Finite Element Method Results Research Articles

Investigation of ambient vibration sources for direct energy harvesting by optimizing resonant frequency using proof mass

Ambient mechanical sources typically vibrate below the frequency of 200 Hz, posing challenges for thin film piezoelectric sensors, including low power, high resonant frequency, and small bandwidth. To optimize the electrical energy harvesting from the ambient sources, it is crucial to reduce the resonant frequency of the energy harvester to match that of the ambient sources. In this study, the energy harvester’s resonant frequency dependency on proof mass is thoroughly investigated using the finite element method (FEM). Further, the FEM results are experimentally validated through a custom-designed vibration set-up. Different ambient vibration energy sources, their vibrating frequencies, and accelerations are examined to harness direct mechanical energy and convert it into electric energy using the piezoelectric sensor. Further, the effective proof mass and position are determined to achieve the targeted frequency obtained from ambient sources. Consequently, the harvester is utilized for direct energy harvesting from the ambient sources. The addition of proof mass can lower the resonant frequency of the harvester from 160 Hz to 40 Hz allowing the harvester to vibrate at maximum amplitude to obtain maximum output voltage. Significant enhancement of output power is observed after the tuning of harvester resonant frequency, harvesting a maximum output power of 19.29 μW when mechanically sourced from the bike mirror, measured at an acceleration of 4.50 g at 43 Hz.

Simplified thermal model for open type 1.5 kW synchronous reluctance motor using thermal equivalent circuit and finite element method

In this study, two thermal models, the thermal equivalent circuit (TEC) and finite element method (FEM), were used for the transient thermal analysis of an open-type synchronous reluctance motor. Both the transient TEC method and the FEM were carried out for longer than 3 h after the motor was driven to reach a steady state. The results were compared with experimental results to verify the feasibility of the two thermal models. When the experimental results were compared with the TEC method results, the average temperature difference of all the parts was within 9 %. When the experimental and FEM results were compared, the average temperature difference of all the parts was within 7 %. Therefore, both thermal models are suitable for predicting the motor temperature.

Three-dimensional theoretical model for effectively describing the effect of craniomaxillofacial structural factors on loading situation in the temporomandibular joint

BackgroundTemporomandibular joint (TMJ) overloading is considered a primary cause of temporomandibular joint disorders (TMD). Accordingly, craniomaxillofacial structural parameters affect the loading situation in the TMJ. However, no effective method exists for quantitatively measuring the loading variation in human TMJs. Clinical statistics, which draws from general rules from large amounts of clinical data, cannot entry for exploring the underlying biomechanical mechanism in craniomaxillofacial system. The finite element method (FEM) is an effective tool for analyze the stress and load on TMJs for several cases in a short period of time; however, it is difficult to generalize general patterns through calculations between different cases due to the different geometric characteristics and occlusal contacts between each case. Methods(1) This study included 88 subjects with 176 unilateral data to measure angle (α) of the distance to the plane of occlusion. The bone destruction score was evaluated for clinical statistics. To rule out effects of the potential factors and ensure the generality of the study, one participant with no obvious bone destruction was selected as the standard case for establishing the three-dimensional (3D) theoretical model and FEM. (2) Three groups of forces, including biting, muscles and joint reaction forces on mandible, were adopted to establish a 3D theoretical model. (3) By modifying the sagittal α and coronal three types of deviation angle (φ) of the original model, nine candidate models were obtained for the FEM studies. Results(1) The static equilibrium equations, were used to establish a 3D theoretical model for describing the loading of the TMJ. The theoretical model was validated by monotonously modifying the structural parameter in comparison to two-dimensional theoretical models reported previously; (2) The force on the TMJ gradually decreased with α, and this trend was validated by both clinic statistics and FEM results; (3) The effects of the three types of deviation angle were different. The results of the case where only rotating biting forces were considered was consistent with clinical statistics, indicating that the side with lower α experiences higher TMJ load. (4) Changing the unilateral proportionality coefficients of biting and muscle force produced opposite effects, wherein the effects of the muscle force were stronger than those of the biting forces. ConclusionsA negative correlation was observed between the joint load and α. Among the three types of asymmetric deformities, occlusal deviations were the primary factors leading to TMD. Unilateral occlusion can result in a greater load on the ipsilateral joint and should be avoided when using the side corresponding to the TMD. This study provides a theoretical basis for the biomechanical mechanism of TMD and also enables the targeted mitigation and treatment of TMD through structural modification.

Physics-informed neural networks based on source term decoupled and its application in discharge plasma simulation

In recent years, the artificial intelligence computing paradigm represented by physics-informed neural networks (PINNs) has received great attention in the field of plasma numerical simulation. However, the plasma chemical system considered in related research is relatively simplified, and the research on solving the more complex multi-particle low-temperature fluid model based on PINNs is still blank. In more complex chemical systems, the coupling relationship between particle densities and between particle densities and mean electron energy become more intricate. Therefore, the applicability of PINNs in dealing with sophisticated reaction systems needs further exploring and improving. In this work, we propose a general PINN framework (source term decoupled PINNs, Std-PINNs) for solving multi-particle low-temperature plasma fluid model. By introducing equivalent positive ions and replacing each particle transport equation with the current continuity equation as a physical constraint, Std-PINN splits the entire solution process into the training processes of two neural networks, realizing the decoupling of the source term of the heavy particle transport equation from the electron density and mean electron energy, which greatly reduces the complexity of neural network training. In this work, the application of Std-PINNs to solving multi-particle low-temperature plasma fluid models is demonstrated through two classic discharge cases with different complexity of reaction systems (low-pressure argon glow discharge and atmospheric-pressure helium glow discharge) and the performance of Std-PINN is compared with that of conventional PINN and finite element method (FEM). The results show that the training results output from the traditional PINN are completely incorrect due to the strong coupling correlation of each physical variable through the source terms of each particle transport equation, while the <i>L</i><sub>2</sub> relative error between Std-PINN and FEM results can reach up to ~10<sup>–2</sup> , thus verifying the feasibility of Std-PINN in simulating multi-particle plasma fluid model. Std-PINN expands the application of deep learning method to modeling complex physical systems and provides new ideas for conducting low-temperature plasma simulations. In addition, this study provides novel insights into the field of artificial intelligence scientific computing: the mathematical form that describes the state of a physical system is not unique. By introducing equivalent physical variables, equations suitable for neural network solutions can be derived and combined with observable data to simplify problems.

Efficient numerical modelling of magnetophoresis in millifluidic systems.

Continuous flow magnetophoresis represents a common technique for actively separating particles within a fluid. For separation systems design, accurately predicting particle behaviour helps to characterise system performance, typically measured by the separation efficiency (SE). While finite element method (FEM) simulations offer high accuracy, they demand extensive computational resources. Alternatively, results can be achieved more quickly with simplified numerical models that use analytical descriptions of fluid flow, magnetic fields, and particle movement. In this research, we model a millifluidic system that separates magnetic particles using magnetophoresis. Therefore, we (1) develop a simple numerical model that can simulate continuous flow magnetophoresis for rectangular channels in two and three dimensions, (2) introduce a novel and simple approach to calculate the SE, and (3) quantify the effects of model assumptions in flow profile and dimensions on SE. Our method for estimating SE considers particle flux variation across the channel's cross-section due to the flow profile. The results are compared to an FEM model developed in COMSOL. The obtained three-dimensional simulation model computes results in seconds, around 180 times faster than the FEM approach, while deviating less than 2% from the FEM results. A comparison of the different two-dimensional and three-dimensional models underscores the significant influence of the flow profile and the SE calculation method on the result. The two dimensional models generally overestimate the SE of up to 15% due to their lower peak flow velocity. However, using a constant flow velocity leads to good agreement for high SE due to the overlap of differences in flow profile and SE calculation.

Experimental and numerical analysis of uni-axial buckling of single-phase functionally graded porous polymeric sandwich plates

The porosity gradient functionally graded material (PFGM) is one of the most popular types of FGM, in which the porosity in the material is made to change in the specified direction. This study looks into the buckling problems of rectangular sandwich plates made of single-phase porous functionally graded materials (PFGMs), commonly used in aircraft structures and biomedical applications. A compression test was performed on the 3D-printed polymeric FG specimens bonded with two thin solid face sheets on the upper and lower surfaces. The critical stress of well-designed and fabricated 3D printed FGM plate samples with various metal core types is determined using a PC installed on universal testing equipment (UTM). The effect of different essential parameters (such as porous ratio, gradient exponent, and aspect ratio) on buckling load and total deformation were explored.The finite element method (FEM) was used to run a numerical simulation on elastic buckling using ANSYS 2021 R1 software to validate the experimental results. The load-displacement relationships and deformed morphologies were investigated using experiments and numerical analysis. The topology arrangement and relative density of the polymer core were examined using the SEM micro-tomography test based on porosity distribution to check the resistance of the sandwich to buckling load. PETG/Al sandwich plates have been found to have critical buckling loads that are 2.52 % higher than PLA/Al sandwich plates, while TPU/Al sandwich plates show increased essential loads of buckling of 5.139 %. The FEM and experiment results show that the existence of porosity in the PLA core in the PFGM plate can reduce the buckling strength tremendously, about 10.52% and 6.8 %, respectively. It was evident that the numerical results show a good agreement with the experimental findings, with a maximum discrepancy of no more than 12 % occurring at the (TPU/Al) sandwich plate with a porosity of 30%.

Heat transfer comparison investigation of the permanent magnet synchronous motor for electric vehicles based on the BEM and the FEM

In the field of heat transfer in permanent magnet synchronous motors (PMSM) for electric vehicles, the boundary element method (BEM) has been applied for the first time to calculate the steady-state temperature of the PMSM with a spiral water-cooled system. In this investigation, the boundary-integration equation for the steady-state heat transfer problem of a water-cooled PMSM is first derived on the basis of thermodynamic theory, and the system of constant coefficient differential equations is obtained by discretizing its boundaries, while the temperature results obtained from the BEM are compared with the finite element method (FEM) results. Furthermore, the temperature distribution and heat transfer characteristics obtained from the FEM and BEM were verified twice using the PMSM prototype and test platform. The results show that the maximum relative error between the temperature calculation results of FEM and BEM is 1.97%, and the maximum relative error between the results of BEM and the test does not exceed 3%, which finally verifies the validity and accuracy of BEM in solving the heat transfer problems of water-cooled PMSM.

Measurement of the roller tilt angle in a double-row tapered roller bearing with strain gauges

Roller tilting can cause severe roller-edge loadings, leading to serious local heat generation or early failure of the rolling element bearing. Therefore, it is significant to obtain the roller tilt angle in real service for evaluating the bearing performance. A measurement method was proposed to determine the roller tilt angle by detecting the variation of the strain amplitude difference between two axial positions of the bearing outer surface with a notched housing. An experimental system was developed to investigate the roller tilt angles in a double-row tapered roller bearing and demonstrate the effectiveness of the proposed measurement method by conducting a comparison with the finite element method (FEM) simulation results. The experimental roller tilt angles agree well with the FEM results. Compared with other measurement methods, the proposed method offers a potential approach to achieve the in-situ and non-destructive measurement of the roller tilt angles for long-term monitoring.

Optimization and realization of a space limited sens-PEH for smart floor applications

IoT and smart home applications demand for simultaneous sensing and energy independence. Here, a screen printable P(VDF-TrFE) cantilever based smart floor design, capable of sensing (sens) and acting as a piezoelectric energy harvester (PEH) is introduced. In a height of only 1 mm, which is well-suited for indoor floor applications, a sens-PEH cantilever optimization is performed by finite element method (FEM) simulations and experimentally validated; bending cantilevers perform much better compared to normal compressions modes with apparent sensitivities to be as high as 98 nC/N for 3.6 mm long cantilevers. This is more than three orders of magnitude larger compared to conventional flat P(VDF-TrFE) film sensors. When energy harvesting is of interest, the simulations show that short cantilevers provide the most energy. Using multilayered screen-printed devices and an active area of 8 cm2, such harvesters provide up to 18.4 µJ of step energy in the experiment, from which 6 µJ is storable in a capacitor. From the combined FEM and experimental results, the optimum sens-PEH geometry, either for sensing or harvesting, can be chosen and implemented in respective smart floor applications.

Vacuum insulated glazing (VIG) units under wind load—part 1: global deformation and stresses on the outer glass surfaces

This paper presents the first part of a study on the effect of wind loads on Vacuum Insulated Glass (VIG) units. The study provides background information on VIG and relevant Standards, explains the numerical modelling process, and discusses the implications of the results in relation to European and North American codes and Standards. The focus of the study is on vertical windows and façade installations in low-rise buildings (< 25 m) commonly found in residential buildings. The mechanical behaviour of VIG was analysed using the Finite Element Method (FEM) with respect to various design parameters such as glass pane size, glass thickness, surface pressure magnitude, and edge boundary conditions. The study analysed global deformation as well as the stresses on the outer glass surfaces. The VIG features such as pillar geometry and contact dynamics, and material non-linear effects, were explicitly modelled. In addition, monolithic glass plates were also modelled, and the FEM results of the monolithic cases were in reasonable agreement with an analytical solution obtained from linear thin plate theory. These results highlight the limit of linear behaviour in monolithic plate bending. The centre-of-pane deflection of the VIG was in good agreement with the FEM and analytical solutions of the equivalent thickness monolithic pane, for sample sizes below 800 × 800 mm. However, for larger glass sizes, a deviation was found, and the VIG exhibited a higher plate stiffness than the equivalent thickness monolithic pane. The simulations also showed that the stresses in the glass panes are highly dependent on the edge of glass boundary condition. Finally, the results demonstrated that with appropriate design choices, the VIG can satisfy the Standards requirements for wind load and glass design in both Europe and North America.

Acoustic emission characteristics of waste fiber‐reinforced recycled high‐density polyethylene composites under damage conditions

AbstractAcoustic emission (AE) technology provides a novel approach for studying the damage and fracture behavior of wood‐plastic composites (WPCs) during their application process. This method allows rapid and precise analysis of the mechanical properties of materials, which is a significant advancement for the emerging WPC industry. Hence, a blend of waste floor sanding powder (FSP) and rice husk (RH) reinforced recycled high‐density polyethylene (Re‐HDPE) was prepared by the extrusion method. The AE technique, scanning electron microscopy, and finite element method (FEM) were used to analyze the damage and fracture behavior of the composite materials under a three‐point bending test. The results indicated that the bearing capacity of RH/Re‐HDPE composites was slightly higher than that of FSP/Re‐HDPE composites. The AE technique can better illustrate the damage evolution and fracture process of composites and determine the damage degree and mode. Matrix cracking and deformation were more pronounced in the FSP/Re‐HDPE composites. Fiber breakage and delamination were more prominent in the RH/Re‐HDPE composites. In addition, repeated loading caused irreversible damage to the composite. The increase in temperature led to the attenuation of the AE signal and a decrease in matrix strength. However, when the temperature reached a certain point, the cumulative AE ringing count increased again because the higher temperature improved the fiber–matrix interface strength, which partially canceled out the negative effect. In addition, Abaqus was used for simulation analysis, and the running results coincided with the experimental value.Highlights RH/Re‐HDPE and FSP/Re‐HDPE are sustainability and low‐cost composites. AE can determine the degree and mode of damage of WPC at different temperatures. The AE curves of RH/Re‐HDPE and FSP/Re‐HDPE all pass through four stages. Heating reduced the matrix strength and improved the interface strength of WPC. The results of FEM were consistent with the experimental results.

A finite-volume implementation of the phase-field model for brittle fracture with adaptive mesh refinement

The phase-field method (PFM) has advantages in modeling crack propagation in rock-like materials by treating the crack surface as a continuous function, therefore avoiding dealing with the sharp discontinuous displacement field. This study presents an implementation of PFM using the finite volume method (FVM) to discretize the governing equations for the conservation of linear momentum and the evolution of phase field. The coupled equations are solved by an iterative staggered scheme. The adaptive mesh refinement (AMR) technique is further employed to improve computational efficiency, which is relatively easy to achieve in FVM as the method can naturally handle unstructured meshes with hanging nodes in both two- and three-dimensional problems, as opposed to the finite element method (FEM). Several classical crack propagation problems, including the single-edge notched tension and shear tests, the L-shaped panel test, and the test on a notched plate with a hole, are simulated and compared with available experimental and FEM results, which demonstrate that the FVM-based PFM can model crack propagation accurately and effectively and could be more efficient than the FEM-based PFM.

Vibration Characteristics and Vibration Mitigation Analysis of End Cap PM Excitation Homopolar Inductor Machine

Vibration analysis and vibration mitigation are very important to ensure the safe and stable operation of electrical machines applied in flywheel energy storage systems (FESS). In traditional integer slot synchronous machines, it is generally believed that there is only the zeroth-mode vibration caused by odd tooth harmonics. However, the homopolar inductor machine (HIM) adopts the end cap excitation structure, which makes its air-gap magnetic field contain the DC component, odd tooth harmonics and even tooth harmonics. The force density distribution on the slotted stator is analyzed by the magnetomotive force-multiplying permeance method. The results show that there are both zeroth- and pth(pole pairs)-mode vibrations caused by tooth harmonics and DC component of the air-gap flux density. Furthermore, to reduce vibration and resonance, and to improve the electromagnetic performance of HIM, it is best to use the stator skew slot, the magnetic slot wedge (MSW) and the sinusoidal rotor slot at the same time. Finally, a 54-slot/6-pole EPE-HIM is tested to validate the analytical and the finite-element method (FEM) results, and the variation trend of the experimental results and FEM results is the same.

Modeling of Current-Driven End-Effect Force Ripple in Air-Cored Linear Synchronous Motor With Multiple Modular Stators

This paper presents a segmented layer model (SLM) for the air-cored linear permanent-magnet synchronous motors (LPMSMs) with multiple modular stators. In the LPMSM with modular stators, the significant current-driven ripples (CDRs) are observed as the PM mover moves across the stator modules, due to the magnetic end effects. Such CDRs become more severe as the current excitation level increases. In order to correctly model the overall motor forces in the LPMSM, the analytic motor model needs to be able to capture the magnetic end effects from both the stator and mover, and also include the current-driven effects. The SLM method presented in this paper divides the analytical region into several segmented layers to be able to accurately capture both the geometric-saliency- and current-driven force ripples by using layers with different fundamental spatial frequencies. The fidelity of the proposed SLM is validated both by the finite element method (FEM) and the experiments. The SLM achieves an accuracy of more than 97 % and 87 % compared to the FEM and experimental results, respectively, while reducing the computation time by two orders of magnitude as compared to the FEM.

Redesign of Split-Hopkinson Tensile bar to eliminate spurious wave and verification of strain signal using FEM

To get the reliable dynamic material properties, the optimization of measuring apparatus such as SHTB (Split-Hopkinson Tensile Bar) is essential. SHTB apparatus is a machine that can acquire the material properties of a specimen for given high-speed situation in [Formula: see text] strain rate region. In this study, a new model improved the waveform is proposed by redesigning and optimizing the initial SHTB equipment. Complete reflected pulse was not obtained, and a spurious wave was measured in original model. To obtain reliable material properties, the spurious wave must be removed or avoided. Factors affecting spurious wave were determined as length of striker bar, length of incident bar, and diameter of striker bar &amp; flange, and verified using LS-DYNA, one of the FEM (Finite-Element Method) software. The modified model was redesigned and manufactured based on FEM results. In FEM results, the incident and reflected pulse were completely improved. Also, overlapping with spurious wave was avoided in redesigned experiment results. Redesigned model was confirmed the residual wave after the incident pulse by a pneumatic launcher and various mechanical components. However, the residual wave didn’t affect the reflected pulse and the waveform was improved than the original model.

Analytic Calculation of External Magnetic Field of High Voltage Toroid-Type Air-Core Reactors

Compared with high voltage column-type air-core reactors (HVCAR), high voltage toroid-type air-core reactors (HVTAR) generate much smaller external magnetic flux density (EMD) during operation, which significantly reduces magnetic clearance (MC), thereby reducing space occupied by air-core reactors. For HVTAR, existing methods such as finite element method (FEM) and numerical integration methods using the Biot-Savart law, cannot reveal the influence of the parameters of reactors on EMD. This paper proposes an analytic formula for EMD, in view of the special winding structure of HVTAR. It can directly reveal the influence of the working current, the number of coil units, n, and the size of the reactor on EMD. The analysis shows that the EMD decays with ρn+1 P, which is much faster than the attenuation rate of ρ3 Pof column-type reactors. The analytic formula is verified by comparison with the results of FEM. The analytic formula is simple and practical to calculate the EMD and MC quickly and guide the electromagnetic design of HVTAR directly.

FEM Investigation of a Multi-Neck Helmholtz Resonator

An increasingly significant area of research with several applications in numerous disciplines is that of multi-neck Helmholtz resonators. This research is set to explore the accuracy and applicability of the finite element method (FEM) for the calculation of the resonance frequency of multi-neck Helmholtz resonators. The FEM is employed for the estimation of the resonance frequency in various cases of multi-neck Helmholtz resonators: with cylindrical or spherical bodies, with unflanged or flanged necks of various dimensions and with various combinations of the above. Also, single neck resonators are examined. The FEM results are compared with the results of a recently proposed theoretical model available in the literature and with the outcome of the lumped element approximation (multi-neck) accounting for the added neck surface area. Comparisons revealed little deviation between the FEM and theoretical model (less than 1.1% error of calculation for every case). On the contrary, in comparison with the lumped element approximation (multi-neck), the error of calculation is significant (up to 40.3% for the cases examined). The FEM will prove useful in expanding our understanding of how multi-neck Helmholtz resonators perform under various conditions and configurations. The present research, which highlights the applicability of the FEM for the calculations of the resonance frequency of multi-neck Helmholtz resonators, goes a step further; this approach can be applied in special cases where it is not trivial to apply an analytical formula. The method can be used for applications of multi-neck Helmholtz resonators for various fields such as acoustic metamaterials, musical acoustics and noise mitigation.

A Study of Combined Graphical Acoustic Computing and the Depth Peeling Technique on Acoustic Backscattering of Multiple-Layered Structures

An efficient graphical acoustic computing (GRACO) method is introduced. Referring to the rendering of semi-transparent objects, the depth peeling (DP) technique is compounded with the GRACO method, forming a combined GRACO and DP method (GRACO–DP) to consider the backscattering of multiple-layered structures in which the contributions of inner structures are included. After that, some examples of GRACO and GRACO–DP are tested to evaluate the accuracy and efficiency of such methods. Through the examples of impedance sphere and benchmark models, GRACO can acquire results with higher efficiency and good consistency compared to the traditional KA method. Also, the TS of a pair of circular plates is determined by GRACO, GRACO–DP, and the finite element method (FEM). The results show that GRACO–DP can fit better with the FEM results. Moreover, the TS of a ribbed double shell is predicted by both GRACO and GRACO–DP; the features of a Bragg wave scattered by the periodically arranged inner ribs can be spotted from the result of GRACO–DP.

Application of Macro Element Method (MEM) for faster automotive crash safety design during concept stage

The non-linear nature of vehicle crash safety simulations using the Finite Element Method (FEM) usually requires a significant investment in computational time and resources, in addition to the requirement of full CAD (Computer Aided Design) data availability, which is challenging to obtain during the concept stage. The current work explores the usage of Macro Element Method (MEM) based on simplified Super Beam Element (SBE), for which the shape functions are developed based on experimental observations rather than algebraic equations. MEM crash analysis run time doesn’t exceed several seconds and is one of the underutilised analytical models which can be used to quickly explore new design directions during the early concept stage of the vehicle and has higher potential to speed up the product development process. In this paper, initially, a calibration study was done to validate the MEM at a component level to compare the results with FEM and experimental values. FEM and MEM simulation models were developed and evaluated for a metallic thin-walled square frusta with varying semi-apical angles subjected to axial compression. The results were compared, and it was found that MEM results show more than 97% correlation compared to FEM and experimental results. Subsequently, a design modification was introduced in the component and the results show close degree of correlation between MEM and FEM. Similar process was followed for full vehicle level simulations, such as full width frontal crash, side pole crash analyses to understand the robustness of the method in applying it to product development environment.

Assessment of the ultimate strength of stiffened panels of ships considering uncertainties in geometrical aspects: Finite element approach and simplified formula

The development of reliability-based assessments has improved the safety design of ship structures by considering the stochastic behaviour of structural components such as stiffened panels. However, previous studies generally considered the variation in the initial geometric imperfection of a stiffened panel as a single deterministic value. To enhance the analysis, in this study, a series of ultimate strength analyses of stiffened panels were carried out by varying and combining the stiffened panel parameters, including variations in the initial geometric imperfection (2.5%, 25%, 50%, 75%, and 100% of the maximum value), modes of the imperfection (column, local, and torsional imperfection modes), slenderness ratios (plate and web slenderness), and span-to-bay ratios (3 and 6). A total of 750 stiffened panel configurations were investigated through a nonlinear finite element method (FEM) analysis using ANSYS APDL incorporated with a code in MATLAB to model the imperfections based on idealised models. The obtained ultimate strength values were then formulated using a nonlinear correlation between the variables in the regression method to yield a new formula for predicting the ultimate strength of the stiffened panel. The results showed that combining the varied parameter values significantly affected the ultimate strength of the stiffened panel and collapse modes. The newly derived formula provided good accuracy with a standard error of only 2.08% when compared with the FEM results of stiffened panels having a significant disparity with the configuration of the idealised models. This confirms the validity of the formula as a reference for evaluating the ultimate strength value with the influence of an initial geometric imperfection.