Finite Difference Model Research Articles

An Integrated Taylor Expansion and Least Squares Approach to Enhanced Acoustic Wave Staggered Grid Finite Difference Modeling

The staggered grid finite difference method has emerged as one of the most commonly used approaches in finite difference methodologies due to its high computational accuracy and stability. Inevitably, discretizing over time and space domains in finite difference methods leads to numerical artifacts. This paper introduces a novel approach that combines the widely used Taylor series expansion with the least squares method to effectively suppress numerical dispersion. We have derived the coefficients for the staggered grid finite difference method by integrating Taylor series expansions with the least squares method. To validate the effectiveness of our approach, we conducted analyses on accuracy, dispersion, and stability, alongside simple and complex numerical examples. The results indicate that our method not only inherits the capabilities of the original Taylor series and least squares methods in suppressing numerical dispersion across small and medium wavenumber ranges but also surpasses the original methods. Moreover, it demonstrates robust dispersion suppression capabilities at high wavenumber ranges.

A dynamic boundary condition finite difference model for predicting pavement profile temperatures: Development and validation

The appearance of ground frost is of vital importance in construction and maintenance of roads in cold climates. Frost often causes ground heave and subsequent road damage, which must be taken into account in designing the road structure. Frost depth, pavement temperature, and freezing/thawing cycles are also important for estimating the frequency of road maintenance and treatment. Various analytical, numerical, and empirical models have been developed to estimate the surface temperature of the pavement and to model the heat flow in the underlying layers. The pavement surface experiences a variety of intricate nonlinear heat transfer mechanisms during winter, making it challenging to accurately model the surface boundary. Dynamic variation of parameters such as cloud cover and traffic density during the modeling period introduces additional complexity. To address this challenge, we have established an experimental setup in Luleå, Sweden, to measure pavement profile temperatures during the winter season. Additionally, we have developed a Finite Difference Model that utilizes local weather data including dynamic cloud cover, and which also takes traffic into account. The experimental and simulation findings demonstrate how the impact of surface temperature fluctuations diminishes and, more or less, vanishes for depths more than 55 [cm] below the pavement surface. The Finite Difference Model presented in this study exhibits the ability to forecast the pavement profile temperatures, including the surface temperature based on weather conditions, with acceptable precision for at least 3 days. As a consequence, a reasonable assessment of pavement layer conditions appears feasible based on local weather conditions, and the model can serve as a useful tool for planning road maintenance and construction in cold regions.

Free Electron Density Gradients Enhanced Biosensor for Ultrasensitive and Accurate Affinity Assessment of the Immunotherapy Drugs.

Accurate affinity assessments play an important role in drug discovery, screening, and efficacy evaluation. Label-free affinity biosensors are recognized as a dependable and standard technology for addressing this challenge. This study constructs a free electron density gradient-enhanced meta-surface plasmon resonance (FED-MSPR) biosensor through a finite-difference time-domain simulation model, the biosensor demonstrates superior detection performance in accurately determining affinity and kinetic rate constants. By controlling the dielectric properties of the metal on the surface of the nanocup arrays, the plasmon resonance effects are easily tuned without changing the nanostructure design. Compared with the single-layer gold chip, the triple-layer FED-MSPR chip demonstrated a four-fold improvement in resolution at the optimal resonance peak. Additionally, the sensitivity and figure of merit (FOM) of the multi-layer chip increased by 3.5 and 7.99 times, respectively. Following modification with high- and low-staggered carboxylation, the noise-signal ratio and baseline stability of the real-time kinetic curves based on these chips are significantly enhanced. The developed carboxylation FED-MSPR platform is successfully used to perform affinity assays for Adalimumab and TNF-α protein, resulting in favorable dynamic curves. These findings validate the proposed FED-MSPR biosensor platform as cost-effective, rapid, sensitive, and label-free, facilitating real-time quality control in drug development.

Acoustics response of granular porous material: experiment and modeling

Granular porous materials such as activated carbon have drawn increasing attention due to their good sound absorption performance at low frequency. However, some unique acoustical behaviors have been observed during sound absorption measurements of granular porous materials in impedance tubes: e.g., circumferential edge-constraint effects, level-dependent absorption behavior, and time-dependent absorption behavior. In order to explain these observations, a 1-dimensional poro-elastic model was first applied to describe the sound propagation in granules stacked in a cylindrical sample holder, as in a standing wave tube. Then this model was extended to a 2-dimensional finite difference (2DFD) model with depth-dependent stiffness of the granule stack considered by combining Janssen's model and Hertzian contact theory. The 2DFD model was validated with the measurements of the granular porous materials, and it was found that the 2DFD model is a powerful tool to analyze the acoustic response of granular porous material, thus providing a means of characterizing the granular materials. In the present work, the above-mentioned experimental observations and the 2DFD model are introduced, and the experiment results are analyzed with the 2DFD model. Finally some future work on granular porous materials is introduced.

Application of bursting pressure and conventional drained behavioural theory to the Brumadinho failure

The Brumadinho tailings dam failure in 2019 has had a profound impact on the way tailings engineers analyse and interpret stability, with the focus of assessments moving away from conventional drained behaviour to consideration of undrained and post-liquefaction resistance. One of the contributing reasons is that the Expert Panel investigation of the failure concluded that, despite the drained factor of safety being adequate, the failure took place in an undrained manner. This paper examines the Brumadinho failure from a conventional drained perspective using two-dimensional limit equilibrium and finite difference modelling. Collectively, these analyses show that the failure can also be explained by a drained triggering mechanism and therefore suggest that several additional lessons may be learnt from the failure.

Visual MODFLOW, solute transport modeling, and remote sensing techniques for adapting aquifer potentiality under reclamation and climate change impacts in coastal aquifer

Global environmental changes, such as climate change and reclamation alterations, significantly influence hydrological processes, leading to hydrologic nonstationarity and challenges in managing water availability and distribution. This study introduces a conceptual underpinning for the rational development and sustainability of groundwater resources. As one of the areas intended for the development projects within the Egyptian national plan for the reclamation of one and a half million acres; hundreds of pumping wells were constructed in the Moghra area to fulfill the reclamation demand. This study investigates the long-term impacts of exploiting the drilled pumping wells under climate change. The approach is to monitor the groundwater levels and the salinity values in the Moghra aquifer with various operational strategies and present proposed sustainable development scenarios. The impact of global warming and climate change is estimated for a prediction period of 30 years by using satellite data, time series geographical analysis, and statistical modeling. Using MODFLOW and Solute Transport (MT3DMS) modules of Visual MODFLOW USGS 2005 software, a three-dimensional (3D) finite-difference model is created to simulate groundwater flow and salinity distribution in the Moghra aquifer with the input of forecast downscaling (2020–2050) of main climatic parameters (PPT, ET, and Temp). The optimal adaptation-integrated scenario to cope with long-term groundwater withdrawal and climate change impacts is achieved when the Ministry of Irrigation and Water Resources (MWRI) recommends that the maximum drawdown shouldn’t be more significant than 1.0 m/ year. In this scenario, 1,500 pumping wells are distributed with an equal space of 500 m, a pumping rate of 1,200 m3/day and input the forecast of the most significant climatic parameters after 30 years. The output results of this scenario revealed a drawdown level of 42 m and a groundwater salinity value of 16,000 mg/l. Climate change has an evident impact on groundwater quantity and quality, particularly in the unconfined coastal aquifer, which is vulnerable to saltwater intrusion and pollution of drinking water resources. The relationship between climate change and the hydrologic cycle is crucial for predicting future water availability and addressing water-related issues.

Effective prediction of drug transport in a partially liquefied vitreous humor: Physics-informed neural network modeling for irregular liquefaction geometry

As the medium for intravitreal drug delivery, the vitreous body can significantly influence drug delivery because of various possible liquefaction geometries. This work innovatively proposes a varying-porosity approach that is capable of solving the pressure and velocity fields in the heterogeneous vitreous with randomly-shaped liquefaction geometry, validated with a finite difference model. Doing so enables patient-specific treatment for intravitreal drug delivery and can significantly improve treatment efficacy. A physics-informed neural network (PINN) model is also established for the simulation, and three cases are used for validation. Despite limited information, the PINN model, together with the varying-porosity approach, captured fluid and drug transport in the partially liquefied vitreous. This opens the possibility for optimizing intravitreal drug delivery based on ultrasonography in clinical practice.

Efficient 3D temperature field prediction and visualization for AFP process analysis in composite structures

AbstractProcess modeling is an indispensable tool for establishing a foundation for process optimization and enhancing the quality of products. Analyzing the thermoforming process of polymer matrix composites presupposes the acquisition of accurate thermal histories. This paper introduces a numerical solution based on the finite difference (FD) model to predict the temperature field of angle‐ply composite structures during automated fiber placement (AFP) process along the machining trajectory. The proposed method incorporates three‐dimensional hybrid boundary conditions and considers temperature‐dependent material properties, significantly improving efficiency compared to the finite element (FE) model and overcoming challenges related to complex boundary conditions and anisotropic heat transfer in analytical solutions. Real‐time temperature measurement experiments were conducted to validate predictions, and further a specific temperature control strategy that meets process requirements was explored. In summary, the paper introduces a novel approach to address the temperature field in complex composite structure manufacturing processes.Highlights A novel 3D finite difference model predicts the temperature field in automated fiber placement. Improved prediction accuracy by managing complex boundary conditions. Effectively handled anisotropic heat transfer in complex composite structures. Superior efficiency over existing numerical methods. Developed a model‐based strategy for optimal temperature control within ±5°C.

Analytical model of three-dimensional concentric ellipsoidal soil arching in geosynthetic-reinforced pile-supported embankments

The geosynthetic-reinforced pile-supported (GRPS) embankment is an effective method for improving soft ground, widely adopted in engineering applications. In this paper, a concentric ellipsoidal soil arching model was proposed to describe the stress distribution within the GRPS embankment. An analytical solution for soil arching efficacy was derived by solving the loads acting on the pile caps and geosynthetics under piles arranged in a squared pattern. Subsequently, finite difference models were established to verify the accuracy of the derived analytical solution. Meanwhile, four field tests were introduced to validate the analytical model. Finally, parametric studies were conducted on the concentric ellipsoidal soil arching model, considering parameters such as the embankment height, the pile spacing, the pile cap width, the unit weight, and the friction angle of fill soil.

Numerical Modeling for Tunnel Lining Optimization

The construction of tunnels excavated by the conventional method in densely populated urban environments requires an adequate characterization of the loads acting on the primary lining during the excavation process, to ensure that the ground is deformed and stresses around the tunnel are relieved, simultaneously complying with the failure and serviceability limits of international standards while minimizing damage to nearby structures. In this paper, common lining design criteria are revisited, through the numerical simulation of an instrumented tunnel section which is part of a 4.5 km long metro line currently under construction in Mexico City. Key needs for improvement in current design approaches are identified. The tunnel was instrumented with load cells, extensometers, and topographical references for convergences and divergences. A three-dimensional finite difference model of the instrumented section was developed, and the load transfer mechanisms between the excavated soil and the primary lining were analyzed. Then, the numerical simulation of the contribution of the secondary lining in the overall stability for sustained load was established, along with the expected ground settlements, which can significantly affect nearby structures. Results gathered from this research are key for updating lining design criteria for urban tunnels built in stiff brittle soils.

Impacts of the mechanical connectors on seismic performance of the restored masonry Plaka Bridge using experimental and numerical studies

The use of clamp and dowel materials in the arch of the masonry bridges holds pivotal significance in enhancing the seismic resilience of these structures, as these elements contribute crucially to the structural integrity and dynamic response of these venerable structures. For this reason, the seismic performance of the restored historic Plaka Bridge, originally constructed in 1866 in Arta, Greece, and later restored in 2015 after flooding, was investigated considering the dowels and clamps in the arch section of this study. A combination of experimental and numerical analyses was conducted and verified to understand the in-situ interface behavior of the arch stones. In the first phase of the study, the mechanical properties of the dowels and clamps were characterized by the experimental tests. Compression and tension tests were conducted on the stone samples containing dowel bars and clamps respectively to determine the mechanical properties, The experimental outcomes were subsequently validated through Three-Dimensional (3D) Finite Difference Models (FDM). It was obtained that the dowels and clamps represent the bi-linear material model at the interfaces. Finally, the idealized interface behavior was constructed with normal and tangential spring stiffness properties in 3D-FD Models. These springs successfully simulate the interface delamination failure of the stones in all directions. In the final phase of the study, a 3D-FD model of the Plaka Bridge was constructed employing the previously validated normal and tangential stiffness elements in the arch of the bridge. A free-field non-reflecting boundary condition was imposed on the base and lateral boundaries of the bridge model. Seismic analyses were conducted utilizing the significant seismic events including 2023 Kahramanmaraş earthquakes. The results revealed significant alterations in the structural behavior of the bridge under considered earthquakes. Furthermore, the displacements, principal stresses and tensile-compressive damages in the main arch significantly reduced with the adopted stiffness values. This study proposes the undeniably realistic and efficient implementation of interface material modeling for the mechanical connections as opposed to the traditionally assumed mortar properties in literature. Furthermore, the combined behavior of the dowel, clamp, and Khorasan mortar proposes non-linear interface material model to the existing literature by invalidating commonly assumed linear interface behavior in masonry structures.

Finite Difference Model of Substrate Channeling in TCA Cycle Enzymes

The Tricarboxylic Acid (TCA) cycle is a highly optimized metabolic pathway. Of particular interest in the TCA cycle is the malate dehydrogenase—citrate synthase (MDH-CS) complex, which accomplishes citrate production from malate with an oxaloacetate intermediate. This complex was studied experimentally by Bulutoglu et al, who showed that mutation of positively-charged residues on the complex surface effected the dynamic response of citrate production as measured by lag time experiments.1 More recently, we have created a Markov State model that was used to predict the transfer efficiency of each complex and determine the reaction pathways on the surface.2 Utilizing the kinetic parameters of both the recombinant and mutant complexes determined experimentally1and Markov state transition matrix produced by previous modeling2, we report a finite difference model to calculate time trajectories of the surface and bulk occupancies by OAA. Solution of a system of differential mass balance equations, based on the transient matrix, provide the occupancy probability at each node in the network. The lag time of MDH-CS was determined computationally to be comparable to experiment for both the recombinant and mutant complex with lag times of 0.44 and 2.1 seconds, respectively.1 Using the model, the dynamics of each of the four possible reaction pathways between the the two source (MDH) active sites and the two sink (CS) sites could be studied independently. This analysis provides a dynamic model for intermediate transport in an electrostatically channeled system, and can be used as a predictive tool to provide mechanistic insight into pathway dominance. We gratefully acknowledge support from the Army Research Office MURI (#W911NF1410263) via The University of Utah. References B. Bulutoglu, K. E. Garcia, F. Wu, S. D. Minteer, and S. Banta, ACS Chemical Biology, 11, 2847–53 (2016). doi:10.1021/acschembio.6b00523Y. Xie, S. D. Minteer, S. Banta, and S. Calabrese Barton, ACS Nanoscience Au, 2, 414-421 (2022). doi:10.1021/acsnanoscienceau.2c00011 Figure 1

A 2D Simulation‐Assisted Investigation of a Low Breakdown Voltage Module

Upscaling perovskites modules from laboratory to market specifications requires modules to resist heat, humidity, and partial shading. When a device is partially covered or damaged, the unbalanced photocurrent generated can create an irreversible damage. Mitigation via bypass diodes can be expensive. This might not be necessary for perovskite‐based modules. After an initial validation of the finite difference model for a simulation‐assisted investigation, a module is systematically damaged and characterized to reproduce an unbalanced photocurrent generation. The device shows a breakdown voltage (< 1 V), which is unexpectedly much lower than for a standalone perovskite cell (≈7 V). The effect is reproducible over time, and it demonstrates that in a reverse bias scenario, electricity can tunnel through the monolithic interconnection (P1P2P3) and operate similar to a low‐voltage bypass diode. Eventually, the current and voltage distribution via electroluminescence is mapped to validate the 2D simulations and reproduce the device behavior at different loads.

How Accurate Numerical Simulation of Seismic Waves in a Heterogeneous Medium Can Be?

ABSTRACT Analysis of equations of motion by Moczo et al. (2022) led to the conclusion that the discrete (grid) representation of the heterogeneous medium must be wavenumber bandlimited up to the Nyquist frequency. This is a consequence of the spatial discretization. Mittet (2021a) reported that if the discrete grid model of medium coincides with the true medium up to some wavenumber, the simulated wavefield is accurate only up to a half of this wavenumber. Here, we present results of the systematic and comprehensive analysis focused on the principal limits of accuracy of numerically simulated wavefields. First, we analyze wavenumber spectra of (1) exact wavefields in a heterogeneous elastic medium, (2) wavenumber bandlimited wavefields, and (3) spatially discretized wavefields. Then, we derive spatial dependence of the frequency spectrum of waves generated by a finite source, and perturbing wavefields due to a small perturbation of the medium and due to a small wavenumber bandlimited perturbation of the medium. We analyze an interaction of an incoming wave with the medium perturbation through a change of phase difference and through wavenumber spectra. We draw conclusions on the wavenumber limitation of wavefields in the wavenumber bandlimited heterogeneous medium. We numerically verify the fundamental finding using exact solutions. The main consequence for the finite-difference (FD) modeling based on spatial discretization of the computational domain is: Due to spatial sampling, the medium must be wavenumber limited up to the Nyquist frequency. Then, the wavefield should not be sampled by less than four spatial grid spacings per shortest wavelength to obtain sufficiently accurate results. This applies to any heterogeneous FD scheme.

Simulation of femtosecond laser-induced periodic surface structures on fused silica by considering intrapulse and interpulse feedback

The formation of laser-induced periodic surface structures (LIPSSs) on fused silica upon irradiation with plane wave, double pulse, spot processing, and scanning processing (pulse duration tp = 35 fs, center wavelength λ = 800 nm, low repetition rate ≈1 kHz) is studied theoretically with an improved three-dimensional finite-difference time-domain plasma model. The model covers both intrapulse feedback under single shot and interpulse feedback under multi-shots, thus enabling better prediction of transient responses during laser–material interaction and the evolution of the ablated morphology and accumulated defects’ density with more shots. In simulations of a single plane wave, a double pulse can modulate LIPSS periodicity. In simulations of spot processing with Gaussian beam, an increase in the number of shots results in a noticeable ablation pattern where high-spatial-frequency LIPSS surrounds low-spatial-frequency LIPSS at a fluence of 2.8 J/cm2. Moreover, simulations of scanning processing with Gaussian beam showcase the broad applicability of this model, revealing that the orientation of the LIPSS depends on the polarization direction rather than the scanning path. This new model provides a powerful tool to simulate the formation of LIPSS on silica, particularly when temporally modulated laser is involved or predicting the evolution of morphology dependent on the number of shots.

Estimating natural groundwater recharge taking into consideration temporal and spatial variability for arid unconfined aquifer

ABSTRACT Water scarcity is one of the major critical affecting arid climate countries. In addition, population growth and climate change are threatening the sustainable development of water resources in the upcoming decades. Therefore, managing groundwater can solve water scarcity such that sustainable development is ensured for future generations. Hence, it is crucial to assess the current quantity of groundwater by analyzing groundwater variables. Thus, natural groundwater recharge could be utilized as an indicator to assess groundwater quantity conditions. This study aims to estimate natural groundwater recharge in an arid unconfined aquifer. El-Qaa plain in Sinai, Egypt was chosen as a case study. Recharge was estimated taking into consideration spatial and temporal variability. To accomplish this, the WetSpass (Water and Energy Transfer between Soil, Plants, and Atmosphere under quasi-Steady State) model was applied. The model was run using morphological, landuse – land cover and climate data from 1986 to 2015 with a monthly time step. To calibrate the model, the isotope signature of δ18O and δ2 H of water was analyzed to determine the natural groundwater recharge of the aquifer. The obtained recharge value was utilized as a reference value for the calibration process. The results were validated using a finite-difference groundwater flow model (MODFLOW). The recharge values were imported to MODFLOW followed via a transient run from 2000 to 2011. The validation was conducted by comparing the simulated groundwater level with the observed ones. After validation process, the inter-annual and seasonal variability of recharge were determined, and a relationship between recharge and other hydrological parameters was deduced. In addition, the recharge distribution was mapped to depict the recharge spatial variability. The results show that the current recharge rate ranges from approximately 8.10 to 16.62 mm/yr, with a mean annual recharge rate of nearly 12.36 mm/yr. The study might be useful for hydrogeologists for determining potential recharge in the study area, groundwater modelers for developing a reliable groundwater flow model, and decision-makers for planning appropriate future sustainable development through water resources management.

КЛИМАТИЧЕСКИЕ ПРОГНОЗЫ. ЧАСТЬ I. СОВРЕМЕННОЕ СОСТОЯНИЕ И ПЕРСПЕКТИВЫ РАЗВИТИЯ

The physical background and main types of climate forecasts issued by the world meteorological centers are considered. It is emphasized that modern forecasting systems have an integrated nature and combine not only data assimilation systems and global numerical atmosphere-ocean-land models, but also support infrastructure for providing forecast products to users. The features of the forecast system of the Hydrometcenter of Russia/North Eurasia Climate Centre (NEACC) and other world meteorological centers are compared. It is noted that, unlike other centers, the forecast system of the Hydrometeorological Center of Russia/NEACC is based on the integrated use of synoptic, statistical, and hydrodynamic methods. It has been revealed that new trends and directions in the development of the forecast system are associated with the emergence of a new version of the SL-AV global semi-Lagrangian finite-difference atmosphere model of the Hydrometcenter of Russia and the Marchuk Institute of Numerical Mathematics of the Russian Academy of Sciences (INM RAS) with an expanded (up to 61 members) forecast ensemble, as well as with using the INR RAS climate model (INM-CM5) and additional applications designed for interaction with various economic sectors. The findings may be useful in determining a vector of future research aimed at developing the Russian climate prediction system.

Effective applications of 1-D finite difference modeling for temperature prediction of concrete structures

This paper introduces a 1-D finite difference model for analyzing the temperature distribution throughout the hardening phase in concrete structures. The model accounts for the degree of hydration-dependent internal heat rate and the solar irradiance on the receiving surface of a concrete structure, allowing for more accurate predictions of early-age behavior of concrete. The research involves the development and validation of the 1-D model using temperature measurements from an on-site concrete footing. Mesh sensitivity studies were conducted to optimize grid size for thermal analysis. The paper also explores the determination of suitable 2-D width-to-height ratios for employing the 1-D model, providing practical guidelines for its application. Major findings indicate that the 1-D model can accurately predict temperature evolution in early-age concrete and offers a cost-effective alternative to more complex 2-D, 3-D finite difference, and finite element models. For preliminary analysis and quick assessments, the 1-D model is advantageous due to its computational speed, and it holds significant potential for concrete temperature control and construction optimization.

A finite difference method on crack resistance of reinforced glass beam with non-linear adhesive

Reinforced glass beam (RG beam) is a type of structural glass member that has been developed in recent years. This RG beam consists of a glass beam to which reinforcement material, such as steel, is adhesively attached in the tensile side to improve crack resistance. The adhesive plays a critical role in the structural performance of RG beam. However, the effect of non-linear shear-slip behavior of the adhesive layer remains unclear and has been neglected in most previous studies, which can result in inaccurate estimations. To address this issue, this paper presents a finite difference model (FDM) that utilizes an explicit step-by-step method and trial-and-error iterative method for interfacial stress analysis. The model predicts the static structural response of a simply-supported RG beam that simulates the adhesive with non-linear stress-slip behavior. Furthermore, the model describes the adherend’s shear deformation in elastic. The FDM results are then compared with the finite element method (FEM), adopting discrete nonlinear connectors (springs) to simulate the adhesion, and available analytical methods. Detailed parametric studies are further conducted to investigate the influences of glass strength, load pattern, reinforcement ratio and height-to-span ratio for proposing design recommendations.

Seismic modeling by combining the finite-difference scheme with the numerical dispersion suppression neural network

Seismic finite-difference (FD) modeling suffers from numerical dispersion including both the temporal and spatial dispersion, which can decrease the accuracy of the numerical modeling. To improve the accuracy and efficiency of the conventional numerical modeling, I develop a new seismic modeling method by combining the FD scheme with the numerical dispersion suppression neural network (NDSNN). This method involves the following steps. First, a training data set composed of a small number of wavefield snapshots is generated. The wavefield snapshots with the low-accuracy wavefield data and the high-accuracy wavefield data are paired, and the low-accuracy wavefield snapshots involve the obvious numerical dispersion including both the temporal and spatial dispersion. Second, the NDSNN is trained until the network converges to simultaneously suppress the temporal and spatial dispersion. Third, the entire set of low-accuracy wavefield data is computed quickly using FD modeling with the large time step and the coarse grid. Fourth, the NDSNN is applied to the entire set of low-accuracy wavefield data to suppress the numerical dispersion including the temporal and spatial dispersion. Numerical modeling examples verify the effectiveness of my proposed method in improving the computational accuracy and efficiency.