Nonlinear Formula Research Articles

Development of a smoothed particle hydrodynamics model for porous media flows with enhanced volume conservation and the revisit of the mass conservation equation

Porous media exist extensively in hydraulic and coastal engineering structures, while the modeling of wave/flow interaction with porous media remains challenging. This work develops a smoothed particle hydrodynamics (SPH) model for accurately simulating wave/flow interaction with porous media. The mass and momentum conservation equations incorporating the mixture theory are adopted. The resistant forces of the solid skeleton of porous media on fluid flows are described by the nonlinear empirical formula. The research contributions of the work lie in two aspects. First, two categories of mass conservation equations for porous media flow are revisited and analyzed to examine the influences of the local time derivative term of fluid volume fraction on simulation results. Second, the Volume Conservation Shifting scheme is, for the first time, introduced into SPH to enhance volume conservation for simulating porous media flows. The developed SPH model is validated by an analytical case of seepage flows in a U-tube with porous media and then applied to study four benchmark examples involving both saturated and unsaturated porous media, i.e., dam-break flow through a crushed stone dam, rapid seepage flow through a rockfill dam, solitary wave propagation over a porous seabed, and solitary wave propagation over a submerged porous breakwater. The morphological features and dynamic pressure heads of the porous media flows have been satisfactorily predicted, demonstrating the good accuracy and enhanced volume conservation of the developed SPH model.

A stacking sequence optimization strategy for process‐induced deformation of multi‐layer thick asymmetric laminates

AbstractThe occurrence of process‐induced deformations of composites laminates is challenging for assembly accuracy and may lead to a service life reduction of parts. However, it can be obviously mitigated through different optimization strategies on the basis of the accurate curing process simulation. In this study, a stacking sequence optimization strategy is proposed and applied to multi‐layer thick asymmetric laminates. The shapes of deformed laminate plates are experimentally investigated in virtue of the three‐dimensional coordinate measuring machine. The thermo‐chemical–mechanical behaviors of plates are first verified through the comparisons of model predicted and experimental process‐induced deformations. Then the nonlinear control formula achieved through the regression model is proposed for the direct relationship between stacking sequences and process‐induced deformations. Finally, the required solutions are generated by solving the control formula. With the comparisons between the average deformations before and after optimizations, it is found that the magnitudes of deformations are significantly reduced, especially when the unoptimizated deformations are large.Highlights The thermo‐chemical–mechanical behavior can be described in the curing process simulation model. The deformed shapes of multi‐layer thick asymmetric laminate plates are measured by a three‐dimensional coordinate measuring machine. The stacking sequence optimization strategy is implemented by introducing a nonlinear control formula.

The Noise Reduction Algorithm May Not Compensate for the Degradation in Output Signal-to-Noise Ratio Caused by Wide Dynamic Range Compression.

Most modern hearing aids (HAs) employ wide dynamic range compression (WDRC) and noise reduction (NR) algorithms. It is known that the nonlinear effects of WDRC and NR cause changes to the output signal-to-noise ratio (SNR) of an HA. However, the relative contributions of WDRC and NR to the nonlinear effects are not fully understood. The current study investigated (a) whether WDRC or NR dominates the nonlinear effects measured at the output of a digital HA and (b) whether the electroacoustic effectiveness of NR depends on WDRC parameters while input SNR and background noise are systematically varied. Test stimuli were Connected Speech Test sentences in multitalker babble noise (2- or 20-talker), presented at input SNRs ranging from -10 to +10 dB. The HA was programmed using multiband WDRC set according to the National Acoustic Laboratories for Nonlinear HA fitting formula 2 prescriptive fits for four standard audiograms and two compression speeds. The NR algorithm of the HA was switched on or off in separate conditions. Nonlinear electroacoustic effects from the WDRC and NR algorithms were assessed by measuring the output SNR of the HA using a phase-inversion technique. To investigate whether there are other factors that may be important besides the output SNR, the Hearing Aid Speech Intelligibility Index and the Hearing Aid Speech Quality Index were applied to the recordings to generate inferences on aided speech intelligibility and perceived speech quality. Results showed that WDRC dominated the net nonlinear effect at low-input SNRs, and the net nonlinear effect of WDRC and NR was reduced at high-input SNRs. Results also showed that the effectiveness of NR depended on compression parameters. The effectiveness of NR was partially explained by the trend of Hearing Aid Speech Intelligibility Index and Hearing Aid Speech Quality Index scores, potentially indicating that the Hearing Aid Speech Intelligibility Index and Hearing Aid Speech Quality Index scores may capture factors that cannot be captured by the output SNR metric. Results suggest that the individual signal-processing stages in an HA should not be considered as independent. Electroacoustic evaluation of WDRC and NR algorithms in isolation is not sufficient to capture the combined nonlinear effect of the two algorithms. https://doi.org/10.23641/asha.25962541.

An Innovative and Efficient Implementation of Matrix-Free Newton Krylov Method for Neutronics/Thermal-hydraulics Coupling Simulation

The core physical behavior of reactors is essentially the result of multi-physical fields coupling feedback. High-fidelity neutronics/thermal-hydraulics (N/TH) analysis can simulate and predict nuclear reactor core phenomena realistically, providing advanced and reliable technical means during the design and safety analysis of nuclear reactor. In this work, an efficient and robustness coupling method using power density as the coupling parameter, Matrix-Free Newton Krylov (MFNK) method, is successfully developed and innovatively implemented in HNET for high-fidelity N/TH coupling simulation. To enhance the efficiency and stability, the multi-level generalized equivalence theory-based CMFD (ML-gCMFD) iterative acceleration method and ML-gCMFD coupling acceleration method are proposed. In addition, the nonlinear preconditioning and hybrid perturbation size formula are implemented to further improve the convergence. Finally, to evaluate the numerical accuracy, convergence, efficiency and stability of MFNK method, a series of representative problems, including a three-dimensional (3D) single fuel pin problem, VERA Benchmark Problem 6, and VERA Benchmark Problem 7, are analyzed by comparing with the current N/TH coupling methods. Numerical results indicate that MFNK method can obtain strong stability, high convergence performance, and relatively high computational efficiency while ensuring high accuracy. It demonstrates that MFNK method has significant performance advantages and potential for high-fidelity N/TH coupling simulation.

Calculation of Porosity and CO2 Content in CO2-Bearing Gas Formations Based on Density and Neutron Logging Forward and Inverse Methods.

When the gas reservoir contains CO2, the logging response tends to be complex. When calculating porosity with neutron and density curves, the accuracy of porosity calculation is reduced due to the uncertainty of fluid parameters, which in turn affects the evaluation of CO2 content. It increases the uncertainty of reserve evaluation. In this paper, a forward modeling model of density logging is established based on the equivalent volume model, and the method of neutron logging is formed by putting forward the nonlinear formula of the reciprocal of the comprehensive deceleration length. The key parameters and their variation rules of neutron logging and density logging forward modeling under different temperature and pressure conditions are obtained by experimental means and Monte Carlo method, respectively. The variation law of the influence of different CO2 content on neutron logging and density logging curves is simulated. The research shows that with the increase of CO2 content, the neutron logging value of gas reservoir decreases, while the density logging value increases, and the greater the porosity of gas reservoir, the more obvious this change trend is. Compared with CH4, CO2 has a more significant impact on the excavation effect of neutron logging. On this basis, combined with the conductivity model of the study area, the porosity and CO2 content of CO2 bearing gas reservoir are solved by the least-square method. The practice shows that the relative error of porosity calculation is within ±4% and the error of CO2 content calculation is within ±10%, which proves the reliability of this method.

Uniform Cubic Spline Interpolation with Some Additional Techniques for Functions with Strong Gradients

Spline interpolation is commonly applied in many applications because several desirable interpolation properties are evident and related to the smoothness and accuracy. It may distort somewhat because of the Gibbs phenomenon though it is less noticeable if there is a gap or if the change from one slope to another is steep. Moreover, the issues of data consistency are needed in some applications, but these peculiarities are not checked automatically by interpolation. Thus, it becomes imperative to apply a few more additional technologies to facilitate the uniformity. Orden and repetitiveness of C^2 are not ensured simultaneously, and thus, the last interpolation is not a spline in actuality. The analysis of the given problem is carried out in the context of developing a sufficient condition for generating a homogeneous cubic spline based on the cubic Hermite interpolator and the identification of two construction methods based on nonlinear formulas. As such, these methods are aimed at reducing the regions where accuracy of order is not maximum or where interpolation is less than C^2 order. We compare the order of accuracy and the intrapolar pattern of the extended and the original method and perform many numbers experiment to check with theory.

Measurement and Calculation of Sediment Transport on an Ephemeral Stream

Sediment transport remains a significant challenge for researchers due to the intricate nature of the physical processes involved and the diverse characteristics of watercourses worldwide. A type of watercourse that is of particular interest for study is the ephemeral streams, found primarily in semiarid and arid regions. Due to their unique nature, a new measurement algorithm was created and a modified bed load sampler was built. Measurement of the bed load transport rate and calculation of the water discharge were conducted in an ephemeral stream in Northeastern Greece, where the mean calculated streamflow rate ranged from 0.019 to 0.314 m3/s, and the measured sediment load transport rates per unit width varied from 0.00001 to 0.00213 kg/m/s. The sediment concentration was determined through various methods, including nonlinear regression equations and formulas developed by Yang, with the coefficients of these formulas calibrated accordingly. The results demonstrated that the equations derived from Yang’s multiple regression analysis offered a superior fit compared to the original equations. As a result, two modified versions of Yang’s stream sediment transport formulas were developed and are presented to the readership. To assess the accuracy of the modified formulas, a comparison was conducted between the calculated total sediment concentrations and the measured total sediment concentrations based on various statistical criteria. The analysis shows that none of Yang’s original formulas fit the available data well, but after optimization, both modified formulas can be applied to the specific ephemeral stream. The results indicate also that the formulas derived from the nonlinear regression can be successfully used for the determination of the total sediment concentration in the ephemeral stream and have a better fit compared to Yang’s formulas. The correlation from the nonlinear regression equations suggests that total sediment transport is primarily influenced by water discharge and rainfall intensity, with the latter showing a high correlation coefficient of 0.998.

Comparative Analysis of Laboratory-Based and Spectroscopic Methods Used to Estimate the Algal Density of Chlorella vulgaris.

Chlorella vulgaris is of great importance in numerous exploratory or industrial applications (e.g., medicals, food, and feed additives). Rapid quantification of algal biomass is crucial in photobioreactors for the optimization of nutrient management and the estimation of production. The main goal of this study is to provide a simple, rapid, and not-resource-intensive estimation method for determining the algal density of C. vulgaris according to the measured parameters using UV-Vis spectrophotometry. Comparative assessment measurements were conducted with seven different methods (e.g., filtration, evaporation, chlorophyll a extraction, and detection of optical density and fluorescence) to determine algal biomass. By analyzing the entire spectra of diluted algae samples, optimal wavelengths were determined through a stepwise series of linear regression analyses by a novel correlation scanning method, facilitating accurate parameter estimation. Nonlinear formulas for spectrometry-based estimation processes were derived for each parameter. As a result, a general formula for biomass concentration estimation was developed, with recommendations for suitable measuring devices based on algae concentration levels. New values for magnesium content and the average single-cell weight of C. vulgaris were established, in addition to the development of a rapid, semiautomated cell counting method, improving efficiency and accuracy in algae quantification for cultivation and biotechnology applications.

Improving Microstructural Estimation in Time-Dependent Diffusion MRI With a Bayesian Method.

Accurately fitting diffusion-time-dependent diffusion MRI (td-dMRI) models poses challenges due to complex and nonlinear formulas, signal noise, and limited clinical data acquisition. Introduce a Bayesian methodology to refine microstructural fitting within the IMPULSED (Imaging Microstructural Parameters Using Limited Spectrally Edited Diffusion) model and optimize the prior distribution within the Bayesian framework. Retrospective. Involving 69 pediatric patients (median age 6 years, interquartile range [IQR] 3-9 years, 61% male) with 41 low-grade and 28 high-grade gliomas, of which 76.8% were identified within the brainstem or cerebellum. 3 T, oscillating gradient spin-echo (OGSE) and pulsed gradient spin-echo (PGSE). The Bayesian method's performance in fitting cell diameter ( ), intracellular volume fraction ( ), and extracellular diffusion coefficient ( ) was compared against the NLLS method, considering simulated and experimental data. The tumor region-of-interest (ROI) were manually delineated on the b0 images. The diagnostic performance in distinguishing high- and low-grade gliomas was assessed, and fitting accuracy was validated against H&E-stained pathology. T-test, receiver operating curve (ROC), area under the curve (AUC) and DeLong's test were conducted. Significance considered at P < 0.05. Bayesian methodology manifested increased accuracy with robust estimates in simulation (RMSE decreased by 29.6%, 40.9%, 13.6%, and STD decreased by 29.2%, 43.5%, and 24.0%, respectively for , , and compared to NLLS), indicating fewer outliers and reduced error. Diagnostic performance for tumor grade was similar in both methods, however, Bayesian method generated smoother microstructural maps (outliers ratio decreased by 45.3% ± 19.4%) and a marginal enhancement in correlation with H&E staining result (r = 0.721 for compared to r = 0.698 using NLLS, P = 0.5764). The proposed Bayesian method substantially enhances the accuracy and robustness of IMPULSED model estimation, suggesting its potential clinical utility in characterizing cellular microstructure. 3 TECHNICAL EFFICACY: Stage 1.

Numerical investigation on the convergence of self-consistent Schrödinger-Poisson equations in semiconductor device transport simulation

Semiconductor devices at the nanoscale with low-dimensional materials as channels exhibit quantum transport characteristics, thereby their electrical simulation relies on the self-consistent solution of the Schrödinger-Poisson equations. While the non-equilibrium Green’s function (NEGF) method is widely used for solving this quantum many-body problem, its high computational cost and convergence challenges with the Poisson equation significantly limit its applicability. In this study, we investigate the stability of the NEGF method coupled with various forms of the Poisson equation, encompassing linear, analytical nonlinear, and numerical nonlinear forms Our focus lies on simulating carbon nanotube field-effect transistors (CNTFETs) under two distinct doping scenarios: electrostatic doping and ion implantation doping. The numerical experiments reveal that nonlinear formulas outperform linear counterpart. The numerical one demonstrates superior stability, particularly evident under high bias and ion implantation doping conditions. Additionally, we investigate different approaches for presolving potential, leveraging solutions from the Laplace equation and a piecewise guessing method tailored to each doping mode. These methods effectively reduce the number of iterations required for convergence.

Modeling and analysis of vehicle stability region based on Lyapunov and coordinated control

The timely intervention of the assisted driving system is the key to improving the handling stability and roll stability of the vehicle, and the vehicle stability region serves as the core basis for determining the intervention timing of the assisted driving system. With the aim of modeling and analyzing the vehicle stability region, a three-degree-of-freedom (3-DOF) vehicle dynamics model including yaw, roll, and lateral motions, as well as a nonlinear Magic Formula tire model are established in this paper. Based on this, a simplified but improved cubic tire model is developed to accurately fit the tire lateral force of Magic Formula tire model within a larger range of slip angles. Subsequently, using the Lyapunov method, the roll stability region and the yaw stability region are respectively constructed, and the accuracy verification and robustness analysis of the established stability region are conducted in the Matlab/Simulink environment. Finally, a model-free adaptive control method is employed to keep the vehicle state within the stability region, without tracking specific vehicle state objectives. The study in this paper can provide theoretical support for stability boundary determination, formulation of intervention timing for assisted driving stability control, and coordination control of vehicle stability and anti-rollover with local compatibility or even conflicts.

Nonlinear vibration characteristics of axially moving anisotropic membrane with gas-thermal-elastic coupling

In the actual printing production process, the printing electronic motion membrane is susceptible to transverse vibration caused by interference from air drag, temperature changes, and other external conditions, resulting in membrane wrinkles, slippage, and other phenomena in the transmission process. We studied the bifurcation and chaos movement properties of anisotropic membranes under air drag and temperature. According to D’Alembert’s theory and von Kármán’s principle, the nonlinear dynamic differential formulas of axially moving anisotropic membranes with gas-thermal-elastic coupling are established. The Galerkin method is applied to discretize the formulas to obtain the state equation of the system. Finally, numerical simulations are performed by applying the fourth-order Runge–Kutta method to analyze the bifurcation and chaos of the system in terms of orthotropic coefficient, dimensionless air drags, and dimensionless temperature. The bifurcation diagrams, Lyapunov exponent diagrams, displacement time-history diagrams, phase-trajectory plane diagrams, and Poincaré diagrams of the membrane system are obtained. The results show that the anisotropic coefficient, dimensionless air drag, and dimensionless temperature significantly impact the investigated nonlinear dynamic of the anisotropic membrane, which provides a theoretical basis for production efficiency and high-quality printing equipment.

Rationale for the extrapolation procedure in selected configuration interaction.

Selected configuration interaction (SCI) methods have emerged as state-of-the-art methodologies for achieving high accuracy and generating benchmark reference data for ground and excited states in small molecular systems. However, their precision relies heavily on extrapolation procedures to produce a final estimate of the exact result. Using the structure of the exact electronic energy landscape, we provide a rationale for the common linear extrapolation of the variational energy as a function of the second-order perturbative correction. In particular, we demonstrate that the energy gap and the coupling between the so-called internal and external spaces are the key factors determining the rate at which the linear regime is reached. Starting from the first principles, we also derive a new non-linear extrapolation formula that improves the post-processing of data generated from SCI methods and can be applied to both ground- and excited-state energies.

Nonlinear Landauer formula for thermal transport of the electrons

The Landauer formula, originally formulated in the context of linear transport, has been a powerful tool for studying quantum devices. However, recent research has shown that extending its application to nonlinear and nonreciprocal transport is crucial for a more comprehensive understanding. In this work, we develop a nonlinear Landauer formula for thermal transport of the electrons and apply it to investigate thermal transport in graphene. Our study reveals intriguing phenomena especially in the presence of large temperature gradients and at low system temperatures. At these conditions, higher-order nonlinear currents emerge, indicating the significance of nonlinear effects in thermal transport. Unlike thermoelectric conductivity, thermal conductivity can be decomposed into intrinsic and extrinsic terms. This decomposition is based on whether the contributions rely on the derivative of the transmission coefficient with respect to energy. This nonlinear Landauer formula presented here serves as a valuable tool for future investigations into the intricate interplay between temperature gradients, system temperatures, and thermal transport in quantum devices.

An asymptotic expansion of the solution of a semi-linear partial differential equation implied by a nonlinear Feynman–Kac formula

This paper introduces an asymptotic expansion for the smooth solution of a semi-linear partial differential equation. Our scheme is based on Itô’s formula, Taylor’s expansion, nonlinear Feynman–Kac formula and some algebras.

Linear and nonlinear analytical equations for fast design of three types of triangular-amplified compliant mechanisms

Compliant amplification mechanisms based on the triangular principle have attracted considerable applications in precision and other engineering fields due to their compactness and efficient amplification capacity. However, a fast engineering design for geometric nonlinearity of large strokes is still much challenging. We report herein a series of analytical equations of nonlinear amplification ratio, input and output stiffness for three commonly-used triangular-amplified compliant mechanisms, namely the rhombus, diamond and bridge types. The pronounced geometric nonlinearities of axially-loaded stiffening and kinematics-arching effects are explicitly formulated. The proposed nonlinear formulas can be directly degenerated as the linear models by vanishing the nonlinear terms, which enables a comprehensive analysis of linear and nonlinear kinetostatics, and hence offers a straightforward way for the fast performance evaluation and size synthesis/optimization. This relieves an engineer's experience and knowledge for an ab initio modeling process. The prediction error is discussed and some insights into the linear and nonlinear characteristics of the three commonly-used triangular-amplified compliant mechanisms are outlined that confirms the advantage of analytical equations bringing to a fast engineering analysis and design.

Magneto-Mechanical Coupling Study of Magnetorheological Elastomer Thin Films for Sensitivity Enhancement.

Magnetorheological elastomer thin films (MREFs) exhibit remarkable deformability and an adjustable modulus under magnetic fields, rendering them promising in fields such as robotics, flexible sensors, and biomedical engineering. Here, we fabricated MREF by introducing magnetostrictive particles (MSPs) and evaluated the magneto-mechanical coupling effect on the enhancement of sensitivity. The saturation magnetization (Ms) in a parallel anisotropic TbDyFe-PDMS MREF was 5.8 emu/g, and the initial tensile modulus was 55% greater than that of an Iso MREF. We propose a nonlinear magnetorheological formula on the magnetostriction effect, incorporating magnetic dipole interactions and the nonlinear prestress of magnetic particles. This formula highlights the complex nonlinear relationship between the external magnetic field (H) and the key parameters that affect the enhanced MR effect of MSPs-MREF, such as saturation magnetization, remanence (Mr), magnetostriction constant (λs) and stress deviator in ferromagnetic particles (Sed) in the magnetic chain structure. Furthermore, we validate the influence of the key parameters of the rectified magnetorheological formula on a nonlinear magneto-mechanical behavior of MSPs-MREF in PDMS-based MSPs-MREF models by using finite-element simulations. Finally, we developed a biosensor based on MSPs-MREF to detect human serum albumin at low concentrations in human urine samples. There is a 4-fold increase in sensitivity, a lower detection of limit (0.442 μg/mL), and a faster response time (15 min) than traditional biosensors, which in the future might provide an effective way of detecting biomolecules of low concentrations.

Nonlinear Kodaira-Bochner formulae and their applications

Nonlinear Kodaira-Bochner formulae and their applications

Mathematical models for resolving the nonlinear formula for solar cell

&lt;span&gt;Accurate representation of a photovoltaic solar cell requires a comprehensive assessment of modeling factors that are unique to the individual device being studied. In the context of the single diode model, it is necessary to ascertain five distinct parameters, namely Rs, Rsh, Iph, Io, and n. In general, analytical or numerical methods may be used to calculate these values. In this paper, two alternative iterative approaches to solving nonlinear problems in solar cells without temperature are described and analyzed. The new iterative approach has several instances that have been quantitatively tested. This novel approach can be seen as a potential option for solving nonlinear equations. Additionally, a comparison between the suggested method, classic chord formula (CCM), and predictor-corrector type reveals that it is better and has the lowest evaluation. This is supported by an examination of accuracy and efficiency (as evaluated by function evaluations) false position method (FPM).&lt;/span&gt;

Problems of modeling elements with shape memory alloys of military equipment

Shape memory alloys (SMAs) are used in various fields, including military equipment, to create means with autonomous or improved properties. It can be especially useful in structural elements that are capable of self-healing after deformations or damage as a result of explosions, impacts or other loads that occur during military operations.&#x0D; SMAs are materials that have the ability to remember and restore the accumulated deformation under certain conditions. This makes them particularly useful in robotics, where they can be used to create moving parts, sensors, actuators and other components.&#x0D; One of the main applications of SMA in robotics is the creation of moving elements. For example, they can be used to create bendable segments in jobs that require flexibility and mobility. SMAs can be used to create manipulators that allow a robot to easily move and perform tasks in different environments. One of the most important properties of SMAs is their ability to retain their shape during deformation. This makes them useful for building sensors that can detect differences in the shape of an object. SMAs can be used to create pressure sensors that respond to a change in shape under the influence of external pressure. These sensors can be used to develop robots that can recognize objects and distinguish them by shape. SMA can also be used to create actuators. Actuators are components that convert electrical, mechanical, or other types of energy into motion. SMA are materials that have the ability to remember and restore the accumulated deformation under certain conditions. This is achieved due to the peculiarities of their microstructure and thermoelastic effect. When SMA is subjected to thermal influence in the temperature zone where it can transition between two states: austenitic (higher temperature) and martensitic (lower temperature), under an existing external load, the phase strain increases, and with the degree of load, the temperature of the beginning and end of the emerging of straine changes. Each of these states has its own crystal structure and characteristics determined by its chemical composition. The article considers the issue of taking into account the shift of characteristic temperatures under the influence of an external load of SMA in terms of the synthetic theory of irreversible deformation and proposes a non-linear formula for finding the characteristic temperatures of metals with memory during loading. The obtained ratio was used to find the proportion of the new phase and the rate of formation of the new phase as a function of the magnitude of the external load, the rate of loading, the absolute temperature, the rate of temperature change and the characteristic temperatures in terms of the effective temperature.