Finite Element Method In Order Research Articles

Comparison of different elastic strain definitions for largely deformed SEI of chemo-mechanically coupled silicon battery particles

Amorphous silicon is a highly promising anode material for next-generation lithium-ion batteries. Large volume changes of the silicon particle have a critical effect on the surrounding solid-electrolyte interphase (SEI) due to repeated fracture and healing during cycling. Based on a thermodynamically consistent chemo-elasto-plastic continuum model we investigate the stress development inside the particle and the SEI. Using the example of a particle with SEI, we apply a higher order finite element method together with a variable-step, variable-order time integration scheme on a nonlinear system of partial differential equations. Starting from a single silicon particle setting, the surrounding SEI is added in a first step with the typically used elastic Green–St-Venant (GSV) strain definition for a purely elastic deformation. For this type of deformation, the definition of the elastic strain is crucial to get reasonable simulation results. In case of the elastic GSV strain, the simulation aborts. We overcome the simulation failure by using the definition of the logarithmic Hencky strain. However, the particle remains unaffected by the elastic strain definitions in the particle domain. Compared to GSV, plastic deformation with the Hencky strain is straightforward to take into account. For the plastic SEI deformation, a rate-independent and a rate-dependent plastic deformation are newly introduced and numerically compared for three half cycles for the example of a radial symmetric particle.

2D Magnetotelluric Inversion Using Linear Finite Element Methods and a Discretize‐Last Strategy With First and Second–Order Anisotropic Regularization

AbstractWe present a new inversion scheme for 2D magnetotelluric data. In contrast to established approaches, it is based on a mesh‐free formulation of the Quasi‐Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS) iteration which uses the cost function gradient to implicitly construct approximations of the Hessian inverse to update the unknown conductivity. We introduce conventional first–order regularization as well as second–order regularization where inversions based on the latter are more appropriate for sparse data and can be read as maximum likelihood estimation of the unknown conductivity. We apply first–order finite element method (FEM) discretizations of the inversion scheme, forward and adjoint problems, where the latter is required for the construction of the cost function gradients. We allow for unstructured first–order triangular meshes supporting an enhanced ground level resolution including topographical features and coarsening at the far field leading to significant reduction in computational costs from using structured mesh. Formulating the inversion iteration in continuous form prior to discretization eliminates bias due to local refinements in the mesh and gives way for computationally efficient sparse matrix techniques in the implementation. A keystone in the new scheme is the multi‐grid approximation of the Hessian of the regularizations to construct efficient preconditioning for the inversion iteration. The method is applied to the Commeni4 benchmark and two field data sets. Tests show that for both first and second–order regularization an anisotropic approach is important to address the vast differences in horizontal and vertical spatial scale which in conventional approaches is implicitly introduced through the elongated shape of grid cells.

Derivation of Equivalent Material Coefficients of 2-2 Piezoelectric Single Crystal Composite.

Piezoelectric composites, which consist of piezoelectric materials and polymers, are widely employed in various applications such as underwater sonar transducers and medical diagnostic ultrasonic transducers. Acoustic transducers based on piezoelectric composites can have high sensitivity with broad bandwidth. In recent studies, it is demonstrated that 2-2 composites based on single crystals provide further increased sensitivity and wide bandwidth. In order to utilize a 2-2 composite in acoustic sensors, it is required to demonstrate the full material coefficients of the 2-2 composite. In this study, we investigated an analytic solution for determining equivalent material coefficients of a 2-2 composite. Impedance spectrums of the single-phase resonators with equivalent material coefficients and 2-2 composite resonators were compared by the finite element method in order to verify the analytic solutions. Furthermore, the equivalent material coefficients derived from the analytic solution were also verified by comparing the measured and the simulated impedance spectrums. The difference in resonance and anti-resonance frequencies between the measured and simulated impedance spectrums was around 0.5% and 1.2%. By utilizing the analytic solutions in this study, it is possible to accurately derive full equivalent material coefficients of a 2-2 composite, which are essential for the development of acoustic sensors.

Modeling and simulation of chemo-elasto-plastically coupled battery active particles

AbstractAs an anode material for lithium-ion batteries, amorphous silicon offers a significantly higher energy density than the graphite anodes currently used. Alloying reactions of lithium and silicon, however, induce large deformation and lead to volume changes up to 300%. We formulate a thermodynamically consistent continuum model for the chemo-elasto-plastic diffusion-deformation behavior of amorphous silicon and it’s alloy with lithium based on finite deformations. In this paper, two plasticity theories, i.e. a rate-independent theory with linear isotropic hardening and a rate-dependent one, are formulated to allow the evolution of plastic deformations and reduce occurring stresses. Using modern numerical techniques, such as higher order finite element methods as well as efficient space and time adaptive solution algorithms, the diffusion-deformation behavior resulting from both theories is compared. In order to further increase the computational efficiency, an automatic differentiation scheme is used, allowing for a significant speed up in assembling time as compared to an algorithmic linearization for the global finite element Newton scheme. Both plastic approaches lead to a more heterogeneous concentration distribution and to a change to tensile tangential Cauchy stresses at the particle surface at the end of one charging cycle. Different parameter studies show how an amplification of the plastic deformation is affected. Interestingly, an elliptical particle shows only plastic deformation at the smaller half axis. With the demonstrated efficiency of the applied methods, results after five charging cycles are also discussed and can provide indications for the performance of lithium-ion batteries in long term use.

Deep Learning-Based Simultaneous Temperature- and Curvature-Sensitive Scatterplot Recognition.

Since light propagation in a multimode fiber (MMF) exhibits visually random and complex scattering patterns due to external interference, this study numerically models temperature and curvature through the finite element method in order to understand the complex interactions between the inputs and outputs of an optical fiber under conditions of temperature and curvature interference. The systematic analysis of the fiber's refractive index and bending loss characteristics determined its critical bending radius to be 15 mm. The temperature speckle atlas is plotted to reflect varying bending radii. An optimal end-to-end residual neural network model capable of automatically extracting highly similar scattering features is proposed and validated for the purpose of identifying temperature and curvature scattering maps of MMFs. The viability of the proposed scheme is tested through numerical simulations and experiments, the results of which demonstrate the effectiveness and robustness of the optimized network model.

Simulation of coupled elasticity problem with pressure equation: hydroelastic equation

PurposeThe main aim of the current paper is to find a numerical plan for hydraulic fracturing problem with application in extracting natural gases and oil.Design/methodology/approach First, time discretization is accomplished via Crank-Nicolson and semi-implicit techniques. At the second step, a high-order finite element method using quadratic triangular elements is proposed to derive the spatial discretization. The efficiency and time consuming of both obtained schemes will be investigated. In addition to the popular uniform mesh refinement strategy, an adaptive mesh refinement strategy will be employed to reduce computational costs.FindingsNumerical results show a good agreement between the two schemes as well as the efficiency of the employed techniques to capture acceptable patterns of the model. In central single-crack mode, the experimental results demonstrate that maximal values of displacements in x- and y- directions are 0.1 and 0.08, respectively. They occur around both ends of the line and sides directly next to the line where pressure takes impact. Moreover, the pressure of injected fluid almost gained its initial value, i.e. 3,000 inside and close to the notch. Further, the results for non-central single-crack mode and bifurcated crack mode are depicted. In central single-crack mode and square computational area with a uniform mesh, computational times corresponding to the numerical schemes based on the high order finite element method for spatial discretization and Crank-Nicolson as well as semi-implicit techniques for temporal discretizations are 207.19s and 97.47s, respectively, with 2,048 elements, final time T = 0.2 and time step size τ = 0.01. Also, the simulations effectively illustrate a further decrease in computational time when the method is equipped with an adaptive mesh refinement strategy. The computational cost is reduced to 4.23s when the governed model is solved with the numerical scheme based on the adaptive high order finite element method and semi-implicit technique for spatial and temporal discretizations, respectively. Similarly, in other samples, the reduction of computational cost has been shown.Originality/valueThis is the first time that the high-order finite element method is employed to solve the model investigated in the current paper.

Analysis and simulation of the (2 + 1)-dimensional Fisher’s reaction-diffusion equation with higher order finite element method

In this article, a finite element analysis of Fisher’s reaction-diffusion equation is performed, by using quadratic shape elements. The higher-order elements in the finite element method (FEM) are known to yield better results in the approximation of solutions. However, such elements are less explored in the literature. Therefore, quadratic shape functions are taken into consideration for the finite element analysis of Fisher’s reaction-diffusion equation. A priori error estimates for the semi-discrete solution are derived by applying the Galerkin approximation. For the fully discrete solution, a priori error estimates are derived using the Crank-Nicolson method and the predictor-corrector scheme. Also, the stability of the time-discrete scheme is analyzed by the energy technique. Third-order convergence for space discretization in L 2 ( Ω ) norm and second-order convergence in H 1 ( Ω ) norm are observed. Furthermore, second-order convergence for time discretization is also found. Finally, we validate our theoretical results by considering a variety of numerical examples in one and two-dimensional spaces, which also illustrate the reliability of the developed finite element algorithm. Through comparison tables, 3D plots, and convergence plots, the results of the current study are compared with the results from the existing literature.

Electrodynamic Forces in Main Three-Phase Busbar System of Low-Voltage Switchgear—FEA Simulation

This paper concerns the effects of electrodynamic forces that act on current paths that are part of high-grade industrial distribution switchgear. This work is composed of experimental and simulation sections. In the experimental section, the short-circuit tests are presented and the occurrence of electrodynamic forces are shown in a visible way. The formation of electrodynamic forces in the current circuits of electrical energy distribution systems is related to the flow of high currents, but mostly it is related to short-circuit currents. In order to highlight these phenomena, the detailed specification of the parameters during tests is displayed. In the simulation section, the physical phenomenon of electrodynamic forces is being captured by employing a detailed real-scale model of switchgear and current paths. Therefore, the authors proposed employment of the FEM (finite element method) in order to obtain values of electrodynamic forces acting on the current paths by executing the detailed 3D coupled simulation. The analysis of the results and aftermath effects of their interactions provided interesting conclusions that concerned the operation of such power distribution layouts in critical short-circuit conditions.

A Comparative Study on the Structural Response of Multi-Linked Floating Offshore Structure between Digital Model and Physical Model Test for Digital Twin Implementation

A digital twin is a virtual model of a real-world structure (such as a device or equipment) which supports various problems or operations that occur throughout the life cycle of the structure through linkage with the actual structure. Digital twins have limitations as a general simulation method because the characteristic changes (motion, stress, vibration, etc.) that occur in the actual structure must be acquired through installed sensors. Additionally, it takes a huge computing cost to output changes in the structure’s characteristics in real time. In particular, in the case of ships and offshore structures, simulation requires a lot of time and resources due to the size of the analysis model and environmental conditions where the wave load acts irregularly, so the application of a different simulation methodology from existing ones is required. The order reduction method, which accurately represents the system’s characteristics and expresses them in a smaller model, can significantly reduce analysis time and is an effective option. In this study, to analyze the applicability of the order reduction method to the development of digital twins for offshore structures, the structural responses of a multi-connected floating offshore structure were estimated by applying the order reduction method based on distortion base mode. The order reduction method based on the distortion base mode predicts the responses by constructing an order-reduced conversion matrix consisting of the selected distortion base mode, based on the mode vector’s orthogonality and autocorrelation coefficients. The predicted structural responses with the reduced order model (ROM) were compared with numerical analysis results derived using the higher order boundary element method and finite element method with in-house code owned by the Korea Research Institute of Ship & Ocean Engineering and measured responses with a model test. When compared with the numerical analysis results, the structural responses were predicted with high accuracy in the wave direction and wave frequency band of the selected distortion base mode, but there are differences due to changed characteristics of the structure when compared with the results of the model test. In addition, differences were also seen in reduced order model evaluation with different sensor locations, and it was confirmed that the more similar the extracted distortion base modes of input sensor location set is to the distortion base modes of predicted location set, the higher accuracy is in predicting the structural responses. As a result, the performance of the reduced order model is determined by the distortion base mode selection method, the locations of the sensor, and the prediction for the structural response.

I+II mixed mode fatigue crack propagation under combination of tensile loading and corrosion in a HSS specimen

Both fatigue and corrosion are two phenomena that can be very often found in metallic components. Thus, it is necessary to study their mutual effect. In this paper, a high strength steel specimen was modelled via finite element method in order to investigate the interaction of a fatigue crack and a corrosion pit. Particularly, a rectangular specimen with a corrosion pit and a nearby angled crack under remote tensile loading was modelled and the crack deflection angle was investigated via MTS and SED criteria for various crack lengths and various crack inclination angles. The results obtained clearly show how the crack behavior is affected by the presence of the corrosion pit.

Fatigue crack propagation near a corrosion pit in a HSS specimen

Corrosion and fatigue are very common forms of metal degradation, and thus an aspect of the complex issues involved is analysed within this work. A rectangular specimen with a surface fatigue crack and a surface corrosion pit subjected to remote tensile loading was modelled via the finite element method in order to assess the crack behaviour for various configurations of both the crack and the pit. A perpendicular crack was considered along with an angled crack, and the basic fracture parameters have been determined and discussed regarding the influence of the corrosion pit on further crack propagation. Additionally, selected fracture criteria have been applied in their classical one-parameter as well as generalised multi-parameter form to estimate the angle of further crack propagation. The results obtained are discussed thoroughly, and it is proved that the larger the corrosion pit, and the closer it is to the crack, the higher its effect on the fatigue crack behaviour is.

A split-explicit second order Runge–Kutta method for solving 3D hydrodynamic equations

Numerical models of marine hydrodynamics have to deal with processes exhibiting a wide range of timescales. These processes include fast external gravity waves and slower internal fully three-dimensional motions. In order to be both time-efficient and numerically stable, the time stepping scheme has to be chosen carefully to cope with the characteristic time scale of each phenomenon. An usual approach is to split the fast and slow dynamics into separate modes. The fast waves are modeled with a two-dimensional system through depth averaging while the other motions, where characteristic times are much longer, are dealt with in three dimensions. However, if the splitting is inexact, for instance in projecting the fields in a new 3D mesh, this procedure can lead to improper results regarding to the physical properties such as mass conservation and tracer consistency. In this work, a new split-explicit Runge–Kutta scheme is adapted and developed for the Discontinuous-Galerkin Finite Element method in order to obtain a new second-order time stepping, yielding to more accurate results. This method combines a three-stage low-storage Runge–Kutta for the slow processes and a two-stage low-storage for the fast ones. The 3D iterations are not affecting the surface elevation, hence an Arbitrary Lagrangian Eulerian implementation is straightforward. Water volume and tracers are conserved. A set of test cases for baroclinic flows as well as a laboratory application demonstrate the performance of the scheme. They suggest that the new scheme has little numerical diffusion.

Isogeometric multi-resolution full waveform inversion based on the finite cell method

Full waveform inversion (FWI) is an iterative identification process that serves to minimize the misfit of model-based simulated and experimentally measured wave field data. Its goal is to identify a field of parameters for a given physical object. For many years, FWI is very successful in seismic imaging to deduce velocity models of the earth or of local geophysical exploration areas. FWI has also been successfully applied in various other fields, including non-destructive testing (NDT) and biomedical imaging. The inverse optimization process of FWI relies on forward and backward solutions of the (elastic or acoustic) wave equation, as well as on efficient computations of adequate optimization directions. Many approaches employ (low order) finite element or finite difference methods, using parameterized material fields whose resolution is chosen in relation to the elements or nodes of the discretized wave field. In our previous paper (Bürchner et al., 2023), we investigated the potential of using the finite cell method (FCM) as the wave field solver. The FCM offers the advantage that highly complex geometric models can be incorporated easily. Furthermore, we demonstrated that the identification of the model’s density outperforms that of the velocity — particularly in cases where unknown voids characterized by homogeneous Neumann boundary conditions need to be detected. The paper at hand extends this previous study in the following aspects: The isogeometric finite cell analysis (IGA-FCM) – a combination of isogeometric analysis (IGA) and FCM – is applied as the wave field solver, with the advantage that the polynomial degree and subsequently also the sampling frequency of the wave field can be increased quite easily. Since the inversion efficiency strongly depends on the accuracy of the forward and backward wave field solutions and of the gradients of the functional, consistent, and lumped mass matrix discretization are compared. The resolution of the grid describing the unknown material density – thus allowing to identify voids in a physical object – is then decoupled from the knot span grid. Finally, we propose an adaptive multi-resolution algorithm that locally refines the material grid using an image processing-based refinement indicator. The developed inversion framework allows fast and memory-efficient wave simulations and object identification. While we study the general behavior of the proposed approach using 2D benchmark problems, a final 3D problem shows that it can also be used to identify void regions in geometrically complex spatial structures.

Custom Transient Finite Element Method and Ray Tracing Hybridization Strategies for Ultrasonic Testing Modelling

The transient spectral element method (SEM) is a specific high order finite element method that is particularly accurate and fast with a low memory footprint, hence enabling 3D ultrasonic testing simulations on a standard PC. However, particular care in its settings and hybridization with a semi-analytical solution remains a major challenge when seeking for both practicality and performance. A natural hybrid strategy consists of coupling a field calculated by ray tracing in the defect-free object to a SEM subdomain wherein the perturbation is enclosed, then synthesizing the signal with a reciprocity argument. The difficulty is to define the suited coupling strategy when the flaw response interacts with the back wall. In a highly heterogeneous medium, it may even become necessary to widen the numerical domain to the entire thickness of the inspected object and, therefore, to ensure the performance of the SEM through a custom discretization strategy. Here we present different customized uses of the SEM for practical ultrasonic inspection applications and discuss how they complement the ray tracing solution.

Analysis of effects of moving wall direction in a one sided semi-circular porous cavity on mixed magneto-bioconvection in the presence of hybrid nanofluid with oxytactic bacteria

A computational analysis on the effects of direction of wall movement in a one sided semi-circular porous cavity on mixed magneto-bioconvection of hybrid nanofluid in the presence of oxytactic bacteria has been performed. The working fluid is the hybrid nanofluid. The enclosure is cooled from semi-circle side and heated from moving lid under the effect of uniform magnetic field in the horizontal direction. The numerical solution of governing equations is obtained by using the higher order finite element method (FEM). The reliability of designed solver is tested through the experimental and numerical data available in the literature. Two cases of the boundary conditions are considered related to the tendency of right vertical lid of the cavity. Other parameters are Richardson number, Darcy number, Hartmann number, Peclet number and porosity. It is found that moving lid direction is the most effective parameter for the flow field, isotherms, oxygen and microorganism distributions. Moreover, higher heat transfer rate is obtained in Case-I for different values of Darcy number. Around 10% increment is occurred on mass transfer with different Pe number in case of downward moving lid due to increment of convection mode of heat transfer inside the cavity.

Improving the convergence rate of Newman’s battery model using 2nd order finite element method

While battery cycling experiments last for years, battery modelling can save time and is environment friendly, to study the degradation mechanisms of lithium-ion batteries. However, battery models are being challenged by issues of long-time calculation, poor accuracy and bad convergence. In this work, a Julia-based framework (Jubat) has been developed for Newman’s battery model. This framework benefits from the high execution efficiency of Julia language, so the mean computational time is lower than its counterpart PyBaMM written in Python. Governing equations are solved by the finite element method, and good accuracy is obtained. The convergence rate is improved by 2nd order elements, which capture better flux variation within elements than the finite volume method and the finite element method using linear elements, with slightly higher computation cost. Numerical examples show easy use of this framework and ability to solve practical problems like drive cycles. This work provides a compact, readable and fast framework for battery modelling and is useful for the rapid development of advanced battery management system. The code is open-source at https://github.com/weilongai/JuBat.

FEM simulation of thixo‐viscoplastic flow problems: Error analysis

AbstractThis note is concerned with the essential part of Finite Element Methods (FEM) approximation of error analysis for quasi‐Newtonian modelling of thixo‐viscoplastic (TVP) flow problems. The developed FEM settings for thixotropic generalized Navier‐Stokes equations is based on a constrained monotonicity and continuity for the coupled system, which is a cornerstone for an efficient monolithic Newton‐multigrid solver. The manifested coarseness in the energy inequality by means of proportional dependency of its constants on regularization, nonoptimal estimate for microstructure, and extra regularity requirement for velocity, is due to the weak coercivity of microstructure operator on one hand and the modelling approach on the other hand, which we dealt with stabilized higher order FEM. Furthermore, we show the importance of taking into consideration the thixotropy inhabited in material by presenting the numerical solutions of TVP flow problems in a 4:1 contraction configuration.

High-order, unconditionally maximum-principle preserving finite element method for the Allen–Cahn equation

In this paper, based on the mass-lumping finite element space discretization, we incorporate the integrating factor Runge–Kutta method and stabilization technique to develop a class of temporal up to the fourth-order unconditionally structure-preserving schemes for the Allen–Cahn equation and its conservative forms. The proposed methods are linear, without requiring any post-processing or limiters, and unconditionally preserve the maximum principle and mass conservation law. Several numerical experiments verify the high-order temporal accuracy of the proposed schemes, as well demonstrate the ability to preserve the maximum principle, mass conservation, and energy stability over long periods. Moreover, by the aid of numerical simulation, we show that the proposed schemes also have good performances in terms of structure-preserving with high order finite element method.

Nonlinear interactions between solitary waves and structures in a steady current

Interactions between solitary waves and structures in a steady current are studied based on a fully nonlinear wave potential theory, and a higher order finite element method (FEM) with a mesh of 8-node quadrilateral isoperimetric elements is employed to simulate the interaction in two-dimensions. Numerical examples are given by solitary waves propagating over an underwater rectangular cylinder in a steady current in a tank and solitary waves acting on single- and twin-rectangular cylinders in a steady current on free surface. Waves and hydrodynamic forces are obtained at different current speeds. It is found that the peak of diffracted wave due to the first and second reflections by an underwater cylinder clearly increase in following current. Furthermore, a packet of periodic waves with constant peak and trough are found to appear when the absolute value of the Froude number becomes large enough in single- and twin-cylinder cases. For the twin-cylinder cases, the maximum wave and horizontal force are clearly affected by the current especially in larger incident wave amplitudes and smaller spacings. In addition, the nonlinearity of wave or force becomes stronger at larger Froude numbers and larger incident wave amplitudes.

Evaluation of macroscopic yield surfaces of periodic porous microstructures employing transformation field analysis and high order finite element method

Evaluation of macroscopic yield surfaces of periodic porous microstructures employing transformation field analysis and high order finite element method