Spatial Step Research Articles

Compact ADI Difference Scheme for the 2D Time Fractional Nonlinear Schrödinger Equation

In this paper, we will introduce a compact alternating direction implicit (ADI) difference scheme for solving the two-dimensional (2D) time fractional nonlinear Schrödinger equation. The difference scheme is constructed by using the L1−2−3 formula to approximate the Caputo fractional derivative in time and the fourth-order compact difference scheme is adopted in the space direction. The proposed difference scheme with a convergence accuracy of O(τ1+α+hx4+hy4)(α∈(0,1)) is obtained by adding a small term, where τ, hx, hy are the temporal and spatial step sizes, respectively. The convergence and unconditional stability of the difference scheme are obtained. Moreover, numerical experiments are given to verify the accuracy and efficiency of the difference scheme.

A 7T fMRI investigation of hand and tool areas in the lateral and ventral occipitotemporal cortex.

Previous studies demonstrated the existence of hand and tool areas in lateral and ventral occipitotemporal cortex (OTC), as well as an overlap between them. We reinvestigated this organization using 7T fMRI, benefiting from a higher signal-to-noise ratio than 3T. This enabled us to include a wider array of categories to achieve a more holistic perspective, and to omit certain spatial preprocessing steps. Despite these improvements, univariate analysis confirmed the existence of hand-tool overlap across OTC, which is striking given the omission of the spatial preprocessing steps that can influence overlap. There was significantly more overlap between hands and tools, compared to other overlap types in the left hemisphere of OTC. The overlap was also larger in the left lateral OTC as compared to the right lateral OTC. We found in all hand areas a differentiation between tools and other types of objects, although they still responded more to bodies than to tools. Regarding the tool areas, we observed a differentiation between hands and other categories such as faces and non-tool objects. Left hemisphere tool areas also differentiated between hands and bodies. When excluding the overlapping voxels from the hand and tool areas, they still showed a significant response to tools or hands (compared to objects or faces) respectively. Multi-voxel pattern analysis indicated that neural representations in the hand areas showed greater similarity between hands and tools than between hands and other objects. In the tool areas, the neural representations between tools and hands and between tools and other type of objects were all equally similar. To summarize, capitalizing on the benefits of 7T fMRI, we further substantiate the evidence in favor of hand-tool overlap in several regions of occipitotemporal cortex.

Temporally and spatially segregated discretization for a coupled electromechanical myocardium model

Abstract In this paper, we propose a novel temporally and spatially segregated numerical scheme to discretize the coupled electromechanical model of myocardium. We perform several numerical experiments with activation of a myocardial slab with structural inhomogeneity and evaluate the dependence of numerical errors on the size of spatial and temporal discretization steps. In our study, we show that the spatial step for the mechanical equations h m ⩽2.5 mm yields reasonable results with noticeable errors only in the region of myocardial inhomogeneity. We also show that time step τ m ⩽1 ms can be used for temporal discretization of mechanical equations, and the propagation velocity of the activation and contraction fronts differs from the reference one by no more than 1.3%for such time step. Finally, we show that the increase of time discretization steps of the mechanical equations τ m and the monodomain equation τ e leads to phase errors with opposite signs, and we can compensate these errors by tuning the relationship between the time steps.

Research on spatial correlation structure of war heritage based on field theory. A case study of Jinzhai County, China

Abstract The value of war heritage is usually reflected in the historical process constituted by the correlation among many heritages rather than a single heritage itself. Traditional heritage protection strategies often focus on ontological protection of heritage, while the inherent relevance between heritage due to their joint participation in historical events is often ignored. Therefore, it is extremely necessary to connect the scattered heritage sites spatially through certain logic, to show the historical process as a whole, and to maximize the integration and protection of the entire resources of the heritage protection mode. Taking the war heritage of Jinzhai as the object, this article analyzes the overall distribution characteristics of the war heritage and the formation process of the war heritage field on the basis of the field theory, and then puts forward the spatial sorting steps based on historical events. The research shows that it is feasible to use key historical events to organize the space of many war heritages to obtain a kind of integrated structure, which can be used as the basic framework for the future use of heritage protection, tourism development, heritage display, and utilization.

Collocation Finite Element Method for the Fractional Fokker–Planck Equation

ABSTRACTIn this study, the approximate results of the fractional Fokker–Planck equations have been investigated. First, finite element schemes have been obtained using collocation finite element method based on the trigonometric quintic B‐spline basis functions. Then, the present method is tested on two fundamental problems having appropriate initial conditions. The newly obtained numerical results contained the error norms and for various temporal and spatial steps are compared with the exact ones and other solutions. More accurate results have been obtained for large numbers of spatial and temporal elements.

Soliton solutions of derivative nonlinear Schrödinger equations: Conservative schemes and numerical simulation

In this paper, we numerically study soliton solutions of derivative nonlinear Schrödinger equations based on several conservative finite difference methods. All schemes own second-order accuracy with the convergence order O(τ2+h2) in the discrete L∞-norm, where h denotes the spatial step size and τ denotes the temporal step size. We show that difference schemes preserve some discrete counterparts of continuous conservation laws, and all these schemes are solvable. Extensive numerical examples with soliton solutions are carried out to verify the theoretical results. These results manifest that our schemes have potential application to soliton propagation in optical fibers.

High-order approximation of Caputo–Prabhakar derivative with applications to linear and nonlinear fractional diffusion models

Abstract In this study, we devise a high-order numerical scheme to approximate the Caputo–Prabhakar derivative of order α ∈ (0, 1), using an rth-order time stepping Lagrange interpolation polynomial, where 3 ≤ r ∈ N $3\le r\in \mathbb{N}$ . The devised scheme is a generalization of the existing schemes developed earlier. Further, we adopt the discussed scheme for solving a linear time fractional advection–diffusion equation and a nonlinear time fractional reaction–diffusion equation with Dirichlet type boundary conditions. We show that the discussed method is unconditionally stable, uniquely solvable and convergent with convergence order O(τ r+1−α , h 2), where τ and h are the temporal and spatial step sizes, respectively. Without loss of generality, applicability of the discussed method is established by illustrative examples for r = 4, 5.

Three-Dimensional Direct-Implicit Particle-in-Cell Model Using Trilinear Anisotropic Immersed-Finite-Element for Plasma Propulsion

Efficiently and accurately calculating the plasma transport process is one of the difficulties in aerospace plasma application simulation, especially in the magnetic sail spacecraft applications that have a huge size. This paper develops a three-dimensional trilinear anisotropic immersed-finite-element direct-implicit particle-in-cell (IFE-DIPIC) model to solve the problem of large-scale, long-term evolution plasma with complex interfaces. The model uses the DIPIC method to track the motion of particles in the plasma while simulating the anisotropic electric field containing an interface by using a modified trilinear anisotropic IFE method. Compared to the previous models, the developed model in this paper allows for the use of larger spatial and time steps in the Cartesian meshes without inducing numerical divergence. Using an interface-independent mesh avoids redundant interpolation in the PIC method, further improving efficiency. These advantages significantly improve the efficiency in solving actual complex three-dimensional plasma physics problems. The accuracy, efficiency, stability, and applicability of the proposed model are proved through numerical examples and the application in magnetic sail. The simulation results indicate that the developed model can efficiently simulate the actual working conditions of magnetic sails. The performance is significantly influenced by both the direction and magnitude of the magnetic moment.

Hierarchical stacked spatiotemporal self-attention network for sea surface temperature forecasting

Sea surface temperature (SST) is a highly complex spatiotemporal variable, which stems from its susceptibility to non-linear dynamical processes and substantial spatiotemporal variability. In particular, accurately forecasting small-scale SST is a formidable challenge due to the compounded effects of diverse physical processes spanning across various scales. In this study, we employ deep learning methods to mine the ocean’s evolutionary patterns, as the ocean’s dynamic mechanisms are inherently embedded in spatiotemporal data. We propose a hierarchical stacked spatiotemporal self-attention mechanism (HSSSA) network architecture. The hierarchical stacked encoder–decoder architecture provides the capability for feature fusion and extraction at different scales. The spatial self-attention and temporal self-attention modules simultaneously focus on information from different spatial locations and time steps, allowing the exploration of spatiotemporal patterns in the complex dynamics of the ocean. The experiments are conducted on a high-resolution East China Sea dataset (1/10°×1/10°) to demonstrate the forecast performance of the proposed model for refined ocean variables. The 15-day forecasts indicate that the HSSSA method outperforms the EOF-ARIMA and CNN-Transformer methods.

Stability and Convergence Analysis of the Discrete Dynamical System for Simulating a Moving Bed

The efficiency of controlling the simulated moving bed (SMB) has long been a critical issue in the chemical engineering industry. Most existing research relies on finite element methods, which often result in lower control efficiency and are unable to achieve online control. To enhance control over the SMB process, this paper employs the Crank–Nicolson method to develop a discrete dynamical model. This approach allows for the investigation of system stability and convergence, fundamentally addressing the sources of error. During the discretization of partial differential equations (PDEs), two main types of errors arise: intrinsic errors from the method itself and truncation errors due to derivative approximations and the discretization process. Research indicates that for the former, the iterative process remains convergent as long as the time and spatial steps are sufficiently small. Regarding truncation errors, studies have demonstrated that they exhibit second-order behavior relative to time and spatial steps. The theoretical validation shows that the iteration works effectively, and simulations confirm that the finite difference method is stable and performs well with varying SMB system parameters and controller processes. This provides a solid theoretical foundation for practical, real-time online control.

A spatial sixth-order numerical scheme for solving fractional partial differential equation

In this paper, a spatial sixth-order numerical scheme for solving the time-fractional diffusion equation (TFDE) is proposed. The convergence order of the constructed numerical scheme is O(τ2+h6), where τ and h are the temporal and spatial step sizes, respectively. The stability and error estimation of proposed scheme are given by using Fourier method. Some numerical examples are studied to demonstrate the correctness and effectiveness of the scheme and validate the theoretical analysis.

Structural Mapping of Protein Aggregates in Live Cells Modeling Huntington's Disease

AbstractWhile protein aggregation is a hallmark of many neurodegenerative diseases, acquiring structural information on protein aggregates inside live cells remains challenging. Traditional microscopy does not provide structural information on protein systems. Routinely used fluorescent protein tags, such as Green Fluorescent Protein (GFP), might perturb native structures. Here, we report a counter‐propagating mid‐infrared photothermal imaging approach enabling mapping of secondary structure of protein aggregates in live cells modeling Huntington's disease. By comparing mid‐infrared photothermal spectra of label‐free and GFP‐tagged huntingtin inclusions, we demonstrate that GFP fusions indeed perturb the secondary structure of aggregates. By implementing spectra with small spatial step for dissecting spectral features within sub‐micrometer distances, we reveal that huntingtin inclusions partition into a β‐sheet‐rich core and a ɑ‐helix‐rich shell. We further demonstrate that this structural partition exists only in cells with the [RNQ+] prion state, while [rnq−] cells only carry smaller β‐rich non‐toxic aggregates. Collectively, our methodology has the potential to unveil detailed structural information on protein assemblies in live cells, enabling high‐throughput structural screenings of macromolecular assemblies.

Structural Mapping of Protein Aggregates in Live Cells Modeling Huntington's Disease.

While protein aggregation is a hallmark of many neurodegenerative diseases, acquiring structural information on protein aggregates inside live cells remains challenging. Traditional microscopy does not provide structural information on protein systems. Routinely used fluorescent protein tags, such as Green Fluorescent Protein (GFP), might perturb native structures. Here, we report a counter-propagating mid-infrared photothermal imaging approach enabling mapping of secondary structure of protein aggregates in live cells modeling Huntington's disease. By comparing mid-infrared photothermal spectra of label-free and GFP-tagged huntingtin inclusions, we demonstrate that GFP fusions indeed perturb the secondary structure of aggregates. By implementing spectra with small spatial step for dissecting spectral features within sub-micrometer distances, we reveal that huntingtin inclusions partition into a β-sheet-rich core and a ɑ-helix-rich shell. We further demonstrate that this structural partition exists only in cells with the [RNQ+] prion state, while [rnq-] cells only carry smaller β-rich non-toxic aggregates. Collectively, our methodology has the potential to unveil detailed structural information on protein assemblies in live cells, enabling high-throughput structural screenings of macromolecular assemblies.

A three-step, "brute-force" approach toward optimized affine spatial normalization.

The first step in spatial normalization of magnetic resonance (MR) images commonly is an affine transformation, which may be vulnerable to image imperfections (such as inhomogeneities or "unusual" heads). Additionally, common software solutions use internal starting estimates to allow for a more efficient computation, which may pose a problem in datasets not conforming to these assumptions (such as those from children). In this technical note, three main questions were addressed: one, does the affine spatial normalization step implemented in SPM12 benefit from an initial inhomogeneity correction. Two, does using a complexity-reduced image version improve robustness when matching "unusual" images. And three, can a blind "brute-force" application of a wide range of parameter combinations improve the affine fit for unusual datasets in particular. A large database of 2081 image datasets was used, covering the full age range from birth to old age. All analyses were performed in Matlab. Results demonstrate that an initial removal of image inhomogeneities improved the affine fit particularly when more inhomogeneity was present. Further, using a complexity-reduced input image also improved the affine fit and was beneficial in younger children in particular. Finally, blindly exploring a very wide parameter space resulted in a better fit for the vast majority of subjects, but again particularly so in infants and young children. In summary, the suggested modifications were shown to improve the affine transformation in the large majority of datasets in general, and in children in particular. The changes can easily be implemented into SPM12.

Unconditional optimal [formula omitted]-norm error estimate and superconvergence analysis of a linearized nonconforming finite element variable-time-step BDF2 method for the nonlinear complex Ginzburg–Landau equation

In this paper, a linearized fully discrete scheme combining the variable-time-step two-step backward differentiation formula (VSBDF2) in time and the nonconforming finite element methods (FEMs) in space is constructed and analyzed for the nonlinear complex Ginzburg–Landau equation. A novel convergence analysis approach is proposed, which shows that the H1-norm error of this scheme can reach optimal order in space and sharp second-order convergence in time. The key ingredients of our analysis are temporal–spatial error splitting approach, a new positive definiteness property of discrete orthogonal convolution (DOC) kernels with respect to complex sequences, an energy projection operator and a discrete Laplace operator Δh related to the nonconforming element space, and a relationship between Δh and classical L2 projection operator in complex space. Furthermore, through a specific treatment of the error equation and the combination technique of interpolating operator and projection operator, we prove that the proposed scheme has the global superconvergence result of order O(τ2+h2) in H1-norm for the first time, which further improves the efficiency of numerical computation. Here τ and h are the maximum time and spatial step sizes, respectively. It should be pointed out that all these error results are rigorously established under mild assumptions of the time step, and get rid of the usual grid ratio constraints between temporal and spatial step sizes in linearized scheme. At last, numerical experiments are provided to verify the theoretical analysis.

Development of a sectionalizing method for simulation of large-scale complicated natural gas pipeline networks

By 2025, the total mileage of natural gas pipelines in China is expected to reach 163,000 km, placing urgent demands on the simulation technology for increasingly large-scale and more complicated pipeline networks. This paper describes the development of a new sectionalizing method, which is based on optimization strategy, for simulating large-scale and complicated natural gas pipeline networks. Based on analysis of the topological structures, the entire pipeline networks are decomposed into multiple sub-networks according to the connecting nodes between the adjoined pipelines. The sectionalizing method is then implemented by minimizing the residuals of conservation for continuity and momentum at the connecting nodes in isothermal mode. The hydraulic variables in each sub-network are obtained after the hydraulic variables at connecting nodes are updated. The calculation accuracy and efficiency of this sectionalizing method are demonstrated by comparison with a commercial software through three different test cases. It is shown that the maximum relative deviation of pressure of the method developed is within 2.5% of that calculated using the commercial software. In a 5 × 104 km test case with variable spatial step sizes, the computational efficiency is 2.1 to 3.2 times of that of the commercial software. The results of the three case studies suggest that this sectionalizing method is promising and suitable for application to large-scale complicate natural gas network simulation.

Nonstandard finite difference methods for a convective predator-prey pursuit and evasion model

One standard and two nonstandard finite difference methods are used to solve a convective predator-prey model consisting of cross-diffusion terms for which no exact solution is known. The model involves a system of strongly coupled nonlinear parabolic partial differential equations. Through some numerical experiments, we observe that reasonable numerical solutions are obtained when the temporal step size satisfies k ≤ 0.0005 with spatial step size h = 0.1 . We construct a nonstandard method using ideas from Anguelov et al. and the scheme is denoted by NSFD1. We obtain four conditions for NSFD1 to replicate the positivity property of the continuous model, but these conditions consist of many parameters. Therefore, an unconditionally positivity-preserving scheme is considered which is denoted as NSFD2. However, we found that NSFD2 is in general not consistent. We obtain conditions under which NSFD2 is both consistent and bounded and some numerical results are presented when these conditions are satisfied.

Flexible anisotropic magnetoresistive sensors for novel magnetic flux leakage testing capabilities

Rigid magnetic field sensors such as anisotropic magnetoresistive (AMR), giant magnetoresistive (GMR) and Hall sensors have been used for years and have become industry standard for electromagnetic non-destructive testing (NDT). Recent technological developments in the field of flexible electronics allow for the fabrication of reshapeable magnetic field sensors on flexible substrates via thin-film deposition or printing. The magnetic properties of these sensors have comparable characteristics to industry-standard rigid magnetic field sensors, with the added ability of adapting to the surface of complex components and scanning in contact with the sample surface. This improves defect detectability and magnetic signal strength by minimizing the scanning lift-off (LO) distance. In this article flexible AMR sensors mounted on a rotative mechanical holder were used to scan a semi-circular ferromagnetic sample with 3 reference defects via magnetic flux leakage (MFL) testing, thus demonstrating the applicability of this type of sensors for the scanning of curved samples. In order to benchmark the performance of these sensors in comparison to industry standard rigid magnetic field sensors, a ferromagnetic sample with 10 reference defects of different depths was scanned employing flexible AMR and rigid GMR sensors. Defects with depths ranging from 110μm up to 2240μm were detected with an signal-to-noise ratio (SNR) of 2.7 up to 27.9 (for flexible AMR sensors) and 6.2 up to 72.3 (for rigid GMR sensors), respectively. A 2D magnetometer mapping of the sample with a spatial scanning step of 10×50μm2 (flexible AMR) and 16×100μm2 (rigid GMR) was obtained. The results show that this type of sensor can be used for high-resolution and high-detail mapping of defects on the surface of planar and non-planar ferromagnetic samples since the scanning lift-off distance is equal to the substrate thickness of 20μm for in-contact scanning. The SNR comparison between flexible and rigid sensors shows that the performance of the flexible AMR sensors employed is not very far behind the performance of the rigid GMR sensors used.

Time two-grid fitted scheme for the nonlinear time fractional Schrödinger equation with nonsmooth solutions

In this paper, we delve into the regularity properties and the development of an efficient numerical strategy for addressing the nonlinear time-fractional Schrödinger equation. Initially, we embark on an examination of the solution’s regularity, followed by an investigation into regularity enhancement through the application of the Laplace and finite Fourier sine transforms. Subsequently, after the decomposition of u-u0, we introduce an innovative time two-grid fitted scheme designed for the equation’s resolution, which demonstrates unconditional stability and convergence in the L2 norm, achieving the global convergence order of O(τF2+τC4+h2). Here, τF and τC denote the temporal step sizes on the fine and coarse grids, respectively, while h represents the spatial step size. Notably, this is the inaugural application of such an approach for solving the nonlinear time fractional Schrödinger equation. Ultimately, our numerical experiments validate that this method not only retains computational efficiency but also significantly enhances convergence accuracy.

Workflow of a Preclinical Robotic Magnetic Resonance Imaging-guided Focused Ultrasound Body System

Abstract Background: Establishing an efficient workflow is crucial for the success of magnetic resonance-guided focused ultrasound (MRgFUS) procedures. The current study provides a comprehensive description of the workflow of a customized MRgFUS robotic body device for preclinical use and accompanied software through experiments in excised porcine tissue. Methods: The employed system comprises a single-element spherically focused transducer of 2.6 MHz that can be moved along four PC-controlled axes. A detailed description of essential software functionalities and its integration with a 3T Siemens magnetic resonance imaging (MRI) scanner through Access-I for interactive remote control of the scanner and real-time access to imaging data is provided. Following treatment planning on preoperative MR images, porcine tissue samples were sonicated in rectangular and irregular grid patterns with varying ultrasonic parameters and spatial step under software-based monitoring. Results: MRgFUS ablations of ex vivo porcine tissue were successfully performed utilizing a multimodal monitoring approach combining MRI-based temperature, thermal dose, and necrotic area mapping, thus demonstrating an efficient procedural workflow. The simulated necrotic regions were in excellent agreement with the actual lesions revealed upon tissue dissection and highly consistent with the planned sonication patterns. The software’s ability to accurately identify regions where necrosis did not occur and indicate to the user the specific points to be re-sonicated was demonstrated. Conclusion: Overall, the study highlights critical aspects in accurately planning and executing preclinical MRgFUS protocols within an efficient workflow. The provided data could serve as the basis for other researchers in the field.