multi-Gaussian Model Research Articles

Phase Coexistence in Hamiltonian Hybrid Particle-Field Theory Using a Multi-Gaussian Approach.

This study introduces an implementation of multiple Gaussian filters within the Hamiltonian hybrid particle-field (HhPF) theory, aimed at capturing phase coexistence phenomena in mesoscopic molecular simulations. By employing a linear combination of two Gaussians, we demonstrate that HhPF can generate potentials with attractive and steric components analogous to Lennard-Jones (LJ) potentials, which are crucial for modeling phase coexistence. We compare the performance of this method with the multi-Gaussian core model (MGCM) in simulating liquid-gas coexistence for a single-component system across various densities and temperatures. Our results show that HhPF effectively captures detailed information on phase coexistence and interfacial phenomena, including microconfiguration transitions and increased interfacial fluctuations at higher temperatures. Notably, the phase boundaries obtained from HhPF simulations align more closely with those of LJ systems compared to the MGCM results. This work advances the hybrid particle-field methodology to address phase coexistence without requiring modifications to the equation of state or introducing additional interaction energy functional terms, offering a promising approach for mesoscale molecular simulations of complex systems.

Improving the morphology of terrace-like transfer film to reduce wear rate of graphite/PTFE composites via filling low content silica

Purpose The purpose of this paper is to fill low content of silica to improve the wear resistance of graphite/polytetrafluoroethylene (PTFE) composites, and to discuss the effect of low content of silica on the morphology of terrace-like transfer film. Design/methodology/approach The tribological properties of silica/graphite/PTFE composites were tested by rotating pin-on-disk friction and wear tester, and the morphology of terrace-like transfer film was observed by 3D laser scanning microscope and quantitatively analyzed through the multi-Gaussian function linear combination model. Findings The results showed that adding 3 wt.% silica in graphite/PTFE composites can reduce the wear rate by 87%. More importantly, the morphology of terrace-like transfer film changed significantly after filling a small amount of silica, including the increase of the number of layers and the coverage rate of transfer film. Originality/value This work is of great significance for further understanding the transfer film and achieving its morphology control to optimize the wear of high performance PTFE composites.

Industrial image anomaly detection based on multi Gaussian discriminant model and robust core set

Abstract To address the issue of false positive (FP) detections in image anomaly detection caused by the loss of low-frequency features when dealing with high-dimensional feature distributions, we propose the multi-layer Gaussian discriminant anomaly detection model (MGAD). This model utilizes distance metrics based on multiple normal distributions to perform anomaly detection. By mining multi-layer feature combinations from normal samples and incorporating a Gaussian mixture model strategy for pixel-by-pixel probability density estimation, a weighting mechanism is designed to emphasize the role of low-frequency features in Gaussian space. This approach effectively models data collections that do not follow a single normal distribution as a mixture of several Gaussian distributions, thereby reducing false detections. Additionally, we propose a method for calculating the minimum Mahalanobis distance based on the estimation of the minimum covariance determinant. By identifying a subset with the smallest covariance matrix determinant, this method enhances the robust estimation of the data’s central position and spread, thereby reducing the impact of outliers. On the MVTec-AD dataset, MGAD demonstrates outstanding performance with an anomaly detection area under the receiver operating characteristic curve (AUROC) of 98.8%, the anomaly localization AUROC of 98.2%, and the per-class true negative rate for normal samples of 93.1%. Compared with the state-of-the-art models, MGAD improves the detection accuracy for normal samples by 3.6%, demonstrating the best performance among all models. These results highlight the model’s excellent capability in anomaly recognition and reduction of FPs.

Joint multi-Gaussian mixture model and its application to multi-model multi-bernoulli filter

The existing multi-model Gaussian mixture-multi-Bernoulli (MM-GM-MB) filter lacks the ability to maintain parallel state estimates under different models. As a result, the change of model-related likelihoods lags behind target maneuvers. To address this problem, a joint multi-Gaussian mixture MB (JMGM-MB) filter is proposed. We first propose a JMGM model. Each single-target state estimate is expressed as a set of parallel model-related Gaussian functions with model probabilities, and a weight characterizes the probability of this state estimate. Based on the Bayesian rule, we derive the updating methods of the weights, model probabilities and model-related means, covariances that are collectively called JMGM components. Then, we approximate the state densities of MB filter using the JMGM model. Based on the interactive multi-model (IMM) method, we derive the interacting, prediction and estimation methods of JMGM components. Then, we provide matched association and merging methods for JMGM components. In nonlinear tracking scenarios where sensors simultaneously carry out translations and rotations, we derive a method based on the derivative rule for composite functions to compute the linearized observation matrix. Simulations demonstrate that whether in active or passive tracking, the proposed JMGM-MB filter overcomes the likelihood lag problem and owns higher accuracy of uncertain maneuvering target tracking than the existing MM-GM-MB filter, requiring lower computational expense.

Reliability modeling of pre-stressed composite structures subject to interlayer slip failure under multi-axial random impact and vibration loads

A time-dependent reliability model for pre-stressed metal/polymer/metal composite structures subject to interlayer slip failure is developed in this study, allowing for predicting the probability of failure of the whole structure under multi-axial random multi-peak impact loads and transport vibration. A multi-Gaussian envelope model with uniform modulations is proposed to model random multi-peak impact loads, and the inverse sampling method is employed to model random transport vibration. An interlayer slip physics model is developed based on the contact surface friction mechanism under multi-axial vibration load and the stress-relaxation constitutive model. By implementing the interlayer slip physics model in the finite element method framework via the user-material routine, the geometry effect of the structure can be dealt with, and the reliabilities at an arbitrary location of the structure can be obtained. The proposed method is demonstrated using engineering examples. The results show that the proposed reliability assessment method provides a viable and systematical procedure of computing the time-dependent reliability of composite structures as a whole under multi-axial impact loads and vibrations.

Improved detection of nonlinear waves in thin solid materials using a pulse-echo method based on nonlinear modeling of wave fields

The nonlinearity parameter is a powerful tool for nondestructive evaluation, as it provides a measure of the state of damage of structural components. When carrying out nonlinearity parameter measurements of solid structures, the pulse-echo method is more suitable than the through-transmission method; however, a planar stress-free boundary is known to destructively alter the nonlinear wave generation process, including the second harmonic component, meaning that the pulse-echo nonlinear method is limited for practical applications, especially for thin solids. In this work, an improved detection method is proposed for the enhancement of nonlinear wave measurement in thin solids using a pulse-echo method with a dual-element transducer consisting of an outer ring transmitter and an inner disk receiver. A multi-Gaussian beam model is developed for the ring transmitter to fast simulate the pulse-echo nonlinear wave fields, and based on the results, improvements for the received nonlinear waves are achieved by optimizing the driving frequency. The advantages of the proposed method are that solid samples with different thicknesses can be evaluated using a dual-element transducer, and a high measurement accuracy can also be achieved for thin solid samples. Experiments are performed to investigate the harmonic waves measured in fatigue-damaged thin steel samples, and the results verify the effectiveness of the proposed method.

Automatic Quantitative Assessment for Diagnostic and Therapeutic Response in Rodent Myocardial Infarct Model.

The purpose of this study was to investigate the most appropriate methodological approach for the automatic measurement of rodent myocardial infarct polar map using histogram-based thresholding and unsupervised deep learning (DL)-based segmentation. A rat myocardial infarction model was induced by ligation of the left coronary artery. Positron emission tomography (PET) was performed 60 min after the administration of 18F-fluoro-deoxy-glucose (18F-FDG), and PET was performed after injecting 64Cu-pyruvaldehyde-bis(N4-methylthiosemicarbazone). Single photon emission computed tomography was performed 60 min after injection of 99mTc-hexakis-2-methoxyisobutylisonitrile and 201Tl. Delayed contrast-enhanced magnetic resonance imaging was performed after injecting Gd-DTPA-BMA. Three types of thresholding methods (naive thresholding, Otsu's algorithm, and multi-Gaussian mixture model (MGMM)) were used. DL segmentation methods were based on a convolution neural network and trained with constraints on feature similarity and spatial continuity of the response map extracted from images by the network. The relative infarct sizes measured by histology and estimated R2 for 18F-FDG were 0.8477, 0.7084, 0.8353, and 0.9024 for naïve thresholding, Otsu's algorithm, MGMM, and DL segmentation, respectively. DL-based method improved the accuracy of MI size assessment.

Health prognostics of lithium-ion batteries based on universal voltage range features mining and adaptive multi-Gaussian process regression with Harris Hawks optimization algorithm

In the field of battery prognostics and health management (PHM), accurate State-of-Health (SOH) estimation is important for the safe and reliable operation of lithium-ion batteries (LIBs). To improve the applicability and accuracy of the SOH estimation model, a new SOH estimation framework is proposed. Firstly, considering the different usage habits of users, to meet the needs of more users, a new health indicator (HI) was put forward, which involves only a small universal voltage range of charging data. Secondly, to alleviate the negative impact of the inherent cell inconsistency on SOH estimation accuracy, an adaptive Multi-Gaussian Process Regression model based on Harris Hawks Optimizer (HHO-MGPR) was proposed. The core idea is to use the HHO algorithm to automatically generate the optimal training subset combination for multiple GPR base learners so as to improve the model generalization ability. Finally, four open-source datasets with different battery types, temperature and operating conditions are employed for performance validation. The experimental results show that in all cases, the proposed HI can effectively characterize the battery degradation and the developed HHO-MGPR can improve the model generalization ability, indicating the proposed framework has wide applicability.

Pyrolytic performance and kinetics study of epoxy resin in carbon fiber reinforced composites: Synergistic effects of epoxy resin and carbon fiber

Pyrolysis is recognized as a sustainable approach for recycling carbon fiber reinforced polymers (CFRP), ensuring enhanced process efficiency and resource utilization. To unveil the pyrolysis mechanism of epoxy resin and its synergistic interaction with carbon fiber in different scenarios, we conducted pyrolysis experiments on pure epoxy resin diglycidyl ether of bisphenol A cured with 4,4'-Diaminodiphenylmethane (C-epoxy), as well as three kinds of carbon fiber-resin mixtures, using TG and Py-GC/MS techniques. Isoconversional analysis (Friedman method) of C-epoxy degradation revealed three reaction stages within the temperature range of 280–540 °C, each corresponding to activation energies of 158 kJ/mol at α = 5%, 179–190 kJ/mol at the plateau of α = 10–60%, and 190–236 kJ/mol for α values surpassing 60%. The multi-Gaussian distributed activation energy model (DAEM) aptly characterized C-epoxy’s thermal decomposition, featuring two Gaussian peaks with similar contributions of 0.58 and 0.42, and activation energy distributions of 217.76 ± 3.92 kJ/mol and 233.36 ± 21.42 kJ/mol, respectively, which are close to the activation energies from isoconversional analysis at α = 10–60% and α > 60%, respectively. The presence of carbon fiber significantly influenced the kinetic parameters of epoxy resin pyrolysis, with the extent of change linked to the mixing methodology applied. Notably, carbon fiber decreased the activation energy of the reaction while accelerating the rate of weight loss. However, the synergistic effects vary depending on different synergistic scenarios. By analyzing the product distribution obtained through an analytical pyrolyzer coupled with a gas chromatography–mass spectrometry set-up (Py-GC/MS), we proposed a pyrolysis mechanism for epoxy resin, encompassing three key stages: cleavage of N–C and NC–COH bonds, phenolic O–C and vicinal C–C bonds, and disruption of the isopropylidene bridge on bisphenol A. The influence of carbon fiber on epoxy resin pyrolysis product distribution was relatively modest, manifesting predominantly within the mixture obtained by premixing before curing. Here, carbon fiber exhibited a dual effect, inhibiting oxygen-containing diphenyl products while promoting oxygen-containing monophenyl products. Ultimately, our study provides essential kinetic data for multidimensional simulation and theoretical guidance pertaining to the pyrolysis recycling of widely used carbon fiber-reinforced epoxy resin composites.

STRUCTURE OF MASS DISTRIBUTIONS OF PHOTOFISSION PRODUCT YIELDS OF 238U AT 17.5 MeV BREMSSTRAHLUNG ENERGY

The values of 29 relative yields of products belonging to 26 mass chains of photofission of 238U were obtained at a maximum bremsstrahlung energy of 17.5 MeV (near the second chance fission threshold). The stimulation of the 238U photofission reaction was performed at the electron accelerator of the Institute of Electron Physics of the National Academy of Sciences of Ukraine, the M-30 microtron. The GEANT4 toolkit was used to model the components of the bremsstrahlung spectrum. The presence of a fine structure in the mass distribution of heavy product yields was found to be localized in the mass range 133-134, 138-139, and 143-144. A multi gaussian model (superposition of three Gaussians) was used to theoretically describe the structure. The calculated values of product yields using the GEF code describe in general terms and predict the fine structure of the mass distribution of photofission products of 238U.

Ultrasonic Water Immersion Nondestructive Testing for Nylon Bars Based on a Multi-Gaussian Beam Model

Ultrasonic Water Immersion Nondestructive Testing for Nylon Bars Based on a Multi-Gaussian Beam Model

Early in vivo Radiation Damage Quantification for Pediatric Craniospinal Irradiation Using Longitudinal MRI for Intensity Modulated Proton Therapy

Proton vertebral body sparing craniospinal irradiation (CSI) treats the thecal sac while avoiding the anterior vertebral bodies in an effort to reduce myelosuppression and growth inhibition. However, robust treatment planning needs to compensate for proton range uncertainty, which contributes unwanted doses within the vertebral bodies. This work aimed to develop an early in vivo radiation damage quantification method using longitudinal magnetic resonance (MR) scans to quantify the dose effect during fractionated CSI. Ten pediatric patients were enrolled in a prospective clinical trial of proton vertebral body sparing CSI, in which they received 23.4 to 36 Gy. Monte Carlo robust planning was used, with spinal clinical target volumes defined as the thecal sac and neural foramina. T1/T2-weighted MR scans were acquired before, during, and after treatments to detect a transition from hematopoietic to less metabolically active fatty marrow. MR signal intensity histograms at each time point were analyzed and fitted by multi-Gaussian models to quantify radiation damage. Fatty marrow filtration was observed in MR images as early as the fifth fraction of treatment. Maximum radiation-induced marrow damage occurred 40 to 50 days from the treatment start, followed by marrow regeneration. The mean damage ratios were 0.23, 0.41, 0.59, and 0.54, corresponding to 10, 20, 40, and 60 days from the treatment start. We demonstrated a noninvasive method for identifying early vertebral marrow damage based on radiation-induced fatty marrow replacement. The proposed method can be potentially used to quantify the quality of CSI vertebral sparing and preserve metabolically active hematopoietic bone marrow.

Temperature-dependent cross section spectra for thulium-doped fiber lasers.

An investigation on the temperature dependence of spectroscopic parameters of trivalent thulium ions is important for the design of high-power, thulium-doped fiber lasers and amplifiers. In this Letter, the thulium absorption/emission cross sections are determined in the spectral range 700-2200 nm and in the temperature range from -196°C to 300°C. The spectra are obtained from the absorption and emission measurements of a thulium-doped fiber and from measured thulium concentration profiles. Attempts were made to estimate the temperature dependence of the spectra where the measurements are not accessible. Firstly, the spectra are fitted to a multi-Gaussian model with temperature dependent parameters. Secondly, a physically motivated model of the cross section spectra is proposed and analyzed.

Convolutional – recurrent neural network proxy for robust optimization and closed-loop reservoir management

Production optimization under geological uncertainty is computationally expensive, as a large number of well control schedules must be evaluated over multiple geological realizations. In this work, a convolutional-recurrent neural network (CNN-RNN) proxy model is developed to predict well-by-well oil and water rates, for given time-varying well bottom-hole pressure (BHP) schedules, for each realization in an ensemble. This capability enables the estimation of the objective function and nonlinear constraint values required for robust optimization. The proxy model represents an extension of a recently developed long short-term memory (LSTM) RNN proxy designed to predict well rates for a single geomodel. A CNN is introduced here to processes permeability realizations, and this provides the initial states for the RNN. The CNN-RNN proxy is trained using simulation results for 300 different sets of BHP schedules and permeability realizations. We demonstrate proxy accuracy for oil-water flow through multiple realizations of 3D multi-Gaussian permeability models. The proxy is then incorporated into a closed-loop reservoir management (CLRM) workflow, where it is used with particle swarm optimization and a filter-based method for nonlinear constraint satisfaction. History matching is achieved using an adjoint-gradient-based procedure. The proxy model is shown to perform well in this setting for five different (synthetic) ‘true’ models. Improved net present value along with constraint satisfaction and uncertainty reduction are observed with CLRM. For the robust production optimization steps, the proxy provides O(100) runtime speedup over simulation-based optimization.

Accurate Measurements of Droplet Volume With Coherence Scanning Interferometry

Accurate volume measurement of droplets in inkjet printed organic light-emitting diode (OLED) is important to achieve defect-free manufacturing. Although coherence scanning interferometry (CSI) has been widely used in the 3-D reconstruction of microscale/nanoscale, most existing CSI-based methods pay little attention to the processing of fringe signals with overlap caused by the transparency of the sample. In this article, we propose an effective and accurate volume measurement method for transparent droplets, i.e., wavelet transform-based CSI (WTCSI). First, the envelopes of fringe signals with or without overlap are obtained by wavelet transform (WT). Then, an adaptive multi-Gaussian model (AMGM) is proposed to describe the envelopes. In addition, AMGM uses the zero-crossing count method (ZC-CM) to adaptively distinguish the fringe signals with or without overlap, and the mask of the overlap region corresponding to the droplet region is available as well. Therefore, the upper and lower surfaces of the droplet are separated via weighted least-squares (WLS) on the envelope. In addition, the zero-order fringe peak is identified for a higher accuracy measurement of the droplet profile by combining the envelope peak and phase information. Finally, the volume of the droplet is calculated by integrating the previously measured mask and height. The experimental results demonstrate that the proposed method is effective in the separation of overlapped fringe signals. In addition, the volume measurement error of the proposed method is ±2%, while the relative deviation is better than 0.5%.

Electron energy loss spectroscopy database synthesis and automation of core-loss edge recognition by deep-learning neural networks

The ionization edges encoded in the electron energy loss spectroscopy (EELS) spectra enable advanced material analysis including composition analyses and elemental quantifications. The development of the parallel EELS instrument and fast, sensitive detectors have greatly improved the acquisition speed of EELS spectra. However, the traditional way of core-loss edge recognition is experience based and human labor dependent, which limits the processing speed. So far, the low signal–noise ratio and the low jump ratio of the core-loss edges on the raw EELS spectra have been challenging for the automation of edge recognition. In this work, a convolutional-bidirectional long short-term memory neural network (CNN-BiLSTM) is proposed to automate the detection and elemental identification of core-loss edges from raw spectra. An EELS spectral database is synthesized by using our forward model to assist in the training and validation of the neural network. To make the synthesized spectra resemble the real spectra, we collected a large library of experimentally acquired EELS core edges. In synthesize the training library, the edges are modeled by fitting the multi-Gaussian model to the real edges from experiments, and the noise and instrumental imperfectness are simulated and added. The well-trained CNN-BiLSTM network is tested against both the simulated spectra and real spectra collected from experiments. The high accuracy of the network, 94.9%, proves that, without complicated preprocessing of the raw spectra, the proposed CNN-BiLSTM network achieves the automation of core-loss edge recognition for EELS spectra with high accuracy.

Research on ultrasonic testing method of rotating component based on a 6-DOF robot

Some rotating components have the structural characteristics of variable thickness and variable curvature. The conventional ultrasonic testing is inappropriate for these components because of bad coupling and too many uncertainties. In order to meet the testing requirements of high-volume, high-efficiency and high-precision, the corresponding ultrasonic automatic detection program is proposed. A 6-DOF industrial robot is used to realize the precise control of the transducer position and posture, and combines the independent turntable to complete the spiral fast scan of the specimen. Based on the multi-Gaussian beam model and the internal flat-bottom hole flaw scattering model of specimen, the flaw ultrasonic measurement model is established. The system parameters, component material parameters and component surface parameters are all introduced to the ultrasonic measurement model to generate a series of flaw-sizing curves. An accurate flaw sizing method based on DGS(Distance-Gain-Size)curve predicted by regular reflector is proposed. The scanning experiment results of the reference specimen show that the minimum Ø 0.8mm artificial flat-bottom-hole flaw in the workpiece can be detected effectively, and the sizing error is less than Ø 0.3mm equivalent.

Ultrasonic field analysis with local interface curvature effect based on the multi-Gaussian beam model

In ultrasonic testing, the amplitude of echoes is determined by the sound pressure along the acoustic axis. When the workpiece has a local curved surface, the ultrasonic field redistribution has a close relationship with the curvature of the incident local region, which raises questions of testing validity and accuracy. In order to develop the curvature adaptation window for a given test condition, the influence of local curved interface on the ultrasonic transmission and reflection field is analyzed quantitatively in this paper. Based on the multi-Gaussian beam model, the distribution of transmitted and reflected sound pressure in the local curved region is calculated by using matrix transformation technique. Furthermore, the relationship between interface curvature and sound pressure along the acoustic axis is obtained. To validate the accuracy of proposed calculation model, developed relationship has been compared against a finite element method computed using COMSOL Multiphysics. It is indicated that the proposed calculation model is more efficient and the regular changes of sound pressure are basically consistent with that of finite element method.

LEGA-C: Analysis of Dynamical Masses from Ionized Gas and Stellar Kinematics at z ∼ 0.8

Abstract We compare dynamical mass estimates based on spatially extended stellar and ionized gas kinematics (M dyn,* and M dyn,eml, respectively) of 157 star-forming galaxies at 0.6 ≤ z < 1. Compared with z ∼ 0, these galaxies have enhanced star formation rates, with stellar feedback likely affecting the dynamics of the gas. We use LEGA-C DR3, the highest-redshift data set that provides sufficiently deep measurements of a K s -band limited sample. For M dyn,*, we use Jeans anisotropic multi-Gaussian expansion models. For M dyn,eml, we first fit a custom model of a rotating exponential disk with uniform dispersion, whose light is projected through a slit and corrected for beam smearing. We then apply an asymmetric drift correction based on assumptions common in the literature to the fitted kinematic components to obtain the circular velocity, assuming hydrostatic equilibrium. Within the half-light radius, M dyn,eml is on average lower than M dyn,*, with a mean offset of –0.15 ± 0.016 dex and galaxy-to-galaxy scatter of 0.19 dex, reflecting the combined random uncertainty. While data of higher spatial resolution are needed to understand this small offset, it supports the assumption that the galaxy-wide ionized gas kinematics do not predominantly originate from disruptive events such as star formation–driven outflows. However, a similar agreement can be obtained without modeling from the integrated emission line dispersions for axis ratios q < 0.8. This suggests that our current understanding of gas kinematics is not sufficient to efficiently apply asymmetric drift corrections to improve dynamical mass estimates compared with observations lacking the signal-to-noise ratio required for spatially extended dynamics.

Small Angle X-ray Scattering Sensing Membrane Composition: The Role of Sphingolipids in Membrane-Amyloid β-Peptide Interaction.

Simple SummaryThe early impairments in Alzheimer’s disease are related to neuronal membrane damage. Different lipids are present in biological membranes, playing relevant physiological roles. Some of them, such as sphingomyelin, cholesterol, and ganglioside GM1, interact with each other and, importantly, with the Aβ peptide. Here, these interactions are studied using small angle X-ray scattering in model membrane systems, such as large unilamellar liposomes. This technique gives information on the width of the bilayer and reveals structural differences due to the different lipid compositions, as well as some small differences due to the presence of the Aβ peptide. The analysis highlights the concentration-dependent effect of GM1 on membrane thickness and the interaction with the Aβ-peptide, together with the inhibiting effect that the presence of sphingomyelin has on the GM1–Aβ interaction.The early impairments appearing in Alzheimer’s disease are related to neuronal membrane damage. Both aberrant Aβ species and specific membrane components play a role in promoting aggregation, deposition, and signaling dysfunction. Ganglioside GM1, present with cholesterol and sphingomyelin in lipid rafts, preferentially interacts with the Aβ peptide. GM1 at physiological conditions clusters in the membrane, the assembly also involves phospholipids, sphingomyelin, and cholesterol. The structure of large unilamellar vesicles (LUV), made of a basic POPC:POPS matrix in a proportion of 9:1, and containing different amounts of GM1 (1%, 3%, and 4% mol/mol) in the presence of 5% mol/mol sphingomyelin and 15% mol/mol cholesterol, was studied using small angle X-ray scattering (SAXS). The effect of the membrane composition on the LUVs–Aβ-peptide interaction, both for Aβ1–40 and Aβ1–42 variants, was, thus, monitored. The presence of GM1 leads to a significant shift of the main peak, towards lower scattering angles, up to 6% of the initial value with SM and 8% without, accompanied by an opposite shift of the first minimum, up to 21% and 24% of the initial value, respectively. The analysis of the SAXS spectra, using a multi-Gaussian model for the electronic density profile, indicated differences in the bilayer of the various compositions. An increase in the membrane thickness, by 16% and 12% when 2% and 3% mol/mol GM1 was present, without and with SM, respectively, was obtained. Furthermore, in these cases, in the presence of Aβ40, a very small decrease of the bilayer thickness, less than 4% and 1%, respectively, was derived, suggesting the inhibiting effect that the presence of sphingomyelin has on the GM1–Aβ interaction.