Accuracy Of Finite Element Method Research Articles

The essentials of the finite element method to solve differential equations: an illustrative case in physics

Here we review the basic steps of the modern and accurate finite element method in its application to solve ordinary differential equations in physics. To exemplify this well-in-vogue numerical technique, we have chosen the second order Poisson-Boltzmann equation, which is a classic equation of colloid science. Aiming to formulate a viable, but didactic, implementation of the finite element technique, we have combined a linear basis of functions, the Galerkin weighted residuals method, the Swartz-Wendroff approximation and the Picard iteration algorithm. In summary, the finite element method transforms a differential equation into a simpler system of algebraic equations for the coefficients of the approximate solution in terms of a set of basis functions. We describe the full computational realisation of the finite element procedure and, also, we examine the corresponding Poisson-Boltzmann numerical predictions for various representative conditions.

Vibration response characteristics of Jujube trees based on finite element method and structure-from-motion

It is an indispensable step to choose optimal vibration parameters in jujube fruits vibration harvesting. Excessive vibration frequency will cause jujube tree damage, otherwise, the separation of fruit and fruit stalk cannot be achieved, resulting in harvesting failure. In our work, finite element method and vibration tests are applied to find optimal vibration parameters. To improve the accuracy of finite element method, a low-cost 3D reconstruction method is proposed to build jujube tree model. Experimental result showed that the phenotypic information of the reconstructed jujube tree and the real jujube tree have a high similarity, and their determination coefficient R2 reach to 0.984. Moreover, the natural frequencies of jujube tree are analyzed by modal analysis, transient analysis is applied to analyze vibration transfer characteristics. The simulation and experimental results show that the optimal vibration frequency range of jujube tree is 25∼30 Hz, the optimal vibration excitation location is on the main trunk and as high as possible close to the secondary branches, which can obtain maximum vibration response for optimum harvesting efficiency. The average determination coefficient is 0.66 between test and simulation results at different frequencies. This study can provide references for accurate 3D modeling of jujube trees, selection of appropriate vibration parameter range and the development of jujube harvesting machinery.

Research on modified lumped parameter method in torsional vibration analysis of reciprocating compressor shaft

Abnormal vibrations often occur during the operation of reciprocating compressors under different load conditions. To efficiently and accurately calculate the torsional vibration response of the shafting under various working conditions, a modified lumped parameter method (MLPM) based on finite element mode modification is proposed. MLPM is derived from the lumped parameter method (LPM) and incorporates corrections to the stiffness matrix based on the multi-degree-of-freedom modal calculation using a finite element model. This allows MLPM to combine the high accuracy of finite element method (FEM) with the fast computing speed of LPM. To demonstrate the effectiveness of MLPM, an abnormal vibration compressor equipped with a stepless capacity control system is considered. MLPM is utilized to calculate the torsional natural frequency (TNF) and torsional torque response of the compressor’s shafting. The computed 1st-order TNF error between MLPM and FEM is only 0.20%. In the time domain, the torque response error of MLPM is less than 3.07%, while in the frequency domain, it is less than 0.57%. Moreover, MLPM exhibits a significantly increased calculation speed of more than 95% compared to FEM. By comparing MLPM results with torsional vibration test data, the 1st-order TNF error is found to be less than 0.50%, and the frequency-domain torque response error is no more than 8.91%. Furthermore, MLPM is effectively applied to solve the abnormal vibration problem of the compressor. It proposes a modification plan for the shafting, which successfully avoids torsional resonance and reduces the peak value of the torque response by 54.77%. These results demonstrate that MLPM is highly suitable for practical engineering applications due to its combination of high accuracy and fast computing speed.

A new crack-tip element for the logarithmic stress-singularity of Mode-III cracks in spring interfaces

AbstractA new crack-tip finite element able to improve the accuracy of Finite Element Method (FEM) solutions for cracks growing along the Winkler-type spring interfaces between linear elastic adherents is proposed. The spring model for interface fracture, sometimes called Linear-Elastic (perfectly) Brittle Interface Model (LEBIM), can be used, e.g., to analyse fracture of adhesive joints with a thin adhesive layer. Recently an analytical expression for the asymptotic elastic solution with logarithmic stress-singularity at the interface crack tip considering spring-like interface behaviour under fracture Mode III was deduced by some of the authors. Based on this asymptotic solution, a special 5-node triangular crack-tip finite element is developed. The generated special singular shape functions reproduce the radial behaviour of the first main term and shadow terms of the asymptotic solution. This special element implemented in a FEM code written in Matlab has successfully passed various patch tests with spring boundary conditions. The new element allows to model cracks in spring interfaces without the need of using excessively refined FEM meshes, which is one of the current disadvantages in the use of LEBIM when stiff spring interfaces are considered. Numerical tests carried out by h-refinement of uniform meshes show that the new singular element consistently provides significantly more accurate results than the standard finite elements, especially for stiff interfaces, which could be relevant for practical applications minimizing computational costs. The new element can also be used to solve other problems with logarithmic stress-singularities.

How to assess the accuracy of volume conduction models? A validation study with stereotactic EEG data.

Volume conduction models of the human head are used in various neuroscience fields, such as for source reconstruction in EEG and MEG, and for modeling the effects of brain stimulation. Numerous studies have quantified the accuracy and sensitivity of volume conduction models by analyzing the effects of the geometrical and electrical features of the head model, the sensor model, the source model, and the numerical method. Most studies are based on simulations as it is hard to obtain sufficiently detailed measurements to compare to models. The recording of stereotactic EEG during electric stimulation mapping provides an opportunity for such empirical validation. In the study presented here, we used the potential distribution of volume-conducted artifacts that are due to cortical stimulation to evaluate the accuracy of finite element method (FEM) volume conduction models. We adopted a widely used strategy for numerical comparison, i.e., we fixed the geometrical description of the head model and the mathematical method to perform simulations, and we gradually altered the head models, by increasing the level of detail of the conductivity profile. We compared the simulated potentials at different levels of refinement with the measured potentials in three epilepsy patients. Our results show that increasing the level of detail of the volume conduction head model only marginally improves the accuracy of the simulated potentials when compared to in-vivo sEEG measurements. The mismatch between measured and simulated potentials is, throughout all patients and models, maximally 40 microvolts (i.e., 10% relative error) in 80% of the stimulation-recording combination pairs and it is modulated by the distance between recording and stimulating electrodes. Our study suggests that commonly used strategies used to validate volume conduction models based solely on simulations might give an overly optimistic idea about volume conduction model accuracy. We recommend more empirical validations to be performed to identify those factors in volume conduction models that have the highest impact on the accuracy of simulated potentials. We share the dataset to allow researchers to further investigate the mismatch between measurements and FEM models and to contribute to improving volume conduction models.

Rapid patient-specific FEM meshes from 3D smart-phone based scans

Objective. The objective of this study was to describe and evaluate a smart-phone based method to rapidly generate subject-specific finite element method (FEM) meshes. More accurate FEM meshes should lead to more accurate thoracic electrical impedance tomography (EIT) images. Approach. The method was evaluated on an iPhone® that utilized an app called Heges, to obtain 3D scans (colored, surface triangulations), a custom belt, and custom open-source software developed to produce the subject-specific meshes. The approach was quantitatively validated via mannequin and volunteer tests using an infrared tracker as the gold standard, and qualitatively assessed in a series of tidal-breathing EIT images recorded from 9 subjects. Main results. The subject-specific meshes can be generated in as little as 6.3 min, which requires on average 3.4 min of user interaction. The mannequin tests yielded high levels of precision and accuracy at 3.2 ± 0.4 mm and 4.0 ± 0.3 mm root mean square error (RMSE), respectively. Errors on volunteers were only slightly larger (5.2 ± 2.1 mm RMSE precision and 7.7 ± 2.9 mm RMSE accuracy), illustrating the practical RMSE of the method. Significance. Easy-to-generate, subject-specific meshes could be utilized in the thoracic EIT community, potentially reducing geometric-based artifacts and improving the clinical utility of EIT.

A high-order residual-based viscosity finite element method for incompressible variable density flow

In this paper, we introduce a high-order accurate finite element method for incompressible variable density flow. The method uses high-order Taylor-Hood velocity-pressure elements in space and backward differentiation formula (BDF) time stepping in time. This way of discretization leads to two main issues: (i) a saddle point system that needs to be solved at each time step; (ii) a stability issue when the viscosity of the flow goes to zero or if the density profile has a discontinuity. We address the first issue by using Schur complement preconditioning and artificial compressibility approaches. We observed similar performance between these two approaches. To address the second issue, we introduce a modified artificial Guermond-Popov viscous flux where the viscosity coefficients are constructed using a newly developed residual-based shock-capturing method. Numerical validations confirm high-order accuracy for smooth problems and accurately resolved discontinuities for problems in 2D and 3D with varying density ratios.

Grounded source transient electromagnetic 3D forward modeling with the spectral-element method and its application in hydraulic fracturing monitoring

A long wire with large current source transient electromagnetic (TEM) monitoring, with a large detection depth, low cost, safety, and environmental protection, has unique advantages in the testing and identification of unconventional reservoir fluid and the evaluation of stimulated reservoir volume. So, the TEM 3D forward modeling method has become a research hotspot. Although the finite-element method (FEM) is a type of numerical algorithm that has been widely applied in three-dimensional (3D) electromagnetic field forward modeling, the efficiency and accuracy of FEM require further improvement in order to meet the demand of fast 3D inversion. By increasing the order of the basis function and adjusting the principle of mesh discretization, the precision of the mixed-order spectral-element (SEM) result will be increased. The backward Euler scheme is an unconditionally stable technique which can ignore the impact of the scale of the time step. To achieve a better description of the nonlinear electromagnetic (EM) response of the grounded source TEM method and to optimize the efficiency and accuracy/precision of the 3D TEM forward modeling method significantly, we proposed the use of 3D TEM forward modeling based on the mixed-order SEM and the backward Euler scheme, which can obtain more accurate EM results with fewer degrees of freedom. To check its accuracy and efficiency, the 1D and 3D layered models are applied to compare the SEM results with the semi-analytical and FEM solutions. In addition, we analyzed the accuracy and efficiency of the SEM method for different types of order basis functions. Finally, we calculated the long-wire source TEM response for a practical 3D earth model of a shale gas reservoir for fracturing monitoring and tested the feasibility of the TEM method in a hydraulic fracturing monitoring area to further demonstrate the flexibility of the SEM method.

An accurate and robust Eulerian finite element method for partial differential equations on evolving surfaces

An accurate and robust Eulerian finite element method for partial differential equations on evolving surfaces

Natural Convection Couette Flow in the Presence of Magnetic Field and Thermal Property

This article examines the natural convection Couette flow in a hydrodynamic viscous fluid that is electrically conductive due to the thermal radiation effect. The governing flow in this study is modelled in the form of partial differential equations (PDEs) in dimensional form with initial and boundary conditions and the Couette fluid model is also be used to characterize the fluid behavior. Then, using suitable non-dimensional quantities, the governing non-linear PDEs are transformed. Since the flow governing equations of the problem under study are extremely complex and complicated, techniques that complement experimental and theoretical fluid dynamics by providing alternative potentially cheaper means of testing fluid flow systems is employed. Therefore, the Finite Element Method (FEM) is used after discretization of the PDEs. With Graphs and tables, the effects of embedded thermo physical parameters of engineering interests associated with the flow quantities viz. velocity, temperature, concentration of the fluid were examined through series of numerical experiments and discussed. This research also analyzes and compares the results obtained by Omokhuale and Jabaka (2022b). It is interesting to report that an excellent agreement was established, thereby authenticating and validating the accuracy of FEM as a strong tool. According to the results of this study, the actions of thermal radiation on the thermal and momentum boundary layers for increasing values are significant, also, increasing the magnetic field parameter impedes the fluid movement due to the Lorentz force action.

Diffuse interface model for cell interaction and aggregation with Lennard-Jones type potential

This study introduces a phase-field model designed to simulate the interaction and aggregation of multicellular systems under flow conditions within a bounded spatial domain. The model incorporates a multi-dimensional Lennard-Jones potential to account for short-range repulsion and adhesive bonding between cells. To solve the governing equations while preserving energy law, a second-order accurate C0 finite element method is employed. The validity of the model is established through numerical tests, and experimental data from cell stretch tests is utilized for model calibration and validation. Additionally, the study investigates the impact of varying adhesion properties in red blood cells. Overall, this work presents a thermodynamically consistent and computationally efficient framework for simulating cell–cell and cell–wall interactions under flow conditions.

Experimental and numerical study on the high-velocity hail impact performance of carbon fiber aluminum alloy laminates

The experiment and finite element method (FEM) simulation of high-velocity hail impact on carbon fiber reinforced aluminum alloy laminates (CARALL) were carried out in this paper. CARALL were prepared via an autoclave process after surface anodizing. The properties of the specimens were verified by mechanical properties testing. The light gas gun was employed to conduct the hail impact tests with different impact velocities, impact angles and ply sequences. The surface damage, displacement of the impact center and deformation depth in different test schemes were compared. The FEM model of the high-velocity impact of hail on CARALL was established using ABAQUS/Explicit. The strain-rate dependent model was adopted for hail while the Johnson-Cook damage constitutive model was adopted for the aluminum alloy layer. The 3D Hashin criterion damage model based on fracture energy was applied for the CFRP layer via the VUMAT subroutine. The cohesive element was inserted between the laminates to investigate the interlaminar damage. The accuracy of FEM was verified by comparing with the results obtained by tests. Combined with the energy absorption and internal damage mechanism, the effects of different parameters on the hail impact resistance of CARALL were studied.

A symmetrized parametric finite element method for simulating solid-state dewetting problems

We propose an accurate, area/mass-conservative and energy-dissipative parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting with arbitrary anisotropic surface energy. The model is governed by the anisotropic surface diffusion equation with suitable boundary conditions at the contact points, and could be used to describe the kinetic evolution of the film/vapor interface with contact line migration. The sharp-interface continuum model is firstly reformulated into a conservative form by introducing a symmetric positive definite matrix Zk(n) that depends on the Cahn–Hoffman ξ-vector and a stabilizing function k(n). A new symmetrized variational formulation with arbitrary anisotropic surface energy is built and then its area/mass conservation and energy dissipation properties are proved. By using piecewise linear elements in space, we propose the spatial semi-discretized scheme, and using backward Euler in time with an appropriate approximation of the unit normal vector, we further derive the fully discretized parametric finite element approximation. We show, theoretically, that the semi-discretized and fully discretized schemes are both area/mass-conservative and energy-dissipative. Finally, numerical experiments are reported to show the accuracy and efficiency of the numerical method as well as the anisotropic effects on the morphological evolution processes for solid-state dewetting of thin films.

Asymmetrical cutting-edge design of broaching tool based on FEM simulation

Finite element method (FEM) based model of broaching Inconel 718 was established to optimize an asymmetrical cutting-edge by the commercial software AdvantEdge in this paper. The asymmetrical cutting-edge form by two circles was proposed. The power-law (P-L) material constitutive model, including strain hardening, strain rate hardening and thermal softening was developed by split Hopkinson pressure bar (SHPB) test and high-temperature Vickers hardness test. The parameters of thermal properties were measured by laser flash diffusivity apparatus. The friction model between TiN (The outer layer coating material of the tool) and Inconel 718 was obtained by ball on disc friction test. The simulated and experimental results have been compared comprehensively in the items of chip formation and tool wear width. The results showed that the simulated results were consistent with the experimental results. The average errors of chip curl radius, chip thickness, wear width of rake face and flank face were 7.68%, 3.56%, 13.64% and 12.24%, respectively, which proved the accuracy of the simulation model. Then the optimal asymmetrical cutting-edge of broach were obtained through orthogonal test based on the accurate FEM model. The rise per tooth of the optimal broach tool is 0.04 mm/tooth, the rake angle is 13°, the circle radius R1 is 0.13 mm, and the circle radius R2 is 0.02 mm.

Rapid estimation of cortical neuron activation thresholds by transcranial magnetic stimulation using convolutional neural networks

BackgroundTranscranial magnetic stimulation (TMS) can modulate neural activity by evoking action potentials in cortical neurons. TMS neural activation can be predicted by coupling subject-specific head models of the TMS-induced electric field (E-field) to populations of biophysically realistic neuron models; however, the significant computational cost associated with these models limits their utility and eventual translation to clinically relevant applications. ObjectiveTo develop computationally efficient estimators of the activation thresholds of multi-compartmental cortical neuron models in response to TMS-induced E-field distributions. MethodsMulti-scale models combining anatomically accurate finite element method (FEM) simulations of the TMS E-field with layer-specific representations of cortical neurons were used to generate a large dataset of activation thresholds. 3D convolutional neural networks (CNNs) were trained on these data to predict thresholds of model neurons given their local E-field distribution. The CNN estimator was compared to an approach using the uniform E-field approximation to estimate thresholds in the non-uniform TMS-induced E-field. ResultsThe 3D CNNs estimated thresholds with mean absolute percent error (MAPE) on the test dataset below 2.5% and strong correlation between the CNN predicted and actual thresholds for all cell types (R2 > 0.96). The CNNs estimated thresholds with a 2–4 orders of magnitude reduction in the computational cost of the multi-compartmental neuron models. The CNNs were also trained to predict the median threshold of populations of neurons, speeding up computation further. Conclusion3D CNNs can estimate rapidly and accurately the TMS activation thresholds of biophysically realistic neuron models using sparse samples of the local E-field, enabling simulating responses of large neuron populations or parameter space exploration on a personal computer.

An energy-stable finite element method for nonlinear Maxwell's equations

In this paper, we consider the three-dimensional (3D) nonlinear Maxwell's equations in the optical media characterized by linear Lorentz dispersion, nonlinear Kerr effect and delayed Raman scattering. Using Crank-Nicolson time discretization and special treatment of the nonlinearity, we propose a second order accurate (in time) finite element method for the nonlinear problem. Fully discrete energy stability analysis is established to show that the scheme is unconditionally stable. The nonlinear system is solved by Newton-Krylov method. In order to efficiently solve the linearized algebraic system in each Newton sub-iteration, we further develop an effective preconditioner by using suitable approximations to the nonlinearity and the auxiliary space preconditioning technique. Numerical experiments are provided to examine the accuracy, the energy stability, the parallel scalability and the robustness in preconditioning of our proposed scheme. We then apply the scheme to simulate the spatial soliton propagation in 3D nonlinear glass. We also demonstrate the performance of the scheme in handling complex geometry through simulating the wave scattering by a spherical hole in the nonlinear optical glass.

Automatic quality control of the numerical accuracy of EEG lead fields

Precise individualized EEG source localization is predicated on having accurate subject-specific Lead Fields (LFs) obtained from their Magnetic Resonance Images (MRI). LF calculation is a complex process involving several error-prone steps that start with obtaining a realistic head model from the MRI and finalizing with computationally expensive solvers such as the Boundary Element Method (BEM) or Finite Element Method (FEM). Current Big-Data applications require the calculation of batches of hundreds or thousands of LFs. LF. Quality Control is conventionally checked subjectively by experts, a procedure not feasible in practice for larger batches. To facilitate this step, we introduce the Lead Field Automatic-Quality Control Index (LF-AQI) that flags LF with potential errors. We base our LF-AQI on the assumption that LFs obtained from simpler head models, i.e., the homogeneous head model LF (HHM-LF) or spherical head model LF (SHM-LF), deviate only moderately from a "good" realistic test LF. Since these simpler LFs are easier to compute and check for errors, they may serve as "reference LF" to detect anomalous realistic test LF. We investigated this assumption by comparing correlation-based channel ρmin(ref,test)and source τmin(ref,test) similarity indices (SI) between "gold standards," i.e., very accurate FEM and BEM LFs, and the proposed references (HHM-LF and SHM-LF). Surprisingly we found that the most uncomplicated possible reference, HHM-LF had high SI values with the gold standards—leading us to explore further use of the channel ρmin(HHM−LF,test)and source τmin(HHM−LF,test)SI as a basis for our LF-AQI. Indeed, these SI successfully detected five simulated scenarios of LFs artifacts. This result encouraged us to evaluate the SI on a large dataset and thus define our LF-AQI. We thus computed the SI of 1251 LFs obtained from the Child Mind Institute (CMI) MRI dataset. When ρmin(HHM−LF,test)and source τmin(HHM−LF,test) were plotted for all test subjects on a 2D space, most were tightly clustered around the median of a high similarity centroid (HSC), except for a smaller proportion of outliers. We define the LF-AQI for a given LF as the log Euclidean distance between its SI and the HSC median. To automatically detect outliers, the threshold is at the 90th percentile of the CMI LF-AQIs (-0.9755). LF-AQI greater than this threshold flag individual LF to be checked. The robustness of this LF-AQI screening was checked by repeated out-of-sample validation. Strikingly, minor corrections in re-processing the flagged cases eliminated their status as outliers. Furthermore, the "doubtful" labels assigned by LF-AQI were validated by neuroscience students using a Likert scale questionnaire designed to manually check the LF's quality. Item Response Theory (IRT) analysis was applied to the questionnaire results to compute an optimized model and a latent variable θ for that model. A linear mixed model (LMM) between the θ and LF-AQI resulted in an effect with a Cohen's d value of 1.3 and a p-value <0.001, thus validating the correspondence of LF-AQI with the visual quality control. We provide an open-source pipeline to implement both LF calculation and its quality control to allow further evaluation of our index.

Multistrategy Cooperation Particle Swarm Optimization for FEM Model Update of the Continuous Warren Truss Steel Railway Bridge

It is very difficult to obtain an accurate finite element method (FEM) model to further analyze structural mechanical properties. Therefore, as the main means of establishing accurate models, the model update has become a research hotspot in the dominion of bridge engineering. Particle swarm optimization (PSO) has the characteristics of being easy to implement, but it is easy to fall into the local optimum. Therefore, multistrategy cooperation particle swarm optimization (MCPSO) that balances exploration and exploitation of particle swarm is proposed. This algorithm achieves the goal of balancing exploration and exploitation by adopting different combinations of particle swarm velocity update strategies in different iteration stages. The application effects of MCPSO in the FEM model update of the continuous Warren truss steel railway bridge are compared and analyzed, and the results show that the algorithm proposed in this paper outperforms the standard PSO (SPSO) algorithm. This paper provides a more effective algorithm for bridge model updates.

The effect of mesh density and element type on the analysis of 3D objects

This paper focuses on the usage of mesh density and element type in the analysis of 3D objects, which are crucial components of 3D object Finite Element Method (FEM) analysis. The document is based on theoretical principles and will be supported by practical numerical examples using the FEM software Abaqus. In order to conduct an accurate FEM analysis, it is vital to employ an appropriate mesh that is easily convertible for calculations and free from imperfections. The suitability of the mesh depends on two key parameters: density, which is determined by the material being used, and element type, such as tetrahedrons, hexahedrons, polyhedrons, and others. Different types of elements and mesh densities in FEM software often involve multiple settings that primarily impact the accuracy of the model, as well as the calculation time and convergence quality. This article is divided into three sections: the first part focuses on the theoretical foundations, the second part presents a practical example, and the third part offers conclusions and comparisons.

Time-efficient surrogate models of thermal modeling in laser powder bed fusion

Two time-efficient surrogate models are proposed to emulate the nonlinear heat equation in the context of laser powder bed fusion, the performance of which is compared in accuracy and online execution time. Fast-computed numerical solvers are critical in implementing the digital twin framework in the additive manufacturing process addressing one of its main open problems: lack of quality assurance. The first surrogate model is the reduced Gaussian process emulator. It is a data-driven model equipped with a nonlinear dimension reduction scheme and manages to predict temperature profiles almost instantly (around 0.036s on average) with an accuracy of 95% for 99.38% of tests. Another surrogate model is the sketched emulator with local projection. It projects the accurate but high-dimensional finite element method solution on a low-dimensional basis and then bypasses the majority of costly computations for the temperature-dependent matrices in the projected model by randomized sketching. It has higher accuracy (97.78% of tests with relative errors below 1%) while spending comparably more time online (around 42.23s on average). Although compared with the finite element model both surrogates promote time efficiency with some minor controlled compromise in accuracy, the reduced Gaussian process emulator enables real-time implementation while the sketched emulator with local projection offers comparably higher levels of accuracy. A series of numerical experiments are carried out, which assumes a three-layer printing process with a fixed laser beam trajectory using a small number of printing control parameters as inputs, namely the laser power, scan speed, and time coordinates. Both surrogates are also principally feasible in other thermal-driven additive manufacturing to obtain better quality assurance with techniques like uncertainty management and closed-loop control.