Finite Element Method Research Articles

Analysis of Influence of Excitation Source Direction on Sound Transmission Loss Simulation Based on Alloy Steel Phononic Crystal

As a type of locally resonant phononic crystal, alloy steel phononic crystals have achieved notable advancements in vibration and noise reduction, particularly in the realm of low-frequency noise. Their exceptional band gap characteristics enable the efficient reduction of vibration and noise at low frequencies. However, the conventional transmission loss (TL) simulation of finite structures remains the benchmark for plate structure TL experiments. In this context, the TL in the XY-direction of phononic crystal plate structures has been thoroughly investigated and analyzed. Given the complexity of sound wave incident directions in practical applications, the conventional TL simulation of finite structures often diverges from reality. Taking tungsten steel phononic crystals as an example, this paper introduces a novel finite element method (FEM) simulation approach for analyzing the TL of alloy steel phononic crystal plates. By setting the Z-direction as the excitation source, the tungsten steel phononic crystal plate exhibits distinct responses compared to excitation in the XY-direction. By combining energy band diagrams and modes, the impact of various excitation source directions on the TL simulations is analyzed. It is observed that the tungsten steel phononic crystal plate exhibits a more pronounced energy response under longitudinal excitation. The TL map excited in the Z-direction lacks the flat region present in the XY-direction TL map. Notably, the maximum TL in the Z-direction is 131.5 dB, which is significantly lower than the maximum TL of 298 dB in the XY-direction, with a more regular peak distribution. This indicates that the TL of alloy steel phononic crystals in the XY-direction is closely related to the acoustic wave propagation characteristics within the plate, whereas the TL in the Z-direction aligns more closely with practical sound insulation and noise reduction engineering applications. Therefore, future research on alloy steel phononic crystal plates should not be confined to the TL in the XY-direction. Further investigation and analysis of the TL in the Z-direction are necessary. This will provide a novel theoretical foundation and methodological guidance for future research on alloy steel phononic crystals, enhancing the completeness and systematicness of studies on alloy steel phononic crystal plates. Simultaneously, it will advance the engineering application of alloy steel phononic crystal plates.

Application of the Finite Element Method for Analyzing the Vibration Characteristics of a Spindle System and Fault Diagnosis

The performance of a spindle will gradually decline, the vibration intensity will increase, and the temperature rise will become abnormal with the accumulation of service time. Consequently, the accuracy of the machining product will not meet its production requirements. Studies on the variation characteristics of the spindle unit have clarified the reasons for its abnormal vibration and temperature rise, in principle, to help enterprises conduct preventive maintenance before any serious failure occurs and improve production efficiency. Based on the Timoshenko beam model and rotor dynamics theory, this study uses the finite element method (FEM) to analyze the vibration characteristics of the spindle rotor system. Moreover, it thoroughly analyzes the influence of the spindle bearing heat on the stiffness of the spindle system and identifies the resonance conditions of the spindle rotor within the power frequency range. This research has identified that when the vibration frequency of the spindle operates at a speed of 12,000 r/min, if its vibration frequency is lower than 200 Hz, it will be affected by abnormal vibrations during startup, which will weaken the health status of the spindle and reduce its service life. Similarly, when the working speed of the spindle is 30,000 r/min, if its vibration frequency is lower than 50 Hz, abnormal vibration will occur during startup and operation, thereby reducing its service life.

A comprehensive study of beam modal functions in the free vibration analysis of cylindrical shells: Critical examination on the applicability to the clamped-free boundary condition

Over the past few decades, approximate methods that can provide solutions of sufficient accuracy have received considerable attention in the free vibration analysis of cylindrical shells, where a great deal of studies adopted the beam modal functions as the trial functions for the axial mode shapes of cylindrical shells. Nevertheless, most studies were restricted to the application of single term beam modal function and failed to simulate elastic boundary conditions of cylindrical shells, while the accuracy of the corresponding methods has recently sparked significant controversy, especially for cylindrical shells under the clamped-free boundary condition. This paper presents a comparative study of three forms of beam modal functions in the free vibration analysis of cylindrical shells, one of which is proposed for the first time to simulate elastic boundary conditions of cylindrical shells. A unified model is developed using the general Rayleigh–Ritz method, incorporating the breathing modes with circumferential orders being zero, and four types of commonly used thin shell theories, namely the Donnell, Reissner, Love, and Sanders theories. From both perspectives of natural frequencies and mode shapes, numerical results are validated by comparison with those existing in the literature and those calculated from the finite element method (FEM). The results not only clarify the distinction of different forms of beam modal functions used in the Rayleigh-Ritz method, but also provide explanations for the controversy raised in recent studies. Furthermore, the unified formulations can be extended to vibration analysis of various forms of shell structures, and can also be helpful to the vibration analysis of beams and plates with elastic boundary conditions.

Optimization of high performance hybrid composite under ballistic test and performance of the hybrid composite under IED blast

Abstract In this investigation a novel material has been developed comprising of Silicon Carbide (ceramic), Aluminium honeycomb (metal) and Poly Ether Ketone (PEK)—Poly Phynelene Benzobisoxaole (PBO) fiber based blast proof composite. The experimental results proved that this polymeric composite is highly suitable for 9 mm pistol bullet with 430 m s−1 at distance of 5 m as well as for improvised explosive devices blast to absorb the sharpnels/splinters. This investigation also highlights an efficient back layer for a blast-proof sandwich composite. Layer optimization was carried out by finite element method (FEM) analysis of blast effect on the multi-layer polymer composite panel by ANSYS Explicit Dynamics (LS DYNA Solver). The resultant optimal layers of PEK-PBO composites then tested to validate results of FEM analysis. Experimentally it is proved that polymeric composite with 10 layers of PBO fabric when test fired under 9 mm pistol bullet with 430 m s–1 at distance of 5 m, the bullet penetrates, however, when the polymeric composite is made with a total of 20 layers of PEK-PBO fabric, all the test fired bullets are stopped. Therefore, under SNANAG 4569 blast test, the polymeric composite was essentially made with 20 layers of PBO fabric and it is experimentally proved that PEK-PBO composite is highly efficient as the back layer of hybrid composite and all the sharpnels of IED blast are completely absorbed.

Analysis and Design of a Recyclable Inductive Power Transfer System for Sustainable Multi-Stage Rocket Microgrid with Multi-Constant Voltage Output Characteristics—Theoretical Considerations

After a traditional one-time rocket is launched, most of its parts will fall into the atmosphere and burn or fall into the ocean. The parts cannot be recycled, so the cost is relatively high. Multi-stage rockets can be recovered after launch, which greatly reduces the cost of space launches. Moreover, recycling rockets can reduce the generation of waste and reduce pollution and damage to the environment. With the reduction in rocket launch costs and technological advances, space exploration and development can be carried out more frequently and economically. It provides technical support for the sustainable use of space resources. It not only promotes the sustainable development of the aerospace field but also has a positive impact on global environmental protection, resource utilization, and economic development. In order to adapt to the stage-by-stage separation structure of the rocket, this paper proposes a new multi-stage rocket inductive power transfer (IPT) system to power the rocket microgrid. The planar coil structure is used to form wireless power transfer between each stage of the rocket, reducing the volume of the magnetic coupling structure. The volume of the circuit topology structure is reduced by introducing an auxiliary coil. An equivalent three-stage S/T topology is proposed, and the constant voltage output characteristics of multiple loads are analyzed. A multi-stage coil structure is proposed to supply power to multiple loads simultaneously. In order to eliminate undesired magnetic coupling between coils, ferrite cores are added between coils for effective electromagnetic shielding. The parameters of the magnetic coupling structure are optimized based on the finite element method (FEM). A prototype of the proposed IPT system is built to simulate a multi-stage rocket. A series of experiments are conducted to verify the advantages of the proposed IPT system, and the three-stage rocket system efficiency reached 88.5%. This project is theoretical. Its verification was performed only in the laboratory conditions.

Dynamic response analysis of a floating bridge structure under combined actions from seismic loading and waves at different incident angles

Abstract This paper investigates the complex coupled response of a cross-sea floating bridge under the combined actions of seismic loading and wave at different incident angles. The bridge structure is modeled using the finite element method, considering the interaction between the girder, pier, pontoon, and mooring chains. Wave forces are calculated based on potential theory, and an added mass approach is applied to simulate the hydrodynamic forces induced by the earthquake. Simulations were conducted using a one-year random wave at three different incident angles of 0o, 45o, and 90o, combined with three-dimensional seismic loading of varying magnitudes: 0.3 g, 0.5 g, and 1 g. The simulation results indicate that the direction of the incident wave significantly influences the bridge's response. The coupling effect between wave and seismic forces on the floating bridge's response is not simply additive. Generally, when the earthquake magnitude is low, the wave loading significantly impacts the bridge's dynamics. However, as the earthquake magnitude increases, the bridge's response becomes dominated by the seismic forces, rendering the influence of wave forces negligible.

Structural Health Monitoring and Failure Analysis of Large-Scale Hydro-Steel Structures, Based on Multi-Sensor Information Fusion

Large-scale hydro-steel structures (LS-HSSs) are vital to hydraulic engineering, supporting critical functions such as water resource management, flood control, power generation, and navigation. However, due to prolonged exposure to severe environmental conditions and complex operational loads, these structures progressively degrade, posing increased risks over time. The absence of effective structural health monitoring (SHM) systems exacerbates these risks, as undetected damage and wear can compromise safety. This paper presents an advanced SHM framework designed to enhance the real-time monitoring and safety evaluation of LS-HSSs. The framework integrates the finite element method (FEM), multi-sensor data fusion, and Internet of Things (IoT) technologies into a closed-loop system for real-time perception, analysis, decision-making, and optimization. The system was deployed and validated at the Luhun Reservoir spillway, where it demonstrated stable and reliable performance for real-time anomaly detection and decision-making. Monitoring results over time were consistent, with stress values remaining below allowable thresholds and meeting safety standards. Specifically, stress monitoring during radial gate operations (with a current water level of 1.4 m) indicated that the dynamic stress values induced by flow vibrations at various points increased by approximately 2 MPa, with no significant impact loads. Moreover, the vibration amplitude during gate operation was below 0.03 mm, confirming the absence of critical structural damage and deformation. These results underscore the SHM system’s capacity to enhance operational safety and maintenance efficiency, highlighting its potential for broader application across water conservancy infrastructure.

Lateral Response Analysis of a Large‐Diameter Pile Under Combined Horizontal Dynamic and Axial Static Loads in Nonhomogeneous Soil

ABSTRACTThe large diameter piles are widely used in structures such as offshore wind turbines due to their superior lateral load‐bearing capacity. To explore the lateral response of a large‐diameter pile under combined horizontal dynamic and axial static loads in nonhomogeneous soil, a simplified analytical model of the pile–soil interaction is developed. This model represents the pile as a Timoshenko beam resting on the Pasternak foundation, incorporating the double‐shear effect by considering both pile and soil shear. The governing matrix equations for the pile elements are derived from the principle of virtual work. Further, the pile's lateral deformations and internal forces are obtained using the modified finite beam element method (FBEM) and then validated through existing analytical solutions. Finally, the contribution of various properties of pile, soil, and applied load to the pile's lateral vibration response are performed. It is found that both pile and soil shear effects significantly impact the lateral dynamic response of a large‐diameter pile. Additionally, in nonhomogeneous soil, decreasing surface soil strength and dimensionless frequency lead to increased lateral displacements and bending moments of the pile, which are significantly affected by the P‐Δ effect under increasing axial load.

Integral equation based wellbore preconditioner for 3D electromagnetic response modeling

Modeling subsurface controlled-source electromagnetic (CSEM) responses using the finite-element (FE) method is challenging in the presence of highly conductive wellbore casing. The very large conductivity contrast between the casing and the host formation leads to increased computation time and potentially unstable solutions. We address this difficulty by preconditioning an FE solver with an integral equation (IE) primary solution that captures the CSEM response of a realistic-sized steel wellbore casing. Our hybrid IE-FE approach determines the primary field using 2D integral-equation forward modeling and then interpolates the IE-computed solution onto the nodes. Then using an existing FE simulator, we solve for the secondary electric and magnetic fields. This approach removes the need for an ultra-fine FE mesh around the wellbore, thereby improving FE solution stability while greatly reducing FE computation time. Our method is illustrated by modeling the CSEM responses of idealized fluid-bearing zones.

Optimization design of a two-step microreactor with spiral structure for high-performance catalytic reactions

Abstract In order to enhance the efficiency of diphenyldimethoxysilane preparation in microreactors, this study utilized the computational fluid dynamics simulation based on the finite element method to explore the impact of the internal structural parameters of the spiral two-step microreactor (STMR) on the reaction outcomes, with the aim of optimizing its structure for high-performance catalytic reactions. By designing a microreactor based on the Archimedean spiral shape and introducing two ribbed obstacles, the structure was optimized through adjusting the relevant ratios. The effects of different-sized structures and obstacles within the reaction zone and non-reaction zone on the product concentration and reaction results were discussed. The results demonstrate that lower obstacle heights and smaller aspect ratios (P = 2:7, R = 5:6) are beneficial for improving the reaction efficiency and product concentration. This study offers a theoretical foundation for microreactor design and is anticipated to further drive the development of microreactor technology.

Analysis of Vibration Characteristics for Rotating Braided Fiber-Reinforced Composite Annular Plates with Perforations

In the current study, a comprehensive numerical model for analyzing the vibrational characteristics of braided fiber-reinforced composite (BFRC) rotating annular plate with perforations under diverse boundary constraints was introduced. This model employs the differential quadrature finite element method (DQFEM), which was developed based on the first-order shear deformation theory (FSDT) and coordinate transformation approach. The BFRC material, specifically a two-dimensional biaxial orthogonal fabric, was utilized to fabricate the annular plate with two distinct types of holes: circular and sector-shaped. The model’s convergence, accuracy, numerical stability, and reliability were confirmed through comparative assessments utilizing data from the literature, from ABAQUS software, and from experimental findings. The analysis focuses on studying the influences of structural properties, material parameters, and boundary restraints on the frequencies of vibration for BFRC rotating annular plates with holes. This theoretical model helps provide scientific basis and technical guidance for the stability and lightweight design of rotating annular plates, such as rotor structures in aircraft engines.

Simulation of melt pool dynamics including vaporization using the particle finite element method

Simulation of melt pool dynamics including vaporization using the particle finite element method

Linear transverse flux generator for wave energy conversion: design optimization and analysis

Abstract The electric power industry impacts each state’s economy significantly, driven by increasing electricity consumption that necessitates expanding power plants and finding alternative energy sources. Among alternative energy sources, ocean and sea wave energy converters can be distinguished as a separate class. Wave energy converters transform wave energy into mechanical and then electrical energy. The purpose of the study is to analyze and optimize the magnetic system of a transverse flux machine (TFM) linear generator and to determine the influence of the distance between the stator cores on the efficiency of the generator. This research included conducting 3D modeling and analysis to identify this rational distance. The methods for investigating the magnetic system and calculating the magnetic field pattern are divided into analytical and numerical. Thanks to advanced software for solving such tasks, numerical calculation methods based on the finite element method play a decisive role. Meanwhile, analytical calculations of the magnetic circuit are performed using Kirchhoff’s second law for preliminary analysis. The article discusses a two-phase linear TFM generator with a U-shaped core and permanent magnets. The results of numerical modeling show that the distance between the stator cores should have a specific size and requires detailed selection when designing the magnetic system in each particular case. In the design studied, it was calculated that 6 mm between the stator cores increases the machine’s performance. 3D modeling is necessary for accurate analysis, considering the axial magnetic flux to minimize stray fields and their mutual demagnetization. Future research will explore an E-shaped core TFM design.

Analysis of bolt bonding surfaces by the finite element method based on the equivalent Iwan model

This study establishes a finite element model based on the Iwan model to accurately characterize the force–displacement relationship and energy dissipation in bolted joints under small-amplitude tangential displacement. The force–displacement behavior of the Iwan model is transformed into a stress–strain relationship using a hybrid hardened intrinsic model. A subroutine developed with UMAT and UHARD is used to simulate the elastic–plastic behavior of the Iwan model during surface contact in the finite element model. Three-dimensional modeling and joint simulation are conducted using ABAQUS software. Experimental data verify the model's validity, and the effects of multiple factors on the bolted bonding surface are analyzed. The influence of friction coefficient, cyclic displacement amplitude, and preload on the force–displacement relationship of the bolted bonding surface is explored. Polynomial interpolation of the loading and unloading curves is applied to investigate the energy dissipation phenomenon. Results show that the finite element method can effectively reflect the hysteresis and energy dissipation behavior of the bonding surface. Increased friction coefficient and preload improve residual stiffness and energy dissipation capacity, while larger cyclic displacement amplitudes increase energy dissipation but have a minimal effect on residual stiffness.

Enhancing efficiency in hot stamping of polymethyl methacrylate foils: computer analysis and modeling

ABSTRACT This work is leading the way in developing a new method to evaluate the efficacy of hot stamping on polymethyl methacrylate (PMMA) foils. The art of hot stamping, which involves stamping, heat, and pressure to produce elaborate images on flexible foil surfaces, has long been prized for its artistry. A thorough examination of the embossing mechanism is complicated by the fact that different process factors might result in different levels of quality for embossed patterns. This study stands out for its creative approach to hot stamping process optimisation. Traditionally, resource-intensive and time-consuming empirical studies are used to determine the ideal process conditions. By using computer analytic techniques, this research, on the other hand, presents a novel way for quickly and precisely determining the most effective process parameters. Furthermore, by exploring the evolution of the embossing phenomena utilising cutting-edge methodologies like the finite element method (FEM) and creative reproduction approaches, this work explores the unexplored area. A thorough analytical model is carefully created, accounting for temperature-specific stress characteristics, strain rates, stress distribution, and the softening characteristics of the elastic foil. The research makes a unique contribution by precisely characterising the stress hardening, deformation, and thermal expansion of PMMA materials. This paves the way for extrusion analysis utilising customised processes.

Microstructural analysis of AA8079 alloy sheets formed by friction stir incremental forming

Friction Stir Incremental Forming (FSIF) is an innovative solid-state metal-forming technique that uses frictional heat and mechanical deformation to create complex shapes without dyes. It has found applications in industries like aerospace, automotive, and consumer goods. In this study, FSIF was applied to AA8079 alloy sheets to form cone and pyramid geometries. A detailed investigation of process parameters such as spindle speed, table feed, step size, and coatings was conducted to assess their effects on surface roughness and forming forces. Optimal parameters were identified using the Taguchi L27 orthogonal array and the TOPSIS method, which determined the best settings as a wall angle of 20°, a step size of 1 mm, a spindle speed of 1500 rpm, and a table feed of 2400 mm/min. Scanning electron microscopy was employed to study surface roughness, texture, and defects, revealing the influence of these parameters on surface quality. Additionally, Finite Element Method (FEM) analysis, performed using ABAQUS software, was used to predict formability and stress distribution during the forming process. Key results showed successful formation of cone and triangular pyramid shapes on AA8079 sheets, with FEM accurately predicting stress levels. This comprehensive study provides valuable insights into optimizing FSIF for improved material performance and structural integrity. The findings emphasize the importance of fine-tuning FSIF parameters to enhance surface finish and reduce defects, offering potential improvements for various industrial applications.

Optimization and Structural Analysis of Automotive Battery Packs Using ANSYS

The development of new energy vehicles, particularly electric vehicles, is robust, with the power battery pack being a core component of the battery system, playing a vital role in the vehicle’s range and safety. This study takes the battery pack of an electric vehicle as a subject, employing advanced three-dimensional modeling technology to conduct static and dynamic analyses. Through weight reduction and structural optimization, an innovative power battery pack design scheme is proposed, aiming to achieve a more efficient and lighter electric vehicle power system. The main research tasks are as follows: Firstly, we designed the main load-bearing components of a certain electric vehicle’s power battery pack and established a three-dimensional (3D) model. Then, the model was simplified according to the actual stress conditions of the power battery pack of the electric vehicle and imported into finite element analysis (FEA) software. Next, based on the fundamental principles of the finite element method (FEM), we conducted static analyses under three conditions: bumpy road sharp left turn, bumpy road sharp right turn, and bumpy road emergency braking. The analysis results indicate that the strength of the battery pack meets the allowable requirements, suggesting that the lower housing design has significant redundancy, providing guidance for subsequent optimization. Finally, through modal analysis, we extracted the first six modes of the power battery box, with the first mode frequency being 33.69 Hz. This suggests that the battery pack may experience resonance during actual operation. Based on the static and modal analysis results, we proposed a structural optimization and lightweight design solution for a certain electric vehicle battery pack and compared it with the pre-optimization data.

Stress-relief annealing friendly Fe-based metallic glass-reinforced Al-Si-Mg-Zr alloy fabricated by laser powder bed fusion

ABSTRACT To enhance the thermal stability of L-PBF Al-Si-Mg alloy, Fe-based metallic glass powder was mixed with Al-Si-Mg-Zr powder to fabricate a novel alloy via L-PBF. The study systematically investigated the effects of process parameters on processability and the impact of stress-relief annealing (300 ∘ C–2h) on the microstructure and mechanical properties. Results showed that metallic glass addition improved surface quality. Plastic deformation was analysed using finite element and molecular dynamics methods. The as-built alloy had a cellular substructure with Al 3 Zr, Si, AlFeMgSi, and AlFeSi phases. Si nanoparticles and stacking faults were observed in α-Al cells. The as-built samples had yield strength (YS) of 249±4 MPa, ultimate tensile strength (UTS) of 434±6 MPa, and 6.3±0.3% elongation. Post-annealing, the alloy retained high strength and plasticity with YS of 255±3 MPa, UTS of 449±15 MPa, and 7.2±0.3% elongation, outperforming similar L-PBF Al-Si and Al-Si-Mg alloys.

Utilization finite element and machine learning methods to investigation the axial compressive behavior of elliptical FRP-confined concrete columns

Nowadays, elliptical sections are among the geometric shapes that are becoming more and more popular in the architecture world. Finite element analysis (FEA) software, such as ABAQUS, is used to simulate elliptical concrete columns confined with fiber reinforced polymer (FRP) jackets based on experimental data published in the literature. The validity of the finite element modeling (FEM) approach was confirmed by comparing it to experimental data obtained from 45 specimens in existing studies, demonstrating a favorable agreement. Additionally, a parametric study was also carried out, which included simulating 40 more specimens, resulting in a total of 85 specimens for analysis. The effect of various test variables such as sectional aspect ratio, the amount of FRP layers, and concrete strength on the elliptical column’s behavior is investigated. The axial compressive strength is predicted using current models from the literature. The outcomes revealed that the higher unconfined concrete strength decreased FRP confinement efficacy. Insufficient confinement with post-peak softening is more likely in FRP-confined high strength concrete columns, especially if the confinement was not stiff enough. Also, by adding more FRP layers, the stress and strain capabilities of elliptical concrete columns confined with FRP composites are improved. Physical experiments require a significant investment of time and resources, but they can produce valuable results. Finite element analysis, on the other hand, depends heavily on the modeler's competence and can minimize the number of experiments required by employing computer simulation, but it also requires high computer settings. In recent years, there has been a major advancement in the usage of machine learning (ML) algorithms in the applications of FRP composites with different concrete components. Taking this into consideration, this investigation is extended to estimate the compressive strength of elliptical FRP-confined columns using four tree-based ML algorithms including Decision Tree, Random Forest, Gradient Boosting and XGBoost. The XGBoost model achieved the highest predictive accuracy with the R2 values of 0.95 for both the compressive strength and axial strain targets.

A novel dimension reduction model based on POD and two-grid Crank–Nicolson mixed finite element methods for 3D nonlinear elastodynamic sine–Gordon problem

Earthquake, petroleum and gas extraction, and underground nuclear test all could cause nonlinear tectonic deformation. It is of great significance to effectively predict and evaluate the disasters and impacts of these geological structure deformation. In this end, a three-dimension (3D) nonlinear elastodynamic sine–Gordon model that can be used to describe nonlinear tectonic deformation including 3D displacement vector, 3 × 3 symmetric stress tensor matrix, nonlinear sin term, and singular initial value functions is first proposed. Then, a new time semi-discrete mixed Crank–Nicolson (TSDMCN) scheme for the 3D nonlinear elastodynamic sine–Gordon model is developed, and the existence, stability, and error estimates of the TSDMCN solutions are proved. Next, a new two-grid mixed finite element Crank–Nicolson (TGMFECN) method with unconditional stability for the 3D nonlinear elastodynamic sine–Gordon model is developed, and the existence, stability, and error estimates of the TGMFECN solutions are proved. Thenceforth, it is the most important thing is that a novel reduced-dimensional iterative TGMFECN (RDITGMFECN) method in matrix form is established by resorting to proper orthogonal decomposition only to lower the unknown TGMFECN solution coefficient vectors and keep TGMFECN basis functions unchanged, which can ensure that the RDITGMFECN method has the same accuracy as the usual TGMFECN method, but can greatly lower the dimension of the unknown TGMFECN solution coefficient vectors so as to mitigate calculated workload, save CPU operating-time, improve computing efficiency, and improve real-time calculating accuracy. In theory, the existence, stability, and error estimates of RDITGMFECN solutions are demonstrated by matrix analysis such that the theoretical analysis becomes very intuitive and easy to be understood by the public, which is a new attempt of theoretical analysis. In application, two numerical examples are used to simulate the 3D nonlinear tectonic deformation caused by earthquake and to verify the correctness of our theoretical results and the effectiveness of the RDITGMFECN method.