Order Finite Element Research Articles

Finite element analysis of a generalized Robin boundary value problem in curved domains based on the extension approach

Abstract A theoretical analysis of the finite element method for a generalized Robin boundary value problem, which involves a second-order differential operator on the boundary, is presented. If $\varOmega $ is a general smooth domain with a curved boundary, we need to introduce an approximate domain $\varOmega _{h}$ and to address issues owing to the domain perturbation $\varOmega \neq \varOmega _{h}$. In contrast to the lift approach used in existing studies, we employ the extension approach, which need not assume that boundary nodes of $\partial \varOmega _{h}$ lie exactly on $\partial \varOmega $. Assuming that approximate domains and function spaces are given by isoparametric finite elements of order $k$, we prove the optimal rate of convergence in the $H^{1}$- and $L^{2}$-norms. A numerical example is given for the piecewise linear case $k = 1$.

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.

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.

Counting conjugacy classes of elements of finite order in exceptional Lie groups

Counting conjugacy classes of elements of finite order in exceptional Lie groups

A Priori error estimates of Runge–Kutta discontinuous Galerkin schemes to smooth solutions of fractional

We give a priori error estimates of second order in time fully explicit Runge–Kutta discontinuous Galerkin schemes using upwind fluxes to smooth solutions of scalar fractional conservation laws in one space dimension. Under the time step restrictions τ ≤ ch for piecewise linear and τ ≲ h4/3 for higher order finite elements, we prove a convergence rate for the energy norm ‖⋅‖Lt∞Lx2+|⋅|Lx2Hxλ/2 that is optimal for solutions and flux functions that are smooth enough. Our proof relies on a novel upwind projection of the exact solution.

Eventually fixed points of endomorphisms of virtually free groups

ABSTRACT We consider the subgroup of points of finite orbit through the action of an endomorphism of a finitely generated virtually free group, with particular emphasis on the subgroup of eventually fixed points, $\text{EvFix}(\varphi)$: points whose orbit contains a fixed point. We provide an algorithm to compute the subgroup of fixed points of an endomorphism of a finitely generated virtually free group and prove that finite orbits have cardinality bounded by a computable constant, which allows us to solve several algorithmic problems: deciding if φ is a finite order element of $\text{End}(G)$, if φ is aperiodic, if $\text{EvFix}(\varphi)$ is finitely generated and if $\text{EvFix}(\varphi)$ is a normal subgroup. In the cases where $\text{EvFix}(\varphi)$ is finitely generated, we also present a bound for its rank and an algorithm to compute a generating set.

Effect of the incorporation of plastic waste on the mechanical properties of composite materials

Through this scientific research, we have tried to study the effect of the incorporation of plastic waste on the mechanical properties of concrete, in order to obtain good concrete with high resistance at a lower cost. To carry out this work, we adopted the following steps: Knowledge of the properties of concrete, which contains plastic waste of High Density Polyethylene (HDPE). Thus, the natural aggregate was replaced by concrete formulated with plastic waste in partial substitution varying between 0%, 10% and 20%. In all the concrete mixtures, the components, water, cement, gravel 3/8 and 8/15 and sand 0/3, remained constant while the HDPE waste varied according to the substitution rate. Through this process, the mechanical properties of concrete in fresh and hardened states were determined. The analysis of the results of the study allowed us to know the incorporation of plastic waste and its effect on the behaviour of the manufactured concrete. The results showed that the density of concrete made from plastic waste is lighter than the reference concrete without waste (C0), which contains only natural aggregates, in a fresh state. In the case of hardening, the results showed a decrease in the compressive strength of composite concrete produced from plastic waste compared to (C0) the reference concrete. In the study's second part, a numerical model for precision and effectiveness is constructed using the finite element (FE) method. Furthermore, manufacturing experiments are scheduled to use computer simulations that account for labour, materials, tests, and time. The nonlinear stress-strain relationship for time-dependent concrete deformations and tension cracks presents a challenge for concrete modelling. ANSYS software is used to use three-dimensional nonlinear finite elements in order to determine this kind of complex mechanical behavior.

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.

Transient topology optimization for efficient design of actively cooled microvascular materials

Microvascular materials containing internal microchannels are able to achieve multi-functionality by flowing different fluids through vasculature. Active cooling is one application to protect structural components and devices from thermal overload, which is critical to modern technology including electric vehicle battery packaging and solar panels on space probes. Creating thermally efficient vascular network designs requires state-of-the-art computational tools. Prior optimization schemes have only considered steady-state cooling, rendering a knowledge gap for time-varying heat transfer behavior. In this study, a transient topology optimization framework is presented to maximize the active-cooling performance and mitigate computational cost. Here, we optimize the channel layout so that coolant flowing within the vascular network can remove heat quickly and also provide a lower steady-state temperature. An objective function for this new transient formulation is proposed that minimizes the area beneath the average temperature versus time curve to simultaneously reduce the temperature and cooling time. The thermal response of the system is obtained through a transient Geometric Reduced Order Finite Element Model (GRO-FEM). The model is verified via a conjugate heat transfer simulation in commercial software and validated by an active-cooling experiment conducted on a 3D-printed microvascular metal. A transient sensitivity analysis is derived to provide the optimizer with analytical gradients of the objective function for further computational efficiency. Example problems are solved demonstrating the method’s ability to enhance cooling performance along with a comparison of transient versus steady-state optimization results. In this comparison, both the steady-state and transient frameworks delivered different designs with similar performance characteristics for the problems considered in this study. This latest computational framework provides a new thermal regulation toolbox for microvascular material designers.

An effective higher order finite element computation method for analyzing the eddy current losses in induction motors using subparametric transformations

An enhanced sub-parametric finite element method is presented for analyzing the eddy current losses in induction motors using higher order sub-parametric transformations. The purpose of this study is to construct a very efficient, simple, and exact higher order sub-parametric approach employing a 2D automated mesh generator implemented in JuliaFEM. Alternating current motors are employed in a wide range of industrial and household applications on a frequent basis such as in fans, water heaters, ovens, pumps and many more. High efficiency of the motors is one of the most desirable parameters of these motors. Eddy current losses are predominant in these motors which are responsible for a decline in the efficiency. Consequently, the use of eddy current analysis is vital in evaluating the behaviour of electric devices under various loads. This paper proposes a novel finite element method approach to minimize the 2D eddy current phenomenon by taking the diffusion equation into account. Eddy currents are the primary source of torque in various types of equipment, such as induction motors. In other cases, it is important to analyse the eddy currents to estimate the losses, reactance, and resistance.

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.

Statically Condensed Iterated Penalty Method for High Order Finite Element Discretizations of Incompressible Flow

Statically Condensed Iterated Penalty Method for High Order Finite Element Discretizations of Incompressible Flow

Uniform preconditioners for high order finite element approximations of planar linear elasticity

A new overlapping Additive Schwarz preconditioner is developed for high order finite element approximation of planar linear elastic problems on triangular meshes. The new preconditioner results in a condition number that is bounded independently of the degree p p , the mesh-size h h and the ratio λ / μ \lambda /\mu . The resulting condition number is reduced to roughly 6.0 6.0 for all values of the parameters and discretization parameters on standard test problems. Crucially, the overall cost of the new preconditioner is comparable to the cost of applying standard domain decomposition based preconditioners.

Retraction: Mixed convective heat transfer in a power-law fluid in a square enclosure: higher order finite element solutions

Retraction: Mixed convective heat transfer in a power-law fluid in a square enclosure: higher order finite element solutions

New procedure for soil-structure interaction problems incorporating free-field effects by 2D quadratic elements

Following several decades of tragic seismic events, current national regulations emphasise the importance of considering seismic soil-structure interaction (SSSI) in the design of foundations, particularly for deep foundations in loose soils. However, incorporating the free field into the SSSI calculation model can be challenging when high order finite or infinite elements are involved.This paper presents a new procedure for considering the effect of the free field in a seismic soil-structure interaction (SSSI) model when using 8-node finite elements bounded by compatible absorbing infinite elements.The proposal, based on a new formulation of three-node column elements paired with a modification of the Cundall method, allows the calculation of the free field effects to be applied at the level of the lateral boundaries of the SSSI model, represented here by the finite-infinite element interfaces. This modification of the dissipative system, considering the kinematic effect at each time step, allows an incremental solution of the SSSI problem, taking into account the nonlinearities of the materials constituting the geotechnical profile. The test validation of the method, performed on two cases of shear wave propagation problems in soft soils, shows good results, demonstrating the usefulness of the 2D approach for SSSI problems.

Algebraic concordance order of almost classical knots

Torsion in the concordance group [Formula: see text] of knots in [Formula: see text] can be studied with the algebraic concordance group [Formula: see text]. Here, [Formula: see text] is a field of characteristic [Formula: see text]. The group [Formula: see text] was defined by Levine, who also obtained an algebraic classification when [Formula: see text]. While the concordance group [Formula: see text] is abelian, it embeds into the non-abelian virtual knot concordance group [Formula: see text]. It is unknown if [Formula: see text] admits non-classical finite torsion. Here, we define the virtual algebraic concordance group [Formula: see text] for Seifert surfaces of almost classical knots. This is an analogue of [Formula: see text] for homologically trivial knots in thickened surfaces [Formula: see text], where [Formula: see text] is closed and oriented. The main result is an algebraic classification of [Formula: see text]. A consequence of the classification is that [Formula: see text] embeds into [Formula: see text] and [Formula: see text] contains many nontrivial finite-order elements that are not algebraically concordant to any classical Seifert matrix. For [Formula: see text], we give a generalization of the Arf invariant.

Investigations on adapted interpolation orders for a mixed isogeometric plate formulation

AbstractIn order to overcome locking effects that especially occur for lower order finite element formulations, different methods can be employed. This can be conducted using mixed formulations or adapted approximation orders, for instance. Hence, in order to tackle shear locking that is caused by non‐matching interpolation degrees in the shear strain equation, an irreducible and a mixed Reissner‐Mindlin plate formulation with accordingly adapted conforming discretizations are derived within the scope of this contribution. In addition, non‐uniform rational B‐splines (NURBS) are employed therefore, in order to benefit from the properties and refinement strategies offered by isogeometric analysis (IGA) and to achieve more accurate results. The effect of various combinations of interpolation orders on the convergence behavior and the ability to alleviate locking is investigated for both the irreducible and the mixed isogeometric plate formulation and examined for a benchmark example. This is also supplemented by investigations on the stability of the considered variants, tested by the existence of the correct number of zero‐energy modes.

Using Deep Neural Networks for Detecting Spurious Oscillations in Discontinuous Galerkin Solutions of Convection-Dominated Convection–Diffusion Equations

Standard discontinuous Galerkin finite element solutions to convection-dominated convection–diffusion equations usually possess sharp layers but also exhibit large spurious oscillations. Slope limiters are known as a post-processing technique to reduce these unphysical values. This paper studies the application of deep neural networks for detecting mesh cells on which slope limiters should be applied. The networks are trained with data obtained from simulations of a standard benchmark problem with linear finite elements. It is investigated how they perform when applied to discrete solutions obtained with higher order finite elements and to solutions for a different benchmark problem.