Standard Finite Element Research Articles

Multiscale Finite Element Method for Ventilation Panels

In the context of electromagnetic compatibility ventilation panels with small quasi-periodic openings exist. The standard finite element (FE) method leads to a large number of unknowns for ventilation panels. An efficient multiscale FE approach is introduced to cope with the multiple scale problem. To this end, special functions are computed by solving cell problems for a single aperture with FEs of higher order to enhance the FE space. The proposed technique is able to cope with varying thickness es of the metallic sheet of the ventilation panel and arbitrary shapes of apertures, as well as lossy materials and frequencies in a wide range. The introduced method improves essentially the accuracy of coarse FE simulations for ventilation panels in 3-D.

Stable generalized finite element method (SGFEM) for three-dimensional crack problems

This paper proposes a stable generalized finite element method (SGFEM) for the linear 3D elasticity problem with cracked domains. Conventional material-independent branch functions serve as singular enrichments. We prove that the proposed SGFEM with the geometric enrichment scheme yields the optimal order of convergence in the energy norm, O(h), for fully 3D elasticity planar crack problems; h is the mesh parameter. To improve the conditioning of SGFEM, two stability techniques have been employed, namely, (a) a cubic polynomial has been used as the PU (partition of unity), instead of the standard FE hat-functions, to address the possible almost linear dependence between the PU functions and the enrichments, and (b) a local principal component analysis (LPCA) has been implemented to address the local bad conditioning produced by multi-fold enrichments at a node. The scaled condition number for the proposed SGFEM is shown to be O(h^{-2}) (same as that of a standard Finite Element Method), for various relative positions of crack surface and mesh. The robustness of the scaled condition number for the proposed SGFEM, with respect to the relative positions of the crack-surface and the element boundaries, has been observed numerically. The numerical experiments for both the planar and fully 3D planar crack problems are presented to show the efficiency of the proposed SGFEM.

Identifying dominant components of vibrational energy flow in U-rib plates of bridge based on structural intensity

Stiffened U-rib plates are the elementary components of steel box girders in bridges that directly bear the input external energy before transferring it to other components. Research on the dynamic characteristics of stiffened U-rib plates is the basis for examining vibration and noise in steel box girders. Most relevant studies have focused on the vibrational modes and single-frequency responses in this context, and few researchers have sought to optimize the design of stiffened U-rib plates to reduce responses due to vibrations and noise. This study identifies the global and local laws of vibrational transmission in stiffened U-rib plates subjected to harmonic loads through structural intensity (SI) analysis that is carried out by jointly using standard finite element software and a post-processing code created by the authors. The contributions of each component of vibrational waves are represented to reflect the dominant roles of the bending wave in the baseplate and the in-plane wave in the U-ribs, respectively. The effects of parameters of the U-rib plates on SI are also quantitatively investigated. SI analysis can be used as a new criterion for designing and optimizing bridges as well as formulating measures for vibration reduction.

A cell-based smoothed finite element model for non-Newtonian blood flow

Smoothed Finite Element Method (S-FEM) has drawn increasing attention in the field of computational fluid dynamics (CFD) and the present work seeks to make further contribution to this growing field of S-FEM by simulating for the non-Newtonian blood flow. This investigation took the form of Streamline Upwind Petrov-Galerkin in conjunction with Stabilized Pressure Gradient Projection (SUPG/SPGP) to alleviate the spatial oscillation and instability problems. The validation of the cell-based S-FEM (CS-FEM) combined with SUPG/SPGP was carried out by the blood flow over a backward-facing step. The performances of the presented method were explored by the blood flow in the carotid bifurcation and blood flow in the intracranial segment of internal carotid artery. Impressively, the results exhibit good features of the CS-FEM on solving severely distorted mesh vis-a-vis the standard finite element method (FEM). The presented method could realize the accurate prediction in the mentioned complex blood flows as the same as Finite Volume Method software STAR-CCM+.

The identification of voids and inclusions based on the parameter level set method using boundary data

Using the temperature information obtained from the boundary to identify internal defects of the component is significant to ensure the reliability and safety of the structure. Based on the standard finite element method and the parameter level set method (PLSM), a new inversion theoretical framework for identifying voids and inclusions is established. A heat transfer solution model for multi-material interpolation with inclusion defects is constructed to avoid the iterative process of mesh reconstruction when the defect configuration changes. The strong nucleation ability of the parameter level set and the good optimization performance of the Levenberg-Marquardt method are fully utilized. The simple and complex defects are systematically discussed in the example section. By discussing different influencing factors such as the number of measuring points, the number of knots, the measurement error, it is found that even in the case of no some prior information, the proposed method can quickly obtain relatively stable and accurate inversion results.

Penalty Virtual Element Method for the 3D Incompressible Flow on Polyhedron Mesh.

In this paper, a penalty virtual element method (VEM) on polyhedral mesh for solving the 3D incompressible flow is proposed and analyzed. The remarkable feature of VEM is that it does not require an explicit computation of the trial and test space, thereby bypassing the obstacle of standard finite element discretizations on arbitrary mesh. The velocity and pressure are approximated by the practical significative lowest equal-order virtual element space pair which does not satisfy the discrete - condition. Combined with the penalty method, the error estimation is proved rigorously. Numerical results on the 3D polygonal mesh illustrate the theoretical results and effectiveness of the proposed method.

On the modeling of flywheel rotor systems via unified formulation: Viability, practicalities, and experimental validation

Recent times have seen a great interest on environmental issues and efficient, sustainable systems. This interest has required the employment of advanced composites for a myriad of industrial machines and innovative equipments. Among these applications, Flywheel Energy Storage Systems – FESS – represent a group of machines that are being re-invented through this process. Modeling composite flywheels has proven to be a complex task, which current Finite Element models fail to fulfill in a number of design contexts. This demand to model complex composite geometries and systems induced the proposition of new methods, aiming to capture the various physical effects existing in the problem. In the present contribution, the authors consider that it is viable to model the dynamic behavior of a Flywheel Energy Storage System via an adapted Carrera Unified Formulation, both in terms of accuracy and computational cost, for practical applications. The present work presents and explores a Carrera Unified Formulation model with extended capabilities dedicated to rotordynamics applications. The differences from standard Finite Elements models are presented, evidencing advantages and drawbacks of the proposed methodology over more traditional approaches. A case study is then presented, modeled, and the results are compared with those stemming from established formulations.

A novel isogeometric layerwise element for piezoelectric analysis of laminated plates with straight/curvilinear fibers

This study presents an isogeometric layerwise element, L-IGA based on the principle of virtual displacement theory to model the bending behavior of laminated smart composite plates integrated with piezoelectric layers. Instead of using Lagrangian or Hermitian type polynomials encountered in standard finite element technology, L-IGA utilizes high-order Non-Uniform Rational B-Splines (NURBS) functions for both in-plane and through-the-thickness discretization of the geometry and the kinematic variables. Additionally, it allows different numbers of NURBS degrees and elements to be used for each patch through the thickness of the plate. In this way, exact geometry, and highly accurate solutions with rapid convergence for the benchmark electromechanical problems in the literature have been guaranteed by using L-IGA analysis. The precision of the results is meticulously verified by 3-D Ansys SOLID226 finite elements and analytical solutions. Thus, the L-IGA element can be adopted for computationally efficient and accurate static analysis of laminated plates having straight/curvilinear fibers for piezoelectric actuator and sensor configurations.

Domain decomposition methods for 3D crack propagation problems using XFEM

The extended finite element method (XFEM) has been successfully implemented in solving crack propagation problems by enriching the polynomial basis functions of standard finite elements with specialized non-smooth functions. The resulting approximation space can be used to solve problems with moving discontinuities, such as cracks, without the need of remeshing in the vicinity of the crack. The enrichment of the displacement field in XFEM inflicts a substantial increase in the ellipticity of the discretized problem. As a consequence, the resulting algebraic systems become strongly ill-conditioned, rendering the convergence of iterative solvers very slow. On the other hand, direct solvers may become inefficient in 3D problems, due to the increased bandwidth of the system matrix. In this paper, two of the most efficient domain decomposition solvers, namely the FETI-DP and P-FETI-DP, are proposed for solving the linear systems resulting from XFEM crack propagation analysis in large-scale 3D problems. Both solvers are amenable to parallelization and can be implemented in modern parallel computing environments, with multicore processors and distributed memory systems, following appropriate modifications, to achieve a drastic reduction of memory and computing time in computationally intensive problems.

A stable CS-FEM for the static and seismic stability of a single square tunnel in the soil where the shear strength increases linearly with depth

A numerical procedure using a stable cell-based smoothed finite element method (CS-FEM) is presented for estimation of stability of a square tunnel in the soil where the shear strength increases linearly with depth. The kinematically admissible displacement fields are approximated by uniform quadrilateral elements in conjunction with the strain smoothing technique, eliminating volumetric locking issues and the singularity associated with the Mohr–Coulomb model. First, a rich set of simulations was performed to compute the static stability of a square tunnel with different geometries and soil conditions. The presented results are in excellent agreement with the upper and lower bound solutions using the standard finite element method (FEM). The stability charts and tables are given for practical use in the tunnel design, along with a newly proposed formulation for predicting the undrained stability of a single square tunnel. Second, the seismic stability number was computed using the present numerical approach. Numerical results reveal that the seismic stability number reduces with an increasing value of the horizontal seismic acceleration (αh), for both cases of the weightless soil and the soil with unit weight. Third, the link between the static and seismic stability numbers is described using corrective factors that represent reductions in the tunnel stability due to seismic loadings. It is shown from the numerical results that the corrective factor becomes larger as the unit weight of soil mass increases; however, the degree of the reduction in seismic stability number tends to reduce for the case of the homogeneous soil. Furthermore, this advanced numerical procedure is straightforward to extend to three-dimensional (3D) limit analysis and is readily applicable for the calculation of the stability of tunnels in highly anisotropic and heterogeneous soils which are often encountered in practice.

Two-grid finite element methods for nonlinear time fractional variable coefficient diffusion equations

In this article, an efficient two-grid finite element method is proposed for solving the nonlinear time fractional variable coefficient diffusion equations. This algorithm firstly solves a nonlinear system to get the numerical solution uHn on the coarse grid with size H, then based on the initial iterative solution uHn on the coarse grid, the linearized finite element system is solved on the fine grid with size h to get the numerical solution Uhn, in which the temporal direction is approximated by the L2−1σ scheme. Besides, the stability and priori error estimates of standard finite element method and two-grid method are given. Finally, the validity and efficiency of the two-grid algorithm are verified by two numerical experiments.

A second-order accurate numerical scheme for a time-fractional Fokker–Planck equation

Abstract A second-order accurate time-stepping scheme for solving a time-fractional Fokker–Planck equation of order $\alpha \in (0, 1)$, with a general driving force, is investigated. A stability bound for the semidiscrete solution is obtained for $\alpha \in (1/2,1)$ via a novel and concise approach. Our stability estimate is $\alpha $-robust in the sense that it remains valid in the limiting case where $\alpha $ approaches $1$ (when the model reduces to the classical Fokker–Planck equation), a limit that presents practical importance. Concerning the error analysis, we obtain an optimal second-order accurate estimate for $\alpha \in (1/2,1)$. A time-graded mesh is used to compensate for the singular behavior of the continuous solution near the origin. The time-stepping scheme scheme is associated with a standard spatial Galerkin finite element discretization to numerically support our theoretical contributions. We employ the resulting fully discrete computable numerical scheme to perform some numerical tests. These tests suggest that the imposed time-graded meshes assumption could be further relaxed, and we observe second-order accuracy even for the case $\alpha \in (0,1/2]$, that is, outside the range covered by the theory.

Finite Element Model to Simulate Two-Phase Fluid Flow in Naturally Fractured Oil Reservoirs: Part I.

Naturally fractured reservoirs host more than 20% of the world’s total oil and gas reserves. To produce from such reservoirs efficiently, a good understanding of the reservoir behavior at various conditions is essential. This allows us to predict the reservoir performance in advance and assess its economic feasibility. However, the production from such reservoirs is challenging due to (a) uncertainty associated with the fracture map, (b) complex physics phenomena of fluid and rock interaction, and (c) lack of comprehensive knowledge of the extent, orientation, and permeability sensitivity of the fracture network. This paper addresses the abovementioned challenges by presenting a three-dimenisonal (3-D) two-phase fluid flow model in a poroelastic environment. The model is based on a hybrid methodology by combining single continuum and discrete fracture network approaches. Also, the capillary pressure effect, saturation, and relative permeability variations are considered. A mathematical formulation for three-dimensional, two immiscible fluid flows including rock deformation for the fracture network and the rock matrix is presented. A standard Galerkin-based finite element method is applied to discretize the poroelastic governing equations in space and time. The characteristic Galerkin discretization method is used to stabilize the solution of the convection equation in a finite element approach. The 3-D model is validated against IMEX commercial software and finite element package to test its robustness. The results show that the developed model has the ability to predict the two-phase flow behavior precisely, which can be used to assess the production performance of naturally fractured reservoirs. Numerical results of fluid flow profiles for single and multiple fractures with different orientations that match experimental oil drainage tests and field case studies are presented that proves the reliability of the developed multiphase flow model. Results of simulated well production data show an excellent match with the field production data.

A multiscale finite element method for coupled heat and water transfer in heterogeneous soils

Large-scale modeling of coupled heat and water transfer (CHWT) is challenging due to the spatial variabilities of soil hydraulic and thermal properties. A multi-scale finite element method (MsFEM) has been designed for simulating liquid water fluxes in unsaturated soils. In this study, the MsFEM approach is expanded as a new scheme that can handle CHWT in soils. Two groups of MsFEM basis functions are constructed to incorporate the heterogeneities in soil hydraulic conductivity and thermal conductivity, and a Petrov-Galerkin formulation is adopted to implement the proposed MsFEM scheme. The MsFEM scheme is also exploited as a sequential solver when heat transfer and water transfer are expressed in a partially coupled formulation (Wang et al., 2022a). Numerical examples illustrate that, without essentially increasing the computing load, the MsFEM scheme can improve the accuracy by up-to 30% compared to the standard finite element method (FEM), especially for thermally driven water transfer. Some fine scale spatial variations in soil temperature can only be revealed with the MsFEM scheme. If the MsFEM sequential solver is applied, the coupled heat and water transfer model can be rewritten into a sequence of modules, where liquid water transfer, heat transfer and vapor transfer are solved step-by-step, but the thermally driven liquid water is omitted due to its relatively small value (Wang et al., 2022a). With the MsFEM sequential solver, a flexible modeling architecture can be achieved at the cost of a relatively small increase of error (<5%). Therefore, the MsFEM scheme presented in this study is an effective numerical approach to simulating CHWT in heterogeneous soils.

Numerical simulation of fracture in layered and sandwich structures: A systematic literature review

A systematic literature review on numerical strategies suitable to simulate defects and discontinuities in layered and sandwich structures is presented here. Particular care is given to studies whose scope concerns the so-called Moving Mesh (MM) or Isogeometric Analysis (IGA) methods. A general overview of the peculiarities of each approach is also provided. A total of fifteen and twenty-six journal/conference articles were summarised and categorised for MM and IGA methods, respectively. Based on the available literature, it can be stated that MM approaches generally allow a lower computational burden due to the reduced number of re-meshing steps required when compared to standard finite element approaches. Conversely, IGA approaches bring strong advantages in the geometric description of curved shell structures and, due to a non-uniform rational b-spline interpolation function, ensure higher numerical accuracy in stress analysis problems.

An Interpolated Galerkin Finite Element Method for the Poisson Equation

We develop a new approach to construct finite element methods to solve the Poisson equation. The idea is to use the pointwise Laplacian as a degree of freedom followed by interpolating the solution at the degree of freedom by the given right-hand side function in the partial differential equation. The finite element solution is then the Galerkin projection in a smaller vector space. This idea is similar to that of interpolating the boundary condition in the standard finite element method. Our approach results in a smaller system of equations and of a better condition number. The number of unknowns on each element is reduced significantly from \((k^2+3k+2)/2\) to 3k for the \(P_k\) (\(k\ge 3\)) finite element. We construct bivariate \(P_2\) conforming and nonconforming, and \(P_k\) (\(k\ge 3\)) conforming interpolated Galerkin finite elements on triangular grids; prove their optimal order of convergence; and confirm our findings by numerical tests.

TT-M Finite Element Algorithm for the Coupled Schrödinger–Boussinesq Equations

In this article, the coupled Schrödinger–Boussinesq equations are solved numerically using the finite element method combined with the time two-mesh (TT-M) fast algorithm. The spatial direction is discretized by the standard Galerkin finite element method, the temporal direction is approximated by the TT-M Crank–Nicolson scheme, and then the numerical scheme of TT-M finite element (FE) system is formulated. The method includes three main steps: for the first step, the nonlinear system is solved on the coarse time mesh; for the second step, by an interpolation formula, the numerical solutions at the fine time mesh point are computed based on the numerical solutions on the coarse mesh system; for the last step, the linearized temporal fine mesh system is constructed based on Taylor’s formula for two variables, and then the TT-M FE solutions can be obtained. Furthermore, theory analyses on the TT-M system including the stability and error estimations are conducted. Finally, a large number of numerical examples are provided to verify the accuracy of the algorithm, the correctness of theoretical results, and the computational efficiency with a comparison to the numerical results calculated by using the standard FE method.

A fully discrete two-grid method for the diffusive Peterlin viscoelastic model

We present in this paper a fully discrete two-grid finite element method to solve the diffusive Peterlin viscoelastic model on the basis of studying the fully discrete standard finite element method. The fully discrete two-grid method involves solving small nonlinear coarse spatial grid systems and linear fine spatial grid systems with the same time step. Under the reasonable assumptions, it is proven theoretically that the accuracy of fully discrete two-grid method is the same as the fully discrete finite element method on the fine spatial grid. Numerical examples are given to validate the accuracy and convergence order of the standard finite element method and two-grid method.

Consistent pressure Poisson splitting methods for incompressible multi-phase flows: eliminating numerical boundary layers and inf-sup compatibility restrictions

For their simplicity and low computational cost, time-stepping schemes decoupling velocity and pressure are highly popular in incompressible flow simulations. When multiple fluids are present, the additional hyperbolic transport equation in the system makes it even more advantageous to compute different flow quantities separately. Most splitting methods, however, induce spurious pressure boundary layers or compatibility restrictions on how to discretise pressure and velocity. Pressure Poisson methods, on the other hand, overcome these issues by relying on a fully consistent problem to compute the pressure from the velocity field. Additionally, such pressure Poisson equations can be tailored so as to indirectly enforce incompressibility, without requiring solenoidal projections. Although these schemes have been extended to problems with variable viscosity, constant density is still a fundamental assumption in existing formulations. In this context, the main contribution of this work is to reformulate consistent splitting methods to allow for variable density, as arising in two-phase flows. We present a strong formulation and a consistent weak form allowing standard finite element spaces. For the temporal discretisation, backward differentiation formulas are used to decouple pressure, velocity and density, yielding iteration-free steps. The accuracy of our framework is showcased through a wide variety of numerical examples, considering manufactured and benchmark solutions, equal-order and mixed finite elements, first- and second-order stepping, as well as flows with one, two or three phases.

A study on the approximation power of NURBS and the significance of exact geometry in isogeometric pre-buckling analyses of shells

We present a comprehensive study on the approximation power of NURBS and the significance of exact geometry in stability analyses of shells. Pre-buckling analyses are carried out to estimate the critical load levels and the initial buckling patterns. Various finite element solutions obtained with the commercial code ANSYS are compared with solutions from the isogeometric version of the finite element method, using our in-house code NumPro. In some problem setups, the isogeometric shell elements provide superior accuracy compared to standard (as opposed to isogeometric) shell finite elements, requiring only a fractional amount of degrees of freedom for the same level of accuracy. The present study systematically investigates the sources of this superior accuracy of the isogeometric approach. In particular, hypotheses are tested concerning the influence of exact geometry and smoothness of splines.