Nonlocal Model Research Articles

Numerical solutions of nonlocal heat conduction technique in tumor thermal therapy

The heat transport that takes place in living tissue during magnetic tumor hyperthermia is described in this study using the nonlocal bioheat model in spherical coordinates. In magnetic fluid hyperthermia, it is crucial to regulate the therapeutic temperature. This paper suggests a hybrid numerical approach that employs the Laplace transforms, change of variables, and modified discretization techniques, coupled with nonlocal hyperbolic shape function, to tackle the present problem. This study investigates the impacts of nonlocal parameter and the disparity in thermophysical properties between diseased and healthy tissue. A graph is displayed to represent the numerical temperature results. The validity of the numerical findings is demonstrated by comparing them with the results reported in previous literature.

Extreme load analysis of flexible wave energy converters utilising nonlocal continuum damage mechanics

In recent years, there has been a notable increase in interest towards Flexible Wave Energy Converters (FlexWECs). These flexible energy harvesters solve structural design challenges faced by rigid-body WECs by responding to external loading by changing shapes. Typically, the structures are made from rubber-like materials which pose few challenges from a material modelling point of view. Firstly, the material is in the finite strain regime requiring a hyperelastic modelling approach, but more critically the material response is expected to change during the operational lifetime. There is softening from both time-dependent viscoelasticity and micro-void growth caused by fatigue loading. The goal of this paper is to understand the latter mechanism and how it manifests within a membrane. To account for this damage accumulation, the gradient-enhanced nonlocal damage model is coupled to a hyperelastic Neo-Hookean constitutive law. The framework has been implemented in the commercial finite element software ABAQUS by exploiting its fully coupled thermo-mechanical formulation. A parametric study is performed on two FlexWEC archetypes: a submerged pressure differential and a floating bulge wave attenuator. The performance evaluation of these devices is carried out by analysing the evolution of the pressure–volume relation and pressure-stretch relation, respectively. The results show that the nonlocal aspects of damage in the pressure differential FlexWECs are small due to membrane action, but the saturation of damage does affect the pressure–volume function of each membrane. However, in the case of attenuator, the damage regularisation plays a crucial role in its behaviour due to the steep stress gradient from the crest of the wave. The outcomes from these analyses suggest FlexWEC design is advantageous from a fatigue loading perspective as it always reaches an equilibrium state which minimises the stress-differential, reducing the likelihood of localised crack growth.

Fulfilling modern astrophysical observations and lattice QCD constraints based on a nonlocal NJL model with 3D form factor

AbstractIn the present work, we employ a nonlocal Nambu–Jona–Lasinio (NJL) model with a Gaussian form factor that is dependent on the spatial components of the momentum (3D‐FF). Focusing on the temperature‐baryon chemical potential plane, we investigate some aspects of the phase diagram. Initially, we propose an assumption that the range of interactions in momentum space may be modified by temperature, allowing us to obtain the critical temperature values based on lattice QCD (LQCD) predictions. Subsequently, we consider this model within a hybrid framework to examine the effects of temperature, together with neutrino trapping, in compact object configurations.

An implicit electrolyte model for plane wave density functional theory exhibiting nonlinear response and a nonlocal cavity definition.

We have developed and implemented an implicit electrolyte model in the Vienna Ab initio Simulation Package (VASP) that includes nonlinear dielectric and ionic responses as well as a nonlocal definition of the cavities defining the spatial regions where these responses can occur. The implementation into the existing VASPsol code is numerically efficient and exhibits robust convergence, requiring computational effort only slightly higher than the original linear polarizable continuum model. The nonlinear + nonlocal model is able to reproduce the characteristic "double hump" shape observed experimentally for the differential capacitance of an electrified metal interface while preventing "leakage" of the electrolyte into regions of space too small to contain a single water molecule or solvated ion. The model also gives a reasonable prediction of molecular solvation free energies as well as the self-ionization free energy of water and the absolute electron chemical potential of the standard hydrogen electrode. All of this, combined with the additional ability to run constant potential density functional theory calculations, should enable the routine computation of activation barriers for electrocatalytic processes.

A nonlocal strain gradient model for buckling analysis of laminated composite nanoplates using CLPT and TSDT

A nonlocal strain gradient model for buckling analysis of laminated composite nanoplates using CLPT and TSDT

A nonlocal elasticity theory to model the static behaviour of edge-cracked nanobeams

In the present paper, the mechanical behaviour of edge-cracked nanobeams under Mixed-Mode loading is analytically investigated by means of the Stress-Driven nonlocal model. Firstly, the proposed formulation is outlined, and then applied to the case of a cantilever edge-cracked nanobeam. A parametric study is performed by varying both the crack depth and the crack position along the beam axis. Finally, the above formulation is applied to simulate some experimental tests available in the literature.

Energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn model

We present a new energy dissipation and maximum bound principle preserving scheme for solving a nonlocal ternary Allen-Cahn (NtAC) model, where the standard Laplace operator is deliberately replaced with a spatial convolution term that aims at describing long-range interactions among particles and the structure-preserving scheme is obtained based on the operator splitting method. The discrete maximum bound principle and global convergence of the new scheme are analyzed rigorously. Moreover, the scheme is strictly dissipative for a modified energy, which coincides with the original energy up to O(τ), where τ is the time step. To the best of our knowledge, this is the first work to show that the operator splitting method can guarantee the maximum bound principle for a ternary Allen-Cahn model. Various 2D/3D numerical experiments including the NtAC model with mass conservation are tested to verify the bound-preserving and energy-stable properties of the resulted scheme.

Simulation of hydraulic fracture propagation based on implicit gradient non-local model

Fracture extension simulation is an important tool for studying hydraulic fracture characteristics. To solve the problem of mesh sensitivity of strain and fluid pressure during hydraulic crack propagation, the equivalent strain in the damage evolution equation was non-localized, and implicit gradient was performed. A hydraulic-mechanical coupling model based on implicit gradient non-local damage was established. Numerical simulation program was further developed. Comparison of simulation results with experimental data on ‘L’-brittle material verified the validity of the program. The fracture propagation process of hydraulic fracturing in a perforated well was simulated. On this basis, the influence of internal length scale parameters and mesh size of the non-local model on the calculation results was analyzed. The influence of mesh size on hydraulic fracture morphology and damage variables is reduced by introducing an implicit gradient nonlocal model, which effectively alleviates the localization effect in hydraulic-mechanical coupling damage simulation. This new model achieves the nonlocalization of the fluid field by introducing the relationship among porosity-permeability and deformation and damage, which avoids the mesh sensitivity of equivalent strain, damage, fluid pressure, and improves the numerical stability. It provides a new method for simulating hydraulic fracture propagation and studying the failure behavior of porous media materials.

Non-local eddy diffusivity model based on turbulent energy density in scale space

Recently, a non-local eddy diffusivity model for the turbulent scalar flux was proposed to improve the local model and was validated using direct numerical simulation (DNS) of homogeneous isotropic turbulence with an inhomogeneous mean scalar (Hamba, J. Fluid Mech., vol. 950, 2022, A38). The non-local eddy diffusivity was assumed to be proportional to the two-point velocity correlation that was expressed in terms of the energy spectrum. Because the Fourier transform of velocity in the homogeneous directions was used to define the energy spectrum, it is not yet understood whether the proposed model can be applied to inhomogeneous turbulence. Thus, this study aimed to improve the non-local model using the scale-space energy density instead of the energy spectrum. First, the scale-space energy density based on filtered velocities was examined using the DNS data of homogeneous isotropic turbulence to obtain its simple form corresponding to the Kolmogorov energy spectrum. Subsequently, the two-point velocity correlation was expressed in terms of the scale-space energy density. Using these expressions, a new non-local eddy diffusivity model was proposed and validated using the DNS data. The one-dimensional non-local eddy diffusivity obtained from the new model agrees with the DNS value. The temporal behaviour of the three-dimensional non-local eddy diffusivity was improved compared with the previous model. Because the scale-space energy density was already examined in turbulent channel flow, it is expected that the new non-local model can also be applied to inhomogeneous turbulence and is useful for gaining insight into turbulent scalar transport.

A re-look into the modeling aspects of Eringen’s strain-driven nonlocal Euler-Bernoulli nanobeam bending problems

There has long been debate on the applicability of Eringen’s strain-driven nonlocal model to the study of nanostructures. Previous studies based on Eringen’s differential and integral nonlocal models to simulate the static bending behavior of nanobeams have shown shortcomings in nanobeam elastostatics solutions. The solution methodologies employed to solve the differential and integro-differential equations arising from Eringen’s strain-driven nonlocal constitutive model have shown inconsistencies in ensuring the fulfillment of essential and natural boundary conditions. This study aims to conduct an in-depth investigation of the modeling aspects of Euler-Bernoulli nanobeams under different loading and boundary conditions. Dirac-delta identity and generalized functions-based approach shall be utilized for the same. A rigorous mathematical proof establishing the equivalence between Eringen’s differential and integral nonlocal models is presented here for the first time.

High-Fidelity compressive spectral image reconstruction through a novel Non-Convex Non-Local Low-Rank tensor approximation model

Low-quality reconstruction is one main factor that greatly limits the development of snapshot compressive imaging (SCI) systems. The common algorithms usually unfold a 3D hyperspectral image (HSI) into a vector or matrix which inevitably corrupts the intrinsic structure of the HSI and insufficiently discovers its prior information. To fully exploit the underlying structures and internal correlations of the HSIs, this paper proposes a novel non-convex model based on Non-Local low-rank Tensor Approximation with Hyper-Laplacian priori (HL), named HL-NLTA, which effectively combines local and non-local similarity information from fused spectral and spatial perspectives. Specifically, a non-local low-rank tensor model based on the minimax concave plus (MCP) penalty and log sum (LS) penalty separately is developed which can simultaneously exploit two intrinsic properties of HSI, i.e., global correlation along the spectral domain (GCAS) and non-local self-similarity along the spatial domain (NSAS). Furthermore, the Hyper-Laplacian priori regularized through a non-convex ℓp (0 < p < 1) norm can preserve well the spectral and spatial structure. An optimization algorithm based on the alternating direction multiplier method (ADMM) framework is designed and accelerated by the Generalized shrinkage/thresholding (GST) algorithm and the fast Fourier transform (FFT) to solve the new nonconvex model. Extensive simulation results on public datasets and experimental results on our real CASSI system indicate that the proposed method enables high-fidelity HSI reconstructions in terms of simultaneous spatial structure detail recovery and spectral preservation. These results verify the effectiveness of the proposed model for snapshot compressive imaging (e.g., Coded aperture snapshot spectral imaging CASSI).

A generalized functions based approach for stress-driven nanobeam bending problems subjected to point loads

The research community has widely accepted the stress-driven nonlocal integral model to study static and dynamic characteristics of nanostructures. Although closed-form solutions for stress-driven point-loaded nanobeam bending problems that find potential for applications in the design of real-world MEMS/NEMS-based sensors have been explored recently, the static bending behavior of stress-driven Bernoulli-Euler and Timoshenko nanobeams subjected to point-loading conditions in various end configurations are studied in the present work wherein, analytical solutions for the same are arrived at by expressing the input bending and shear fields in terms of generalized functions for the first time. Equipped with the Helmholtz bi-exponential kernel function to model the nonlocality, the obtained solutions have demonstrated their ability to capture size-effects consistently. An overall stiffening effect on the nanobeam is observed as a consequence of these size-effects.

Free vibration analysis of functionally graded nanobeams based on surface stress-driven nonlocal model

A surface stress-driven nonlocal model is employed in this manuscript to study the coupled effects of the long-range interaction and surface energy on the free vibrations of nano-beams made of metal-ceramic functionally graded material. The nanobeam theory is formulated based on the Bernoulli-Euler kinematics and surface effects include surface elasticity, surface residual stresses, surface density and rotary inertia. Hamilton’s principle is applied to derive the size-dependent governing equation. The main results of a parametric investigation, carried out considering four different kinematic boundary conditions, i.e. Cantilever, Simply-Supported, Clamped-Pinned and Doubly-Clamped, are also presented and discussed, varying the nonlocal parameter and the material gradient index. The results show how the proposed model is able to capture surface energy effects on the overall dynamic behavior of functionally graded Bernoulli-Euler nanobeams and provides a cost-effective method for the design and the optimization of nano-scaled structures.

Effects of multiple edge cracks, shear force, elastic foundation, and boundary conditions on bucking of small-scale pillars

The buckling instability of micro- and nanopillars can be an issue when designing intelligent miniaturized devices and characterizing composite materials reinforced with small-scale beam-like particles. Analytical modeling of the buckling of miniaturized pillars is especially important due to the difficulties in conducting experiments. Here, a well-posed stress-driven nonlocal model is developed, which allows the calculation of the critical loads and buckling configurations of the miniaturized pillars on an elastic foundation and with arbitrary numbers of edge cracks. The discontinuities in bending slopes and deflection at the damaged cross-sections due to the edge cracks are captured through the incorporation of both rotational and translational springs. A comprehensive analysis is conducted to investigate the instability of pillars containing a range of one to four cracks. This analysis reveals interesting effects regarding the influence of crack location, nonlocality, and elastic foundation on the initial and subsequent critical loads and associated buckling configurations. The main findings are: (i) the shielding and amplification effects related to a system of cracks become more significant as the dimensions of pillars reduce, (ii) the influence of the shear force at the damaged cross-section related to the translational spring must not be neglected when dealing with higher modes of buckling and long cracks, (iii) an elastic foundation decreases the effects of the cracks and size dependency on the buckling loads, and (iv) the effects of the edge cracks on the critical loads and buckling configurations of the miniaturized pillars are highly dependent on the boundary conditions.

Wave breaking phenomenon in the unidirectional non-local wave model

In this paper, we investigate wave breaking phenomenon in the unidirectional non-local wave model. Making use of the L2-conservation law and fine structure of the model, we show a local-in-space wave breaking condition for the unidirectional non-local wave model.

Nonlinear thermo-mechanical stress-driven modeling of nano arches augmented by higher order double-scaled kernel

Modeling the nonlocal behavior of small-scale structures is almost controversial. The recently developed approach, known as the stress-driven nonlocal model, in which the stress is the input of nonlocal integral, has been proved to pose as a remedy for the ill-posing behavior of strain-driven solutions. Though it is still in its early stages, stress-driven modeling of the nonlinear behavior of nano-scaled problems needs further thought. The coupled displacement field, in particular, makes nano-arches more challenging. The stress-driven formulation, enhanced by constitutive boundary conditions, reveals the thermo-mechanical buckling and post-buckling features of shallow nano-arches for the first time in this research. Furthermore, the single-scaled Helmholtz kernel is replaced with the more appropriate double-scaled kernel known as the bi-Helmholtz averaging nonlocal kernel. Higher-order continuity of the bi-Helmholtz kernel and, hence, constitutive boundary conditions and higher-order differential equations corresponding to the stress-driven scheme hold the key to efficiency. The arch is subjected to uniform temperature and radial pressure at the same time. To demonstrate the response of the nano-arch during buckling and post-buckling, the nonlinear equilibrium path, whether for mechanical or thermal loading, is given for the first time by prevailing on the challenges of the higher-order problems. Moreover, it would be beneficial if one could have the chance to recognize how and when the arch buckles. These valuable data are available in this study to give the limiting parameters at the moment of buckling, including limiting temperature, nonlocal parameter, and geometry.

Dynamic stability analysis of porous functionally graded beams under hygro-thermal loading using nonlocal strain gradient integral model

Dynamic stability analysis of porous functionally graded beams under hygro-thermal loading using nonlocal strain gradient integral model

Coupled continuum modelling of size segregation driven by shear-strain-rate gradients and flow in dense, bidisperse granular media

Dense granular systems that consist of particles of disparate sizes segregate based on size during flow, resulting in complex, coupled segregation and flow patterns. The ability to predict how granular mixtures segregate is important in the design of industrial processes and the understanding of geophysical phenomena. The two primary drivers of size segregation are pressure gradients and shear-strain-rate gradients. In this work, we isolate size segregation driven by shear-strain-rate gradients by studying two dense granular flow geometries with constant pressure fields: gravity-driven flow down a long vertical chute with rough parallel walls and annular shear flow with rough inner and outer walls. We perform discrete element method (DEM) simulations of dense flow of bidisperse granular systems in both flow geometries, while varying system parameters, such as the flow rate, flow configuration size, fraction of large/small grains and grain-size ratio, and use DEM data to inform continuum constitutive equations for the relative flux of large and small particles. When the resulting continuum model for the dynamics of size segregation is coupled with the non-local granular fluidity model – a non-local continuum model for dense granular flow rheology – we show that both flow fields and segregation dynamics may be simultaneously captured using this coupled, continuum system of equations.

In-Plane Wave Propagation Analysis of Human Breast Lesions Using a Higher-Order Nonlocal Model and Deep Learning

The wave propagation characteristics of biological tissues are of high importance in improving healthcare technologies and can be used as an early clinical indicator of many diseases. However, the current mathematical models that describe the mechanical properties of biological tissues do not account for the difference in softening and hardening observed at different scales and this limits their utility in biomedical imaging. In this paper, a higher-order nonlocal model is developed to study in-plane wave propagation in healthy, benign, and cancerous breast tissues. To verify the mathematical approach, finite element simulations are conducted. Furthermore, a sequential deep neural network model of feedforward type with multiple hidden layers is developed to understand the intrinsic in-plane wave characteristics of breast tissues. The deep learning algorithm shows potential in accurately extracting the frequencies and phase velocities of breast lesions under in-plane waves even when there is a limited number of clinical samples. Using the higher-order nonlocal model, significant differences between healthy fibroglandular tissue and early breast cancer in the form of ductal carcinoma in situ have been found. The combination of nonlocal and strain gradient parameters allows for the concurrent incorporation of stiffness hardening and softening, solving the rigid-tumour–soft-cell paradox of cancer biomechanics.

Optimal Distributed Control for a Viscous Non-local Tumour Growth Model

In this paper, we address an optimal distributed control problem for a non-local model of phase-field type, describing the evolution of tumour cells in presence of a nutrient. The model couples a non-local and viscous Cahn–Hilliard equation for the phase parameter with a reaction-diffusion equation for the nutrient. The optimal control problem aims at finding a therapy, encoded as a source term in the system, both in the form of radiotherapy and chemotherapy, which could lead to the evolution of the phase variable towards a desired final target. First, we prove strong well-posedness for the system of non-linear partial differential equations. In particular, due to the presence of a viscous regularisation, we can also consider double-well potentials of singular type and cross-diffusion terms related to the effects of chemotaxis. Moreover, the particular structure of the reaction terms allows us to prove new regularity results for this kind of system. Then, turning to the optimal control problem, we prove the existence of an optimal therapy and, by studying Fréchet-differentiability properties of the control-to-state operator and the corresponding adjoint system, we obtain the first-order necessary optimality conditions.