Inhomogeneous Case Research Articles

Simulation of the response of a magnetic polymersome in an inhomogeneous magnetic field

Purpose. Identification of the behavior of a magnetic polymersome placed in an inhomogeneous field of a point dipole using coarse-grained molecular dynamics simulations.Methods. The system under the study is represented as a set of interacting particles of two types: polymeric particles simulating a double layer of an amphiphilic membrane, and magnetic nanoparticles located in the membrane layer. Polymeric particles interact through elastic potentials designed to maintain an equilibrium spherical vesicular geometry. Magnetic nanoparticles interact with each other and external fields as point dipoles. The excluded volume and steric interaction of magnetic particles with polymeric walls are modeled in the form of soft repulsion. The entire system is under isothermal conditions, and the model parameters are chosen according to typical interaction energies relative to thermal fluctuations. A model polymersome with a diameter of about 100 nm in an aqueous solution at 25°C is considered. An inhomogeneous magnetic field is created by a dipole of constant direction located at a fixed distance. The dynamics of the system is monitored by numerical solution of the equations of particles motion with introduced interactions and imposed conditions.Results. In numerical experiments, the response of the polymersome to a magnetic field was obtained for three values of the parameter describing the inhomogeneity of the field. The problem with a gradual increase in the strength (and its gradient) of the magnetic field near the polymersome is considered. Changes in the magnetization of the system are shown, and the redistribution of the concentration of nanoparticles is analyzed. A simulation of the situation when the center of the polymersome at the initial moment was placed at a point with a fixed value of the magnetic field for three cases of inhomogeneity of the field was carried out. A significant restructuring of the magnetic layer of the vesicle combined with movement of the entire capsule was detected. Estimates are made about the average velocity based on the displacement of the object.Conclusion. The model allows to evaluate the features of the combined magnetic, structural and mechanical response of the polymersome to an inhomogeneous field in the context of potential applications for controlled delivery of contents into cells.

An adaptive lattice Green's function method for external flows with two unbounded and one homogeneous directions

We solve the incompressible Navier-Stokes equations using a lattice Green's function (LGF) approach, including immersed boundaries (IB) and adaptive mesh refinement (AMR), for external flows with one homogeneous direction (e.g. infinite cylinders of arbitrary cross-section). We hybridize a Fourier collocation (pseudo-spectral) method for the homogeneous direction with a specially designed, staggered-grid finite-volume scheme on an AMR grid. The Fourier series is also truncated variably according to the refinement level in the other directions. We derive new algorithms to tabulate the LGF of the screened Poisson operator and viscous integrating factor. After adapting other algorithmic details from the fully inhomogeneous case [1], we validate and demonstrate the new method with transitional and turbulent flows over a circular cylinder at Re=300 and Re=12,000, respectively.

Cancelling the effect of sharp notches or cracks with graded elastic modulus materials

Recent technologies permit to build materials which have elastic spatially varying modulus which can also imitate solutions adopted in Nature to optimize some structures. It has been shown that for example the stress concentration due to a hole in an infinite plate can be cancelled with a radially varying modulus making it similar to load-bearing bones which seem to resist structural failures even in the presence of blood vessel holes (foramina). Here, we attempt to study the classical problem of a sharp wedge (which includes the important case of a crack) looking for stresses varying as power law of the distance from the notch tip, σ∼rα, with a modulus varying as E∼rβ. In the inhomogeneous case the order of singularity of the LEFM case decreases if β>0, as confirmed by FEM investigations. Hence, we can remove stress singularities, which suggests an interesting alternative to the “rounding” of the notch. More in general, since for many materials it has been found that both strength and modulus are power laws of the density, using the so called strength-modulus exponent ratio we can obtain optimal design by keeping the asymptotic stress constantly equal to the strength. The present investigation paves the way for a new optimization strategy in the problems which eliminates size-scale effects due to singular stress fields, with potentially very wide applications.

Tempered nonlocal integrodifferential equations with nonsmooth solution data

This work on the time discretization for the solution of the tempered nonlocal integro-differential equation with smooth and nonsmooth initial data are extended. We find the difficulty arising from theoretical analysis can be overcome by Laplace transform technique. A mount of proof skills have been provided based on Laplace transform technique for this kind of equations. The Laplace transform method is employed effectively to show that the proposed interpolating quadrature scheme is convergence for smooth and non-smooth initial data in both homogeneous and inhomogeneous cases.

A Study on Linear Prabhakar Fractional Systems with Variable Coefficients

The focus of this paper is on addressing the initial value problem related to linear systems of fractional differential equations characterized by variable coefficients, incorporating Prabhakar fractional derivatives of Riemann–Liouville and Caputo types. Utilizing the generalized Peano–Baker series technique, the state-transition matrix is acquired. The paper presents closed form solutions for both homogeneous and inhomogeneous cases, substantiated by illustrative examples.

Flame self-interactions in MILD combustion of homogeneous and inhomogeneous mixtures

Flame self-interaction (FSI) events in Moderate or Intense Low-Oxygen Dilution (MILD) combustion of homogeneous and inhomogeneous mixtures of methane and oxidiser have been analysed using three-dimensional Direct Numerical Simulations (DNS). The simulations have been conducted at the same global equivalence ratio (⟨ϕ⟩=0.8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\phi \\rangle = 0.8$$\\end{document}) for different levels of O2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathrm {O_2}$$\\end{document} concentration (dilution) and initial turbulence intensities. It has been reported that both homogeneous and inhomogeneous mixture MILD combustion cases exhibit significant occurrences of FSI events, with the peak frequency of FSI events occurring towards the burned gas side in all cases. Moreover, the frequency of FSI events increases with increasing dilution level and turbulence intensity, but the presence of mixture inhomogeneity leads to a reduction in total FSI events. In all cases, the cylindrical FSI topologies (i.e. tunnel formation and tunnel closure) were found to have a higher likelihood of occurrence compared to spherical FSI topologies (i.e. unburned and burned gas pockets). The geometries of FSI topologies were also analysed using the mean and Gaussian curvatures. It has been shown that the inward propagating spherical FSI topologies (i.e. unburned gas pockets) are associated with negative mean curvature, while outward propagating spherical FSI topologies (i.e. burned gas pockets) are associated with positive mean curvature. Moreover, tunnel formation (tunnel closure) FSI topologies predominantly exhibit either elliptic geometries with positive (negative) mean curvature or hyperbolic saddle geometries with negative (positive) mean curvature. It has been shown for the first time in MILD combustion that the mean values of kinematic restoration and dissipation terms in the transport equation of the magnitude of the reaction progress variable conditional upon the reaction progress variable tend to cancel each other in the vicinity of the critical points associated with cylindrical topologies. Thus, the singular contributions in these terms, which are obtained from analytical descriptions in the vicinity of tunnel formation and tunnel closure topologies, do not affect the balance equation of the magnitude of the gradient of the reaction progress variable. Consequently, there is no need for a separate model treatment for singularities in modelling approaches based on the magnitude of the gradient of the reaction progress variable. The FSI events in the reaction dominated and propagating flame regions of MILD combustion have also been analysed for the first time. It has been found that more FSI events occur in the reaction dominated region, particularly towards the burned gas side. However, the majority of spherical FSI topologies are found in the propagating flame region. The findings from this study indicate that turbulence intensity, dilution level and mixture inhomogeneity effects need to be considered in any attempt to extend flame surface-based modelling approaches to MILD combustion.

A General Approximate Solution for the Slightly Non-Axisymmetric Normal Contact Problem of Layered and Graded Elastic Materials

We present a general approximate analytical solution for the normal contact of layered and functionally graded elastic materials for almost axisymmetric contact profiles. The solution only requires knowledge of the corresponding contact solution for indentation using a rigid cylindrical flat punch. It is based on the generalizations of Barber’s maximum normal force principle and Fabrikant’s approximation for the pressure distribution under an arbitrary flat punch in an inhomogeneous case. Executing an asymptotic procedure suggested recently for almost axisymmetric contacts of homogeneous elastic media results in a simple approximate solution to the inhomogeneous problem. The contact of elliptical paraboloids and indentation using a rigid pyramid with a square planform are considered in detail. For these problems, we compare our results to rigorous numerical solutions for a general (bonded or unbonded) single elastic layer based on the boundary element method. All comparisons show the quality and applicability of the suggested approximate solution. Based on our results, any compact axisymmetric or almost axisymmetric contact problem of layered or functionally graded elastic materials can be reduced asymptotically to the problem of indenting the material using a rigid cylindrical flat punch. The procedure can be used for different problems in tribology, e.g., within the framework of indentation testing or as a tool for the analysis of local features on a rough surface.

Wave equation for Sturm-Liouville operator with singular potentials

The paper is denoted to the initial-boundary value problem for the wave equation with the Sturm-Liouville operator with irregular (distributive) potentials. To obtain a solution to the equation, the separation method and asymptotics of the eigenvalues and eigenfunctions of the Sturm-Liouville operator are used. Homogeneous and inhomogeneous cases of the equation are considered. Next, existence, uniqueness, and consistency theorems for a very weak solution of the wave equation with singular coefficients are proved.

Strain-gradient solution to elastodynamic scattering from a cylindrical inhomogeneity

In this paper, Mindlin’s first strain-gradient theory is adopted for the first time to come up with a solution to the problem of the elastic scattering of an anti-plane time-harmonic wave incident upon an isolated circular cylindrical inhomogeneity embedded in an infinite body. The effect of micro-inertia is also captured in this analysis, which alongside the strain-gradient effect enhances the adopted theory to account for different aspects of the size effect on the elastodynamic fields developed in the medium. The general case of an elastic inhomogeneity and two special cases of a rigid immovable obstacle and a cavity are addressed and the analytical expressions for the corresponding displacement fields are obtained. These solutions reveal the existence of some kind of decaying standing waves in the medium, which cannot be predicted by the classical elastodynamic theory. Moreover, the analytical expressions for the differential and total scattering cross-sections of the cylindrical inhomogeneity are determined. The results demonstrate that, if the cross-sectional radius of the inhomogeneity or the wave-length of the incident wave is comparable with the characteristic lengths of the medium, then the size effect will be significant and the discrepancies between the solutions obtained by the adopted theory and their counterparts in the classical elastodynamic theory will be pronounced.

Decreasing the memory of the discontinuous Galerkin volume integral equation method for scattering from inhomogeneous dielectric objects.

A memory-efficient implementation scheme for the discontinuous Galerkin volume integral equation method (DGVIE) using Schaubert-Wilton-Glisson (SWG) basis functions is proposed to analyze electromagnetic scattering from inhomogeneous dielectric objects. For this proposed scheme, almost no half-SWG basis functions are needed for the elements separating nonconformal meshes, while these half-SWG basis functions are indispensable for the conventional DGVIE-SWG method. This is realized by applying the divergence-free condition of the electric displacement vector explicitly for nonconformal meshes separating neighboring subdomains of an inhomogeneous dielectric body. Therefore, the number of unknowns of the conventional DGVIE method can be further reduced. As a result, the memory of the proposed DGVIE method is only about half of the conventional one for inhomogeneous dielectric problems. Meanwhile, the total solution time has been reduced by the use of the proposed scheme. Particularly, the proposed DGVIE-SWG method is efficient in memory usage not only for inhomogeneous dielectric cases with high contrast ratio but also for cases with relatively low contrast ratio.

Explicit analytical solutions of an incommensurate system of fractional differential equations in a fuzzy environment

Fuzzy fractional models have attracted considerable attention because of their comprehensive and broader understanding of real-world problems. Analytical studies of these models are often complex and difficult. Therefore, it is beneficial to develop a suitable and comprehensive technique to solve these models analytically. In this paper, an explicit analytical technique for solving two-dimensional incommensurate linear fuzzy systems of fractional Caputo differential equations (FLSoCFDEs) considering generalized Hukuhara differentiability (gH-differentiability) is presented and demonstrated. This extracted explicit solution is presented for different classes of such systems, including homogeneous and non-homogeneous cases with commensurate and incommensurate fractional orders. Moreover, the potential solution of FLSoCFDEs in terms of the Mittag-Leffler function involving double series is presented. The originality of the proposed technique is that the fuzzy Cauchy problem is transformed into a system of fuzzy linear Volterra integral equations of second kind and then the solution is extracted using the iterative Picard scheme based on the Banach fixed point theorem. Moreover, several interesting results are derived from FLSoCFDEs in terms of the Mittage-Leffler function for both homogeneous and inhomogeneous cases. To understand the proposed technique, we solve a diffusion process problem (a biological model) and several mass-spring systems as applications. Their graphs are analyzed to illustrate and support the theoretical results.

Non-intersecting Path Constructions for TASEP with Inhomogeneous Rates and the KPZ Fixed Point

We consider a discrete-time TASEP, where each particle jumps according to Bernoulli random variables with particle-dependent and time-inhomogeneous parameters. We use the combinatorics of the Robinson–Schensted–Knuth correspondence and certain intertwining relations to express the transition kernel of this interacting particle system in terms of ensembles of weighted, non-intersecting lattice paths and, consequently, as a marginal of a determinantal point process. We next express the joint distribution of the particle positions as a Fredholm determinant, whose correlation kernel is given in terms of a boundary-value problem for a discrete heat equation. The solution to such a problem finally leads us to a representation of the correlation kernel in terms of random walk hitting probabilities, generalizing the formulation of Matetski et al. (Acta Math. 227(1):115–203, 2021) to the case of both particle- and time-inhomogeneous rates. The solution to the boundary value problem in the fully inhomogeneous case appears with a finer structure than in the homogeneous case.

Stochastic control with inhomogeneous regime switching: Application to consumption and investment with unemployment and reemployment

This paper presents a stochastic control problem with an inhomogeneous regime switching and applies it to a consumption and investment model. We prove that the inhomogeneous Markov chain is a semimartingale, providing a basis to extend the HJB equations with regime switching to inhomogeneous cases. Explicit solutions are obtained by solving the corresponding HJB equations. Additionally, we study the impact of different levels of unemployment income and intensities of unemployment and reemployment on consumption and value function. Our findings suggest that improving unemployment security and reemployment intensity can increase overall happiness and improve the total value function.

Bethe ansatz diagonalization of the Heun–Racah operator

The diagonalization of the Heun–Racah operator is studied with the help of the modified algebraic Bethe ansatz. This operator is the most general bilinear expression in two generators of the Racah algebra. A presentation of this algebra is given in terms of dynamical operators and allows the construction of Bethe vectors for the Heun–Racah operator. The associated Bethe equations are derived for both the homogeneous and inhomogeneous cases.

Fractional evolution equation with Cauchy data in L^{p} spaces

In this paper, we consider the Cauchy problem for fractional evolution equations with the Caputo derivative. This problem is not well posed in the sense of Hadamard. There have been many results on this problem when data is noisy in L^{2} and H^{s}. However, there have not been any papers dealing with this problem with observed data in L^{p} with p neq 2. We study three cases of source functions: homogeneous case, inhomogeneous case, and nonlinear case. For all of them, we use a truncation method to give an approximate solution to the problem. Under different assumptions on the smoothness of the exact solution, we get error estimates between the regularized solution and the exact solution in L^{p}. To our knowledge, L^{p} evaluations for the inverse problem are very limited. This work generalizes some recent results on this problem.

Implicit dynamic buckling analysis of thin‐shell isogeometric structures considering geometric imperfections

AbstractIn continuation of our previous work on nonlinear stability analysis of trimmed isogeometric thin shells, this contribution is an extension to dynamic buckling analyses for predicting reliably complex snap‐through and mode‐jumping behavior. Specifically, a modified generalized‐ time integration scheme is used and combined with a nonlinear isogeometric Kirchhoff–Love shell element to provide second‐order accuracy while introducing controllable high‐frequency dissipation. In addition, a weak enforcement of essential boundary conditions based on a penalty approach is considered with a particular focus on the inhomogeneous case of imposed prescribed displacements. Moreover, we propose a least‐squares B‐spline surface fitting approach and corresponding error measures to model both eigenmode based and measured geometric imperfections. The imperfect geometries thus obtained can be naturally integrated into the framework of isogeometric nonlinear dynamic shell analysis. Based on this idea, the different modeling methods and the influence of the appropriately considered geometric imperfections on the dynamic buckling behavior can be investigated systematically. Both perfect and geometrically imperfect shell models are considered to assess the performance of the proposed method. We compare our method with established developments in this field and demonstrate superior achievements with regard to solution quality and robustness.

An Anisotropic Elastic Solid With an Elliptical Inhomogeneity Under a Non-Uniform In-Plane And Anti-Plane Remote Loading

Summary We consider the case of an anisotropic elastic elliptical inhomogeneity embedded in an infinite anisotropic elastic matrix subjected to a non-uniform remote loading described by remote stresses and strains which are linear functions of the two in-plane coordinates. The internal stresses and strains within the elliptical inhomogeneity are found to be linear functions of the two in-plane coordinates. In addition, we obtain explicit real-form solutions describing the elastic field inside the inhomogeneity as well as hoop stress vectors and hoop stresses on the matrix side and on the inhomogeneity side. We also obtain the corresponding solutions for the two limiting cases in which the elliptical inhomogeneity takes the form of a hole or a rigid inhomogeneity. The solution method presented here can be extended to accommodate the more general scenario in which the remote applied stresses and strains are arbitrary-order polynomials of the two in-plane coordinates.

Two-temperature thermodynamics and a viscoplasticity model for FCC metals with grain boundary effect

A non-equilibrium thermodynamic route is pursued to model viscoplasticity, including the evolving grain boundary, in face-centered cubic (FCC) metals. The thermodynamic description includes two subsystems: the configurational subsystem describing the relatively slower motion of dislocations and grain boundaries, and a fast-evolving kinetic-vibrational (K-V) subsystem depicting atomic vibrations about equilibrium positions in the lattice. Grain boundary and dislocation densities as well as their interactions are incorporated within the model. The model is initially developed for homogeneous plastic deformation, which is then generalized to the inhomogeneous case via spatial variations in the thermodynamic states over a wide range of strain rate and temperature. The model is implemented as a user material subroutine (VUMAT) in ABAQUS to simulate a problem of industrial interest, viz. deep drawing on an AA 7075 aluminum alloy. Finally, the thermo-viscoplasticity model is used to predict negative strain rate sensitivity in the AA 2219 aluminum alloy at room temperature, for which an experiment is performed at a facility of the VSSC. The predictive quality of the model is demonstrated by a comparison of numerical simulations with the experimental evidence.

Image temperature calculation for gas and particle system by the mid-infrared spectrum using DRESOR and SNBCK model

The balance between spatial and spectral resolutions plays a key role in image temperature measurement using Imaging Fourier Transform Infrared Spectroscopy (IFTS). In this study, the performance of different combinations of spatial and spectral resolutions was investigated in a two-dimensional (2D) rectangular enclosure filled with gas and particle. The distributions of ratios of energy scattered or reflected (DRESOR) method combined with the statistical narrow band correlated-k (SNBCK) model was developed to obtain the radiative intensity image. Then the image temperature was calculated by the spectral thermometry. The accuracy of the DRESOR-SNBCK method was examined by comparison with the finite volume method (FVM)-line by line (LBL) method. The results showed that the average relative deviation of the image temperatures calculated by the spatial resolutions of 1 × 57 pixel and 1 × 140 pixel was <0.4%, indicating the accuracy of image temperatures was not improved by the increment of spatial resolution. However, the average relative error increased from 4.3% to 7.0% with the number of spectral bands increasing from 18 to 36. In addition, the large relative error of DRESOR-SNBCK method occurred in the low image temperature region due to the scattering effects. The average relative deviation of the homogeneous and inhomogeneous cases was <0.5%, indicating that the image temperatures were not affected by the gas homogeneity.

A robust adiabatic constant amplitude spin-lock preparation module for myocardial T1ρ quantification at 3T.

T1ρ quantification has the potential to assess myocardial fibrosis without contrast agent. However, its preparation spin-lock pulse is sensitive to B1 and B0 inhomogeneities, resulting in severe banding artifacts in the heart region, especially at high magnetic field such as 3T. We aimed to design a robust spin-lock (SL) preparation module that can be used in myocardial T1ρ quantification at 3T. We used the tan/tanh pulse to tip up and tip down the magnetization in the spin-lock preparation module (tan/tanh-SL). Bloch simulation was used to optimize pulse shape parameters of the tan/tanh with a pulse duration (Tp ) of 8, 4, and 2ms, respectively. The designed tan/tanh-SL modules were implemented on a 3-T MR scanner. They were evaluated in phantom studies under three different cases of B0 and B1 inhomogeneities, and tested in cardiac T1ρ quantification of healthy volunteers. The performance of the tan/tanh-SL was compared with the composite SL preparation pulses and the commonly used hyperbolic secant pulse for spin-lock (HS-SL). Feasible pulse shape parameters were obtained using Bloch simulation. Compared with HS-SL, the quantification error of tan/tanh-SL was reduced by 27.7% for Tp = 8ms (mean ∆Q = 126.15 vs. 174.42) and 75.6% for Tp = 4ms (mean ∆Q = 136.65 vs. 559.53). In the phantom study, tan/tanh-SL was less sensitive to B1 and B0 inhomogeneity compared with composite SL pulses and HS-SL. In cardiac T1ρ quantification, the T1ρ maps using tan/tanh-SL showed fewer banding artifacts than using composite SL pulses and HS-SL. The proposed tan/tanh-SL preparation module greatly improves the robustness to B0 and B1 field inhomogeneities and can be used in cardiac T1ρ quantification at 3T.