the proposed functions is the possibility of having closed-form analytical solutions for the unified crack growth law. Moreover, the mean stress dependence is the immediate consequence of the proposed law. The corresponding formulas for crack length over the number of cycles are derived.Design/methodology/approachIn this paper, the method of representation of crack propagation functions through appropriate elementary functions is employed. The choice of the elementary functions is motivated by the phenomenological data and covers a broad region of possible parameters. With the introduced crack propagation functions, differential equations describing the crack propagation are solved rigorously.FindingsThe resulting closed-form solutions allow the evaluation of crack propagation histories on one hand, and the effects of stress ratio on crack propagation on the other hand. The explicit formulas for crack length over the number of cycles are derived.Research limitations/implicationsIn this paper, linear fracture mechanics approach is assumed.Practical implicationsShortening of evaluation time for fatigue crack growth. Simplification of the computer codes due to the elimination of solution of differential equation. Standardization of experiments for crack growth.Originality/valueThis paper introduces the closed-form analytical expression for crack length over number of cycles. The new function that expresses the damage growth per cycle is also introduced. This function allows closed-form analytical solution for crack length. The solution expresses the number of cycles to failure as the function of the initial size of the crack and eliminates the solution of the nonlinear ordinary differential equation of the first order. The different common expressions, which account for the influence of the stress ratio, are immediately applicable.

A proposal for unification of fatigue crack growth law

In the present paper, the new fractional-differential dependences of cycles to failure for a given initial crack length upon the stress amplitude in the linear fracture approach are proposed. The anticipated unified propagation function describes the infinitesimal crack length growths per increasing number of load cycles, supposing that the load ratio remains constant over the load history. Two unification fractional-differential functions with different number of fitting parameters are proposed. An alternative, threshold formulations for the fractional-differential propagation functions are suggested. The mean stress dependence is the immediate consequence from the considered laws. The corresponding formulas for crack length over the number of cycles are derived in closed form.