Stochastic Multiscale Model Research Articles

Stochastic Multiscale Modeling of Electrical Conductivity of Carbon Nanotube Polymer Nanocomposites: An Interpretable Machine Learning Approach

This study introduces an interpretable machine learning (ML) framework for efficiently predicting the electrical conductivity of carbon nanotube (CNT)/polymer nanocomposites. A stochastic multiscale numerical model based on representative volume element (RVE) is employed to generate a representative dataset. This dataset is used to train three ML models, including random forest, XGBoost, and artificial neural networks (ANN). The dataset includes six input features: CNT length, aspect ratio, intrinsic CNT conductivity, number of CNT conduction channels, energy barrier height, and volume fraction, with the electrical conductivity of the nanocomposites as the output feature. The findings highlight the exceptional accuracy of the ANN model in predicting electrical conductivity at significantly lower computational costs. Furthermore, the use of Shapley additive explanations (SHAP) enhances the interpretability of these ML models, identifying the volume fraction, energy barrier height, and intrinsic CNT conductivity as the most influential factors affecting conductivity. This approach sets the stage for rapid and efficient modeling of CNT/polymer nanocomposites facilitating the design of materials with tailored electrical properties for diverse applications.

Investigation of P. vivax elimination via mass drug administration: A simulation study

Plasmodium vivax is the most geographically widespread malaria parasite. P. vivax has the ability to remain dormant (as a hypnozoite) in the human liver and subsequently reactivate, which makes control efforts more difficult. Given the majority of P. vivax infections are due to hypnozoite reactivation, targeting the hypnozoite reservoir with a radical cure is crucial for achieving P. vivax elimination. Stochastic effects can strongly influence dynamics when disease prevalence is low or when the population size is small. Hence, it is important to account for this when modelling malaria elimination. We use a stochastic multiscale model of P. vivax transmission to study the impacts of multiple rounds of mass drug administration (MDA) with a radical cure, accounting for superinfection and hypnozoite dynamics. Our results indicate multiple rounds of MDA with a high-efficacy drug are needed to achieve a substantial probability of elimination. This work has the potential to help guide P. vivax elimination strategies by quantifying elimination probabilities for an MDA approach.

A stochastic multi-scale thermal conductivity numerical model for 2D C/SiC-Ti3SiC2 composite

Thermal conductivity and thermal response are the essential indexes of C/SiC composites in the thermal protection systems (TPS). A stochastic multi-scale finite element model is put forth for predicting the above parameters based on the microstructure characteristics of the plain woven (2D) C/SiC-Ti3SiC2 composite. The reliability of the proposed stochastic multi-scale model is verified by the experimental data of thermal conductivity as well as the temperature history obtained from the continuous-wave laser irradiation experiment. The effects of porosity, the carbon fiber and the Ti3SiC2 MAX phase volume fractions on thermal conductivity and temperature field are analyzed. The results show that the porosity significantly reduces the thermal conductivity and elevates the temperature. The mesoscale thermal conductivity rises with the increase of the Ti3SiC2 MAX phase. Increasing the volume fraction of Ti3SiC2 can decreases the temperature at the laser irradiation center zone. The study provides an efficient tool in the design of the thermal response of MAX phase-modified ceramic matrix composites.

Bottom-up stochastic multiscale model for the mechanical behavior of multidirectional composite laminates with microvoids

This study for the first time develops a coherent and efficient bottom-up stochastic multiscale modeling approach, which aims at revealing the impact of microvoids on the mechanical properties and damage mechanisms of multidirectional (MD) carbon fiber-reinforced polymer (CFRP) composites. To this end, void defects within the composites are characterized using X-ray micro-computed tomography to construct a high-fidelity representative defective volume (RDV). Subsequently, the influence of voids on the microscale failure modes and macroscopic mechanical properties are analyzed by using micromechanics-based finite element (FE) method. Finally, a stochastic multiscale analysis method is developed to predict the mechanical response of MD CFRP laminates and validated by tensile and three-point bending experiments. The results indicate that the proposed framework can accurately predict the mechanical behavior and damage mechanisms of MD CFRP composites. This framework is believed to be a promising method to provide valuable guidance for reliability design and optimization of CFRP composites.

Effect of carbon nanotube type and length on the electrical conductivity of carbon nanotube polymer nanocomposites

Carbon nanotube (CNT) type and length are two key factors that affect the electrical behavior of CNT/polymer nanocomposites. However, numerical studies that consider these two factors simultaneously are limited. This paper presented a stochastic multiscale numerical model to predict the electrical conductivity and percolation threshold of polymer nanocomposites containing single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). The combined effects of CNT type and length on the electrical conductivity and percolation threshold of the polymer nanocomposites were investigated. The model predictions were validated against experimental data of commercially available CNTs. Our results showed that the effect of CNT type varied based on both the length and aspect ratio of the CNTs. Long SWCNTs exhibited the greatest enhancement of the polymer’s electrical conductivity with the lowest percolation threshold among all the CNT types studied.

Multiscale damage modelling of notched and un-notched 3D woven composites with randomly distributed manufacturing defects

This work proposes a stochastic multiscale computational framework for damage modelling in 3D woven composite laminates, by considering the random distribution of manufacturing-induced imperfections. The proposed method is demonstrated to be accurate, while being simple to implement and requiring modest computational resources. In this approach, a limited number of cross-sectional views obtained from micro-computed tomography (µCT) are used to obtain the stochastic distribution of two key manufacturing-induced defects, namely waviness and voids. This distribution is fed into a multiscale progressive damage model to predict the damage response of three-dimensional (3D) orthogonal woven composites. The accuracy of the proposed model was demonstrated by performing a series of finite element simulations of the un-notched and notched tensile tests (having two different hole sizes) for resin-infused thermoplastic (Elium®) 3D woven composites. Excellent correlation was achieved between experiments and the stochastic finite element simulations. This demonstrates the effectiveness of the proposed stochastic multiscale model. The model successfully captured the stochastic nature of tensile responses (ultimate tensile strength and stiffness), damage modes (matrix damage and fibre failure), and initiation and propagation of transverse cracks in thermoplastic 3D woven composites, consistent with experimental observation. The stochastic computational framework presented in this paper can be used to guide the design and optimization of 3D textile composite structures.

Stochastic higher-order three-scale strength prediction model for composite structures with micromechanical analysis

Stochastic multiscale modeling and analysis for the strength prediction of composite structures with complex multiscale random configurations remain a challenging problem. This is mainly due to the high-dimensional physical properties, the non-linear and non-Gaussian features, and the fact that many repeated evaluations of the corresponding stochastic multiscale model are often required. In this paper, we develop a stochastic higher-order three-scale strength prediction (SHTSP) model for composite structures, which is designed to overcome the limitations of prohibitive computation involving the microscale, the mesoscale, and the macroscale. By virtue of asymptotic homogenization theory and micromechanical analysis, the SHTSP model is established from the detailed stochastic higher-order three-scale homogenization analysis implemented with analytic solutions of typical composite structures subjected to tension, bending, and twist loads. The SHTSP model represents strength anisotropy through different strength criteria for evaluating the yield state of the different component materials of composite structures, which are induced by material interface deboning or matrix cracking in multiple scales. Moreover, these are constructed and calibrated from the high-accuracy mechanical analysis with help of two classes of mesoscopic and microscopic auxiliary cell functions, respectively. The corresponding numerical algorithm of the SHTSP model and a preprocessing multilevel strategy is designed to improve the computational efficiency. Finally, the numerical experiments in 3D cases illustrate the outstanding performance of the proposed SHTSP model, and the proposed method can significantly reduce the computational time.

Multiple morphogens and rapid elongation promote segmental patterning during development.

The vertebrate hindbrain is segmented into rhombomeres (r) initially defined by distinct domains of gene expression. Previous studies have shown that noise-induced gene regulation and cell sorting are critical for the sharpening of rhombomere boundaries, which start out rough in the forming neural plate (NP) and sharpen over time. However, the mechanisms controlling simultaneous formation of multiple rhombomeres and accuracy in their sizes are unclear. We have developed a stochastic multiscale cell-based model that explicitly incorporates dynamic morphogenetic changes (i.e. convergent-extension of the NP), multiple morphogens, and gene regulatory networks to investigate the formation of rhombomeres and their corresponding boundaries in the zebrafish hindbrain. During pattern initiation, the short-range signal, fibroblast growth factor (FGF), works together with the longer-range morphogen, retinoic acid (RA), to specify all of these boundaries and maintain accurately sized segments with sharp boundaries. At later stages of patterning, we show a nonlinear change in the shape of rhombomeres with rapid left-right narrowing of the NP followed by slower dynamics. Rapid initial convergence improves boundary sharpness and segment size by regulating cell sorting and cell fate both independently and coordinately. Overall, multiple morphogens and tissue dynamics synergize to regulate the sizes and boundaries of multiple segments during development.

Stochastic multiscale model for HfO2-based resistive random access memories with 1T1R configuration

In this paper, we propose a stochastic model for the resistive switching of ReRAM devices with 1T1R configuration. We work with the fact that the switching occurs in the narrowest zone of the conductive filament due to changes caused by the electric field. This active region is represented by a net of vertical connections, each one composed of three electrical elements: two of them are always low resistive (LR) while the third one acts as a breaker and can be low or high resistive (HR). The breaker can change its state according to a switching probability (Ps), which depends on the voltage drop in the breaker and the threshold voltage, Vset or Vreset for the set or reset process, respectively. This approach gives the model the stochastic behavior and generates the variability observed in the current–voltage curves of most of ReRAM devices. Further, we add another resistor in series in the circuit to represent the electrical signal of the transistor. By comparing measured and simulated IV curves of HfO2-based ReRAM devices of two different scales, nm2 and µm2, we validated the model. The flexibility and straightforward implementation of this resistive switching model make it a powerful tool for studying ReRAM memories and other stochastic devices.

A stochastic multiscale peridynamic model for corrosion-induced fracture in reinforced concrete

Concrete fracture caused by corrosion of reinforcing bars may cause subsequent structure failure. To better predict this process, we introduce a partially-homogenized stochastic peridynamic model with the simplest constitutive relation (linear elastic with brittle failure). The model links microscale information (phase volume fractions of mortar, aggregates, interfaces) to macroscale fracture behavior, while costing the same as a fully homogenized model. We explain why a fully-homogenized peridynamic model fails to capture the correct concrete fracture modes/patterns, while the new model succeeds. The multiscale model predicts the evolution of fracture in reinforced concrete caused by corrosion products expansion in samples with a single or multiple rebars. Non-uniform expansion of corrosion products is enforced here as preset, incremental radial displacements. The computed fracture patterns and the order in which various cracks develop match what is seen in experiments. The model’s robustness is tested under different stochastic realizations and discretization grid types.

A Low-Rank Approximated Multiscale Method for Pdes With Random Coefficients

This work presents a stochastic multiscale model reduction approach to solve PDEs with random coefficients. An ensemble-based low-rank approximation method is proposed to approximate multiscale bas...

A second-order multiscale approach for viscoelastic analysis of statistically inhomogeneous materials

An effective second-order multiscale approach is developed in this work to discuss viscoelastic properties of statistically inhomogeneous materials. In these materials, the sophisticated microscale structure of inclusions, including related shape, orientation, size, spatial distribution, volume fraction and so on, results in varying of the macroscale properties. At first, the Laplace transform is applied to the linear viscoelastic problems, and expected relaxation modulus in Laplace domain for the materials is given. Also, the second-order multiscale formulas for evaluating the viscoelastic problems of statistically inhomogeneous materials are derived. Next, the stochastic multiscale algorithm is proposed, and the expected relaxation moduli in time domain are obtained by the least-square and inverse Laplace transform. The salient features of the proposed approach are the asymptotic high-order homogenizations that do not require high-order continuity of the coarse-scale (or macroscale) solutions and can handle the random materials with complicated microstructure at a fraction of computational cost. Finally, some examples for the composites are computed by the effective algorithm, and compared with the data by the theoretical models and experimental results. The comparison illustrates that the stochastic multiscale model is efficient for determining the viscoelastic properties of the materials and shows their potential application in practical engineering calculation.

Swelling and Diffusion during Methanol Sorption into Hydrated Nafion

Diffusion within polymer electrolyte membranes is often coincident with time-dependent processes such as swelling and polymer relaxation, which are factors that limit their ability to block molecular crossover during use. The solution-diffusion model of membrane permeation, which is the accepted theory for dense polymers, applies only to steady-state processes and does not address dynamic internal structural changes that can accompany permeation. To begin discovery of how such changes can be coupled to the permeation process, we have constructed a stochastic multiscale reaction-diffusion model that examines time-dependent methanol uptake into and swelling of hydrated Nafion. Several potential mechanisms of diffusion and polymer response are tested. The simulation predictions are compared to real-time Fourier transform infrared attenuated total reflectance spectroscopy (FTIR-ATR) absorbance reported in the literature [ Hallinan , D. T. , Jr. ; Elabd , Y. A. J. Phys. Chem. B 2007 , 111 , 13221 - 13230 ]. Of the proposed polymer response mechanisms, only one, a reaction-limited, local response to increasing methanol concentration that takes the entire experimental time frame of 600 s, produces simulated FTIR-ATR data consistent with experiment. The simulations show that water diffusion out of the membrane is minimal during methanol sorption and that changes in the measured infrared absorbances are due primarily to the increase in methanol concentration accompanied by dilution of water during swelling. Swelling involves densification of the polymer structure even as there is an overall volume expansion of the film. Potential connections between the polymer densification and molecular-level structural changes of Nafion in methanol are discussed. These results indicate that the interaction between methanol and Nafion serves to increase Nafion's capacity to accommodate large volumes of methanol-water solutions, facilitating increased permeation across the membrane relative to pure water.

A Novel Stochastic Multi-Scale Model of Francisella tularensis Infection to Predict Risk of Infection in a Laboratory.

We present a multi-scale model of the within-phagocyte, within-host and population-level infection dynamics of Francisella tularensis, which extends the mechanistic one proposed by Wood et al. (2014). Our multi-scale model incorporates key aspects of the interaction between host phagocytes and extracellular bacteria, accounts for inter-phagocyte variability in the number of bacteria released upon phagocyte rupture, and allows one to compute the probability of response, and mean time until response, of an infected individual as a function of the initial infection dose. A Bayesian approach is applied to parameterize both the within-phagocyte and within-host models using infection data. Finally, we show how dose response probabilities at the individual level can be used to estimate the airborne propagation of Francisella tularensis in indoor settings (such as a microbiology laboratory) at the population level, by means of a deterministic zonal ventilation model.

Estimation of the effective bone-elasticity tensor based on μ CT imaging by a stochastic model. A multi-method validation

Abstract In previous work, we proposed a stochastic model to describe the elasticity of bone matrix (so-called ultrastructure, US) based on basic statistical information on the tissue mineral density ( TMD ). This information was obtained by analyzing high-resolution images of a human femoral neck realized by means of synchrotron radiation micro-computed tomography ( SR − μ CT ). In this paper, we extend this study by focusing at the upper scale where cortical bone is described as a two-phase mixture made up of water-filled Haversian pores (HP) embedded in the surrounding solid US. The goal of this paper is to develop a stochastic model of cortical bone elasticity accounting for the effect of uncertainty affecting both phases, the US via the TMD and the HP. Experimental information was assumed to be given in terms of mean values and dispersions of the average TMD (denoted TMD ¯ ) and HP at the millimeter scale. To this aim, SR − μ CT images were used to extract several representative volume elements (RVEs) spanning the whole cortical tissue which, in turn, were analyzed to obtain the required statistical information on TMD ¯ and HP. This information has been used for constructing a stochastic multiscale model of cortical bone based on the Maximum Entropy (denoted MaxEnt) principle. This stochastic multiscale model is used in the estimation of the effective elastic properties of cortical bone (CB-S μ M) based on continuum micromechanics ( μM). In parallel, a deterministic nominal multiscale model of cortical bone (CB-NμM) was developed by using as input data the mean values of TMD ¯ and HP. The elastic moduli estimated with CB-NμM has been compared with the average value of the same counterpart obtained with the CB-SμM model. The two estimates differ for less than 1 % , proving the robustness of the CB-NμM model to uncertainties. Moreover, the accuracy of the SμM stochastic model has been tested. Estimations of the cortical bone elasticity of each RVE with the μ M (cylindrical HP and homogeneous US), has been compared with other estimation obtained through others homogenization techniques: finite elements (FEM) and fast Fourier transform (FFT), both able to account for the real morphology of Havesian porosity and the heterogeneity of the ultrastructure. Results show that (1) the transverse isotropic (TI) elastic μM model correctly approximates the FEM and FFT estimates (∼98% TI); (2) the μM model accurately estimates the axial modulus ( Y 3 ) in longitudinal direction and the lateral contraction ratio ν 31 .

Stochastic Multiscale Model of MEMS Stiction Accounting for High-Order Statistical Moments of Non-Gaussian Contacting Surfaces

Stiction is a failure mode of microelectromechanical systems (MEMS) involving permanent adhesion of moving surfaces. Models of stiction typically describe the adhesion as a multiple asperity adhesive contact between random rough surfaces, and they thus require a sufficiently accurate statistical representation of the surface, which may be non-Gaussian. If the stiction is caused primarily by multiple asperity adhesive contact in only a small portion of the apparent area of the contacting surfaces, the number of adhesive contacts between asperities may not be sufficiently statistically significant for a homogenized model to be representative. We have proposed a probabilistic multiscale model of multiple asperity adhesive contact that can capture the uncertainty in a stiction behavior. Whereas the previous paper considered Gaussian random rough surfaces, the aim of this paper is to extend this probabilistic multiscale model to non-Gaussian random rough surfaces whose probabilistic representation accounts for the high-order statistical moments of the surface height. The probabilistic multiscale model thus obtained is validated by means of a comparison with experimental data of stiction tests of cantilever beams reported in the literature. [2017–0215]

Coarse-graining and hybrid methods for efficient simulation of stochastic multi-scale models of tumour growth

The development of hybrid methodologies is of current interest in both multi-scale modelling and stochastic reaction–diffusion systems regarding their applications to biology. We formulate a hybrid method for stochastic multi-scale models of cells populations that extends the remit of existing hybrid methods for reaction–diffusion systems. Such method is developed for a stochastic multi-scale model of tumour growth, i.e. population-dynamical models which account for the effects of intrinsic noise affecting both the number of cells and the intracellular dynamics. In order to formulate this method, we develop a coarse-grained approximation for both the full stochastic model and its mean-field limit. Such approximation involves averaging out the age-structure (which accounts for the multi-scale nature of the model) by assuming that the age distribution of the population settles onto equilibrium very fast. We then couple the coarse-grained mean-field model to the full stochastic multi-scale model. By doing so, within the mean-field region, we are neglecting noise in both cell numbers (population) and their birth rates (structure). This implies that, in addition to the issues that arise in stochastic-reaction diffusion systems, we need to account for the age-structure of the population when attempting to couple both descriptions. We exploit our coarse-graining model so that, within the mean-field region, the age-distribution is in equilibrium and we know its explicit form. This allows us to couple both domains consistently, as upon transference of cells from the mean-field to the stochastic region, we sample the equilibrium age distribution. Furthermore, our method allows us to investigate the effects of intracellular noise, i.e. fluctuations of the birth rate, on collective properties such as travelling wave velocity. We show that the combination of population and birth-rate noise gives rise to large fluctuations of the birth rate in the region at the leading edge of front, which cannot be accounted for by the coarse-grained model. Such fluctuations have non-trivial effects on the wave velocity. Beyond the development of a new hybrid method, we thus conclude that birth-rate fluctuations are central to a quantitatively accurate description of invasive phenomena such as tumour growth.

A Stochastic Multiscale Model That Explains the Segregation of Axonal Microtubules and Neurofilaments in Neurological Diseases.

The organization of the axonal cytoskeleton is a key determinant of the normal function of an axon, which is a long thin projection of a neuron. Under normal conditions two axonal cytoskeletal polymers, microtubules and neurofilaments, align longitudinally in axons and are interspersed in axonal cross-sections. However, in many neurotoxic and neurodegenerative disorders, microtubules and neurofilaments segregate apart from each other, with microtubules and membranous organelles clustered centrally and neurofilaments displaced to the periphery. This striking segregation precedes the abnormal and excessive neurofilament accumulation in these diseases, which in turn leads to focal axonal swellings. While neurofilament accumulation suggests an impairment of neurofilament transport along axons, the underlying mechanism of their segregation from microtubules remains poorly understood for over 30 years. To address this question, we developed a stochastic multiscale model for the cross-sectional distribution of microtubules and neurofilaments in axons. The model describes microtubules, neurofilaments and organelles as interacting particles in a 2D cross-section, and is built upon molecular processes that occur on a time scale of seconds or shorter. It incorporates the longitudinal transport of neurofilaments and organelles through this domain by allowing stochastic arrival and departure of these cargoes, and integrates the dynamic interactions of these cargoes with microtubules mediated by molecular motors. Simulations of the model demonstrate that organelles can pull nearby microtubules together, and in the absence of neurofilament transport, this mechanism gradually segregates microtubules from neurofilaments on a time scale of hours, similar to that observed in toxic neuropathies. This suggests that the microtubule-neurofilament segregation can be a consequence of the selective impairment of neurofilament transport. The model generates the experimentally testable prediction that the rate and extent of segregation will be dependent on the sizes of the moving organelles as well as the density of their traffic.

Stochastic multiscale modelling of cortical bone elasticity based on high-resolution imaging.

Accurate and reliable assessment of bone quality requires predictive methods which could probe bone microstructure and provide information on bone mechanical properties. Multiscale modelling and simulation represent a fast and powerful way to predict bone mechanical properties based on experimental information on bone microstructure as obtained through X-ray-based methods. However, technical limitations of experimental devices used to inspect bone microstructure may produce blurry data, especially in in vivo conditions. Uncertainties affecting the experimental data (input) may question the reliability of the results predicted by the model (output). Since input data are uncertain, deterministic approaches are limited and new modelling paradigms are required. In this paper, a novel stochastic multiscale model is developed to estimate the elastic properties of bone while taking into account uncertainties on bone composition. Effective elastic properties of cortical bone tissue were computed using a multiscale model based on continuum micromechanics. Volume fractions of bone components (collagen, mineral, and water) were considered as random variables whose probabilistic description was built using the maximum entropy principle. The relevance of this approach was proved by analysing a human bone sample taken from the inferior femoral neck. The sample was imaged using synchrotron radiation micro-computed tomography. 3-D distributions of Haversian porosity and tissue mineral density extracted from these images supplied the experimental information needed to build the stochastic models of the volume fractions. Thus, the stochastic multiscale model provided reliable statistical information (such as mean values and confidence intervals) on bone elastic properties at the tissue scale. Moreover, the existence of a simpler "nominal model", accounting for the main features of the stochastic model, was investigated. It was shown that such a model does exist, and its relevance was discussed.

Atomistically informed stochastic multiscale model to predict the behavior of carbon nanotube-enhanced nanocomposites

A comprehensive, point-information-to-continuum-level analysis framework is presented in this paper to accurately characterize the behavior of carbon nanotube (CNT)-enhanced composite materials. Molecular dynamics (MD) simulations are performed to study sub-nanoscale interactions of the CNT with the polymeric phase of the nanocomposite. The effect of cross-linking between the epoxy resin and the hardener on the mechanical properties of the polymer is investigated; furthermore, the effect of CNT weight fraction on the probability distribution of polymer cross-linking degree is also studied through stochastic models. The stochastic distributions obtained from MD simulations provide a basis to simulate local variations in the matrix properties in the continuum model at the microscale. The inclusion of an atomistically informed elastic–plastic model at the microscale reveals a significant deviation of the mechanical properties from those obtained based on classical homogenization approaches. Microstructural variability arising from heterogeneous cross-linking degree in the polymer phase and variations in fiber geometry and spacing is also found to cause deviations in the mechanical response when compared to the assumption of a perfectly ordered fiber–matrix microstructure.