Efficient Finite Element Method Research Articles

Finite element Soret Dufour effects on an unsteady MHD heat and mass transfer flow past an accelerated inclined vertical plate

AbstractThe present study is focused on the Soret and Dufour effects on magnetohydrodynamics unsteady fluid flow over an accelerated inclined vertical plate with thermal radiation and heat source. Solution of the nondimensional governing differential equations are worked out by the efficient Galerkin finite element method. The influence of several relevant flow parameters on velocity, temperature, and concentration distributions, as well as the numerical results, are studied and graphically displayed. The nondimensional skin friction and the rate of heat and mass transfer parameters are presented in the Tables 1‐3 below. Raising the Soret number results in growing concentrations, but the converse is true for the Schmidt number. Skin friction reduces when Soret and Dufour numbers increase. The present simulations apply to the processing of magnetic materials in the chemical and metallurgical industries.

Ultra-high negative dispersion compensating modified square shape photonic crystal fiber for optical broadband communication

In this work, a modified square photonic crystal fiber (MS-PCF) architecture is exposed with ultra-high negative dispersion for communication window. The validation of the model is successfully performed by applying an efficient full-vector finite element method (FV-FEM) with anisotropic perfectly matched layers (PMLs) for precise simulation of PCFs. Additionally, waveguide dispersion engineering is accomplished by tuning the structural parameters of the MS-PCF. From the numerical investigation, negative dispersion behavior permits remarkable suppression rather than positive dispersion. It offers the high negative dispersion of −2357.54 ps/nm/km and Kerr nonlinearity (γ) of 74.68 W−1 km−1 simultaneously at the operating wavelength, λ = 1550 nm. Nevertheless, the designed fiber ensures the mono mode operation in the entire band of interest from λ = 1340 nm to λ = 1580 nm. MS-PCF with these outstanding optical performances has distinguished potency to be practiced as a dispersion compensating fiber in optical communication systems.

Dynamic analysis of flat and folded laminated composite plates under hygrothermal environment using a nonpolynomial shear deformation theory

In this paper, the effect of the hygrothermal environment on the dynamic analysis of one-fold and two-fold folded laminated composite plates using a nonpolynomial shear deformation theory (NPSDT) is investigated. A computationally efficient C0 finite element method (FEM) is applied to examine the free vibration characteristic, transient behavior, and steady-state response of laminated composite plates. The formulation employs the total Lagrangian approach for analysis, and the initial stress generated due to hygrothermal load is taken into account through the Green–Lagrange strain displacement relation. The Newmark’s method is employed to integrate the spatial–temporal partial differential governing equations. The effect of the thermal environment on the natural frequency of folded and flat composite plates has been illustrated through various examples. The transient displacement and stresses are plotted for various loads. Moreover, transient damped analysis under the thermal load is carried out by considering the Rayleigh damping model. Further, linear harmonic analysis under the hygrothermal environment has also been carried out, and the responses in the neighborhood of natural frequency are plotted. The present model has been compared and validated by the existing literature and the obtained ANSYS APDL solutions, and is found to be performing better for NPSDT.

Mechanical analysis of asphalt pavement based on bimodulus elasticity theory

The bimodulus elasticity theory was introduced into pavement mechanical analysis for the first time in this study. By developing an efficient finite element method of bimodulus structural calculation for asphalt pavement, the mechanical analysis of typical asphalt pavement structure in China was carried out. The results indicated that the mechanical properties of asphalt pavement based on the bimodulus theory are nonlinear and anisotropy. Besides, the deviation between the mechanical response calculation based on bimodulus elasticity theory and the traditional linear case was more than 50%. Moreover, a great deviation on the normal stress and strain was reported, but in the case of the shear stress and shear strain, a small error was found. Especially the mechanical response of the key points in pavement design was greatly affected by the difference in tensile and compressive moduli of surface layer and base cemented mixture, which had its maximum deviation is up to 280%. Due to the significant and inevitable difference in the tensile and compression modulus of pavement materials, the bimodulus elasticity theory is suggested to be used in the mechanical calculation of asphalt pavement design.

A multi-scale method for predicting the properties of 3D braided composite under three-point bending load

A more versatile and efficient multi-scale coupling finite element method for researching the mechanical response of 3D braided composites under three-point bending load is represented in this paper. In the mesoscale, the multiphase representative unit-cell models are established to describe the mesoscopic structure which consists of braiding yarns and matrix. In the macroscale, the unit-cells are regarded as homogeneous material, and the load and constraint conditions are applied on the macroscopic structure model. The multi-scale homogenization theory is introduced to calculate the equivalent stiffness matrixes of mesoscopic unit-cells and build the mathematical relationships between the mesoscopic stress fields and the macroscopic strain fields. According to the element damage criterion, the bending modulus and ultimate load-bearing ability of 3D braided composites are predicted by simulating the progressive damage process of unit-cells Comparing with the experimental result, the predicted result satisfies the required precision for engineering.

An Efficient Finite Element Method and Error Analysis for Schrödinger Equation with Inverse Square Singular Potential

We provide in this study an effective finite element method of the Schrödinger equation with inverse square singular potential on circular domain. By introducing proper polar condition and weighted Sobolev space, we overcome the difficulty of singularity caused by polar coordinates’ transformation and singular potential, and the weak form and the corresponding discrete scheme based on the dimension reduction scheme are established. Then, using the approximation properties of the interpolation operator, we prove the error estimates of approximation solutions. Finally, we give a large number of numerical examples, and the numerical results show the effectiveness of the algorithm and the correctness of the theoretical results.

A numerical scheme with divergence free H-div triangular finite element for the Stokes equations

In this short paper, we introduce an efficient finite element method for solving the Stokes equations based on a divergence free H(div) element. The strong curl of C0-Pk+1 basis functions forms a basis of the divergence-free subspace of the Pk-H(div) (i.e. BDMk) finite element space. The [Pk+1]2×2 weak gradients are employed on these non-H1 functions to obtain a conforming weak formulation. A strength of this approach is that the saddle point problem is replaced by a symmetric positive definite system with much less unknowns, which can be solved efficiently. Numerical examples are investigated to test accuracy and robustness of the method.

Coupling Peridynamics with the Classical Methods for Modeling Hydraulic Fracture Growth in Heterogeneous Reservoirs

Summary Hydraulic fracture (HF) modeling is a multiscale and multiphysics problem. It should capture various effects, including those of in-situ stresses, poroelasticity, and reservoir heterogeneities at different length scales. A peridynamics (PD)-based hydraulic fracturing simulator has been demonstrated to reproduce this physics accurately. However, accounting for such details leads to a reduction in computational speed. In this paper, we present a novel coupling of the PD-based simulator with numerically efficient finite element methods (FEMs) and finite volume methods (FVMs) to achieve a significant improvement in computational performance. Unlike classical methods, such as FEM and FVM that solve differential equations, PD uses an integral formulation to circumvent the undefined spatial derivatives at crack tips. We implemented four novel coupling schemes of our PD-based simulator with FEM and FVM: static PD region scheme, dynamic PD region scheme, adaptive mesh refinement scheme, and dynamic mesh coarsening scheme. PD equations are solved using a refined mesh close to the fracture, whereas FE/FV equations are solved using a progressively coarser mesh away from the fracture. As the fracture grows, a dynamic conversion of FE/FV cells to PD nodes and adaptive mesh refinement are incorporated. To improve the performance further, the dynamic mesh coarsening scheme additionally converts the fine PD nodes back to coarse FE/FV cells as the HF grows in length. The coupling schemes are verified against the Kristianovich-Geertsma-de Klerk (KGD) fracture propagation problem. No spurious behavior is observed near the transition between PD and FE/FV regions. In the first three coupling schemes, the computational runtime for single fracture propagation is reduced by up to 10, 20, and 50 times, respectively, compared to a pure PD model. Laboratory experiments on the interaction of an HF with a natural fracture (NF) are revisited. The model captures complex fracture behavior, such as turning in the case of low stress contrast and low angle of interaction, kinking for higher stress contrast or higher angle of interaction, and fracture crossing for near-orthogonal NFs. Moreover, several previously reported phenomena, including fracture propagation at an angle to the principal stress directions, competing fracture growth from multiple closely spaced clusters, and interaction with layers of varying mechanical properties are successfully modeled. Thus, the coupling of PD with FEM and FVM offers an innovative and fundamentally comprehensive solution to alleviate the high computational costs typically associated with the pure PD-based hydraulic fracturing simulations. At the same time, these coupling schemes retain the versatility of the nonlocal PD formulation at modeling the evolution of arbitrary material damage, commonly observed during HF propagation in complex heterogeneous reservoirs.

Large deformation failure analysis of slopes using the smoothed particle finite element method

Slope instability and landslides can be catastrophic events leading to loss of lives and properties. To prevent and assess the risks of slope failures, it is often desired that the dynamic process of the slope failure can be simulated, which is difficult with the classic Finite Element Method (FEM). In this study, the smoothed particle finite element method is developed based on the popular and efficient FEM. A numerical example with a slope model is employed to demonstrate the capacity of the proposed approach. An elastoplastic material model based on the Mohr–Coulomb yield criterion is used. The run out distance and failure mass is recorded which paves a way of being able to better quantify slope failure consequence and risk.

Efficient Theoretical and Numerical Methods for Solving Free Vibrations and Transient Responses of a Circular Plate Coupled With Fluid Subjected to Impact Loadings

Abstract This paper presents an easy-to-use theoretical method and an efficient numerical method for solving free vibrations and transient responses of a circular plate coupled with fluid subjected to impact loadings and provides insights into various coupling cases with these developed methods. The Kirchhoff plate theory, Mindlin–Reissner plate theory, and the linear velocity potential function are used. The wet mode of the coupled system is described as the superposition of dry modes of the plate, which has been considered in few studies. The natural frequencies and corresponding mode shapes are solved using the orthogonality of dry modes. The transient responses of the plate are then solved using the superposition of the wet modes and the orthogonality of dry modes. To validate the theoretical results, an efficient and flexible finite element method is proposed and verified by comparing with commercial software. The four-node mixed interpolation of the tensorial component quadrilateral plate finite element (MITC4) and the eight-node acoustic pressure element are used to model the plate and the fluid, respectively. The theoretical and numerical methods provide reliable and accurate results. Parametric studies are performed to investigate the influence of geometric sizes, plate material properties, and fluid properties on the natural frequencies of the coupled system. A coupling parameter of fluid–structure interaction is proposed. The nondimensional added virtual mass incremental (NAVMI) factor decreases as the coupling parameter increases. Besides, the influence of fluid on wet modes of the plate decreases with the order.

Three-dimensional magnetotelluric modeling using the finite element model reduction algorithm

This study presents a fast and efficient vector finite element method, based on the model reduction algorithm, to simulate the 3-D magnetotelluric response. Firstly, we derive a finite element governing equation based on electric field vector from Maxewell's equations Then we express the electric field by a function of frequency parameter to get a transfer function that can be approximated by the Krylov subspace model reduction algorithm. The transfer function after model reduction has far lower dimensions than the original one, and it can be directly solved by the Pardiso solver. Therefore, the computational efficiency is significantly improved. For validation, we compare the proposed algorithm with the 3-D direct finite element method (3DFEM) by a three-layer model, and the results show that the computation time of the former is less than 1/10 of the latter for the same accuracy. We also compare the electromagnetic response obtained by the proposed method with that obtained by two classical models, COMMEMI 3D-2A Model and Dublin Test Model 1. These results agree well, indicating that the proposed method is feasible and efficient for 3-D magnetotelluric forward modeling.

Identification of Elastic Properties of Composites by Inversion of Ultrasonic Guided Wave Data

Given the inherent manufacturing variabilities and potential for in-service damage of composite parts, the identification of the elastic properties of composites is important to ensuring the safety and the proper performance of the part. The primary objective of this manuscript is to determine, nondestructively, the elastic properties of composite parts, whether as in-situ components that are part of a larger system, or as laboratory coupons. The proposed technique is based on multimode and dispersive ultrasonic guided waves propagating along a single direction, and the inversion of their phase velocity dispersion curves. The inversion procedure utilizes an efficient Semi-Analytical Finite Element method to solve the forward problem, and a Simulated Annealing algorithm as the optimization tool. The method is particularly well-suited for the characterization of composite laminates. In particular, the manuscript presents experimental evidence of the effectiveness of this technique, that was suggested earlier in a solely numerical work previously conducted by the authors. The test results show that reasonable accuracy can be obtained in the identification of four in-plane and three out-of-plane engineering constants of a quasi-isotropic laminate and a highly anisotropic laminate utilizing the single wave propagation direction. Non-obvious sensitivities of certain wave modes to particular constants are explained on the basis of stress coupling phenomena that are revealed by the SAFE wave propagation models. The study gives experimental evidence of the suitability of ultrasonic guided wave inversion schemes to identify the engineering constants of laminated composites, with the potential to properly characterize parts in-situ, because of the insensitivity of guided waves to boundary conditions located outside of the transmitter–receiver path.

Superconvergence analysis of a nonconforming MFEM for nonlinear Schrödinger equation

In this paper, an efficient nonconforming mixed finite element method (MFEM) is studied with element and zero-order Raviart–Thomas element for a generalized nonlinear Schrödinger equation. On the one hand, by introducing as an intermediate variable, we split the equation into two low-order equations and present a semi-discrete scheme for nonconforming MFEM and prove its existence and uniqueness. And the superconvergence results for original variable u in broken -norm and flux in norm are derived by using the proved properties of the above nonconforming MFEs. On the other hand, a linearized Crank–Nicolson fully discrete scheme is constructed and the superclose and superconvergence results with order for above variables are also derived. The keys to our analysis are the following two skills: one is an important novel property for the above MFEs (see Lemma 2.3) and another is a splitting technique for nonlinear terms, while previous literature always only obtain the convergent estimates in a routine way. Finally, two numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and τ is the time step.

Low-loss and dispersion flattened terahertz fiber

A porous core photonic crystal fiber (PCF) for terahertz (THz) wave propagation is reported in this paper. The core structure is positioned at ninety degrees to the cladding lattice which allows us to extend the porosity of the fiber to 70 %. Numerical results obtained from an efficient finite element method (FEM) show that the fiber has a very low effective absorption loss (0.06 cm−1) at the operating frequency f = 1 THz and a flattened dispersion of ±0.06 ps/THz/cm within a wide frequency band of 0.45–1.25 THz. The fiber can be manufactured using drilling and drawing methods. It is anticipated that the proposed structure would be useful in a wide range of THz appliances.

An Efficient Parallel Hybrid Method of FEM-MLFMA for Electromagnetic Radiation and Scattering Analysis of Separated Objects

In order to meet the rapidly increasing demand for accurate and efficient analysis of complex radiating or scattering structures in the presence of electrically large objects, a finite element method (FEM)-multilevel fast multipole algorithm (MLFMA) hybrid method that based on the Finite Element-Iterative Integral Equation Evaluation (FE-IIEE) mesh truncation technique is proposed in this paper. The present method makes use of FEM for the regions with small and complex features and MLFMA for the analysis of the electrically large objects, which ensure the accuracy and applicability of the method are better than most commonly adopted FEM-high frequency technique (HFT) hybrid method. The mutual interactions between regions are taken into account in a fully coupled way through iterative near filed computation process. In order to achieve an excellent performance, both algorithms have been implemented together from scratch, being able to run over multi CPU cores. An efficient parallel FEM domain decomposition method (DDM) solver with exploiting geometrical repetitions is included to drastically reduce memory requirements and computational time in the calculation of large array antenna. Also, the parallel MLFMA is adopted to expedite the near-field information exchange between regions. Through numerical example, the effect of distance between regions on the convergence of the proposed hybrid method is studied, and it is shown that the proposed method converge well even if the distance is equal to 0.05λ. Through comparisons with an in-house higher order method of moments (HOMoM) code and commercial software FEKO, the accuracy and effectiveness of the implemented parallel hybrid method are validated showing excellent performance.

An efficient finite element method and error analysis for eigenvalue problem of Schr\xf6dinger equation with an inverse square potential on spherical domain

We provide an effective finite element method to solve the Schrödinger eigenvalue problem with an inverse potential on a spherical domain. To overcome the difficulties caused by the singularities of coefficients, we introduce spherical coordinate transformation and transfer the singularities from the interior of the domain to its boundary. Then by using orthogonal properties of spherical harmonic functions and variable separation technique we transform the original problem into a series of one-dimensional eigenvalue problems. We further introduce some suitable Sobolev spaces and derive the weak form and an efficient discrete scheme. Combining with the spectral theory of Babuška and Osborn for self-adjoint positive definite eigenvalue problems, we obtain error estimates of approximation eigenvalues and eigenvectors. Finally, we provide some numerical examples to show the efficiency and accuracy of the algorithm.

Efficient Galerkin finite element methods for a time-fractional Cattaneo equation

In this paper, we develop two efficient fully discrete schemes for solving the time-fractional Cattaneo equation, where the fractional derivative is in the Caputo sense with order in (1, 2]. The schemes are based on the Galerkin finite element method in space and convolution quadrature in time generated by the backward Euler and the second-order backward difference methods. Error estimates are established with respect to data regularity. We further compare our schemes with the L2-1_{sigma } scheme. Numerical examples are provided to show the efficiency of the schemes.

Efficient CUF-based FEM analysis of thin-wall structures with Lagrange polynomial expansion

Thin-walled structures have been widely adopted in engineering projects because of the advantages of good load-bearing capacity, light weight and low cost. In this paper, a refined finite element method (FEM) based on the Carrera Unified Formula (CUF) is performed to analyze the free vibration of thin-walled beams with the variables cross-section, length and boundary. The low-dimensional FEM method could improve the efficiency of numerical analysis and reduce the time consumption. The one-dimensional (1D) CUF is employed in which the models are assumed to be a beam-like axis-oriented structures. In this case, the geometry of the thin-walled beam can be discretized as a limited number of 1D beam elements along the axis, while the Lagrange polynomial expansion may be used to approximate the displacement field on the cross-beam surface. Thus, FEM matrices and vectors can be written in terms of fundamental nuclei whose forms are independent of beam theories. The validity and capabilities of the method presented are examined in some numerical examples, and a comparative study is carried out between the method proposed and the three-dimensional element method with and without warping solutions. The results obtained by CUF 1D models are in close agreement with the reference solutions. In addition, it has been argued that the innovative approach presented in this paper can be used as a precise tool for structural analysis for complex cross-section thin-walled beams to reduce computational costs.

Efficient second-order semi-implicit finite element method for fourth-order nonlinear diffusion equations

We focus here on a class of fourth-order parabolic equations that can be written as a system of second-order equations by introducing an auxiliary variable. We design a novel second-order fully discrete mixed finite element method to approximate these equations. In our approach, we propose new techniques using the second-order backward differentiation formula for the time derivative and a special technique for the approximation of nonlinear terms. The use of the proposed technique for nonlinear terms makes the developed numerical scheme efficient in terms of computational cost since the proposed method only deals with a linear system at each time step and no iterative resolution is needed. A numerical convergence study is performed using the method of manufactured and analytical solutions of the system where we investigate different boundary conditions. With respect to the spatial discretization, convergence rates are found to at least match a priori error estimates available for linear problems. The convergence analysis is completed with an investigation of the temporal discretization where we numerically demonstrate the second-order time-accuracy of the proposed scheme using the method of reference solution. We present a series of numerical tests to demonstrate the efficiency and robustness of the proposed scheme.

Flexural wave propagation and vibration isolation characteristics of sandwich plate-type elastic metamaterials

Sandwich structures are widely used in the fields of aerospace, automobile as well as ship and offshore structures because of their excellent mechanical performances such as lightweight, high specific strength and high specific stiffness. In this study, the flexural wave band gaps and vibration isolation characteristics of sandwich plate-type elastic metamaterials are numerically and experimentally investigated. The proposed sandwich plate-type elastic metamaterials are constituted of local resonant stubs periodically deposited on a sandwich plate with periodic thin-wall aluminium tube cores. An efficient finite element method combined with a solid-shell coupling method and Bloch periodic boundary conditions is presented and validated by experimental measurements to calculate the dispersion relations and the acceleration frequency responses of sandwich plate-type elastic metamaterials. The influences of geometric parameters on the flexural wave band gaps are performed and discussed. Results show that the proposed sandwich plate-type elastic metamaterials can yield flexural wave band gaps in the low-frequency range and show significant performance on the flexural vibration isolation. Moreover, the flexural wave band gaps can be effectively modulated by the geometric parameters.