Hybrid Finite Element Method Research Articles

Twofold Domain Decomposition Method for the Analysis of Multiscale Composite Structures

A nonconformal twofold domain decomposition method (TDDM) based on the hybrid finite element method-boundary element method (FEM-BEM) is proposed for analyzing 3-D multiscale composite structures. The proposed TDDM starts by partitioning the composite object into a closed exterior boundary domain and an interior volume domain. The interior and exterior boundary value problems are coupled to each other through the Robin transmission conditions (RTCs). Both domains are then independently decomposed into subregions to facilitate computation. Specifically, FEM-DDM with the second order transmission conditions (SOTCs) is employed for the interior domain, and BEM-discontinuous Galerkin (BEM-DG) based on the combined field integral equation (CFIE) is applied for the exterior boundary domain. The proposed TDDM allows for nonconformal discretization between any touching subdomains. Without the introduction of a stabilization term that relies on a line integral over intersection of nonmatching meshes and relevant terms involving surface-line integrals, the proposed TDDM provides an effective domain decomposition (DD) preconditioner for the global system. Numerical examples are presented, and the comparisons of the simulation results with FEM-BEM confirm the validity and accuracy of TDDM. Moreover, its ability to model practical large-scale and multiscale targets is also demonstrated.

Numerical investigation on the stability of deforming fractured rocks using discrete fracture networks: a case study of underground excavation

The stability of a fractured rock mass around a subsurface opening is critical to tunnel excavation. The traditional underground excavation analysis is based on a continuum description of randomly distributed discrete pre-existing fractures. In contrast to this, we have developed an improved hybrid finite element method (FEM) to investigate the stability of fractured rocks around an excavation by incorporating the outputs into the FEM codes. The proposed model consists of a discrete fracture network (DFN) model and cohesive zone model (CZM). The DFN model automatically generates a fracture network with a given fracture opening distribution and provides grid generation strategy to the FEM. The CZM captures material failure and intersection and surface contacts through the fracture element with different constitutive laws. As a case study, a DFN model was formulated for the underground excavation at Jinping Hydropower Station, in light of the input requirements of our model, and its deformation analysis was performed. The comparison analysis of rock deformation was made for excavation in both intact rock mass and fractured rock mass under the same boundary conditions. The numerical results show that two different modes of rock failure exist in these two rock masses and that intense deformation at the fractures intersected by the tunnel is responsible for fractured rock mass instability. The proposed approach was verified using data from field investigations. A larger displacement can be produced if the rock mass is weakened by a key block. A sensitivity analysis was carried out to investigate the effects of different model parameters on deformation variations. This study provides an insightful understanding of the deformation of fractured rock mass during tunnel excavation.

Micromechanically motivated constitutive model embedded in two-dimensional polygonal finite element framework for magnetostrictive actuators

A two-dimensional constitutive model based on micromechanical domain rotation events is presented in this work to demonstrate the nonlinear actuator behavior of magnetostrictive materials, in particular, Galfenol. The model constructed upon thermodynamic principles accounts back fields which resist or aid the domain rotation events inside a grain due to external magnetomechanical loading. The developed model is then incorporated into the polygonal finite element technique that combines Voronoi-based discretization with the hybrid finite element method. In this approach, the stress and magnetic flux density are treated as approximate functions inside the element, but the mechanical displacement and magnetic potential, which act as degrees of freedom, are defined only along the element boundary. This approach allows each randomly generated Voronoi polygon in the plane discretization to act as a single finite element mimicking an individual magnetomechanical grain in a polycrystalline Galfenol, eliminating the need for further subdiscretization of the Voronoi polygon. This coupled framework simulates the nonlinear actuator characteristics of the magnetostrictive material under complex magnetomechanical loading conditions in line with the experimental observations reported in the literature.

Reflection and transmission characteristics of Lamb waves at an adhesive lap joint of plates

The reflection and transmission characteristics of Lamb waves at an adhesive single lap joint of plates are examined theoretically by the hybrid finite element method. The adhesive joint is modeled by a linear spring-type interface, which is characterized by normal and tangential stiffnesses. For the incidence of the lowest-order antisymmetric (A0) Lamb mode in a low frequency range, it is shown that the reflection and transmission coefficients of the A0 mode take local maxima and minima at multiple frequencies. This behavior results from the interference of waves originating from the lowest-order antisymmetric guided wave mode in the overlap region. The peak frequencies of the transmission coefficient increase monotonically with increasing tangential stiffness, but are almost invariant with the normal stiffness of the adhesive joint. Furthermore, time-domain numerical simulation using the finite element method is carried out to discuss the theoretical results. As a result, for the A0 mode incidence, the reflection and transmission waveforms of the A0 mode from the lap joint are found to show long-oscillation tails. The spectral analysis for the obtained waveforms shows that these tails are necessary to identify the frequencies at which the reflection and transmission coefficients take local maxima and minima.

Space-angle discontinuous Galerkin method for plane-parallel radiative transfer equation

The radiative transfer equation (RTE) for a plane-parallel problem involving scattering, absorption and radiation is solved using the discontinuous Galerkin (DG) finite element method (FEM). Both space and angle directions are discretized by the DG method. Thus, while the method has a higher accuracy in angle direction than hybrid FEM-Discrete Ordinate (SN) and FEM-Spherical Harmonic (PN) methods, it removes the continuity constraint implied by the form of basis function for continuous FEMs in space and angle. The discrete formulation of the problem is presented for nonzero phase function and a variety of boundary conditions. The numerical results demonstrate a p+1 convergence rate when the intensity is interpolated by an order p polynomial in both space and angle. The method is validated against exact solutions, and compared with other space-angle and hybrid FEMs for a few benchmark problems. The appropriateness of the DG formulation for problems with discontinuous solution is demonstrated by solving a problem with delta source term, where an in-element averaging of the source term eliminates negative intensity values for high order elements. Finally, a problem with an angular-line source term and a convergence study where the solution order is zero in angle are used to further demonstrate the advantages of the high order space-angle DG formulation; for the convergence study problem, the error was reduced by about 13 binary orders of magnitude, by increasing the order in both space and angle, rather than in space only.

Thin Conductor Modelling Combined with a Hybrid Numerical Method to Evaluate the Transferred Potential from Isolated Grounding System

Grounding systems are essential parts of substations and power generation stations. The evaluation of transferred potentials from an active grounding system to other passive ones or to any near conductors is an important aspect to be considered, because transferred potentials may cause serious and fatal events. Moreover, it is an intrinsic issue of the Smart Grid where the ground systems of the power and ICT systems could be close to each other. Therefore, the estimation of the transferred potential is necessary at grounding system design stage for people safety and electric components safeguard. Numerical methods are the best choice to perform a truthful estimation, especially when large and complex grounding systems have to be designed. However, this task is complicated by the “unbounded” nature of the electromagnetic field and by the presence of components of extremely different size in the analysis domain. In this paper, an efficient hybrid finite element method is applied for the accurate and fast computation of transferred earth potentials from grounding systems. Moreover, the small dimensions of the components in the analysis domain are taken into account by the use of one-dimensional finite elements inserted in the tetrahedral mesh. It is worth mentioning the additional advantage of obtaining the electric potential on the earth surface without any post-processing operation.

The multigrid method for the combined hybrid elements of elasticity mechanical problem

The combined hybrid finite element method is a kind of stable finite element discrete method, which can provide more looser optimization spaces for the discretization of the field variables. In this paper, the multigrid method is developed for the combined hybrid elements approximation of the elasticity problem. Theoretical convergence analysis of the method has been obtained for the energy norm. The numerical example is presented to support the theoretical results and illustrate the efficiency of the method.

A Relaxation Projection Analytical–Numerical Approach in Hysteretic Two-Phase Flows in Porous Media

Hysteresis phenomenon plays an important role in fluid flow through porous media and exhibits convoluted behavior that are often poorly understood and that is lacking of rigorous mathematical analysis. We propose a twofold approach, by analysis and computing to deal with hysteretic, two-phase flows in porous media. First, we introduce a new analytical projection method for construction of the wave sequence in the Riemann problem for the system of equations for a prototype two-phase flow model via relaxation. Second, a new computational method is formally developed to corroborate our analysis along with a representative set of numerical experiments to improve the understanding of the fundamental relaxation modeling of hysteresis for two-phase flows. Using the projection method we show the existence by analytical construction of the solution. The proposed computational method is based on combining locally conservative hybrid finite element method and finite volume discretizations within an operator splitting formulation to address effectively the stiff relaxation hysteretic system modeling fundamental two-phase flows in porous media.

Thermal Analysis of a Functionally Graded Coating/Substrate System Using the Approximated Transfer Approach

As a heterogeneous material, functionally graded material (FGM) behaves as continuously changed material properties in certain directions from one composition to another, and hence it has received much more attention for biomedical applications and thermal protections to achieve innovative functions that conventional homogeneous material cannot accomplish. However, due to the particularly small thickness ratio of coating to substrate in practice, the conventional mesh discretization of the coating region is inefficient. To simplify the meshing procedure and increase the efficiency of analysis, the approximated transfer algorithm based on the concept of finite difference is developed for transferring boundary conditions applied on the coating surface to the interface of coating and substrate. As a result, only the substrate with transferred convection boundary conditions needs to be solved numerically, i.e., by the fundamental-solution based hybrid finite element method (HFS-FEM) with high accuracy and feasible polygonal element construction, in which only integrals along the element boundary are evaluated because of the application of fundamental solutions of the problem as kernel functions of interior approximated fields. Finally, numerical experiments including the single-layered, multi-layered and functionally graded coatings are carried out to verify the accuracy and applicability of the present method.

An integrated approach for jointed rock slope stability analysis using stochastic discrete fractures: A case study in mountains area

Stochastic discrete fractures are ubiquitous in practicing engineering and contribute much to the stability for geological mediums. In contrast to the traditional treatments based on continuum mechanics, we propose an integrated approach using discrete fractures based on an improved hybrid finite element method (FEM). A typical jointed rock slop is selected as a case study and then was simulated in the light of geological conditions for the study site. The basic equations of fracture domains are derived to generate stochastic discrete fractures which render a fundamental to the model. To implement this procedure through computer program, discrete fractures generating algorithm was developed. In the aspect of discretization, we presented a meshing scheme and proposed an algorithm to create fracture elements. To evaluate rock slope stability in a quantitative way, safety factor is calculated by strength reduction method. Numerical simulations were carried out showing the mechanical behaviors of jointed rock slope under different boundary conditions.

Multidimensional mixed–hybrid finite element method for compositional two-phase flow in heterogeneous porous media and its parallel implementation on GPU

A general multidimensional numerical scheme, primarily designed to simulate two-phase compositional flow in porous media, is presented with serial and parallel implementations suitable for solving problems with degenerate (capillary) diffusion or capillary barrier effect in heterogeneous porous materials. The numerical scheme is based on the mixed–hybrid finite element method with the semi-implicitapproach for the time discretization in order to obtain a system of linear equations in each time step. The scheme is implemented in serial for CPU and in parallel for CPU and GPU using TNL that provides an efficient abstraction layer for accessing various parallel hardware architectures. In order to demonstrate the applicability of the numerical scheme, a numerical analysis is presented for problems of two-phase flow in 1D, 2D, and 3D for which exact (semi-analytical) solutions are known and a series of benchmark problems for two-phase flow in heterogeneous porous media is discussed to show correct simulation of the capillary barrier effect. The efficiency and accuracy of the implementations on CPU and GPU are discussed. Moreover, we construct an analytical solution and use it to demonstrate convergence of the numerical scheme for two-phase compositional flow problems in porous media.

A unified study of continuous and discontinuous Galerkin methods

A unified study is presented in this paper for the design and analysis of different finite element methods (FEMs), including conforming and nonconforming FEMs, mixed FEMs, hybrid FEMs,discontinuous Galerkin (DG) methods, hybrid discontinuous Galerkin (HDG) methods and weak Galerkin (WG) methods. Both HDG and WG are shown to admit inf-sup conditions that hold uniformly with respect to both mesh and penalization parameters. In addition, by taking the limit of the stabilization parameters, a WG method is shown to converge to a mixed method whereas an HDG method is shown to converge to a primal method. Furthermore, a special class of DG methods, known as the mixed DG methods, is presented to fill a gap revealed in the unified framework.

Parasitic Oscillations in Smooth-Wall Circular Symmetric Gyrotron Beam Ducts

In order to study parasitic oscillation that may occur in a realistic beam duct upstream to the gyrotron cavity, the self-consistent linear and spectral code TWANGlinspec has been modified. The large inhomogeneities in the smooth-wall beam duct geometry or in the magnetic field profile required the implementation of a numerical approach using a hybrid finite element method. The new model permits to characterize a large number of potentially spurious TE modes. Compared to previous studies on gyrotron beam duct instabilities, an extended interaction space including also the gyrotron cavity has been considered. The role of the connecting part between the beam duct and the cavity, called spacer, is highlighted and it is shown that the gyro backward-wave TE modes excited in this region generally have their minimum starting current. The sensitivity of the minimum starting current on electron beam velocity spread is also evaluated.

A hybrid finite-element and integral-equation method for forward modeling of 3D controlled-source electromagnetic induction

We have developed a hybrid solver that combines the finite-element and integralequation method for 3D CSEM modeling based on unstructured meshes. To avoid the source singularity, the secondary field is used in the modeling framework. The primary electromagnetic field from an electric dipole source in a layered medium is calculated based on the magnetic vector potential method. The inhomogeneities of the computational region are discretized by a vector-based finite-element mesh with boundaries at finite distance from the inhomogeneities by using the dyadic Green’s function, reducing the truncation boundary effect and the solution region. The electric and magnetic Green’s function is used in data postprocessing to reduce the numerical errors owing to inaccurate gradients because of unstructured meshes; thus, the electromagnetic field is more accurately calculated. Finally, the proposed algorithm is applied to a block and a disc model, and we assess the topography effect on the field components.

Modeling of shallow water waves with variable bathymetry in an irregular domain by using hybrid finite element method

An efficient numerical model is constructed with the combination of analytical and Hybrid Finite Element Method (HFEM) based on mild slope equation for shallow water waves to analyze the wave induced oscillation in an irregular geometrical domain. The domain of interest is divided in two regions as bounded and unbounded region. The solution of the mild slope equation is obtained in the bounded region with the consideration of variable bathymetry and partially reflected boundary using Hybrid element method. Further, an analytical approach is utilized in an unbounded region based on Fourier bessel series solution of scattered waves. Hybrid mesh elements is considered for bounded region to enhance the numerical accuracy. The numerical validation is conducted by the comparison of simulation results with analytical approximations and convergence analysis for the rectangular domain is also obtained. The amplification factor is obtained for T-shaped and TT-shaped domain to analyze the resonance modes. Finally, the current numerical model is applied on a realistic Pohang New Harbor (PNH) under the resonance conditions. Thus, the current numerical model can be utilized efficiently for redesigning and constructing artificial ports or harbors in the coastal regions with variable bathymetries.

Wave Induced Oscillation in an Irregular Domain by using Hybrid Finite Element Model

Harbors are designed to provide the safe loading, unloading and sheltering for the moored vessels. In seasonal weather conditions, semi-enclosed harbors experienced high resonance, which amplifies the surface wave amplitude in the interior of the harbor due to combined effect of wave refraction, diffraction and partial reflection from the solid harbor walls. An accurate description boundary of the harbor is required to analyze the impact of resonant frequency waves over surface ocean wave height. The fluid domain is divided into two regions as bounded and open sea region. The bounded region consists of harbor boundary and interior of the harbor and open sea region consists of ocean area outside the bounded region. Firstly, the mild slope equation (MSE) is derived for both the regions in terms of a potential function using the energy conservation principle. The total wave energy in the bounded region is estimated by using a mathematical model based on Hybrid Finite Element Method (HFEM) is used to formulate the mild slope equation. In HFEM model, the finite element method is coupled with the boundary element method to solve the mild slope equation in both the region. Further, the present HFEM model is validated with existing studies lead by Lee (1971) and Ippen and Goda (1963). The current numerical model is implemented on realistic Pohang New Harbor (PNH), which is situated on the southeast coast of South Korea. The present numerical model can be used as an efficient engineering tool for planning and designing of the artificial industrial harbor and predict the incident wave response under the resonance conditions.

3D numerical modelling of rock fracture with a hybrid finite and cohesive element method

To investigate the rock fracture, penetration force, and chip shape during rock breaking with a conical pick, a 3D numerical model was developed by inserting cohesive elements into finite elements. Mixed-mode cohesive traction response was applied to describe the micromechanics of rock cementation, which was also able to reflect the rock damage process and crack propagation associated with the zero-thickness elements. Simulations of Brazilian disc tests and uniaxial compression tests were used to verify the proposed numerical model and obtain rock micro-parameters, and a comparison of the penetration forces in the experimental and simulation results confirmed the accuracy and efficiency of the proposed method. Similar to the experimental phenomena of rock fracture with a conical pick, the rock crushing zone, microscopic radial cracks, macroscopic dominating crack, and main chip were produced successfully with the numerical model. Quantitative analysis of the rock fracture features (penetration force and chip shape) and breaking parameters of the conical pick revealed some valuable relationships between them, which can provide useful information for the optimal design of conical picks or breaking devices with multiple conical picks for mechanical excavation.

On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

SummaryFinite element formulations for second‐order elliptic problems, including the classic H1‐conforming Galerkin method, dual mixed methods, a discontinuous Galerkin method, and two primal hybrid methods, are implemented and numerically compared on accuracy and computational performance. Excepting the discontinuous Galerkin formulation, all the other formulations allow static condensation at the element level, aiming at reducing the size of the global system of equations. For a three‐dimensional test problem with smooth solution, the simulations are performed with h‐refinement, for hexahedral and tetrahedral meshes, and uniform polynomial degree distribution up to four. For a singular two‐dimensional problem, the results are for approximation spaces based on given sets of hp‐refined quadrilateral and triangular meshes adapted to an internal layer. The different formulations are compared in terms of L2‐convergence rates of the approximation errors for the solution and its gradient, number of degrees of freedom, both with and without static condensation. Some insights into the required computational effort for each simulation are also given.

A stabilized hybrid mixed DGFEM naturally coupling Stokes–Darcy flows

The coupling of free fluid and porous medium flows, governed by the Stokes and Darcy systems in their mixed velocity–pressure formulations, is addressed in two space dimensions. A stabilized mixed hybrid finite element method is proposed using discontinuous finite element spaces for the velocity and pressure approximations. Lagrange multipliers, associated with the traces of the velocity fields, are introduced to weakly impose the transmission conditions on the edges of the elements in both domains. The Beavers–Joseph–Saffman interface conditions on the interface between Stokes and Darcy domains also are naturally imposed by the same Lagrange multipliers. By condensing the velocity and pressure degrees-of-freedom at the element level the global system is assembled involving only the degrees-of-freedom associated with the multipliers. Stability of the proposed formulation is proved for ample choices of the finite element spaces including equal order of approximations for all fields on triangle and quadrilateral elements. Numerical experiments are performed illustrating flexibility, robustness and optimal rates of convergence.

Multiscale Numerical Simulations of Branched Polymer Melt Viscoelastic Flow Based on Double-Equation XPP Model

The double-equation extended Pom-Pom (DXPP) constitutive model is used to study the macro and micro thermorheological behaviors of branched polymer melt. The energy equation is deduced based on a slip tensor. The flow model is constructed based on a weakly-compressible viscoelastic flow model combined with DXPP model, energy equation, and Tait state equation. A hybrid finite element method and finite volume method (FEM/FVM) are introduced to solve the above-mentioned model. The distributions of viscoelastic stress, temperature, backbone orientation, and backbone stretch are given in 4 : 1 planar contraction viscoelastic flows. The effect of Pom-Pom molecular parameters and a slip parameter on thermorheological behaviors is discussed. The numerical results show that the backbones are oriented along the direction of fluid flow in most areas and are spin-oriented state near the wall area with stronger shear of downstream channel. And the temperature alongy=-1is little higher in entropy elastic case than one in energy elastic case. Results demonstrate good agreement with those given in the literatures.