Hierarchical Finite Element Research Articles

Fluid-structure interactions of internal pressure pipeline using the hierarchical finite element method

We study the influence of the fluid with the structure in vibration between fluid and structure of a cylinder of circular section granted by the phenomenon of the interaction fluid structure of a conditioned flow of laminar nature and incompressible in the form of the macrostructure. These two phenomena by the mechanical relations of stresses according to displacements, modelled by a cylinder. The analysis of the vibrations of cylinders filled with fluid is studied with limiting conditions of fluid and the solid with the coupling conditioned by its limits of action-reaction in forces. The problem of the cylindrical pipe is formulated by deriving the deformation and the kinetic energies of the vibrating cylinder with and its fluid to have different natural frequencies, we use the principle of Hamilton change the problem in the expression of the equation cylindrical differential which gives three displacement functions in a system of partial differential equations of the cylindrical coordinate of circular section which meet the limiting conditions imposed at both ends. Let us apply the Navier-Stocks equation in cylindrical coordinates, with the fluid continuity equation, for the solid equation of mechanical behaviour of stresses in terms of displacement by strain. To obtain the results of natural frequencies we use the Galerkin method for solid and for Galerkin-time fluid. Where the liquid influences the inner surface of the circular cylinder, depending on the condition of the coupling that the stresses of the solid are equal to the stresses fluid. The modelling is done by a computer language (MATLAB), the hierarchical finite element method is presented by a Legendre polynomial with double integral of Rodrigues, to arrive at the final formula of the mass-rigidity matrix, which dissects on three parts (fluid, coupling and structure). Based on a comparison with experimental results. We continue to study some geometrical and physical parameters which influence the natural frequencies, in a proportional or inversely proportional way.

Three-dimensional free flexural vibrations of fluid-filled functionally graded circular cylindrical shell with curvilinear radius variation

The three-dimensional free flexural vibrations of functionally graded circular cylindrical shell with curvilinear radius variation fully or partially filled with incompressible and inviscid fluid are investigated for the first time using a coupled axisymmetric solid-fluid hierarchical finite element method based on curved elements. Curved edges are described exactly by means of blending functions. The method is used in conjunction with the three-dimensional elasticity theory and Laplace’s equation to obtain the governing equations. The material properties follow the power law distribution in the thickness direction. The effectiveness of the method is illustrated using examples of clamped-free and clamped-guided functionally graded circular cylindrical shells with quadratic and cubic radius variations. The method is validated through convergence test and comparison study. New results for the cycle frequencies of the lowest two flexural modes are presented, which may serve as benchmarks for future studies. Moreover, the effects of several geometrical and mechanical parameters on the cycle frequencies, modal deflections, and modal hydrodynamic pressures are explored.

An Differential Quadrature Finite Element and the Differential Quadrature Hierarchical Finite Element Methods for the Dynamics Analysis of on Board Shaft

In this paper the dynamic analysis of a shaft rotor whose support is mobile is studied. For the calculation of kinetic energy and stiffness energy, the beam theory of Euler Bernoulli was used, and the matrices of elements and systems are developed using two methods derived from the differential quadrature method (DQM). The first method is the Differential Quadrature Finite Element Method (DQFEM) systematically, as a combination of the Differential Quadrature Method (DQM) and the Standard Finite Element Method (FEM), which has a reduced computational cost for problems in dynamics. The second method is the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) which is used by expressing the matrices of the hierarchical finite element method in a similar form to that of the Differential Quadrature Finite Element Method and introducing an interpolation basis on the element boundary of the hierarchical finite element method. The discretization element used for both methods is a three-dimensional beam element. In the differential quadrature finite element method (DQFEM), the mass, gyroscopic and stiffness matrices are simply calculated using the weighting coefficient matrices given by the differential quadrature (DQ) and Gauss-Lobatto quadrature rules. The sampling points are determined by the Gauss-Lobatto node method. In the Differential Quadrature Hierarchical Finite Element Method (DQHFEM) the same approaches were used, and the cubic Hermite shape functions and the special Legendre polynomial Rodrigues shape polynomial were added. The assembly of the matrices for both methods (DQFEM and DQHFEM) is similar to that of the classical finite element method. The results of the calculation are validated with the h- and hp finite element methods and also with the literature.

Derivative-orthogonal non-uniform B-Spline wavelets

This paper attempts to merge the concept of hierarchical finite element analysis (FEA) into isogeometric analysis (IGA). The proposed methodology replaces the traditional grid refinement of IGA with custom enrichment functions. The enrichment functions are properly designed B-Spline wavelets tailored to eliminate scale-coupling terms in the stiffness matrix. In this way, the refined solution is synthesized from contributions of smaller independent problems. The proposed approach has two obvious benefits: (1) the calculations performed at each resolution are not discarded when proceeding to a finer one, and (2) it has less computational requirements since the solution is divided into smaller systems. Numerical results on an elasticity problem demonstrate superior performance and accuracy compared to traditional FEA and IGA schemes.

Hybrid integral transform and p-version FEM for thermo-mechanical analysis of a functionally graded piezoelectric hollow cylinder subjected to asymmetric loads

As the first endeavor, a combination of fast Fourier transform and p-version of finite element method is proposed for electro-thermo-elastic analysis of a thick hollow cylinder under asymmetric thermal loadings. Especially in shells of revolution, the proposed FFT-pFE method is accompanied by a significant decrease in the computational costs. Due to the problem periodicity in such structures, the fast Fourier transform technique is used to discretize the governing equations into a set of harmonics in circumferential direction. Each harmonic is then partitioned using higher order finite elements. Hierarchical finite elements based on Legendre polynomial interpolation functions are utilized to discretize 2D governing equations of a functionally graded piezoelectric (FGP) cylinder. 3D governing equations of a FGP hollow cylinder are then discretized by using the higher-order Lagrangian finite elements. The effects of FFT grid-size as well as the order of the interpolation functions are investigated on convergence behavior of the proposed mixed FFT-pFE method. The piezoelectric material properties, with the exception of the Poisson’s ratio, are considered to vary along the radius of the cylinder and pursue the power function. The governing equations are derived using the principle of virtual displacements. For a 3D FGP hollow cylinder, the influence of ...

Computational analysis of the effects of carbon nanotubes on the bending, buckling, and vibration characteristics of carbon fabric/polymer hybrid nanocomposite plates

In this paper, the effects of adding carbon nanotubes (CNTs) on the bending, buckling and free vibration characteristics of carbon fabric/polymer hybrid nanocomposite plates (HNCPs) are investigated using a computational approach. First, a hierarchical finite element micromechanics approach is proposed to estimate the hybrid composite properties. The formation of interfacial region due to the non-bonded van der Waals interactions among the CNTs and polymer is considered. Then, finite element analysis is carried out to compute the mechanical responses of the CNT/carbon fabric hybrid composite structures. It is found that incorporating the CNTs into the carbon fabric-reinforced polymer materials results in an enhancement of buckling capacity and natural frequencies of the HNCPs. The maximum deflection of the traditional carbon fabric/polymer composite plates is decreased by adding the CNTs. A parametric study is accomplished to evaluate the influences of amount and aspect ratio of CNTs, material properties and thickness of interfacial region as well as the thickness and various shapes of plate on the flexural deformation, buckling and vibration characteristics of the HNCPs. The predictions are compared with those available in the literature to verify the validity of the presented computational model.

Dynamic characteristics of multi-layered, viscoelastic beams using the refined zig-zag theory

In the present paper the displacement based refined zig-zag theory is used, for the first time, to solve dynamic problems of composite and sandwich beams made of both elastic and VE layers. The linear constitutive relations are used to model physical properties of the VE layers. The rheological models can be in the classical or fractional versions, but the main focus is on the fractional Zener model. The dynamic characteristics of considered beams are evaluated. The hierarchical finite element and the Laplace transformation are used to derive the nonlinear eigenvalue problem and its solutions yield the required dynamic characteristics of beams. The continuation method is adopted to solve this problem numerically. Accuracy, effectiveness and efficiency of the proposed method are verified in several examples. The results of the presented calculations are summarized with some conclusions.

Hierarchical finite element-based multi-scale modelling of composite laminates

This paper presents a hierarchic finite element-based computational framework for the multi-scale modelling of composite laminates. Hierarchic finite elements allow changing the approximation order locally or globally without changing the underlying finite element mesh. Both micro- and macro-level structures are discretised with these elements. The macro-level structures of composite laminates are divided into several blocks during the pre-processing stage, and approximation orders are assigned to each block independently. Due to a sharp increase in the interlaminar stresses, higher approximation orders are used in the vicinity of free edges as compared to the rest of the problem domain. This freedom of assigning approximation orders independently to each block provides an efficient and accurate way for modelling composite laminates. The computation framework can either accept the user-defined ply-level homogenised elastic material properties or calculates these directly from the underlying representative volume element consisting of matrix and fibre using the computational homogenisation. The model developed for the computational homogenisation has the flexibility of unified imposition of representative volume element boundary conditions, which allows convenient switching between linear displacement, uniform traction and periodic boundary conditions. The computational framework has additional flexibly of high-performance computing and makes use of state-of-the-art computational libraries including Portable, Extensible Toolkit for Scientific Computation (PETSc) and the Mesh-Oriented datABase (MOAB). Symmetric cross-ply, angle-ply and quasi-isotropic laminates subjected to uniaxial loading are used as test cases to demonstrate the correct implementation and computational efficiency of the developed computational framework.

A parametric analysis of oilfield design factors affecting the detectability and characterization of electrically conductive hydrofracks

Electrical responses in the vicinity of energized steel-cased well sources offer significant potential for monitoring induced fractures. However, the high complexity of well-fracture-host models spanning multiple length scales compels analysts to simplify their numerical models due to enormous computational costs. This consequently limits our understanding regarding monitoring capabilities and the limitations of electrical measurements on realistic hydraulically fracturing systems. Here, we use the hierarchical finite element approach to construct geoelectric models in which geometrically complex fractures and steel-cased wells are discretely represented in 3D conducting media without sacrificing the model realism and computation efficiency. We have discovered systematic numerical analyses of the electrical responses to evaluate the influences of borehole material conductivity and the source type as well as the effects of well geometry, conductivity contrast, source location, fracture growth, and fracture propagation. The numerical results indicate that the borehole material property has a strong control on the electrical potentials along the production and monitoring wells. The monopole source located at a steel-cased well results in a current density distribution that decays away from the source location throughout the well length, whereas the dipole source produces a current density that dominates mainly along the dipole length. Moreover, the conductivity contrast between the fractures and host does not change the overall pattern of the electrical potentials but varies its amplitude. The fracture models near different well systems indicate that the well geometry controls the entire distribution of potentials, whereas the characteristics of the voltage difference profiles along the wells before and after fracturing are insensitive to the well geometry and the well in which the source is located. Further, the hydraulic-fracturing models indicate that the voltage differences along the production well before and after fracturing have strong sensitivity to fracture growth and fracture set propagation.

Free vibration of shear deformable symmetric VSCL elliptical plates by a curved rectangular p-element

The p-version of the finite element method based on the first-order shear deformation theory (FSDT) is applied, to the free vibration of symmetric variable stiffness composite laminated (VSCL) elliptical plates, with curvilinear fibers. A curved hierarchical rectangular finite element is developed and used to model the elliptical plate. The geometry is described accurately by the blending function method. The element stiffness and mass matrices are integrated numerically using the Gauss-Legendre quadrature. The equations of motion are derived employing Lagrange’s method. The numerical results are validated by means of convergence tests and compared with available data in the literature, for isotropic and constant stiffness composite laminated (CSCL) circular and elliptical plates. The contour plots of the fundamental frequency as a function of the fiber orientation angles are presented for clamped and soft simply supported symmetric VSCL elliptical plates. The effect of thickness ratio, aspect ratio, boundary conditions, and mechanical parameters on the frequencies of the lowest five modes is investigated and discussed. The mode shapes are examined by considering the effect of the fiber orientation angles and stacking sequences.

Hierarchical beam finite elements for geometrically nonlinear analysis coupled with Asymptotic Numerical Method

This paper couples the Carrera's Unified Formulation (CUF) with the Asymptotic Numerical Method (ANM) for investigating geometrically nonlinear behaviors of beam structures. On the basis of CUF, a family of advanced one-dimensional beam models is firstly established by deriving a fundamental nucleus. The ANM that is an efficient and robust nonlinear solver is then used to solve the resulted nonlinear systems. Several typical nonlinear problems in thin/thick beam structures are carried out, and numerical results demonstrate the validity and efficiency of the proposed framework. Besides, the importance of using high-order kinematics for beam structures undergoing large deformations is emphasized.

The coordinate ascent hierarchical infinite element method for the three-dimensional free flexural vibration analysis of water-column interaction systems

A simple and highly accurate numerical method is presented for the free flexural vibration analysis of water-column interaction systems. The three-dimensional elasticity theory is used to predict accurately local stresses in the flexible circular column. The effects of water compressibility and gravity surface waves are considered. Small amplitude free vibrations are assumed. A hierarchical finite element is developed to describe the displacements. A coordinate ascent hierarchical infinite element based on the wave equation is developed to describe the hydrodynamic pressure. The method is illustrated with the example of a tapered cantilever circular column. Examples of stepped and parabolic cantilever circular columns are also shown. The method is validated through convergence study and comparison with analytical results for a prismatic cantilever circular column in full contact with water. The cycle frequencies predicted by the Euler-Bernoulli beam theory show deviation from the three-dimensional solutions. New results for the cycle frequencies of tapered, single-stepped, and parabolic cantilever circular columns in full contact with water are provided, which may serve as a benchmark for future research. A parametric study is performed showing the effect of the taper ratio on the cycle frequencies, modal deflections, modal bending stresses, and modal hydrodynamic pressures.

A three-dimensional hierarchic finite element-based computational framework for the analysis of composite laminates

A three-dimensional hierarchic finite element-based computational framework is developed for the investigation of inter-laminar stresses and displacements in composite laminates of finite width. As compared to the standard finite elements, hierarchic finite elements allow to change the order of approximation both locally and globally without modifying the underlying finite element mesh leading to very accurate results for relatively coarse meshes. In this paper, both symmetric cross-ply and angle-ply laminates subjected to uniaxial tension are considered as test cases. Tetrahedral elements are used for the discretisation of laminates and uniform or global p-refinement is used to increase the order of approximation. Each ply within laminates is modelled as a linear-elastic, homogenous and orthotropic material. With increasing the order of approximation, the developed computational framework is able to capture the complex profiles of inter-laminar stresses and displacements very accurately. Results are compared with reference results from the literature and found in a very good agreement. The computational model is implemented in our in-house finite element software library Mesh-Oriented Finite Element Method (MoFEM). The computational framework has additional flexibly of high-performance computing and makes use of the state-of-the-art computational libraries including Portable, Extensible Toolkit for Scientific Computation (PETSc) and the Mesh-Oriented datABase (MOAB).

Effect of cortical bone micro-structure in fragility fracture patients on lamellar stress

This work investigates how changes in cortical bone microstructure alter the risk of fragility fractures. The secondary osteons of non-osteoporotic (by DXA) women with fragility fractures have reduced lamellar width and greater areas of birefringent brightness in transverse sections, a pathological condition. We used hierarchical finite element (FE) models of the proximal femur of two women aged 67 and 88 (younger and older) during one-legged stance. At specific locations of the anterior-inferior neck (ROI), we analyzed micro-models containing osteons comprised of alternating birefringent extinct and bright lamellae. The plane of lamellar isotropy (XY) was transverse to the osteon longitudinal axis (Z) which was parallel to the femoral neck axis. To evaluate changes in fracture risk with changes in microstructure, we investigated principal and von Mises stresses, and planar stress measures that accounted for transverse isotropy. For both younger and older femurs, 48% to 100% of stress measures were larger in models with healthy architecture than in models with pathological architecture, while controlling for type of lamella and osteon. These findings suggest that bone adaptation reduces stress at most pathological lamellar sites. However, in the bright lamellae of the younger femur, the pathological tensile, compressive and distortional stresses in the transverse plane and distortional stress in the longitudinal planes were larger than the non-negligible corresponding stresses in 6 of the 28 osteon models with healthy architecture, in 5 of the 7 locations. Therefore, a minority of sites with pathological architecture present greater stress, and therefore, greater fracture risk.

Generalized viscoelastic model for laminated beams using hierarchical finite elements

The formulation of an enriched hierarchical one-dimensional finite element suitable to analyze the rheological behavior of thick arbitrarily laminated beams is presented. The formulation is based on the Equivalent Single Layer (ESL) theory and was developed to allow the use of any high-order beam shear deformation theory (HBST) in a unified approach. A generalized Maxwell model was implemented to analyze the time-dependent behavior of the composite. The finite element employs local Lagrange and Hermitian support functions enriched with orthogonal Gram-Schmidt polynomials and is free of shear locking. The enriched macro elements can be used with very coarse meshes and the precision can be controlled without generating a new mesh. The formulation has been validated with numerical examples of symmetric and non-symmetric laminated beams using the three-dimensional finite element program PLCD.

Dynamic Study of a Functionally Graded Material Rotating Conical Shaft Based on a New Model of Variation by Slice in the Material Properties

This article is concerned with the dynamic behavior study of a rotating conical shaft bi-supported in functionally graded material (FGM), using the Timoshenko beam theory (FSDBT), with consideration of the gyroscopic effect. The material properties are varied through the thickness direction of the shaft based on a new model which defines the variation per slice in the volume fraction of the two extreme materials according to the exponential law distribution (E-FGM). This dynamic system is examined from a mathematical development and a hierarchical finite element modeling p-version. To validate our model, the numerical results obtained are compared with those available in the literature. Several examples are treated and a discussion is held to determine the influence of the internal properties (FGM), the conicity (α) and the slenderness ratio on the eigen-frequencies and consequently on the dynamic behavior of FGM rotating conical shafts.

Vibration analysis of functionally graded tapered rotor shaft system

Abstract This investigation deals with the vibration analysis of a rotating tapered shaft in Functionally Graded Material (FGM). The dynamic system is modeled using the Timoshenko beam theory (FSDBT) with consideration of gyroscopic effect and rotary inertia. The equations of motion are expressed by the hierarchical finite element method based on bi-articulated boundary conditions. The material properties are continuously varied in the thickness direction of a hollow shaft according to the exponential law function (E-FGM). The presented model is validated by comparing the numerical results found with the available literature. Various analyses are carried out to determine the influence of taper angle and material distribution of the two extreme materials on the dynamic behavior of FGM conical rotors system.

Free vibration of functionally graded sandwich shallow shells in thermal environments by a differential quadrature hierarchical finite element method

This paper presents a differential quadrature hierarchical finite element method (DQHFEM) for dynamic analyses of functionally graded material (FGM) sandwich shallow shells in thermal and non-thermal environments. A layer-wise theory based on the first-order shear deformation theory (FSDT) for each layer was adopted. Effective material properties of the FGM are estimated according to Voigt’s rule of mixture (ROM) and/or Mori–Tanaka (MT) scheme. For the shells in thermal environment, a nonlinear temperature distribution in thickness direction is considered and the elastic properties are assumed to be temperature dependent. The results obtained from the proposed formulation are validated with those available in literatures. Natural frequencies obtained from Sander’s, Love’s and Donnell’s shell theories for different geometric and boundary conditions are compared with each other first to assess the performance of different shell theories for FGM sandwich shells under non-thermal environment. Then the effects of volume fraction index, core thickness and temperature gradient on natural frequencies of FGM sandwich shells are investigated. The presented DQHFEM is much like the fixed interface mode synthesis method but does not need modal analysis and is of high accuracy. This work is the first application of the method to functionally graded sandwich shells in thermal environments.

Evaluation of shear and membrane locking in refined hierarchical shell finite elements for laminated structures

Shear and membrane locking phenomena are fundamental issues of shell finite element models. A family of refined shell elements for laminated structures has been developed in the framework of Carrera Unified Formulation, including hierarchical elements based on higher-order Legendre polynomial expansions. These hierarchical elements were reported to be relatively less prone to locking phenomena, yet an exhaustive evaluation of them regarding the mitigation of shear and membrane locking on laminated shells is still essential. In the present article, numerically efficient integration schemes for hierarchical elements, including also reduced and selective integration procedures, are discussed and evaluated through single-element p-version finite element models. Both shear and membrane locking are assessed quantitatively through the estimation of strain energy components. The numerical results show that the fully integrated hierarchical shell elements can overcome the shear and membrane locking effectively when a sufficiently high polynomial degree is reached. Reduced and selective integration schemes can help with the mitigation of locking on lower-order hierarchical shell elements.

Free Vibration of Curved beams with Hierarchical Finite Element Method

Free Vibration of Curved beams with Hierarchical Finite Element Method