Non-uniform Model Research Articles

PSNet: A non-uniform illumination correction method for underwater images based pseudo-siamese network

Autonomous underwater vehicles (AUVs) based on visual perception play an important role in maritime operations. However, underwater environment often suffers from poor lighting conditions, making the utilization of artificial light sources necessary. This reliance on artificial lighting frequently results in non-uniform illumination. Furthermore, the absorption and scattering effects of water cause further degradation, such as color distortion and blurring of details. To address these challenges, we propose a pseudo-siamese network, named PSNet, designed for underwater optical image enhancement. PSNet separates the non-uniformly illuminated layer from the optimally uniformly illuminated image and utilizes a cascading iteration strategy to enhance the image details. To achieve a better balance prediction quality, we introduce structure loss and residual reconstruction loss as additional guides for model learning. Additionally, we incorporate a color consistency loss to mitigate color distortion. To address the lack of training data, we develop a non-uniform illumination model and generate a dataset that includes both non-uniformly illuminated layers and uniformly illuminated images. Through comprehensive experimental evaluations, PSNet significantly enhances the visual quality of underwater optical images and consistently outperforms state-of-the-art approaches in multiple performance metrics.

Energy efficient strategy for heterogeneous truck platooning based on a non-uniform platooning model

To investigate the energy-saving advantages of truck platoons, this study focused on two, three, and six-axle trucks as research subjects. A non-uniform platoon model of trucks was constructed based on braking performance differences. The aerodynamic characteristics of the non-uniform platoon fleet were analyzed in depth using numerical simulation methods, and the fuel consumption rate of the fleet was calculated. In a heterogeneous three-vehicle platoon, the 3-2-6 platooning approach exhibited the most effective reduction in fuel consumption, whereas the 3-6-2 platooning approach was the least effective. For a four-vehicle heterogeneous platoon comprising a three-axle truck, a six-axle truck, and two two-axle trucks, the 3-2-2-6 platooning approach yielded the best fuel efficiency. In a four-vehicle heterogeneous platoon with a two-axle truck, a six-axle truck, and two three-axle trucks, the 3-2-3-6 platooning approach proved to be the most effective in reducing fuel consumption. In a four-vehicle heterogeneous fleet composed of two-axle trucks, three-axle trucks, and two six-axle trucks, the 3-2-6-6 platooning approach achieved the best fuel efficiency. The conclusions of this study offer a fundamental scheme for truck formation, laying a crucial theoretical foundation for energy conservation and emissions reduction in transport systems.

Photoelectric sensor array-based measurement method for the key motion parameters of the small flying object considering non-uniform curve trajectory

Purpose The paper aims to accurately measure the key motion parameters, such as velocity, azimuth and pitch angle, of the small flying object with a non-uniform curve trajectory. It proposes a measurement method and its calculation model of non-uniform curve trajectory using a photoelectric sensor array. Design/methodology/approach First, the basic composition of the measurement system and mechanism of photoelectric sensor array are described, respectively. Second, a non-uniform curve mathematical measurement model is constructed differently from the traditional linear trajectory, taking into account the influence of gravity and air resistance. Third, the measurement error of the system is analyzed through numerical simulation. Finally, the accuracy and feasibility of the approach are verified by live-ammunition experiments. Findings The results show that the systematic error of the hitting point coordinates can be reduced by 9% compared to the traditional linear measurement model. Consequently, this method can meet the higher measurement requirement for the key motion parameters of the small flying object under the non-uniform curve trajectory. Research limitations/implications (if applicable)- although the approach itself is generalizable, the method is unable to detect the motion parameters of multiple small flying objects. Research limitations/implications Although the approach itself is generalizable, the method is unable to detect the motion parameters of the multiple small flying objects. Practical implications It is evident that the proposed non-uniform curve measurement model is more precise in quantifying the essential characteristics of the small flying object, particularly in consideration of the environmental conditions. Social implications The precise measurement of the key motion parameters of the small flying object can facilitate the enhancement of the protective performance of protective materials. Originality/value A novel approach to measurement is proposed, which differs from the conventional uniform trajectory model. To this end, the space construction of the photoelectric sensor array is optimized. The number of the sensors is revised.

Numerical Modeling, Trim, and Linearization of a Side-by-Side Helicopter in Hovering Conditions

In the present paper, a flight dynamics model is adopted to represent the trim and stability characteristics of a side-by-side helicopter in hovering conditions. This paper develops a numerical representation of the rotorcraft behavior and proposes a set of guidelines for trimming and linearizing the highly coupled rotor dynamics derived by the modeling approach. The trim algorithm presents two nested loops to compute a solution of the steady-state conditions averaged around one blade’s revolution. On the other hand, a 38-state-space linear representation of the helicopter and rotor dynamics is obtained to study the effects of flap, lead–lag, and inflow on the overall stability. The results are compared with an analytical framework developed to validate the rotorcraft stability and compare different modeling approaches. The analysis showed that non-uniform inflow modeling led to a coupled longitudinal inflow–phugoid mode which made the vehicle prone to dangerous instabilities. The flap and lead–lag dynamics introduced damping in the system and can be considered beneficial for rotor dynamics.

Bifurcation control and analysis of traffic flow model based on driver prediction effect.

Based on the measured data, in this paper, we find that there are significant differences in the ideal speed of drivers with different attributes during their driving process. Through wavelet analysis, it is proved that the regularity of the car-following behavior of the driver is poor in a short time, and it cannot be predicted and analyzed at a small time scale. As one of the important participants in the traffic system, it is necessary to consider the driving behavior factors in the traffic flow model. Therefore, in this paper, we propose a nonuniform continuous traffic flow model. Based on the difference in the expected headway of the driver, the model considers the self-stabilization effect of the prediction time of the driver on the optimal speed difference and considers control theory to further improve the model. Through this model, the bifurcation theory can be applied to the stability analysis of the traffic system to study the stability mutation behavior of the traffic system at the bifurcation point. Through linear and nonlinear analysis methods, the stability conditions of the model can be derived, and the type of equilibrium point of the model can be judged. In addition, the random function is applied to the traffic model, and the bifurcation control is carried out for the bifurcation behavior in the traffic; that is, the bifurcation characteristics of the traffic system are changed by designing linear and nonlinear random feedback controllers. In this paper, we theoretically prove the existence conditions and types of Hopf bifurcation at the equilibrium point of the model and analyze the internal causes of the stability mutation of the traffic flow through the Hopf bifurcation point. Then a feedback controller is designed to control the amplitude of the Hopf bifurcation and the limit cycle formed by the Hopf bifurcation. Finally, the theoretical derivation results are verified by experimental numerical simulation. In this paper, we show that, by adjusting the control parameters of the feedback controller, the Hopf bifurcation of the stochastic system can be delayed or even eliminated, and the amplitude of the limit cycle formed by the Hopf bifurcation can be adjusted to achieve the purpose of controlling the stability of the traffic system and prevent or alleviate traffic congestion.

Simplified site response analysis for regional seismic risk assessments

Seismic risk assessment on a regional scale requires an estimation of ground motion intensity and its spatial variation over large geographical areas. This task is particularly challenging in the presence of very soft soil deposits where significant amplification occurs at specific frequencies that are often referred to as local resonances. It is often not feasible to conduct a nonlinear site response analysis at thousands of sites of interest across a region such as an urban area due to the computational effort involved and the required detailed information on soil profiles at each site. Thus, in regional seismic risk analyses the hazard is typically represented by a field of response spectral ordinates estimated by ground motion models. This study proposes a simplified site response analysis procedure to improve the characterization of spectral ordinates to be used in regional seismic risk assessments. In the proposed procedure, reduced-order models are used to conduct site response analysis using very few parameters to transform response spectra at rock outcrop into response spectra at the surface of each of the sites. A particular emphasis is given to soft-soil sites characterized by low shear-wave velocities overlaid on rock or firm soils. A non-uniform continuous shear beam model with a parabolic variation of shear modulus along the depth and modal damping is used in combination with Random Vibration Theory. It is shown that the proposed approach provides better estimates than those of ground motion models or physics-based ground motion simulation models which often cannot capture the local resonances. Site-specific validation of the shear beam model and overall simplified site response analysis is performed at four soft-soil sites in San Francisco, California. Despite its simplicity, it is shown that the proposed procedure adequately captures the amplitudes and spectral shapes of the motions. It is also shown that in conjunction with ground motion models or physics-based models, the proposed simplified site response analysis procedure can be used to efficiently simulate the seismic hazard on a regional scale through an example of a regional seismic hazard analysis of 160 softer soil sites in Downtown San Francisco during the Loma Prieta earthquake.

Partial recovery and weak consistency in the non-uniform hypergraph stochastic block model

Abstract We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $\gamma$ fraction of the vertices classified correctly, where $\gamma \in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati ((2017) Ann. Stat.45(1) 289–315.) for non-uniform HSBMs. Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularised adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularisation of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

Simulation and analysis of proppant transportation and distribution rules under multi-cluster perforation in horizontal wellbore

Unconventional reservoirs require segmented multi-cluster fracturing technology for commercial development. Due to the settling and inertial effects of proppant particle flow, even if the fracturing fluid is evenly distributed among each perforation, the proppant may still be unevenly distributed. However, the uniform distribution of proppant in multi-fracture is a favorable condition for high-yield gas wells. Based on the proppant dynamics model in the wellbore, coupled with the non-uniform fracture propagation model, a mathematical model of proppant wellbore transportation flow under multi-fracture is established. Through numerical calculations, the proppant transport and distribution mechanism are studied. The study shows that due to gravity, distribution in the upstream is more uniform, and the proppant in the downstream gradually settles and most particles flow toward the bottom. Proppant distributed in each cluster is affected by the flow rate of slurry to a certain extent, and is also controlled by its own properties. The turning efficiency determines the inflow of proppant in perforations. Increasing the pump rate can alleviate gravity but reduce turning efficiency. The perforation azimuth primarily regulates the flow of proppant into downstream perforations. The research results can provide theoretical guidance for the design of proppant transport and cluster perforation in hydraulic fracturing.

Robust Direction Estimation of Terrestrial Signal via Sparse Non-Uniform Array Reconfiguration under Perturbations

DOA (Direction of Arrival), as an important observation parameter for accurately locating the Signals of Opportunity (SOP), is vital for navigation in GNSS-challenged environments and can be effectively obtained through sparse arrays. In practical application, array perturbations affect the estimation accuracy and stability of DOA, thereby adversely affecting the positioning performance of SOP. Against this backdrop, we propose an approach to reconstruct non-uniform arrays under perturbation conditions, aiming to improve the robustness of DOA estimation in sparse arrays. Firstly, we theoretically derive the mathematical expressions of the Cramér–Rao Bound (CRB) and Spatial Correlation Coefficient (SCC) for the uniform linear array (ULA) with perturbation. Then, we minimize CRB as the objective function to mitigate the adverse effects of array perturbations on DOA estimation, and use SCC as a constraint to suppress sidelobes. By doing this, the non-uniform array reconstruction model is formulated as a high-order 0–1 optimization problem. To effectively solve this nonconvex model, we propose a polynomial-time algorithm, which can converge to the optimal approximate solution of the original model. Finally, through a series of simulation experiments utilizing frequency modulation (FM) signal as an example, the exceptional performance of this method in array reconstruction has been thoroughly validated. Experimental data show that the reconstructed non-uniform array excels in DOA estimation accuracy compared to other sparse arrays, making it particularly suitable for estimating the direction of terrestrial SOP in perturbed environments.

Research on eccentric-compressive behavior of steel stub columns with localized corrosion

To explore the impact of localized corrosion on the mechanical properties of hollow steel columns under eccentric loads, a specialized localized immersion-accelerated corrosion apparatus was developed. This device facilitated the creation of 17 locally corroded specimens, all subjected to eccentric loading. A systematic examination was conducted on various corrosion parameters including corrosion length, corrosion angle, localized corrosion rate, and corrosion location as well as loading conditions, which encompassed eccentricity and loading angle. Surface corrosion morphology and statistical parameters of the specimens were obtained using a 3D scanner. Subsequently, a nonuniform corrosion model, based on element displacement, was formulated through finite element analysis. Both experimental findings and numerical simulations indicate that localized corrosion does not alter the failure mode of locally corroded circular steel tubes (LCCSTs), which predominantly undergo local buckling in the compressed region. However, it was noted that localized corrosion leads to the elimination of the ascending segment in the load-displacement curve of the LCCSTs, thereby reducing both the ultimate load and ductility. Interestingly, the ultimate load of the LCCSTs was found to be relatively unaffected by variations in corrosion length and location but significantly influenced by factors such as localized corrosion rate, corrosion angle, eccentricity, and loading angle. Furthermore, through a comparative analysis of existing calculation models for the ultimate load of LCCSTs in major specifications, a practical model that accounts for localized corrosion is proposed. This model demonstrates high accuracy and is expected to enhance the understanding and prediction of the ultimate load capacity of LCCSTs in real-world applications.

Non-uniform distribution model of rust layer around steel bar circumference and its effect on corrosion-induced concrete cracking

This paper presents a critical review of non-uniform distribution models of rust layer around a reinforcing bar’s circumference and a numerical investigation into the effect of non-uniform distribution of rust on (a) crack patterns and (b) development of concrete surface crack width. Different shapes of rust layer simulated by mathematical models were established based on statistical distribution functions which were compared with the transport-electrochemical model and published testing results. In addition, an advanced 2D finite element model was implemented to simulate crack propagation in concrete caused by rebar corrosion. It is found that non-uniform rust distribution shape around the rebar not only affected concrete surface crack width evolution, but also crack length and crack pattern. Transport-electrochemical model and Gaussian distribution-based model could reasonably simulate rust distribution and associated corrosion-induced cracks. In addition, the authors proposed a simplified semi-empirical model that incorporated an analytical chloride transport model and a Gaussian model to simulate time-dependent advancement of non-uniform rust layer. Furthermore, a parametric study of the effect of elastic stiffness of the rust layer on crack patterns and surface crack width development was also performed.

A Jacobi-Ritz Approach for Aeroelastic Analysis of Swept Distributed Propulsion Aircraft Wing

Abstract This paper presents a Jacobi-Ritz approach for conducting flutter and divergence analysis of a complex distributed propulsion aircraft wing like NASA X-57. The general orthogonal Jacobi polynomials are used to approximate the bending displacement and torsional rotation angle in the Ritz method based structural and aeroelastic analysis. The Jacobi polynomials satisfy the orthogonality condition using weight functions which are easily modified to satisfy different essential and natural boundary conditions. Compared to simple polynomials, Jacobi polynomials can eliminate the well-known ill-conditioning numerical issues when considering higher-order polynomials. The Jacobi-Ritz method is also found to alleviate mode switching often encountered in tracking the changes of modes with the varying airspeed. The Jacobi-Ritz method is later used to investigate the flutter and divergence speeds under distributed propulsor mass and their locations, nonuniform aerodynamic model in the presence of multiple propulsors, and the sweep angle. Results show that placing the distributed propulsors on the wing's leading edge increases the flutter speed even though the bending and torsion modal frequencies are decreased compared to those of the wing without propulsors. The presence of pods for the middle high-lift motors causes an extra aerodynamic moment which reduces the flutter speed. Parametric studies also show that the divergence speed is lower than the flutter speed for a uniform and straight distributed propulsor wing. Using swept-back wing configuration and placing the tip propulsor near the wing's leading edge can help to increase both flutter and divergence speeds.

Rapid prediction of transient particle transport under periodic ventilation using a non-uniform state Markov chain model

Predicting transient particle transport is crucial to address the risks posed to human health by indoor contaminants and to improve the design and control of ventilation systems. The Markov chain model with a coarse matrix has proven to be efficient in this regard, demonstrating faster speeds than traditional methods and higher computational accuracy when based on non-uniform states. However, its application under unsteady airflow conditions is limited, and discussions of how to determine non-uniform Markov states and their corresponding transient transition probabilities when the flow field changes are limited. Therefore, this study developed an application method of Markov chain model based on fixed non-uniform states in unsteady airflow fields. First, this model rapidly provides various Markov state schemes based on airflow field velocity and clustering methods. It then selects the optimal scheme through frequency and finally obtains the transition matrix through Lagrangian tracking for particle transport prediction. This model reduces the computational cost of transient particle transport when the wind speed changes while ensuring high computational accuracy. The proposed method was validated using published flow field experiments and simulation data. The computational speed of the proposed method was compared with computational fluid dynamics (CFD), and in the case studies considered in this research, the normalized root-mean-square deviation of the model was less than 10% and the total computation time was more than 75% faster than that of CFD.

Automatic heliostat learning for in situ concentrating solar power plant metrology with differentiable ray tracing

Concentrating solar power plants are a clean energy source capable of competitive electricity generation even during night time, as well as the production of carbon-neutral fuels, offering a complementary role alongside photovoltaic plants. In these power plants, thousands of mirrors (heliostats) redirect sunlight onto a receiver, potentially generating temperatures exceeding 1000°C. Practically, such efficient temperatures are never attained. Several unknown, yet operationally crucial parameters, e.g., misalignment in sun-tracking and surface deformations can cause dangerous temperature spikes, necessitating high safety margins. For competitive levelized cost of energy and large-scale deployment, in-situ error measurements are an essential, yet unattained factor. To tackle this, we introduce a differentiable ray tracing machine learning approach that can derive the irradiance distribution of heliostats in a data-driven manner from a small number of calibration images already collected in most solar towers. By applying gradient-based optimization and a learning non-uniform rational B-spline heliostat model, our approach is able to determine sub-millimeter imperfections in a real-world setting and predict heliostat-specific irradiance profiles, exceeding the precision of the state-of-the-art and establishing full automatization. The new optimization pipeline enables concurrent training of physical and data-driven models, representing a pioneering effort in unifying both paradigms for concentrating solar power plants and can be a blueprint for other domains.

Performance evaluation of multivariate deep-time convolution neural architectures for short-term electricity forecasting: Findings and failures

Deep learning (DL) models hold great promise in enhancing the decision-making abilities of electricity market participants and system operators in the short term, as they excel at capturing the intricate interdependencies and uncertainties in electricity consumption. Nevertheless, currently, the adoption of DL models for electricity prediction is not keeping up with the advancements in DL research. Within this framework, the aims of this study are to examine the probabilistic, multivariate, and multihorizon time series prediction of deep time convolution neural (DTCN) models. The effectiveness of DTCN in forecasting electricity consumption over daily, weekly and monthly time horizons is emperically assessed. We analyze both probabilistic and deterministic predictions. The evaluation data comprises real residential electricity consumption, average temperature and humidity with hourly granularity, collected over one year from a residential building. For evaluation, we considered accuracy and uncertainties, sensitivity to hyperparameters, the impact of exogeneous variables, as well as computational efficiency. Results show that the probabilistic models offer greater advantages when it comes to forecasting on multihorizons, especially for diverse datasets with non-uniform time series models. Unlike traditional models, the DTCN models demonstrated superior performance in capturing complex patterns and reducing forecast errors in short-term scenarios. However, their performance can be influenced by the sensitivity of hyperparameters. These findings can act as a benchmark for quantitatively assessing the performance of novel multivariate DL models in predicting electricity demand.

Dynamics of axially moving spinning microbeams composed of tri-directional graded porous materials with axisymmetric cross-sections and piezoelectric layers in complex fields

The current article appraises the vibration and stability of tri-directional functionally graded porous microscale beams with rectangular cross-sections integrated with piezoelectric layers under spinning and axial movements in complex environments. The microbeam is surrounded by a three-parameter Winkler-Pasternak-Hetenyi medium, and its material characteristics are graded in thickness, width, and longitudinal spatial directions by considering non-uniform and uniform porosity models. Dynamic equations, vibration frequencies, and stability criteria of the system are determined with the aid of the Galerkin approach and Laplace transform. The Campbell diagram and stability maps are drawn. Frequency and stability analyses, as well as comparison and parametric analyses, are conducted. The impacts of piezoelectric voltage, magneto-hygro-thermal fields, axial and tangential distributed follower forces, substrate characteristics, scale parameter, aspect ratio, porosity factor, and material gradation on flutter and divergence instability boundaries are assessed in detail. It is deduced that instability regions are condensed, and the instability threshold is enhanced by fine-adjusting the porosity and material gradient. It is discovered that destructive environmental effects can be alleviated by regulating the piezoelectric voltage. In addition, compared with the case of a square cross-section, the divergence/flutter instability region of the microbeam with a rectangular cross-section is smaller/larger. The outcomes of the present research can be helpful in the design of next-generation bi-gyroscopic systems.

Seismic response of mountain tunnel induced by fault slip

With the rapid development of Chinese transportation networks, such as the Sichuan-Tibet railway, numerous tunnels are under construction or planned in mountainous regions. Some of these tunnels must traverse or be situated near active fault zones, which could suffer damage from fault slip. In this study, the seismic response of a mountain tunnel subjected to coseismic faulting was analyzed using a fault-structure system in a two-step process. Firstly, a nonuniform slip model was proposed to calculate the ground deformations and internal displacements induced by a specific active fault on a geological scale, considering nonuniform slips on the fault plane. The 1989 Loma Prieta and 2022 Menyuan earthquakes were chosen as case studies to validate the proposed slip model. Secondly, the calculated displacement of the Menyuan earthquake was used as the input load for the discrete–continuous coupling analysis of the Daliang tunnel on an engineering scale. The simulated deformation of the Daliang tunnel aligned with the on-site damage observations following the Menyuan earthquake. Lastly, the effects of different fault conditions on the tunnel seismic response were investigated. The results indicate that the distribution of the peak longitudinal strain of the lining is governed by fault mechanisms, and the degree of fault slip significantly influences the response of the tunnel. A tunnel passing through an active fault with a wider fault fracture zone and smaller dip angle experience less damage.

Efficient infrared nonuniform correction method considering optical attenuator and integration time

With the increasing number of optical attenuators and integration times in infrared imaging systems, this paper proposes an efficient nonuniform correction method that addresses drawbacks of complex operations and coefficient storage with traditional two-point nonuniform correction methods. First, the efficient nonuniform correction model was derived through the response model of the infrared imaging system with variable attenuators and integration times, which can achieve nonuniform correction of all attenuators and integration times using a correction equation. Based on the model, the efficient nonuniformity correction algorithm using a small number blackbody images was proposed. Finally, the experimental results show that the method has excellent nonuniformity correction performance. Compared with the two-point correction method, the maximum nonuniformity is reduced from 3.7% for the two-point method to 2.2% for the proposed method. In the meantime, the proposed method can greatly reduce the correction time and the number of blackbody temperature sampling points. Moreover, for the infrared imaging system with five integral times and three attenuators in this paper, the two-point method requires 30 correction coefficients, while the proposed method only requires eight correction coefficients. It reduces the required correction coefficients by about three times, achieving efficient and fast nonuniform correction.

Unlocking slip-mediated bending in multilayers: Efficient modeling and solutions with high precision and simplicity

This work introduces a novel approach to understanding the bending behavior of multilayer structures with weak interfaces. Despite the existence of various theoretical models, achieving high accuracy and computational efficiency remains a challenge. To address these limitations, we propose a Non-uniform Slip Model (NSM) governed by two essential dimensionless parameters: the number of layers and the shear factor. Utilizing the Semiparametric Hybrid Variational (SHV) method, we derive theoretical solutions that require less computational effort and offer enhanced accuracy. We further simplify the NSM through homogenization to the Uniform Slip Model (USM), yielding clear and concise analytical solutions for deflection curves and effective bending stiffness with high precision. The USM is also extended to account for nonlinear slipping interfaces with continuous shearing flow, providing an analytical stiffness expression that includes the bending angle. This extension explains the experimental observation of multilayered graphene’s bending stiffness, which decreases from cubic to linear with increasing number of layers due to continuous slip and dislocation between atomic layers. Our study delivers theoretical insights for analyzing slip-mediated bending in laminated materials, paving the way for the design and optimization of such structures.

Advanced Simulation and Analysis of Anisotropic Warp Fields with Positive Energy

This paper presents a comprehensive theoretical model for a warp bubble that enables faster-than-light travel using positive energy densities, an advancement over traditional models that rely on exotic negative energies. Building on the framework proposed by Eric Lentz, we introduce a non-uniform energy distribution model to enhance control and efficiency in warp field generation. The research validates the stability and feasibility of a warp bubble sustained by positive energy through detailed numerical simulations and analysis. Key findings include the discovery of anisotropic behavior in the warp bubble's structure, which allows for directional tuning of the warp field, thereby optimizing space- time manipulation and reducing energy demands. These results represent a significant step forward in the theoretical foundations of warp drive technology and its potential practical applications.