Inhomogeneous Problem Research Articles

A Localized Collocation Method Based on Nonsingular Fundamental Solutions for Steady-State Heat Conduction Analysis in Heterogeneous Materials

This paper further develops a localized nonsingular method of fundamental solutions (LNMFS) for effectively solving heat conduction problems in functionally graded materials (FGMs). In this numerical framework, the LNMFS approach is proposed as a solution for the inhomogeneous boundary value problems, where a pseudo-spectral Chebyshev collocation scheme (CCS) is employed for the approximation of the corresponding particular solutions. In the proposed LNMFS, the artificial boundary that is present in the existing method of fundamental solutions (MFS) is eliminated. Accordingly, the singularities are replaced by the normalized area integrals of the fundamental solution over small circular disks, which encompass the source nodes that intersect with the collocation nodes. The proposed method inherits the high accuracy of the nonsingular method of fundamental solutions (NMFS). Additionally, in contrast to the NMFS, the LNMFS not only possesses the capability to achieve solutions with a notable level of accuracy in nonharmonic boundary condition but also overcomes the constraint of exclusively simulating heat conduction issues in FGMs that exhibit specific spatial variations. The feasibility and accuracy of the recently developed method are demonstrated by means of comparing it with analytical solutions for several numerical examples.

Improved techniques for the solution of several circular dielectric and hollow layers in a straight rectangular waveguide

The first objective of this research is to develop two effective techniques for solving inhomogeneous problems in a rectangular waveguide with several circular dielectric and hollow layers. The cross section consists of a four-layer periodic circular structure with alternating dielectric and hollow layers. We assume that the dielectric profile is different in each circular dielectric layer. The first technique enables the calculation of the dielectric profile and the elements of the matrices of circular dielectric layers by using the image method. The second alternative technique improves the first technique by using the ω ε function as well as the image method. The second objective is to examine the influence of this discontinuous cross section on the output field. The contribution of the proposed techniques is important to improve the mode model method especially in the case of a large number of circular dielectric and hollow layers in the cross section. The application is useful for an inhomogeneous cross section with several circular dielectric and hollow layers in millimeter regime.

Development and validation of an analytical method for the simultaneous quantitation of posaconazole Form I and Form-S in oral suspensions

Any polymorphic conversion of an active pharmaceutical ingredient (API), even if partial, is likely to lead to changes in its efficiency and safety. Posaconazole, an antifungal drug, is detected as Form-S in the commercially available oral suspensions. However, a mixture of Form-S and the initial Form I is likely to coexist depending either on the manufacturing process of the suspensions and/or to the storage conditions of the suspension. The simultaneous quantitation of these crystal forms in suspensions is challenging because of the dose inhomogeneity and the instability of Form-S at ambient conditions. Although X-ray Powder Diffraction (XRPD) was initially employed, preferred orientation issues in the suspension inhibited the successful quantitative determination of the polymorphs. In order to circumvent the problem Raman spectroscopy was selected for addressing this challenge, while the additional application of a home-made rotation system reduced the inhomogeneity problems. A separate calibration curve was generated for each polymorph using a conventional linear regression. The combination of these two relations led to the formation of a comprehensive equation relating the characteristic Raman peak intensities of posaconazole Form-S and Form I with the concentrations thereof. The method was validated and the results were confirmed through a partial least square regression (PLSR). The detection limits for Form-S and Form I were determined equal to 1.9 mg/mL and 2.2 mg/mL, respectively.

Eshelby's inhomogeneity model within Mindlin's first strain gradient elasticity theory and its applications in composite materials

Eshelby's inhomogeneity problem is solved within the second form of Mindlin's first strain gradient elasticity theory for the prediction of the effective elastic properties of composites. Considering Green's function technique, an integral equation is established for an ellipsoidal inhomogeneity embedded in a homogeneous elastic medium and subjected to non-uniform boundary conditions. Within isotropic elasticity, the mean strain inside a spherical inhomogeneity is detailed to provide analytical results. In addition to the elastic properties of the inhomogeneity and the matrix, the strain localization depends on five gradient elastic constants, introduced by the first strain gradient elasticity theory. The effective bulk and shear moduli of a two-phase composite are predicted through Mori-Tanaka's scheme. The strain localization and the effective elastic moduli are then expressed within some simplified gradient elasticity theories. To test the relevance of the developed model, its predictions are compared with those of some investigations and the effective elastic properties are analyzed for a metal matrix composite. Finally, some comparisons with experimental data are performed to estimate the characteristic length scale parameters and gradient elastic constants of local phases.

Plate and beam moment-field formulation

Governing equations of the linear elastic Euler–Bernoulli beam and pure bending Kirchhoff–Love plate are developed in a pure stress form with the evolution of the bending moments fields. In the case of elastodynamics, the governing equations are used to find mode restrictions of vibration within both structures. These vibration restrictions are then compared with the solutions found through the displacement formulation finding both forms to give the same solution under the same boundary conditions. The waves are assumed to be harmonic with only spatial contributions being examined, validating the stress approach. More advanced cases involving anisotropic and non-classical boundary conditions are also explored to demonstrate further implications of following the stress evolution. Elastostatic plates are examined under the compatibility conditions to be met for moment fields to possess uniqueness and to show the CLM (Cherkaev–Lurie–Milton) invariance of the inhomogeneous plane-stress problem.

The uniqueness of minimizers for L2$$ {L}^2 $$‐subcritical inhomogeneous variational problems with a spatially decaying nonlinearity

By constructing various Pohozaev identities, we study the uniqueness of minimizers for ‐subcritical inhomogeneous variational problems with spatially decaying nonlinear terms, which contains as a singular point.

A finite difference informed random walker (FDiRW) solver for strongly inhomogeneous diffusion problems

In nature, many complex multi-physics coupling problems exhibit strong diffusivity inhomogeneity. For instance, in the context of radionuclide absorption by porous wasteform materials within a flowing waste stream, the difference of species’ diffusivity in solid and liquid phases spans by 3 ∼ 8 orders of magnitude. To solve the diffusion equations with strongly inhomogeneous diffusivity, traditional discretization-based methods, such as the Finite Difference Method (FDM), require infinitesimally small-time steps (<10-10) as high spatial resolutions are employed in most microstructure evolution processes, leading to prohibitively high computational costs. This work developed an integrated numerical approach (FDiRW: Finite Difference informed Random Walk) to tackle this challenge. The idea is that utilizing the Random Walk concept, the fast diffusion is modelled as a superposition of point source’s solution for a concentration distribution while FDM is used to obtain the point source’s solution at each node. A mesh-coarsening algorithm is developed to generate an exclusive coarse mesh for FDiRW approach to maximize its efficiency. The effectiveness of the coarse mesh-based FDiRW approach is validated by benchmarking Finite Difference solutions. Numerical results demonstrated that FDiRW achieves a remarkable 1000x computational efficiency improvement over FDM while preserving desired accuracy for a medium-sized model of 192×192×192 grids. As models scale up, a floating-point operations (PLOPs) analysis of the FDiRW algorithm reveals that its computational complexity grows quadratically in terms of the number of nodes employed in computation.

Existence and regularity of strict solutions for a class of fractional evolution equations

AbstractWe study the existence and Hölder regularity of solutions for fractional evolution equations of order . By means of an analytic resolvent, we construct an interpolation space, which can effectively lower the regularity of initial data. By virtue of the interpolation space and some properties of the analytic resolvent, we derive the existence and Hölder regularity of strict solutions for an inhomogeneous problem, as well as the existence and Hölder regularity of a nonlinear problem.

In vivo imaging of the human brain with the Iseult 11.7-T MRI scanner.

The understanding of the human brain is one of the main scientific challenges of the twenty-first century. In the early 2000s, the French Atomic Energy Commission launched a program to conceive and build a human magnetic resonance imaging scanner operating at 11.7 T. We have now acquired human brain images in vivo at such a magnetic field. We deployed parallel transmission tools to mitigate the radiofrequency field inhomogeneity problem and tame the specific absorption rate. The safety of human imaging at such high field strength was demonstrated using physiological, vestibular, behavioral and genotoxicity measurements on the imaged volunteers. Our technology yields T2 and T2*-weighted images reaching mesoscale resolutions within short acquisition times and with a high signal and contrast-to-noise ratio.

Critical Temperature and Order Parameter in Superconductor/Inhomogeneous Ferromagnet Heterostructures

A method for solving a self-consistent boundary value problem for the linearized Usadel equation is proposed. The method makes it possible to find the sample-normalized distribution of the superconducting order parameter and the critical temperature as functions of the problem parameters and to solve relatively complex spatially inhomogeneous problems, e.g., superconducting heterostructures containing inhomogeneous magnetic layers. Within this approach, layered structures containing superconducting and domain-split ferromagnetic layers are considered. The theory is compared with experiment for the Fe 20 Å/V 340 Å/Fe 8 Å/Cr/Fe 8 Å system.

DAUP: Enhancing point cloud homogeneity for 3D industrial anomaly detection via density-aware point cloud upsampling

The use of 3D information in industrial anomaly detection tasks has been shown to enhance performance by uncovering unseen abnormal patterns in the RGB modality. Despite the focus on detection pipeline design and multimodal fusion schemes in previous approaches, explorations of dataset characteristics were often overlooked. In contrast to RGB images where pixels form regular grids, point clouds intrinsically lack order and exhibit inhomogeneous densities across regions, thereby adversely affecting the feature extraction process. In this work, we propose a learning-based density-aware point cloud upsampling module (DAUP) to address the inhomogeneous problem. A learning-based neural shape function is developed to generate a local representation of the surface for point upsampling purposes. Utilizing the points generated by the neural shape function, we devise a density-aware resampling mechanism aimed at selecting a diverse number of points from varied regions to facilitate adaptive upsampling within regions of varying densities. DAUP can substantially reducing the misclassification rate for off-the-shelf anomaly detection pipelines. Extensive experiments confirm the effectiveness of our upsampling method on the benchmark dataset MVTec 3D-AD. Notably, our method surpasses previous state-of-the-art methods in terms of image-level AUROC based on the feature bank-based anomaly detection pipeline.

Generation of nonlinear gravity waves based on the harmonic polynomial cell method incorporated with a mass source

A nonlinear numerical wave tank is established using the Harmonic Polynomial Cell (HPC) method, which is incorporated with a mass source. The numerical wave tank consists of the mass source wave generation region and the working region, with sponge layers at both ends for wave absorption. In the mass source region, a generalized HPC method is applied to solve the inhomogeneous elliptic boundary value problems. The Poisson equation's special solution is represented by a bi-quadratic function. In the remaining domains, the HPC method is employed with harmonic polynomials to solve the problems governed by the Laplace equation in each grid cell. The free surface is tackled by the immersed boundary method (IB-HPC), and the kinematic and dynamic conditions of the free surface are described using a semi-Lagrangian approach. A variety of waves propagation are simulated, including Second-order Stokes wave, fifth-order Stokes wave, solitary wave, fifth-order Fenton stream function wave, random wave and the interaction with a straight vertical wall. The numerical solutions are compared with theoretical solutions. The numerical simulation results demonstrate that the present method can generate arbitrary two-dimensional wave fields by specifying an appropriate source function. Additionally, the reflected waves can propagate through the wave generation region, ensuring that the process of wave generation is not affected by the reflections.

A multiscale stochastic particle method based on the Fokker-Planck model for nonequilibrium gas flows

The Fokker-Planck (FP) model characterizes the intermolecular collisions as continuous stochastic processes in velocity space. Compared to the Direct Simulation Monte Carlo (DSMC) method, which is grounded in the Boltzmann equation, the stochastic particle method based on the FP model, referred to as the SP-FP method, demonstrates a relative improvement in computational efficiency. However, the traditional SP-FP method, utilizing operator splitting scheme, encounters analogous limitations to DSMC, thereby limiting the application of large temporal-spatial discretization for simulating near-continuum and continuum flows. In this study, two critical advancements are introduced to improve the traditional SP-FP method. First, the exact integration scheme of the ellipsoidal-statistical FP (ESFP) model is derived, enabling the replication of the theoretical solutions for homogeneous relaxation with any time step. Furthermore, leveraging the integration scheme of the ESFP model, we propose a Multiscale Stochastic Particle (MSP) method. The MSP method quantifies the numerical errors stemming from the operator splitting scheme and corrects them in the relaxation step. This capability allows the MSP method to be employed with larger temporal-spatial discretization for inhomogeneous problems, achieving second-order accuracy while retaining implementation simplicity. Application of the MSP method to various benchmark problems, including Couette flow, Fourier flow, Poiseuille flow, Sod tube flow, Lid-driven cavity flow, and flow through a slit, demonstrates its promise for simulating multiscale nonequilibrium gas flows with high computational efficiency and accuracy.

Улучшение работы алгоритма Алексеева при решении минимаксных неоднородных задач

Улучшение работы алгоритма Алексеева при решении минимаксных неоднородных задач

The effect of inhomogeneous terms for asymptotic profiles of solutions to second-order abstract evolution equations with time-dependent dissipation

The inhomogeneous problem of second-order evolution equations with a time-dependent dissipation term of the form u″+Au+b(t)u′=g(t) in a real Hilbert space H is considered, where A is a general nonnegative selfadjoint operator in H. The result is the explicit description of asymptotic profile of solutions of such a problem when the dissipation b provides a diffusive structure with a suitable restriction. The proof is a basic energy method with a scope of well-behaved quantity for the diffusive structure.

Numerical approximations of a lattice Boltzmann scheme with a family of partial differential equations

In this contribution, we address the numerical solutions of high-order asymptotic equivalent partial differential equations with the results of a lattice Boltzmann scheme for an inhomogeneous advection problem in one spatial dimension. We first derive a family of equivalent partial differential equations at various orders, and we compare the lattice Boltzmann experimental results with a spectral approximation of the differential equations. For an unsteady situation, we show that the initialization scheme at a sufficiently high order of the microscopic moments plays a crucial role to observe an asymptotic error consistent with the order of approximation. For a stationary long-time limit, we observe that the measured asymptotic error converges with a reduced order of precision compared to the one suggested by asymptotic analysis.

Adomian Modification Methods via Orthogonal Polynomials: A Comparative Study

The present manuscript proposes different modification procedures for the standard Adomian Decomposition Method (ADM). These procedures are based on the application of orthogonal polynomials that play vital parts in approximation theories. Moreover, the study also scrutinizes four test nonlinear inhomogeneous initial-value problems, and distinctively examines their respective absolute error differences. Remarkably, different computational benefits of the proposed modification are noted with regard high-level of accuracy and fewer computational steps.

Simulation of melting characteristics of thermal washing of the wax layer in the pipeline based on lattice Boltzmann method

ABSTRACT The escalation of transportation costs and the obstruction of pipelines due to wax deposition significantly impact the secure and efficient production of oil fields. The thermal washing method, characterized by its simplicity and cost-effectiveness, is widely employed for dewaxing crude oil gathering pipelines. The investigation of the melting mechanism of the wax layer during thermal washing is extremely important. In this paper, the two-zone composite model of multiphase flow-porous medium in the mushy zone is introduced into the solid-liquid phase change lattice Boltzmann model based on the enthalpy method, and the problem of circumferential inhomogeneity in the melting of the wax layer and the important influence of flow and heat transfer in the mushy zone on the thermal washing are systematically investigated. The demarcation point (γ tr) between the high and low liquid fraction zones is measured to be 0.81 by microscopic experiment. The impacts of varying γ tr and Ste numbers on wax melting at different circumferential locations are analyzed in detail. The results show that the melting efficiency of wax exhibits a nonlinear enhancement from − 90° to 90°. The melting efficiency of wax decreases with increasing γ tr for wax layers between 90° and − 45°. The melting efficiency of wax increases with higher Ste numbers at all angles. The Ste number increases from 0.42 to 5.0, and the melting efficiency of the wax layer at angles ranging from 90° to − 90° increases by factors of 2.16, 2.16, 2.20, 2.61, and 3.33 respectively. The growth of the Ste number improves the warming rate and shortens the generation time of convection at the center point of the wax layer. The research findings serve as a fundamental reference for formulating a rational and efficient thermal washing scheme.

Solvability analysis for the Boussinesq model of heat transfer under the nonlinear Robin boundary condition for the temperature.

We consider the new boundary value problem for the generalized Boussinesq model of heat transfer under the inhomogeneous Dirichlet boundary condition for the velocity and under mixed boundary conditions for the temperature. It is assumed that the viscosity, thermal conductivity and buoyancy force in the model equations, as well as the heat exchange boundary coefficient, depend on the temperature. The mathematical apparatus for studying the inhomogeneous boundary value problem under study based on the variational method is being developed. Using this apparatus, we prove the main theorem on the global existence of a weak solution of the mentioned boundary value problem and establish sufficient conditions for the problem data ensuring the local uniqueness of the weak solution that has the additional property of smoothness with respect to temperature. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Hypercomplex operator calculus for the fractional Helmholtz equation

In this paper, we develop a hypercomplex operator calculus to treat fully analytically boundary value problems for the homogeneous and inhomogeneous fractional Helmholtz equation where fractional derivatives in the sense of Caputo and Riemann–Liouville are applied. Our method extends the recently proposed fractional reduced differential transform method (FRDTM) by using fractional derivatives in all directions. For the special separable case in three dimensions, we obtain completely explicit representations for the fundamental solution. This allows us to interpret and to understand the appearance of spatial steady‐state solutions or spatial blow‐ups of the fractional Helmholtz equation in a better way. More precisely, we were able to present explicit conditions for the parameters in the representation formulas of the fundamental solutions under which we obtain bounded or spatial decreasing steady‐solutions and when spatial blow‐ups occur. We also illustrate this with some representative numerical examples. Furthermore, we show that it is possible to recover the recently studied cases as well as the classical cases as particular limit cases within our more general setting. Using the hypercomplex operator approach also allows us to factorize the fractional Helmholtz operator and obtain some interesting duality relations between left and right derivatives, Caputo and Riemann–Liouville derivatives, and eigensolutions of antipodal eigenvalues in terms of a generalized Borel–Pompeiu formula. This factorization, in turn, allows us to tackle inhomogeneous fractional Helmholtz problems.