High-Accuracy Positivity-Preserving Finite Difference Approximations of the Chemotaxis Model for Tumor Invasion.

Numerical simulation of the complex evolution process for tumor invasion plays an extremely important role in-depth exploring the bio-taxis phenomena of tumor growth and metastasis. In view of the fact that low-accuracy numerical methods often have large errors and low resolution, very refined grids have to be used if we want to get high-resolution simulating results, which leads to a great deal of computational cost. In this paper, we are committed to developing a class of high-accuracy positivity-preserving finite difference methods to solve the chemotaxis model for tumor invasion. First, two unconditionally stable implicit compact difference schemes for solving the model are proposed; second, the local truncation errors of the new schemes are analyzed, which show that they have second-order accuracy in time and fourth-order accuracy in space; third, based on the proposed schemes, the high-accuracy numerical integration idea of binary functions is employed to structure a linear compact weighting formula that guarantees fourth-order accuracy and nonnegative, and then a positivity-preserving and time-marching algorithm is established; and finally, the accuracy, stability, and positivity-preserving of the proposed methods are verified by several numerical experiments, and the evolution phenomena of tumor invasion over time are numerically simulated and analyzed.

Finite Difference Modeling and Experimental Investigation of Cyclic Thermal Heating in the Fused Filament Fabrication Process.

Fused filament fabrication (FFF) is one of the most popular additive manufacturing (AM) processes due to its simplicity and low initial and maintenance costs. However, good printing results such as high dimensionality, avoidance of cooling cracks, and warping are directly related to heat control in the process and require precise settings of printing parameters. Therefore, accurate prediction and understanding of temperature peaks and cooling behavior in a local area and in a larger part are important in FFF, as in other AM processes. To analyze the temperature peaks and cooling behavior, we simulated the heat distribution, including convective heat transfer, in a cuboid sample. The model uses the finite difference method (FDM), which is advantageous for parallel computing on graphics processing units and makes temperature simulations also of larger parts feasible. After the verification process, we validate the simulation with an in situ measurement during FFF printing. We conclude the process simulation with a parameter study in which we vary the function of the heat transfer coefficient and part size. For smaller parts, we found that the print bed temperature is crucial for the temperature gradient. The approximations of the heat transfer process play only a secondary role. For larger components, the opposite effect can be observed. The description of heat transfer plays a decisive role for the heat distribution in the component, whereas the bed temperature determines the temperature distribution only in the immediate vicinity of the bed. The developed FFF process model thus provides a good basis for further investigations and can be easily extended by additional effects or transferred to other AM processes.

Numerical simulation on transient electromagnetic response of separation layer water in coal seam roof.

Mining stress induces deformation and fracture of the overlaying rock, which will result in water filling the separation layer if the aquifer finds access to abscission space along the fracture channels. Accurate detection is crucial to prevent water hazards induced by water-bearing fractures. The 3-D time-domain finite-difference method with Yee's grid was adopted to calculate full-space transient electromagnetic response; meanwhile, a typical geologic and geophysical model with a water-bearing block in an separation layer was built according to regional tectonics and stratigraphic developments. By using numerical simulation, the induced voltage and apparent resistivity for both vertical and horizontal components were acquired, and then an approximate inversion was carried out based on the "smoke ring" theory. The results indicate that the diffusion velocity of induced voltage is significantly affected by the water-bearing body in the fracture, and the horizontal velocity of induced voltage is lower than the vertical one. The induced voltage curves indicate that the horizontal response to an anomaly body is stronger than the vertical one, leading to a high apparent resistivity resolution of conductivity contrast and separation layer boundary in the horizontal direction. The results of 3-D simulation making use of a measured data model also demonstrate that the horizontal component of apparent resistivity can reflect the electrical structure in a better way; however, its ability to recognize the concealed and fine conductor is rather weak. Accordingly, the observation method or numerical interpolation method needs to be further improved for data processing and interpretation.

Probing the cw-Laser-Induced Fluorescence Enhancement in CsPbBr3 Nanocrystal Thin Films: An Interplay between Photo and Thermal Activation.

Perovskite nanocrystals hold significant promise for a wide range of applications, including solar cells, LEDs, photocatalysts, humidity and temperature sensors, memory devices, and low-cost photodetectors. Such technological potential stems from their exceptional quantum efficiency and charge carrier conduction capability. Nevertheless, the underlying mechanisms of photoexcitation, such as phase segregation, annealing, and ionic diffusion, remain insufficiently understood. In this context, we harnessed hyperspectral fluorescence microspectroscopy to advance our comprehension of fluorescence enhancement triggered by UV continuous-wave (cw) laser irradiation of CsPbBr3 colloidal nanocrystal thin films. Initially, we explored the kinetics of fluorescence enhancement and observed that its efficiency (φph) correlates with the laser power (P), following the relationship φph = 7.7⟨P⟩0.47±0.02. Subsequently, we estimated the local temperature induced by the laser, utilizing the finite-difference method framework, and calculated the activation energy (Ea) required for fluorescence enhancement to occur. Our findings revealed a very low activation energy, Ea ∼ 9 kJ/mol. Moreover, we mapped the fluorescence photoenhancement by spatial scanning and real-time static mode to determine its microscale length. Below a laser power of 60 µW, the photothermal diffusion length exhibited nearly constant values of approximately (22 ± 5) µm, while a significant increase was observed at higher laser power levels. These results were ascribed to the formation of nanocrystal superclusters within the film, which involves the interparticle spacing reduction, creating the so-called quantum dot solid configuration along with laser-induced annealing for higher laser powers.

Numerical and Experimental Analysis of Buckling and Post-Buckling Behaviour of TWCFS Lipped Channel Section Members Subjected to Eccentric Compression.

The paper presents a static analysis of the buckling and post-buckling state of thin-walled cold-formed steel (TWCFS) lipped channel section beam-columns subjected to eccentric compression. Eccentricity is taken into consideration relative to both major and minor principal axes. An analytical-numerical solution to the buckling and post-buckling problems is described. The solution is based on the theory of thin plates. Equations of equilibrium of section walls are derived from the principle of stationary energy. Then, to solve the problem, the finite difference (FDM) and Newton-Raphson methods are applied. Linear (buckling) and nonlinear (post-buckling) analyses are performed. As a result, pre- and post-buckling equilibrium paths are determined. Comparisons of the obtained numerical results, FE simulation results, and experimental test results are carried out and presented in comparative load-shortening diagrams. Additionally, a comparison of the buckling forces and buckling modes obtained from theoretical analysis and experiments is presented.

An efficient numerical approach for singularly perturbed time delayed parabolic problems with two-parameters.

OBJECTIVES: The main objective of this work is to design an efficient numerical scheme is proposed for solving singularly perturbed time delayed parabolic problems with two parameters. RESULTS: The scheme is constructed via the implicit Euler and non-standard finite difference method to approximate the time and space derivatives, respectively. Besides, to enhance the accuracy and order of convergence of the method Richardson extrapolation technique is employed. Intensive numerical experimentation has been done on some model examples. Further, the layer behavior of the solutions is presented using graphs and observed to agree with the existing theories. Finally, error analysis of the scheme is done and observed that the proposed method is parameter uniform convergent with the order of convergence ( Δ t ) 2 + h 2 .

Flow and Heat Transfer of CoFe2O4-Blood Due to a Rotating Stretchable Cylinder under the Influence of a Magnetic Field.

The flow and heat transfer of a steady, viscous biomagnetic fluid containing magnetic particles caused by the swirling and stretching motion of a three-dimensional cylinder has been investigated numerically in this study. Because fluid and particle rotation are different, a magnetic field is applied in both radial and tangential directions to counteract the effects of rotational viscosity in the flow domain. Partial differential equations are used to represent the governing three-dimensional modeled equations. With the aid of customary similarity transformations, this system of partial differential equations is transformed into a set of ordinary differential equations. They are then numerically resolved utilizing a common finite differences technique that includes iterative processing and the manipulation of tridiagonal matrices. Graphs are used to depict the physical effects of imperative parameters on the swirling velocity, temperature distributions, skin friction coefficient, and the rate of heat transfer. For higher values of the ferromagnetic interaction parameter, it is discovered that the axial velocity increases, whereas temperature and tangential velocity drop. With rising levels of the ferromagnetic interaction parameter, the size of the axial skin friction coefficient and the rate of heat transfer are both accelerated. In some limited circumstances, a comparison with previously published work is also handled and found to be acceptably accurate.

Sensitivity of Electrocardiogram on Electrode-Pair Locations for Wearable Devices: Computational Analysis of Amplitude and Waveform Distortion.

An electrocardiogram (ECG) is used to observe the electrical activity of the heart via electrodes on the body surface. Recently, an ECG with fewer electrodes, such as a bipolar ECG in which two electrodes are attached to the chest, has been employed as wearable devices. However, the effect of different geometrical factors and electrode-pair locations on the amplitude and waveform of ECG signals remains unclear. In this study, we computationally evaluated the effects of body morphology, heart size and orientation, and electrode misalignment on ECG signals for 48 scenarios using 35 bipolar electrode pairs (1680 waveforms) with a dynamic time warping (DTW) algorithm. It was observed that the physique of the human body model predominantly affected the amplitude and waveform of the ECG signals. A multivariate analysis indicated that the heart-electrode distance and the solid angle of the heart from the electrode characterized the amplitude and waveform of the ECG signals, respectively. Furthermore, the electrode locations for less individual variability and less waveform distortion were close to the location of electrodes V2 and V3 in the standard 12-lead. These findings will facilitate the placement of ECG electrodes and interpretation of the measured ECG signals for wearable devices.

Enhancement of skin tumor laser hyperthermia with Ytterbium nanoparticles: numerical simulation.

Laser hyperthermia therapy (HT) has emerged as a well-established method for treating cancer, yet it poses unique challenges in comprehending heat transfer dynamics within both healthy and cancerous tissues due to their intricate nature. This study investigates laser HT therapy as a promising avenue for addressing skin cancer. Employing two distinct near-infrared (NIR) laser beams at 980 nm, we analyze temperature variations within tumors, employing Pennes' bioheat transfer equation as our fundamental investigative framework. Furthermore, our study delves into the influence of Ytterbium nanoparticles (YbNPs) on predicting temperature distributions in healthy and cancerous skin tissues. Our findings reveal that the application of YbNPs using a Gaussian beam shape results in a notable maximum temperature increase of 5 °C within the tumor compared to nanoparticle-free heating. Similarly, utilizing a flat top beam alongside YbNPs induces a temperature rise of 3 °C. While this research provides valuable insights into utilizing YbNPs with a Gaussian laser beam configuration for skin cancer treatment, a more thorough understanding could be attained through additional details on experimental parameters such as setup, exposure duration, and specific implications for skin cancer therapy.

Analysis of thermomechanical stresses in dual compound thick cylinders under asymmetric loads: An analytical and numerical method.

This study presents a 2D comprehensive analytical and numerical analysis of the thermomechanical stresses in an unsymmetric dual compound thick cylinder under steady-state conditions. By employing mathematical analysis, this research aims to investigate the effectiveness of a 2D compound cylinder in reducing elastic and thermoelastic stresses. The temperature and displacement fields are thought to be dependent on the radial and circumferential directions, subject to asymmetric thermal and mechanical boundary conditions on the inner and outer surfaces. In this scenario, the Poisson ratio is considered to be a constant. The techniques of variable separation and complex Fourier series are employed analytically in the solution of heat conduction and Navier equations. The results obtained from the developed analytical method are compared and validated against those obtained from a finite difference method (FDM). The findings of this study suggest that the clamping of the outer layer has a significant influence on stress distribution in the structure, and the impact of tangential stress on the behavior of a compound cylinder is highly dominant. Furthermore, changes in temperature significantly influence hoop stress compared to variations in internal pressure levels. Moreover, the influence of internal pressure is relatively attenuated when a pressure vessel is fabricated utilizing different metals. In addition, the findings indicated that the configuration of layers and the location of the highest temperature had a significant impact on the performance of the vessel. Nevertheless, the technology provided has sufficient robustness to effectively address the complexities associated with the design of multilayered graded materials (GM) in additive manufacturing applications.

Re-constructing the river bed using the streamline-generation method.

River bed reconstruction plays an essential part in supporting the hydrodynamic simulation and understanding the morphological processes of a river. The streamlines can be solved using Laplace equations. The equation is first numerically solved in a computational environment and then adapted to the whole considered physical field to solve the resulting streamlines in a physical domain. One of the goals of this research is to determine the bottom line of the riverbed, and doing this through a field survey is very expensive, the methods presented in this research help a lot in reducing costs. By determining the concave line, the path of the flood in the river bed is determined, so one of the practical achievements and efficiencies of this research is flood trending at a low cost. In the present study, the bottom line is determined for the meandering Qinhe River, a distributary of the Yellow River, China, and Gaz River, located in Khuzestan Province, Iran. The method is based on the following steps:•Reconstruction of 2D river based on the Laplace equation.•Use the Finite Element Method to solve streamlines in the physical domain.•Use the Finite Difference Method to solve streamlines in the physical domain.

The dropper stress characteristics under different moving load speeds for a high-speed railway.

Dropper failure seriously threatens the operation safety of a high-speed railway. In this work, for a simple chain suspension catenary, one span with five droppers is performed to establish a model and thus the effects of the moving load speed on dropper stress are investigated. First, the partial differential vibration equation of dropper is obtained through the mechanical analysis and converted into the finite difference equation. Then, we consider contact line as a beam element to obtain its motion equation. Furthermore, the boundary and initial conditions of five droppers are determined. Finally, the stresses of five droppers are numerically calculated and the effects of the moving load speed on dropper stress are investigated by writing a MATLAB code. The results suggest that the dropper location significantly affects its stress. Compared with other droppers, droppers II and IV have much more severe vibration amplitudes. Different moving load speeds could cause different stress change of each dropper. With the increasing speed, dropper experiences longer bending compression stage and the bending amplitude increases. The impact of the moving load speed on dropper stress is significant.

Mathematical analysis and numerical simulation for fractal-fractional cancer model.

The mathematical oncology has received a lot of interest in recent years since it helps illuminate pathways and provides valuable quantitative predictions, which will shape more effective and focused future therapies. We discuss a new fractal-fractional-order model of the interaction among tumor cells, healthy host cells and immune cells. The subject of this work appears to show the relevance and ramifications of the fractal-fractional order cancer mathematical model. We use fractal-fractional derivatives in the Caputo senses to increase the accuracy of the cancer and give a mathematical analysis of the proposed model. First, we obtain a general requirement for the existence and uniqueness of exact solutions via Perov's fixed point theorem. The numerical approaches used in this paper are based on the Grünwald-Letnikov nonstandard finite difference method due to its usefulness to discretize the derivative of the fractal-fractional order. Then, two types of stabilities, Lyapunov's and Ulam-Hyers' stabilities, are established for the Incommensurate fractional-order and the Incommensurate fractal-fractional, respectively. The numerical results of this study are compatible with the theoretical analysis. Our approaches generalize some published ones because we employ the fractal-fractional derivative in the Caputo sense, which is more suitable for considering biological phenomena due to the significant memory impact of these processes. Aside from that, our findings are new in that we use Perov's fixed point result to demonstrate the existence and uniqueness of the solutions. The way of expressing the Ulam-Hyers' stabilities by utilizing the matrices that converge to zero is also novel in this area.

Modeling a Fluid-Coupled Single Piezoelectric Micromachined Ultrasonic Transducer Using the Finite Difference Method.

A complete model was developed to simulate the behavior of a circular clamped axisymmetric fluid-coupled Piezoelectric Micromachined Ultrasonic Transducer (PMUT). Combining Finite Difference and Boundary Element Matrix (FD-BEM), this model is based on the discretization of the partial differential equation used to translate the mechanical behavior of a PMUT. In the model, both the axial and the transverse displacements are preserved in the equation of motion and used to properly define the neutral line position. To introduce fluid coupling, a Green's function dedicated to axisymmetric circular radiating sources is employed. The resolution of the behavioral equations is used to establish the equivalent electroacoustic circuit of a PMUT that preserves the average particular velocity, the mechanical power, and the acoustic power. Particular consideration is given to verifying the validity of certain assumptions that are usually made across various steps of previously reported analytical models. In this framework, the advantages of the membrane discretization performed in the FD-BEM model are highlighted through accurate simulations of the first vibration mode and especially the cutoff frequency that many other models do not predict. This high cutoff frequency corresponds to cases where the spatial average velocity of the plate is null and is of great importance for PMUT design because it defines the upper limit above which the device is considered to be mechanically blocked. These modeling results are compared with electrical and dynamic membrane displacement measurements of AlN-based (500 nm thick) PMUTs in air and fluid. The first resonance frequency confrontation showed a maximum relative error of 1.13% between the FD model and Finite Element Method (FEM). Moreover, the model perfectly predicts displacement amplitudes when PMUT vibrates in a fluid, with less than 5% relative error. Displacement amplitudes of 16 nm and 20 nm were measured for PMUT with 340 µm and 275 µm diameters, respectively. This complete PMUT model using the FD-BEM approach is shown to be very efficient in terms of computation time and accuracy.

Dynamics of chemically reactive Carreau nanomaterial flow along a stretching Riga plate with active bio-mixers and Arrhenius catalysts.

Nanomaterial flow has fascinated the concern of scientists across the globe due to its innovative applications in various manufacturing, industrial, and engineering domains. Bearing aforementioned uses in mind, the focal point of this study is to examine the Carreau nanofluid flow configured by the Riga surface with Arrhenius catalysts. Microorganisms are also suspended in nanofluid to strengthen the density of the regular fluid. Time-dependent coupled partial differential equations that represent the flow dynamics are modified into dimensionless patterns via appropriate non-dimensional variables, and handled through an explicit finite difference approach with stability appraisal. The performances of multiple flow variables are examined graphically and numerically. Representation of 3D surface and contour plots for heat transportation and entropy generation are also epitomized. The findings express that the modified Hartmann number strengthens the motion of nanomaterial. Reverse outcomes for heat transport rate and entropy are seen for the radiation variable. Concentration diminishes for chemical reaction variable. Activation energy enhances the concentration of nanomaterial, whereas reduction happens in the movement of microbes for bio-Lewis number. Greater Brinkman variable heightens the entropy.

Exploring of Diverse Plant Communities and Adaptation to Drought Conditions Based on Advanced Logistics Model with Variable Growth.

Bio reciprocal symbiosis is very common in nature, such as soybeans providing food for rhizobia, which uses atmospheric nitrogen to synthesize nitrogen to provide nutrients to soybeans. This paper proposes an advanced Logistic model that adjusts to changes in precipitation and an environmental capacity parameter that varies with the level of symbiosis. The aim is to precisely depict the symbiotic relationship between plants and the interplay among symbiosis, competition, and independent growth of each population in the plant community, as precipitation changes by adapting finite difference method and tertiary Hermit interpolation. The model in this paper offers a comprehensive understanding of how plant populations interact with one another, providing valuable insights into the dynamics of plant growth and development. This paper finally finds that a combination of woody and herbaceous plants had the highest growth rate and total biomass, while herbaceous-only plants required 7 times longer to reach environmental capacity. This paper also reveals that irregular weather patterns, and different levels of species biomass can have different impacts on the recovery time of plant communities after drought or damage, and different types of pollution can have various effects on the community's regeneration, while the effect of overgrazing is the smallest.

A fourth-order arithmetic average compact finite-difference method for nonlinear singular elliptic PDEs on a 3D smooth quasi-variable grid network.

The analysis of nonlinear elliptic PDEs representing stationary convection-dominated diffusion equation, Sine-Gordon equation, Helmholtz equation, and heat exchange diffusion model in a battery often lacks in closed-form solutions. For the long-term behaviour and to assess the quantitative behaviour of the model, numerical treatment is necessary. A novel numerical approach based on arithmetic average compact discretization employing a quasi-variable grid network is proposed for a wide class of nonlinear three-dimensional elliptic PDEs. The method's key benefit is that it applies to singular models and only needs nineteen-point grids with seven functional approximations. Additionally, the suggested method disseminates the truncation error across the domain, which is unrealistic for finite-difference discretization with a fixed step length of grid points. Often, small diffusion anticipates strong oscillation, and tuning the grid stretching parameter helps error dispersion over the domain. The scheme is examined for maximal error bounds and convergence property with the help of a monotone matrix and its irreducible character. The metrics of solution accuracies, mainly root-mean-squared and absolute errors alongside numerical convergence rate, are inspected by different types of variable coefficients, singular and non-singular 3D elliptic PDEs appearing in a convection-diffusion phenomenon. The performance of the numerical solution corroborates the fourth-order convergence on a quasi-variable grid network.

Bearing Capacity and Deformation of the Tandem Compound Piles Improved Foundation: A Parametric Study.

The tandem compound piles are a combination of a granular column in the deep section and a concrete pile in the shallow section. This method effectively utilizes the consolidation and densification effects of the granular column, as well as the cementation strength of the concrete material. The granular column acts as a consolidation path, aiding in the densification of the surrounding soil. On the other hand, the concrete pile prevents the bulging deformation that commonly happens in granular columns during field construction. To study the bearing capacity and deformation of the improved foundation with tandem compound piles, a coupled continuum-discrete numerical model was developed in this study. The accuracy of the model was confirmed by comparing its results with experimental measurements. Additionally, a parametric study was conducted, considering three influential factors: (1) cushion thickness and modulus, (2) length, modulus, diameter, and spacing of the tandem compound pile, and (3) soil modulus. The results indicated that reducing the cushion thickness and increasing the cushion modulus allowed the pile to bear more loads. Moreover, increasing the length and modulus of the deep section of the pile reduced deformation and improved the bearing capacity. The pile modulus, however, had a limited effect on enhancing the bearing capacity. It is important to maintain a critical pile spacing of at least twice the pile diameter. Finally, a high modulus of the underlying stratum led to higher vertical and radial stresses in the pile.

A multisource mass transfer model for simulating VOC emissions from paints.

Indoor decoration generates a large number of volatile organic compounds (VOCs), which are simultaneously released from different paints. Nevertheless, the interaction mechanism of pollutant diffusion from multisource building materials (such as primer and finish) needs to be examined. In this paper, a multisource mass transfer model for VOC emissions from different combinations of paints is established, and the analytical solution is derived. The finite difference method is used to simulate the experimental results of VOC release in the environmental chambers, and its convergence and stability are verified. Using the optimization parameters of the single-source model and the law of conservation of mass, the key parameters of the multisource mass transfer model are obtained. The results show that the established model is in excellent agreement with both experimental data and literature data. In addition, the Little number Lt is used to analyse the change trend from the initial released concentration in the single-source and multisource models.

Efficient Numerical Simulation of Biochemotaxis Phenomena in Fluid Environments.

A novel dimension splitting method is proposed for the efficient numerical simulation of a biochemotaxis model, which is a coupled system of chemotaxis-fluid equations and incompressible Navier-Stokes equations. A second-order pressure correction method is employed to decouple the velocity and pressure for the Navier-Stokes equations. Then, the alternating direction implicit scheme is used to solve the velocity equation, and the operator with dimension splitting effect is used instead of the traditional elliptic operator to solve the pressure equation. For the chemotactic equation, the operator splitting method and extrapolation technique are used to solve oxygen and cell density to achieve second-order time accuracy. The proposed dimension splitting method splits the two-dimensional problem into a one-dimensional problem by splitting the spatial derivative, which reduces the computation and storage costs. Finally, through interesting experiments, we show the evolution of the cell plume shape during the descent process. The effect of changing specific parameters on the velocity and plume shape during the descent process is also studied.