Hybrid Finite Element Method Research Articles

Modeling one-dimensional tension stiffening of plain and fiber reinforced concrete with discrete cracks

In this paper, a mechanical model is developed to simulate the behavior of 1D plain reinforced concrete (RC) and fiber reinforced concrete (FRC) members subjected to uniaxial tension, considering the development of discrete cracks. The force bridged over the crack is a function of the crack width, fracture energy, and fiber volume. The model is formulated and based on the variational principle of the virtual work, it is more general than previous models and it can utilize any loading scheme, constitutive laws or boundary conditions. It is solved numerically using a special hybrid finite element method. It is validated by comparisons with available examples and test results. Results obtained by the model demonstrate its capabilities of simulating FRC members and incorporating the contribution of the fibers in bridging residual tensile forces over the cracks. The influence of the fiber volume on the global structural behavior of FRC members through the bridging forces in the cracks is discussed.

A Hybrid Finite Element Method–Analytical Model for Classifying the Effects of Cracks on Gear Train Systems Using Artificial Neural Networks

Rotating machinery is fundamental in industry, gearboxes especially. However, failures may occur in their transmission components due to regular usage over long periods of time, even when operations are not intense. To avoid such failures, Structural Health Monitoring (SHM) techniques for damage prediction and in-advance detection can be applied. In this regard, correlations between measured signal variations and damage can be inspected using Artificial Intelligence (AI), which demands large numbers of data for training. Since obtaining signal samples of damaged components experimentally is currently unviable for complex systems due to destructive test costs, model-based numerical approaches are to be explored to solve this problem. To address this issue, this work applied an innovative hybrid Finite Element Method (FEM)–analytical approach, reducing computational effort and increasing performance with respect to traditional FEM. With this methodology, a system can be simulated with accuracy and without geometrical simplifications for healthy and damaged cases. Indeed, considering different positions and dimensions of damages (e.g., cracks) on the tooth roots of gears can offer new ways of damage investigation. As a reference to validate healthy systems and damage cases in terms of eigenfrequencies, a back-to-back test rig was used. Numerical simulations were performed for different cases, resulting in vibrational spectra for systems with no damage, with damage, and with damage of different intensities. The vibration spectra were used as data to train an Artificial Neural Network (ANN) to predict the machine state by Condition Monitoring (CM) and Fault Diagnosis (FD). For predicting the health and the intensity of damage to a system, classification and multi-class classification methods were implemented, respectively. Both sets of classification results presented good prediction agreement.

On the convergence of the primal hybrid finite element method on quadrilateral meshes

We consider the approximation of a second order elliptic equation (Darcy problem) by using the Primal Hybrid Finite Element Method on quadrilateral meshes. We present new results in terms of sufficient, and in some cases also necessary, conditions to obtain the optimal convergence rates on convex quadrilaterals obtained from bilinear mappings. Numerical experiments are performed to illustrate the theoretical results.

Thermal-Mechanical Coupling Model Based on the Hybrid Finite Element Method for Solving Bipolar the Plate Deformation of Hydrogen Fuel Cells

New energy is the focus of attention all over the world, and research into new energy can inject new vitality into the industrial system. Hydrogen fuel cells are not only environmentally friendly, but also rich in reserves that can be used as a strategic resource for the entire country. The difficulty lies in the safe design of application equipment and the batch generation and storage of hydrogen. In addition, fuel cells have the disadvantage of a slow start-up. Based on the above problems, this paper proposes a hybrid-element method to solve the thermal-mechanical coupling model of fuel cell plate, which can effectively solve the thermal stress change, temperature field distribution and displacement change of the battery plate when working. Firstly, the hybrid-element algorithm is given for 2D plate deformation. Then, the deformation application of a 3D fuel cell plate is given. The 2D numerical results show that the hybrid finite element method (FEM) is more flexible for realizing the flexible combination of sub-mesh and finite element basis functions, and has a better mesh quality compared to the traditional constant strain triangular element (CST) adaptive FEM and quadrilateral isoparametric element (Q4) adaptive FEM. This method achieves a balance between numerical accuracy and solving efficiency for the multi-porous elastic plate. In addition, a deformation control formula is given which can display the displacement deformation and stress merge to same graph, since it is convenient to quickly compare the regions where the displacement and stress extremum appear. In short, the hybrid finite element method proposed in this paper has good mesh evaluation results, and when the number of discrete elements is equivalent, the hybrid element converges faster and the solution efficiency is higher. This paper also provides a good numerical theory and simulation reference for industrial mechanics and new energy applications.

Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations

This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincaré type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Peclét numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.

Hybrid Grey Wolf and Finite Element Method (GWO-FEM) Algorithm for Enhancing High Voltage Insulator String Performance in Wet Pollution Conditions

The presence of wet pollution on the upper surface of a string insulator increases the electric field on the insulator surface, especially at the triple junction (pin-cement, cement-porcelain) as well as at the surrounding air of the insulator. The rise of the electric field leads to the ionization of the air surrounding the insulator. This phenomenon, called corona discharge, is accompanied by several consequences that are harmful to the electricity transmission network, such as electromagnetic interferences, energy losses, visible light, audible noise, and the destruction of materials by erosion. If favorable conditions are gathered, it may even cause the flashover of the insulator. Designing an optimal insulator shape that reduces this electric field value at the triple junction will be an important achievement in enhancing the performance of electrical grids. The objective of this paper is to evolve a hybrid algorithm based on the GWO-FEM for optimizing the shape and the electrical performance of a string insulator. To achieve this purpose, this work is structured in four parts. First, modeling of the insulator string geometry is conducted in Comsol-multiphysics, then FEM computation of the electric field on the polluted surface of the string insulator is completed, and the maximum electric field value at the triple junction is saved as the fitness function that will be sent to the GWO algorithm to be optimized (minimized). The third part of the work is devoted to the coding of the constrained (electrical and geometrical constraints) GWO algorithm in a Matlab interface, and finally, coupling the GWO code with the FEM code. This study is achieved in wet polluted conditions. The results are given in both 2D and 3D representations. From the obtained results we can confirm that the developed GWO-FEM hybrid algorithm for optimizing insulator strings is a very promising tool for designing and enhancing the shape and the electrical performance of insulators.

Artificial Neural Network Approach for Prediction of Leakage Current of polymeric insulator under Non-Uniform Fan-shaped Contamination

The accumulation of environmental pollution in symmetrical and asymmetrical categories conditions can lead to the aging of polymeric insulators through leakage current enhancement. In this paper, real asymmetric contamination, namely fan-shaped, is investigated accurately. However, the main approaches are that the real configuration of fan-shaped pollution is not normally uniform, and asymmetrically non-uniform fan-shaped pollution is detected. In particular, the segment angle, thickness, and electrical conductivity of the contamination layer are three major effective parameters in the insulation analysis of a fan-shaped polluted insulator. Finite element method (FEM) simulation and the proposed hybrid FEM combined with the method of moment (MOM) of a 20 kV polymeric insulator under asymmetric non-uniform fan-shaped contamination through leakage current calculations illustrate that variation in non-uniformity of related parameters of contamination layer are so crucial in the insulation situation of insulator. Also, leakage current is predicted using an artificial neural network (ANN) in processing obtained data from proposed methods, by applying effective parameters of non-uniform fan-shaped contamination. Experimental tests have been performed to validate the results and evaluate the leakage current of the fan-shaped polluted insulator. The results of the predicted values show that the difference is less than 5% compared to the actual values.

Coupled thermal-hydrodynamic-mechanical–chemical numerical simulation for gas production from hydrate-bearing sediments based on hybrid finite volume and finite element method

Gas production from hydrates induced by depressurization is a complex thermal-hydrodynamic-mechanical–chemical (THMC) coupled process. In this paper, we present a THMC coupled model to simulate the fluid flow in hydrate-bearing sediments (HBS) and the geomechanical behavior of HBS. The model is made of two subsystems, which are the fluid part of non-isothermal multi-phase flow with hydrate kinetic and solid part of geomechanical deformation. It accounts for two-way coupling effects between these two subsystems, i.e. the effect of pore pressure and hydrate dissociation on the solid mechanical behavior and the effect of stress on the hydraulic behavior. A new numerical method based on the hybrid control volume finite element method (CVFEM)-finite element method (FEM) is developed to solve the mathematical models. The local conservative CVFEM is used for the fluid part, and the standard FEM for the solid part. In the framework of hybrid CVFEM-FEM, the local conservation is reserved and the primary variables for the two subsystem are co-located. A multi-point flux approximation (MPFA) is adopted without orthogonal meshes so that it is very flexible to build complex geometrical models. The accuracy and reliability of the newly developed simulator QIMGHyd-THMC are tested by comparing with two experimental examples and a large-scale benchmark problem of other popular simulators.

Partial differential equations on hypergraphs and networks of surfaces: Derivation and hybrid discretizations

We introduce a general, analytical framework to express and to approximate partial differential equations (PDEs) numerically on graphs and networks of surfaces – generalized by the term hypergraphs. To this end, we consider PDEs on hypergraphs as singular limits of PDEs in networks of thin domains (such as fault planes, pipes, etc.), and we observe that (mixed) hybrid formulations offer useful tools to formulate such PDEs. Thus, our numerical framework is based on hybrid finite element methods (in particular, the class of hybrid discontinuous Galerkin methods).

Hybrid Finite Element Method to Thermo-Elastic Interactions in a Piezo-Thermo-Elastic Medium under a Fractional Time Derivative Model

In this work, the effect of the fractional time derivative on the piezo-thermo-elastic medium is studied, using the hybrid Laplace transform and finite element methods (LFEM). The generalized fractional piezoelectric–thermoelastic basic equations for piezo-thermo-elastic medium are formulated. The Laplace transforms are used for the time derivatives, and the finite element method is used to discretize for the space derivatives. The inversions process is performed using the Stehfest numerical technique. The finite element approach is used to obtain the solutions of complex coupled formulations of this problem. The effects of fractional time derivative and thermal relaxation time on piezoelectric–thermoelastic mediums are studied. It can be seen from the distribution that the thermal-induced displacement, the temperature and the stress of the medium increase at a high fractional parameter.

Learning constitutive relations of plasticity using neural networks and full-field data

Neural networks (NNs) have demonstrated strong capabilities of learning constitutive relations from big data. However, most NN-based constitutive models require experimental data from a considerable number of stress–strain paths that are expensive to collect. Here, we develop a hybrid finite element method - NN (FEM-NN) framework for learning the constitutive relations from full-field data. As a result, the non-uniform displacement field from a deformed sample with geometrical inhomogeneities can be used for training NNs. Such full-field data have the advantage of providing many different stress–strain paths at different locations in the sample by a single test, thereby enabling the highly efficient training of NNs. We apply FEM-NN simulations to learn the constitutive relations of several model materials characterized by rate-independent J2 plasticity. These FEM-NN studies demonstrate that the trained NNs produce the constitutive relations of plasticity with high accuracy and efficiency.

Simulation of residual stress in thick thermal barrier coating (TTBC) during thermal shock: A response surface-finite element modeling

The current study demonstrates a well-designed response surface methodology (RSM), based on the generated dataset of finite element method (FEM) to establish an integrated model for simulation of residual stress distribution in a thick thermal barrier coating (TTBC). In this study, typical TTBCs were applied on Hastelloy X Nickel-based superalloy using air plasma spray technique followed by thermal cycling. The recorded stress data of Raman spectroscopy was employed to verify the proposed FEM model. A relatively good agreement was obtained between predicted residual stresses and measured ones. Verified FEM model was used to carry out the parametric studies to evaluate the effects of such various parameters as interface amplitude, wavelength, thermally grown oxide thickness and preheating temperature on the stress distribution in the TTBC during the thermal cycling. The computed data were subsequently used for the development of RSM model. In conclusion, experimentally verified numerical data was used to construct a statistical model based on RSM and successfully used to predict the residual stress distribution field in TTBC during thermal cycling. The obtained results of hybrid FEM- RSM model were in acceptable conformity with Raman spectroscopy measurements.

Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

<p style='text-indent:20px;'>This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Peclét numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.</p>

A Flexible FEM-BEM-DDM for EM Scattering by Multiscale Anisotropic Objects

In this article, a nonconformal domain decomposition method (DDM) based on the hybrid finite element method (FEM) and boundary element method (BEM) is proposed to solve arbitrary 3-D anisotropic (including bianisotropic and single anisotropic) multiscale composite objects. The object is divided into interior and exterior regions. The interior region is solved by the bianisotropic finite element domain decomposition method (FEM-DDM), and the exterior region is solved by the discontinuous Galerkin (DG) method based on the combined field integral equation (CFIE). In order to decouple the electric and magnetic fields of the bianisotropic object, a new finite element–boundary element–domain decomposition method (FEM-BEM-DDM) formulation is derived. To ensure the continuity of electric current and electric field between anisotropic FEM subdomains, a new auxiliary current is introduced into the interface of different subdomains of FEM and the interface of FEM subdomains with BEM subdomains. The former uses second-order transmission conditions (SOTCs) to ensure continuity, whereas the latter uses Robin transmission conditions (RTCs) to ensure continuity. The BEM dense matrix–vector multiplication is accelerated by a multilevel fast multipole algorithm (MLFMA). The proposed FEM-BEM-DDM allows for nonconformal discretization between any touching subdomains and provides an effective preconditioner. In numerical examples, the accuracy and effectiveness of the method are verified.

Hydro-mechanical modelling of gas transport in clayey host rocks for nuclear waste repositories

Host rocks, as the final impediment to waste migration, play a significant role in the nuclear waste repositories. Modelling of gas migration in saturated host rocks as well as its coupled hydro-mechanical (HM) behaviors is of importance to the assessment of geological disposal facilities. A comprehensive literature review is carried out on the models for simulating gas migration in saturated rock materials. Several aspects have been provided and discussed, including the material properties and experimental interpretations, governing equations, constitutive models for hydraulic and mechanical processes, fracture propagation models. Specifically, these models are discussed in detail with respect to their performance in describing the recorded experimental behaviors. It is found that the visco-capillary two phase flow model with enriched intrinsic permeability is commonly used to describe the advective gas transport in saturated host rocks. The embedded fracture model (EFM) or enriched EFM seems to be the most favored model as it accounts for the fracturing mechanism which is more representative to implicitly simulate the preferential gas pathways. To describe the mechanical behavior of rocks during the gas migration process, linear/nonlinear elastic, elastoplastic and damage models have been included in the mechanical processes. The HM models incorporating plasticity or damage may not be applicable in most experimental studies, which are not competent to simulate the key experimental behavior associated with the development of gas preferential pathways. Current models that can explicitly describe the gas induced micro-fracturing are seldomly reported, the existed ones are not able to represent all the key experimental behaviors related to preferential gas flow. Advanced approaches, i.e., phase field (PF) method, extended finite element method (XFEM), discrete element method (DEM), hybrid finite discrete element method (FDEM) can be integrated with cohesive zone model (CZM) to provide some promising aspects in representing the development of preferential gas pathways. Lastly, the conclusions and recommendations for future modelling are given.

Modeling variable-density flow in saturated-unsaturated porous media: An advanced numerical model

Modeling variable-density flow in unconfined aquifers is a challenging task because of the nonlinear coupling between variably saturated flow and contaminant transport. This results in a highly nonlinear system since the strongly nonlinear Richards flow equation is, in addition, coupled to the advection-dispersion transport equation by viscosity and density variation. The solution of such a nonlinear system is often subject to convergence issues and can be very expansive in terms of computational time, especially for large-scale problems. Conventional numerical algorithms based on the sequential approach and the classical finite difference or finite element methods with the first-order backward Euler time integration scheme are generally inefficient and/or do not provide satisfactory results. In this work, we develop a new efficient and accurate 2D numerical model for the transport of dense contaminants in unsaturated porous media that allows for the simulation of large-scale problems. This research describes a new model that combines advanced spatial discretization methods (mixed hybrid finite element method, discontinuous Galerkin finite element method, and multipoint flux approximation method) with higher-order time integration techniques via the method of lines (MOL). The latter allows one to adapt the time step's size and the order of the time integration to improve the computational efficiency while maintaining accuracy. The robustness and accuracy of the new model are shown by comparison against a widely used commercial code based on the standard finite element method. The applicability of the developed model to a large-scale problem is then investigated by simulating saltwater intrusion under a climate change projection and long-term pumping regimes for the Akkar coastal aquifer in Lebanon using a simplified 2D conceptual model.

Phase‐Field Simulation of Hydraulic Fracturing by CO2, Water and Nitrogen in 2D and Comparison With Laboratory Data

AbstractHydraulic fracturing of tight subsurface formations has mainly been conducted using water. Waterless hydraulic fracturing by CO2 may have advantages over fracturing by water. The challenge has been numerical simulation of the process. In this work, we conduct a comprehensive study on 2D hydraulic fracturing simulation by water, CO2, and nitrogen. The simulations are based on the phase‐field; the results are compared with the lab data. We first present advances in simulation of fluid exchange between the fractures and the rock matrix. The conventional nodal‐based finite element method may give rise to unphysical negative pressure around the tip and fracture notch. We demonstrate that the mixed hybrid finite element method gives non‐negative pressure distribution. Next, we examine the effect of inertial term on fracture configuration and observe branching by CO2 under dynamic formulation with a low critical energy release rate. Lastly, we find from our simulations that in shale rocks, the critical energy release rate is the lowest for CO2, followed by nitrogen, and then water. Consequently, CO2 provides the highest fracture surface area and fracture intensity. In a sandstone sample we find nitrogen has a lower energy release rate than water. As a result, the fracture surface area is higher for nitrogen than water. Our simulation results confirm that the phase‐field method may incorporate fluid‐rock surface free energy without the need for various parameter adjustments.

Investigation of hydraulic fracture branching in porous media with a hybrid finite element and peridynamic approach

Simulation of complex fracture patterns in porous media can help understand and improve hydraulic fracturing processes, with potential for significant impact on enhancing oil and gas recovery. In this paper, a fully coupled hydraulic fracture propagation simulation method employing a hybrid finite element method (FEM) and peridynamic (PD) approach is presented. Considering the ability of PD in solving discontinuous problems, the area where cracks can potentially occur is discretised by PD and the crack-free area is discretised by FEM. The solid deformation and fracture propagation are captured by PD and FEM, while the fluid flow in both the reservoir and fracture is simulated with FEM. The whole process is solved in a monolithic way with an implicit scheme. The presented method demonstrates the capability of modelling complex dynamic crack propagation via benchmark examples. Branching phenomenon is then investigated with the proposed model. It is found that faster loading rate, lower-energy release rate, and more brittle and impermeable media will cause crack branching more easily.

Low-Computational-Cost Hybrid FEM-Analytical Induction Machine Model for the Diagnosis of Rotor Eccentricity, Based on Sparse Identification Techniques and Trigonometric Interpolation.

Since it is not efficient to physically study many machine failures, models of faulty induction machines (IMs) have attracted a rising interest. These models must be accurate enough to include fault effects and must be computed with relatively low resources to reproduce different fault scenarios. Moreover, they should run in real time to develop online condition-monitoring (CM) systems. Hybrid finite element method (FEM)-analytical models have been recently proposed for fault diagnosis purposes since they keep good accuracy, which is widely accepted, and they can run in real-time simulators. However, these models still require the full simulation of the FEM model to compute the parameters of the analytical model for each faulty scenario with its corresponding computing needs. To address these drawbacks (large computing power and memory resources requirements) this paper proposes sparse identification techniques in combination with the trigonometric interpolation polynomial for the computation of IM model parameters. The proposed model keeps accuracy similar to a FEM model at a much lower computational effort, which could contribute to the development and to the testing of condition-monitoring systems. This approach has been applied to develop an IM model under static eccentricity conditions, but this may extend to other fault types.

An Approximate Thermo-Mechanical Solution of a Functionally Graded Cylinder Using Hybrid Integral Transform and Finite Element Method

This article introduces a novel mixed method that combines the Fast Fourier Transform technique and a conventional Finite Element Method for investigating thermo-mechanical behavior of a thick functionally graded cylinder under asymmetric loadings. Material properties are assumed to vary along the radial direction according to a power function. Thermo-elastic governing equations of the cylinder are derived using principle of virtual work in cylindrical coordinates. Plane strain assumption is considered for a long cylinder during the analysis. Fast Fourier Transform technique is utilized in circumferential direction to discretize equations and related boundary conditions. Finite element method is then applied to remaining equations. For convergence study, the results obtained from this method are compared with those extracted from exact and complete FE solutions. It is observed from the results that the method has a super algebraic convergence behavior in circumferential direction. Influence of the mesh refinement is also investigated in the radial direction. According to ability of the mixed FFT-FE method for asymmetric analyzing, two kinds of loadings are considered here and results are presented. In thermo-elastic analyzing of the long cylinder, it’s obvious that the present method benefits from some features such as fast convergence and low computational cost in comparison with FE solution.